Friction-Induced Vibration and Sound Principle and Application

FRICTION-INDUCEDVIBRATIONSand SOUND

Principles and Applications

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CRC Press is an imprint of theTaylor & Francis Group, an informa business

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FRICTION-INDUCEDVIBRATIONSand SOUND

Principles and Applications

GANG SHENG

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Sheng, Gang.Friction-induced vibrations and sound : principles and applications / by Gang Sheng.

p. cm.Includes bibliographical references and index.ISBN-13: 978-1-4200-5178-0ISBN-10: 1-4200-5178-41. Vibration--Mathematical models. 2. Friction--Mathematical models. 3. Machinery--Sound. 4.

Machinery--Vibration. 5. Machinery--Noise. I. Title.

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Dedication

In Memory of My Father

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ContentsPreface ........................................................................................................................................... xiiiAuthor ............................................................................................................................................ xv

Chapter 1 Introduction.................................................................................................................. 1

1.1 Definition of Friction, Acoustics, and Friction-Induced Vibrations and Sound ..................... 11.2 Significances and Challenges of Studies on Friction-Induced Vibrations and Sound ............ 21.3 Organization of the Book ........................................................................................................ 3References ......................................................................................................................................... 4

Chapter 2 Vibrations and Sound .................................................................................................. 7

2.1 Introduction .............................................................................................................................. 72.2 Linear Vibrations ..................................................................................................................... 7

2.2.1 Vibration Motion ......................................................................................................... 72.2.1.1 Sinusoidal Periodic Motion .......................................................................... 82.2.1.2 Complex Periodic Motion............................................................................. 92.2.1.3 Transient Nonperiodic Motion...................................................................... 9

2.2.2 Linear Single-Degree-of-Freedom (SDOF) System .................................................. 102.2.2.1 Free Vibration of an SDOF System with Viscous Damping ..................... 102.2.2.2 Free Vibration of an SDOF System with Both Viscous

and Friction Damping ................................................................................. 122.2.2.3 Forced Vibration of an SDOF System with Viscous Damping ................. 132.2.2.4 Forced Vibration of an SDOF System with Friction Damping.................. 16

2.2.3 Linear Multiple-Degree-of-Freedom (MDOF) System.............................................. 172.2.3.1 Eigenvalues and Eigenvectors .................................................................... 182.2.3.2 Forced Vibration Solution of an MDOF System ....................................... 22

2.2.4 Vibration of Continuous Systems.............................................................................. 232.2.4.1 Transverse Vibrations of String.................................................................. 232.2.4.2 Wave Equation............................................................................................ 272.2.4.3 Longitudinal Vibration of Rods.................................................................. 272.2.4.4 Torsional Vibration of Shafts ..................................................................... 282.2.4.5 Transverse Vibration of Beams .................................................................. 292.2.4.6 Galerkin Method ......................................................................................... 332.2.4.7 Vibrations of Disk Plates ............................................................................ 34

2.3 Nonlinear Vibration Systems................................................................................................. 402.3.1 Perturbation Method, Duffing Equation, and Van der Pol’s Equation...................... 402.3.2 Method of Variation of Parameter............................................................................. 452.3.3 Phase Plot, Limit Cycles, Self-Excited Oscillations, and Chaos............................... 472.3.4 Stability of Equilibrium ............................................................................................. 482.3.5 Parametrically Excited System and Mathieu’s Equation,

Nonstationary Vibrations ........................................................................................... 522.3.6 Multiple-Degree-of-Freedom Systems....................................................................... 54

2.4 Random Vibrations ................................................................................................................ 542.4.1 Probability Density Function and Gaussian Random Process .................................. 552.4.2 Autocorrelation Function ........................................................................................... 57

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2.4.3 Response of an SDOF System to an Arbitrary Function Input................................. 582.4.4 Power Spectral Density Function .............................................................................. 612.4.5 Joint Probability Density Function ............................................................................ 632.4.6 Cross-Correlation Function........................................................................................ 632.4.7 Response of an SDOF System to a Random Input ................................................... 642.4.8 Response of MDOF Systems to Random Inputs....................................................... 65

2.5 Fundamentals of Sound ......................................................................................................... 682.5.1 Airborne Sound and Measures .................................................................................. 682.5.2 Noise and Spectrum Analysis of Nonstationary Sound ............................................ 692.5.3 Sound Perception and Weighting Curves.................................................................. 712.5.4 Sound Wave Equation ............................................................................................... 722.5.5 Sound Radiation of Structures ................................................................................... 752.5.6 Sound Interference ..................................................................................................... 762.5.7 Stress Wave Propagation in Elastic Solid Medium................................................... 78

Bibliography.................................................................................................................................... 83

Chapter 3 Contact and Friction .................................................................................................. 85

3.1 Introduction ............................................................................................................................ 853.2 Surface Roughness................................................................................................................. 85

3.2.1 Characteristics of Random Rough Surfaces .............................................................. 853.2.2 Surface Roughness Parameters .................................................................................. 86

3.3 Contact between Two Solid Surfaces .................................................................................... 873.3.1 Spherical Contact ....................................................................................................... 873.3.2 Multiple Asperity Contacts ........................................................................................ 88

3.3.2.1 Contact of Identical Asperities ................................................................... 893.3.2.2 Statistical Analysis of Contact of Nominally Flat Rough Surfaces ........... 90

3.3.3 Plastic Deformation.................................................................................................... 923.4 Friction ................................................................................................................................... 93

3.4.1 Adhesion .................................................................................................................... 933.4.1.1 Solid–Solid Adhesion ................................................................................. 933.4.1.2 Liquid-Mediated Adhesion ......................................................................... 97

3.4.2 Dry Friction.............................................................................................................. 1013.4.2.1 Friction Mechanisms................................................................................. 1013.4.2.2 Energy Dissipation during Friction........................................................... 1073.4.2.3 Rubber Friction ......................................................................................... 1083.4.2.4 Friction Transitions ................................................................................... 1103.4.2.5 Static Friction, Hysteresis, Time, and Displacement Dependence ........... 1123.4.2.6 Effect of Environmental and Operational Condition................................ 114

3.4.3 Wet Friction ............................................................................................................. 1153.4.3.1 Stribeck Curve .......................................................................................... 1153.4.3.2 Unsteady Liquid-Mediated Friction.......................................................... 1173.4.3.3 Negative Slope of Friction–Velocity Curve ............................................. 120

3.5 Friction–Vibration Interactions ............................................................................................ 1223.5.1 Stick–Slip ................................................................................................................. 1223.5.2 Effect of Normal Oscillations .................................................................................. 125

3.6 Friction in Nano- and Molecular Scales .............................................................................. 127References ..................................................................................................................................... 133

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Chapter 4 Friction-Induced Vibrations and Sound .................................................................. 141

4.1 Introduction .......................................................................................................................... 1414.2 Vibrations of Single-Degree-of-Freedom Systems with Friction ........................................ 143

4.2.1 Friction Law............................................................................................................. 1434.2.2 Self-Excited Vibrations and Forced Vibrations....................................................... 151

4.2.2.1 Free Vibrations.......................................................................................... 1524.2.2.2 Self-Excited Vibrations............................................................................. 1524.2.2.3 Forced Vibrations...................................................................................... 1534.2.2.4 Combined Self-Excited Vibration and Forced Vibration

V 6¼ 0, F0 6¼ 0 ........................................................................................... 1534.2.2.5 Closed Form Solution for Self-Excited Vibrations .................................. 1564.2.2.6 Closed Form Solution for Combined Self-Excited Vibration

and Forced Vibration: Dither Effect ......................................................... 1594.2.2.7 Variable Normal Force ............................................................................. 162

4.2.3 Contact Vibrations and Vibro-Impacts of System with Friction............................. 1634.2.3.1 Basic Features ........................................................................................... 1644.2.3.2 Frequency Response ................................................................................. 1674.2.3.3 Friction System with Variable Normal Force and Vibro-Impact ............. 168

4.3 Vibrations of Multi-Degree-of-Freedom Systems with Friction ......................................... 1694.3.1 Negative Damping Instability Due to Negative Slope

of Friction–Velocity Curve...................................................................................... 1694.3.2 Internal Combination Resonance Due to Velocity-Dependent Friction

and Variable Normal Force ..................................................................................... 1734.3.3 Mode-Coupling Instability Due to Proximity of Modes of Subsystems

with Constant Friction ............................................................................................. 1754.3.4 Complex Modal Analysis ........................................................................................ 1794.3.5 Nonlinear Numerical Analysis................................................................................. 186

4.3.5.1 Numerical Approaches.............................................................................. 1864.3.5.2 Numerical Example................................................................................... 189

4.4 Vibrations and Sound of Continuum Systems with Friction .............................................. 1944.4.1 Longitudinal Vibrations of Rod with Velocity-Dependent Friction ....................... 1944.4.2 Beam Transverse Vibrations Due to Bowing Effects.............................................. 1954.4.3 Beam Transverse Vibrations Due to Axial Stick–Slip Excitation........................... 1974.4.4 Beam Transverse Vibrations with Modal Coupling Due to

Geometry Constraints .............................................................................................. 1984.4.5 Weak Contact and Strong Contact: Random Interactions of Friction..................... 200

4.4.5.1 Weak Contact and Strong Contact ........................................................... 2004.4.5.2 Random Interactions of Friction ............................................................... 200

4.4.6 Sprag–Slip Excited Vibrations................................................................................. 2034.4.7 Rubbing Sound Wave under Weak Contact............................................................ 2064.4.8 Instability of Waves under Frictional Sliding of Strong Contact............................ 210

4.5 Friction-Induced Vibrations and Sound in Social Life, Nature, Science,and Engineering ................................................................................................................... 2154.5.1 Friction Music Sound and Friction Sound Synthesis .............................................. 215

4.5.1.1 Music Sound of Bowed String Instruments ............................................. 2154.5.1.2 Friction Sound Synthesis .......................................................................... 220

4.5.2 Friction Sound in Nature ......................................................................................... 2204.5.2.1 Creature Friction Sound............................................................................ 2204.5.2.2 Nature Sand Sound ................................................................................... 222

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4.5.3 Friction-Induced Instability and Sound in Geological Scienceand Engineering ....................................................................................................... 2244.5.3.1 Stick–Slip and Dynamic Instability in Earthquakes ................................. 2244.5.3.2 Characterization of Sediment Grain Using Friction Sound...................... 226

4.5.4 Friction-Induced Vibrations in MEMS Devices and Equipment ............................ 2294.5.4.1 Ultrasonic Motor ....................................................................................... 2294.5.4.2 Stick–Slip in Contact Mode AFM............................................................ 231

4.5.5 Friction-Induced Vibrations and Noise in Vehicles ................................................ 2324.5.5.1 Friction-Induced Noise and Vibrations in Clutches ................................. 2324.5.5.2 Friction-Induced Gear Noise..................................................................... 2344.5.5.3 Piston Stick–Slip Noise in Engine............................................................ 2374.5.5.4 Friction-Induced Wiper Blade Noise ........................................................ 2374.5.5.5 Friction Noises of Tire–Road Interaction and Wheel–Rail

Interaction ................................................................................................. 2384.5.5.6 Squeak and Rattle in Automobiles ........................................................... 239

References ..................................................................................................................................... 242

Chapter 5 Friction-Induced Vibrations and Acoustic Emission and Applicationsin Hard Disk Drive System ..................................................................................... 249

5.1 Introduction .......................................................................................................................... 2495.2 Contact and Friction-Induced Vibrations of Slider in Magnetic Hard Disk Drive ............. 251

5.2.1 Slider–Disk Interface ............................................................................................... 2515.2.2 Nanometer Spacing Interface and Budgets.............................................................. 2515.2.3 Slider–Disk Contact Model and Interactions........................................................... 252

5.2.3.1 Intermolecular Force ................................................................................. 2565.2.3.2 Electrostatic Force..................................................................................... 2565.2.3.3 Meniscus Forces........................................................................................ 257

5.2.4 Nonlinear Dynamics Near-Contact Air-Bearing Slider and theFriction-Induced Vibrations ..................................................................................... 2585.2.4.1 Nonlinearity of Air Bearing...................................................................... 2595.2.4.2 Contact Vibrations .................................................................................... 2605.2.4.3 Friction-Induced Vibrations with Effect of Intermolecular Forces .......... 2635.2.4.4 Friction-Induced Vibrations with the Effect of Meniscus Force .............. 2645.2.4.5 Experimental Observations ....................................................................... 266

5.3 Modeling of Friction-Induced Acoustic Emission............................................................... 2725.3.1 Empirical Model ...................................................................................................... 2735.3.2 SL Model ................................................................................................................. 275

5.4 Identification of Interface Contact and Force Using Acoustic Emission Signal ................. 2815.4.1 Contact Locations .................................................................................................... 281

5.4.1.1 Force Identification ................................................................................... 2845.4.1.2 Method of Direct Calibration.................................................................... 2865.4.1.3 Method of System Identification .............................................................. 287

5.5 Disk Surface Screening Using Acoustic Emission Techniquefor Mass Production............................................................................................................. 2895.5.1 Glide Testing for Disk Surface Finish Certification................................................ 2895.5.2 Nanometer Clearance Certification and Calibration ................................................ 291

References ..................................................................................................................................... 295

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Chapter 6 Friction-Induced Noise in Power Transmission Belt Systems................................ 301

6.1 Introduction .......................................................................................................................... 3016.2 Belt Friction ......................................................................................................................... 305

6.2.1 Laws of V-Belt Friction and Belt Effective Friction............................................... 3056.2.2 Belt Dynamic Friction Testing ................................................................................ 3066.2.3 Belt Friction Features............................................................................................... 307

6.3 Belt Vibrations ..................................................................................................................... 3106.3.1 Belt Vibrations Analysis .......................................................................................... 3106.3.2 Belt Vibrations Measurements................................................................................. 313

6.4 Misalignment Belt Noise ..................................................................................................... 3146.4.1 Experimental Observation of V-Belt Radial Motion Due to Misalignment ........... 3156.4.2 Chirp and Squeal of Automotive V-Ribbed Belt .................................................... 3176.4.3 Influence of Rib and Pulley Profile ......................................................................... 3216.4.4 Modeling of Radial Stick–Slip Motion of V-Belt ................................................... 322

6.5 Dry Belt Slip Noise.............................................................................................................. 3296.5.1 Dry Belt Slip and Friction ....................................................................................... 3296.5.2 Dry Belt Friction and Friction Noise....................................................................... 329

6.6 Wet Belt Friction and Friction-Induced Noise in Automotive AccessoryBelt Drive System................................................................................................................ 3356.6.1 Wet Friction ............................................................................................................. 3356.6.2 Wet Belt Slip Noise ................................................................................................. 338

6.7 Timing Belt Friction-Induced Noise.................................................................................... 3396.7.1 Timing Belt Friction ................................................................................................ 3396.7.2 Friction-Induced Noise ............................................................................................ 340

References ..................................................................................................................................... 342

Chapter 7 Friction-Induced Vibrations and Noise in Vehicle Brake System .......................... 347

7.1 Brake Structure, Materials, Vibration, and Noise................................................................ 3477.1.1 Brake Structure and Materials ................................................................................. 3477.1.2 Friction-Induced Vibration and Noise in Brake System ......................................... 3487.1.3 Groan and Judder..................................................................................................... 3497.1.4 Low-Frequency Squeal ............................................................................................ 3517.1.5 High-Frequency Squeal ........................................................................................... 352

7.2 Numerical Approaches and Analysis of Brake Noise ......................................................... 3547.2.1 Real Modal Analysis................................................................................................ 3547.2.2 Complex Eigenvalue Analysis................................................................................. 3547.2.3 Transient Analysis ................................................................................................... 3577.2.4 Interactive Analysis.................................................................................................. 359

7.3 Test Technique..................................................................................................................... 3617.3.1 Modal Test and Operational Deflection Shape Test................................................ 3617.3.2 Dynamometer Test and Vehicle Test ...................................................................... 3637.3.3 Noise Evaluation...................................................................................................... 3677.3.4 Tribology Testing .................................................................................................... 369

7.4 Features Associated with Brake Noise ................................................................................ 3727.4.1 Interface Aging Effect.............................................................................................. 3727.4.2 Friction–Velocity Dependence and Hysteresis ........................................................ 3747.4.3 Friction–Pressure Dependence and Hysteresis ........................................................ 3757.4.4 Temperature Dependence ........................................................................................ 376

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7.4.5 Humidity Effect ..................................................................................................... 3777.4.6 Microtribology Perspective.................................................................................... 3787.4.7 Contact Stiffness .................................................................................................... 3807.4.8 Damping and Young’s Modulus Effect ................................................................. 3827.4.9 Structural Effect ..................................................................................................... 3837.4.10 Acoustic Radiation Efficiencies ............................................................................. 3867.4.11 Uncertainty............................................................................................................. 387

7.5 Prevention ............................................................................................................................ 3877.5.1 Friction Screening.................................................................................................. 387

7.5.1.1 Groan....................................................................................................... 3877.5.1.2 Judder ...................................................................................................... 3887.5.1.3 Squeals .................................................................................................... 388

7.5.2 Stiffness Optimizations .......................................................................................... 3897.5.3 Damping................................................................................................................. 3907.5.4 Geometry Optimization.......................................................................................... 3917.5.5 Other Approaches .................................................................................................. 395

References ..................................................................................................................................... 396

Index ............................................................................................................................................. 401

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PrefaceIn the last two decades, a large amount of industrial efforts and academic research has been directedto friction-induced vibrations and sound, but hardly any of this can be found in any existing bookrelated to this area. Some previously published books consist of relevant chapters that onlyemphasized exclusively treated particular systems. These include the book of Guran, Pfeiffer, andPopp (1996), book chapter by Seireg (1997), as well as survey papers (Ibrahim, 1992; Akay, 2002).However, some of the most important engineering applications such as the friction-induced acousticemission techniques applied in the data storage industry and the friction-induced noise controltechniques applied in the automotive industry have not been touched, and a number of recentdevelopments have not been included. The primary purpose of this book is to fill this void.

The intent of this book is to present a fundamental understanding of friction-induced vibrationsand sound in a unified theoretical framework. Emphasis is also given to the engineering applicationsincluding the data storage industry and the automotive industry, where the control of friction-induced vibrations and sound has been one of the most critical techniques. Being an interdiscip-linary area, the important aspects of friction-induced vibrations and sound have been difficult tocover in a single book of interest to various groups of readers, including active engineers in industry,researchers in academia, and students in universities. To prepare such a wide-ranging book, I havetried to harness my knowledge and experience gained over the years in several industries includingthe computer data storage industry, the automotive industry, the consumer electronics industry, thegovernment research lab, as well as academia.

Most of the first four chapters in the book are concerned with introducing basic concepts andanalytical methods. The first chapter introduces the entire book. The second chapter briefs the readeron vibrations and sound principles. The third chapter presents the principles of contact and frictions.The fourth chapter presents general principles on how friction induces vibrations and sound, andalso touches on various friction-induced vibrations and sound in nature, in everyday life, as well asin science and engineering applications.

In the remaining three chapters the concepts and methods are extended to some of the mostcritical engineering applications in the data storage industry and in the automotive industry. Thefifth chapter presents the friction-induced acoustic emission technique and applications for datastorage. The sixth chapter discusses friction-induced noise and control in automotive belt drivesystems. The last chapter examines friction-induced noise and control in vehicle brake systems.

This book is set out as the first comprehensive reference of this kind for engineers, researchers,as well as classroom use; it is self-contained. An extensive bibliography is included. I attempted toreference every paper that appeared in an archive journal and related to the materials in the book;however, omissions could unavoidably occur.

I hope this book will transmit some of the beauty which I found is inherent in the subject of theproblem of friction-induced vibrations and sound, and which I have gathered from my work inacademia and industry. Some of the topics presented here are not fully understood at present. It ismy sincere hope that the book will stimulate some readers to pursue and develop some new ideas onthe problems mentioned here.

I wish to thank my mentors, my former and present colleagues, my friends, and my family whohave contributed to my learning of friction-induced vibrations and sound.

I would like to thank Professors Zhenghuang Luo and Zhiyan Chen at Shanghai Jiao TongUniversity, Professors Shuzi Yang and Hanming Shi at Huazhong University of Science andTechnology, and Professors Tat-Ching Fung and Sau-Cheong Fan at Nanyang Technological

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University, who introduced me to the fields of acoustics, dynamics, and tribology, and guided me topursue the related research. Professor Zhenghuang Luo directly stimulated my long-term interest inthis topic. Having worked in the U.S. industry in the 1950s, Professor Luo has a diverse array ofexperiences. When I had him as my mentor in 1984, he introduced me to many problems that werestill unsolved.

Special thanks to Dr Jerzy Otremba, Keming Liu, Les Brown, Dr Lance Miller, Dr Bob South,Dr Shawn Wu, Charlie Ochoa, Scott Ciemniecki and John Bohenick at Gates Corporation, Dr KeqinXu at Acopolo, Dr Zhiwei Zhong at TRW, Dr Hua He, Dr Powell Bob, Dr Xiandi Zeng at GeneralMotors, Professor Rao Dekkati at Fairfield University, Dr Mahamad, Qatu, Dr Jiang Pang at FordMotor Company, Dr Qi Wan at AMC Consultant, Dr Shanlin Duan, Dr Li Tang, Dr Xiaoping Bian,Dr Ragger Woods at Technical Division of IBM Corporation (now Hitachi Global Storage),Dr Okada Kanzo, Kok Leong Tong, Takada Akio, Jianbing Zhao at Sony Singapore ResearchLab, Professor June Wang at Northwestern University, Professor Kato Ono at Tokyo Institute ofTechnology, Professor Hashimoto Tagawa at Kesai University, Qisuo Chen at Seagate Technology,Dr Yufeng Li at Western Digital, Dr Wei Hua, Dr Yiaolong Zhu, Dr Bo Liu, Dr Jun Zhang, andDr Yiansheng Ma at Singapore Data Storage Institute, from whom I immensely benefited. I am alsopleased to acknowledge my indebtedness to a large number of friends not mentioned here forvaluable suggestions. I owe many thanks to editor Michael Slaughter, project coordinator MarshaPronin, and project editor Richard Tressider of Taylor & Francis and project manager SuryakalaArulprakasam of SPi, who have expended much time and effort in editing the manuscript, and fortheir many helpful suggestions concerning the subject matter.

My special thanks go to my daughter Hanlu, who has always given me something to lookforward to, and who has been forbearing when I spent many weekends in preparation of this book.Last, but not least, I wish to thank my mother who has supported my career over the years.

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AuthorGang Sheng is a research scientist at Gates Technical Center (a Tomkins company) in RochesterHills, Michigan where he has served since 2002. His areas of expertise are in acoustics, tribology,and dynamics.

Gang Sheng received his BS and MS in mechanical engineering from Shanghai JiaotongUniversity in 1984 and 1987, respectively, and his PhD from Nanyang Technological Universityin Singapore in 1997. Gang Sheng once worked for Huazhong University of Science and Technol-ogy in China, Singapore Data Storage Institute, Sony Singapore Research Lab, and IBM TechnicalDivision in San Jose, California.

Gang Sheng has accumulated ten years worth of research and teaching experience in the fields ofacoustics, dynamics, and tribology as they relate to automotive engineering. He also has seven yearsof research experience in the fields of dynamics, tribology, and instrumentation in high-tech areasincluding information storage devices, mechatronics, and micro=nanosystems. Gang Sheng has lednumerous research projects and made substantial contributions to basic research, automotive system,information storage system, and other engineering systems. He has published over 400 industrialreports, over 100 research papers (more than half are refereed journal papers), 5 books=bookchapters, and holds 5 patents. He has received many academic and industrial awards including the2006 Author Achievement Award from the Gates Corporation.

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1 Introduction

In this introductory chapter, the definition of friction, acoustics, and the friction-induced vibrationsand sound is described. Then the industrial significance of friction-induced vibrations and sound ispresented. In the last section, the organization of the book is introduced.

1.1 DEFINITION OF FRICTION, ACOUSTICS, AND FRICTION-INDUCEDVIBRATIONS AND SOUND

When the surfaces of two objects are placed in contact and allowed to slide, there is a resistanceto the motion. This resistance is the friction that is experienced whenever one solid body movesover another. Friction is one of the most important problems in tribology, that is, the science andtechnology of interacting surfaces in relative motion concerned with friction, wear, and lubrica-tion in interface. Vibration is the oscillation motion of objects and the forces associated with it.The friction and interface interaction often gives rise to diverse forms of waves and oscillationswithin solids which frequently lead to radiation in the surrounding media. Sound is the vibrationin media including both solid and liquid. Acoustics is the science and technology of vibrationsand sound.

The contact and friction event between the two contact surfaces is an energy transition processin which the kinetic energy of the moving body is transferred into the energy of irregularmicroscopic motion of the interface asperities, particles, and atoms. The friction usually gives riseto vibrations, and the vibration propagating in media is sound, either its propagation in sliding partsor radiation to air.

Up to now, there is no universally accepted friction model or theory to cover general frictionphenomena. Different models have been developed for individual conditions. Part of the reason isbecause friction is a complex process in which forces are transmitted, mechanical energy isconverted, surface topography is altered, interface material could be removed or formed, andphysical and even chemical change could occur. Friction coefficients are not intrinsic propertiesof materials; they depend on the properties of the contact surfaces, their operational conditions, theirtime history, environmental conditions, and even their interactions.

Tribology and acoustics used to be two distinct fields. However, with the recent rapiddevelopments in friction acoustics related areas, especially in the application of automotive engin-eering and information storage engineering, engineers are turning to combine tribology andacoustics for efficient methods to handle and analyze the vast amounts of practical cases. Theexciting subarea known as friction acoustics has subsequently evolved.

This book, Friction-Induced Vibrations and Sound: Principles and Applications, offers acombined treatment of the modeling, analysis, and testing of many problems application thatengineers and scientists are trying to solve. After delineating these mathematical characterizations,it presents several applications in use today for analyzing friction-induced vibrations and sound.Emphasis is on the contemporary knowledge of friction-induced vibrations and sound.

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1.2 SIGNIFICANCES AND CHALLENGES OF STUDIESON FRICTION-INDUCED VIBRATIONS AND SOUND

Friction-induced vibrations and sound occur in our everyday life, in numerous science andengineering systems. Just to name a few: bowed string instrumental music sound, brake squealsof automobile and motorcycle, wheel squeals of rail vehicles in narrow curves, elevator rail rollersqueaks, robotic joint squeaks, chattering of machine tools, creaking of door hinges, chalk onblackboard, etc. The examples of friction-induced vibrations and sound cover numerous phenomenain science and engineering. Even in the interior of an automobile, from time to time, we may perceptthe clutch shudder, wiper squeal, belt drive chirp, instrument panel squeak, tire road friction noise,etc. Friction-induced vibrations and sound extend beyond music and industrial noise; they includenumerous phenomena in science and nature. This book considers friction-induced vibrations andsound in its broader meaning yet concentrates on fundamentals and the engineering applications.

To give some examples of the problems treated in the book, let us consider the immense effortsthat are being put into dealing with friction-induced vibrations and sound in information storageindustry and automotive industry.

We are living in an information age. The needs for information storage systems are tremen-dously high and ever-increasing. There are a variety of information storage systems with varyingdegrees of development and commercialization. To date, magnetic information storage technology,particularly hard disk drive, is most widely used. We are all familiar with computers in whichthe hard disk is one of the key components. The worldwide hard disk drive revenue had reached$40 billion. Magnetic hard disk drives are based on the same fundamental principles of magneticrecording which involves a recording head and a recording medium. The former is on a suspension-supported slider, the latter is on a spinning disk. The slider is flying on the spinning disk with airgap. The operation of hard disk drive is based on a self-pressurized air bearing between the sliderand the spinning disk, which maintains a constant separation called flying height. The state-of-the-art flying height is on the order of below 10 nm, while the relative speed between slider and disk isextremely high (10 m=s or higher). The mechanical spacing between the slider and the disk has to befurther reduced to less than 5 nm in order to achieve an areal density of 1 Tbit=in.2. In these regimes,the interface friction and contact induced vibration=stability has been the most challenging andcritical problem for the products. On the other hand, over the last few decades, the friction-inducedacoustic emission testing technique has been the most important technique for slider disk interfaceinvestigation and for disk certification in mass production.

Our life is highly dependent on various machines and vehicles. Many methods of mechanicalpower transmission are in use today to drive industrial, automotive, agricultural, and domesticequipment and machinery, but by far the most common means of power and motion transmission isthe flexible belt. Belts have played an important role in the industrial development of the world formore than 100 years; for instance, about one-third of the electric motor transmissions used in theindustrial and commercial sectors use belt transmissions. Flat belts are the purest form of a frictiondrive while V-belts and V-ribbed belts rely on coefficient of friction but have a multiplying effectthrough the V or wedging characteristic. The most familiar and widely used type of belt in theindustrial is the V-belt. The jointed industrial V-belts and V-ribbed belts are the extension of V-belt.A properly designed belt transmission system provides high efficiency, cleanliness, and low noise,does not require lubrication, and can have low maintenance requirements. In the automotiveindustry, currently V-ribbed belts have been widely used for most of automotive accessory drives.With the reduction of engine noise over the years, the accessory belt drive system (ABDS) noise hassurfaced as one of the major noise and vibration concerns to engine and vehicle designers and a topchallenging issue to ABDS suppliers. Historically, academics and industrial researchers had pub-lished hundreds of publications of belt vibrations and dynamics; by now, the complex dynamicproblem has become a secondary issue and commercial software has been used as a standard designtool to deal with most of dynamic problems in industry. On the other hand, the noise issue has been


2 Friction-Induced Vibrations and Sound: Principles and Applications

the top critical issue challenging automotive accessory belt drive suppliers and vehicle systemmanufacturers.

Most of our vehicles like automotives, motorcycles, rail vehicles, and aircrafts have used brakesystem for stopping or urgent braking. In friction brake system there are both a principal functional=safety performance factor and a potential cause of undesirable noise and vibrations. The structuresand principles of varied brake systems in different vehicles are analogous and similar, but the noisehas been considered as an unsolved problem for the last half century. Friction-induced vibrationsand noise impact the reliability and quality of brake systems in many ways. For instance, the brakesqueal has been the number one challenging issue in automotive brake system as it has been equal tothe quality of products percepted by customers. It was estimated that the warranty work of noise andvibration of automotive brake systems costs approximately $1 billion a year in the Detroit areaalone. Even in aircraft braking systems, friction-induced vibration has been a critical issue, itsfriction-induced torque oscillations can lead to excessively high loads in the landing gear and brakestructure, and this results in passenger discomfort and=or component failure and thereby warrantyclaims.

Understanding the nature of tribology and acoustics and these interactions and solving thetechnological problems associated with the friction-induced vibrations and sound are the essence ofthese fields.

Modeling of friction-induced vibrations and sound in mechanical and other systems requires anaccurate description of friction. Unfortunately, there is no universally accepted friction model ortheory to cover general friction phenomena due to the reasons mentioned in the last section. On theother hand, the resultant vibrations often exhibit various nonlinear, transient, and unsteady features;the resultant sound is usually nonstationary with interested range from tens of hertz to ultrasonic.Moreover, small changes in interfacial parameters could have a significant effect on the resultantvibrations and sound; thus the scales of influencing factor span from macro-, micro-, to nanometerlevels. The boundary condition of the problems is not fixed or given in prior; actually it is dependenton environmental conditions, operation conditions, system interactions, and is also time-dependent.Because of the complexity of the problem of friction-induced vibrations and sound, it has beenconsidered to be an unsolved problem in many engineering applications. Since the modeling and thepredictions are not very reliable, the ‘‘try and error’’ approach has been extensively used.

The recent extensive efforts on modeling, analytical, and experimental investigation have mademany substantial progresses in many practical applications. Many techniques like laser Dopplervibration measurement, time–frequency spectrum analysis, and wavelet analysis have been used asefficient means to address nonstationary motions; it enables to efficiently quantify the friction-induced vibrations and sound. The emergence and applications of scanning tunneling microscopeand the atomic force microscope have allowed systematic investigations of interfacial problems withhigh resolution, which have led to the development of the insight of friction in micro-, nano-,molecular, or even atomic-scale level.

The research in friction-induced vibrations and sound has many purposes; just to name a few: todevelop fundamental understanding of friction-induced vibrations and sound as well as theirinteractions from various scales; to make use of its principle to realize some physical processeslike ultrasonic motor; to make use of the vibrations and sound to explore phenomena in complexprocesses where other means are not accessible, like interface friction monitoring using friction-induced acoustic emission; and understandably to reduce and eliminate the instability and noise inengineering systems caused by frictions.

1.3 ORGANIZATION OF THE BOOK

The book has been set out with a twofold aim. The first aim is to give a general introduction to thetheory of friction, vibration, and sound, by offering a physical picture of the fundamental theory.The second aim is to give a series of examples of the applications of the theoretical approaches.


Introduction 3

The author is expected to provide contemporary coverage of the primary concepts and techniques inthe treatment of friction-induced vibrations and sound.

This book consists of seven chapters. The basic principles of friction-induced vibrations andsound are introduced in the first four chapters. Chapter 1 introduces the whole book. Chapter 2provides a comprehensive introduction to the analysis of vibrations and sound, from vibrations oflinear systems, random excited systems to nonlinear systems, by covering all the required areas andapplications. The fundamentals of sound in air and elastic media are also described. Chapter 3describes the friction principles. Chapter 4 deals with friction-induced vibrations and soundcovering single-degree-of-freedom systems, multi-degree-of-freedom systems, and continuum sys-tems. Moreover, in viewing that the rapid growth of tribology and acoustics has addressed manyproblems of friction-induced vibrations and sound by various communities, Chapter 4 also toucheson many applications in nature, our everyday life, science, and engineering.

The remaining three chapters present critical engineering applications including the applicationsin information storage industry and automotive industry, where the rigorous mathematical analysisis ignored; actually, once we are able to grasp the physical picture behind the phenomena, readerscan themselves add what extra generality and mathematical rigor they may need. Chapter 5 isdevoted to acoustic emission techniques used in computer hard disk drive systems. Chapter 6is dedicated to the vibration and noise control of automotive belt drive systems and the last chapter,Chapter 7, deals with the vibration and noise control of automotive brake systems.

Complete references given in the book provide comprehensive perspective on the developmentsin friction-induced vibrations and sound, as well as coverage for various applications. For didacticalreasons, the text is not interrupted by the inclusion of references. However, at the end of eachchapter, the relevant published literatures are cited.

REFERENCES

1. Akay, A., Acoustics of friction, Journal of Acoustical Society of America, 111, 4, 1525, 2002.2. Seireg, A.A., Friction-induced sound and vibration, in Friction and Lubrication in Mechanical Design,

Marcel Dekker, New York, 1998, chap. 11.3. Guran, A., Pfeiffer, F., and Popp, K., Dynamics with Friction, World Scientific Publishing, New Jersey,

1996.4. Ibrahim, R.A., Friction-induced vibration, chatter, squeal, and chaos. Part 1: Mechanics of contact and

friction, Applied Mechanics Reviews, 47, 7, 209, 1994.5. Ibrahim, R.A., Friction-induced vibration, chatter, squeal, and chaos. Part 2: Dynamics and modeling,

Applied Mechanics Reviews, 47, 7, 227, 1994.6. Adams, G.G., Concepts in contact recording, ASME, TRIB-Vol 3, 43, 1992.7. Talke, F.E., A review of ‘contact recording’ technology, Wear, 207, 118, 1997.8. Sheng, G., Liu, B., and Zhu, Y.L., Vibrations in contact magnetic recording systems, in Advance in

Information Storage System, Bhushan, B. and Ono, K., Eds., World Scientific Publishing, 1998.9. Sheng, G., Brown, L., Liu, K., and Otremba, J., Advance of Noise, Vibration and Harshness Technology

of Automotive Accessory Belt Drive System, Proceedings of Global Powertrain Congress, Novi,Michigan, 2006.

10. Sheng, G. and Wang, Q., Brake NVH technology, AMC Report, 2002.11. Kinkaid, N.M., O’Reilly, O.M., and Papadopoulos, P., Automotive disc brake squeal, Journal of Sound

and Vibration, 267, 1, 105, 2003.12. Bhushan, B., Tribology and Mechanics of Magnetic Storage Devices, Springer, New York, 1996.13. Bhushan, B., Principles and Applications of Tribology, John Wiley & Sons, New York, 1999.14. Armstrong-Helouvry, B., Dupont, P., and Canudas De Wit, C., A survey of models, analysis tools and

compensation methods for the control of machines with friction, Automatica, 30, 7, 1083, 1994.15. Oden, J.T. and Martins, J.A.C., Models and computational methods for dynamic friction phenomena,

Computer Methods in Applied Mechanics and Engineering, 52, 527, 1985.16. Nayfeh, A.H. and Mook, D.T., Nonlinear Oscillations, Wiley, New York, 1979.



17. Den Hartog, J.P., Mechanical Vibrations, 4th edition, McGraw-Hill, New York, 1956.18. Morse, P.M. and Ingard, K.U., Theoretical Acoustics, McGraw-Hill Book Company, Inc., New

York, 1968.19. Morse, P.M., Vibration and Sound, 2nd edition, McGraw-Hill, New York, 1948.20. Pierce, A.D., Acoustics: An Introduction to Its Physical Principles and Applications, McGraw-Hill Book

Company, Inc., New York, 1981.


Introduction 5


2 Vibrations and Sound

2.1 INTRODUCTION

In this chapter we present the fundamentals of vibrations and sound. The first part of the chapter isdevoted to the analysis of vibrations, which includes three sections. The first section deals with theanalysis of linear systems, including single-degree-of-freedom (SDOF), multiple-degree-of-freedom(MDOF), and continuous systems. In the second section, a general analysis of one-degree-of-freedom nonlinear system is given in order to provide a perspective on nonlinear problems. Thethird section describes the treatment of linear system under stationary random excitation. Thesecond part of the chapter is dedicated to the fundamentals of sound. After introducing the basicconcepts of air sound, the noise and nonstationary sound are described. The sound perceptions andweighting curves are presented. The wave equation of air sound, the sound radiation of structure,and sound interference are discussed briefly. Finally, the fundamental of stress wave propagation insolid is introduced, with an emphasis on the surface wave associating with interface interaction.

2.2 LINEAR VIBRATIONS

2.2.1 VIBRATION MOTION

Vibration is the oscillatory motion of a body or a system. Vibration occurs when a system isdisplaced from its stable equilibrium position, and is under the action of restoring forces. A physicalsystem undergoing a time-varying interchange or dissipation of energy among or within itselementary storage or dissipative devices leads to the vibrationary state.

A dynamic system composed of a finite number of elements is said to be lumped or discrete,while a system containing elements, whose variables in a physical system are functions of locationas well as time, is called continuous. The analytical description of the dynamics of the discrete caseis a set of ordinary differential equations, while for the continuous case it is modeled by a set ofpartial differential equations.

In general, real systems are continuous and their parameters are distributed. However, in mostcases, it is possible to approximate the distributed characteristics of a system by discrete ones.

If the dependent variables in the system differential equation appear to be the first power only,then the system is called linear. If there are fractional or higher powers, then the system is nonlinear.Some system differential equations with dependent variables exhibit to be the first power only, but ifthere exists some variable discontinuity or nonsmooth, the system is also a nonlinear system. Thepresence of a time-varying coefficient does not make a model nonlinear, but this kind of systemusually needs to employ nonlinear system analysis approach for efficient solution. Models withconstant coefficients are known as time-invariant or stationary models, while those with variablecoefficients are time-variant or nonstationary.

The independent coordinates required to specify completely the configuration of a dynamicsystem are called generalized coordinates. The number of generalized coordinates is called thenumber of degrees of freedom of a dynamic system. The number of degrees of freedom of avibratory system is the number of independent spatial coordinates necessary to define its configur-ation. A configuration is defined as the geometric location of all the masses of the system. A rigid

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7

body in space requires six coordinates for its complete identification, in which three coordinates todefine the rectilinear positions and three for the angular positions. Generally speaking, the masses ina system are constrained to move only in a certain manner. Thus, the constraints limit the degrees offreedom of a system.

The excitation varies in accordance with a prescribed function of time. The vibratory behaviorof dynamic systems is characterized by the motions caused by these excitations and is referred to asthe system response. Forced vibrations under excitation may be either deterministic or random. Ifthe vibratory motion is periodic, the system repeats its motion at equal time intervals. The minimumtime required for the system to repeat its motion is called a period; this is the time to complete onecycle of motion. Frequency is defined as the number of times that the motion repeats itself per unittime. Aperiodic motion is the motion that does not repeat itself at equal time intervals. The responseof systems to initial conditions or excitations is known as free vibrations. Free vibrations describethe natural behavior or the natural modes of vibration of a system. A discrete model of a dynamicsystem possesses a finite number of degrees of freedom, whereas a continuous model has an infinitenumber of degrees of freedom. For linear systems, their response is proportional to input andsuperposition is applicable. They closely approximate the behavior of many real dynamic systems.Their response characteristics can be obtained from the form of system equations without a detailedsolution. A closed-form solution is often possible. The numerical analysis techniques have beenwell developed, and they serve as a basis for understanding more complex nonlinear systembehaviors. It should, however, be noted that in most nonlinear problems it is impossible to obtainclosed-form analytic solutions for the equations of motion.

Many systems need to be treated as damped systems. An external force that acts on the systemcauses forced vibrations. In this case, the exciting force continuously supplies energy to the systemto compensate for that dissipated by damping. Forced vibrations may be either deterministic orrandom. The differential equations of motion of the dynamic systems considered in this chapter areall deterministic, i.e., the parameters are not randomly varying with time. However, the excitingforce considered may be either a deterministic or a random function of time. In deterministicvibrations, the amplitude and frequency at any designated future time can be completely predictedfrom the past history; whereas random forced vibrations are defined in statistical terms, and only theprobability of occurrence of designated magnitudes and frequencies can be predicted. Self-excitedvibration is a kind of periodic and deterministic oscillations, for which the energy required to sustainthe vibration is obtained from a nonalternating power source, and this vibration creates the periodicforce that excites the vibrations themselves. Under certain conditions, the equilibrium state in such avibration system becomes unstable, and any disturbance causes the perturbations to grow until someeffect limits any further growth.

2.2.1.1 Sinusoidal Periodic Motion

We start by considering sinusoidal periodic motion expressed by a function,

x(t) ¼ X sin(2pf0t þ u) (2:1)

in which X is amplitude, f0 is cyclical frequency in cycles per unit time, u is initial phase angle withrespect to the time origin in radians, x(t) is instantaneous value at time t.

The time interval required for one full fluctuation or cycle of sinusoidal data is called period TP.The number of cycles per unit time is called the frequency f0. The frequency and period are related by

TP ¼ 1f0

(2:2)

Such spectra are discrete spectra or line spectra.



2.2.1.2 Complex Periodic Motion

Complex periodic motion can be defined mathematically by a time-varying function whose wave-form exactly repeats itself at regular intervals such that

x(t) ¼ x(t � nTP), n ¼ 1, 2, 3, . . . (2:3)

The complex periodic motion can be expanded into a Fourier series as

x(t) ¼ a02þX1n¼1

(an cos 2pnf1t þ bn sin 2pnf1t) (2:4)

in which f1¼ 1=TP

an ¼ 2TP

ðTP0x(t) cos 2pnf1t dt, n ¼ 0, 1, 2, . . .

bn ¼ 2TP

ðTP0x(t) sin 2pnf1t dt, n ¼ 0, 1, 2, . . .

Another way of expressing the Fourier series for complex periodic data is

x(t) ¼ X0 þX1n¼1

Xn sin(2pnf1t � un) (2:5)

in which X¼ a0=2, Xn ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2n þ b2n

p, un¼ tan�1(an=bn), n¼ 1, 2, . . . . Equation 2.5 implies that

complex periodic data consists of a static component, X0, and an infinite number of sinusoidalcomponents called harmonics, which have amplitudes Xn and phase un. The frequencies of theharmonic components are all integral multiples of f1. The phase angles are often ignored whenperiodic data is analyzed in practice. For this case, Equation 2.5 can be characterized by a discretespectrum.

2.2.1.3 Transient Nonperiodic Motion

Transient motion is defined as all nonperiodic motions other than the almost-periodic data discussedabove. Transient motions include all motions, which can be described by some suitable time-varying function. Physical phenomena which produce transient data are numerous and diverse.The important characteristic of transient motion is its continuous spectral representation. Transientdata can be obtained in most cases from a Fourier integral given by

X( f ) ¼ð1

�1X(t)e�j2pft dt (2:6a)

The Fourier spectrum X( f ) is generally a complex number, which can be expressed in complexpolar notation as

X( f ) ¼ X( f )j je�ju( f ) (2:6b)

wherejX( f )j is the magnitude of X( f )u( f ) is the argument


Vibrations and Sound 9

2.2.2 LINEAR SINGLE-DEGREE-OF-FREEDOM (SDOF) SYSTEM

Next we discuss the simplest linear vibration dynamic system. We consider an SDOF model of alinear dynamic system as shown in Figure 2.1. From Newton’s third law we obtain

F(t)� Fs(t)� Fd(t) ¼ mx€(t) (2:7)

whereF(t), Fs(t), and Fd(t) are the exciting, spring, and damping forces, respectivelym denotes the mass of the bodyx€(t) is its acceleration

Because Fs(t)¼ kx(t) and Fd(t)¼ cx_(t), Equation 2.7 becomes

mx€(t)þ cx_(t)þ kx(t) ¼ F(t) (2:8)

where c and k are the viscous damping and stiffness coefficients, respectively.Equation 2.8 is the equation of motion of the linear SDOF and is a second-order linear

differential equation with constant coefficients.

2.2.2.1 Free Vibration of an SDOF System with Viscous Damping

In the case of the free vibration of an SDOF system, the exciting force F(t)¼ 0 and the equation ofmotion is

mx€(t)þ cx_(t)þ kx(t) ¼ 0 (2:9)

If we define v2n ¼ k=m and j¼ c=2mvn, Equation 2.9 can be written as

x€(t)þ 2jvnx_(t)þ v2nx(t) ¼ 0 (2:10)

To solve Equation 2.10, we assume

x(t) ¼Aest (2:11)

whereA is a constants a parameter to be determined

By substituting (2.11) into (2.10), we obtain

(s2 þ 2jvnsþ v2n)Ae

st ¼ 0 (2:12)

m F (t )

k

c

x(t )

FIGURE 2.1 Linear single-degree-of-freedom system.



Since Aest 6¼ 0, then

s2 þ 2jvnsþ v2n ¼ 0 (2:13)

Equation 2.13 is known as the characteristic equation of the system. This equation has the followingtwo roots:

s1, s2 ¼ �j �ffiffiffiffiffiffiffiffiffiffiffiffiffij2 � 1

q� �vn (2:14)

For Case a, j< 1 (Underdamped Condition):


q� �vn

x(t) ¼ A exp(�ivnt) cos vn

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� j2

qt � f

� �(2:15)

x(t) ¼ A exp(�ivnt) cos(vdt � f) (2:16)

wherevn is the natural circular frequencyj is the damping factor

vd ¼ vn


pis the damped frequency of the system

Constants A and f are determined from the initial conditions.

For Case b, j> 1 (Overdamped Condition):


q� �vn

x(t) ¼ A1 exp �j þffiffiffiffiffiffiffiffiffiffiffiffiffij2 � 1

q� �vnt þ A2 exp �j �

ffiffiffiffiffiffiffiffiffiffiffiffiffij2 � 1

q� �vnt (2:17)

The motion is aperiodic and decays exponentially with time. Constants A1 and A2 are determinedfrom the initial conditions.

For Case c, j¼ 1 (Critically Damped Condition):

s1 ¼ s2 ¼ �vn

x(t) ¼ (A1 þ A2) exp(�vnt) (2:18)

Equation 2.18 represents an exponentially decaying response. The constants A1 and A2 depend onthe initial conditions. For this case, the coefficient of viscous damping has the valuecc ¼ 2mvn ¼ 2

ffiffiffiffiffiffiffikm

p.

Hence,

j ¼ c=cc (2:19)

The locus of the roots s1 and s2 can be represented on a complex plane, as shown in Figure 2.2. Thispermits an instantaneous view of the effect of the parameter j on system response. For an undampedsystem with j¼ 0, the imaginary roots are �ivn. For a system 0< j< 1, the roots s1 and s2 are



complex conjugates that are located symmetrically with respect to the real axis on a circle of radiusvn. For j¼ 1, s1¼ s2¼�vn, and as j ! 1, s1 ! 0, and s2 ! �1.

We further consider the undamped condition in which t1 and t2 denote the times correspondingto the consecutive displacements x1 and x2, measured one cycle apart. By using Equation 2.16, wecan write

x1x2

¼ A exp(�ivnt1) cos(vdt1 � f)

A exp(�ivnt2) cos(vdt2 � f)(2:20)

Since t2¼ t1þ T¼ t1þ 2p=vd, then cos(vdt1�f)¼ cos(vdt2�f). Equation 2.20 then reduces to

x1=x2 ¼ exp(jvnT) (2:21)

We define logarithmic decrement as

d ¼ ln(x1=x2) ¼ jvnT ¼ 2pj=ffiffiffiffiffiffiffiffiffiffiffiffiffi1� j2

q(2:22)

To determine the amount of damping in the system, it is sufficient to measure any two consecutivedisplacements x1 and x2 and obtain j from the equation

j ¼ d� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

(2p)2 þ d2q

(2:23)

For small damping, d is a small quantity, and Equation 2.23 becomes

j ffi d=2p (2:24)

2.2.2.2 Free Vibration of an SDOF System with Both Viscous and Friction Damping

Next we consider an SDOF system with friction. The motion equation is described as

mx€(t)þ cx_(t)þ kx(t)þ F(x_(t)) ¼ 0 (2:25)

in which Coulomb friction force is

F(x_(t)) ¼ ff sign(x_(t)) (2:26)

0 < x < 1

wn

x = 0

x = 0 –wn

Re

Im

0s1 s2

wn

s1

s2

x > 1

1 − x2wn

FIGURE 2.2 Complex planar representation of roots s1 and s2.



whereff is constant friction forcesign(.) is sign function

Equation 2.25 is piecewise solvable. We can further write Equation 2.25 as

x€(t)þ 2jvnx_(t)þ v2nx(t) ¼ �v2

nxf (t) for x_(t) > 0 (2:27)

x€(t)þ 2jvnx_(t)þ v2nx(t) ¼ v2

nxf (t) for x_(t) < 0 (2:28)

where xf¼ ff=k. Assume we start with x(t0)¼X0> xf, x_(t0)¼ 0, then the motion starts with x_< 0, andthe equation has the following solution:

x(t) ¼ (X0 � xf ) exp(jvn(t � t0))(cosvd(t � t0)þ bsinvd(t � t0))þ xf (2:29)

in which b ¼ j=ffiffiffiffiffiffiffiffiffiffiffiffiffi1� j2

p¼ d=(2p). This solution is valid until x_(t)¼ 0 at which time t¼ t1¼ t0 þ

p=vd and X1¼ x(t1)¼�e�bp X0 þ (e�bp þ 1)xf. If X1<�xf then the mass will reverse directionand continue to slide with x_(t)> 0 according to the motion equation. The solution for this interval ofmotion is

x(t) ¼ (X0 þ xf ) exp(�jvn(t � t1))(cosvd(t � t1)þ bsinvd(t � t1))� xf (2:30)

which is valid until x_(t) ¼ 0, at which time t¼ t2¼ t1 þ p=vd and

X2 ¼ x(t2) ¼ e�bpX1 � (e�bp þ 1)xf (2:31)

If X2> xf, motion will continue and so on until xf� Xn��xf. At this moment the motion stops. Thisiterated process results in a recursive relation for the successive peaks and valleys in the oscillatoryresponse:

Xi ¼ �e�bpXi�1 � (�1)i�1(e�bp þ 1)xf , i ¼ 1, 2, . . . , n (2:32)

From this evolution of decaying peaks and valleys, the viscous effect can be separated, and thenextract the Coulomb effect. A sum of consecutive extreme displacement values cancels out the dry-friction contribution. Taking the ratio between successive sum gives

(Xi þ Xiþ1)=(Xi�1 þ Xi) ¼ �e�bp (2:33)

2.2.2.3 Forced Vibration of an SDOF System with Viscous Damping

We now consider the response of an SDOF system to a harmonic excitation, for which the equationof motion is

mx€(t)þ cx_(t)þ kx(t) ¼ F0 cos vt (2:34)

whereF0 is the amplitudev the frequency of the excitation

Equation 2.34 can be simplified as

x€(t)þ 2jvnx_(t)þ v2nx(t) ¼ (F0=k)v

2n cos vt (2:35)



The solution of Equation 2.35 consists of two parts, the complementary function, which is thesolution of the homogeneous equation, and the particular integral. The complementary function diesout with time for j> 0 and is often called the transient solution, whereas the particular solution doesnot vanish for a large t and is referred to as the steady-state solution to the harmonic excitation. Weassume a solution of the form

x(t) ¼ X cos(vt � f) (2:36)

where X and f are the amplitude and phase angle of response, respectively.By substituting Equation 2.36 into Equation 2.35, we obtain

X½(v2n � v2) cos(vt � f)� 2jvnv sin(vt � f)� ¼ (F0=k)v

2n cosvt (2:37)

By developing the terms in above equation, and equating the coefficients of cos vt and sin vt onboth sides of Equation 2.37, we obtain

X½(v2n � v2) cosfþ 2jvnv sinf� ¼ (F0=k)v

2n (2:38a)

X½(v2n � v2) sinf� 2jvnv cosf� ¼ 0 (2:38b)

By solving Equation 2.38, we get

X=(F0=k) ¼ f½1�(v=vn)2�2 þ ½2j(v=vn)�2g�1=2

(2:39)

and

f ¼ tan�1f½2j(v=vn)�=½1� (v=vn)2�g (2:40)

Equations 2.39 and 2.40 indicate that the nondimensional amplitude X=(F0=k) and the phase anglef are functions of the frequency ratio v=vn and the damping ratio j. For v=vn� 1, both theinertia and damping forces are small, and this results in a small phase angle f, with X=(F0=k)ffi 1.However, for v=vn� 1, the phase angle f! 1808 and X=(F0=k)! 0. For v=vn¼ 1, the phase anglef¼ 908 and X=(F0=k)¼ 1=2j. In summary, the complete solution of Equation 2.35 is given as

x(t) ¼ A1 exp(�ivnt) cos(vdt þ f1)þF0

k

cos(vt � f)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� (v=vn)

2� �2þ 2j(v=vn)½ �2

q (2:41)

where the constants A1 and f1 are determined by the initial conditions.Let us reconsider Equation 2.35 and represent the excitation by the complex form

(F0=k)v2ne

�ivt ¼ Xsv2ne

�ivt (2:42)

where Xs¼F0=k and is referred to as a static response. We assume a solution in the form

x(t) ¼ Xe�ivt (2:43)

By substituting Equation 2.43 into Equation 2.35, we get

½v2n � v2 � 2ijvnv�Xe�ivt ¼ Xsv

2ne

�ivt (2:44)



X=Xs ¼ ½1� (v=vn)2 � 2ij(v=vn)��1 ¼ H(v) (2:45)

where H(v) is known as the complex frequency response function. Its magnitude jH(v)j refers to amagnification factor and is given as

H(v)j j ¼ f½1� (v=vn)2�2 þ ½2j(v=vn)�2g�1=2

(2:46)

The phase angle f will be

f ¼ tan�1 2j(v=vn)

1� (v=vn)2

� �(2:47)

The excitation considered thus far has been a simple harmonic force. We can generalize the resultswhen the exciting force is periodic, because periodic force can be expanded in terms of the Fourierseries as follows:

F(t) ¼ a1 sinvt þ b1 cosvt þ a2 sin 2vt þ b2 cos 2vt þ � � � þ an sin nvt þ bn cos nvt (2:48)

where an and bn are the coefficients of the Fourier series expansion and it has been assumed that theconstant b0¼ 0. Because

an sin nvt þ bn cos nvt ¼ fn sin(nvt þ an) (2:49)

wherefn ¼ a2n þ b2n

� �1=2an¼ tan�1(bn=an)

it follows that

F(t) ¼ f1 sin(vt þ a1)þ f2 sin(2vt þ a2)þ � � � þ fn sin(nvt þ an) (2:50)

Because superposition is valid, we can consider each term on the right-hand side of Equation 2.50 asa separate forcing function and get the steady-state response by adding individual responses due toeach forcing function acting separately. Hence, it follows that

x(t) ¼ X1 cos(vt þ a1 � f1)þ X2 cos(2vt þ a2 � f2)þ � � � þ Xn cos(nvt þ an � fn) (2:51)

in which

Xn ¼ fn=k

1� (nv=vn)2

� �2þ 2j(nv=vn)½ �2n o1=2

and

fn ¼ tan�1 2j(nv=vn)

1� (nv=vn)2

� � , n ¼ 1, 2, . . .

Hence, the steady-state response is also periodic, with the same period as the forcing function, butwith different amplitude and an associated phase lag.

We now discuss an alternative method for the analysis of the forced vibrations of an SDOFdamped system. The approach uses a frequency domain analysis and is based on the concept of



transfer and harmonic response functions. By taking the Laplace transformation of Equation 2.35,we get

m½s2x�(s)� x(0)s� x_(s)� þ c½sx�(s)� x(0)� þ kx�(s) ¼ F�(s) (2:52)

and it follows that

x�(s) ¼ F�(s)

ms2 þ csþ kþ (msþ c)x(0)þ mx_(0)

ms2 þ csþ k(2:53)

Equation 2.53 can be written as

x�(s) ¼ A(s)=B(s) (2:54)

whereA(s) and B(s) are polynomials with B(s) of a higher order

The response x(t) is found by taking the inverse Laplace transformation of Equation 2.54. Ifonly the forced solution is considered, we can define impedance transform as

Z(s) ¼ ½F�(s)=x�(s)� ¼ ms2 þ csþ k (2:55)

A transfer function H(s) is defined by

H(s) ¼ 1=Z(s) ¼ (ms2 þ csþ k)�1 (2:56)

and relates the input to the output of the system as follows:

x�(s) ¼ H(s) F�(s) (2:57)

The inverse Laplace transformation of x�(s) in Equation 2.57 yields

x(t) ¼ 1mvd

ðt0

F(t) exp �jvn(t � t)½ � sinvd(t � t) dt (2:58)

2.2.2.4 Forced Vibration of an SDOF System with Friction Damping

Next we consider the forced vibration of an SDOF system with friction damping. The equation ofmotion can be written as

mx€þ cx_ þ kxþ Ff (x_) ¼ F(t) (2:59)

wherem, c, and k are the mass, damping, and spring constantF(t)¼ kY0 cos(vt þ f) is the base excitation motionFf(x_)¼Ff sgn(x_) is the Coulomb friction force

For the case of x_< 0, we have

x€þ c

mx_ þ v2

n(x� xf ) ¼ v2nY0 cos(vt þ f) (2:60)



where xf¼Ff=k denotes the equivalent friction displacement. There exist nonsticking responses withpeak value x0, which occurs at the turning points. The value of x0 is given by

x0Y0

¼ �GxfY0

� �þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1q2

� H2xfY0

� �2s

(2:61)

whereG andH are parameter functions independent of theCoulomb friction, and can be represented as

G ¼ sinh(jbp)� j=ffiffiffiffiffiffiffiffiffiffiffiffiffi1� j2

psin(vdp=v)

cosh(jbp)þ cos(vdp=v)(2:62)

H ¼ b=ffiffiffiffiffiffiffiffiffiffiffiffiffi1� j2

psin(vdp=v)

cosh(jbp)þ cos(vdp=v)(2:63)

and q is the inverse response function with pure viscous damping:

q ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� v

vn

� �2" #

þ 2jv

vn

� �2vuut (2:64)

In the above expression, j is the nondimensional damping ratio, b¼vn=v, and vd ¼ vn


prepresents the frequency of damped oscillation.

Note that G and H in Equations 2.62 and 2.63 are functions of j, vn, and v only. Given a systemwith fixed parameter values, these parameter functions can be treated as constants.

On the other hand, note that by allowing the work per cycle done by the friction to be equal tothat of a equivalent damping force, we can get the equivalent damping constant c for friction Ff,c¼ 4Ff=pvx0, then we get another expression.

x0 ¼ Y0k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4Ff

pY0

� �s

1� v2

v2n

(2:65)

2.2.3 LINEAR MULTIPLE-DEGREE-OF-FREEDOM (MDOF) SYSTEM

Next we consider the MDOF discrete system shown in Figure 2.3, its general equations of motionare written as

½m�f€xg þ ½c�f _xg þ ½k�fxg ¼ fF(t)g (2:66)

m1

k1

c1

x1

f1

mi

ki

ci

ki +1

ci +1 cn +1

kn +1

mn… …

xi

fi

xn

fn

FIGURE 2.3 Linear multi-degree-of-freedom system.



where {F(t)} denotes the externally applied force. In Equation 2.66, [m], [c], and [k] are n3 n mass,damping, and stiffness matrices, respectively. For linear systems, these matrices are constant,whereas for nonlinear systems, the elements of these matrices are functions of generalized displace-ments and velocities that are time dependent.

The response {x(t)} of Equation 2.66 consists of two parts: first, {xh(t)} the homogeneoussolution which is the transient response; and, second, {xp(t)} the particular solution which is thesteady-state or forced response.

2.2.3.1 Eigenvalues and Eigenvectors

2.2.3.1.1 Undamped SystemNext we discuss the simplest case, the equations of motion for the free vibration of an undampedMDOF system, with focus on its eigenvalues and eigenvectors. By setting [c] and {F(t)} in Equation2.66 to be zero, it follows that

½m�f€xg þ ½k�fxg ¼ f0g (2:67)

We use a linear transformation to replace {x} by

fxg ¼ ½f�fyg (2:68)

where [f] is a constant nonsingular square matrix to be specified in the following analysis. It isreferred to as a transformation matrix,

f€xg ¼ ½f�f€yg (2:69)

By substituting Equations 2.68 and 2.69 into Equation 2.67, we obtain

½m�½f�f€yg þ ½k�½f�fyg ¼ f0g (2:70)

Premultiply both sides of Equation 2.70 by [f]T to yield

½f�T ½m�½f�f€yg þ ½f�T ½k�½f�fyg ¼ f0g (2:71)

From Equation 2.71, it follows that

½M*�f€yg þ ½K*�fyg ¼ f0g (2:72)

where [M*] and [K*] are diagonal matrices known as the generalized mass and the stiffnessmatrix, respectively. Equation 2.72 refers to the uncoupled homogeneous equations of motion ofthe system. It follows that the uncoupled equation of motion for the ith degree of freedom is

y€i þ v2i yi ¼ 0 (2:73)

where vi is the frequency corresponding to the ith mode of vibration. The solution of Equation 2.73is given as

yi(t) ¼ Ai sinvit þ Ai* cosvit (2:74)

where arbitrary constants Ai and Ai* are determined by the initial conditions xi(0) and x_i(0).



We now consider Equation 2.67 and premultiply both sides by [m]�1 to yield

½m��1½m�f€xg þ ½m��1½k�fxg ¼ f0g (2:75)


½I�f€xg þ ½D�fxg ¼ f0g (2:76)

where[I] is the unit matrix[D]¼ [m]�1[k] which is known as the dynamic matrix

Let us assume a harmonic motion so that

fxg ¼ fAgeivt (2:77)

Equation 2.77 yields

f€xg ¼ �v2fAgeivt ¼ �lfxg (2:78)

where l¼v2. Substituting Equation 2.78 into Equation 2.76 results in

½½D� � l½I��fxg ¼ f0g (2:79)

The characteristic equation of the system is then the determinant being zero, i.e.,

½D� � l½I�j j ¼ 0 (2:80)

The roots li of the characteristic equation are called eigenvalues. The natural frequencies of thesystem are determined from

li ¼ v2i (2:81)

By substituting li into the matrix Equation 2.79, we obtain the corresponding mode shapes, whichare called the eigenvectors. Thus, for an n-degree-of-freedom system, there are n eigenvalues and neigenvectors.

Let us consider two distinct solutions corresponding to the rth and the sth modes, respectively,v2r , {f

(r)} and v2s , {f

(s)} of the eigenvalue problem. Because these solutions satisfy Equation 2.67,it follows that

½k�ff(r)g ¼ v2r ½m�ff(r)g (2:82)

and

½k�ff(s)g ¼ v2s ½m�ff(s)g (2:83)

We premultiply both sides of Equation 2.82 by {f(s)}T and both sides of Equation 2.83 by {f(r)}T

to obtain

ff(s)gT ½k�ff(r)g ¼ v2rff(s)gT ½m�ff(r)g (2:84)



ff(r)gT ½k�ff(s)g ¼ v2sff(r)gT ½m�ff(s)g (2:85)

Now we take the transpose of Equation 2.85 to get

ff(s)gT ½k�ff(r)g ¼ v2sff(s)gT ½m�ff(r)g (2:86)

By subtracting Equation 2.86 from Equation 2.84, we obtain

(v2r � v2

s )ff(s)gT ½m�ff(r)g ¼ 0 (2:87)

Because vr 6¼ vs, we conclude that

ff(s)gT ½m�ff(r)g ¼ 0, r 6¼ s (2:88)

Equation 2.88 represents the orthogonality condition of modal vectors. It can also be shown that

ff(s)gT ½k�ff(r)g ¼ 0, r 6¼ s (2:89)

Thus, [f] is composed of {f(i)}, i¼ 1, 2, . . . , n. If each column of the modal matrix [f] is dividedby the square root of the generalized mass Mi*, the new matrix ½�f� is called the weighted modalmatrix. It can be easily be seen that

�f� �T

m½ � �f� � ¼ I½ � (2:90)

and

k½ � �f� � ¼ m½ � �f

� �v2� �

(2:91)

Premultiplying (2.91) by ½�f�T results in

�f� �T

k½ � �f� � ¼ �f

� �Tm½ � �f� �

v2� � ¼ v2

� �(2:92)

2.2.3.1.2 Damped SystemNext we consider the case with damping. The equation of motion for the free vibration of a dampeddiscrete MDOF system is given as

½m�f€xg þ ½c�f _xg þ ½k�fxg ¼ f0g (2:93)

We define a vector

fyg ¼ _xf gxf g

and use an identity ([m]{ _x} � [m]{ _x}¼ 0) to obtain

0½ � j m½ �� m½ � j c½ �

24

352n 2n

€xf g_xf g

2n 1

þ� m½ � j 0½ �� 0½ � j k½ �

24

352n 2n

_xf gxf g

2n 1

¼ 0f g0f g

(2:94)



Equation 2.94 can be rewritten as

½A�f _yg þ ½B�fyg ¼ f0g (2:95)

where matrices [A] and [B] are defined as

½A� ¼0½ � j m½ �

�� m½ � j c½ �

24

35, ½B� ¼

� m½ � j 0½ �� 0½ � j k½ �

24

35

By premultiplying both sides of Equation 2.95 with [A]�1, we obtain

f _yg � ½H�fyg ¼ f0g (2:96)

where [H]¼�[A]�1[B].Let us assume a solution of Equation 2.96 as

fyg ¼ fCgegt (2:97)

in which g is a complex number and {C} a modal vector with complex elements.Substituting Equation 2.97 into Equation 2.96 yields

g I½ � � H½ �½ �fCg ¼ f0g (2:98)

where [I ] is the unit matrix. Hence, the characteristic equation of the system is

g I½ � � H½ �j j ¼ 0 (2:99)

The roots gi of the characteristic equation represent 2n eigenvalues, which are complex conjugates.By substituting gi into Equation 2.98, we can obtain corresponding eigenvectors, which are alsocomplex conjugates. The modal matrix [c] is given as

c½ � ¼ fCg1fCg2 � � � fCg2n½ � (2:100)

By employing simple matrix manipulations, the orthogonality condition of eigenvectors can beestablished, and it can be shown that the following relationships are valid:

fCgTr A½ �fCgs ¼ 0 for r 6¼ s (2:101a)

fCgTr B½ �fCgs ¼ 0 for r 6¼ s (2:101b)

From Equations 2.101a,b, it follows that

½C�T ½A�½C� ¼ ½U� (2:102a)

½C�T ½B�½C� ¼ ½V � (2:102b)

where [U ] and [V ] are the diagonal matrices.



2.2.3.2 Forced Vibration Solution of an MDOF System

We consider Equation 2.66 and first solve the undamped free-vibration problem to obtain theeigenvalues and eigenvectors, which describe the normal modes of the system and the weightedmodal matrix ½�f�. Let

fxg ¼ �f� �fyg (2:103)

Substituting Equation 2.103 into Equation 2.66 yields

½m�½ �f�f€yg þ ½c�½ �f�f _yg þ ½k�½ �f�fyg ¼ fF(t)g (2:104)

Premultiply both sides of Equation 2.104 by ½�f�T to obtain

½ �f�T ½m�½ �f�f€yg þ ½ �f�T ½c�½ �f�f _yg þ ½ �f�T ½k�½ �f�fyg ¼ ½ �f�TfF(t)g (2:105)

Notice that the matrices ½�f�T [m]½�f� and ½�f�T [k]½�f� on the left-hand side of Equation 2.105 arediagonal matrices that correspond to the matrices [I] and [v2], respectively. However, the matrix½�f�T [c]½�f� is not a diagonal matrix. If [c] is proportional to [m] or [k] or both, the ½�f�T [c]½�f� becomesdiagonal, in which case we can say that the system has proportional damping. The equations ofmotion are then completely uncoupled, and the ith equation will be

y€i þ 2jiviy_i þ v2i yi ¼ f�i tð Þ, i ¼ 1, 2, . . . , n (2:106)

where �fi(t) ¼ f�f(i)gfF(t)g. Thus, instead of n-coupled equations, we will have n-uncoupledequations.

Let [c]¼a[m] þ b[k], in which a and b are proportionality constants. Then we have,

½ �f�T ½c�½ �f� ¼ ½ �f�T a½m� þ b½k�ð Þ½ �f� ¼ a½I� þ b v2� �

(2:107)

This will yield the uncoupled ith equation of motion as

y€i þ (aþ bv2i )y_i þ v2

i yi ¼ f�i(t) (2:108)

and the modal damping can be defined as

2jivi ¼ aþ bv2i (2:109)

The solution of Equation 2.106 is obtained by using Equation 2.58 with initial conditions yi(0)and y_i(0),

yi(t) ¼ 1vd

ðt0

f�i(t) exp �jivi(t � t)½ � sinvdi(t � t)dt þ yi(0) exp(�jivit)

(1� j2i )1=2

cos(vdit � ci)þy_i(0)vdi

exp(�jivit) sinvdit (2:110)

wherevdi ¼ 1� j2i

� �1=2vi

ci ¼ tan�1 ½ji= 1� j2i� �1=2�



Similarly, the contribution from each normal mode is calculated and substituted in Equation2.103 to obtain the complete response of the system. This is known as the normal mode summationmethod. The contributions of the higher vibration modes to the system response are often quitesmall, and for all practical purposes may be ignored in the summation procedure by consideringfewer modes of vibration.

2.2.4 VIBRATION OF CONTINUOUS SYSTEMS

So far we have discussed discrete systems where elasticity and mass are modeled as discreteproperties. Discrete systems have a finite number of degrees of freedom specifying system finiteconfiguration. Continuous systems are distributed systems such as strings, cables, rods (bars), andbeams, as well as plates where elasticity and mass are distributed parameters. We consider thecontinuous distribution of elasticity, mass, and damping and assume each of the infinite number ofelements of the system can vibrate.

The displacement of these elements is described by a continuous function of position and time.The governing equations of motion for discrete systems are ordinary differential equations whereasthe governing equations are partial differential equations for the continuous systems, and exactsolutions can be obtained for only a few special configurations. For the vibration analysis of systemswith distributed elasticity and mass, it is necessary to assume that the material is homogeneous andisotropic and follows Hooke’s law.

2.2.4.1 Transverse Vibrations of String

Consider a stretched flexible string of mass r per unit length having its end points attached to fixedsurfaces. The string is free to vibrate in a vertical (x–y) plane as shown in Figure 2.4a. Thecoordinate of y is a function of both position along the string x and time t or

y ¼ y(x, t) (2:111)

The equilibrium position of the string is shown by the thick black line in Figure 2.4a and itsdifferential element in any possible position of motion is shown in Figure 2.4b. In order to developthe governing equation of motion for the string, the following assumptions are made: resistance ofair and internal friction and gravitational forces are neglected in comparison with the tension in the

(b)

F

Fdxdx

∂x 2

∂

2y+∂x∂y

∂x∂y

(a)

x

y

l

FIGURE 2.4 Schematic of (a) the string and (b) the differential element.



string which is quite large. The displacement of any point in the string is very small and occurs onlyin the x–y plane.

Denoting the tension in the string as T and the change in slope as @2y=@x2� �

dx and consideringthe vertical motion of the differential element in Figure 2.4b, we can write Newton’s second law as

�Tsin@y

@xþ Tsin

@y

@xþ @2y

@x2dx

� �¼ r dx

@2y

@t2(2:112)

Since the slopes are very small, we use the approximation sin u u and Equation 2.112 becomes

�T@y

@xþ T

@y

@xþ @2y

@x2dx

� �¼ r dx

@2y

@t2(2:113)

or

T@2y

@x2¼ r

@2y

@t2(2:114)

Equation 2.114 is a linear, second-order, partial differential equation with constant coefficientsand represents the governing equation for transverse motion of the string. Rewrite Equation 2.114 as

c2@2y

@x2¼ @2y

@t2(2:115)

where c ¼ ffiffiffiffiffiffiffiffiT=r

p. Equation 2.115 is called the one-dimensional wave equation and the constant c is

called the wave speed. We assume the solution as

y(x,t) ¼ F(x)G(t) (2:116)

Then substitute the above equation to Equation 2.115, we have

c2@2F

@x2

F ¼ @2G

@t2

G ¼ m (2:117)

in which m is a constant. The initial conditions are

y(x,0) ¼ F(x)G(0) ¼ F(x)G0 (2:118a)

@y(x,0)@t

¼ F(x) _G(0) ¼ F(x) _G0 (2:118b)

The boundary conditions are

y(0,t) ¼ F(0)G(t) ¼ 0 (2:119)

y(‘,t) ¼ F(‘)G(t) ¼ 0 (2:120)

The solution of Equation 2.117 depends on the value of m. It can easily be verified that m< 0 is theonly possibility which satisfies both the differential equations and the boundary conditions. Hencefor m< 0, we assume m¼�v2, Equation 2.117 can be rewritten as



€Gþ v2G ¼ 0 (2:121)

F00 þ v2

c2F ¼ 0 (2:122)

which have the solutions

G(t) ¼ A sinvt þ B cosvt (2:123)

F(x) ¼ C sinv

cxþ D cos

v

cx (2:124)

The displacements are then given by

y(x,t) ¼ (A sinvt þ B cosvt) C sinv

cxþ D cos

v

cx

� �(2:125)

Now applying the boundary conditions given by Equations 2.119 and 2.120 will give

sinv

c‘ ¼ 0 (2:126)

or

v

c‘ ¼ p, 2p, . . . , np, . . . (2:127)

Equation 2.127 will give an infinite number of natural frequencies

vn ¼ npc

‘¼ np

‘

ffiffiffiffiT

r

s, n ¼ 1, 2, 3, . . . (2:128)

In Equation 2.128 each vn corresponds to a principal mode having the harmonic mode shapesin(np=‘). The general solution can be written as

y(x,t) ¼X1n¼1

(An sinvnt þ Bn cosvnt) sinvn

cx (2:129)

whereAn¼ACBn¼BD for each mode

Now consider the case of initially deforming the string into the shape of a half sine curve andreleasing it from rest. The initial conditions are then

y(x,0) ¼ y0 sinp

‘x (2:130)

and

@y(x,0)@t

¼ 0 (2:131)

Now, applying the initial conditions (2.130) and (2.131) to Equation 2.129 gives

An ¼ 0 for all n

B1 ¼ y0Bn ¼ 0, n 6¼ 1 (2:132)



The displacement of every point on the string is then given by

y(x,t) ¼ y0 cospc

‘t sin

p

‘x (2:133)

Equation 2.133 represents the vibration of the string in its fundamental mode and in this modeevery point moves harmonically with the amplitude y0 sin(px=‘). The shape of the string is ahalf sine wave at all times. The ratio of the amplitude of any two points is always the same. Theentire string goes through the equilibrium position simultaneously. There is no apparent progress inaxial direction and the string appears to move up and down. Such motion is referred to as astanding wave. The wavelength l is defined as the length of one complete sine wave, sin(npx=‘).That is, for

np l

‘¼ 2p (2:134)

or

l ¼ 2‘n

where l is the wavelength. Figure 2.5 shows the standing waves for the first three principalmodes.

Mode 1: n = 1 w =

lp

ρ

Txl= 2l

Mode 2: n = 2 w =

l2p

ρ

Txl= l

Mode 3: n = 3 w =

l3p

ρ

Txl =

3

2l

3

x

yx = l

y0

x

yx = l

y0

x = 2

l

x =

32l

x

yx = l

y0

x = l

FIGURE 2.5 Natural modes of string.



2.2.4.2 Wave Equation

It can be showed that the wave Equation 2.115 has an explicit solution given by

y(x,t) ¼ yr(x� ct)þ y1(xþ ct) (2:135)

where yl and yr are interpreted as left-going and right-going traveling waves, respectively. Therefore,it is possible to consider the solution to the wave equation as a sum of traveling waves, through whicha single traveling wave is investigated and simulated using other approach instead of modal analysis.

Traveling waves can be described by a simple harmonic solution at each point on the wave. Ateach point waves still have a frequency or period, which describes their periodicity in time.However, waves also have a wavelength which determines how often they repeat in space. Wetherefore need to introduce two new quantities to describe a wave, one is the above-mentioned wavevelocity c, and the other one is wavelength l,

l ¼ c=f ¼ 2pc=v (2:136)

This is the most important equation in wave motion and Equation 2.134 is a special case of thisequation. There are other quantities related to the wavelength and containing the same concepts, thewave number k¼ 2p=l is often used instead of the wavelength.

There are two broad classes of waves, transverse waves and longitudinal waves. The mostfamiliar waves are the transverse waves, such as the waves on a string as above discussed and thewaves at the surface of liquids such as water. They are called transverse waves because thedisplacements are perpendicular or transverse to the direction of wave motion. The height of atransverse traveling wave is given by

y(x,t) ¼ A cos(kx� vt) (2:137)

For each situation where a wave occurs, the frequency, wavelength, and velocity are related to theproperties of the material. Radio waves, light, and many other waves are also transverse waves. Analternative type of wave is a longitudinal wave. The most notable example of this type of wave is asound wave. Sound waves are really oscillations in the pressure inside the material. Sound wavescan occur in gases, liquids, and solids and in each case the sound velocity is different as we shall seein the last section of this chapter.

When two waves meet, they interfere and the following two cases illustrate what can happen. Atthe same time, if two waves have crests at the same place, they get added and this is calledconstructive interference, and the waves are said to be in phase. If two waves are out of phase, sothat one wave has a maximum just where the other has a minimum, the two waves interferedestructively. In fact, if the two waves have exactly the same amplitude, they annihilate eachother. The property of interference is important and many scientific and engineering devices rely onunderstanding and controlling wave interference, like active noise control. When a wave hits abarrier, it can reflect. If the barrier is hard, then the wave inverts on reflection.

In contrast to the modal analysis detailed above, the wave approach to solve wave equation is anoption as an efficient technique suitable for real-time implementation by using digital waveguidemodels. It has been so developed that the representations of structural response in terms of dampedwaves are described by complex wavenumbers. For quasi-harmonic systems such as violin strings,digital waveguides are the preferred choice. For systems which show a strongly inharmonicspectrum, usually modal analysis is preferred.

2.2.4.3 Longitudinal Vibration of Rods

Consider the vibration of a uniform slender rod. The rod can execute axial or longitudinal vibrationswhen the equilibrium condition is disturbed axially due to the axial forces. Figure 2.6 shows the



free-body diagram of a differential element of this rod of length dx. The equilibrium position of theelement is denoted by x and the deformed position is u. That is if u is the displacement at x, then thedisplacement at xþ dx is uþ @u=@x dx. In other words, the deformed length of the element1þ @u=@xð Þ dx is greater than the original length. Application of Newton’s second law to thedifferential element gives

�sAþ s þ @s

@xdx

� �A ¼ rA dx

@2u

@t2(2:138)

whereA is cross-sectional area of the rodr is mass density of the material of the rods is stress

For elastic deformations, Hooke’s law gives the ratio of the unit stress to unit strain which isequal to the modulus of elasticity E of the material of the rod. Thus

s ¼ E « ¼ E@u

@x(2:139)

Now combining Equations 2.138 and 2.139 gives

E@2u

@x2¼ r

@2u

@t2(2:140)


c2@2u

@x2¼ @2u

@t2(2:141)

where c ¼ ffiffiffiffiffiffiffiffiE=r

pis the velocity of propagation of the displacement or stress wave in the rod. This

equation is same as Equation 2.115.

2.2.4.4 Torsional Vibration of Shafts

Next we consider the torsional vibrations of elastic circular shaft in Figure 2.7a. Considering thedifferential shaft elements of length dx as shown in Figure 2.7b, where Ip is the polar moment ofinertia of a shaft with outside radius r and density r

Ip ¼ pr4

2r dx (2:142)

P +∂x∂P dx

P

dxxu

u +∂x∂u dx

dx +∂x∂u

dx

FIGURE 2.6 Displacement of the differential element of a rod.



Applying Newton’s second law to an element dx as shown in Figure 2.7b, we obtain

T þ @T

@xdx� T ¼ pr4r

2dx

@2u

@t2(2:143)

or

@T

@x¼ pr4r

2@2u

@t2(2:144)

Considering the analysis to elastic deformations only, we use the basic relation

t ¼ Gg ¼ Gr@u

@x¼ Tr

J0(2:145)

to obtain torque,

T ¼ GJ0@u

@x(2:146)

In Equation 2.145, t is the maximum shear stress, g¼ r(@u=@x) is the corresponding shear strain, Gis the shear modulus of the material, and J0¼pr4=2 is the polar moment of inertia of the cross-sectional area of the shaft. Equation 2.144 can now be written as

G@2u

@x2¼ r

@2u

@t2(2:147)

or, as the one-dimensional wave equation

c2@2u

@x2¼ @2u

@t2(2:148)

where c ¼ ffiffiffiffiffiffiffiffiffiG=r

pis the wave speed. This equation is of the same form as that of longitudinal

vibration of rods (Equation 2.141), except for that u and G=r replace u and E=r, respectively.

2.2.4.5 Transverse Vibration of Beams

If an elastic beam such as the fixed-hinged beam shown in Figure 2.8a is deformed elastically andthen released, transverse or lateral oscillation occurs. Assuming only elastic deflections occur, thedisplacement of any point on the beam is small and motion occurs only in a direction normal tothe axis of the beam. We neglect the inertia effects of rotation and shear effect of any section ofthe beam.

The lateral displacement at any point of the beam is represented as

y ¼ y(x,t)

x dx

lq

q

dx

T T +∂x∂T

dx

(b)(a)

FIGURE 2.7 Torsional vibration of long shaft: (a) circular shaft and (b) differential element dx.



From the strength of materials, the beam curvature and the moment M are related by

M ¼ EI@2y

@x2(2:149)

whereEI is the flexural stiffness of the beamM is the bending moment at any transverse sectionE is the modulus of elasticity of the beam materialI is the moment of inertia of the cross-sectional area of the beam about the axis of bending

Figure 2.8b shows an isolated beam section with bending moment M, shear force Q, andexternal load per unit length q(x,t).

Taking the sum of the moments about the left end of the section in Figure 2.8b, we get

@M

@xdx� Q dx� @Q

@x(dx)2 � q

(dx)2

2¼ 0 (2:150)

Neglecting the higher-order terms containing (dx)2, we obtain

Q ¼ @M

@x(2:151)

If the mass of the beam per unit length is denoted as r, then the equation of motion in the verticaldirection as given by Newton’s second law is

@Q

@xdxþ q dx ¼ r dx

@2y

@t2(2:152)

Using Equations 2.149 and 2.151, Equation 2.152 becomes

@2

@x2�EI

@2y

@x2

� �þ q ¼ r

@2y

@t2(2:153)

or

@2

@x2EI

@2y

@x2

� �þ r

@2y

@t2¼ q (2:154)

0

yy

x

dxx

l dxQ

M

q (x,t )

M +∂x

∂Mdx

Q +∂x

∂Qdx

(b)(a)

FIGURE 2.8 Transverse vibration of beam: (a) a fixed-hinged beam and (b) differential element dx.



Assuming the properties of the beam are constant along its length, Equation 2.154 becomes

EI@4y

@x4þ r

@2y

@t2¼ q (2:155)

The case of free vibration is obtained by setting q(x,t)¼ 0. Hence

EI@4y

@x4þ r

@2y

@t2¼ 0 (2:156)

Equation 2.156 can be rewritten as

�c2@4y

@x4¼ @2y

@t2(2:157)

where c ¼ ffiffiffiffiffiffiffiffiffiffiEI=r

p. We consider a separable solution of the form

y(x,t) ¼ F(x)G(t) (2:158)

Then Equation 2.156 is equivalent to the two ordinary differential equations

d2G

dt2þ v2G ¼ 0 (2:159)

and

d4F

dx4� v2

c2F ¼ 0 (2:160)

where v is to be determined. The solution for Equation 2.159 can be written by inspection as

G(t) ¼ A1 sinvt þ A2 cosvt (2:161)

where A1 and A2 are constants. The solution for Equation 2.160 is assumed as

F(x) ¼ Aesx (2:162)

where A and s are constant. Substituting the assumed solution into the governing equation gives

s4 � v2

c2

� �Aest ¼ 0 (2:163)

from which we obtain the four roots. The solution is then

F(x) ¼ C1elx þ C2e

ilx þ C3e�lx þ C4e

�ilx (2:164)


F(x) ¼ A3 sinh lxþ A4 cosh lxþ A5 sin lxþ A6 cos lx (2:165)



where we have redefined the constants as

C1 ¼ (A3 þ A4)=2, C2 ¼ (A6 � iA5)=2, C3 ¼ (A4 � A3)=2 (2:166)

and

C4 ¼ (A6 þ iA5)=2

The solution to Equation 2.156 is then

y(x,t) ¼ (A1 sinvt þ A2 cosvt)(A3 sinh lxþ A4 cosh lxþ A5 sin lxþ A6 cos lx) (2:167)

with

v ¼ l2c (2:168)

The constants in the solution and the natural frequencies are determined by applying the boundaryconditions for the beam and the initial conditions of the motion.

For a simply supported beam, the boundary conditions are

y(0,t) ¼ y(‘,t) ¼ 0 (2:169)

and

@2y

@x2

��x¼0

¼ @2y

@x2

��x¼‘

¼ 0 (2:170)

Equation 2.170 expresses the absence of bending moment at each end of the beam. Applying theboundary conditions to Equation 2.167 gives

A4 þ A6 ¼ 0

A3 sinh l‘þ A4 cosh l‘þ A5 sin l‘þ A6 cos l‘ ¼ 0 (2:171a)

A4 � A6 ¼ 0

A3 sinh l‘þ A4 cosh l‘� A5 sin l‘� A6 cos l‘ ¼ 0 (2:171b)

Equations 2.171a,b will be satisfied if

A3 ¼ A4 ¼ A6 ¼ 0 (2:172)

and

A5 sin l‘ ¼ 0

which is the frequency equation. The frequency equation will be satisfied and nontrivial solutionsobtained, if

l‘ ¼ np, n ¼ 1, 2, 3, . . . (2:173)

Combining Equations 2.173 and 2.168, we obtain the natural frequencies



vn ¼ n2p2

‘2c ¼ n2p2

ffiffiffiffiffiffiffiEI

r‘4

s(2:174)

Each of the frequencies in Equation 2.174 corresponds to a principal mode of vibration.The free response of the beam is given by the superposition of the principal modes. Hence

y(x,t) ¼X1n¼1

(An sinvnt þ Bn cosvnt) sinnp

‘x (2:175)

By applying the initial conditions, we obtain the two constants An and Bn in Equation 2.175.Theoretically, the beam has infinite modes. However, the higher the modes, the lower the accuracy

of the mode with respect to real system due to the basic assumption associated with the theory.At high speeds during the lateral motion of beams, a considerable rotary acceleration associated

with a rotary inertia force must be resisted. Also, for shorter beams carrying lateral forces, there is aconsiderable shear deformation. The combined effect of these two factors: rotary inertia and sheardeformation cannot be neglected, and the thick beam theory or Timoshenko beam theory shouldbe used.

2.2.4.6 Galerkin Method

In the approximation solution of continuous system, Galerkin method has been widely used.Galerkin method can be applied for converting the partial differential equations to the ordinarydifferential equations, and can be applied for converting the ordinary differential equations to theproblems of linear algebra equations. It relies on the weak formulation of an equation or works inprinciple by restricting the possible solutions as well as the test functions to a smaller space than theoriginal one. These small systems are easier to solve than the original problem, but their solution isonly an approximation to the original solution.

The method assumes a solution of the partial differential equations like Equation 2.115 as

y(x,t) ¼Xni¼1

fi(x)qi(t) (2:176)

wherefi(x) are trial functionsqi(t) are generalized coordinates

This solution is used in conjunction with the differential equation of continuous system andweak form to obtain an approximate solution. For instance, the wave equation of Equation 2.115 canbe put on the following standard form by rearrangement:

M y€½ � þ L y½ � ¼ 0 (2:177)

where M and L are differential operators for mass and stiffness. With these operators the system canbe discretized directly into matrix form using the Galerkin procedure with elements of mass andstiffness matrices,

mij ¼ðD

fiM fj

� �dD (2:178)



kij ¼ðD

fiL fj

� �dD (2:179)

where fi, i¼ 1, 2, . . . , N, is a set of trial functions describing the assumed modes of the system. Ifthese trial functions are chosen as the eigenfunctions of the component, only a small number offunctions are needed in order for the solution to converge. The damping matrix elements can also becalculated, based on an assumption of stiffness proportional damping,

cij ¼ 2jijkij=vn (2:180)

wherecij is the ratio of critical damping for the given componentvn is a characteristic frequency

The original wave equation system is then converted to be a set of ordinary differentialequations similar to Equation 2.66, except for {x} being replaced by {q}.

2.2.4.7 Vibrations of Disk Plates

Next we present the vibrations of disk plate. Let us consider a circular plate structure with a centralhole as shown in Figure 2.9. The plane structure is assumed isotropic and symmetric, and weassume the plate is thick and no further structure simplification will be assured. A cylindricalcoordinate system is chosen as (w,r,u). For the convenience of description, we will call the motionsin the w direction the ‘‘out-of-plane’’ (OP) motions since they involve mostly the bending (ortransverse) deflections of the circular disk. Accordingly, the motions in both r, u directions arecalled the ‘‘in-plane’’ (IP) motions due to the fact that they involve mostly in the shear deflections inthe disk plane both circumferentially and radially.

Mathematically, the circular disk OP vibrations are completely decoupled with the IP vibrationssince these two kinds of vibrations are described by two separate sets of equations of motion. TheOP motion and IP motion of the disk are dynamically disintegrated, or decoupled, and thus can bedescribed independently. Exact solutions of natural frequencies and modes can be sought for its OPbending vibrations, and for its IP circumferential and radial vibrations.

1. Out-of-plane vibrationsConsidering a circular and elastic disk, outer radius a, inner radius b, and thickness h, theequation of motion in terms of transverse displacement w(r,u,t) with respect to the stationarycoordinate system (r,u) can be written as

rh@2w

@t2þ Dr4w ¼ 0 (2:181)

w

r

q

FIGURE 2.9 Schematic of circular disk.



where

D ¼ Eh3

12(1� v2)

r2 ¼ @2

@r2þ 1

r

@

@rþ 1r2

@2

@u2

in which r is mass density, E is Young’s modulus, and v is Poisson’s ratio. Assume the generalsolution has the form of

w(r,u,t) ¼ R(r) sin(nu) sin(vnt) (2:182)

R(r) ¼ AnJn(lr)þ BnYn(lr)þ CnIn(lr)þ DnKn(lr) (2:183)

wherel4 ¼ rhv2

n=D, Jn(lr), Yn(lr) are the nth order Bessel functionIn(lr), Kn(lr) are the nth order modified Bessel function

Boundary conditions are defined as follows. Outer radius is free at r¼ a

@2w

@r2þ v

1r

@w

@rþ 1r2

@2w

@u2

� �¼ 0 (2:184)

@

@rr2wþ 1� v

r2@2

@u2@w

@r� w

r

� �¼ 0 (2:185)

Inner radius is clamped at r¼ b

w ¼ 0 (2:186)

@w

@r¼ 0 (2:187)

Substituting Equation 2.182 into Equations 2.184 through 2.187, we obtain the characteristicequations

AnJn(lb)þ BnYn(lb)þ CnIn(lb)þ DnKn(lb) ¼ 0 (2:188)

AnJ0n(lb)þ BnY

0n(lb)þ CnI

0n(lb)þ DnK

0n(lb) ¼ 0 (2:189)

An J 00n (la)þv

laJ 0n(la)�

vn2

(la)2Jn(la)

� �þ Bn Y 00

n (la)þv

laY 0n(la)�

vn2

(la)2Yn(la)

� �

þ Cn I 00n (la)þv

laI 0n(la)�

vn2

(la)2In(la)

� �þ Dn K 00

n (la)þv

laK 0n(la)�

vn2

(la)2Kn(la)

� �¼ 0

(2:190)



An J 000n (la)þ1la

J 00n (la)�1

(la)2J 0n(la)�

(2� v)n2

(la)2J 0n(la)þ

(3� v)n2

(la)3Jn(la)

� �

þ Bn Y 000n (la)þ

1la

Y 00n (la)�

1

(la)2Y 0n(la)�

(2� v)n2

(la)2Y 0n(la)þ

(3� v)n2

(la)3Yn(la)

� �

þ Cn I 000n (la)þ1la

I 00n (la)�1

(la)2I 0n(la)�

(2� v)n2

(la)2I 0n(la)þ

(3� v)n2

(la)3In(la)

� �

þ Dn K 000n (la)þ

1la

K 00n (la)�

1

(la)2K 0n(la)�

(2� v)n2

(la)2K 0n(la)þ

(3� v)n2

(la)3Kn(la)

� �¼ 0

(2:191)

Writing Equations 2.188 through 2.191 into the matrix form,

G(n)11 G(n)

12 G(n)13 G(n)

14

G(n)21 G(n)

22 G(n)23 G(n)

24

G(n)31 G(n)

32 G(n)33 G(n)

34

G(n)41 G(n)

42 G(n)43 G(n)

44

2666664

3777775

An

Bn

Cn

Dn

8>>>><>>>>:

9>>>>=>>>>;

¼ 0 (2:192)

in which

G(n)11 ¼ Jn(lb), G(n)

12 ¼ Yn(lb), G(n)13 ¼ In(lb), G(n)

14 ¼ Kn(lb)

G(n)21 ¼ J 0n(lb), G(n)

22 ¼ Y 0n(lb), G(n)

23 ¼ I 0n(lb), G(n)24 ¼ K 0

n(lb)

G(n)31 ¼ J 00n (la)þ

v

laJ 0n(la)�

vn2

(la)2Jn(la)

G(n)32 ¼ Y 00

n (la)þv

laY 0n(la)�

vn2

(la)2Yn(la)

G(n)33 ¼ I 00n (la)þ

v

laI 0n(la)�

vn2

(la)2In(la)

G(n)34 ¼ K 00

n (la)þv

laK 0n(la)�

vn2

(la)2Kn(la)

G(n)41 ¼ J 000n (la)þ

1la

J 00n (la)�1

(la)2J 0n(la)�

(2� v)n2

(la)2J 0n(la)þ

(3� v)n2

(la)3Jn(la)

G(n)42 ¼ Y 000

n (la)þ1la

Y 00n (la)�

1

(la)2Y 0n(la)�

(2� v)n2

(la)2Y 0n(la)þ

(3� v)n2

(la)3Yn(la)

G(n)43 ¼ I 000n (la)þ

1la

I 00n (la)�1

(la)2I 0n(la)�

(2� v)n2

(la)2I 0n(la)þ

(3� v)n2

(la)3In(la)

G(n)44 ¼ K 000

n (la)þ1la

K 00n (la)�

1

(la)2K 0n(la)�

(2� v)n2

(la)2K 0n(la)þ

(3� v)n2

(la)3Kn(la)

Then, the solution of natural frequency is to find the zero of determinant of following equation:

G(n)11 G(n)

12 G(n)13 G(n)

14

G(n)21 G(n)

22 G(n)23 G(n)

24

G(n)31 G(n)

32 G(n)33 G(n)

34

G(n)41 G(n)

42 G(n)43 G(n)

44

��

��¼ 0 (2:193)



and the mode shape is given by

G(n)11 G(n)

12 G(n)13

G(n)21 G(n)

22 G(n)23

G(n)31 G(n)

32 G(n)33

264

375

An

Bn

Cn

8><>:

9>=>; ¼ �

G(n)14

G(n)24

G(n)34

8><>:

9>=>;Dn (2:194)

where

An ¼

�G(n)14D G(n)

12 G(n)13

�G(n)24D G(n)

22 G(n)23

�G(n)34D G(n)

32 G(n)33

��

��G(n)

11 G(n)12 G(n)

13

G(n)21 G(n)

22 G(n)23

G(n)31 G(n)

32 G(n)33

��

��

, Bn ¼

G(n)11 �G(n)

14D G(n)13

G(n)21 �G(n)

24D G(n)23

G(n)31 �G(n)

34D G(n)33

��

��G(n)

11 G(n)12 G(n)

13

G(n)21 G(n)

22 G(n)23

G(n)31 G(n)

32 G(n)33

��

��

, Cn ¼

G(n)11 G(n)

12 �G(n)14D

G(n)21 G(n)

22 �G(n)24D

G(n)31 G(n)

32 �G(n)34D

��

��G(n)

11 G(n)12 G(n)

13

G(n)21 G(n)

22 G(n)23

G(n)31 G(n)

32 G(n)33

��

��where Dn is defined by the following normalization condition:ð

SRmn(r) cos(nu)j j2dm ¼ rh

ðSRmn(r) cos (nu)j j2r dr du

¼ 2rhpÐ ab Rmn(r)j j2r dr ¼ Id, n ¼ 0

rhpÐ ab Rmn(r)j j2r dr ¼ Id, n 6¼ 0

((2:195)

or

Dn ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiId

2rhpÐ ab R(An=DnJn(lr)þ Bn=DnYn(lr)þ Cn=DnIn(lr)þ Kn(lr))j j2r dr

s, n ¼ 0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiId

rhpÐ ab R(An=DnJn(lr)þ Bn=DnYn(lr)þ Cn=DnIn(lr)þ Kn(lr))j j2r dr

s, n 6¼ 0

8>>>><>>>>:

(2:196)

Thus we obtain the mode shape of the out-of-plane modes of the disk:

W(r,u) ¼ (AnJn(lr)þ BnYn(lr)þ CnIn(lr)þ DnKn(lr)) sin(nu) (2:197)

The node lines of mode shapes are the locations where zero deflections are found. Let the aboveequation be equal to zero and the node lines are therefore defined. In the situation where the centralhole is assumed constrained, these node lines will be one or more concentric circles, defined by R(r),mixed with one or more diametric lines, defined by sin(nu). For convenience, we refer the set ofnode lines the (r,u) lines.

Figure 2.10 shows some of the natural modes of these (r,u) modes. In general, there exist threetypes of OP bending modes. The first type is the diametric modes or called the OPD modes,represented by the (0,n) mode lines and its first two modes are shown in the first and second plots inFigure 2.10. Mode (0,0) and Mode (0,1) are both rigid body modes and are excluded here. Thesecond type is the disc twisting modes or the OPT modes, represented by the (1,n) mode lines andthe first two of them are shown in the third and fourth plots in Figure 2.10. The third type is the diskumbrella modes or the OPU modes, represented by (n,0) mode lines, and the first two of them areshown in the fifth and the sixth plots in Figure 2.10.



2. In-plane vibrationsFor the IP vibrations, there exist two types of motions that are described as the circumferentialvibration, defined by coordinate uu, and the radial vibration, defined by coordinate ur in Figure 2.9.These two IP vibrations are dynamically coupled. The IP free vibrations are expressed by a pair ofmutually coupled equations. A simplified form is presented by the following two equations:

EI

d4@3ur@u3

� @2uu@u2

� �� EA

d2@2uu@u2

þ @ur@u

� �þ rA

@2uu@t2

¼ 0 (2:198)

EI

d4@4ur@u4

� @3uu@u3

� �þ EA

d2@uu@u

þ ur

� �þ rA

@2ur@t2

¼ 0 (2:199)

whereEI and EA are stiffness constantsrA is mass constantd is geometry constant

Specifically, Equation 2.198 is for the circumferential vibration and Equation 2.199 is for theradial vibration.

The eigen equations for an eigen frequency of Equations 2.198 and 2.199 therefore become:

EI

d4� d3Ur

du3þ @2uu

du2

� �þ EA

d2d2Uu

du2þ dUr

du

� �þ rAv2Uu ¼ 0 (2:200)

EI

d4� d4Ur

du4þ d3Uu

du3

� �� EA

d2dUu

duþ Ur

� �þ rAv2Ur ¼ 0 (2:201)

Since the disk is in circular shape, the solutions for Equations 2.200 and 2.201 have to be in types of

Ur,n(u) ¼ An cos n(u� w) (2:202)

Uu,n(u) ¼ Bn sin n(u� w) (2:203)

where w is a phase constant since a nodal line does not show a preference for the orientation of itsmodes and there are no external forces applied. Substitution of Equations 2.202 and 2.203 intoEquations 2.200 and 2.201, we have

rhv2n � k11,n �k12,n�k21,n rhv2

n � k22,n

� �An

Bn

¼ 0

0

(2:204)

where kii,n is a constant related to mode n. Thus, a nontrivial solution leads to

(0,2) (0,3) (1,1) (1,2) (1,0) (2,0)

FIGURE 2.10 Out-of-plane natural modes of circular disk.



v4n � K1,nv

2n þ K2,n ¼ 0 (2:205)

K1,n ¼ n2 þ 1

d6rA

n2EI

a2þ EA

� �

K2,n ¼ n2(n2 � 1)2

d6(rA)2EAð Þ(EI)

(2:206)

v2n,1 ¼

K1

21�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4

K2

K21

s !(2:207)

v2n,2 ¼

K1

21þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4

K2

K21

s !(2:208)

wherevn,1 is the mode frequency for radial vibrationvn,2 is the mode frequency for circumferential vibration

Substituting Equation 2.207 or Equation 2.208 into Equation 2.204 will generate the modeshape for that mode frequency, defined by

An,i

Bn,i¼

rAv2n,i �

n

d

� �2 EI

d2þ EA

� �n

d2

� � n

d

� �2EI þ EA

� � , i ¼ 1, 2 (2:209)

Figure 2.11 shows the natural modes of the IP modes. At frequency vn,1, the mode shapes ofEquation 2.209 define the IP radial mode shape where n¼ 0, 1, 2, 3, . . . . When n¼ 0, this mode iscalled the breathing mode. When n¼ 1, it is a rigid mode with an offset center. When n� 2, they arecalled the nth order of breathing modes, the first three modes are shown in the first row in Figure2.11. Frequency vn,2 is another type of IP frequency that defines the circumferential vibration. It isanalog to the longitudinal vibrations of a beam and sometimes called the longitudinal mode. Itsflexible modes start with an index of n � 1. The first three modes are shown in the second row inFigure 2.11.

FIGURE 2.11 In-plane natural modes of circular disk.



2.3 NONLINEAR VIBRATION SYSTEMS

In vibration systems, nonlinear relationships generally exist, particularly when large deformationexists. In addition to the nonlinear behavior of inertia force, damping force, and stiffness force,nonlinearity could be due to geometry. A system with friction is usually nonlinear. One of the mainreasons for modeling a system as a nonlinear one is that some phenomena occurring in nonlinearsystems are not predictable by linear theory. These could consist of one or more of the followingaspects: the frequency of vibration may be dependent on the amplitude of vibration; when thefrequency of excitation is increased, the amplitude of vibration may have a significant jump; when aharmonic force excites a system, the response will not only have the basic harmonic component butmay consist of superharmonic or subharmonics, as well as chaos; the system can become self-excited, and amplitudes of vibration may grow even without any external disturbance; the systemcan become unstable under certain conditions.

In this section, we will introduce some basic analytical methods for nonlinear systems, includ-ing the perturbation method and the method of variation of parameters. We present these methodsby using several types of typical nonlinear equations, including Duffing equation, Van der Polequation, and Mathieu equation, with focusing on illustrating the basic characteristics associatedwith these nonlinear systems.

2.3.1 PERTURBATION METHOD, DUFFING EQUATION, AND VAN DER POL’S EQUATION

Considering an undamped SDOF under free conditions has restoring force defined by a cubicfunction, the equation of motion can be written as

md2x

dt2þ axþ bx3 ¼ 0 (2:210)

Let

p ¼ffiffiffiffiffiffiffiffiffia=m

p, T ¼ pt, dt ¼ dT=p, « ¼ b=mp2 (2:211)

then Equation 2.210 becomes

€xþ xþ «x3 ¼ 0 (2:212)

Equation 2.212 is known as Duffing’s equation and it is nonlinear with respect to the restoring forcedefined by the cubic function, which is shown in Figure 2.12. Next we describe how to useperturbation method to deal with it. It is not possible to obtain an exact solution to Equation2.212 as in the case of a linear system.

Hence, we attempt an approximation solution. Assume the initial conditions for the system aregiven by

x(0) ¼ A and x_(0) ¼ 0 (2:213)

Let x0(t) be the linear solution (b¼ 0). We can then perturb the solution about the linear solution in aseries form as

x(T) ¼ x0(T)þ «x1(T)þ «2x2(T)þ � � � (2:214)

The coefficient « in Equation 2.214 is a nonlinear parameter and the solution assumed in Equation2.214 converges fast if «� 1. Now substituting Equation 2.214 in Equation 2.212, we get



(x€0 þ «x€1 þ «2x€2 þ � � � )þ (x0 þ «x1 þ «2x2 þ � � � )þ «(x0 þ «x1 þ «2x2 þ � � � )3 ¼ 0 (2:215)

Since « is a parameter, sorting out the terms of «0, «1, «2, etc., we obtain

x€0 þ x0 ¼ 0

x€þ x1 ¼ �x03

. . .

(2:216)

Since the first of Equation 2.216 is the linear part and in view of the initial conditions (2.213) has asolution of the form

x0 ¼ a1 cos T þ b1 sinT ¼ A cos T (2:217)

Substituting Equation 2.217 in the second equation of 2.216, we get

x€1 þ x1 ¼ �A3 cos3 T ¼ �34A3 cos3 T � 1

4A3 cos 3T (2:218)

From Equation 2.217, we observe that there is an exciting term with third harmonic of the naturalfrequency and thus the solution (2.214) contains higher harmonic terms. Separating Equation 2.218into two parts, one belonging to the harmonic solution and the second one belonging to the higherharmonic, thus

x€11 þ x11 ¼ �34A3 cos T

x€12 þ x11 ¼ �14A3 cos 3T

x1 ¼ x11 þ x12

(2:219)

It may be noted here that Equation 2.219 is not a classical forced vibration problem of a system, andthe right-hand terms in Equation 2.219 are due to the perturbation of the nonlinear free-vibrationproblem. Hence, we must consider both the transient and particular integral parts of equation inEquation 2.219 for the final solution. Noting that the first equation in Equation 2.219 is excited byforcing term at resonant frequency, we write the following:

x11 ¼ T(a2 cos T þ b2 sinT)

e > 0 Hardening spring

e = 0 Linear spring

e< 0 Softening spring

x

x + ex 3

FIGURE 2.12 Hardening and softening spring characteristics.



Substituting the above in the first equation of (2.219), we get

x11 ¼ �38A3T sinT (2:220)

Similarly for the higher harmonic, we have

x12 ¼ a3 cos T þ b3 sinT � A3

32cos 3T

Since

x12 ¼ 0; x_12 ¼ 0 at T ¼ 0

we have

x12 ¼ � 132

A3(cos T � cos 3T) (2:221)

Hence, the complete solution for x1 is given by

x1 ¼ � 38A3T sinT � 1

32A3(cos T � cos 3T) (2:222)

The first term on the right-hand side of the above equation grows with time. Such a term is calledsecular term. It is not physically possible to have a growing solution for the system in Figure 2.12 asthe amplitude of vibration should be limited. This difficulty is overcome by introducing frequency–amplitude interaction by Lindstedt–Poincare perturbation method as follows.

For the linear system with «¼ 0, in Equation 2.212, the solution becomes periodic with period2p. For « 6¼ 0, we introduce an unspecified function v as follows:

t ¼ vT (2:223)

We choose v such that the secular term in Equation 2.222 can be avoided. With the help of theabove equation, we can now write Equation 2.212 as

v2x€þ xþ «x3 ¼ 0 (2:224)

As before, we write the solution to Equation 2.224 as

x(t) ¼ x0(t)þ «x1(t)þ «2x2(t)þ � � � (2:225)

and in addition choose the function for v in series as

v ¼ v0 þ «v1 þ «2v2 þ � � � (2:226)

Therefore

(v0 þ «v1 þ «2v2 þ � � � )2(x€0 þ «x€1 þ «2x€2 þ � � � )þ (x0 þ «x1 þ «2x2 þ � � � )þ «(x0 þ «x1 þ «2x2 þ � � � )3 ¼ 0 (2:227)



As before, we now separate terms of «0, «1, «2, etc., to obtain

v20x€0 þ x0 ¼ 0

v20x€1 þ x1 ¼ �2v0v1x€0 � x€30

v20x€2 þ x2 ¼ �(2v0v2 þ v2

1)x€0 � 2v0v1x€1 � 3x20x1

. . .

(2:228)

In addition to the initial condition (2.213), we now prescribe

xi(t þ 2p) ¼ xi(t) (2:229)

since the solution should be periodic in t with time period 2p.The solution of the first equation in (2.228) gives

x0 ¼ a1 cost

v0þ b1 sin

t

v0

In view of the initial conditions (2.213) and the periodic condition (2.229), the above equationgives

x0 ¼ A cost (2:230)

v0 ¼ 1 (2:231)

With the help of the above, the second equation in (2.228) becomes

x€1 þ x1 ¼ �2v1A cos t � A3 cos3 t ¼ 2v1A� 34A3

� �cos t � 1

4A3 cos 3t (2:232)

The bracketed term in the above equation gives rise to secular term in the solution. Hence, in orderto avoid the secular term in the solution, we select

v1 ¼ 38A2 (2:233)

Now the solution of Equation 2.232 is governed by the third harmonic excitation term and it canbe written as

x1 ¼ A3

32(�cos t þ cos 3t) (2:234)

Substituting Equation 2.234 in the third equation of (2.228) we determine v2 and x2. Consideringthese three terms only, the solution of Equation 2.224 is given by

x(t) ¼ A� 132

«A3 þ 231024

«2A5

� �cos t þ 1

32«A3 � 3

128«2A5

� �cos 3t þ 1

1024«2A5 cos 5t

(2:235)

v ¼ 1þ 38«A2 � 21

256«2A4 (2:236)



Equation 2.236 shows the dependency of frequency v on amplitude A given initially. If «> 0, wehave a hardening spring. If «< 0, then the spring is of softening type as shown in Figure 2.12.Equation 2.236 is plotted in Figure 2.13 for both the cases.

The free-vibration response of the Equation 2.210 is therefore given by

x tð Þ ¼ A� 132

«A3 þ 231024

«2A5

� �cosv

ffiffiffiffia

m

rt þ 1

32«A3 � 3

128«2A5

� �

cos 3v

ffiffiffiffia

m

rt þ 1

1024«2A5 cos 5v

ffiffiffiffia

m

rt (2:237)

Next we consider the amplitude frequency dependent and jump phenomenon of damped Duffing’sequation under external harmonic excitation,

x€þ cx_ þ axþ bx3 ¼ F cos(vt þ f) ¼ A1 sinvt þ A2 cosvt (2:238)

Assume the first approximation to be

x1 ¼ A cosvt (2:239)

where A is assumed fixed and v to be found. It can be derived that

(a� v2)Aþ 34bA3

� �2þ (cvA)2 ¼ A2

1 þ A22 ¼ F2 (2:240)

The response curves from Equation 2.240 are depicted in Figure 2.14. It shows that the nonlinearsystem exhibits surprising phenomena that do occur in linear systems: the amplitude of vibration ofthe system described by Equation 2.238 could increase or decrease suddenly as the excitationfrequency v is increased or decreased gradually. This behavior is known as the jump phenomenon.It is clear that there could exist two amplitudes of vibration for a given forcing frequency, asdepicted in the bend regions of the curves of Figure 2.14 (cases b 6¼ 0).

As another application example of the perturbation method, we consider a self-excited systemdescribed by Van der Pol’s equation,

x€� «(1� x2)x_1 þ v2x ¼ 0 (2:241)

the initial condition is x(0)¼ 0, x_(0)¼Av0,

e > 0e = 0

e < 0

A

ww = 1

FIGURE 2.13 Amplitude–frequency relations of nonlinear system.



Applying Lindstedt–Poincare method gives

x(t) ¼ x0(t)þ «x1(t)þ «2x2(t)þ � � �v2 ¼ v2

0 þ «v21 þ «2v2

2 þ � � �(2:242)

By inserting the above series into Equation 2.241, and equating coefficients of like terms, the resultto the order «2 is

x(t) ¼ 2� 29«2

96v20

� �sinv0t þ «

4v0cosv0t þ «

4v0

3«4v0

sin 3v0t � cos 3v0t

� �� 5«2

124v20

sin 5v0t

(2:243)

In the next sections, we will further elaborate the features of Van der Pol’s equation.

2.3.2 METHOD OF VARIATION OF PARAMETER

The perturbation method gives a steady-state solution while the method of variation of parameterdescribed below allows small changes in amplitude and phase angle in the time solution. Considerthe equation of a nonlinear system given by

x€þ v20xþ «f(x,x_,t) ¼ 0 (2:244)

wheref(x,x_,t) is the nonlinearity« is a constant

If we neglect f (x,x_,t) the solution corresponding to the linear equation x€þ v20x ¼ 0 is

x ¼ A cos(v0t þ u) (2:245)

x_ ¼ �v0A sin(v0t þ u) (2:246)

Assume A and u are time-dependent functions rather than constants. Differentiating Equation 2.245with respect to time t gives

x_ ¼ A_ cosc� (A sinc)(v0 þ _u) (2:247)

|A|

0 w

F = 0

F1

F2F2

F1

|A|

0

(c)(b)(a)

w

F = 0

F1

F2

|A|

0 w

F = 0

F1

F2

a a a

FIGURE 2.14 Response curves: (a) b¼ 0; (b) b > 0; and (c) b < 0.



wherec¼v0t þ u

Subtracting Equation 2.246 from Equation 2.247, we obtain

A_ cosc� _uA sinc ¼ 0 (2:248)

Differentiating Equation 2.246 with respect to t gives

x€¼ �v0A_ sinc� (v0A cosc)(v0 þ _u) (2:249)

Substituting Equations 2.245 and 2.246 into Equation 2.244 and subtracting the resultant equationfrom Equation 2.249, we get

�A_v0 sinc� _uv0A cosc ¼ �«f(A cosc,� v0A sinc,t) (2:250)

Solving Equations 2.248 and 2.250 for A_ and _u, we obtain

A_ ¼ «

v0sincf(A cosc,� v0A sinc,t)

and_u ¼ «

v0Acoscf(A cosc,� v0A sinc,t) (2:251)

Thus the second-order differential equation is transferred into a first-order differential equation.An approximate solution of Equation 2.251 is obtained by assuming « is small and that A and u

do not change rapidly. The expression of A_, _u can be expanded into a Fourier series. Since themotion is over a single cycle and the terms under the summation signs in the series are of the sameperiod and consequently vanish, and also if the change in A and u is small over one cycle, thenapproximately the average values of A_ and _u equal to the instantaneous values. Hence,

A_ «

2pv0

ð2p0

sincf(A cosc,� v0A sinc,t) dc (2:252a)

_u «

2pv0A

ð2p0

coscf(A cosc,� v0A sinc,t) dc (2:252b)

where A is assumed constant in the integrand.Next we apply the method of variation of parameters to solve the following Rayleigh’s equation,

x€� (a� bx_2)x_ þ v2x ¼ 0 (2:253)

by applying the above procedures, we have the following relationships:

A_ ¼ �12pv

ð2p0

cos2c(�aþ bA2v2 cos2 c)Av dc ¼ A(a� 3bA2v2=4)2

(2:254)



The above equation can be integrated directly

t ¼ 2ðAA0

dA

A(a� gA2)¼ 1

aln

A2

a� gA2(2:255)

solving for A

A ¼ a

g

1

1þ a

gA20

� 1

� �e�at

2664

37751=2

(2:256)

in which

g ¼ 3b2v=4 (2:257)

Actually, the Rayleigh equation can be transferred to Van der Pol equation through propertransformation. We will further elaborate the features of Rayleigh equation.

2.3.3 PHASE PLOT, LIMIT CYCLES, SELF-EXCITED OSCILLATIONS, AND CHAOS

In addition to the closed-form approximate solutions derived above, graphical method is very usefulin determining the qualitative information about the behavior of the nonlinear dynamic systems. AnSDOF system requires two parameters to describe the state of motion completely. The twoparameters are the displacement and velocity of the system. When these two parameters are usedas coordinate axes, then the resulting graphical representation of the motion is called the phase planerepresentation. Here each point in the phase plane represents a possible state of the system. As timechanges, the state of the system changes. A typical representative point in the phase plane movesand traces a curve known as the trajectory. The trajectory shows how the solution of the systemvaries with time. Figure 2.15 is the phase plot of the solutions of Rayleigh’s equation for two giveninitial conditions.

x

x•

FIGURE 2.15 Limit cycle for Rayleigh’s equation, the broken line is limit cycle.



The limit cycles are the trajectories due to different initial conditions, which form a family ofconcentric closed curves in a phase plane. The broken line in Figure 2.15 shows a stable limit cycle.For some of the initial points inside or outside the closed curves, they could approach to and thenstay on the closed curve. In contrast with the forced vibration, the nonlinear system is autonomous.A limit cycle is a nonconservative and nonlinear phenomenon. As shown in Figure 2.15, there isonly one closed trajectory, and this is independent of the initial conditions. The initial conditionscould be inside or outside of the closed trajectory. Once initiated, the system will eventually ‘‘lock’’itself into a limit cycle with constant amplitude and frequency. A limit cycle occurs if, over onecycle, the net energy input from the excitation is equal to the energy dissipation within the system.Self-excitation is a function of the motion itself, such as displacement and=or velocity. If theexcitation force in a function of velocity is Fx_, we get

mx€þ cx_ þ kx ¼ Fx_

or

mx€þ (c� F)x_ þ kx ¼ 0

The equation is autonomous. If (c � F)> 0, the system is stable. If (c � F)< 0, we have negativedamping and the amplitude increases with each oscillation. If (c � F)¼ 0, sustained oscillation ispossible. The oscillation occurs near the natural frequency of the system.

Chaos represents the behavior of a system that is inherently unpredictable. In other words,chaos refers to the dynamic behavior of a system whose response, although described by adeterministic equation, becomes unpredictable because the nonlinearities in the equation enor-mously amplify the errors in the initial conditions of the system. Another characteristic of chaoticvibration is that, if the numerical solution (and, presumably, the physical system it represents) isstarted twice at nearly identical initial conditions, the two solutions will diverge exponentiallywith time. Some of chaos phenomena associated with friction system will be further illustratedin Chapter 4.

2.3.4 STABILITY OF EQUILIBRIUM

A nonlinear system can have more than one equilibrium position. Therefore, it is necessary to definethe equilibrium positions and determine the stability of the system about each equilibrium. Thecondition {x_, x€}¼ {x_1, x_2}¼ {0, 0} is an equilibrium. An equilibrium can also be expressed asdx2=dx1¼ 0=0. Hence, an equilibrium position is called a singular point. The other points in thephase plane are called regular points.

To determine the equilibrium positions, let us consider a one-degree-of-freedom systemdescribed by two first-order differential equations

x_1 ¼ X1(x1,x2) and x_2 ¼ X2(x1,x2) (2:258)

where X1 and X2 are functions of (x1,x2). Let (s1,s2) be an equilibrium. It follows that

x_1 ¼ X1(s1,s2) ¼ 0 and x_2 ¼ X2(s1,s2) ¼ 0 (2:259)

The values of s1 and s2 can be found from Equation 2.259.If the origin of the equations be transferred to (s1,s2) by a coordinate translation.

x1 ¼ y1 þ s1 and x2 ¼ y2 þ s2 (2:260)



then Equation 2.258 becomes

x_1 ¼ y_1 ¼ X1(s1 þ y1, s2 þ y2)

x_2 ¼ y_2 ¼ X2(s1 þ y1, s2 þ y2)

Expanding X1 and X2 about (s1,s2) by Taylor’s series, we obtain

y_1 ¼ a11y1 þ a12y2 þ f1(y1,y2)

y_2 ¼ a21y1 þ a22y2 þ f2(y1,y2)

or

y_1y_2

� �¼ a11 a12

a21 a22

� �y1y2

� �þ f1

f2

� �(2:261)

This can be written as

fy_g ¼ Afyg þ ffg (2:262)

where (y1,y2) are the state variables referred to the origin at (s1,s2), the elements in the matrix A areconstants, and the elements in {f} are the nonlinear terms. Hence, the linearized equations of thesystem about the equilibrium at (s1,s2) are

fy_g ¼ Afyg (2:263)

The stability of the system about (s1,s2) can be examined by the trajectory of Equation 2.263 in aphase plane. Applying linear theory, the solution of Equation 2.263 is

fyg ¼ fy0gelt (2:264)

wherel is an eigenvalue{y0} a vector of constants

Substituting Equation 2.264 in Equation 2.263, and simplifying, we get

½lI � A�fy0g ¼ f0g (2:265)

where I is a 23 2 unit matrix. The values of l are obtained from the characteristic equation

D(l) ¼ lI � Aj j ¼ 0 (2:266)

There exist two eigenvalues, l1 and l2. The system is stable about the equilibrium (s1,s2) only if theroots l1 and l2 from Equation 2.266 have zero or negative real parts.

Assuming a similarity transformation

fyg ¼ Bfzg ory1y2

� �¼ b11 b12

b21 b22

� �z1z2

� �(2:267)



where B is a nonsingular matrix, and the equations in the variables (z1,z2) are uncoupled. Substi-tuting Equation 2.267 in Equation 2.263 and premultiplying by B�1, the resulting uncoupledequations are

fz_g ¼ B�1ABfzg (2:268)

or

z_1z_2

� �¼ l1 0

0 l2

� �z1z2

� �(2:269)

or

z_1 ¼ l1z1 and z_2 ¼ l2z2 (2:270)

The stability of the system about an equilibrium (s1,s2) can be examined in the (y1,y2) or (z1,z2) phaseplane. The system stability can be examined in either phase plane. The eigenvalues of Equations 2.263and 2.268 are identical because the matrices A and B�1AB possess the same determinant.

Recalling jB�1j ¼ jBj�1, we have jB�1ABj ¼ jB�1j jAj jBj ¼ jBj�1 jAj jBj ¼ jAj.Here, the trajectories are mapped from one plane to the other. Consider now the stability of the

four cases:

Case 1. Node: l1,2 real and same signA node occurs if l1 and l2 are real and of the same sign. From Equation 2.270, we have

dz2dz1

¼ l2z2l1z1

ordz2z2

¼ l2l1

dz1z1

which can be integrated directly to yield

z2 ¼ Czl2=l11 (2:271)

where C¼ constant. The exact plot depends on the values of C and the ratio l2=l1. For instance, ifl2=l1¼ 2 andC¼ 1, we get z2¼ z1

2. If l2¼ l1, the trajectories are simply radial lines from the origin.The directions of the trajectories can be deduced from Equation 2.270. If both l1 and l2 are

negative, the system is stable and the trajectories converge towards the equilibrium. Conversely, ifl1,2 are positive, the system is unstable and the trajectories point away from equilibrium.

Note that the trajectories in the (z1,z2) plane are symmetrical, but those in the (y1,y2) plane aregoverned by the characteristics discussed earlier.

Note that since (y1,y2) are obtained from (x1,x2) by a coordinate translation, the x’s and y’s havethe same physical interpretation. The characteristics of phase trajectories, as discussed in the lastsection, do not apply to the plots in the (z1,z2) phase plane.

Case 2. Saddle point: l1,2 real and opposite signA saddle point occurs if l1,2 are real and have opposite signs. Since one of the l’s is positive, thesystem in the neighborhood of a saddle point is always unstable. Since the ratio l2=l1 is negative,Equation 2.271 can be expressed as

z2 ¼ Cz� l2=l1j j1 or z2z

l2=l1j j1 ¼ C (2:272)

This clearly shows that the trajectory is a hyperbola.



Case 3. Vortex: l1,2 imaginaryA vortex, or center, occurs if l1,2¼�j b are imaginary, where j ¼ ffiffiffiffiffiffiffi�1

pand b¼ constant. From

Equation 2.270, we get

z_1 ¼ jbz1 and z_2 ¼ �jbz2

or

z1 ¼ z10ejbt and z2 ¼ z20e

�jbt

where (z10,z20) are constants. The factor e�jbt represents a harmonic motion of unit magnitude with

circular frequency b. Therefore, the resulting motion in the (z1,z2) plane is a combination of twoharmonic motions, which is an ellipse. The motion in the neighborhood of equilibrium in the (z1,z2)or (y1,y2) plane forms a closed curve. Hence by definition, the system is stable.

Case 4. Focus: l1,2 complex conjugatesA focus occurs if l1,2¼a � jb are complex conjugates, where j ¼ ffiffiffiffiffiffiffi�1

pand a and b are constants.

From Equation 2.270, we get

z_1 ¼ (aþ jb)z1 and z_2 ¼ (a� jb)z2 (2:273)

or

z1 ¼ (z10eat)ejbt and z2 ¼ (z20e

at)e�jbt

where (z10,z20) are constants. The factor e�jbt represents a harmonic motion of unit magnitude with

circular frequencyb as before. Ifa> 0, eat increases exponentially with time t and the trajectory in the(z1,z2) plane is a divergent logarithmic spiral. Hence for a> 0, the equilibrium is unstable. If a< 0,the trajectory is a convergent logarithmic spiral and the system is therefore asymptotically stable.

Summarizing, the stability about an equilibrium at (s1,s2) can be examined from the roots of thecharacteristic equation. Substituting A¼ [aij] from Equation 2.261 in Equation 2.266, the charac-teristic equation and the roots l1,2 are

l2 � (a11 þ a22)lþ (a11a22 � a12a21) ¼ 0 (2:274)

l1,2 ¼ 1=2f(a11 þ a22)� ½(a11 þ a22)2 � 4(a11a22 � a12a21)�1=2g ¼ 0 (2:275)

Introducing the parameters u and v as

u ¼ (a11 þ a22) and v ¼ (a11a22 � a12a21) (2:276)

Thus, Equations 2.274 and 2.275 become

l2 � ulþ v ¼ 0 (2:277)

l1,2 ¼ 1=2fu�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 � 4v

pg (2:278)

The four cases above can be summarized as follows:

Case 1. Node: l1,2 real and same sign. This requires u2> 4v and v> 0, with u2¼ 4v as alimiting case. The node is stable if u< 0 and unstable if u> 0.



Case 2. Saddle point: l1,2 real and opposite sign. This requires u2> 4v and v< 0. This isalways unstable.

Case 3. Center: l1,2 imaginary. This requires u2< 4v and v¼ 0. By definition, the system isstable.

Case 4. Focus: l1,2 complex conjugates. This requires u2< 4v and v 6¼ 0. The system is stableif u< 0 and unstable if u> 0.

The stability of the cases can be mapped in the (u,v) plane. The demarcation between the nodesand foci is given by

u2 ¼ 4v (2:279)

Whereby the center is mapped in the positive real axis and the stable regions are given in the fourthquadrant of the map plot.

2.3.5 PARAMETRICALLY EXCITED SYSTEM AND MATHIEU’S EQUATION,NONSTATIONARY VIBRATIONS

Next we consider parametrically excited system. The simplest case of parametrically excited systemcan be described by the following equation:

€xþ v2(lþ g cosvt)x ¼ 0 (2:280)

in which the stiffness of the system is a harmonic function. Equation 2.280 is known as Mathieu’sequation.

For convenience, let v¼ 2, the above equation reduces to the standard form of Mathieu’sequation,

€xþ (dþ 2« cos 2t)x ¼ 0, « � 1 (2:281)

The dynamic stability properties of the equation can be investigated by using parameter group (d,«).Next we seek the solution by Lindstedt–Poincare method, assuming the solution has the form

x(t) ¼ x0(t)þ «x1(t)þ «2x2(t)þ � � � (2:282)

and in addition choose the function for d in series as

d ¼ d0 þ «d1 þ «2d2 þ � � � ,ffiffiffiffiffid0

p¼ 0, 1, 2, . . . (2:283)

Inserting the above two into Mathieu equation, and equating coefficient of like power of « to zero,we obtain the sets of equations

€x0 þ d0x0 ¼ 0 (2:284a)

€x1 þ d0x1 ¼ �(d1 þ 2 cos 2t)x0 (2:284b)

€x2 þ d0x2 ¼ �(d1 þ 2 cos 2t)x1 � d2x0,ffiffiffiffiffid0

p¼ 0, 1, 2, . . . (2:284c)

. . .

This should be solved sequentially for the various values offfiffiffiffiffid0

p(¼ 0, 1, 2, . . . ). The zero-order

approximation is given by



x0 ¼ cosffiffiffiffiffid0

pt

sinffiffiffiffiffid0

pt

,

ffiffiffiffiffid0

p¼ 0, 1, 2, . . . (2:285)

The stability boundary curve is obtained by introducing the solution x0 ¼ cosffiffiffiffiffid0

pt and

x0 ¼ sinffiffiffiffiffid0

pt (

ffiffiffiffiffid0

p ¼ 0, 1, 2, . . . ) into Equation 2.284a and allowing the xi(t) (i¼ 1, 2, . . . ) beingperiodic. It yields an infinite number of solution pairs, one pair for every value of

ffiffiffiffiffid0

pexcept forffiffiffiffiffi

d0p ¼ 0. Considering the case

ffiffiffiffiffid0

p ¼ 0, in which case x0¼ 1, Equation 2.284b yields to

€x1 ¼ �d1 � 2 cos 2t (2:286)

Since x1 should be periodic, we must have d1¼ 0, then the solution is

x ¼ (1=2) cos 2t

Thus Equation 2.284b becomes

€x2 ¼ (�2 cos 2t)((1=2) cos 2t)� d2 ¼ �(1=2þ d2)� (1=2) cos 4t (2:287)

for x2 periodic, the constant term in the above equation should be set equal to zero, this givesd1¼�1=2. Thus there is the following equation for

ffiffiffiffiffid0

p ¼ 0:

d ¼ �(1=2)«2 þ � � � (2:288)

It is a parabola passing through the origin of the parameter plane. Next we considerffiffiffiffiffid0

p ¼ 1, usingsimilar approach, we obtain two transition curves,

d ¼ 1� «� (1=8)«2 þ � � � corresponding to x0 ¼ cos t

d ¼ 1þ «� (1=8)«2 þ � � � corresponding to x0 ¼ sin t(2:289)

Furthermore, we obtain two transition curves,

d ¼ 4þ (5=12)«2 þ � � � corresponding to x0 ¼ cos 2t

d ¼ 4� (1=12)«2 þ � � � corresponding to x0 ¼ sin 2t(2:290)

Following the same pattern, transition curves can be obtained for x0 ¼ cosffiffiffiffiffid0

pt and

x0 ¼ sinffiffiffiffiffid0

pt (

ffiffiffiffiffid0

p ¼ 0, 1, 2, . . . ). The curves are plotted in Figure 2.16. For pairs of parametersthat are inside shaded areas, the motion is unstable.

0.5

2

e

d3 4

−0.5

0 1

FIGURE 2.16 Stability chart for Mathieu’s equation.



We have discussed nonlinear vibration system with constant frequency and amplitude as well asthe parametric vibrations with time-dependent stiffness. In friction-involved system, nonstationaryphenomena could occur due to the dynamic transition of coupling of two components under externaloperational conditions or under system interactions. For simplification, we here only illustrate thecase with an excitation that has time-dependent frequencies and amplitude. The motion equation ofthe most simplified case can be represented as

€xþ v20x ¼ «f (x,x_)þ 2«a(«t) cos (u(t,«t)) (2:291)

This system leads to nonstationary or nonperiodic resonance. This could be a representative of thecases where a modification of the response near the resonance condition or a passage throughresonance takes place.

2.3.6 MULTIPLE-DEGREE-OF-FREEDOM SYSTEMS

The general form of the equations of motion for a nonlinear multi-degree-of-freedom system isgoverned by

½m�f€xg þ ½c(x_,x)�fx_g þ ½k(x_,x)�fxg ¼ fF(t)g (2:292)

where the internal set of forces opposing the displacement and velocity is assumed to be nonlinearfunctions of {x} and {x_}. These types of equations are usually solved by numerical approaches. Inorder to find the displacement and velocity vector {x}, {x_} that satisfy the nonlinear equilibrium inEquation 2.292, it is necessary to perform an equilibrium iteration sequence in each time step.

There are several numerical methods available for the solution of the response of linear ornonlinear multi-degree-of-freedom systems.

There are many explicit and implicit integration schemes available. Typical explicit schemesinclude the central difference methods, two-cycle iteration with trapezoidal rule and the fourth-orderRunge–Kutta method. The implicit schemes include Wilson-u method, Newmark-b method, andhigh order methods. Newmark-b method and high order methods will be detailed in Chapter 4 withthe applications illustrated for friction involved nonlinear dynamic system.

2.4 RANDOM VIBRATIONS

If a vibration motion or the associated force does not exhibit any obvious pattern, the vibration iscalled a random vibration. This is different from what we usually meet, the deterministic process,where the records obtained are always alike if an identical experiment is performed several times. Onthe contrary, when all conditions in the experiment remain unchanged but the records are continuallychanging, we then consider the process to be random, or nondeterministic. In this situation, a singlerecord is not sufficient to provide a statistical description of the totality of possible records.

In a random process, instead of one time history, a whole family or an ensemble of possible timehistories is described, as shown in Figure 2.17. Any single individual time history that belongs to theensemble is referred to as a sample record.

Let xk(t1) be the value of random variable x(t) at time t1, obtained from the kth record. Then theexpected value (average or mean) E[x(t1)] of x

k(t1) for a fixed time t1, obtained from all records, i.e.,k¼ l, 2, 3, . . . , n is

E½x(t1)� ¼ limn!1

1n

Xnk¼1

xk(t1) (2:293)

If E[x(t1)] is independent of t, i.e., E[x(t1)]¼E[x(t1 þ t)] for all t, then the random process is calledstationary. If in addition to this property, each record is statistically equivalent to any other record suchthat E[x(t1)] in Equation 2.293 can be replaced by a time average of a sample representative record x(t)



x�(t) ¼ E(x) ¼ limT!1

1T

ðT0

x(t) dt (2:294)

then the stationary process is ergodic. For many applications this assumption is fairly reasonable.The variance s2

x of x(t) is given by

E x� E(x)½ �2¼ limT!1

1T

ðT0

x� E(x)½ �2 dt (2:295)

For the special case with E(x)¼ 0, the variance s2x of x becomes its mean square value and is given

by x2(t), where

x2(t) ¼ E x2(t)� � ¼ lim

T!11T

ðT0

x2(t) dt (2:296)

2.4.1 PROBABILITY DENSITY FUNCTION AND GAUSSIAN RANDOM PROCESS

The probability density function of random data is the probability that the data will assume a valuewithin some defined range at any instant of time. We consider a sample time history, as shown inFigure 2.18. The probability that x(t) will occur within x and x þ Dx can be obtained from the ratioTx=T, where Tx indicates the total amount of time for which x(t) falls within the range of x andx þ Dx, i.e., Tx ¼

Pki¼1 Dti and T is observed time. We define

Prob½x < x(t) � xþ Dx� ¼ P(x) ¼ limT!1

(Tx=T) (2:297)

xn(t)

t1 t1+t

x3(t)

t1 t1+t

x2(t)

t1 t1+t

x1(t)

t1 t1+t

t

t

t

t

FIGURE 2.17 Ensemble of sample functions forming a random process.



For small Dx, a probability density function p(x) can be defined as

Prob½x < x(t) � xþ Dx� ¼ P(x) ¼ p(x)Dx (2:298)

More precisely,

p(x) ¼ limDx!0

P(x)

Dx¼ lim

Dx!0

1Dx

limT!1

TxT

� �(2:299)

It is evident from Equation 2.299 that p(x) is the slope of the cumulative probability distribution P(x).The area under the probability density curve between any two values of x represents the probability ofthe variable being in this interval. Also, the probability of x(t) being between x¼�1 is

P 1ð Þ ¼ð1

�1p(x) dx ¼ 1:0 (2:300)

The mean value x�(t) coincides with the centroid of the area under the probability density curve p(x), asshown in Figure 2.19. Therefore, in terms of the probability density p(x), the mean value is given by

x� tð Þ ¼ð1

�1xp(x) dx (2:301)

Likewise, the mean square value x2(t) is determined from the second moment to be

0

x

tT

∆t1 ∆t2 ∆t3 ∆t4

FIGURE 2.18 Probability measurement.

X–x 0 x x + ∆

p (x)

p (x)∆x

FIGURE 2.19 Probability density curve.



x2 tð Þ ¼ð1

�1x2p(x) dx (2:302)

The variance s2x , previously defined as the mean square value about the mean, is

s2x ¼

ð1�1

(x� x�)2p(x) dx

¼ð1

�1x2p(x) dx� 2x�

ð1�1

xp(x) dxþ (x�)2ð1

�1p(x) dx

¼ x�2 � 2(x�)2 þ (x�)2

¼ x�2 � (x�)2 (2:303)

The standard deviation sx is the positive square root of s2x .

The most widely used statistical distribution for modeling random processes is the Gaussianor normal random process. The probability density function x(t) of a Gaussian random process isgiven by

p(x) ¼ 1ffiffiffiffiffiffi2p

psx

e�1=2 x��xsx½ �2 (2:304)

where x� and sx are the mean value and standard deviation of x. By defining a standard normalvariable z as

z ¼ (x� x�)=sx:

Equation 2.304 becomes

p(x) ¼ 1ffiffiffiffiffiffi2p

p e�1=2z2 (2:305)

The probability of x(t) in the interval �ks and þks assuming x�¼ 0 is

Prob½�ks � x(t) � ks� ¼ðks

�ks

1ffiffiffiffiffiffi2p

pse�1=2 x2

s2dx (2:306)

where k is the positive number. Figure 2.20 shows the Gaussian probability density function that is abell-shaped curve symmetric about the mean value.

2.4.2 AUTOCORRELATION FUNCTION

The autocorrelation function of a stationary random process is defined as the average value of theproduct x(t) and x(t þ t). The process is sampled at time t and then again at time t þ t, as shown inFigure 2.21.

Rx(t) ¼ limT!1

1T

ðT0

x(t)x(t þ t) dt (2:307)



The quantity Rx(t) is always a real-valued even function with a maximum occurring at t¼ 0, i.e.,

Rx(0) ¼ limT!1

1T

ðT0

x2(t) dt ¼ x2(t) (2:308)

For very large time intervals, with t!1, the random process will be uncorrelated, and in this case,

Rx(1) ¼ x�(t)½ �2

i.e.,

x�(t) ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiRx(1)

p(2:309)

The mean value of x(t) is equal to the positive square root of the autocorrelation as the timedisplacement becomes very long.

2.4.3 RESPONSE OF AN SDOF SYSTEM TO AN ARBITRARY FUNCTION INPUT

Next we consider the response of an SDOF vibrations system to arbitrary function input.The equation of motion is given by

y€þ 2zvny_ þ v2ny ¼ x(t) (2:310)

wherex(t)¼F(t)=mvn ¼ ffiffiffiffiffiffiffiffiffi

k=mp

z ¼ c=Cc

Cc ¼ 2 km

x0

p(x )

prob [– ks ≤x (t ) ≤ ks ]

–c s +c s

FIGURE 2.20 Gaussian probability distribution.

t

Tt

x (t)

0 t

FIGURE 2.21 Autocorrelation measurement.



The arbitrary function x(t) is plotted in Figure 2.22. There are two ways to obtain the solution ofEquation 2.310. The first one is the impulse response method.

In the method, we assume the forcing function x(t) to be made up of a series of impulses ofvarying magnitude as shown in Figure 2.23 and the amplitude applied at time t is x(t) dt. Also ify(t)¼ h(t � t) denotes the response to the unit impulse excitation d(t � t), it is called the impulseresponse function. Note that the unit impulse applied at t¼ t is denoted by

x(t) ¼ d(t � t)

where d(t � t) is called the Dirac delta function with

d(t � t) ! 1 as t ! t (2:311)

d(t � t) ¼ 0 for all t except at t ¼ t and

ð1�1

d(t � t) dt ¼ 1

The total response of the system can be obtained by superimposing the responses to the impulses ofx(t) dt applied at different values of t¼ t. Hence, the response to the total excitation is given by thesuperposition or convolution integral:

y(t) ¼ðt

�1x(t)h(t � t) dt (2:312)

where x(t)h(t � t) is the response to the excitation x(t) dt. The typical impulse response function isshown in Figure 2.24.

x(t )

x(t)

0t

t t + d t

FIGURE 2.22 Forcing function in the form of a series of impulses.

x(t )

0tt

∞

d (t – t)

Unit

FIGURE 2.23 Unit impulse excitation at t¼ t.



It should be noted here that since h(t � t)¼ 0 when t< t or t> t, the upper limit of integrationof Equation 2.312 can be replaced by 1. That is,

y(t) ¼ð1

�1x(t)h(t � t) dt (2:313)


y(t) ¼ð1

�1x(t � u)h(u) du (2:314)

where we have changed the variable from t to u¼ t � t in Equation 2.313. Another approach isfrequency response method. The transient response is

x(t) ¼ 12p

ð1v¼�1

X(v)eivt dt (2:315)

Equation 2.315 indicates the superposition of components of different frequencies v. If the forcingfunction of unit modulus is

�x(t) ¼ eivt (2:316)

then its response is given by

�y(t) ¼ H(v)eivt (2:317)

where H(v) is the complex frequency response function.The total response of the system by superposition principle gives

y(t) ¼ H(v)x(t) ¼ð1

�1H(v)

12p

X(v)eivt dv ¼ 12p

ð1�1

H(v)X(v)eivt dv (2:318)

or

y(t) ¼ 12p

ð1�1

Y(v)eivt dv (2:319)

where Y(v) is the Fourier transform of the response function y(t). We note from Equations 2.318 and2.319 that

tt

x(t )

0

h(t – t)

FIGURE 2.24 Impulse response function.



Y(v) ¼ H(v)X(v) (2:320)


y(t) ¼ h(t) ¼ 12p

ð1�1

X(v)H(v)eivt dv (2:321)

where X(v) is the Fourier transform of x(t)¼ d(t).

X(v) ¼ð1

�1x(t)e�ivt dt ¼

ð1�1

d(t)e�ivt dt ¼ 1 (2:322)

Now, since d(t)¼ 0 everywhere except at t¼ 0 where it has a unit area and e�ivt¼ 1 at t¼ 0,Equations 2.321 and 2.322 give

h(t) ¼ 12p

ð1�1

H(v)eivt dv (2:323)

or

H(v) ¼ð1

�1h(t)e�ivt dt (2:324)

2.4.4 POWER SPECTRAL DENSITY FUNCTION

The power spectral density (PSD) of a random process provides the frequency composition of thedata in terms of the spectral density of its mean square value. The mean square value of a sampletime-history record, in a frequency range v and vþDv, can be obtained by passing thesample record through a bandpass filter with sharp cutoff features and computing the average ofthe squared output from the filter. The average square value will approach an exact mean squarevalue as T ! 1, i.e.,

C2x(v,Dv) ¼ lim

T!11T

ðT0

x2(t,v,Dv) dt (2:325)

where x(t,v,Dv) is the portion of x(t) in the frequency range v and v þ Dv. For a small value ofDv, a PSD function Sx(v) is defined as

C2x(v,Dv) ffi Sx(v)Dv (2:326)

i.e.,

Sx(v) ¼ limDv!0

C2x(v,Dv)Dv

¼ limDv!0

1Dv

limT!1

1T

ðT0

x2(t,v,Dv) dt (2:327)

The quantity Sx(v) is always real-valued nonnegative function.



In experimental work, a different unit of PSD is often used. The experimental spectral density isdefined by W( f ), where f denotes the frequency in cycles per unit time. The relation between S(v)and W( f ) is

W( f ) ¼ 4pS(v) (2:328)

In which v¼ 2pf. For a stationary random process, the autocorrelation and PSD functions arerelated by a Fourier transform as

R(t) ¼ð1

�1S(v)eivt dv (2:329)

S(v) ¼ 12p

ð1�1

R(t)e�ivt dt (2:330)

In a limiting case, where t¼ 0,

R(0) ¼ E x2(t)� � ¼ ð1

�1S(v) dv (2:331)

i.e., the mean square value is equal to the sum over all frequencies of S(v) dv; therefore S(v) may beinterpreted as a mean square spectral density. Probability density functions, autocorrelation func-tions, and power spectral densities for four sample time-history records are shown in Figure 2.25.

x(t)

x(t)

x(t)

x(t)

P(x)

P(x)

P(x)

P(x)

Rx(t)

Rx(t)

Rx(t)

Rx(t)Sx

Sx

Sx

t

t

Narrow band

Wideband random

t

t

Sx

t

−x x0

0

0

0

t

tt

t

t

t

t

FIGURE 2.25 Probability density, autocorrelation, and power spectral density functions for four sample timehistories.



2.4.5 JOINT PROBABILITY DENSITY FUNCTION

The joint probability density p(x,y) of two random variables is the probability that both variablesassume values within some defined pair of ranges at any instant of time. If we consider two randomvariables x(t) and y(t), the joint probability density has this property: the fraction of ensemblemembers for which x(t) lies between x and x þ dx and y(t) lies between y and y þ dy is p(x,y) dx dy.The joint probability densities are positive, and the probabilities of mutually exclusive events areadditive. Also,

ð1�1

ð1�1

p(x,y) dx dy ¼ 1 (2:332)

When two variables are statistically independent, the joint probability density is given by

p(x,y) ¼ p(x)p(y) (2:333)

2.4.6 CROSS-CORRELATION FUNCTION

The cross-correlation function of two random variables indicates the general dependence of onevariable on the other. The cross-correlation function of the time-history records x(t) and y(t) isshown in Figure 2.26 and given as

Rxy(t) ¼ limT!1

1T

ðT0

x(t)y(t þ t) dt (2:334)

The function Rxy(t) is always a real-valued function and can be either negative or positive.Also, Rxy(t) does not necessarily have a maximum at t¼ 0, nor is Rxy(t) an even function. However,Rxy(t) is symmetric, i.e.,

Rxy(�t) ¼ Rxy(t) (2:335)

when Rxy(t)¼ 0; then x(t) and y(t) are said to be uncorrelated.

x(t)

0T

t

y(t)

0T

t

t

FIGURE 2.26 Cross-section measurements.



We have previously defined the complex frequency response function H(v), whose magnitudeis equal to the amplitude ratio and whose ratio of imaginary to real parts is equal to the tangent of thephase angle f. The Fourier transforms of the response and excitation, i.e., X(v) and F(v), are relatedthrough the frequency response function as

X(v) ¼ H(v)F(v) (2:336)

This relation is valid for any arbitrary excitation f(t). If the excitation is a stationary random process,then the response will also be a stationary random process. By using mathematical manipulations, itcan be shown that for a linear system the response mean square spectral density Sx(v) and the meansquare spectral density Sf (v) of the excitation are related as

Sx(v) ¼ H(v)j j2Sf (v) (2:337a)

The mean square value of the response can be obtained as

Rx(0) ¼ E x2(t)� � ¼ 1

2p

ð1�1

H(v)j j2Sf (v) dv (2:337b)

From Equations 2.336 through 2.337, it is evident that for a linear system the response mean squarespectral density and mean square value can be calculated from the mean square spectral density ofthe excitation and the magnitude of the complete frequency response function H(v), respectively.

If the excitation has a Gaussian probability distribution and the system is linear, then theresponse will also be Gaussian. This implies that for the stationary process the probability distri-bution of the response is completely defined by the mean and mean square values of the response.

2.4.7 RESPONSE OF AN SDOF SYSTEM TO A RANDOM INPUT

As an example, we can consider an SDOF system as shown in Figure 2.27, which is a simple modelof a quarter vehicle traveling on rough surface. The equations of motion are

my€þ c(y_ � z_)þ k(y� z) ¼ 0 (2:338)

it can be converted to be the following form:

my€þ 2jvny_ þ v2ny ¼ v2

n f (t)

m Vy (t )

c k / 2 k / 2

z (t )

FIGURE 2.27 Schematic of a model of a quarter of vehicle traveling over a rough road.



wheref (t)¼ (2jvn)z_ þ zt¼ x=V, x is the horizontal coordinate

The PSD of the response is given by

Sy(v) ¼ H(v)j j2Sz(v) (2:339)

where

H(v)j j2¼n½1� (v=vn)

2�2 þ ½2j(v=vn)�2o�1

(2:340)

If Sz(v)¼ S0 (a constant), which represents a track input corresponding to white noise, then fromEquations 2.339 and 2.340 we obtain

Sy(v) ¼ S0

1� (v=vn)2

� �2þ 2j(v=vn)½ �2(2:341)

The mean square value of the response will be

E y2(t)� � ¼ S0

2p

ð1�1

dv

1� (v=vn)2

� �2þ 2j(v=vn)½ �2(2:342)

An integration of Equation 2.342 can be performed by using the residue theorem of complexvariables, which gives

E½y2(t)� ¼ S0vn=4j (2:343)

Because the random process is Gaussian with zero mean value, the mean square value equationdescribed by Equation 2.343 is sufficient to determine the shape of the response probability densityfunction; this makes it possible to evaluate the probability that the response y(t) might exceed agiven displacement.

2.4.8 RESPONSE OF MDOF SYSTEMS TO RANDOM INPUTS

We have already seen from Equation 2.106 that the uncoupled equation of motion of the rth mode ofa dynamic system that has proportional damping is given by

y€r þ 2jrvry_r þ v2r yr ¼ f�r(t) ¼ �f rð Þ

n oTF(t)f g (2:344)

where �f rð Þn o

represents the weighted rth modal vector of the undamped system. We introduce thefollowing Fourier transforms of yr(t) and f�r(t), respectively, in the form:

Yr(v) ¼ð1

�1yr(t) e

�ivt dt

Fr(v) ¼ð1

�1f�r(t) e

�ivt dt ¼Xnj¼1

�fj(r)

ð1�1

Fj(t) e�ivt dt

(2:345)



Then, we obtain Fourier transforms of both sides of Equation 2.344 as

Yr(v) �v2 þ i2jrvrvþ v2r

� � ¼ v2r Fr(v), r ¼ 1, 2, . . . , n (2:346)

Equation 2.346 can be solved for Yr(v) as

Yr(v) ¼ Hr(v)Fr(v), r ¼ 1, 2, . . . , n (2:347)

where

Hr(v) ¼ ½1� (v=vr)2 þ i2jr(v=vr)��1, r ¼ 1, 2, . . . , n (2:348)

The response correlation matrix [Rx(t)] is given as

Rx(t)½ � ¼ limT!1

1T

ðT=2�T=2

x(t)f g x(t þ t)f gT dt (2:349)

Because the vector x(t)f g ¼ �f� �

y(t)f g, we can write Equation 2.349 as

Rx(t)½ � ¼ limT!1

1T

ðT=2�T=2

�f� �

x(t)f g x(t þ t)f gT �f� �T

dt ¼ �f� �

Ry(t)� �

�f� �T

(2:350)

where

Ry(t)� � ¼ lim

T!11T

ðT=2�T=2

y(t)f g y(t þ t)f gT dt (2:351)

is the response correlation matrix associated with generalized coordinates yr(t) (r¼ 1, 2, . . . , n).If [H(v)] is the diagonal matrix of the frequency response function and [H*(v)] is its conjugate,

then the correlation matrix is

Ry(t)� � ¼ 1

2p

ð1�1

H*(v)½ � Sf (v)� �

H(v)½ � eivt dv (2:352)

where [Sf (v)] is an n3 n excitation matrix associated with the generalized forces f�r(t). Now[Sf (v)] can be expressed in terms of the fourier transform of the excitation correlationmatrix [Rf (t)] associated with f�r(t) as

Sf (v)� � ¼ ð1

�1Rf (v)� �

e�ivt dt (2:353)

and [Rf (t)] has the form

Rf (t)� � ¼ lim

T!11T

ðT=2�T=2

�f(t)� �

�f(t þ t)� �T

dt (2:354)



where {�f(t)} is the vector of generalized forces f�r(t)Therefore

�f(t)� � ¼ �f

� �F(t)f g (2:355)

and

�f(t þ t)� �T¼ F(t þ t)f gT �f

� �TBy substituting Equation 2.355 into Equation 2.354, we obtain

Rf (t)� � ¼ �f

� �RF(t)½ � �f

� �T(2:356)

where

RF(t)½ � ¼ limT!1

1T

ðT=2�T=2

F(t)f g F(t þ t)f gT dt (2:357)

Introducing Equation 2.356 into Equation 2.353, we obtain

Sf (v)� � ¼ �f

� � ð1�1

RF(t)½ � e�ivt dt �f� �T¼ �f

� �SF(v)½ � �f

� �T(2:358)

where

SF(v)½ � ¼ð1

�1RF(v)½ � e�ivt dt (2:359)

is the excitation spectral matrix associated with the forces Fi(t) (i¼ 1, 2, . . . , n).The response correlation matrix is obtained by substituting Equations 2.352 and 2.358 into

Equation 2.350

Rx(t)½ � ¼ 12p

�f� � ð1

�1H*(v)½ � �f

� �SF(v)½ � �f

� �TH(v)½ � eivt dv �f

� �T(2:360)

and the autocorrelation function associated with the random response process xi(t) is

Rxi(t)½ � ¼ 12p

�fi

� � ð1�1

H*(v)½ � �f� �

SF(v)½ � �f� �T

H(v)½ � eivt dv �fi

� �T(2:361)

where �fi

� �is the ith row matrix, i.e., �fi

� � ¼ �f1ð Þi , �f

2ð Þi , . . . , �f

nð Þi

h i, which for t¼ 0 yields the mean

square value

Rxi(0)½ � ¼ 12p

�fi

� � ð1�1

H*(v)½ � �f� �

SF(v)½ � �f� �T

H(v)½ � dv �fi

� �T(2:362)



2.5 FUNDAMENTALS OF SOUND

In this section, we introduce the fundamentals of sound with emphasizing on air sound. We willdescribe the basic concepts of physical sound, sound perception and weighting functions, soundwave equation, sound radiation of structures, sound reflection and refraction. Finally, we discuss inbrief the solid sound, or stress wave propagation in elastic solid media, with focus on surface stresswave associated with friction.

Sounds are pressure variations that propagate in an elastic medium like the liquid, gaseousmaterials, and solid. The words ‘‘sound’’ and ‘‘vibration’’ are often connected. The generation ofsound is usually attributed to the vibration of solid objects or liquid disturbance, and sound could beconsidered as the vibrations occur in elastic medium.

The sound pressure variation propagates in the form of sound waves. The simplest type of waveis a sine wave that is a periodic wave. The time it takes to complete a complete cycle is the period ofthe wave, and is equal to the inverse of frequency. The wavelength is the distance that sound travelsin the interval of one period; it can be expressed as l¼ c=f¼ 2pc=v in terms of Equation 2.136. Itequals to the speed of sound divided by the frequency.

2.5.1 AIRBORNE SOUND AND MEASURES

Our ears are essentially very sensitive to wide range of airborne sound with variation of amplitudeand frequency. The audible frequency ranges from 20 Hz to 20 kHz. The sound with frequencyvariations within this range and with proper amplitude can be perceived by brain through ear. Thegeneration, propagation, and perception of airborne sound are of essential importance. The soundsbelow or above the frequency limit are referred to as infrasound and ultrasound.

Airborne sound exhibits as the fluctuation (compressions and rarefactions) of air pressure.Sound pressure can be regarded as an alternating component superposed to constant atmosphericpressure. Air pressure P(t) in a given point of space can be expressed as the sum of the atmosphericpressure Po and the sound pressure p(t).

P(t) ¼ Poþ p(t) (2:363)

In dry air at 208C the speed of sound c is 344 m=s. The dependence of the speed of sound on variousparameters can be described as

c ¼ffiffiffiffiffiffiffiffiffiffiffigRTa

p(2:364)

in which g is thermal ratio, R is gas constant, and Ta is thermal temperature.For arbitrary complex sound waves, Fourier analysis can be used to transform it to as the sum of

a number of sine waves of different frequencies, amplitudes, and phases, as detailed in the previoussection. The spectrum of a sound wave is a two-dimensional representation showing the amplitudesof sine waves of different frequencies that comprise the wave. Each spike on a spectral plotcorresponds to a sine wave.

The standardized unit of sound pressure is the Pascal (1 Pa¼ 1 N=m2), note that the atmosphericpressure is about 100,000 Pa.

The maximum sensitivity of the ear is around 3 kHz. The pressure fluctuations which can besensed by ear are in the range from slightly below 23 10�5 Pa of the nominal threshold of hearingto 200 Pa of the threshold of pain, and above. In this wide range of pressure, the human response isnonlinear, typically a tribling of actual pressure fluctuations is perceived as a doubling of theloudness of the sound.

This nonlinearity and the very wide pressure range have led to the use of a logarithmic scale tomake objective measurements more feasible. We commonly use the decibel scale to measure sound



levels with the measurement units being decibels (dB). The quantitative value is called soundpressure level (SPL),

SPL ¼ 10 log 10Prms

Pref

� �2

(dB) (2:365a)

where Prms is root mean square (rms) value of the measured pressure fluctuations p(t), in pascals, PaPref is the standard reference pressure of 23 10�5 Pa, and this is the level of the 1 kHz sinusoidalsound just yet audible by the average human ear. Note that the sound pressure level is based on thesquare of the pressure ratio to a given reference.

The threshold of pain (140 dB) corresponds in air to pressure fluctuations of Prms¼ 200 Pa.Sound can also be characterized by the sound power level (SWL),

SWL ¼ 10 log 10W

Wref

(dB) (2:365b)

where W is the sound power for the element of the measurement surface, watts, Wref is the standardreference power of 10�12 W.

Sound can also be characterized by the acoustic power that comes through a unit of area. Thisquantity is called sound intensity and its value can be expressed as the product of sound pressure andparticle velocity. The sound intensity I is defined as the energy flux (power per surface area)corresponding to sound propagation. Intensity is usually given in related form in dB, too. Intensityis expressed as the intensity level (IL),

IL ¼ 10 log 10I

Iref(dB) (2:365c)

The reference intensity level Iref¼ 10�12 W=m2. For progressive plane waves, I ¼ p2rms=r0c0, wherer0c0¼ 43 102 kg=m2 s for air under atmospheric conditions.

2.5.2 NOISE AND SPECTRUM ANALYSIS OF NONSTATIONARY SOUND

Noise is unwanted sound in air. It is usually made up of energy at many different frequencies. Theextreme case is the flat spectrum and rough waveform that characterizes white noise, which containsequal amounts of all frequencies. In application, the noise might be band limited, containing energyin a certain band of frequencies.

Conventional noise classifications include the steady noise and unsteady noise. The unsteadynoise can be subdivided into fluctuating noise, intermittent noise, and impulse noise.

According to ISO 2204, the impulse noise can be characterized as a burst sound or continuousburst sounds with a duration time less than one second.

It is not straightforward to classify the friction sound in terms of the above classification.Usually friction noise could be intermittent type chirp or continuous squeal, and it could be theimpulse type of sound like single chirp or tick.

As discussed in preceding sections, Fourier transform-based techniques are effective as long asthe frequency contents of the signal do not change with time. However, when the frequency contentsof the vibration and sound signal evolve over an observation period, time–frequency or wavelettransforms should be considered. This is the case for addressing much friction-induced noise.Specifically, the time–frequency transform is suited for signals with slow frequency changes (narrowinstantaneous bandwidth), such as sounds heard during an engine run-up or run-down, whereas thewavelet transform is suited for signals with rapid changes (wide instantaneous frequency bandwidth),



such as sounds associated with engine ticking or knocking. The success of applications of the time–frequency and wavelet transforms is largely based on understanding their fundamentals.

The friction-induced sounds are usually the nonstationary sounds. We have already indicatedthat most sounds are made up of spectral components, or partials, of different frequencies.Typically, the relative amplitude of these partials does not remain constant throughout the durationof a sound. That is, it is usually incorrect to think of the envelope simply as a function that controlsthe scaling of a waveforms amplitude over time. A more accurate picture of what happens can beobtained by thinking of a waveform in the frequency domain, and then using a separate envelope todescribe each partial. This type of representation is known as a time-varying spectral plot.

Time–frequency analysis and wavelet are two approaches to provide time-varying spectral plot.Time–frequency analysis is based on the simultaneous representation of a signal in both time

and frequency domain. The time signal is first windowed into small intervals and the fast Fouriertransform is taken for each interval. By changing the window function, the time–frequencyresolution can be enhanced.

The basis functions of the Fourier series are sine and cosine functions, whereas for the wavelettransforms any zero mean function that is transformable and invertible can be used as a basisfunction. One of advantage of the wavelet transform is the proportional decrease of the window sizewith increasing frequency. A short window at high frequencies allows excellent time resolution butresults in a somewhat poor frequency resolution. The high time resolution at high frequencies ofwavelet transforms makes it possible to resolve multiple short consecutive events that occur at thefriction-induced nonstationary sound.

Linear time–frequency analysis can be used to deal with a signal in both the time and thefrequency domain where the amplitude of the signal is analyzed. The bilinear or quadratic time–frequency analysis is where the energy of the signal is analyzed. The short-time-Fourier transformsdeal with linear time–frequency distributions, whereas the spectrograms are concerned with thequadratic form of the time–frequency analysis.

Linear time–frequency analysis or short-time–frequency analysis is based on the expansion ofthe signal into a set of weighted frequency modulated Gaussian functions where the weighsdescribe the signal behavior in the local time and frequency domain. A linear time–frequencyanalysis does not contain cross-term interference, it is given by

STFT(t, f ) ¼ðx(t)h*(t � t)e�2pift dt (2:366a)

wherex(t) is the measured signalh*(t � t) is the window function

Quadratic (bilinear) time–frequency analysis has the following typical quadratic time–frequencydistributions:

W(t,v) ¼ðs*(u� t=2)s(uþ t=2)e�itv dt (2:366b)

wheres is the signals* is the complex conjugate of the measured signal

Wavelet transforms are based on time scale analysis rather than on time–frequency analysis. Forwavelet transforms the signal is projected onto a family of zero mean functions namely wavelets.The advantage of using wavelets is that no cross-term interference exists and high time resolution athigh frequencies is obtained. The continuous wavelet transform is defined as



WT xð Þx (t, f ) ¼

ðt0

x(t0)ffiffiffiffiffiffiffiffiffiffiffiffif =f0j j

pg*

f

f0(t0 � t)

� �dt0 (2:367)

whereg * is the analyzing waveletx(t0) is the measured signalf is the frequency. Wavelets use frequency-dependent filtering windows

In the low frequency regime, the analyzing window size is large allowing high frequencyresolution. With increasing signal frequency, the analyzing window size is reduced and an improvedtime resolution is achieved. The bandwidth is proportional to frequency, thus, wavelet transformsare best suited to detect short consecutive high frequency components of a signal. Similar to theshort-time–frequency analysis, the wavelet transform is a linear time–frequency transformation. Thesquared wavelet transform is called a scalogram.

A single scalogram can easily cover the normal audible frequency range of 20 Hz to 20 kHz(with octave bandwidth) over a record length of several seconds, with a time resolution ofapproximately 0.1 ms for the high frequency components. This wide dynamic range in bothfrequency and time makes the scalogram suitable for such varied tasks as obtaining an overviewof a complex sound recording, such as quantifying the friction noise in automobiles for which awide range of frequency analysis is needed.

2.5.3 SOUND PERCEPTION AND WEIGHTING CURVES

Human’s hearing is not only nonlinear with respect to amplitude, but also with respect to frequency.Sounds of different frequencies at the same level are not heard as equally loud.

Loudness depends on frequency. In general, tones at given amplitude with frequencies betweenabout 1000 and 5000 Hz are perceived as louder than those that are higher or lower. More exactfunctions of loudness with frequency can be found in the form of equal-loudness contours. Theequal loudness contours for sine waves, also known as the Fletcher–Munson loudness curves, areillustrated in Figure 2.28, which illustrate human perception to sounds at various frequencies andamplitudes.

Sound loudness is therefore a subjective term describing the strength of the ear’s perception to asound and is given in units of phons. The phon is a unit dependent on frequency and amplitude.

dB

130 130

Phon

110

90

70

50

30

10

110

90

70

50

30

10

20 200 1 k 20 k

FIGURE 2.28 Fletcher–Munson loudness curve.



Sound duration is a variable that is not accounted for in this empirical unit. The equal soundloudness curves are based on continuous pure tones for comparative phon levels.

When a more detailed understanding of the human response is required, loudness curves or‘‘sound quality’’ analysis systems are employed. For subjective judgment of the sound level, theterm of loudness level was introduced. Loudness level of an arbitrary sound is as many phons thatmany dBs which is the sound pressure level of the 1 kHz tone of the same loudness.

To evaluate the resulting level of simultaneous sounds, the term loudness has been introduced.Loudness is denoted as N and its unit is called sone. Loudness is the sensation most directlyassociated with the physical property of sound intensity. Loudness is measured in sones (aninternational standard defines the loudness of a 1000 Hz tone at 40 dB as 1 sone).

In addition to frequency, there are also additional dependences, like the place theory of pitch,which states that sound’s frequencies must be separated enough for the human ear to distinguish. Ifthe sounds are within a critical band for the ear, which is additionally frequency dependent, thecombined sound loudness will only be slightly louder than the individual sounds.

Masking is another term related to the simultaneous presence of two different sounds. Maskingmeans covering the weaker sound with a stronger sound when each has a different frequency. Highfrequency sounds are easier to mask.

Spatial parameters of sound are also very important. First of all, direction of the sound has to bementioned. In the horizontal plane, the sound is localized by the difference of the sound pressures atour ears. At low frequencies the phase difference is detected while on higher frequencies adifference in intensity arises due to the shadowing caused by the head. To perceive direction inthe vertical plane, the head has to be moved up and down.

Friction-induced sound is usually transient with short duration. Loudness depends on duration.For sounds shorter than about a second, loudness increases with duration. For example, for eachhalving in duration, the intensity of noise bursts below 0.2 s must be increased by about 3 dB inorder to maintain constant loudness. However, for sounds longer than a second, loudness remainsconstant. Loudness is affected by bandwidth. Within a critical band, energy is summed; outside of acritical band, loudness is summed. Sounds with a large bandwidth are louder than those with anarrow bandwidth, even if they have the same energy.

The apparent loudness of a sound (i.e., the subjective evaluation made by the human ear) varieswith frequency as well as with sound pressure. Compared with the response to sounds in thefrequency range 250 to 8000 Hz, the ear is less sensitive at other frequencies, particularly below100 Hz. To align the frequency response of measuring systems to be similar to that of the humanear, several weighting curves were proposed. The ‘‘A’’ weighted curve was designed to approximateto the human response at low sound pressure levels (<55 dB). Similarly, the ‘‘B’’ and ‘‘C’’weightings were intended for use at sound pressure levels of 55 to 85 dB and above 85 dB,respectively. A-weighting is used in perceived loudness measurements. C-weighting is used forsound spectral analysis. The trend is towards the exclusive use of the ‘‘A’’ weighting curve for noisemeasurements associated with engineering application. Figure 2.29 shows the weighting curves ofA-weighting and C-weighting.

2.5.4 SOUND WAVE EQUATION

Sound pressure variation propagates in the form of sound waves. Sound wave travels through threespace dimension and time. A direct way is to identify all the parts of the wave which have the samephase relationship. We call these like parts with surface as wavefront, which is perpendicular to thewave propagation direction. If wavefront is planar surface, the wave is a plane wave that propagatesin one direction only.

A point source is a source whose size is small compared to distance from receiver. A soundwave propagates away from a point source the waves spread outward in all directions. The wavefronts are the surfaces of a series of concentric expanding sphere. The sound power produced by the



source is a constant. The intensity of the sound depends on the source power divided by the areaover which the power is spread. The intensity decreases with distance from the source.

Considering a differential element in air as shown in Figure 2.30, the continuum, isentropic,and frictionless hypothesis are assumed. For plane wave, as it propagates in one direction x, itsequation is

@2p

@x2¼ 1

c2@2p

@t2(2:368)

in which p is sound pressure and c is the sound velocity. We have discussed this type of waveequation in dealing with the string vibrations by using modal analysis in Section 2.2.4.1 and wehave also briefed other approaches in Section 2.2.4.2.

The general solution of Equation 2.368 is

p(x,t) ¼ Ae�jkx þ Be jkx� �

e jvt (2:369)

in which k¼v=c is wave number, the first term in the equation represents the wave propagatingalong positive direction of x-axis, whereas the second term in the equation represents the wavepropagating along negative direction of x-axis. If we consider the wave propagation in infinitemedium where there is no reflection, then the wave propagates in one direction and we have

p(x,t) ¼ pAej(vt�kx) (2:370)

dB

10

0

20 200

A-Weighting

C-Weighting

1 k 20 k

−10

−20

−30

−40

−50

FIGURE 2.29 Weighting curves: A-weighting is used in perceived loudness measurements. C-weighting isused for sound spectral analysis.

P dy dz dy dzdxx

PP )( ∂

∂

dy

dz

dx

+

FIGURE 2.30 Schematic of differential element in analysis of airborne sound.



where pA is the pressure amplitude. We have particle velocity

v(x, t) ¼ � pAr0c

e j(vt�kx) ¼ vAej(vt�kx) (2:371)

The factor r0c is defined as the characteristic impedance. Specific acoustic impedance is defined asthe ratio of the complex pressure to the complex particle velocity,

Zs ¼ p

v(2:372)

Sound power is the average energy per unit time passing wavefront. For plane wave,

W ¼ 12p2Ar0c

S (2:373)

in which S is forefront area. Sound intensity is the average sound power per unit area. It is a realvector quantity and indicates the amount of energy per unit area per unit time in a given direction,for plane wave,

I ¼ p2A2r0c

¼ p2er0c

(2:374)

W ¼ IS (2:375)

The spherical wave is the wave with spherical wavefront. The pressure is only relevant to sphericalcoordinate r, and the wave equation is

@2p

@r2þ 2

r

@p

@r¼ 1

c2@2p

@t2(2:376)

its solution is

p(r,t) ¼ A

re�jkr þ B

re jkr

� �e jvt (2:377)

The first term stands for the wave radiation from source, whereas the second term represents thewave with reversal direction. If we consider the case of infinite medium radiation without reflection,we get

p(r,t) ¼ A

re j(vt�kr) (2:378)

from motion equation we can get velocity

v(x,t) ¼ A

rr0c1þ 1

jkr

� �e j(vt�kx) (2:379)

The specific acoustic impedance is

Zs ¼ p

vr¼ r0c

jkr

1þ jkr(2:380)



It is noted that if kr� 1 the specific acoustic impedance is equal to that of plane wave. Actually, thismeans that if the location is far away from source then the spherical wave can be assumed to be asplane wave.

Consider the area of spherical wave wavefront is

S ¼ 4pr2

so the sound power is

W ¼ 4pr2p2er0c

(2:381)

2.5.5 SOUND RADIATION OF STRUCTURES

Next we discuss the sound radiation from ideal simple source then extend to structures. Thecomplex sound source can be decomposed to be the summation of ideal simple sources. Thesimplest sound source of elastic body is the pulsating elastic ball. Assume the surface of the ballexperiences a harmonic motion and the vibration velocity on the ball surface is

u(a,t) ¼ uAei(vt�ka) (2:382)

in which k¼ 2p=l, a is ball radius. If the surface radiates sound wave, it is a spherical wave. Thepressure in infinite medium is

p(r,t) ¼ A

rei(vt�kr) (2:383)

the particle velocity in medium is

vr(r,t) ¼ A

rr0c1þ 1

jkr

� �ei(vt�kr) (2:384)

on the surface, if the particle has the same velocity as the surface, then we have

A ¼ ika2r0cuA1þ ika

¼ Aj jeiu (2:385)

and

p(r,t) ¼ Aj jrei(vt�krþu) (2:386)

As such, the radiated pressure is proportional to the vibrationary frequency and velocity as well asthe dimension of ball.

The sound intensity on the wavefront can be derived as

I ¼ 12r0cu

2A

a

r

� �2 k2a2

1þ k2a2(2:387)

and the sound power is

W ¼ 4pr2I ¼ 2pa2r0cu2A

k2a2

1þ k2a2(2:388)



Define sound radiation efficiency of vibration resource as

srad ¼ W

r0cShV2i (2:389)

in which hV2i is the time average of the square of surface vibration velocity, S is surface area, thenfor the pulsating ball, we have

srad ¼ (ka)2

1þ (ka)2(2:390)

which indicates that higher the frequency, the higher the radiation efficiency.If the radius is smaller enough, ka � 1, it becomes a point sound resource. In infinite medium,

the pressure due to point sound resource is

p(r,t) ¼ ikr0cuAS

4prei(vt�kr) (2:391)

Complex resource can be approximated by the superposition of the point sound resource.To give some sense to the sound radiation of structures, we consider the sound radiation of an

infinite plate. Assume the propagation velocity of the radiation wave is c, the wavelength is l, andthe wave number is k¼v=c. Considering the bending vibration of plate, from its wave equation, thepressure solution can be derived by combining with the vibration velocity, and the radiationefficiency of plate can be derived as

srad ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� k2P=k

2p (2:392)

in which k2P ¼ vffiffiffiffiffiffiffiffiffiffiffirh=D

pis the wave number of the transverse wave. This shows that the condition

for plate to radiate sound is k> kP, and the radiation achieves its maximum at the value k kP. Wedefine the corresponding threshold frequency as

fc ¼ c2

2p

ffiffiffiffiffirh

D

r(2:393)

The plate will not radiate sound wave unless the vibration frequency is larger than the thresholdfrequency. For plate with finite dimension, the radiation of sound could occur at all frequencyranges.

2.5.6 SOUND INTERFERENCE

The process of interference that results from the linear superposition of wave disturbances underliesmany commonly encountered acoustic phenomena. Superposition applies to any linear sound field,whether continuous or transient; whether generated by discrete compact sources, such as smallimpact, or by complex sound sources that are extended in space, such as vibrating machines; orcreated by the sound field radiated by a source together with reactions of that field by surroundingobstacles. Interference occurs even if sources are random in time and broadband in frequency.Reflection and refraction are the basic phenomena of sound interference.

A mechanical wave is reflected by an encounter with an interface between the wave-supportingmedium and some other medium having different dynamic properties, particularly the soundvelocity. As a result a wave travels back into the air, and this is the reflected wave. Reflection isnot perfect, and portion of the energy of the incident wave is transmitted into the second media. If a



reflecting surface is smooth and extensive compared with the wavelength of the incident sound, thereflected wave will obey Snell’s law of optical reflection: the angle of incidence equals the angle ofreflection.

Generally, waves bend and change direction of propagation when they impinge at an angle thatis not perpendicular to the second media. With different speeds of propagation, the waves willrefract.

Considering a sound wave is transmitted to the second medium where speed changes, itsdirection of travel changes. Figure 2.31 shows the schematic of the sound incident and refraction,where ui is the angle of incidence, ut is the angle of transmission, and c1, c2 are the propagationspeed in the first and second medium, respectively.

Snell’s law relates the transmitted beam direction to the incident beam direction and the speedsof propagation in the two media,

sin utsin ui

¼ c1c2

(2:394)

After the reflection and refraction, the sound frequency remains the same in the reflection andrefraction portions.

For simplification, let us consider the perpendicular incident sound and its reflection andrefraction as shown in Figure 2.32.

Assume the incident sound, reflection sound, and refraction or transmitted sound are,respectively,

pi(x,t) ¼ Piej(vt�k1x) (2:395a)

Incident beam

Slow medium

Fast mediumRefracted beam

qi

qt

C1

C2

FIGURE 2.31 Schematic of sound refraction.

Plane of discontinuity

Transmitted wave

x+

xT

x −

Incident and reflectedwaves

FIGURE 2.32 Perpendicularly incident wave.



pr(x,t) ¼ Pr ej(vtþk1x) (2:395b)

pt(x,t) ¼ Pt ej(vt�k2x) (2:395c)

k1 ¼ v=c1, k2 ¼ v=c2 (2:396)

The sound pressure of the waves in medium 1 and 2 are, respectively,

p1(x,t) ¼ pi(x,t)þ pr(x,t) ¼ Pi ej(vt�k1x) þ Pr e

j(vt�k1x) (2:397)

p2(x,t) ¼ pt(x,t) ¼ Pr ej(vt�k2x)

The particle velocities in medium 1 and 2 can be represented as

u1(x,t) ¼ 1z1

Pi ej(vt�k1x) þ Pr e

j(vt�k1x)� �

(2:398)

u2(x,t) ¼ Pt

z2e j(vt�k2x) (2:399)

Considering the pressure continuity and velocity equality across the interface, we have

pt(0,t) ¼ p2(0,t) (2:400)

Pi þ Pr ¼ Pt (2:401)

u1(0,t) ¼ u2(0,t) (2:402)

Pi � Pr

Z1¼ Pt

Z2(2:403)

We get the reflection coefficient

R ¼ Pr

Pi¼ Z1 � Z2

Z1 þ Z2(2:404)

and refraction coefficient

T ¼ Pt

Pi¼ 2Z2

Z1 þ Z2(2:405)

in which z1¼ r1c1, z2¼ r2c2 are specific acoustic impedance.

2.5.7 STRESS WAVE PROPAGATION IN ELASTIC SOLID MEDIUM

Next we discuss the stress wave propagates in elastic solid medium, which can be considered assolid sound to some extent. Of particularly interested is the acoustic emission, which is a stress wavein solid, generally caused by contact, bending, torsion, and crack formation. Acoustic emission willbe further discussed in Chapter 5 with focus on the application in detecting contact phenomena. Inthis chapter we introduce the basic concepts of stress wave propagates in elastic solid medium.

Considering the motion transmitted in solid medium, the vibrations concern with the motionwith time, whereas the waves concern with the motion over whole medium with respect to space andtime domain. In an infinite size medium, assuming a sine wave propagate along x direction, its wavecan be represented as



j ¼ a sin(kx� vt) (2:406)

Considering the specific position in the medium x0, its vibration can be represented as

j ¼ a sin(kx0 � vt) (2:407)

a wave is the propagation of a disturbance through a medium without actual displacement of themedium itself.

Waves show a behavior that can be described mathematically, i.e., moving waves carry energyfrom one point to another. In elastic solid, there exist both volume deformation and shearingdeformation. Longitudinal wave is the wave whose motion direction is on the same direction ofpropagation. The shear wave (transversal wave) is the wave whose propagation direction isperpendicular to the wave propagation direction. Longitudinal wave corresponds to the volumedeformation and shear wave corresponds to the shearing deformation. Next we introduce the basicconcepts and equation of elastic stress and propagation in elastic solid medium.

Consider elastic deformation in infinite elastic body; assume the stress state is defined by sx, sy,sz, gxy, gzy, gxz. Consider the deformation displacement (u,v,w) of a space point (x,y,z); assume itsnearby displacement is u þ du, v þ dv, w þ dw; and we have the following relationship for rotationangles,

du ¼ �vzdyþ vydzþ «xdxþ gxydyþ gzxdz

dv ¼ �vxdzþ vzdxþ gxydxþ «ydyþ gyzdz

dw ¼ �vydxþ vxdyþ gzxdxþ gzydyþ «zdz

(2:408)

vx ¼ 12

@w

@y� @v

@z

� �

vy ¼ 12

@u

@z� @w

@x

� �

vz ¼ 12

@v

@x� @u

@y

� � (2:409)

The elongation ratio or strain is given by

«x ¼ @u

@x

«y ¼ @v

@y

«z ¼ @w

@z

(2:410)

The shear strain is given by

gyx ¼12

@u

@yþ @v

@x

� �

gyz ¼12

@w

@yþ @v

@z

� �

gzx ¼12

@w

@xþ @u

@z

� � (2:411)



Then we consider strain and stress. Considering small deformation and isotropic elastic medium,Hook’s law holds, we have expansion ratio:

sx ¼ l�þ 2m«x

sy ¼ l�þ 2m«y (2:412)

sz ¼ l�þ 2m«z

tyx ¼ 2mgyz

tzy ¼ 2mgzy (2:413)

tyx ¼ 2mgyx

� ¼ «x þ «y þ «z (2:414)

in which l, m are Lame constants and m is shear elastic constant.

m ¼ E

2 1þ sð Þ (2:415)

in which s is Poisson’s ratio. The external forces per unit volume are X, Y, Z. The fundamentaldifferential equation governing elastic dynamic problem is

@sx

@xþ @txy

@yþ @txz

@zþ r X � @2u

@t2

� �¼ 0

@sy

@yþ @tyx

@xþ @tyz

@zþ r Y � @2v

@t2

� �¼ 0 (2:416)

@sz

@zþ @tzx

@xþ @tzy

@yþ r Z � @2w

@t2

� �¼ 0

in which r is the density. Consider no external force case and assume that the elastic constants areuniform in the medium; the above equation can be simplified as

r@2u

@t2¼ (lþ m)

@�

@xþ mr2u

r@2v

@t2¼ (lþ m)

@�

@yþ mr2v (2:417)

r@2w

@t2¼ (lþ m)

@�

@zþ mr2w

Apply @=@x, @=@y, @=@z to the above three equations, respectively, then make summarization:

@2�

@t2¼ (lþ 2m)

rr2� (2:418)

Apply @=@z to the second equation of Equation 2.417, then minus the result of applying @=@y to thethird equation of Equation 2.417,

@2vx

@t2¼ m

rr2v (2:419)



Similarly,

@2vy

@t2¼ m

rr2v (2:420)

@2vz

@t2¼ m

rr2v (2:421)

All of the above four equations can be represented as the following compact form:

@2j

@t2¼ cr2j (2:422)

This is the wave equation in elastic medium, c is the wave velocity.The air sound wave can be considered as the special case of the solid sound wave with the

condition that m¼ 0.We consider the elastic body wave, the simplest solution of j(x,y,z,t) has the following form:

j ¼ f (x� vt) (2:423)

It is a plane wave, which is independent of y, z; its phase is identical on the same plane perpendicularto x-axis. Assume r ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2 þ z2

pthe following solution is applicable for spherical wave:

j ¼ 1rf (r � vt) (2:424)

The surface of wave with same phase at a moment is the wavefront. The wavefront of plane wave isa plane and the wavefront of spherical wave is a sphere.

Based on Equation 2.418, Q wave propagates in terms of velocity

Vp ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(lþ 2m)=r

p(2:425)

it is called P wave. The compressive wave or longitudinal wave is P wave, whose motion directionis same as that of propagation direction.

Based on Equations 2.419 through 2.421, vx, vy, vz waves propagate in terms of velocity

Vs ¼ffiffiffiffiffiffiffiffiffim=r

p(2:426)

it is called S wave. The shear wave or transversal wave is S wave, whose motion direction isperpendicular to the propagation direction.

In solid, the wave is the superposition of the both. They are generally named as body wave. Theratio of the speed of P wave to that of S wave is only dependent on the Poisson ratio:

Vp=Vs ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2(1� s)=(1� 2s)

p(2:427)

Consider P wave propagates in plane wave; assume x axle is the wave propagation direction; and xaxle is also the wave motion direction, then the displacement can be represented as

u ¼ u(x,t), v ¼ w ¼ 0 (2:428)

based on motion equation, the general solution is



u ¼ f1(x� Vpt)þ f2(xþ Vpt) (2:429)

In infinite elastic medium, when wave is propagated in radioactive manner, the body wave energydecreases proportionally with the power of distance.

Consider S wave propagates in plane wave; assume x axle is the wave propagation direction;and z axle is the wave motion direction, then the displacement can be represented as

w ¼ w(x,t), v ¼ w ¼ 0 (2:430)

based on motion equation, the general solution is

w ¼ f1(x� Vst)þ f2(xþ Vst) (2:431)

If wave propagates in the proximity of free surface and propagates along the free surface, it is asurface wave. The energy reduction of surface wave is proportional to the distance from source, andthe magnitude of the surface wave decreases with the depth rapidly. Assume the free surface is xy(z¼ 0), the solution of surface wave is

j ¼ aeaz sin(kx� vt) (2:432)

k2 � a2 ¼ v2=v2 (2:433)

Considera> 0, the downward direction is negative z direction, then it exists on the surface with speed

C ¼ v=k ¼ Vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� (a=k)2

q(2:434)

For the vx¼vy¼vz¼ 0 condition, we have

u ¼ Aeaz cos(kx� vt) (2:435)

w ¼ a

kAeaz sin(kx� vt) (2:436)

k2 � a2 ¼ v2=V2p (2:437)

For Q¼ 0 condition, we have

u ¼ b

kBebz cos(kx� vt) (2:438)

w ¼ Bebz sin(kx� vt) (2:439)

k2 � b2 ¼ v2=V2s

The superposition of Equations 2.435 and 2.436 and Equations 2.438 and 2.439 on the surface ofisotropic, half infinite elastic medium gives rise to the Rayleigh wave,

u ¼ A eaz þ fb

kebz

� �cos(kx� vt) (2:440)

w ¼ Aa

keaz þ febz

� �sin(kx� vt) (2:441)



where A is arbitrary constant and VR¼v=k is Rayleigh wave speed, and the following relationshipholds:

a=k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� (VR=Vp)

2q

(2:442)

b=k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� (VR=Vs)

2q

(2:443)

f ¼ � 2a=k

1þ (b=k)2(2:444)

It is noted that near the surface, it is a backward motion, whereas it is forward motion below thesurface region.

BIBLIOGRAPHY

1. Den Hartog, J.P., Mechanical Vibrations, 4th ed., McGraw-Hill, New York, 1956.2. Hutton, D.V., Applied Mechanical Vibration, McGraw-Hill, New York, 1981.3. Meirovitch, L., Elements of Vibration Analysis, McGraw-Hill, New York, 1986.4. Meirovitch, L., Computational Methods in Structural Dynamics, Sijhoff and Noodhoff International

Publishers, The Netherlands, 1980.5. Clough, R.W. and Penzein, J., Dynamics of Structures, McGraw-Hill, New York, 1975.6. Hurley, W.C. and Rubinstein, M.F., Dynamics of Structures, Prentice-Hall, Englewood Cliffs, NJ, 1963.7. Nayfeh, A.H. and Mook, D.T., Nonlinear Oscillations, Wiley, New York, 1979.8. Timoshenko, S., Young, D.H., and Weaver, W., Vibration Problems in Engineering, Wiley, New York,

1974.9. Soedel, W., Vibrations of Shells and Plates, 2nd ed., Marcel Dekker, Inc., New York, 1993.10. Crandall, S.H. and Mark, W.D., Random Vibration in Mechanical Systems, Academic Press, New York,

1963.11. Rao, S.S., Mechanical Vibrations, Addison-Wesley, Reading, MA, 1986.12. Shabana, A.A., Theory of Vibration—Volume I: An Introduction, Springer-Verlag, New York, 1991.13. Shabana, A.A., Theory of Vibration—Volume II, Springer-Verlag, New York, 1991.14. Steidel, R.F., An Introduction to Mechanical Vibration, Wiley, New York, 1989.15. Thomson, W.T., Theory of Vibrations with Applications, Prentice-Hall, Englewood Cliffs, NJ, 1981.16. Newland, D.E., Random Vibrations and Spectral Analysis, Longman Group Limited, New Jersey, 1964.17. Robson, J.D., Random Vibrations, Edinburgh University Press, Edinburgh, 1964.18. Morse, P.M. and Ingard, K.U., Theoretical Acoustics, McGraw-Hill Book Company, Inc., New York,

1968.19. Morse, P.M., Vibration and Sound, McGraw-Hill, 2nd ed., New York, 1948.20. Pierce. A.D., Acoustics: An Introduction to Its Physical Principles and Applications, McGraw-Hill Book

Company, Inc., New York, 1981.21. Ingard, K.U., Fundamentals of Waves and Oscillations, Cambridge University Press, Cambridge, 1988.22. Shi, H.M., Sheng, G., and Wu, Y., Mechanical Vibration Systems: Analysis, Modeling, Measurement and

Control, HUST Press, Wuhan, 1991.




3 Contact and Friction

3.1 INTRODUCTION

Friction is a tangential resistant force from sliding interface. To investigate friction we need tocharacterize surface roughness to understand its statistical properties. Friction is linked with thesurface topography, and friction investigation is attributed to the determination of the real area ofcontact and the understanding of the mechanism of mating contacts. For instance, adhesion physicsexplains friction in terms of the formation of adhesive junctions by interacting asperities and theirbreakaway by shearing. To address the problem of real sliding asperity contact is quite difficult.The feasible approach is to assume the contact to be of a quasi-static nature. In many applicationswith relatively smooth surfaces the deformation of contacting asperities can be assumed to beelastic. For some problems the contact needs to extend to nonelastic conditions.

In this chapter we present the fundamentals of contact and friction between two contact surfaces.We focus on the mechanics of contact and friction by outlining the mechanical attributes of variousfriction processes in the context of the problems of the friction-induced vibrations. For advancedtopics related to tribological materials, chemistry, and physics, the readers are recommended to referto dedicated tribology references [1–25].

After the introduction, the second section briefs the mathematical description of surfaceroughness. The third section introduces the fundamental contact mechanics including Hertz analy-tical solution of single asperity contact and the treatment of multiple asperity contact between twosolid surfaces. The fourth section presents the basic principles of friction of two contact surfaces.The basic physics of the adhesion of two contact surfaces are detailed. The friction of two drycontact surfaces is presented, and then wet friction or the friction of two liquid-mediated surfaces isdescribed. The fifth section discusses the interactions between friction and vibrations. In the lastsection, we present some progress of investigations on friction in nano- and molecular scales, so asto provide a relatively comprehensive picture of friction.

3.2 SURFACE ROUGHNESS

3.2.1 CHARACTERISTICS OF RANDOM ROUGH SURFACES

There are no topographically smooth surfaces in nature and in engineering practice. All solidsurfaces, however smooth in a given scale, are comparatively rough in a smaller scale. In otherwords, solid surfaces are almost always comprised of random variations in surface height in somescales as shown in Figure 3.1. This feature of surface repeats down to nanoscale. Like a surface thatlooks smooth visually but is actually quite rough under microscale investigation. These surfaceheight deviations are called surface roughness and are characterized by asperities and valleys ofvarious magnitudes. Surface roughness is associated with a length scale. The magnitude of thesurface height deviations can vary at different length scales for the same surface. Usually, surfacefluctuations with long wavelengths are known as the waviness of a surface, whereas those havingshorter wavelengths are commonly associated with the roughness.

Sheng/Friction-Induced Vibrations and Sound: Principles and Applications 51784_C003 Final Proof page 85 1.11.2007 10:21pm

85

3.2.2 SURFACE ROUGHNESS PARAMETERS

Surface roughness is usually quantified by use statistics. Statistic parameters like the distribution ofthe surface heights are used to describe the vertical deviations of surface height with respect to areference plane. Most surface roughness is random and the distribution of the surface heightsusually follows a Gaussian distribution. The most common measures of surface roughness includethe centerline-average Ra characterizing the average roughness and root mean square roughness Rq.We consider a surface height profile z(x) over a length L in which profile heights are measured froma mean line. The centerline-average is defined as

Ra ¼ 1L

ðL0

z� mj j dx (3:1)

in which the mean is given by

m ¼ 1L

ðL0

z dx (3:2)

The root mean square roughness is defined as

Rq ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1L

ðL0

z2 dx

vuuut (3:3)

The variation of the distribution is characterized by the standard deviation of the height of thesurface from the centerline, s, which is defined as

Smooth engineering surface

Microscale observation

Nanoscale observation

Visual observation

FIGURE 3.1 Roughness in different scales.



s ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1L

ðL0

(z� m)2 dx

vuuut ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2q � m2

q(3:4)

A more detailed statistical analysis is given below. If we denote probability density function by p(z)to be the probability that the height of a particular point in the surface will lie between z and z þ dz,then the probability that the height of a point on the surface is smaller than z0 is given by thecumulative probability function:

P(z � z0) ¼ðz0

�1p(z) dz ¼ P(z0) (3:5)

It has been found that many real surfaces exhibit a height distribution that is close to the followingnormal or Gaussian probability function:

p(z) ¼ 1

sffiffiffiffiffiffi2p

p e �(z�m)2

2s2

� �(3:6)

3.3 CONTACT BETWEEN TWO SOLID SURFACES

3.3.1 SPHERICAL CONTACT

When two elastic spheres are brought together in contact and then loaded, local deformation willoccur. The contact point will enlarge into a surface of contact. Its area is a circle when the contactingbodies are spheres. The contact area size and the pressure between them can be determined fromtheir geometry and elastic properties. The solutions for deformation, area of contact, pressuredistribution, and stresses at the contact area were made by conventional Hertz theory. The relatedassumption is made that the deformation is elastic and that the dimensions of the contact area aresmall, relative to the radii of curvature and to the overall dimensions of the bodies. Thus the radii,though varying, may be taken as constant over the very small areas surrounding the contact area.The deflection integral is going to be derived for a plane surface.

Assume two spheres of different material with radii R1 and R2, respectively. Figure 3.2 showsthe spheres after loading, with the radius a of the contact area greatly exaggerated for illustration.

R2

R1

2a

FIGURE 3.2 Contact of two spheres.


Contact and Friction 87

Assuming the effective modulus involving Young’s moduli and Poisson ratios, E1, E2, v1, and v2 aredefined as follows:

E* ¼ (1� v21)

E1þ (1� v22)

E2

� ��1

(3:7)

This expression describes the compliance of the system at a fixed contact area. Furthermore, Hertztheory gives the development of the contact of the two spheres as the two surfaces are brought intocontact under load. It had been derived that the contact radius a is given by

a ¼ 3RP

4E*

� �1=3

(3:8)

in which P is load. Here R is a function of R1 and R2, called the radii of curvature

R ¼ 1R1

þ 1R2

� ��1

Define the contact displacement as

d ¼ R1 þ R2 � d (3:9)

where d is the distance between the centers of the two spheres; the compressive displacement or thenormal approach is given by

d ¼ a2

R(3:10)

The contact area is given by

A ¼ pa2 ¼ pRd (3:11)

The formula can be used for many other cases like one surface being concave, which indicates thatwhen the contact area is on the inside of a surface the numerical value of its radius is to be taken asnegative in all equations.

In the above model, the assumption is that the two surfaces in contact are frictionless, so thatshear stresses cannot be sustained at the interface. This ‘‘frictionless’’ assumption is often appro-priate for very stiff materials where adhesive forces are relatively unimportant, but it is often not thecase for softer materials such as elastomers, where adhesive forces play a very important role. Inthese cases, a ‘‘full-friction’’ boundary condition, where sliding of the two surfaces is not allowed, isoften more appropriate. Hertz model is applicable to nonadhesive cases of the load and displace-ment. Actual values of the load and displacement will generally differ from these values when thecontacting surfaces adhere to one another. There are numerous researches [26–80] dedicated toextend the application of Hertz theory, some of which will be presented in the subsequent section.

3.3.2 MULTIPLE ASPERITY CONTACTS

All engineering surfaces are not smooth. The asperities on the surface of very compliant solids suchas soft rubber could be squashed flat by the contact pressure, and perfect contact could be obtained



through the contact area. It is not usual to flatten rough surfaces by elastic or plastic deformation ofthe asperities.

In general, contact between solid surfaces is discontinuous and the real area of contact is a smallfraction of the nominal contact area as shown in Figure 3.3. The real contact surfaces are actually theasperity contact summation, and are usually much smaller than the apparent surface. Next wediscuss the contact of identical asperities and the random asperities.

3.3.2.1 Contact of Identical Asperities

Although in general all surfaces have roughness, some simplification can be achieved if the contactof a single rough surface with a perfectly smooth surface is considered. Moreover, the problem willbe simplified further by introducing a theoretical model for the rough surface in which the asperitiesare considered as spherical cups so that their elastic deformation characteristics may be defined bythe Hertz theory. It is further assumed that there is no interaction between separate asperities, that is,the displacement due to a load on one asperity does not affect the heights of the neighboringasperities.

Figure 3.4 shows a surface of unit nominal area consisting of an array of identical sphericalasperities all of the same height z with respect to some reference plane xx0. The number of asperity isN. As the smooth surface approaches the rough surface, due to the application of a load, the normalapproach is given by (z � d ), where d is the current separation between the smooth surface and thereference plane.

Clearly, each asperity is deformed equally and carries the same load Pi so that for N asperitiesper unit area the total load P will be equal to NPi. For each asperity, the load Pi and the area ofcontact A are known from the Hertz theory. Thus if R is the asperity radius, then

Pi ¼ 4E*

3R1=2d3=2 (3:12)

N

V

FIGURE 3.3 Contact surfaces.

x �x z

d

Reference plane on rough surface

Smooth surface

FIGURE 3.4 Contact of sphere array against smooth surface.



and Ai¼pRd. The total load is

P ¼ 4E*NA3=2i

3p3=2R(3:13)

The load can be related to the total contact area A¼NAi, by

P ¼ 4E*A3=2

3p1=2N1=2R(3:14)

This result indicates that the real area of contact is related to the two-thirds power of the load, whenthe deformation is elastic.

3.3.2.2 Statistical Analysis of Contact of Nominally Flat Rough Surfaces

It has been discussed that the asperities of real surfaces have different heights characterized by aprobability distribution of their peak heights. Therefore, the simple surface models should bemodified accordingly and the analysis of its contact must now include probability estimation as tothe number of the asperities in contact. If the separation between the smooth surface and thereference plane xx0 is d, there will be a contact at any asperity whose height was originally greaterthan d as shown in Figure 3.5.

The conventional statistical model dealing with contact surfaces with asperities is theGreenwood and Williamson (GW) model, which assumes that the summit of asperities on a surfaceis spherical, the asperity summits on two surfaces have a constant radius, Rp1 and Rp2, respectively,and a composite radius defined below is used to characterize the interface

Rp ¼ 1Rp1

þ 1Rp2

� ��1

(3:15)

The height is assumed to be random and with Gaussian distribution, with standard deviations sp1,sp2, and an equivalent deviation is defined as follows:

sp ¼ (s2p1 þ s2

p2)1=2 (3:16)

For elastic contact in static condition or in dynamic condition without shear stress, the GW modelquantifies the relationship of apparent pressure pa, apparent contact area Aa, mean real pressure pr,real contact area Ar, and number of contact n as a function of separation d. Then the problem reducesto a plate in contact with asperity. If p(z) is the probability density of the asperity peak-heightdistribution, then the probability that a particular asperity has a height between z and zþ dzabove the reference plane will be P(z). Thus, the probability of contact for any asperity ofheight z is

x

z

p(x )x �

Smooth surface

d

FIGURE 3.5 Rough surface.



P(z > d) ¼ð1d

p(z) dz (3:17)

If we consider a unit nominal area of the surface containing asperities, assuming the total asperitynumber is N, the number of contacts n will be given by

n ¼ N

ð1d

p(z) dz (3:18)

Since the normal approach is (z � d) for any asperity, the total area of contact and the expected loadwill be given by

Ar ¼ pNRp

ð1d

(z� d)p(z) dz (3:19)

P ¼ Arpr ¼ Aapa ¼ 43NE*Rp

1=2ð1d

(z� d)3=2p(z) dz (3:20)

It is convenient to use nondimensional variables. According to [4,24,30], the derived relationshipcan be written as

pa

(hRpsp)E*(sp=Rp)1=2

¼ 43F3=2(D) (3:21)

pr

E*(sp=Rp)1=2

¼ 43p

F3=2(D)=F1(D) (3:22)

ArE*(sp=Rp)1=2

paAa¼ 3p

4F1(D)=F3=2(D) (3:23)

(Ar=n)Rpsp ¼ pF1(D)=F0(D) (3:24)

nRpspE*(sp=Rp)1=2

paAa¼ 3F0(D)=4F3=2(D) (3:25)

Using these equations one may evaluate the total real area, load, and number of contact spots for anygiven height distribution. In the above equations, D¼ d=sp is the dimensionless separation,h¼N=Aa is the density of asperity summits per unit area on a surface with smaller density, andFm(D) is a function defined as

Fm(D) ¼ð1D

(s� D)mp*(s) ds (3:26)



where p*(s) is the standardized peak-height probability density function in which the heightdistribution has been scaled to make its standard deviation unit. A commonly used approximationis given by

Ar � 3:2paAa

E*(sp=Rp)1=2

(3:27)

from which the real area of contact is proportional to the load. The relationship between the real areaof contact and the load will be dependent on both the mode of deformation and the distribution ofthe surface profile. More supplicated analysis is based on fractal roughness description andcontinuum plasticity theory. When the asperities deform elastically, depending on the deformationmode within the contact, its real area can be estimated from

Ar ¼ cP

E*

� �n

(3:28)

in which c is constant, 2=3< n< 1; the linearity between the load and the real area of contact occursonly when the distribution approaches an exponential form and this is very often true for manypractical engineering surfaces.

3.3.3 PLASTIC DEFORMATION

When two surfaces of high elastic modulus materials are pressed together, they only come intocontact at the tips of the asperities. The total contact area is Ar and Ar << Aa, where Aa is the nominalarea of contact. For this reason the stress on the asperities is generally large. When the stress exceedsyield pressure, the plastic deformation occurs.

When analyzing the real contact between two engineering surfaces, it is usually assumed thatthey are covered with asperities having random height distribution and deforming elastically orplastically under normal load. The sum of all microcontacts created by individual asperitiesconstitutes the real area of contact which is usually only a tiny fraction of the apparent geometricalarea of contact. There are two groups of properties, namely, deformation properties of the materialsin contact and surface topography characteristics, which define the magnitude of the real contactarea under a given normal load P. Generally, the contact behavior of solid in contact is determinedby a value defined as plasticity index, c,

c ¼ E*

H

!sp

Rp

� �1=2

(3:29)

in which E* is the composite or effective elastic modulus, H is hardness, sp and Rp are thecomposite standard deviation and composite radius of summits (surface heights), respectively.

If the plasticity index c < 0.6, then the deformation is largely elastic. In the case when c >1.0,the predominant deformation mode within the contact zone is plastic deformation. The indexdepends on both the mechanical properties and the surface roughness of the contact surfaces.

The mechanical property ratio E=H and the surface roughness determine the extent of plasticityin the contact region. For metals and ceramics usually E=H > 100, whereas for polymers it is on theorder of 10. Thus the plasticity index for polymer is on the order of one-tenth of that of metals andceramics; therefore, the contact is primarily elastic except for very rough surfaces. On the other



hand, for surface with very small surface roughness, such as the diamond-like carbon-coated surfaceof magnetic disk, the contact is usually elastic.

For plastic contact, we have the following approximation:

Ar ¼ cP

H(3:30)

where c is the proportionality constant. When the asperities deform plastically, the load is linearlyrelated to the real area of contact for any distribution of asperity heights.

The introduction of an additional tangential load produces a phenomenon called junction growthwhich is responsible for a significant increase in the asperity contact areas. The magnitude of thejunction growth of metallic contact can be estimated from the expression

A ¼ Ap0 1þ aFf

P

� �2" #1=2

(3:31)

in which Ap0 is the real area of contact without any shear stress, a is a constant, Ff is the frictionforce, and P is the normal load.

3.4 FRICTION

3.4.1 ADHESION

3.4.1.1 Solid–Solid Adhesion

When bringing two solid surfaces into contact, adhesion or bonding across the interface develops,which leads to the adhesive force that is perpendicular to the surface, in addition to the appliednormal force. Adhesion could occur both in solid–solid contact and liquid involved solid–solidcontact. Two clean solid surfaces tend to create strong bond, whereas contaminated or boundaryfilm covered surfaces tend to yield weak bonds.

The adhesive junctions of asperities on solid–solid contact are caused by interatomic andintermolecular force of attractions. Generally, the adhesion could be chemical or physical. Theformer includes covalent bonds, ionic or electrostatic bond, metallic bonds, and hydrogen bonds;the physical interaction involves the van der Waals bonds.

The adhesion is generally proportional to the normal force. This is because the applied normalforce increases the real contact area that promotes the bonds. On the other hand, the real contact areaalso increases as a result of interatomic attraction. Moreover, an additional shear force usuallyincreases the adhesive force, as the shear effect in addition to the compressive effect tends toincrease the real contact area.

There are many references on the solid–solid friction [26–114]; some of the recent develop-ments incorporate the results of numerical analysis and experimental investigation for the elastic–plastic contact, adhesion, and sliding asperity in modern statistical representation of roughness. Inthe following we describe some of the basic theories.

3.4.1.1.1 Electrostatic ForcesElectrostatic bonding occurs when positively charged cations and negatively charged anions haveinteraction. Electrostatic attractive forces across the interface can arise from a difference in workfunctions or from electrostatic charging of opposed surfaces. Difference in the work function leads tothe formation of an electrical double layer by a net transfer of electrons from one surface to the other.

At small separations the electrostatic pressure between flat surfaces is generally lower than thevan der Waals pressure. It can be assumed that all objects are free of charge before contact.



However, after contacting the objects, contact electrification and triboelectrification occur, andforces due to these charges could occur.

The free energy for electrostatic interaction between two charged atoms or ions with a distanceof x is given by

Es ¼ z1z2e2

4p«0«x(3:32)

The electrostatic force is obtained by differentiating the energy with respect to distance x as

Fs ¼ z1z2e2

4p«0«x2(3:33)

in which e is the charge of a single electron, e¼ 1.63 10�19 C, z1 and z2 are the ionic valences, «0 isthe permittivity of a vacuum, 8.8543 10�12 C2=Nm2, and « is the dielectric constant of the medium,1.00059 in air at 1 atm.

Consider the electrostatic force between two parallel plates with surface charge density of s onone of the plates; the electric field E is given by

E ¼ s

«0«(3:34)

Then the electrostatic force per unit area is given by

Fs ¼ �12««0E

2 (3:35)

If the potential difference between the plates is V, then the electric field is E¼V=x.

3.4.1.1.2 Van der Waals ForcesVan der Waals forces always occur between molecules and are much smaller than the binding forcebetween atoms. The van der Waals forces are effective from comparatively large separation (up to10 nm) down to interatomic spacing (about 0.2 nm). The van der Waals dispersion forces betweentwo bodies are caused by mutual electric interaction of the induced dipoles in the two bodies.Dispersion forces generally dominate over orientation and induction forces except for strongly polarmolecules. van der Waals forces exist for every material in every environmental condition anddepend on the object geometry, material type, and separation distance.

When two atoms are brought close enough, they experience the intermolecular forces interact-ing each other. At the beginning, it is an attraction force, and its strength increases with decreasingdistance until a maximum point is reached; then it decreases with decreasing distance. When thedistance is reduced further, the force becomes repulsive and increasingly stronger. This is reflectedin Figure 3.6, which shows the potential energy between two atoms as a function of their distance.A common expression used to describe this potential is the Lennard–Jones potential, in which theattractive van der Waals potential is modeled as an inverse sixth power term and the repulsivepotential is modeled as an inverse twelfth power term

W ¼ � C

r6þ D

r12(3:36)

in which C¼ 10�77 J m6 and D¼ 10�134 J m12 are two constants for atoms in vacuum.



For interface with a few nanometers spacing, the repulsive potential term can be ignored, and weobtain the purely van der Waals potential

W ¼ � C

r6(3:37)

Equation 3.37 is the potential between two atoms, and it can be integrated over an infinitely long andinfinitely deep half space to obtain the potential between an atom and an infinite plate

W ¼ �pCr16h6

þ pDr145h9

(3:38)

wherer1 is the number density of atoms in the infinite plateh is the distance between the atom and the plate

Equation 3.38 can be further integrated over a volume of material to get the potential between anamount of material and an infinite plate.

To obtain the intermolecular force between an amount of material with surface S and the plate,we need to differentiate the integrated potential in the direction perpendicular to the plate. After that,the intermolecular force between each S of material and the plate can be written as

Fv ¼ dWv

dz¼ A

6p

ð ðS

dx dy

h3� B

45p

ð ðS

dx dy

h9(3:39)

whereA is the Hamaker constantB is the constant related to the repulsive term

The first term on the right-hand side of Equation 3.39 is the attractive van der Waals force, andthe second term is the repulsive intermolecular force. The attractive and repulsive portions of theforce have different acting ranges. The attractive van der Waals force has a much longer actingrange than the repulsive portion.

Based on Equation 3.39, the intermolecular force per unit area can be simplified as

pv ¼ A

6ph3� B

45ph9(3:40)

in which A is the Hamaker constant and B is the repulsive term related constant.

DistancePot

entia

l ene

rgy

FIGURE 3.6 Potential energy of molecules.



The two solid surfaces first experience the attraction force when the distance between them is lessthan about 10 nm. The strength of the attractive force increases with the reduction of the spacing untilha¼ (2B=5A)1=6 reaches maximum attractive value. When the spacing is further reduced, the shortrange repulsive force becomes effective and in the end the intermolecular force is zero ath0¼ (2B=15A)1=6. Below this threshold value the repulsive force will be dominant and other inter-actions like the rearrangement of surface could happen due to the rapid increase of repulsive force.

As an approximation to solids, assume A¼ 10�19 J and B¼ 10�76 J m6, we get ha � 0.3 nm. Anapproximation of the interaction energy, due to van der Waals forces, per unit area between twoparallel plates in the nonretarded regime (h < 20 nm) can be given by the attractive term only

Ev ¼ � A

12ph2

The intermolecular force per unit area between the plates can be approximated by

pv ¼ A

6ph3(3:41)

in which A is the Hamaker constant. For most solids and liquids, the Hamaker constant lies in therange 0.4�43 10�19 J.

3.4.1.1.3 Approximation of Free Surface Energy of AdhesionAn approximation to precise calculation of van der Waals force is to apply the concept of freesurface energy [3,4,7]. Assuming two materials have free surface energies per unit area of g1, g2, g12is the surface energy of the contact interface per unit area, the energy of adhesion per unit area isdefined as

Wa ¼ Dg ¼ g1 þ g2 � g12 (3:42)

in which Dg equals to a reduction in the surface energy of the interface per unit area. It is negativeand it represents the energy that needs to be applied to separate a unit bonded interface. Further-more, it can be approximated as Dg¼C(g1 þ g2), in which C is the compatibility parameter for thetwo materials, and always falls in between 1 and 0.

In most cases, a kind of assumption can be made for microasperities, but an exact solutionwould be more appropriate for nanoasperities. The intermolecular attractive forces depend onatomic spacing and the corresponding surface energies of materials. For super smooth surface, theadhesion can be very large leading to the virtual welding of one surface onto another. For rough orparticles involved interface, its effects can be neglected.

3.4.1.1.4 JKR (Johnson–Kendall–Roberts) ModelFor sufficiently small size contacts, the normal adhesion forces between the surfaces affect thecontact conditions. Various adhesion models, between an elastic sphere and a flat surface, have beenintroduced to extend the Hertz model. The model by Johnson et al. [27] assumes that the attractiveintermolecular surface forces cause elastic deformation beyond that predicted by the Hertz theory,and produce a subsequent increase of the contact area. This model also assumes that the attractiveforces are confined to the contact area and are zero outside the contact area.

When the surface adhesion force is considered, the stress distribution between two surfaces istensile at the edge of the contact area, and remains compressive at the center. JKR model is based onthese considerations.

Consider the energy release rate, w, which has the unit of a surface energy (energy=area), anddescribes the amount of energy that is needed to decrease the contact area, A, by a unit amount



w ¼ @

@A(UE þ UM) (3:43)

whereUE is the elastic energy of the systemUM is the mechanical potential energy associated with the applied load

In terms of the current values of P, R, and a, the contact radius was found to be

a ¼ 3R

4E*Pþ 3pRwþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6pRPwþ (3pRw)2

q� ��1=3"(3:44)

where w is the energy per unit contact area, equal to the thermodynamic work of adhesion. E* is theequivalent elastic modulus of the two spheres. As a result of the surface forces, the contact size islarger than the value in the Hertz model and will be finite for zero external force.

When an external force is applied to pull them apart, the contact displacement will decrease untila critical negative value is reached. When the rate of release of the value is reached the stored elasticenergy and just exceeds the rate of increase of the surface energy from the creation of free surface atthe interface, the surfaces will then separate. The contact force P1 between the surfaces is biggerthan the normal load P:

P1 ¼ Pþ 3pRwþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6pRwPþ (3pRw)2

q(3:45)

The corresponding contact displacement is

d ¼ (3P1)2=3

(4E*)2=3R1=3� 1

2E*4E*

3RP1

!1=3

(P1 � P) (3:46)

which depends nonlinearly on the external force P. The JKR model is based on the analysis atequilibrium conditions and has been verified by static and quasi-static experiments.

Another model of this type assumes that the contact area does not change due to the attractivesurface forces and remains the same as in the Hertz theory; thereby the attractive forces are assumedto act only outside of the contact area. Other models had also been developed to describe continuoustransition between the above two models.

3.4.1.2 Liquid-Mediated Adhesion

Liquid involved interface has extra adhesion in addition to the above-mentioned adhesions acting ondry solids [115–160]. Liquid-mediated adhesion generally consists of meniscus force and viscousforce. Viscous force is rate-dependent and is only significant for high viscous liquid. Meniscus forceis due to surface tension. The presence of the capillary condensation and liquid film couldsignificantly increase the adhesion between surfaces.

Particularly, even for a dry interface, moisture (or other liquid vapor) could condense fromvapor onto the interface as bulky liquid, and exhibit annular-shaped capillary condense around thecontact asperities. The schematic of liquid condensation in interface is shown in Figure 3.7. Thisphenomenon is very common in engineering interfaces, particularly for having exposure in ambientenvironment where water moisture is unavoidable.

A thin liquid layer between two solid plates can work as an adhesive. If the contact anglebetween liquid and solid is less than 90 as shown in Figure 3.8, the pressure inside the liquid will belower than outside and a net attractive force between the plates exists.



Assume u is the contact angle between liquid and solid in air. h is the liquid thickness, and A isthe wet area. F is the external force. The pressure difference Dpla at the liquid–air interface is givenby the Laplace equation

Dpla ¼ gla

r(3:47)

wheregla is the surface tension of the liquid–air interfacer is the radius of curvature of the meniscus (negative if concave)

Consider that the liquid is between the plates and the liquid contacts the solid at the fixed contactangle. From simple geometry it follows that

r ¼ � h

2 cos u(3:48)

In equilibrium, an external force F separating the plates must be applied to counterbalance thecapillary pressure forces

F ¼ �DplaA ¼ 2Agla cos u

h(3:49)

where A is the wetted area. Note that a positive force F corresponds to a negative Laplace pressure.The pressure inside the liquid is lower than outside and the plates are pushed together by pressureforces. This equation can also be used to evaluate the meniscus between a solid sphere contacting aplate with liquid meniscus.

For stiction calculations it is convenient to calculate the surface energy stored at the interfacethat is bridged by a drop of liquid. Consider a drop of liquid placed on a solid, surrounded by air.In equilibrium, the contact angle between liquid and solid is determined by the balance between thesurface tensions of the three interfaces. This balance is expressed by Young’s equation

Liquid condensate

FIGURE 3.7 Liquid condensation in interface.

hθ

A

F

F

FIGURE 3.8 A thin layer liquid working as adhesive between two plates.



gsa ¼ gsl þ gla cos u, 0 < u < p (3:50)

wheregsa is the surface tension of the solid–air interfacegsl is the surface tension of the solid–liquid interface

Young’s equation is also valid for configurations other than the typical one. The contact angle isthe same on a curved or irregular shaped surface.

Consider a general case of solid sphere proximity to a surface with separation of D forming ameniscus. The surface has continuous liquid film h as shown in Figure 3.9 and the meniscus isgiven by

F ¼ �2pRg(1þ cos u) (3:51)

When the surface has no liquid film, or h¼ 0, the meniscus is given by

F ¼ �4pRg cos u=(1þ D=(S� D)) (3:52)

In real interface, stiction can show a large dependence on the relative humidity of air. Frictionmeasurements of silicon and ceramics show a strong dependence of the static friction coefficient onrelative humidity. This is caused by the meniscus force due to capillary condensation. Both thenormal and horizontal components of meniscus force could contribute to the friction. Liquids onsurfaces could spontaneously condense into cracks, pores, and into small gaps surrounding thepoints of contact between the contacting surfaces. At equilibrium the meniscus curvature is equal tothe Kelvin radius

rK ¼ Vgla

RT ln (p=ps)(3:53)

whereV is the molar volume (1.8043 10�5 m3=mol at 208C)p is the vapor pressureps is the saturation vapor pressure

At room temperature, Vgla=RT¼ 0.54 nm for water. The meniscus curvature strongly dependson the relative vapor pressure p=ps. At 100% relative humidity, a water film can grow all over thesurface. The amount of condensed liquid in thermodynamic equilibrium is determined both bythe Kelvin radius and the contact angle. The meniscus curvatures are equal to the Kelvin radius andthe contact angles satisfy Young’s equation.

S

R

r

h

q

f

D

FIGURE 3.9 Schematic of an isolated meniscus in the presence of a liquid film.



Models for the adhesion force due to capillary condensed liquid have been fully developed inhard disk drive tribology. Kinetic meniscus force can be estimated as follows [4]. When a surfaceapproaches another one with the liquid mediation, there is kinetic process leading to the finalequilibrium. During this process, the flow of liquid could increase the meniscus. The drivingpressure can be represented as

p ¼ � gla

r� A

6ph3(3:54)

The first term is due to the Laplace equation, the second term originated from the van der Waalseffect.

The volume of liquid that is flowing in is v, its flow rate is

dv

dt¼ � 2prh3

3hl

@

@r

gla

rþ A

6ph3

� �(3:55)

The radius of projected region of meniscus, neck radius x(t), can be derived as a function of timewith respect to its equilibrium value (x)eq; finally, the meniscus force can be given by

F(t) ¼ 2pRg(1þ cos u)x(t)

(x)eq

� �2(3:56)

A drop of lubricant filling in the gap between the two surfaces will form a toe-dipping regime andthus initiate the stiction. A typical example is the meniscus developed in the slider–disk interface ofhard disk drive. With the increase of the contact time, in addition to the solid–solid asperitymicrodisplacement, the microflow and the diffusion of the lubricant will redistribute the interfacelubricant to form an equilibrium state, and therefore form pillbox regime accompanying themicrodescent process. This is schematically shown in Figure 3.10. Many efforts have been reportedto quantify the meniscus force.

Another meniscus model is given in the following. As shown in Figure 3.9, the hemisphericalsurface of an asperity with radius R on disk is in near contact with (distance D) the flat surface of theslider. The lubricant buildup is with a transient thickness h on the slider flat surface.

The meniscus force can be derived as

Fm ¼ 2pR2g(1þ cos u)�d�

dD(3:57)

Slider SliderLube buildup Lube buildup

Lube LubeDisk Disk

(b)(a)

FIGURE 3.10 Schematic of rough asperities on a disk in contact with slider surfaces. (a) Short-term: toe-dipping regime and (b) long-term: pillbox regime.



whereg is the surface tension of the lubricantu is the contact angle formed by the lubricantD is the distance between the flat plane and the hemispherical surface of the asperity

In addition to the meniscus force, the viscous force also contributes to the adhesive force by thefollowing relationship:

Fv ¼ bhl

ts(3:58)

in which b is a proportional constant, hl is the dynamic viscosity of the liquid, and ts is the time toseparate the two surfaces. Obviously, ts is inversely proportional to the breakaway acceleration andvelocity of the interface.

3.4.2 DRY FRICTION

3.4.2.1 Friction Mechanisms

When there is contact sliding between two bodies under a normal load, P, a resistant force alwaysexists—this force is called frictional force, F. The frictional force acting on a component always actsin a direction opposite to the motion of the component. Two basic facts about friction have beenexperimentally established: (i) the frictional force is a function of the normal load on the contact,F¼mP, where m is the coefficient of friction (cof) and P is the normal load; (ii) the frictional force isindependent of a nominal area of contact. These two statements constitute what is known as the lawsof sliding friction under dry conditions.

Studies of sliding friction have a long history. But there is still no simple model that could be usedby an engineer to calculate the frictional force for a given pair of materials in contact. It is now widelyaccepted that friction results from complex interactions between contacting bodies. This includes theadhesion at the points of contact due to the molecular interaction, the effects of surface asperitydeformation, as well as the plastic deformation of a weaker material. Figure 3.11 shows the schematicof normal contact and slope contact of asperities, the former tends to develop adhesions, the latter candevelop both adhesion and asperity deformation in a horizontal direction. A number of factors, suchas the mechanical and physical–chemical properties of the materials in contact, surface topography,and environment conditions, determine the dominant components in the friction process. At afundamental level there are three major phenomena that determine the friction of dry solids:

(i) The real area of contact(ii) Shear strength of the adhesive junctions formed at the points of real contact(iii) The way in which these junctions break away during relative motion

FIGURE 3.11 Normal contact and the slope contact of asperities.



The friction properties of a given material are not its intrinsic properties, they depend on manyfactors related to a specific application. Quantitative values for friction in the form of frictioncoefficient depend on the following basic groups of parameters:

(i) The components, i.e., the components of interface and their relevant physical andgeometrical properties

(ii) The operational variables, i.e., load (pressure), speed, and time(iii) The environmental variables, i.e., temperature and humidity(iv) The system, the mutual interaction of the system’s components, and their time-dependent

variables (note that the interface is a time-dependent variable due to wear and layer forming)

Moreover, coefficient of friction is also complicated by other factors, like the scales. For instance,for a given interface, coefficient of friction could be inversely proportional to the normal load;however, when the load is down from micro- to nano-Newton level, the coefficient of friction couldhave a different feature.

Friction is always associated with energy dissipation and conversion. Mechanical energy istransformed within the real area of contact, mainly through elastic deformation and hysteresis,plastic deformation, plowing, and adhesion.

Dissipation of mechanical energy takes place mainly through thermal dissipation (heat), storagewithin the bulk of the body by the generation of defects, cracks, and emission like acoustic andthermal generation.

For typical contact interface, the friction is given by

F ¼ A½ata þ (1� a)tl� (3:59)

in which ta, tl are, respectively, the average shear stress of the dry contact and the liquid film. a isthe fraction of dry area. The average shear stress of the liquid film is contributed by both viscosity ofthe liquid and meniscus effect

tl ¼ hlV

hþ tm (3:60)

in which hl is the dynamic viscosity of the liquid, V is the relative sliding velocity, h is the liquid filmthickness, and tm is the shear stress due to meniscus effect. In mixed lubrication conditions or theinterfaces exposed to humid environments, the meniscus and even viscous effect could be significant.

3.4.2.1.1 Dry Friction due to AdhesionWe have shown that the area of real contact between two bodies is usually a very small fraction ofthe apparent geometrical area of contact. However, the friction force is determined almost entirelyby the area of real contact.

Consider two dry surfaces where the upper body moves parallel to the lower body with thevelocity v. The friction force is given by

F ¼ taAr (3:61)

whereAr is the real contact areata is the shear stress

One of the most important components of friction originates from the formation and breakawayof interfacial adhesive bonds. Extensive theoretical and experimental studies have explained thenature of adhesive interaction. The main emphasis was on the electronic structure of the bodies in



frictional contact. From a theoretical point of view, attractive forces within the contact zone includeall those forces which contribute to the cohesive strength of a solid, such as the metallic, covalent,and ionic short-range forces as well as the secondary van der Waals bonds which are classified aslong-range forces. The interfacial adhesion is as natural as the cohesion which determines the bulkstrength of materials.

The coefficient of friction due to adhesion can be approximated as the ratio of the interfacialshear strength of the adhesive junctions to the yield strength of the asperity material

ma ¼F

P¼ Arta

P� ta

pr(3:62)

in which pr is the mean real pressure. Substitute Ar from Equation 3.27 and we obtain

ma �3:2ta

E*(sp=Rp)1=2

(3:63)

For single asperity case, it can be approximated as

ma /1

P1=3(3:64)

For plastic contact

ma �taH

(3:65)

In the elastic contact case such as a diamond-like carbon coated disk against ceramic slider, ma

decreases with an increase in roughness. In the plastic contact case, ma could be independent withroughness in a moderate range. It can be seen that ma tends to be high at very smooth surfaces dueto the growth of real area of contact, and also tends to be high at very rough surfaces due tointerlocking.

Evaluation of ma needs to quantify ta, which can be estimated by a limit analysis. Generally, theshear strength ta cannot substantially exceed the bulk shear strength for plastic contacts.

For most engineering materials this ratio ta=pr is of the order of 0.2 and therefore the dry frictioncoefficient could be of the same order of magnitude. In the case of clean metals, where the junctiongrowth is most likely to take place, the adhesion component of friction may increase to about10–100 times.

The presence of any type of surface oxide layer, lubricant, or water film preventing theformation of the adhesive junction can dramatically reduce the magnitude of the adhesion compon-ent of friction. The real area of contact could be much larger than that from the deformation due toapplied load; this is because of the work of adhesion. Assume the attack angle or the slope of theasperity is u; the modification of Equation 3.62 with adhesion work can be derived by lettingthe work done by the normal force be equal to the work done in deformation and the change in thesurface energy. Using this equilibrium, the above simple model can be supplemented by the surfaceenergy of the contacting components; the friction coefficient is given by

ma ¼tapr

1� 2w tan u

pr

� ��1

(3:66)

where w is the surface energy.



Next we discuss the plastic effect. Suppose the material has a yield stress sy. If the initial area ofcontact is small, the asperities in contact will yield and deform plastically in compression. The areaof contact will grow as these asperities are squashed and others come into contact. Eventually thenormal force P¼Arsy and the true area of contact will be

Ar ¼ P=sy (3:67)

When the sliding asperities pass each other, the frictional force, Fs, relates to average shear stresst¼Fs=Ar acting on the asperities. If the contact at the asperities is intimate and the asperities havedeformed extensively, the asperities may have to shear plastically in order to slide. In this case,t¼ ta, in which ta is the yield stress in shear. Since the yield stress in tension and compression issy¼ 2ta in terms of Tresca criterion, the ta ¼ sy=2, therefore

Fs ¼ Arsy=2

Consider Ar¼P=sy, we have Fs¼ 0.5 P. The approximation of the friction law for asperitydeformation yields a friction coefficient of 0.5. This is close to the value found for dry solidsincluding metals, ceramics, and glasses. More typically, solids like metals oxidize to some extent inair and form a thin oxide layer on their surfaces. The interface between two oxide surfaces may slideat stresses less than ta of the metal, leading therefore to ma < 0.5.

Note that the formation of a strong junction bond needs acting time for asperity creepdeformation. Once sliding occurs, the alternative asperity attach=detach and the oscillations innormal direction are insufficient to achieve full atom-to-atom bonding over the entire contactarea; therefore, the coefficient of friction could decrease in the sliding process. This is one of thereasons that kinetic friction is smaller than dynamic friction.

Actual values of the coefficient of friction may range from as high as 5 to as low as 0.05,depending on the materials brought into contact and the condition of their surfaces. For pure metalsin contact, the local contact areas, ‘‘junctions,’’ between asperities grow as sliding occurs due toextensive plastic deformation around the asperities. This growth in junction area leads to extensivebonding and a coefficient of friction >5. This phenomenon is part of the basis of ‘‘friction welding’’of metals and plastics. If surfaces are rapidly rubbed against each other, the heat generated can meltthe low melting materials such as the resin in bowed string.

For metals with clean surfaces, the interfacial shear force can be approximated by; ts¼ c0þc1p þ c2p

2, where c0, c1, c2 are material constants, and p is normal pressure [109].The above discussion applies to sliding contacts between the metals, where Ar is small but sy

and ta are large so that the coefficient of friction is large. For other kind of sliding pairs, the situationdepends on the real contact area Ar, sy, and ta.

For ceramics on ceramic, e.g., slider–disk interface in hard disk, the hardness and sy are high.The contact area and the tendency to bond across the contact are small; and ta is small and thus thecoefficient of friction is also small. Rigid polymer films on harder materials like Teflon (PTFE) onmetals yield small coefficient of friction; this is because the polymer chains are oriented along thedirection of sliding and only weak van der Waals bonds between chains need to be broken.

In addition to the adhesion, elastic and plastic deformation related friction; the fracture processalso helps develop the friction. Consider the mode of failure due to crack propagation; the frictiondue to the fracture of an adhesion junction was derived

F ¼ cs12dc

n2(PH)1=2(3:68)

in which c is the coefficient, s12 is the interfacial tensile strength, P is the normal load, dc is thecritical crack opening displacement, n is the work-hardening factor, and H is the hardness.



3.4.2.1.2 Dry Friction due to Deformation (Slope)Apart from the microscope interface interaction of smooth surface where asperity normal contactdominated, the micro- and macroscopic interaction of asperity of rough surfaces could occur andconsequently asperity deformation and plowing of soft material by hard material usually occur.

In this sliding process, the mechanical energy is dissipated through the deformations ofcontacting bodies. The slip-line (lines of maximum shear stress) model of continuum mechanicscould be used to analyze the deformation of the single surface asperity. Figure 3.12 shows a slip-linefield–based deformation model of friction based on Prandtl two-dimensional stress analysis. Thematerial in the region ABCDE flows downward and outward as the hard material moves forward.Three distinct regions of plastically deformed material may develop in these regimes. The flow shearstress of the material defines the maximum shear stress which can be developed in these regions[5,15]. The angle c can be chosen so that the velocities of elements of material on the free surface,contact surface, and boundary of the rigid plastic material are consistent. The coefficient of frictionis given by

md ¼ l tan arcsin

ffiffiffi2

p(2þ c)

4(1þ c)

� �� (3:69)

where l is the portion of plastically supported load and is a function of elastic modulus andhardness.

The proportion of load supported by the plastically deformed regions is a complicated functionof the ratio of the hardness to the elastic modulus. For completely plastic asperity contact and anasperity slope of 458, the coefficient of friction is 1.0. It decreases to 0.55 for an asperity slopeapproaching 0.

Another approach to this problem is to assume that the frictional work performed is equal to thework of the plastic deformation during steady-state sliding. This energy-based plastic deformationmodel of friction gives the following expression:

md ¼Ar

Ptm 1� 2

ln 1þ tstm

� �� ts

tm

� �

ln 1� tstm

� �2" #

8>>>><>>>>:

9>>>>=>>>>;

(3:70)

whereAr is the real area of contacttm denotes the ultimate shear strength of a materialts is the average interfacial shear strength

PF B

A

C

D

Ey

FIGURE 3.12 Slip-line theory.



The problem of relating friction to surface topography in most cases reduces to the determin-ation of the real area of contact and studying the shear stress. If one surface is harder than the other,the asperities of the hard one could penetrate into the soft one. This leads to groove if the shearstrength is exceeded. Plowing not only increases friction, it also creates wear particle which changesfriction as well. To maintain the plowing motion, a force is required and it may constitute a majorcomponent of the overall frictional force. The schematic of a plowing of soft material by the hardconical asperity is shown in Figure 3.13a. Assume the effective slope is Q; the formula forestimating the coefficient of friction in this case is as follows:

mp ¼2p

tanQ (3:71)

For most engineering materials, the slope angle of asperity (roughness angle) is very small (58–108),and the plowing component of friction is correspondingly very small. A typical case is that effectiveslope is less than 108, and the coefficient mp is about 0.05 and less. In elastic contact, mp is oftenassumed to be negligibly small.

The frictional force produced by plowing is very sensitive to the ratio of the radius of curvatureof the asperity to the depth of penetration. Consider a spherical asperity of radius R in contact with asofter body as shown in Figure 3.13b; r is the size of the plowing area that is proportional to thedepth of penetration. The coefficient of friction is given by

mp ¼2p

R

r

� �2

sin�1 r

R

�� R

r

� �2

�1

" #1=28<:

9=; (3:72)

The coefficient of friction increases rapidly with an increase of r=R and as the sphere digs deeper.Moreover, plowing brittle materials is associated with the microcracking process; a model

developed in terms of fracture mechanics to quantify the effects of fracture toughness, elasticmodulus, and hardness on the coefficient of friction is given as

m ¼ cK2k

E(HP)1=2(3:73)

in which Kk is the fracture toughness, E is the elastic modulus, and H is the hardness.For rough hard interface or hard interface involved particles, the plowing could be the dominant

factor for energy dissipation. Plastic materials could undergo plastic deformation. Viscoelasticmaterial could exhibit hysteresis process. Figure 3.14 shows the schematic of the plowing of hard

P

F

Θ

R

r

(a) (b)

FIGURE 3.13 Plowing of (a) hard conical and (b) sphere asperity against soft elastic substrates.



sphere asperity against viscoelastic polymer substrate. During the sliding, the softer material ispressed and the stress is released as contact moves on. Each time the contact portion is stressed andenergy is stored, when progressing to the next contact portion, most of the energy is released as thestress is removed. Only a small portion is dissipated as heat in hysteresis losses.

A model for hard asperity sliding on a polymer without adhesional effects is given by

md ¼ khpaE0 tan d (3:74)

in which kh is a constant associated with asperity geometry, pa is the apparent pressure, and E0 andtand are determined from complex modulus.

3.4.2.2 Energy Dissipation during Friction

In engineering interfaces, several friction mechanisms could coexist and interact with each other in acomplicated way. In general, frictional work is dissipated at two different locations within thecontact zone.

The first location is the interfacial region characterized by high rates of energy dissipation andusually associated with an adhesion mechanism of friction. The other one involves the bulk of thebody and the larger volume of the material subjected to deformations where the rates of energydissipation are much lower. Energy dissipation during plowing and asperity deformations takesplace in this second location. The above distinction of two locations is artificial and only for thepurpose of simplification of a very complex problem. Actually, in describing KJL modeling weillustrate the basic energy balance equation which indicates the interaction aspect from a contactpoint of view. The various processes can be briefly characterized by one or several items as follows:

(i) Plastic deformations and microcutting(ii) Viscoelastic deformations and fatigue cracking and tearing, and subsequently excessive

heating and damage(iii) Excessive heating and chemical degradation like polymers(iv) Interfacial shear creating transferred films(v) The propagation of Schallamach waves (elastomers)

P

V

Release Press

Hard Asperity

Shearing

P

V

Release Press

Polymer

Hard asperity

Shearing

Deformation and hysteresis

FIGURE 3.14 Plowing of hard sphere asperity against viscoelastic polymer substrate.



During sliding contact, part of the kinetic energy produces waves and oscillations in the bodies, andpart of it leads to plastic and elastic deformation of asperity tips. Some energy expends throughviscous dissipation, and the balance through adhesion, fracture, chemical reactions, and photoemis-sion. Distribution of energy conversion through this process varies for different applications. Eachof these processes provides a mechanism for converting the original kinetic energy to an interim onein the form of vibration and sound, deformation energy, surface energy, tribochemical energy,and other triboemissions. In the end, part of the initial energy remains stored as potential energy, andpart of it converts to thermal energy, eventually dissipating to the surroundings. Thus, friction canbe viewed as a combination of processes that transforms ordered kinetic energy into either potentialenergy or a disorderedly thermalized state of kinetic energy. It then follows that the friction forcecan be considered as a combination of forces that resist motion during each of these energyconversion processes

Ff ¼ Fadhesion þ Felastic deformation þ Fplastic deformation þ Fviscosity þ Ffracture þ Frandom

The schematic depiction illustrates the contributions to friction at different scales. To characterizethe above process affecting friction, different models from macro- to micro- or even smaller scaleare required, as the various events take place in each scale.

The integration of the contribution from these scales requires expertise from different disciplinesof science and engineering. Usually, a continuum scale is suitable for modeling engineeringproblems. At the atomic scale, the primary problems relate to the dissipation and oscillation ofatoms and will be briefed in a later section.

Many of the current efforts to model friction start at the continuum scale, relating surfaceroughness to friction. The obvious mechanisms that contribute to friction in such models includeelastic=plastic contacts, viscous dissipation, fracture, and adhesion. Each of the processes developsat each true contact region between the surfaces and the true contacts take place between asperitieson the surfaces.

3.4.2.3 Rubber Friction

The above observation is quite different when we consider elastomer (rubber), which is a fullyviscoelastic material. Consider elastomer sliding on rigid surface with adhesion. The combined effectof adhesion and internal damping contributes to friction. During sliding, the adhesion develops in thecontact point; strain also develops in the material leading to the energy to be stored elastically. Whenthe elastic stress exceeds the adhesive force, the breakage of the contact bond takes place and thecontact junction relaxes. The adhesion takes place at a new point, and so on. For rubber sliding onclean glass, Schallamach reported that the waves of detachment have been observed, which traversethe contact area from front to rear, or from compression side to tension side, at a very high speed. Theadhesion appears to be complete between these waves, which move folds in the rubber surface,probably produced by buckling attributed to tangential compressive stresses. The excitation for thewaves of detachment is the tangential stress gradient. There is continuous and alternative deattachand reattach process passing through the contact zone. Friction work is associated with the energylost during this continuous process. Elastomers have the capacity for sustaining very large elasticstrains. Unless surfaces are very rough, elastomers can deform elastically to establish an actual areaof contact that is equal to the nominal area of contact A. As such, many of the above formulae offriction no longer hold for elastomer, for which friction increases with the nominal area of contact A.The area of contact for elastomer is usually large and shear strength ta is in the moderate range;therefore the coefficient of friction is high for elastomer contacting with metals or ceramics.Elastomers can perform well where high friction is needed, like rubber tires, drive belts, and caliperbrake pads.



Rubber exhibits unusual sliding friction. When rubber is slid on a hard, rough substrate, thesurface asperities of the substrate exert oscillating forces on the rubber surface leading to energy‘‘dissipation’’ via the internal friction of the rubber. Because of its low elastic modulus and the highinternal friction, the adhesion of the rubber to the substrate is very important. It has been shown thatthe adhesion force will deform the rubber at the rubber–substrate interface, in such a manner that, atlow sliding velocities, the rubber completely follows the short-wavelength surface roughnessprofile. This gives an additional contribution to the friction force.

The rubber friction, in many cases, is directly related to the internal friction of the rubber. Theexperiments with rubber surfaces sliding on a rigid surface like silicon or glass have shown that thefriction coefficient has the same temperature dependence as that of the complex elastic modulus ofrubber. This suggests that the friction force is likely to be affected by bulk property of the rubber.

The friction force between rubber and a rough hard surface has two contributions commonlydescribed as the adhesion and hysteric components, respectively. The hysteric component resultsfrom the internal friction of the elastomer. During sliding, the asperities of the rough substrate exertalternative compressive forces and release transient on the rubber surface, which leads to cyclicdeformations of the rubber and energy dissipation through the internal damping of the rubber. Theadhesion component is important for very clean elastomer surfaces.

Next, we discuss the hysteric contribution associated with surface waviness of the substrate, andthen we consider the role of adhesion between the rubber and the substrate for which the interactionwill deform the rubber at the surface so that it will follow the roughness profile of the substrate.Finally, we discuss some other aspects of rubber friction and of the friction of other polymers slidingon hard substrates.

The results presented in the previous sections assume that the contacting materials have well-defined elastic or plastic constants. Elastomer material has viscoelastic character, and it is importantto understand how these effects should be taken into account. Viscoelastic effects can be taken intoanalysis to estimate the contribution from the waviness of the substrate to the friction force whenrubber slides on a rough hard substrate. The overall elastic response of the system can be describedby the effective elastic constant with the modulus being time dependent. On the other hand, in thecase where adhesion is present, the stress near the crack tip could be defined by stress intensityfactors that are time dependent. This type of viscoelastic behavior could be referred to as large-scaleviscoelasticity.

The straightforward approach is to estimate the energy dissipation in a viscoelastic material,then to use it to estimate the contribution from the internal friction to the sliding friction. Thefollowing analysis is given [8,110,111]. Consider rubber sliding on a substrate; it results influctuating stresses that is characterized by the frequency v0 � v=l, where v is the sliding velocityand l is a length of order of the diameter of the contact area between a substrate asperity and therubber surface. Based on the energy associated with this process, consider the main part of theenergy dissipation occurring in a volume element; the friction coefficient is derived as

mi � �CIm E(v0)

E(v0)j j (3:75)

whereE(v0) is the complex compressive modulus of elastomerIm(�) denotes the imaginary partC is a number of order unity which depends on the nature of the surface roughnessv0 is the frequency of the fluctuating forces due to the substrate asperities exerting on the rubber

For elastically hard materials as discussed previously, adhesion usually does not manifest itselfon a macroscopic scale. The situation is different for rubber cases where even a weak adhesive



junction between the surfaces could be elongated before breaking by a distance which is larger thanthe height of the surface asperities. During the stage of disattach, a large fraction of the junctionswill be also elongated and exert a force on the block in the direction toward the substrate. Moreover,for an elastically soft solid, the area of real contact is generally much larger than for an elasticallyhard solid under similar conditions.

The adhesive interaction between rubber and substrate will induce additional viscoelasticdeformations of the rubber in such a way that the rubber will completely follow the roughnessprofile of the substrate. Consider the area l2 and the case to allow the rubber to fill out the cavity.Assume that the surface roughness is characterized by the height h and the wavelength l. Considerthe elastic deformations of the rubber in such a manner that the normal force that acts on the surfaceof rubber is approximated as

Pn � 2 Gj jlh (3:76)

in which the G is complex shear modulus. During sliding, the frictional stress is given by

F � �A(h=l)2 ImG(v0) (3:77)

Then the coefficient of friction is given by

m ¼ ch ImG(v0)

Gj jl3 (3:78)

in which c is a constant. Another model of adhesive friction of elastomers is given as follows:

m ¼ a(A=P)t tan d (3:79)

in which a is a constant, A is the contact area, P is the load, t is the shear stress, and d is the tangentmodulus. A similar expression is given by

m ¼ a(1=P)(G0)1=3 tan d (3:80)

where G0 is the dynamic shear modulus. For polymer, m¼ (t0=p0)þa generally holds, whichindicates that the coefficient of friction decreases with the increase of load, but under low load,the trend could be reversed.

Because of its viscoelastic nature, rubber at low sliding velocities tends to flow or creep over asurface just as the viscous liquid flows smoothly around objects. Friction in this creep region tendsto increase with speed. After velocity increases into the sliding region, the mean friction level oftendecreases as the rubber is no longer able to maintain full contact with the counterface that isobserved by the appearance of transverse waves of detachment.

3.4.2.4 Friction Transitions

We discussed above the interface with clean surfaces. For the real engineering surfaces, they are allunlikely to be clean. The oxide films could form on the surface of metal and other engineeringmaterials, yielding a boundary layer. Its thickness could vary from a few atomic thicknesses to afraction of a micron. The boundary layers usually have lower shear strength than the original solid.For many engineering surfaces, the moisture or other chemical vapor could adsorb on the surface toform an adsorbed layer.

The transfer film forms more readily on a roughened surface and can exist in solid state and in alow viscosity or fluid state. A polymeric film is transferred and remains firmly attached to the metalcounterface. Polymers are susceptible to friction transfer when rubbing both against metalsand polymers. It has been observed that polymer is transferred onto metal part surface in theform of flakes of small size. The thickness of the transferred layer increases monotonically up to



a thickness larger than 0.1 mm and then oscillates about a mean value. The amplitude of oscillationsdepends on the test conditions, especially on load and sliding velocity. The transferred polymerfragment may exhibit a wide variety of forms depending on polymer properties and operationconditions.

Friction is always associated with wear. They always coexist and both are the result of one solidrubbing against another and interacting with each other. Wear is the effect of friction on materialsurfaces that rub together. Wear occurs as a result of interaction between two contacting surfaces,and it can be classified as a type of fatigue, adhesion, chemical wear, corrosion, and abrasion. Wearis usually associated with the loss of material from contacting bodies in relative motion. Theformation and removal of interface films due to wear could lead to substantial change and variationin friction. Like friction, wear is controlled by the properties of the material, the environmental andoperating conditions, and the geometry of the contacting bodies. Adhesive wear is invariablyassociated with the formation of adhesive junctions at the interface. Archard’s classical wearmodel states that the rate of wear, _V , is proportional to the pressure, p, which is given by

_V ¼ kpL

3H(3:81)

wherek is the wear coefficientL is the sliding distanceH is the hardness of the softer material in contact

Two materials exhibiting the same friction coefficient can exhibit quite different wear ratesbecause the energy is partitioned differently between and within the materials.

The typical engineering surfaces are always covered by adsorbed film that consists of water,oxygen, or even oil. For metal surfaces, there is usually an oxide layer. Sliding surfaces are alwaysprogressive during the sliding process due to the friction and wear process. The changes in thesurface also impact the friction. For interface, after a certain period of sliding, the friction could leveloff to a steady state. It is called the ‘‘break-in’’ or ‘‘run-in’’ period. After the break-in, the coefficientof friction could increase to a higher plateau or a lower plateau, exhibiting an S- or Z-shaped curve.The break-in process is associated with the peel off of high asperities, surface polish approaching tomore matchable, original film breakage and wear off, and new stable film forming. For somematerial under some limit operation, there could be onset of severe wear, and this could give riseto a salient rise in friction.

Another associated problem is the temperature. When two surfaces slide over each other, theheat Q produced is proportional to the sliding velocity and friction force

Q ¼ðAr

Vt dA

in which Ar is the actual area of contact, t is the shear stress, and V is the velocity. Q depends on thearea of contact Ar, the shear stress, and the velocity, or indirectly depends on normal load andcoefficient of friction. The temperature rise of surface depends on the thermomechanical propertiesof the two contacting bodies as well as Q.

The surface temperature generated in contact areas has a major influence on wear, materialproperties, and material degradation. The friction process converts mechanical energy primarily intothermal energy which results in a temperature rise.

The friction intensity may not be sufficiently large to cause a substantial temperature rise on thebody, but it could be sufficiently large to cause a substantial temperature rise on the surface. The



flash temperature is defined as the temperature rise in the contact area above the bulk temperature ofthe solid as a result of friction energy dissipation. The surface temperature rise can influence localsurface geometry through thermal expansion, causing high spots on the surface which concentrateon the load and lead to severe local wear. The temperature level can lead to physical and chemicalchanges in the surface layers. These changes can lead to substantial transitions in friction mechan-isms and wear phenomena. Consider a simple formulation for the mean flash temperature in acircular area of real contact of diameter 2a. The friction energy is assumed to be uniformlydistributed over the contact region. For stationary heat source, consider one of the sliding compon-ents, component 1; the mean temperature increase above the bulk solid temperature is

DT ¼ Q

4ak1(3:82)

whereQ is the rate of frictional heat supplied to component 1 (Nm=s)k1 is the thermal conductivity of body 1 (W=m 8C)a is the radius of the circular contact area (m)

3.4.2.5 Static Friction, Hysteresis, Time, and Displacement Dependence

Static friction: For contact surfaces, the break force required to initiate motion could be lower,higher, or even equal to the force needed to maintain the surface sliding in the subsequent relativemotion. Therefore, the coefficient of static friction could be smaller, higher, or equal to thecoefficient of kinetic friction. The higher static friction usually appears for clean surface. Forthese cases, the static friction is usually a function of dwell time. The static friction increase withtime for dry interface is considered due to the interface bonding from the interaction of the atoms onthe mating surface and the plastic flow and creep of interface asperities under load. The time-dependent static friction model can usually be represented as

m(t) ¼ m1�(m1�m0) exp(�ats) (3:83)

wherem0 is the initial value of the coefficient of frictionm1 is the limit value of static friction at long-term timets is rest timea is a constant

Other models based on power law have also been developed [86].Hysteresis: During acceleration and deceleration of sliding motion, the friction vs. velocity curvesare usually not identical. There is usually some delay to form hysteresis loop as shown in Figure3.15. The friction exhibits the memory feature. The state variable models have been developed toquantify this type of characteristic, which has critical applications as in rock mechanics [87,88]. Insolid mechanics, the constitutive relations have been used effectively at capturing steady-state andcertain transient effects in a wide variety of materials with interfaces having micron scale roughness.The approach involves expressing the friction in terms of the instantaneous slip speed at theinterface and one or more state variables, for which phenomenological evolution equations arealso introduced.

One of the underlying assumptions is that the interfacial area is large enough to be self-averaging, so that a mean-field-like state variable is sufficient to capture the collective dependenceof the microscopic degrees of freedom on the dynamical variables, including time, displacement,and slip speed, which characterize the motion.



For instance, the earthquake has been considered to be a fault line stick–slip event. Recently therelevant models have been extended to other materials including lubricated metal, Teflon on metal,glass, and plastics. The state variable models contain a dependence on displacement theory. It notonly adopts a steady-state dependence on velocity that incorporates Stribeck curve which will beillustrated in the next section, but it also assumes an instantaneous dependence on velocity whichsuggests that instant velocity change gives instantaneous change in friction. Moreover, it assumes anevolutionary dependence on characteristic sliding distances, which indicates that following a suddenchange in velocity, the steady-state curve is approached through an exponential decay overcharacteristic sliding distances.

Various phenomenological models developed provide such a description, where the coeffi-cients of one or two dynamical equations are fitted to experiment variables and then used todescribe a wide range of observed frictional behaviors, such as the dilation of a liquid under shearand the transition between stick–slip (regular or chaotic) and smooth sliding friction. However,most ‘‘state variables’’ in models are unable to be quantitatively related to physical systemproperties.

In many applications, the displacement dependence of static friction needs to be characterized.The static friction is actually a constraint and it is associated with an elastic or plastic deformationunder traction. The displacement before overcoming static friction can be modeled with an equiva-lent spring, k, which is a function of asperity, normal force, material elasticity, and surface energy.The presliding displacement has long been studied in many engineering problems, like the microslipin magnetic recording tape and the elastic creep in power transmission rubber belts.

To elaborate this, let us consider the predisplacement of a belt wrapping on a pulley with tensionbefore slip motion. The coefficient of friction between belt and pulley is assumed to be m. Thefriction causes a change in belt tension thereby causing the elastic belt to extend and contract overthe contact section on pulley. The resulting relative motion s is a microslip called elastic creep.A similar event has been experimentally observed for belt–roller interface in magnetic taperecording system. The presliding displacement results in wear and the wear debris builds up onthe roller which affects the real coefficient of friction [82].

The microslip is the elongation of belt due to tension before belt–roller relative rigid slippage.s¼ (Thigh � Tlow)=k, in which Thigh and Tlow are, respectively, the tight and slack side tension, andk is the belt longitudinal stiffness. The contact zone is composed of traction zone (in which thetension is constant) and microslip zone. us and ut are, respectively, the slip and traction zone.The microslip occurs only in slip zone, and Euler formula can be used to approximate therelationship

Thigh ¼ Tlowem�s (3:84)

0

0.04

0.08

0.12

0.16

0.2

V (m/s)

0 0.04 0.08 0.12 0.16 0.2

CO

F

FIGURE 3.15 Hysteresis of friction–velocity curve.



where m is coefficient of friction. The schematic representation of the belt–roller interface is shownin Figure 3.16, in which the elastic belt undergoes microslip during the transition of tension in thewrap section from Tlow to Thigh.

However, this theory may not be applicable to inflexible or inextensible industrial belts. Theshear strain in belt window between the reinforce cord and the pulley surface is a large factor in beltmechanics’ behavior. For this consideration, assume the two sections composed of slip section usand adhesion ua where tractive force builds up linearly along the adhesion section. The coefficient offriction is considered to be kinetic mk and static ms, applied on slip section and traction section,respectively. For this case, it was derived as

Thigh ¼ Tlowemk�s=(1� �ams=2) (3:85)

This is based on shear theory.

3.4.2.6 Effect of Environmental and Operational Condition

3.4.2.6.1 Temperature DependenceApart from the temperature rise due to interface friction generated heat, environmental temperaturechange also has a direct impact on friction. For elastomers which have a close relationship betweentheir dynamic mechanical properties and their coefficient of friction, the temperature and velocityvariation of friction should follow a time–temperature superposition principle. The elastomer’smodulus is strongly dependent on the temperature, which can be modeled as E¼E0 exp(�bT).In many cases, the maximum coefficient of friction correlates with the Tg of elastomer. Otherfriction peaks may not correlate with glass transition but are due to the changes in molecularadhesion. For instance, the coefficient of friction of tire–road interface could increase with thedecrease of temperature till around �188C. For the environments below 08C, the frictions couldstrongly be related to the pressure sliding melt of ice film in interface and the meniscus effects.

Humidity dependence: In a humid environment, the amount of condensate water present at theinterface increases with the increase of their relative humidity. The adsorbed water film thickness ondiamond-like carbon-coated magnetic disk can be approximated as follows:

h ¼ h1(RH)þ h2 ea(RH�1) (3:86)

RH is the relative humidity ranging from 0 to 1. h1, h2, a are constants. For instance, some existingexperiments show that the coefficient of friction of lubricated disk increases rapidly when relativehumidity is above a critical value of 60%; this critical humidity is dependent upon the interfaceroughness. Above the critical limit, the static friction increase could be up to five times whereas thekinetic friction could only increase slightly.

qs

qt

Thigh

Tlow

Traction zone

Microslip zone

FIGURE 3.16 Microslip of magnetic recording tape belt.



3.4.2.6.2 Load and Velocity EffectThe oxide films are likely produced on the surface of metals and alloys. This usually gives rise tolower friction than clean surface under low load. When the load increases, plowing could occur, thefilm could get penetrated, and higher friction could be given.

When the load is very high, the friction decreases due to the larger quantity of wear debris andthe surface roughing. The coefficient of friction usually decreases with an increase in velocity. Highspeed motion tends to reduce the formation of asperity junction of interface. High speed tends togenerate heating, softening of asperity, and the oxidation of film. On the other hand, softening couldincrease plowing; high shear rates lead to lower real contact area and this renders the friction higherat high speed.

The load and speed dependence of the friction of ceramics could be related to the changes in thechemical surface films and the extent of fracture. The coefficient of friction is low at low load, butincreases rapidly after the brittle fracture occurs. A decrease in friction with sliding velocity isattributed to the increase in the interface temperature which promotes the formation of tribochemicalfilms.

For polymers, it has been to link the shape of friction–velocity curve to the viscoelasticproperties. The friction peak corresponded velocity is directly related to the frequency where theloss modulus peaks. The shape of friction–velocity curve is similar to that of tan d–frequency curve.Figure 3.17 shows the typical curve of friction of elastomer as a function of slip velocity.

In many cases the friction–velocity peak occurs because of the competing effects of strain ratedependence of the shear strength (proportional to the tensile strength) and the reduced contact areadue to viscoelastic stiffening of the elastomer (storage modulus increases with increasing frequency).

3.4.3 WET FRICTION

3.4.3.1 Stribeck Curve

We have discussed above the interface with dry friction surfaces or surface friction without liquid.For many real engineering interfaces, the water and moisture are mediated, and for lubricationsystems, grease and oil as well as fatty acid type lubricants could be applied on the surfaces to alloweasy sliding over each other. It is well known that there are four regimes of lubrication effects in aninterface with fluid: (i) static friction, (ii) boundary lubrication, (iii) mixed lubrication (partial fluidand partial solid), and (iv) full fluid lubrication. Figure 3.18 is known as the Stribeck curve showingthe regimes.

Consider the contact sliding interface with a liquid film. To discuss the four basic lubricationregimes, let us define a ratio to quantify the relative thickness of liquid and the roughness, l, isdefined as the ratio of the mean liquid film thickness to the combined roughness of both surfaces to

0

0.1

0.2

0.3

0.4

0.5

0.6

−4 −3 −2 −1 0 1 2 3

Log10V (mm/s)

CO

F

FIGURE 3.17 Friction of elastomer as a function of slip velocity.



l ¼ h0=Rq (3:87)

in which h0 is the mean liquid film thickness, and Rq is the average rms roughness of the twosurfaces.

Static friction is the lateral force that must be applied to initiate sliding of one object overanother, while kinetic friction is the lateral force that must be applied to maintain sliding. Strictlyspeaking, static friction is not a friction force but a threshold. Its presence implies that the objectshave locked together into a local energy minimum that must be overcome before motion by theexternal force.

Different mechanisms have been suggested for static friction such as geometric interlocking,adhesion junction elastic instabilities, plastic deformation, and crack propagation, etc. All thesemechanisms may well be applicable, although each one under different circumstances. Basically,the first regime does not depend on velocity. The static contact allows the asperity junctions todeform elastically and plastically and yields static bonding. A traction force is needed to break awaythe bond, or static friction. There is usually a presliding displacement associated with the breakawayprocess as discussed in Section 3.4.2.5. The ratio of the breakaway force to the normal force is thecoefficient of friction. The contacts are compliant in both normal and tangential directions. Thejunction in the surfaces of the continuum system is like an equivalent spring k. Under the traction,the presliding displacement is linearly proportional to the applied force to a critical value equal tothe static friction; then breakaway occurs and the transition to sliding is actually not abrupt. Apartfrom the solid–solid bond, meniscus due to liquid mediation also contributes to the static friction.

The second regime is boundary lubrication which occurs at very low sliding velocity with l< 1.The velocity is not sufficient to establish a film between the solid surfaces, and the liquid lubricationis not important. The entire applied load is actually carried by the surface asperities, and the frictiondepends on the lubrication properties of the molecules on the surfaces. Coefficients of frictionbetween 0.06 and 0.1 are typical when a low shear strength boundary film is present. If no such filmis present, then coefficients ranging between 0.2 and 0.4 can be exhibited, even rising as high as 1.0in some cases. In this regime, atomically flat surfaces could be separated by a few molecular layersof liquid; however, the behavior of the interface becomes qualitatively different from the bulkviscosity which is traditionally associated with lubricant film. The interfacial material can pack intoa solid-like structure due to its confinement and exhibit properties such as a finite yield stressand stick–slip. The boundary lubrication layer is solid-like and will be maintained under the contactand serves to provide lubrication. There is shearing in the liquid in boundary lubrication due to

Static friction

Velocity

Fric

tion

forc

e

Mix

edlu

bric

atio

n

Hyd

rody

nam

iclu

bric

atio

n

Bou

ndar

y lu

bric

atio

n

FIGURE 3.18 Generalized Stribeck curve.



solid-to-solid contact. Boundary lubrication is basically a process of shear in a solid, despite theinvolvement of liquid.

The third regime is mixed lubrication regime and is approximately characterized by 1� l< 3.In these contacts, part of the load is carried by the liquid, and part by the interacting asperities.The rubbing together of asperities will increase friction, and this can be minimized by thecompounds adhering to the surfaces or the tribolayer in the interface. The interface is partiallysupported by hydrodynamic force and partially by asperity contact force. Some liquid is expelled bypressure, but viscosity or wetting effect prevents all of the liquid from escaping and thus a film isformed. The friction process could be dominated by the interaction of liquid viscosity, motionspeed, pressure, and contact geometry. For this case, the greater the viscosity or motion velocity thethicker the fluid film will be, and therefore the lower the coefficient of friction. The surfaceroughness, asperity size, and orientation have essential impact on the characteristics of formedfilms. In some other cases, the friction process could be determined by the tribolayer, speed,pressure, and contact geometry. In this case, the surfaces are so close that the liquid viscosity isrelatively not significant. It is the liquid physical and chemical interaction with the surface thatdominates the friction. The mixed lubrication usually has liquid film thickness in the range of30 nm–3 mm. In tribology practice, the additives are usually used to provide the desired propertiestogether with base liquid that needs low surface tension and a low contact angle. The additives areusually adsorbed on the surface or react with it to form monolayers with low shear strength andtherefore reduce friction as a bottom line to protect the surface wear.

The fourth regime is the full fluid lubrication where the solid-to-solid contact is eliminated.The process is governed by either the hydrodynamic or elasto-hydrodynamic lubrication character-ized by 3< l< 10. In this situation the surfaces are kept apart by a pressurized fluid. The clearancespace is much larger than the average surface roughness, and therefore the surfaces can be consideredsmooth. The pressurization of the fluid is usually achieved by external means in hydrostatic bearings,but is accomplished in hydrodynamic contacts by the relative motion and geometry of the surfaces.

3.4.3.2 Unsteady Liquid-Mediated Friction

In this section we discuss only the first three regimes of lubrication in the interface, which are mostlikely associated with the friction-induced vibration problems in engineering.

For liquid-mediated interface, the static friction or stiction could exhibit strong time depend-ence: the static friction increases with rest time remarkably. The empirical model can also berepresented by Equation 3.83; the power law or divided formula can also be employed. In somecases like stiction in the slider–disk interface of conventional hard disk drive, the static friction startsto rapidly increase after a certain rest time. The long-term friction after days’ dwell could be10 times higher than the static friction without dwell or having short dwell. Moreover, the higherthe acceleration the higher the static friction due to the viscous effects of wet friction.

Figure 3.19 shows the recorded static friction and long-term static friction in an interface of anew disk drive during contact start–stop (CSS) and dwell process. Each CSS cycle consists of thesame operation: 3 s acceleration, 3 s constant speed operation, 3 s deceleration, and 6 s short soakingwith slider parking on disk statically. Long-term (10 h) soaking was introduced every 1000 CSScycles. Figure 3.20 shows similar results for the same interface in the next 5000 CSS operations.It can be seen that the static friction remains the same feature after the ‘‘run-in.’’

The ceramic slider and diamond-like carbon-coated disk pairs are used in the above interface.From Figure 3.19, the long-term stiction is higher than the short-term stiction. This is because long-term soaking will lead to significant redistribution and accumulation of the interface lubricant, dueto the microflow of lubricant around the toe-dipping regimes, in addition to the slider–disk asperitybond creep development.

Generally, there are several possible mechanisms that allow static friction force to increase withthe contact time: the area of real contact increases with the time of stationary contact; the



redistribution and accumulation of the interface liquid, due to the microflow of lubricant around thetoe-dipping regimes, and if the long-chain polymers are involved, chain interdiffusion, alsocontributes to it; moreover, if the liquid film exists dynamical phase transitions from fluid stateduring slip to a solid state during stick whose formation is a nucleation process, then the staticfriction force could also increase with the time of stationary contact.

Figure 3.21 shows the measured static friction coefficient as a function of dwell time for differentlubricants. Many engineering wet interfaces or liquid-mediated interfaces exhibit the unsteady mixedlubrication. The boundary and mixed lubrications are important for friction-induced vibrationsbecause they are usually associated with the stiction and the negative slope of friction vs. velocitycurve. For most wet interfaces, the measured static friction is almost always higher than kineticfrictions, so the stick–slip and=or negative damping could cause concerns. In the mixed lubricationregime, the friction is a function of velocity because the physical process of shear in the junctionchanges with velocity and is due to the liquid film effect. Details of the friction–velocity curvedepend upon the degree of boundary lubrication and the details of mixed lubrication.

Fluid friction arises from the shear of fluid film within contact and can be represented as [100]

Pf ¼ Pf (V ,P,hi,a, h0,E*, r0, T , b) (3:88)

in which V, P, hi, a, h0, E*, r0, T, b are the velocity, load, fluid viscosity, pressure–viscositycoefficient, film thickness, effective elastic modulus, combined radius of curvature, temperature, andfilm length, respectively. On the other hand, the solid friction force can be written in functional form as

Long-term stiction

Short-term stiction

CSS cycles

Stic

tion

forc

e (g

)

1 1001

1

2

3

4

5

6

7

8

02001 3001 4001 5001

FIGURE 3.19 Static friction force vs. CSS operation cycles for a new interface.

1 1001

CSS cycles

Stic

tion

forc

e (g

)

2001 3001 4001 5001

8

7

6

5

4

3

2

1

0

FIGURE 3.20 Static friction force vs. CSS operation cycles for the conditioned interface.



Ps ¼ Ps(P, h0,E*, r0,s, b) (3:89)

in which s is the combined surface roughness. To model the total friction we add the two forces anddivide by the total normal load to acquire friction coefficient

m ¼ Pf þ Ps

P¼ m(V ,P,hi,a, h0,E*, r

0, T ,s, ts, b) (3:90)

For the boundary and mixed lubrication regime, liquid viscosity could have an insignificant effecton the friction characteristics as the interface load is dominated by asperity contacts. Althoughthe liquid film does not control the friction, tribolayer could play a key role in distinguishingfriction properties between wet friction and dry friction; this could be the case for many wetinterfaces.

To acquire some insight, let us consider the contact together with the hydrodynamic lubrication;the minimum film thickness for smooth surfaces with hydrodynamic lubrication can be calculatedfrom the following formula:

h0 ¼ cm0RV

P(3:91)

wherec is constantP is the normal load per unit width of the contactR is the relative radius of curvature of the contacting surfacesm0 is the lubricant viscosity at inlet conditionsV is the relative surface velocity

To estimate whether the interface is in the boundary lubrication, we can calculate the specific filmthickness or the lambda ratio. The total load is shared between the asperity load and the film load.

Consider the particular lubricated concentrated contacts; both the contacting asperities and thelubricating film contribute to supporting the load. Thus

P ¼ Pf þ Ps

0

0.1

0.2

0.3

0.4

0.5

0.6

0.1 1 10 100 1,000 10,000

Time (s)

CO

FLubricant A

Lubricant B

Lubricant C

Lubricant D

FIGURE 3.21 Measured static friction coefficient as a function of dwell time.



whereP is the total loadPf is the load supported by the liquid filmPs is the load supported by the contacting asperities

Load Ps supported by the contacting asperities results in the asperity pressure pa given by

pa ¼ 43(hRpsp)E

*(sp=Rp)1=2F3=2(de=sp) (3:92)

whereFm (D) is a statistical function in the GW model of contact between two real surfacesRp is the relative radius of curvature of the contacting surfacesE* is the effective elastic modulush is the asperity densitysp is the standard deviation of the peaksde is the equivalent separation between the mean height of the peaks and the smooth surface

For the case l � 1, the liquid pressure to total pressure is approximated by

pl¼ 1

l

h0h

� �ap (3:93)

a is a constant, p is the total pressure, h is the mean thickness of the film between two actual roughsurfaces, and h0 is the film thickness with smooth surfaces.

In liquid-mediated interface, there exist strong friction delay-hysteresis phenomena. If there is achange in velocity, the corresponding change in friction has a delay. The friction memory can beattributed to that friction level lags a change in system state [86].

The hysteresis is the separation in friction levels during acceleration and deceleration.An alternative to the state variable models described in Section 3.2.4.5 is the pure time lag

model. In lubricated contacts, simple time delay better describes the effect

F(t) ¼ Fvel( _x(t � Dt)) (3:94)

in which F(t) is the instantaneous friction force, Fvel(�) is friction as a function of steady-statevelocity, and Dt is the lag parameter, the time by which a friction lags a change in velocity. The lagincreases with the increase of liquid viscosity and contact force.

3.4.3.3 Negative Slope of Friction–Velocity Curve

The wet friction occurs in many mechanical and automotive engineering applications. The wetfriction usually is smaller than the dry friction, but there are many reversal results in which thewet friction is higher than the dry friction. Moreover, the trend of negative slope is usuallydetermined by the competing of many factors, and could have larger variations. Next we illustratethis by using several application cases in automotive engineering.

A typical case is the friction exhibited by friction material utilized as a torque converter duringthe operation of wet clutches in which it is submerged in an automatic transmission fluid (ATF).Friction-induced shudder is an issue associated with automatic transmissions, which can be causedby velocity-dependent friction.



For automotive wet clutch interface with same solid clutch friction material, the coefficientof friction could have quite different profiles when different automatic transmission fluids are used.Figure 3.22 plots the coefficient of friction as a function of velocity for three different applied ATF.A is considered to be favorable. B and C may be susceptibly vulnerable to self-excited vibrations.The optimization of the fluid is mainly achieved by regulating the additives in the base liquid. Onthe other hand, the similar profile group could be attained by using the same fluid but with differentsolid friction materials.

Figure 3.23 shows that friction–velocity curve for dry friction tests as well as for tests withtransmission fluids. It shows that the friction material produces a negative coefficient of friction–velocity curve for both dry friction and lubrication with transmission fluids. At low sliding speeds,the coefficient of friction when operating in transmission fluids for the friction materials is greaterthan that under dry friction.

Another example is the complex friction features of an automotive brake friction material pairunder water mediation; the wet coefficient of friction could be higher than the dry coefficient offriction and the transition process is usually complex. The examples for the same friction coupleare shown in Figure 3.24. During the first part of the test, the friction coefficient was measured withthe brakes in a dry condition. The brake disk was then moistened with water. The results show that

Velocity

Fluid A

Fluid B

Fluid CC

oeffi

cien

t of f

rictio

n

FIGURE 3.22 Schematic of friction–velocity curves for three automatic transmission fluids.

Velocity

Dry friction

Lubricated

Coe

ffici

ent o

f fric

tion

FIGURE 3.23 Dry and wet coefficient of friction vs. velocity of a clutch interface.



the friction force rises considerably when the brake disk is moistened. This is caused by a rise in theadhesion forces between the lining and disk due to the water in the contact surface. In addition,the initial increase in the friction coefficient in all cases and the decrease in the friction coefficientduring a further increase in the relative velocity are remarkable.

One more example is the temperature effects on the brake lining-disk pain. The cold wetcoefficient of friction is higher than the hot dry coefficient of friction, the hot dry coefficient offriction is higher than the cold dry coefficient of friction. All have negative slope of friction–velocitycurve, as shown in Figure 3.25. Moreover, high relative air humidity could have obvious effects onthe friction. Figure 3.26 shows the contrary effects of humidity on two different brake pairs.Figure 3.27 shows the typical friction vs. slip velocity of tire under different road conditions.

3.5 FRICTION–VIBRATION INTERACTIONS

3.5.1 STICK–SLIP

So far we have discussed elastic and plastic surface friction interaction with and without liquidinvolvement, where we implicitly assume that the continuum components have no bulk oscillations.Next we consider the effect of system oscillation on the friction.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5

Velocity (m/s)

CO

F

Wet

Dry

FIGURE 3.24 Dry and wet coefficient of friction vs. velocity of a brake interface.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5

Velocity (m/s)

CO

F

Wet cold

Dry, 120°C

Dry cold

FIGURE 3.25 Dry and wet coefficient of friction vs. velocity of a brake interface for different environmentaltemperatures.



Actually, the system oscillation is always associated with the friction process. There are severallevels in which frictional energy is dissipated for a given sliding contact pair.

The first level has frictional energy being used in deforming or breaking the boundary films.The second level has energy being dissipated in deforming the contacting asperities elastically andplastically, and even leads to fracture and wear. The third level allows the interaction of contactsliding components to extend beyond the proximity of the contact interface. This interaction resultsin the wave and vibration response of the surrounding elastic continuum. The fourth level is thereaction of the wave and vibration of the continuum of system to the interface, which altersthe contact and friction process through vibrational deformation. Friction and vibration as well aswaves of elastic interface constitute a feedback relationship. During a sliding contact, friction andexternal excitation results in the vibration and wave; the waves and vibrations could affect frictionby modifying the distribution of the true contact areas. This interaction of vibration and normalmotion of interface at the continuum bodies constitutes a feedback loop between friction andvibrations.

Accurately quantifying friction-excited vibrations needs to account for the coupling of systemdynamics and the simultaneous development of normal contact and friction during the slidingmotion. This involves to combine system dynamics with the friction through the interactions

0

0.1

0.2

0.3

0.4

0.5

0.6

0 20 40 60 80 100Relative humidity (%)

CO

F

Type A

Type B

FIGURE 3.26 Coefficient of friction vs. relative air humidity for two different brake pads.

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5Slip speed (m /s)

CO

F

Dry asphalt

Wet asphalt

Ice

FIGURE 3.27 Coefficient of friction vs. slip velocity of tire under different road conditions.



between microlevel asperities and macrolevel medium motion in terms of the deformation andadhesion of asperities. Forces at each asperity contact depend on the area of that contact which, inturn, depends on the nominal relative position of the surfaces. This kind of dependence betweenthe individual true contact areas and relative position of surfaces relates friction to the motion ofthe surfaces. Normal oscillation couples with the tangential direction. A simple model of this typeneeds to describe vibration motions in both tangential and normal directions, and to relate thetangential and normal forces to the instantaneous properties of the true contact area, to describe thevariation of the true contact area as a function of the motion of the system in both tangential andnormal directions, relative velocity. Theoretically, modeling microlevel asperity and frictiontogether with macrolevel system response simultaneously yields the friction force and systemresponse that affect each other. A more feasible approach for engineering application is to avoidthe coupled friction and vibration analysis; instead, treat them, respectively, with assumptions.To provide some insight and give a bigger picture of the friction=contact vibration interactionanalysis, we will discuss the normal–tangential motion coupling system with specific friction lawsin Chapter 4.

Many engineering systems with sliding friction have noticeable system vibrations associatedwith the interface friction process, in both normal direction and sliding direction, due to a finitesliding mass and limited elastic properties. Next we brief basic features of the case with horizontalcompliance associated with stick–slip, and the case with normal compliance.

Consider a horizontally sliding continuum elasticity or system driven by spring stiffness k. Thespring is not necessarily an external spring but could represent the overall elastic properties ofthe sliding elasticity body. The simplest case is when the free end of the spring moves with aconstant velocity V. The friction force is applied on the surface of the body. The force in the springas a function of time reflects the friction force. The typical motion could be either steady motionor stick–slip motion where the motion alternates between stick and slip. Figure 3.28 shows thespring force as a function of time for two different stick–slip cases, the steady type and thechaotic type.

The stick–slip is a very popular phenomenon widely existing in many systems, which ischaracterized by the sawtooth pattern in the friction vs. time curve. Typically, it is because staticfriction is larger than kinetic friction. It could take place in a random, chaotic fashion, or repetitively.The stick–slip is usually associated with vibration. The stick–slip takes place over a wide range oftimescales; oscillation frequencies can range from several thousands of hertz to 10�9 Hz, or lessthan one slip per couple of hundred years, subject to the friction properties and the systemcompliance characteristics. Stick–slip has been observed experimentally, and numerous theoreticalinvestigations had been given to this problem. Usually, the occurrence of stick–slip is not onlyconcerned with interface friction characteristics, but is also concerned with the continuum systemdynamics. Figure 3.29 illustrates the basic properties of stick–slip, as a function of velocity, forseveral values of spring stiffness.

Fric

tion

forc

e

Fric

tion

forc

e

Time Time

Periodic stick–slip Chaotic stick–slip(a) (b)

FIGURE 3.28 Typical stick–slip: (a) periodic and (b) chaotic.



In Figure 3.29 we see that there are two approaches to eliminate the stick–slip, one is by usingstiffer system and the other is by using high velocity. Slip amplitude is a decreasing function ofvelocity up to an abrupt elimination of stick–slip. We will give more details on the stick–slipmechanism using discrete elastic model and parametric friction model in the next chapter. More-over, we will also brief another stick–slip reduction approach, the dither control, by which a higherfrequency excitation is introduced in traction direction to reduce the stick–slip.

3.5.2 EFFECT OF NORMAL OSCILLATIONS

Next we consider the interface with normal oscillations and their effects on friction. Consider thecoefficient of friction under dynamic situation. If the dynamic normal force is N(t) and the frictionforce is F(t), then the instantaneous friction coefficient is

m(t) ¼ F(t)=N(t) (3:95)

Of particular interest is the interpretation of average friction. Assume the average of friction as thetime average of m(t), denoted by hm(t)i

m(t)h i ¼ðm(t) dt (3:96)

Alternatively, we can also denote an average friction coefficient mav; then

mav ¼ F(t)�

= N(t)�

(3:97)

in which

F(t)� ¼ ð F(t) dt (3:98)

N(t)� ¼ ð N(t) dt

0

0.05

0.1

0.15

0.2

0.25

0.3

−3 −2 −1 0 1

log10V (mm/s)

Stic

k–sl

ip a

mpl

itude

k1 = 5000 N /m

3k1

6k1

30k1

FIGURE 3.29 Stick–slip amplitude as a function of slip velocity for different connecting stiffnesses.



If both the normal force and the friction force remain constant or with small variation, then the twodefinitions are the same, otherwise they are not. Consider an interface with Hertzian contact.Assume an oscillating normal load is applied: N0 (1 þ cos Vt), and also assume it is just enoughto give impending contact loss at one extreme of the motion. Assume friction coefficient is m0 whenthe load is at its mean value N0. If the instantaneous friction force is assumed to be proportionalto the instantaneous real area of contact, it is shown in [89] that mav=m0¼ 0.92, whereashmi=m0¼ 1.84. For proper definition of average friction, mav is usually preferred. In friction testone needs to monitor whether the normal load is constant. If it is not constant, one obtains the‘‘apparent friction’’ coefficient that can be quite different from the actual friction. Particularly,apparent friction sometimes includes stiction or the value corresponds to the loss of contact that doesnot represent friction in the usual sense.

Next we consider the normal motion of a single-degree-of-freedom system with contactinterface under excitation. Assuming Hertz contact, the system equation is given by

m€yþ c _y� f (d) ¼ �N0(1þ a cosVt)� mg for d > 0 (3:99)

whered is the contact compressionf(d) is the restoring force

given by

f (d) ¼ 43ER1=2d3=2 ¼ k1(y0 � y)3=2, y0 ¼ ½(N0 þ mg)=k1�2=3 (3:100)

An approximation y(t) to this equation can be obtained using perturbation technique detailed in thelast chapter. Based on the adhesion theory of friction, the instantaneous friction force is assumed tobe proportional to the area of the contact. Since the contact area is proportional to the compression,we have

F(t)=F0 ¼ A(t)=A0 ¼ ½1� y(t)�=y0 (3:101)

The normal oscillation, y(t), is asymmetrical due to the nonlinear contact stiffness.We also have

hF(t)=F0i¼hA(t)i=A0 ¼ (1� hy(t)i)=y0 (3:102)

As oscillation increases, the average contact area and the average friction are reduced. It is foundthat a reduction in average friction force of up to 10% could occur prior to loss of contact.

Many experiments have shown the normal vibration effect on friction. Figure 3.30 shows atypical coefficient of friction as function of normal vibration acceleration under different frequenciesfor a steel balls–steel beam interface. One can assume that occasionally contact loss begins to occurwhen the normal acceleration reaches amplitude of 1 g for the case without external normal load. Athigher normal acceleration, where there is progressively more intermittent contact loss, a largereduction in friction occurs. Higher frequency tends to reduce friction more. Figure 3.30 shows atypical coefficient of friction as a function of normal vibration acceleration under different frequen-cies. A similar result was obtained by pin-on-disk system test in which the contact modulation isexcited by surface irregularities. In this experiment, the average friction is measured as a function ofvarious sliding speeds. It was found that the normal vibration increases with sliding speed, and thefriction decreases with the increase of sliding speed. It shows that only at speeds well above thevalue that is associated with initial loss of contact, can one obtain large reduction in average frictionof as much as 30%.



It is noted that all experimental setups measure apparent coefficient of friction, which are tosome extent affected by the dynamic characteristics of the apparatus and differ from the realcoefficient of friction. It is highly recommended to use test system with high stiffness and damping,so as to minimize the occurrence of vibrations or even jumps of the samples.

In many other applications, similar experiments have shown that a drastic reduction in the stick–slip amplitude and the friction force F is attained by applying low energy oscillations of highfrequency and small amplitude perpendicular to the sliding direction. This result has been used inmany systems to reduce the friction force or to eliminate stick–slip motion through a stabilization ofdesirable modes of motion.

It has also been known that the use of ultrasonic pulses can reduce dormancy related stiction,particularly in microscopic mechanical contacts. The ultrasonic pulses can release stuck surfacefrom bonds or junctions. The simplest interpretation is that the pulses transfer momentum andvibration energy thereby exciting the surface microjunctions to release them from their deep stuckstate. They also transfer energy to the liquid bridge across the asperities to allow them to havemicroflowing and redistribution. The transferred momentum leads to tensile stress across theasperity bonds resulting in tensile stress that separate the parts of the substrate. The frictionreduction using ultrasonic pulses can also be attributed to the effect of Rayleigh-type surfaceacoustic waves. It is shown that with increasing wave amplitude, friction can be completelysuppressed. The effect of lateral and vertical surface oscillation components of wave motion hasbeen experimentally verified, and it is found that the friction reduction effect is only due to thevertical oscillation component, and does not appear for purely in-plane polarized Love waves.

3.6 FRICTION IN NANO- AND MOLECULAR SCALES

With the advent of the atomic force microscope (AFM) and the surface forces apparatus (SFA) itbecame possible to study individual sliding junctions at the molecular level [162–201], whichinvolves investigations at atomic-length scales and very short timescales. The AFM and SFA areideal tools in nano-, micro-, and macroscopic tribological experiments for measuring the normal andlateral forces, and wear, between a nanometer-radius tip and a substrate surface. It is noted that thereare some substantial differences between the friction characteristics at microlevel and at nanolevel.For instance, recent experimental evidence shows that there is a significant change in the frictionstress acting on a single asperity contact, as the contact area changes from the micro- to nanoscale.Some experiments with friction force measured using an AFM show that the friction stress can bemore than an order of magnitude higher as compared to experiments in which the friction ismeasured with the SFA. A typical contact radius for an AFM tip is estimated to be on the orderof 3–14 nm, whereas for SFA it is on the order of 40–250 mm. The schematic difference of the

0.01

0.1

1

10

0.1 1 10 100

Acceleration (m/s2)

CO

F

f = 20

f = 1000

FIGURE 3.30 Friction as a function of normal vibration acceleration under different frequencies.



relationship between the nondimensional friction stress and the nondimensional contact radius,according to [162–164], is given in Figure 3.31. The contact radius a is normalized by the Burgersvector and the friction stress is normalized by the effective shear modulus G*¼ 2G1G2=(G1 þ G2)where G1 and G2 are the shear moduli of the contacting bodies.

The scale dependence of the friction stress was modeled by Hurtado and Kim (HK model)[163,164], using a micromechanical dislocation model of frictional slip between two asperities for awide range of contact radii. According to the HK model, if the contact radius a is smaller than acritical value, the asperities slide past each other in a concurrent slip process, where the adhesiveforces are responsible for the shear stress; hence the shear stress remains at a high constant value. Onthe other hand, if the contact radius is greater than the critical value, the shear stress decreases forincreasing values of contact radius until it reaches a second constant, but a lower value. In thetransition region between the two critical contact radius values, single dislocation assisted slip takesplace, where a dislocation loop starts in the periphery of the contact region and grows toward thecenter; the shear stress is dominated by the resistance of the dislocation to motion.

For contact radii smaller than a critical value, the friction stress is constant. Above that criticalvalue, the friction stress decreases as the contact radius increases until a second transition occurswhere the friction stress again becomes independent of the contact size.

The single asperity nanocontact model of HK is incorporated into a multiasperity model forcontact and friction which includes the effect of asperity adhesion forces using the Maugis–Dugdalemodel by Adams et al. [162]. The model spans the range from nano- to micro- to macroscalecontacts.

Another remarkable breakthrough is to make use of molecular dynamics simulation technique toinvestigate molecular and atomic level sliding frictions. It is in general unlikely to investigate slidingdynamics of engineering interested region using molecular dynamic calculations. Such a kind ofcalculation is a ‘‘first-principle’’ method and can only be performed for very short time periods, say10�9 s or shorter. On the other hand, sliding dynamics in engineering like stick–slip oscillations usuallyare in the timescale of seconds, and sometimes evenmuch longer such as 100 years for big earthquakes.

However, the molecular dynamics approach can provide many insights and give a much deeperunderstanding of the complex tribology phenomena. These include the tribology of micro-electro-mechanical systems (MEMS), computer hard disk drive devices, and friction reduction by vibrationsystems. The conventional empirical laws of friction do not always hold in such systems due to theirhigh surface-to-volume ratio and the greater importance of surface chemistry, adhesion, surfacestructure, and roughness effect.

Atomistic level molecular dynamics simulations have a wide range of applicability and havereached a high level of rigor and accuracy. They help us to acquire a deeper understanding of the

0.0001

0.001

0.01

0.1

Dim

ensi

onle

ss f

rict

ion

stre

ss Based on AFMmeasurement

Based on SFA measurement

0.01 1 100

Dimensionless contact radius

10,000 1,000,000

Transition region

FIGURE 3.31 Dimensionless friction stress as a function of dimensionless contact radius.



relationship between static and kinetic friction, the nature of transitions between stick–slip andsmooth sliding, slippage at solid–liquid interfaces, shear thinning, and the friction of rough surfaces.But molecular dynamics simulations are currently limited to timescales no greater than tens ofnanoseconds and length scales of tens of nanometers, which are too short for analyzing many realtribological systems. Figure 3.32 shows the schematic of the feasible time and geometry scales formolecular, nano-, micro-, and macrolevel tribology. It constitutes a hierarchy of modeling tribolo-gical behavior that relates atomistic models with engineering models.

In order to solve technological problems using the accurate molecular dynamics simulation, thedifferent levels of investigations can be linked as chain along the ladder from quantum-level studiesto engineering design, as illustrated in Figure 3.32.

Molecular dynamics simulations are able to bring us a step closer to ‘‘seeing’’ what takes placeduring sliding contact. In atomic dynamics simulation, computer algorithm solves Newton’sequation of motion for each single atom under the influence of its neighboring and of outsidesources. But we usually need to study large numbers of atoms, sometimes at least tens of thousands,which in turn means solving large numbers of equations, one for each atomic coordinate. In twodimensions the atom motion equations are solved with mapping according to their kinetic orpotential energy.

Friction exhibits its most fundamental form at the atomic level. During friction between twoatomically flat surfaces, some of the kinetic energy associated with the relative motion propagates asphonons beyond the surfaces into the bodies and dissipates.

Dissipation means the conversion of kinetic energy to thermal energy. The same phenomenonoccurs during passage of a sound wave in a solid when some of its energy converts to thermalenergy. Thermal energy relates to the vibrations of atoms in a solid.

In solids atoms are held in equilibrium with respect to each other by means of electrostaticforces or interatomic potentials, often described as bonds between atoms. As external forces such asa sound wave or friction on a surface excite atoms, their energy levels increase as represented bytheir vibration amplitudes. Models that study thermal energy consider that vibrations of atoms in asolid have a periodic structure of identical mass–spring cells. In such a system, nonlinear springslink the masses together.

The research provides many insights into the friction’s atomic-level origins; for instance,some perspective indicates that sliding friction stems from various unexpected sources, including

Quantum scale

Years

Minutes

Milliseconds

Microseconds

Picoseconds

1 A 1 mm1000 nm1000 A10 A

Molecularscale

Microstructure scale

Continuum scale

Engineering scale

FIGURE 3.32 Time and length scales of friction problems in different contexts.



sound energy. This explained why friction, when probed at a microscopic level, manifests itselfin a very different manner from that which we routinely observe at the macroscopic level.

Friction arising from sound waves, or atomic-lattice vibrations, occurs when atoms close to onesurface are set into motion by the sliding action of atoms in the opposing surface. In this way, someof the mechanical energy needed to slide one surface over the other is converted to sound energy,which is eventually transformed into heat. The amount of mechanical energy transformed into soundwaves depends on the nature of the sliding substances. Solids are much like musical instruments inthat they can only vibrate at certain distinct frequencies; so the amount of mechanical energyconsumed will depend on the frequencies actually excited. If the ‘‘plucking’’ action of the atoms inthe opposing surface resonates with one of the frequencies of the other, then friction arises. But if itis not resonant with any of the surface’s own frequencies, then sound waves are not generated. Thisfeature implies some possibilities that sufficiently small solids, which have relatively few resonantfrequencies, might exhibit nearly frictionless sliding.

It is believed that the source of this friction has both an electronic and a phononic contribution(phonons are vibrations in a crystal lattice, like an atomic sound wave). In addition to conventionaldissipation mechanisms (e.g., photonic and electronic), friction of the nonlinear system can besignificantly affected by the dynamical properties of the sliding system such as, for example, thefluctuations of each individual element from the center of mass motion. A nonlinear system drivenfar from equilibrium can exhibit a variety of complex spatial and temporal behaviors, each resultingin different patterns of motion and corresponding to different friction coefficients

Although a model based on first principles that describes the mechanism by which thisconversion takes place continues to be a research topic, a one-dimensional array of atoms canhelp elucidate the behavior of the response of atoms. Considering an array of oscillators connectedwith identical nonlinear springs with a spring constant. The simple models are useful in illustratingthe origins of static and kinetic friction, and in understanding the results of detailed simulations. Asan application example, consider two clean, flat, crystalline surfaces in direct contact at the planeindicated by the dashed line as shown in Figure 3.33a. The bottom solid is assumed to be rigid, sothat it can be treated as a fixed periodic substrate potential acting on the top solid. In order to makethe problem analytically simple, only the bottom layer of atoms from the top solid is retained, andthe interactions within the top wall are simplified.

(a)

(b)

FIGURE 3.33 Schematic of Frenkel–Kontorova model used for molecular dynamics simulation.



Consider the nearest-neighbor spacing in the bottom and top walls, respectively. The Frenkel–Kontorova model replaces the bottom surface by a periodic potential. It includes masses and springsof stiffness between neighbors in the top wall as shown in Figure 3.33b. The basic equations for thedriven dynamics of a one-dimensional particle array of N identical particles moving on a surface aregiven by a set of coupled nonlinear equations of the form

m€xi þ m _xi ¼ � @U

@xi� @V

@xiþ fi þ g(t), i ¼ 1, . . . ,N (3:103)

wherexi is the coordinate of the ith particlem is its massm is the linear friction coefficient representing the single particle energy exchange with thesubstratefi is the applied external forceg(t) is the Gaussian noise

The particles are subjected to a periodic potential U and interact with each other via a pair-wisepotential V.

Simplify this model to the case where the substrate potential has a simple periodic form, thesame force fi is applied to each particle, and the interparticle coupling is linear. The coupling withthe substrate is strongly nonlinear. For this case, using the dimensionless phase variable f¼ 2px=a,the equation of motion reduces to the dynamic Frenkel–Kontorova model

€fi þ m _fi þ sin(fi) ¼ f þ k(fiþ1 � 2fi � fi�1) (3:104)

In the model the atoms are coupled to nearest-neighbors by springs, and the coupling to the atomsabove is ignored. Due to their simplicity, these models arise in a number of different problems and agreat deal is known about their properties.

Next we use an example to illustrate how to make use of the molecular dynamics simulation tohelp explain some complicated stiction phenomenon in the interface in hard disk drive, where theconventional approaches are invalid. The molecular dynamics computational method is used tosimulate meniscus formation around an asperity in a rough surface represented as a sinusoidal wave.Two assumptions are proposed. First, it is assumed that the meniscus force causes a steep increase ofthe contacting load, thus causing increased friction force. A kinetic meniscus model has beenproposed by Bhushan et al. [118] to predict the resting stiction force between slider and disk causedby the capillary pressure. This model, however, has difficulty to interpret the meniscus formation onan ultrathin thickness of 3–10 nm liquid film. In this case, the continuous assumption is invalid. It isdoubtful that the fluid film can flow as freely as in the bulk state. Another assumption is that it is theattractive forces between the two surfaces and between the surface and the film that cause theincrease of the contacting load, thus causing increased friction force. This model can successfullyexplain the high friction coefficient but it cannot explain why stiction forces are great enough to pulloff the slider. The key problem rests on whether it is possible to form a meniscus around an asperitywith an ultrathin film and on the length of time for such meniscus formation.

Using the Lennard–Jones interatomic potential for argon atoms, the positions of the atoms andthe interactive force between the atoms and the surfaces are calculated [167]. The physical modelused in the calculation is that of a thin film confined between a smooth surface and a rough surface, asshown in Figure 3.10, except for that here we consider the smooth substrate is at the bottom andthe rough substrate is at the top in line with the schematic of the model in Figure 3.33. The lowersurface is not shown explicitly but is assumed a stationary, smooth surface to save computation time.



The upper wall is thick enough to make it appear infinite to the fluid atoms. Its roughness isrepresented as a sinusoidal wave surface.

The Lennard–Jones potential in Equation 3.36 is used to represent the interaction between aparticle of type A and a particle of type B, where subscripts A, B stand for ‘‘solid’’ and ‘‘fluid,’’respectively. The calculated results are shown in Figure 3.34 in which the snapshot of meniscusformation at different time steps is plotted. Figures 3.34a and b show the results for the timeevolution of liquid meniscus formation. Figure 3.34b gives the final state of the fluid atoms confinedbetween a smooth and a rough surface. During the meniscus formation the atoms at the top of thefluid move to the asperity first and bridge the upper rough wall with the substrate. When these fluidatoms move a vacancy appears.

Simulation results show that the meniscus formation depends on the interaction potentialbetween the solid wall and the liquid atoms. For completely and partially dry substrates a meniscuscannot form around an asperity. For partially and completely wet substrates the asperity helps toadsorb the fluid atoms and form a meniscus. These simulation results confirm that if the filmthickness exceeds a critical value, the capillary pressure contributes strongly to stiction.

Molecular dynamics modeling and simulation have also been used to investigate the stick–slipand its control using active vibration. Calculations demonstrated that oscillations of the normal loadcould lead to a transition from a state of high-friction stick–slip dynamics to a low-friction smoothsliding state. Manipulation by mechanical excitations, when applied at the right frequency, ampli-tude and direction, pull the molecules out of their potential energy minima and thereby reducefriction (at other frequencies or amplitudes the friction can be increased).

Friction can be manipulated by applying small adjustments (perturbations) to accessible elem-ents and parameters of the sliding system. It has been shown that friction in thin-film boundarylubricated junctions can be reduced by coupling of small amplitude in the order of 1 Å directionalmechanical oscillations of the confining boundaries to the molecular degree of freedom of thesheared interfacial lubricating fluid. Using a surface force apparatus, modified for measuring frictionforces while simultaneously inducing normal (out-of-plane) vibrations between two boundary-lubricated sliding surfaces, load- and frequency-dependent transitions between a number of ‘‘dynam-ical friction’’ states have been observed. The regimes of vanishingly small friction at interfacialoscillations were found. Significant changes in frictional responses were observed in the two-platemodel by modulating the normal response to lateral motion. In addition, the surface roughness andthe thermal noise are significant factors in making decisions about control on the micro- and thenanoscale. It provides an insight into realizing ultra-low friction in mechanical devices.

(a) (b)N = 480 and I = 0 N = 480 and I = 80,000

20

15

10Z� Z�

5

0

20

15

10

5

0

0 5 10 15 20

Solid particleFluid atom

Solid particleFluid atom

I = 0 I = 80,000

X�

0 5 10 15 20X�

FIGURE 3.34 Snapshot of meniscus formation at different time steps. (From Chen, Y.F., Weng, J.G., Lukes,J.R., Majumdar, A., and Tien, C.L., Appl. Phys. Lett., 79, 9, 27, 2001. With permission.)



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4 Friction-Induced Vibrationsand Sound

4.1 INTRODUCTION

The modeling and analysis of friction-induced vibrations and sound are challenging multidisciplin-ary problems from nano-, micro-, to macrolevels, involving contact mechanics, tribology, nonlineardynamics, and acoustics. The problems are generally concerned with time varying boundaryconditions, wide spatial timescales, as well as wide frequency ranging from tens of hertz toultrasonic. Friction-induced vibrations could exhibit various characteristics, such as steady vibra-tions or self-excited vibrations, deterministic or random vibrations, stationary or nonstationaryvibrations subject to system and conditions.

One of the major challenges of friction-induced vibrations and sound is to establish a propersystem model to predict sound and vibrations’ response subject to friction. It is usually quite difficultto model friction itself as well as predict its trend with certainty. A friction force usually interactswith and often depends on the response of the system on which it depends. Such interaction sets up afeedback between the friction force and the waves on the surfaces (or the vibration). It makes theminterdependent. In addition to the instantaneously dynamic interaction, friction usually sustains achange over a longer time which is associated with the deformation and wear of interface, thechange of environmental conditions including temperature and humidity. Moreover, the systeminvolving friction reverse usually constitutes a nonlinear system that could produce very compli-cated response. Friction sound and vibration are usually nonstationary problems.

Consider two components or systems in contact sliding state; for relatively simple problems, wecan treat the two systems independently with the interface as an external boundary. We can thenassume a proper friction law that is dependent on system oscillation to represent the real interfaceeffect, then the two systems can be addressed, respectively. This can be considered as a weakcontact case where two systems do not have modal coupling or react as dependent systems; in otherwords, the waves and oscillations develop independently in respective systems during sliding. Theeffects usually localize to the interface region, produce responses in each system at its own naturalfrequencies, nearly independent of the other system.

When negative slope of friction–velocity curve is involved, the system could develop self-excited oscillations due to negative damping effect, and the fundamental frequency could lock intoone of the natural modes of the original subsystem.

Weak contacts of rough surface could produce light impacts as asperities come into contact, andproduce a random response with amplification at the natural frequencies of each system or withamplification at the characteristic frequency of roughness. Such contact sliding conditions producesolid sound, referred to as roughness noise or surface noise. Such contact forces excite linear ornonlinear vibration response in each system.

However, for many complicated problems, we have to treat the two systems as coupled withcontact sliding interface as an internal boundary condition, and assume proper friction law as thecoupled condition to solve the equations for the two systems together.

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141

This is the so-called strong contact condition where the influence of contact force reachesbeyond the interface. The friction pair becomes a coupled system and produces a more complexresponse. Under such conditions, instabilities develop and the response frequencies of the systemhave no direct relationship with the original natural frequencies of the two systems. If the strongcoupling takes place in a very short period of time, the transient response frequencies or transientmode could be established momentarily and change with the transition of contact, and the systemexhibit nonstationary features.

However, this classification is not absolute; for a system with given sliding velocity andcontact surface geometry, the weak contact could develop to strong contact if normal load isincreased. The normal force may also be affected by the system interactions. A comparativelycomprehensive model should consist of the coupled equations of detailed interface tribologywith the equations of continuum dynamics, but this is usually too difficult to be tackled inengineering.

For light contact loads, vibration and sound sources are confined to the surface. Other studiesalso relate sound radiation to the effects of surface roughness. In the case of heavy contact loads likeroller bearings, gear, belts, and brakes, the instability problem of friction between elastic half spacescould penetrate to whole system; the steady sliding generates self-excited waves due to destabil-ization of slip waves.

Generally speaking, there are many unique dynamic phenomena and underlying mechanismsassociated with friction-induced vibrations and sound. The most prominent ones could include thefollowing:

. Stick–slip

. Velocity-dependent friction

. Modal couplings of different components

. Nonsmoothness

. Vibro-impact

. Random impact

. Sprag–slip

. Parametric resonance

. Nonstationary

. Uncertainty

Stick–slip motion is an ‘‘attach–detach’’ process. It is generated by the variation of frictioncoefficient and it is due to the effect of the difference between the coefficients of kinetic and staticfriction and special system stiffness. The combined effect of the friction–velocity dependencefeatures and system leads to the motion-induced sustainable vibrations.

Velocity-dependent friction is one the major mechanisms leading to system instability. Negativeslope of friction–velocity curve yields an equivalently negative damping to system dynamic motionand is thereby likely to cause system instability. It helps to form an energy input-accumulatingmechanism to a self-excited system. It usually allows the fundamental oscillating frequency of thesystem to be close to or lock into one of the natural frequencies.

The modal coupling of a system due to structural constraints and friction effect can form astrong energy input-accumulating mechanism. It results in dynamic instability of the system. Itusually causes a strong nonlinear oscillation of the system.

The reverse motion direction of sliding component under low sliding speed or high frequencyoscillations could occur for many cases; it renders the system to have nonlinear attributes ofdiscontinuousness, which cause very rich nonlinear oscillations.

The loss of interfacial contact due to severe oscillation in normal direction usually introducesvibro-impact motions, which complicates the original system of dynamic features with specialpatterns.



In contact sliding motion, the interfacial asperity impact is a form of a time series, whichgenerates a random excitation to the system. It results in random vibration of the system and thenatural modes of original system can be amplified.

Sprag–slip motion is a unique phenomenon in real contact sliding system. Its occurrencedepends on special structural configuration and the interface irregular friction that can cause thechanges of the normal and friction forces. Schematically, it occurs when one of the slidingcomponents is a hinged arm at an acute angle to the moving surface. When a salient asperity ofmoving surface moves toward the arm, the friction force increases the normal load leading tospragging, and the compressing and=or creeping of the contact asperity allows it to pass the arm.Then the motion of the salient asperity away from the arm decreases the normal force. It isessentially a ‘‘digging-in’’ and release process, mainly due to the geometric constraints of config-uration and kinetic constraints in terms of motion and forces. The sprag–slip can occur at constantcoefficient of friction for the above special structures. It could take place at macrolevel or micro-asperity level due to uneven contact, leading to instantaneous sprag–slip. It usually is the transientlocal force concentration and release, due to unbalanced or asymmetric support.

In some cases, the vibration system with friction is associated with parametric excitations due toperiodic normal motion, which could generate parametric resonance.

Friction-induced vibrations usually exhibit nonstationary characteristics. This is mainly due tothe unsteady interactions in the interface and the transition of instantaneous modes. There are manyuncertain factors existing in friction interface. For instance, the generation, transfer, and removal ofinterfacial powder, debris, and film due to wear and other interaction could have substantial effecton interface friction, thus altering the original system features substantially. Moreover, the forma-tion of the third body in interface could even change contact compliance or stiffness through therough surface, thereby resulting in a change of inherent dynamic characteristics of the system. Thiskind of aging effects is not very predictable and therefore the uncertainty is unavoidable.

Unlike many areas where an identical dominant principle governs system general character-istics, like minimum energy principle in elasticity theory, there is no single theory that can be usedto explain every friction-induced vibration and sound phenomenon. In some cases, one mechanismcan give a good interpretation, and in other cases a combination of several mechanisms is needed tooffer a better explanation. Which mechanism is dominant depends on the components, interface, andoperational and environmental conditions.

In this chapter, we will present the vibrations of SDOF and MDOF systems with friction. Thenwe present the continuum systems with friction by discussing friction-induced vibrations and thewaves in solid. Finally we describe a variety of friction-induced vibrations and sound in social life,nature, science, and engineering.

4.2 VIBRATIONS OF SINGLE-DEGREE-OF-FREEDOM SYSTEMSWITH FRICTION

4.2.1 FRICTION LAW

It is in general unlikely to study sliding friction dynamics directly using ‘‘first-principle’’ methodssuch as molecular dynamic calculations. It is also unlikely to study contact sliding dynamics directlyusing multiphysics-based models where Newton’s force and motion equation, asperity contactmechanics, liquid film dynamics, heat transfer, etc. are all included. The multiphysics-based modelsare suitable for fundamental understanding and applications interpretation as well as in-depthinvestigation and research. However, the model is computationally expensive and too complicatedto be implemented compared with empirical models, limiting its applications in engineering andscience. For engineering applications, it is usually needed to construct friction laws which can beused directly in Newton’s equation of motion to predict the dynamic performance of the slidingmacroscopic bodies [1–11].


Friction-Induced Vibrations and Sound 143

It is usually feasible to construct empirical friction laws based on phenomenological observationsto predict the sliding behavior of macroscopic bodies. This kind of model is based on macroscopicquantities observed, the displacement, the state variables, load, temperature, and their history.

When we consider elastic continuum vibrations, the friction description could be from as simpleas Coulomb friction law to microlevel surface tribological description detailed in Chapter 3. Theformer could be too simple for capturing dynamic friction phenomena sufficiently, the latter couldbe too complicated to be brought into the differential equation of dynamics motion. There have beenmany efforts in science and engineering communities to develop trade-off friction law for variousproblems, and numerous friction laws have been developed with phenomenological expressions.

The normal component of contact forces relating to the deformation of the contact areas cangenerally use the Hertz theory for approximation. Consider an interface represented by sphericalcontact, with effective radii of curvature R. Based on Equations 3.7 through 3.10, the contact forceFN(t) can be assumed as a function of elastic deformation in terms of normal displacement yH

FN(t) ¼ kHy3=2H (t) (4:1)

in which

kH ¼ 4E*ffiffiffiR

p=3 (4:2)

If the contact is retained, this type of contact introduces nonlinear stiffness to the system. If thecontact gets lost, then a strong nonlinearity of vibro-impact will occur in the system. On the otherhand, the coefficient of friction as a mathematical function of a variety of parameters can bedeveloped by using the fundamental formulations described in the last chapter.

The most widely used friction law is about the friction–velocity curve characterizing Stribeckfeatures of real systems. The phenomenological expressions for a friction force vs. slip velocity leadto qualitative results that help characterizing underlying physical phenomena. Polynomial andexponential forms of friction laws are most commonly used for capturing these kinds of attributes.One of the exponential expressions has the following form:

m(vr) ¼ sgn(vr) 1� e�bv vrj j� �1þ ( fr � 1)e�av vrj j� �

(4:3)

wherevr is the slip speedav and bv are constantsfr represents the ratio of the static to kinetic friction coefficients

Some applications require a more detailed and more accurate representation of friction to betterpredict system response and the development of instabilities. In these cases, in addition to describingthe Stribeck curve, it is also needed to quantify friction and coefficient of friction in terms of staticfriction transition, hysteresis, and presliding displacement (i.e., displacement that occurs just beforea complete slip takes place). In cases where oscillatory relative motion is involved, particularly thosewith small amplitudes, the modeling requires further details of the friction force in terms of itsdependence on both displacement and velocity. Velocity reversal and the change of friction withdisplacement are important in many applications, where the friction force remains constant duringsliding, but changes with displacement during each reversal of direction. The slope of the frictionchange with displacement can be treated as the equivalent linear tangential stiffness of the bondingforce. Moreover, some modeling needs to take into account the time-dependent characteristics ofstatic friction. The analyses of some dynamic systems even need to represent friction as a functionthat depends on pressure or even temperature in addition to velocity [12–19].



A more general description of friction than in Equation 4.3 is given by the following to includeCoulomb friction, viscous friction, static friction, and the Stribeck effect:

F ¼F(vr)

Fe

Fs sgn(Fe)

8><>:

if vr 6¼ 0

vr ¼ 0 and Fej j < Fs

vr ¼ 0 and Fej j � Fs

(4:4)

whereF(vr) is an arbitrary function that captures the friction profile of the Stribeck curveFs is static frictionFe is externally applied force

Different parameterizations have been proposed for F(vr), one of which is F(vr)¼Nm(vr), whereN is normal force and m(vr) is given in Equation 4.3. A common form for the friction function isgiven by [12]

F(vrÞ ¼ fc þ ( fs � fc)e� vr=vsj jds þ fvvr (4:5)

wherefs, fc, ds are constant characterizing static and dynamic frictionsvs is the Stribeck velocityfv represents viscosity

Another model is the hyperbolic model given by

F(vr, x) ¼F(x) ¼ min ( Fex(x)j j,Fs)sgn Fex(x)½ �, vr ¼ 0

F(vr) ¼ �Fs sgn(vr)

1þ d vrj j , vr 6¼ 0

8<: (4:6)

in which Fex¼ kx is the externally applied force and is displacement dependent. The friction forcein the stick phase is limited by the maximum static friction force

F(x)j j � Fs (4:7)

ADCK (Armstrong–Hélouvry, Dupont, Canudas de Wit, and Karnopp) model is a more generalmodel with more parameters [12,13]. It actually consists of two separate models with one forstiction and one for sliding. The friction is modeled as a stiff spring during stiction

Fl(x) ¼ �klx (4:8)

wherex is the presliding displacementkl is the equivalent stiffness

When sliding, the friction is modeled in the following form including the Coulomb friction andviscous friction as well as the Stribeck friction:

Ff ( _x,t) ¼ Fc þ Fv _xj j þ Fs(g,t2)1

1þ ( _x(t � tL)= _xs)2

� �sgn( _x) (4:9)



in which

Fs(g,t2) ¼ Fa þ (F1 � Fa)t2

t2 þ g(4:10)

in which Ff is the friction force, Fc is the Coulomb friction, Fv is the viscous force, and Fs describesthe varying friction levels at breakaway or static friction. The level of the static friction force Fs

varies with the dwell time t2. The force Fa is the magnitude of the Stribeck friction at the end of theprevious sliding period. The force F1 is the magnitude of the Stribeck friction after dwell. g is anempirical parameter characterizing time dependence of static friction. tL is time delay accounting forthe desired frictional memory. _xs is the characteristic velocity of the Stribeck friction.

It is noted that the inherent characteristics of the velocity dependence of friction actually make itdifficult to observe this phenomenon accurately in experiment. The friction is typically investigatedvia steady-state sliding experiments in friction tester whose response is an elastic rather than viscousto sudden displacement input. A coefficient of friction of negative slope could eliminate thepossibility of steady-state sliding as the dynamics of the tester consist of a negative damping andresults in unstable oscillations. In other words, the apparent phenomenon of a negative slope offriction vs. speed in a steady-state sliding experiment could be the result of the interactions betweenthe system dynamics and the complicated constitutive relationship for friction. As we discussedin the last chapter, the oscillation of the system could reduce interface friction, and systemoscillation due to asperity effect is usually proportional to sliding speed; thus the friction speeddependence could be the result of interaction. Because of these types of considerations, rate andstate friction models have been developed to simulate friction dynamics [20,21]. In these models,the dependence of friction on the relative velocity between the two bodies in contact is modeledusing a differential equation. These models include the presliding displacement. The state variablefriction models have been developed to qualify dynamic friction behavior, including Stribeckfriction, rising static friction, and frictional memory.

A widely used nonlinear model characterizing hysteretic behavior in mechanical systems isthe Bouc–Wen model [22–27]. The model was developed to predict various hardening or softening,and smoothly varying or nearly bilinear hysteretic behavior. This model was extended to charac-terize the hysteretic behavior as shown in Figure 4.1. The model is restricted to rate-independenthysteretic systems in which velocity-dependent damping effects are nearly negligible. The modelhas also been extended to combine with the Coulomb friction and velocity-dependent dampingin order to deal with the rate-dependent damping behavior of some systems like wire cable

F

X

Fc

−Fc

FIGURE 4.1 Friction force as a function of displacement.



isolator. In the Bouc–Wen model, the interfacial force is assumed to consist of a linear part anda nonlinear part

F(t) ¼ kx(t)þ knz(t) (4:11)

in which the friction force is denoted by F(t), the excitation displacement is denoted by x(t), and k isthe linear spring stiffness. The nonlinear force is characterized by z(t). The constant kn is used toscale the nonlinear force, so that here z(t) has the dimension of displacement. The Bouc–Wen modelis represented by the following form:

_z(t) ¼ a _x(t)� b _x(t)j j z(t)j jn�1z(t)� g _x(t) z(t)j jn (4:12)

where the dot denotes the time derivative, and a, b, g, n are the model parameters to be determined.The parameters a, b, g and the exponent n can be estimated by parameter identification techniquesbased on experimental data.

In solid mechanics, various models have been developed to characterize hysteresis. One of suchmodels is the Dahl model [28], which uses the stress–strain curve in classic solid mechanics as a startingpoint for the friction. The stress–strain curve is modeled using the following differential equation:

dF

dx¼ s 1� F

Fcsgn(vr)

� i

(4:13)

wherex is the displacementF is the friction forceFc is the Coulomb friction forces is the contact stiffness coefficientvr is the slip speedthe exponent i determines the shape of the stress–strain curve

Then the following friction model is introduced by including presliding displacement andincorporating tangential compliance:

F ¼ sz, s > 0; _z ¼ _x 1� s

Fcsgn( _x)z

� �i(4:14)

where z specifies the state of strain in the frictional contact. Friction force is determined for alltrajectories. The integer exponent i was used to characterize the transition rate of z so as to attain anoptimal experimental fit. For simplification the exponent i is usually assumed to be 1. The steady-state version of the above model gives Coulomb friction. Dahl’s model is a comparatively simplemodel capturing many phenomena like hysteresis, but it does not cover the Stribeck effect becausethe model assumes friction depending only on displacement.

LuGre model extended the Dahl model to include the Stribeck effect [29–32]. The LuGre modelcaptures many aspects of friction such as stick–slip motion. This model with continuous states isinterpreted as an example of Prandlt’s elasto-plastic material model. The LuGre model has beenused in many complicated engineering problems like tire friction modeling. Moreover, some elasto-plastic models have been proposed to overcome the drawbacks of LuGre model which exhibits driftfor arbitrarily small external force [30]. Schematically, it uses a so-called ‘‘bristle’’ assumption tointerpret friction [32]. Assume two proximity contact surfaces with contacting asperities extendingfrom each. The asperities can be represented by small bristles. The friction between the twosurfaces is assumed to be caused by many engaged bristles. The portion of friction contributed by



each bristle is proportional to the strain of the bristle. When the strain exceeds a certain level, thenthe bristle engagement or bond is broken. As such, bristle behaviors, like stiff spring with damper,each cause microscopic displacements and restore forces. If the displacement becomes too large, theengaged junctions (stick) break. When this breakaway occurs, macroscopic sliding (slip) starts. Thefriction is thus modeled as the average deflection of the bristles. When a tangential force is applied,the bristles deflect like springs-dashpot. Denoting by z the average bristle deflection as the internalstate, the model is given by

F ¼ s0zþ s1 _zþ s2vr, si > 0, i ¼ 0, 1, 2 (4:15)

in which s0, s1, s2 are Coulomb, damping, and viscous friction parameters, respectively, for thetangential compliance, and _z is given by

_z ¼ vr 1� s0

yss(vr)sgn(vr)z

� i

(4:16)

wheres0 can be considered as the stiffness of the bristless1 is the dampingyss (vr) is auxiliary function

One more extension of the models allowing a purely elastic regime is given by the followingformulations [19,30]:

F(z, _z, vr,w) ¼ s0zþ s1 _zþ s2vr þ s3w, si > 0, i ¼ 0, 1, 2, 3 (4:17)

in which

_z(z,vr) ¼ vr 1� a(z,vr)s0 sgn(vr)z

yss(vr)

� �i(4:18)

where a(z,vr) is an adhesion map which controls the rate of change of z in order to avoid drift.The model defines the averaged bristle behavior as a first-order system, with z and _z being

interpreted as the mean bristle displacement and velocity, respectively. vr is the relative velocitybetween the two bodies in contact. The model suggests that the friction force F comes from fourcomponents: an elastic term, an internal dissipation term, a viscosity term, and a noise term. Thefourth component term w(t) is assumed to be a pseudo random function of time series representingthe random effects or uncertainty of interface like surface roughness. The auxiliary functions yss (vr)and a are defined as [30]

yss(vr) ¼ fc þ ( fs � fc)e�(vr=vs)2

h i(4:19)

a(vr,z) ¼0, zj j < zba

am(vr,z), zba < zj j < zmax(vr) if sgn(vr) ¼ sgn(z)1, zj j > zmax(vr)0, if sgn(vr) 6¼ sgn(z)

8>><>>:

(4:20)

in which the function am(vr,z) is parameterized as

am(vr, z) ¼ 12

1þ sin pz� (zmax(vr)þ zba)=2

zmax(vr)� zba

� � �(4:21)



which describes the transition between elastic and plastic behavior. The parameter zba defines thepoint where a starts to take nonzero values, and is called a breakaway displacement.

In general, for the constant normal stress cases, the generalized friction model has the followingform [13–16]:

Ff (t) ¼ f (V , x1, x2, . . . , xn) (4:22)

_xi ¼ gi(V , x1, x2, . . . , xn), i ¼ 1, 2, . . . , n

in which xi is state variable. This model suggests that a sudden velocity change is unable to create asudden change in the state, but changes its time derivative. The state variables can be found in manyphysical interpretations in applications.

The above models consider only the interfacial tangential direction force and deformation. Odenand Marins introduced a constitutive model for frictional interfaces that represent both the nonlinearnormal compliance and sliding resistance of the interface [33,34]. The constitutive equations consistof two parts: a normal interface law and a friction law. For the normal response, the normalcompliance of the interface is modeled as

sN ¼ cNamN þ bNa

IN _a for a � 0 (4:23)

sN ¼ 0 for a < 0 (4:24)

wheresN is the normal stressa is the normal approach on the interface_a is its time derivative

The coefficients cN, bN, mN, IN depend on the properties of the contacting surfaces and thematerials of the two components. The second constitutive equation gives friction law.

When a < 0, then sT ¼ 0 (4:25)

When a � 0, then jsT j � cTamT

And

sTj j < cTamT , then _d ¼ 0sTj j ¼ cTamT , then _d ¼ �lsT (l � 0)

�(4:26)

where _d is the sliding velocity calculated as the time derivative of sliding distance, and the indexT indicates a direction tangential to the contact surface. The friction force is a function of the normalapproach of the two surfaces, which in turn depends on the normal force.

For a given surface profile and response curves, the parameters in Oden–Martins law can bedetermined and the static coefficient of friction can be estimated. The friction law was also modifiedto model velocity-dependence, in which the coefficient of friction is given as a function of therelative tangential velocity at the interface

m ¼ sT

sN

�� ¼ m(vT ) (4:27)

A further extension of the law is that the coefficient of friction depends on the normal load. The loaddependence is often weak for hard materials, but may be appreciable for soft materials such asrubbers and polymers. The rubber friction usually has two contributions commonly described



as adhesion and hysteric. The hysteric component results from the internal friction of the rubbercharacterized by material properties. For instance, the following is a simplified friction model ofrubber to characterize the relationship between coefficient of friction and interface contact pressure:

m ¼ ½4b1c(p=E)n þ b2(t0=H)� tan d (4:28)

wherep is interface contact pressureE is Young’s modulusb1 and n are constants for various asperity shapesc is asperity density factorb2 is proportional constantt0 is shear stressH is hardnesstan d is value characterizing damping

In engineering applications, data fitting–based empirical models have been used to quantifyfriction as the function of operation parameters like velocity, pressure, temperature, etc. If the involveddatabase is very big, and the engagement data are directly gathered or indirectly derived from a varietyof sources, the data can be simply treated as a large look-up table of friction under various conditions.They can be reduced to a compact mathematical representation through a system identificationtechnique or using a typical static model. Static model is most commonly used, in which friction ismeasured as a function of applied pressure, slip speed, temperature, and time.More process parameter,denoted as L, as additional variable may be added. Mathematically, the friction can be written as

F ¼ function(p,V , T , L, t) (4:29)

In practice, an empirical model is often coupled with an algebraic model.Based upon certain system identification techniques, assume a certain functional form f1, . . . , fk

with a set of unknown parameters, ai1, . . . , ain, i¼ 1, . . . , k. Then we can express friction as

F ¼Xk

fi(p,V , T , L, t ai1j , . . . , ain)

It can be fitted to the measured data and solved for a friction coefficient set. Many techniques areavailable to prescribe this functional form and determine its empirical parameters.

A comparatively new technique for system identification of friction model is based on anartificial neural net. The artificial neural net is a tool for general pattern recognition. It empiricallyidentifies input–output relationships of an unknown system based upon available data. The neuralnet is especially applicable for a nonlinear system-identification or used as a system-modeling toolwhen the physical nature of a target system is not well understood. Next we briefly illustrate basicneural net structure and operations for the friction component modeling. The artificial neural net canbe considered as a ‘‘black box’’ that accurately recreates system friction behaviors even if the systemis indescribable with first principles. A two-layer feedforward neural net has the prescribedfunctional form as follows [35]:

yk ¼ f2Xj

Wjk f1Xi

Wijxi

!" #( )(4:30)

wherei¼ index for input variablesj¼ index for first layer neuronsk¼ index for output variables



Neural net weights (Wij,Wjk) represent unknown model parameters. While a set of data flowsthrough the net from left to right, a connector multiplies data with a weight. Then, a node carries outsummation

�P�and a simple linear or nonlinear operation (f1, f2) such as sigmoid transformation. It

is mathematically proven that this simple two-layer neural net is capable of capturing input to outputrelationships of any continuous functions when connector weights are suitably selected. Thus, a keyto construct a successful neural net friction model is to optimize a set of weights using the data froma target system. This process is often called neural net training and is analogous to determiningcoefficients in a conventional regression or system identification technique.

4.2.2 SELF-EXCITED VIBRATIONS AND FORCED VIBRATIONS

Many friction vibration systems have period responses that consist of a fundamental frequency andits harmonics. The vibration systems with friction are usually nonlinear, or even strong nonlinearsystems. The presence of harmonic frequencies to some extent indicates the nonlinear behavior ofthe system, which is actually due to the presence of friction. The periodic response of friction systemexhibits as limit cycle in phase space, or a trajectory on the surface of a three-dimension space, orhigher-dimensional torus.

In single-degree-of-freedom systems with friction, the fundamental frequency of the responsecould occur at a frequency lower than the undamped natural frequency of the system. Its magnitudeis dependent on the strength of the nonlinearity in the system equation. If the nonlinearity is weak,as compared with the linear system, the fundamental frequency is close to the natural frequency ofthe system. If the nonlinearity is significant, the fundamental frequency decreases far away fromnatural frequency, with the increase of the harmonic content of the response spectrum.

To give a basic picture on the friction-induced vibration, we start by considering one-degree-of-freedom system with friction. We will consider the free vibrations, the self-excited vibrations due tosliding friction motion, and the vibrations due to external force excitation. The schematic of thesystem model is plotted in Figure 4.2. We will consider three typical friction laws: Coulombfriction, the stiction friction, and the Stribeck friction. Next we use the numerical cases given byHinrichs et al. [36–38] for the illustration, but the observation does not lose the generality. Theprofiles of friction laws are shown in Figure 4.3.

Consider the SDOF with sliding friction and external excitation in Figure 4.2; assume mass m,spring constant k, displacement of mass x, excitation F(t)¼F0 cosvt¼ ku0 cosvt, dampingconstant c, normal force FN, and base velocity V. The friction force Ff follows the three kinds offriction laws I–III as shown in Figure 4.3.

The equation of motion is given by

m€xþ c _xþ kxþ Ff ( _x� V) ¼ F0 cosvt (4:31)

mk

x(t)

V (t)

c

FN

F0 cosw t

m

FIGURE 4.2 SDOF with sliding friction and external excitation.



Using the normalized time t¼vnt, damping ratio j ¼ c=2ffiffiffiffiffiffikm

p, natural frequency vn ¼

ffiffiffiffiffiffiffiffiffik=m

p, and

frequency ratio r¼v=vn, vr¼vn _x � V, the system equation can be transformed as

€xþ 2j _xþ xþ Ff (vr)=k ¼ uo cos rt (4:32)

In Chapter 2 we have briefed the free vibration and forced vibrations of an SDOF system withfriction. Here we add the motion constraint with sliding friction that can result in self-excitedvibration due to more complicated friction interaction. In the following sections we present the freevibration, self-excited vibration, and forced vibrations, respectively.

4.2.2.1 Free Vibrations

At first, we discuss the system behavior without self-excited and external excitation by assumingV¼ 0, F0¼ 0. The phase portraits of the free vibration system for different initial conditions underfriction laws I–III are plotted in Figure 4.4. It can be seen that with increasing time a point on thetrajectories in the phase plane travels to the right in the upper half plane ( _x > 0) and to the left in thelower half plane ( _x < 0). After a short time the mass sticks on the ground in the region highlightedby the thick line. For frictions II and III there exists a dead zone that is caused by the decrease of thefriction characteristics.

4.2.2.2 Self-Excited Vibrations

Next we consider the case of self-excited vibrations by assuming V 6¼ 0, F0¼ 0. Figure 4.5 shows thephase portraits for the friction laws I–III. The stick–slip could take place is some ranges. For laws I andII, the slip mode solution is represented by arcs of circles corresponding to the solution of the linearequation. Due to the friction decreasing characteristic for friction law III, the self-excited vibration

0

0.1

0.2

0.3

0.4m m m

0 1 2 3 0 1 2 3 0 1 2 30

0.1

0.2

0.3

0.4

0

0.1

0.2

0.3

0.4

nrnrnr

FIGURE 4.3 The friction laws: I: ms¼mk¼ 0.25; II: ms¼ 0.4, mk¼ 0.25; III: m(vr) ¼ 0:3=(1þ 1:42 vrj j)þ0:1þ 0:01v2r .

I

x•

x

II

x•

III

x•

x x

FIGURE 4.4 Phase portraits for free vibrations of the friction oscillator. (Reprinted with permission fromHinrichs, N., Oestreich, M., and Popp, K., J. Sound and Vib., 105, 2, 324, 1998. Copyright Elsevier, 1998.)



instability could occur and there exists a limit cycle. As a typical feature of systems with friction thetransients for the three characteristics decay very fast if the initial values lie outside the limit cycles.

In general, friction oscillators with pure self-excitation result in robust limit cycles. The energyis transferred from the moving surfaces to the oscillator by friction forces with a decreasingcharacteristic. In the later sections we will discuss the case when energy is transferred from themoving surfaces to the oscillator by fluctuating normal forces under constant friction coefficient.

4.2.2.3 Forced Vibrations

Consider the harmonically excited linear oscillator with friction, V¼ 0, F0 6¼ 0. For this case, thesystem exhibits nonsmooth friction characteristics due to possible velocity reversal. Many nonsmoothphysical phenomena such as dry friction, impact, and backlash in mechanical systems can be quantifiedby the similar mathematical models with certain discontinuity. A nonsmooth system is characterized byforce and=or motion characteristics which are not continuous or nondifferentiable. In engineeringapplications, it needs to know whether periodic solutions exist for a certain parameter region and howthese periodic solutions change for varying parameters of the system. Such parameter studies are usuallyconducted by means of path-following techniques where a branch of fixed points or periodic solutionsare followed while varying a parameter. A branch of fixed points or periodic solutions can fold or cansplit into other branches at critical parameter values. This qualitative change in the structural behaviorof the system is called ‘‘bifurcation.’’ It is shown that a bifurcation in nonsmooth continuoussystems can be discontinuous in the sense that an eigenvalue jumps over the imaginary axisunder the variation of a parameter. These kinds of systems exhibit the so-called Hopf bifurcation.

For the forced vibration case, the oscillator could exhibit qualitatively different types ofmotions. For small amplitudes excitation, the response displacements in some range hold stableequilibrium positions. For larger values of the excitation amplitude, the typical responses are shownin Figure 4.6. The friction oscillator could have motions without stick, with one stick, or two sticks,or four sticks, respectively, for different frequency ratio r. The frequency ratio serves as bifurcationparameter here. The stick here means _x(t)¼ 0 for a finite time. The qualitative change of the systembehavior for small changes of one system parameter can be summarized by means of a bifurcationdiagram as shown in Figure 4.7. This plot shows the displacement x of the mass at the stick as afunction of the frequency ratio r. Obviously for r larger than some value, no regions of sticking existin the range investigated. The number of sticks increases with decreasing r.

4.2.2.4 Combined Self-Excited Vibration and Forced Vibration V 6¼ 0, F0 6¼ 0

Next we consider the case of combined self-excited vibration and forced vibration. The systembehaviors are solved for the phase trajectories for a given set of bifurcation parameters F0=k, FN=k, r.Figure 4.8 shows a typical type of phase plane plots for oscillator with friction law III. It shows that

I

x•

x

II

x•

x

III

x•

x

FIGURE 4.5 Phase portrait for self-excited vibrations of the friction oscillator. (Reprinted with permission fromHinrichs, N., Oestreich, M., and Popp, K., J. Sound and Vib., 105, 2, 324, 1998. Copyright Elsevier, 1998.)



the system could have one periodic solution, higher periodic solutions, and even chaotic solutionsfor different parameter conditions.

The Poincaré section contains a large number of discrete points of velocity plotted as a functionof displacement of the chaotic motion where the points are sampled stroboscopically with referenceto a particular phase angle of the forcing periodic function. Rather than a random scatter of points,the Poincaré section generally reveals striking patterns. The Poincaré section is sometimes referredto as an attractor. Chaotic vibration also differs from random motion in that the power frequencyspectrum generally has distinct peaks rather than consisting of broadband noise. There will often benot only synchronous response peaks at the forcing function frequency as in the response of linearsystems, but there will also be a significant asynchronous response peak or peaks at the system’snatural frequency or frequencies.

5

2.5

0

�2.5

�50 0 20 40 60 8015 30

(c) (d)45

t t60 75

x•x•

x•x•

5

2.5

0

�2.5

�5

�15

�10

�5

0

5

10

15

0 10 20 30 40 50t

�10

�5

0

5

10

0 10 20 30 40 50t(a) (b)

FIGURE 4.6 Forced vibrations of oscillator with friction law II: FN=k¼ 10, u0¼ 8. (a) r¼ 0.75; (b) r¼ 0.5;(c) r¼ 0.25; (d) r¼ 0.2. (Reprinted with permission from Hinrichs, N., Oestreich, M., and Popp, K., J. Soundand Vib., 105, 2, 324, 1998. Copyright Elsevier, 1998.)

0

−10

0

10

x

r0.1 0.2 0.3 0.4 0.5 0.6 0.7

FIGURE 4.7 Bifurcation diagram of the oscillator with friction law II: FN=k¼ 10, u0¼ 8. (Reprinted withpermission from Hinrichs, N., Oestreich, M., and Popp, K., J. Sound and Vib., 105, 2, 324, 1998. CopyrightElsevier, 1998.)



For a more global bifurcation diagram showing transition point from stick to slip, the displace-ment x is plotted as a function of the bifurcation parameter r in Figure 4.9 for friction laws II and III.

In order to give an overview of the system behavior depending on two bifurcation parameters,parameter maps were calculated as shown in Figure 4.10. The periodicity of the solutions is visualizedby a grey color code. In the parameter map, the regions of one periodic orbit are marked in light grey,the regions of five or higher periodic orbits including chaos are represented in black. For each set ofbifurcation parameters FN=k, r with u0¼ 0.5, the corresponding system behavior can be determined.

Comparing the results gained for the different friction characteristics shows that the globalbifurcational behavior is similar. For small values of r the limit curves of the lowest two light greyregions are nearly identical. However, for high values of r and FN=k, the portion of high periodicmotions disappear for friction characteristic II.

The one-degree-of-freedom self-excited friction oscillator shows a robust limit cycle. The robustlimit cycle of stick–slip vibrations can be broken up by an external harmonic excitation. Theresulting system behavior can exhibit rich bifurcational behavior and also chaos. For chaotictrajectories of the stick–slip oscillations the energy could have a broadband spectrum.

Next we give more analytical details to the self-excited vibrations and the combined self-excitedvibrations and forced vibrations.

x•

x•

x

xxx

(a) (b) (c)

•

FIGURE 4.8 Phase portraits of oscillator with friction law III: (a) r¼ 0.9, FN=k¼ 10, u0¼ 0.5; (b) r¼ 1.15,FN=k¼ 10, u0¼ 0.5; (c) r¼ 1.915, FN=k¼ 10, u0¼ 1. (Reprinted with permission from Hinrichs, N., Oestreich,M., and Popp, K., J. Sound and Vib., 105, 2, 324, 1998. Copyright Elsevier, 1998.)

r

r

x

(b)

(a)

x

2

2

11 2 3 4 5

10

0

1 2 3 4 5

FIGURE 4.9 Bifurcation diagram of oscillators with friction laws II and III: FN=k¼ 10, u0¼ 0.5. (a) Frictionlaw II; (b) friction law III. (Reprinted with permission from Hinrichs, N., Oestreich, M., and Popp, K., J. Soundand Vib., 105, 2, 324, 1998. Copyright Elsevier, 1998.)



4.2.2.5 Closed Form Solution for Self-Excited Vibrations

For illustration purpose, we use the simplified case worked out by Thomsen and Fidlin [39,40].Consider Equation 4.31 with friction law as shown in Figure 4.11; assume F0¼ 0, the nondimen-sional form equation can be written as

€xþ 2j _xþ xþ m( _x� v0) ¼ 0 (4:33)

Periods

25F

N /

k

20

15

10

5

25

FN

/ k

20

15

10

5

0.0 0.5(a)

(b)

1.0 1.5 2.0 2.5

r

r

3.0 3.5 4.0 4.5 5.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

1 2 3 4 5

FIGURE 4.10 Bifurcation maps of oscillators with friction law II (a) and III (b): u0¼ 0.5. (Reprinted withpermission from Hinrichs, N., Oestreich, M., and Popp, K., J. Sound and Vib., 105, 2, 324, 1998. CopyrightElsevier, 1998.)

m

ms

mm

nm nr

FIGURE 4.11 Friction vs. velocity curve.



in which t¼vnt, vn ¼ffiffiffiffiffiffiffiffiffik=m

p, x¼ x=L, L¼FN=k, v0¼V=vnL. For convenience, the following

equation is used to approximate the friction law:

m(vr) ¼ ms sgn(vr)� a1vr þ a3v3r (4:34)

in which

vr ¼ _x� v0, a1 ¼ 3(ms � mm)=2vm, a3 ¼ (ms � mm)=2v3m

wherems is the coefficient of static frictionvm is the velocity point corresponding to the minimum coefficient of kinetic friction

For convenience, Equation 4.34 is transferred to the following one with the origin near theequilibrium, by using the following transformation with variable u(t):

u(t) ¼ x(t)� ms þ a1v0 � a3v30 (4:35)

It is found that the features of the stationary motion solution depend on the excitation speed ranges.In some regions, the solution exhibits stick–slip, whereas in some other regions the system has pure-slip motion. The analysis shows that the respective regions are determined by the characteristicvelocities v00 and v01

v00 ¼ffiffiffiffiffiffiffiffi4=5

pv01, v01 ¼ vm

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4jvm=3(ms � mm)

p(4:36)

The following are the summarizations of the solved vibration amplitudes and frequencies for thecase of relative friction difference h¼ (ms � mm)=vm in proper small levels.

1. For excitation speed in the range v0 < v00, the system has stick–slip oscillation solution.The stationary response displacement and velocity amplitude A0 and Av0 are, respectively,given by

A0 ¼ (1� pj)v0 þ 3pv04vm

(ms � mm) 1� 54

v0vm

� 2" #

(4:37)

Av0 ¼ ½v0 � _u(tm)�=2 (4:38)

The fundamental frequency vss is given by

vss ¼ 2pv0v0ts þ u(0)� u(ts)

(4:39)

It is slightly less than the linear natural frequency of the system.

2. For excitation speed in the range of v00 < v0 < v01, the system has pure-slip oscillations,with stationary displacement amplitude A1 given by

A1 ¼ 2vm

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� (v0=vm)

2 � 4jvm3(ms � mm)

s¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv201 � v20

q, v00 < v0 < v01 (4:40)

The fundamental frequency vps¼ 1, which is equal to the natural frequency of the originalsystem. For special system that has parameter v00¼ v01, this speed region vanishes and



therefore the pure-slip oscillations cannot occur at all. This could happen when the viscousdamping is sufficiently large, or the friction forces or difference between static and kineticfriction is sufficiently small.

3. For excitation speed, v0> v01, the system has static equilibrium at position x(t)¼ms� a1v0þa3v0

3. This corresponds to a state of steady sliding of the mass.

Figure 4.12a shows the variation of displacement amplitude with excitation speed for typicalparameters. Here A0 indicates the amplitude during stick–slip oscillations for v0< v00 and A1 isthe amplitude for pure-slip oscillations for v0 in the range from v00 to v01. When the excitation speedis increased from zero, stick–slip oscillations occur with increasing amplitude until v0¼ v00 wherepure-slip oscillations take over. These occur only in a narrow range of sliding velocities between v00and v01. Above v01, the oscillations cease and steady slip becomes the stable type of motion.

Figure 4.12b depicts the variation of fundamental frequency vss as a function of excitationspeed. It shows a slight drop in frequency for the lower velocities. Figure 4.13 illustrates thedisplacement amplitude of stick–slip and pure-slip oscillation as a function of excitation speed fordifferent levels of relative friction difference. It suggests that the oscillation amplitude increaseswith both the slip velocity increase and the increase of the level of friction difference. For thesmaller values of friction difference, the relation between stick–slip amplitudes and excitation speedis very close to being linear.

In the above discussed system, for the case of large levels of friction difference, higher-orderapproximation should be used to improve the accuracy. For severe stick–slip, the spring can be

0

0.5

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6

0 0.1 0.2 0.3 0.4 0.5 0.6

A0,

A1

(a)

(b)n0

n0n00 n01

wss

A0A1

FIGURE 4.12 Amplitude and frequency of periodic motions as a function of excitation velocity amplitude Aand base frequency v of stable periodic motions as a function of excitation speed v0.



compressed thereby acting toward the right direction, a reversal of the relative velocity and frictioncan take place. The velocity reversal could occur during stick–slip if the static friction force isseveral times greater than the kinetic friction force subject to friction law. The condition for velocityreversal during the slip phase was given in [42–44], which is independent of the friction model.Assume the moment of ts corresponds to the stick–slip transition from stiction in a stick–slip cycle;the moment for velocity reversal is tr, which is when the relative velocity equals zero for the firsttime during the slip phase, and the damping effect is ignored.

The velocity reversal occurs at tr if the spring force at the moment is larger than the instantan-eous friction force, jkx(tr)j > Ff (tr), or

F2f (ts)

2k>

ðtrts

Ff (vr)vr dt

��þ

F2f (tr)

2k(4:41)

in which Ff (ts)¼ kx(ts) is static friction force.We can infer approximately that the velocity reversal could occur during the slip phase of a

stick–slip cycle if the static friction is sufficiently larger than the friction during the slip phase andthe friction at the moment of velocity reversal.

4.2.2.6 Closed Form Solution for Combined Self-Excited Vibration and ForcedVibration: Dither Effect

Next we consider further the combined self-excited vibration and forced vibration. If there is aperiodic external excitation applying on the system with self-excitation, Equation 4.33 becomes

€xþ 2j _xþ xþ m( _x� v0) ¼ u0 cos rt (4:42)

For this system, the phenomenon of frequency entrainment could occur. If the frequency ratio r, theration of the frequency of external excitation to that of the free oscillation, is sufficiently far awayfrom unity, there is usually the beat phenomenon of the two frequencies. If r approaches sufficientlynear to unity, the beat phenomena disappear suddenly and there remains only one frequency as if thefrequency of the autoperiodic oscillation has been entrained by the external frequency. Theentrainment of frequency may also occur, when the frequency ratio r is in the proximity of aninteger (subharmonics) or a fraction (harmonics). Under these conditions, the frequency of freeoscillation is entrained by a frequency which is an integral multiple or submultiple of the externalexcitation frequency.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6

A0,

A1

n0

h = 0.15

h = 0.3

h = 0.6

FIGURE 4.13 The displacement amplitude as function of excitation speed for different levels of relativefriction difference, h¼ (ms � mm)=vm, j¼ 0.05, vm¼ 0.5, ms¼ 0.4. (Courtesy of Thomsen, J.J. and Fidlin, A.)



Next we give more detailed analysis on an important case in which the excitation frequency ofexternal excitation is greater than system natural frequency, and the amplitude of external excitationis very small. This type of excitation is called dithering. We have known that the vibration systemwith negative slope of friction–velocity and=or discontinuity could lead to instability within limitcycles. Next we will demonstrate that the adding of dithering can eliminate the instability in self-excited systems. The dithering effect can change the original friction effect in the system. Theeffective friction due to dithering may lose the feature of negative slope of friction vs. speed, and thefeature of friction discontinuity. The averaging effect due to dithering causes the effective friction,which could eliminate the self-excitation and the stick–slip motion.

We consider the following equation to illustrate the dithering effect, which was solved byThomsen [40]. It is noted that this equation has some difference from the preceding model

€xþ 2j _xþ xþ g2m( _x� v0) ¼ ar2 cos rt (4:43)

in which g is a constant dependent on normal load. The amplitude of excitation is driving frequencydependent, which represents some applications like the machine with rotating unbalanced massat certain eccentricity. Assume the excitation frequency is much higher than system naturalfrequency, r� 1. We will illustrate how this high-frequency excitation affects the existence andthe character of self-excited oscillations occurring at a much slower frequency. Assume the frictionlaw used is the same as Equation 4.34; for convenience, the friction difference is defined ash¼ 3g2(ms�mm)=2vm. In [40], the method of direct partition of motions is used to separate slowand fast components of motions. This produces an autonomous differential equation for the slowmotions, where the fast excitation is accounted for only by its average influence. The slow equationturns out to be in a form similar to Equation 4.33, and so the results obtained in Equations 4.37through 4.40 for that system can be reused here. Assume the total motion of the mass x(t) is splitinto slow and fast components as follows:

x(t) ¼ z(t)þ r�1w(t,rt) (4:44)

wherez describes slow motions at the timescale of free oscillations of the massr�1w is an overlay of fast motions at the much faster rate of the external excitation

Consider t and rt as the slow and the fast timescales, respectively. The slow motion z is thus ofprimary interest, whereas the detail of the fast overlay w is interesting mainly by its effect on z.

Considering equation as a transform of variables, from x to (z,w), one needs to specify anadditional constraint to make the transform unique. For this one requires that the fast-time-averageof the fast motions be zero:

w(t,rt)h i ¼ 12p

ð2p0

w(t,rt) d(rt) ¼ 0 (4:45)

where h�i defines time-averaging over one period of the fast excitation with the slow time tconsidered fixed.

Based on the above two equations, the first-order stationary solution for w is given by

w(t,rt) ¼ �ar sin(rt)þ O(r�1) (4:46)

Hence, based on Equation 4.44 the total solution is

x(t) ¼ z(t)� a sin(rt)þ O(r�2) (4:47)



The governing equation for the slow component of the motion is

€zþ 2j _zþ zþ g2 �m( _z� V) ¼ 0 (4:48a)

in which �m is defined as the effective friction characteristic in the presence of fast excitation

�m(vr) ¼ m(vr � ar cos(rt))h i (4:48b)

Equation 4.48a for the slow motion is similar in form to Equation 4.33 for the total motion, though,the time-dependent excitation is accounted for by the effective friction characteristic �m instead of theordinary m. Equation 4.48a is autonomous, and thus is much easier to solve than the non-autonomous Equation.

For the friction law given in Equation 4.34, the effective friction was derived as

�m(vr) ¼ms 1� 2

parccos vr=(ar)½ �

� þ a1 þ 3

2a3(ar)

2

� �vr þ a3v3r for vrj j � ar

m(vr)þ 32a3(ar)

2vr for vrj j � ar

8><>: (4:49)

The fast excitation effectively changes the friction characteristics. For relative velocity jvrj � ar,the discontinuity at vr¼ 0 is smoothened, which helps effectively cancel the negative slope of thefriction characteristics, and thus prevent self-excited oscillations. Moreover, for jvrj> ar the effectivefriction coefficient is larger than the true friction coefficient. This is illustrated in Figure 4.14.

For Equation 4.43 if a¼ 0, the system becomes unstable and self-excited oscillations occurwhen the slope of m(vr) at v0 is more negative than a threshold value determined by the systemdamping, or m0(v0) þ 2j=g2 < 0; this still holds for a 6¼ 0 with �m0(v0)þ 2j=g2 < 0. From theabove equation it is inferred that for the case of moderate difference between mm andms(mm=ms > 1� ffiffiffi

6p

=p) under high-frequency excitation of intensity ar, self-excited oscillationscannot occur for velocity v0 � ar. There are also a certain range of velocities where self-excitedoscillations can occur, and the velocity range is given as follows:

ar � v0 < vm

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� (ar=vm)

2=2� 2j=hq

(4:50)

Finally, if the following relationship holds, then self-excited vibration cannot exist for any value ofvelocity v0:

ar > vmffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(2� 4j=h)=3

p(4:51)

nr

ar = 0

0.3

0.5m

FIGURE 4.14 Effective friction characteristic as a function of velocity for different ditherings. (Courtesy ofThomsen, J.J.)



The region of velocity with and without self-excited vibrations as a function of excitation frequencyratio for different damping is plotted in Figure 4.15.

In general, dithering is a periodic external excitation with higher frequency compared withsystem resonant frequency. It can be used to alter nonlinear characteristics of system to stabilize andreduce vibrations. The fundamental principle is to smooth nonlinear characteristics by an averagingprocess and other related inherent mechanisms. After proper dithering, an averaging process isapplied and nonlinearity can be reduced. It can be used to suppress instability of nonlinear dampingor nonlinear stiffness systems.

We have introduced Thomsen’s approach [40], which shown above that the effect of high-frequency excitation is to smoothen the discontinuity of the friction function. This gives thepossibility of eliminating the negative slope of this friction and thus to prevent the occurrence ofself-excited oscillations.

It is noted that the dithering excitation in the normal direction has also been demonstrated tohave the effect of eliminating stick–slip. But this could be attributed to an effective decrease in thefriction force through the effective reduction in contact area, thereby causing a lowering of the meancoefficient of friction. This effect has been discussed in Chapter 3.

4.2.2.7 Variable Normal Force

We illustrate next the cases with variable normal force. For convenience, consider the system withcombined self-excitation and force excitation, and the system has a harmonic normal force. Thesystem equation is given by

€xþ 2j _xþ xþ m( _x� v0)(1þ b sin rt) ¼ a sin rt (4:52)

in which the normal force has a harmonic component whose frequency is the same as the frequencyof the external excitation. Here we consider only the situation where the amplitude of dynamicnormal force is always less than the static normal load. More complicated cases of loss of contactwill be discussed in later sections. It is further assumed that the friction–velocity characteristicsremain unchanged during dynamic loading. For small values of b and r, the results are quite similarto the case under purely transverse excitation. Under these conditions, beat frequencies andsubharmonic entrainment could occur. When the load coefficient b is small, the effect of theperiodic coefficient terms would be negligible and Equation 5.52 would behave much in the sameway as in the case of purely combined self-excitation and force excitation.

With higher load ratio, the external excitation would undoubtedly cause changes in thecontact conditions between the sliding surfaces. Under such circumstances, a very different

0

0.1

0.2

0.3

0.4

0.5

0.6

0 10 20 30 40 50

x = 0.05

x = 0

x =0.1

Self-excitedvibration

No self-excitedvibration

r

n 0

FIGURE 4.15 Region of velocity having self-excited vibrations as a function of excitation frequency.



friction–velocity characteristic might be present. In a realistic situation, even small normal periodicexcitations could cause early breakdown of the asperity contact, and there is likely to be micro-movement in the normal direction. This suggests that the assumption of the friction law being thesame for both constant and variable normal force cases is not valid. Some analyses suggest that thisassumption could result in poor agreement between the analytical and experimental values whenthe load ratio is greater than 0.25. It has been experimentally shown that by increasing the loadratio b¼ 0.16 (with r¼ 3) the stick–slip vibration disappeared. It was also observed that the staticfriction value was reduced, due to changes in the interface interaction, as a result of the dynamicnormal loading.

4.2.3 CONTACT VIBRATIONS AND VIBRO-IMPACTS OF SYSTEM WITH FRICTION

When a vibration system with friction experiences severe oscillations under self-excited vibrationsor external excitations, the separation of two surfaces or the loss of contact due to normal vibrationscould occur transiently for some situations. We discuss next the normal vibrations of contactsystems with focus on the dynamics features of vibrations with loss of contact, or the vibro-impact.We will try to use this to illustrate the basic characteristics of the vibration of friction system withloss of contact.

Consider a contact model of one-degree-of-freedom-system sliding on a rough surface shown inFigure 4.16a. The topography of the moving surface is modeled with a harmonic waviness orroughness, zd0¼A sinvt, in which A is the amplitude and v is the frequency. Following [45–48],we assume the contact characteristics are represented by linear contact stiffness kc and the dampingcc. Assume a preload Fl is applied on the mass. Denoting the vertical displacement of the mass as z,the equation of motion of the mass is given by

m€zþ c _zþ kz ¼ �Fl, z > zd0 ¼ A sinvt (4:53a)

m€zþ c _zþ kz ¼ �Fl � kc(z� zd0)� cc( _z� _zd0), z � zd0 ¼ A sinvt (4:53b)

Assuming x¼ zd0 þ x0 � z, Equation 4.53 can be rewritten as

€xþ 2j1v1 _xþ v21x ¼ �A(v) sin(vt þ a), x < x0 (4:54a)

€xþ 2j2v2 _xþ v22xþ (v2

1 � v22)x0 ¼ �A(v) sin(vt þ a), x � x0 (4:54b)

where

v1 ¼ffiffiffiffiffiffiffiffiffik=m

p, v2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(kc þ k)=m

p¼

ffiffiffiffiffiffiffiffiffiffiffikcs=m

p, x0 ¼ �Fl=k,

Disk surface

z

v

Slider

Force

(a) (b)

kcs = k + kc

x0 x

k

c kF1

m

cc

y

kc

FIGURE 4.16 Schematic of (a) contact vibration model and (b) asymmetric piecewise linear restoring force.



�A(v) ¼ Affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(v2

1 � v2)2 þ (2j1v1v)2

q, j1 ¼ c=(2

ffiffiffiffiffiffimk

p), j2 ¼ (cþ cc)= 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimðk þ kc)

p� �, a ¼ tg�1

½2j1v=(v21 � v2)�. By the above transformation, Equation 4.53 is recast as a harmonic force excited,

unsymmetrical, piecewise-linear, spring-damping system in Equation 4.54. The restoring force of thesystem is illustrated in Figure 4.16b. When x0 < 0, it is a contact system whereas when x0 > 0, it is asystem with gap. In both cases, there are trivial motions in which mass may never leave the surface ornever contact the surface. In both cases there are also periodic impactingmotions for sufficient drivingamplitude. The variables affecting the motion are the equivalent excitation amplitude �A(v), the ratioof the exciting angular frequency to the natural angular frequency of the oscillator system v=v1, thedamping coefficients j1,j2, and the offset value x0. However, if excitation is sufficiently strong, thereare also periodic or other complicated vibrations in both cases. The relative amplitude of excitationb ¼ �A(v)=(v2

1x0) is critical for determining the motion behavior.

4.2.3.1 Basic Features

Many theoretical and experimental efforts have been made to characterize the vibrations ofunsymmetrical, piecewise-linear, spring-damping system under excitation. For these kinds ofsystems, periodic motions, resonance, bifurcations, and chaos could occur subject to the combin-ation of different parameters b, v, j. The stability and bifurcation of the system can be determinedby examining the egienvalues of the Jacobian matrix of a Poincaré map involving time of contactand velocity at each contact. The Poincaré section was taken at the location of contact. The approachproved to be exact in the case of infinite contact stiffness (impact) but is only approximate for finitecontact stiffness.

A unique type of bifurcation, the grazing bifurcation, exists for this type of system, where thelimiting case of an impact with zero velocity is given. For some combinations of these parameters,the system response exhibits both flip and fold bifurcation. Chaotic motion can be observedfor finite stiffness ratio. The periodic and chaotic orbit may be determined for certain regions (b,v=v1, j1, j2) in space. A typical region by using b and v=v1 as parameters and with j1¼ 0.05,j2¼ 0 is illustrated in Figure 4.17. The shaded region in Figure 4.17 is the region of stability forharmonic and subharmonic motions (one-impact period-n orbits) and the curves refer to bifurca-tions. The overlap of the band may lead to hysteresis. For small frequencies and excitationamplitudes the motion is stable. As the damping coefficient tends to unity from current value, thebands of stability could become wider. Chaotic motion may exist in some regions; for example, itcould occur around v=vn¼ 4.5. As stiffness ratio kc=k decreases, the regions could shrink. A typical

4

3

2

1

0

−1

−2

−3

−41 2 3 4 5 6 7 8 9

Frequency ratio

Exc

itatio

n ra

tio

FIGURE 4.17 Region of stability.



resonance response and bifurcation diagram is illustrated in Figure 4.18. The resonance peaks occurat v=vn¼ 2, 4, 6, . . . whereas the bifurcation regions are around v=vn¼ 3, 5, 7, . . . .

Usually, for small effective excitation with b� 1, the mass keeps making contact with thesurface because the preload is larger than the threshold of effective excitation to cause anyseparation. With an increase in the amplitude of the effective excitation, the static predeflectionis overcome and the system could exhibit a complicated and even chaotic motion (for b close to 1or b> 1 under some frequency ratios). This means that increased value b will increase the tendencyof the occurrence of the large suspension vibration. Figure 4.19 illustrates that the effectiveexcitation ratio varies with the frequency ratio and the amplitude ratio. The higher dampingcoefficients j1 and j2 allow the system vibrations to have smaller amplitudes under the same b.The critical value of b which causes chaotic motion increases with damping.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency ratio

Res

pons

e am

plitu

de

FIGURE 4.18 Response amplitudes as the function of exciting frequency.

00.5

11.5

2

0

1

2

30

1

2

3

4

5

6

Frequency ratioAmplitude ratio

Exc

itatio

n ra

tio

FIGURE 4.19 Excitation ratio as the function of amplitude and frequency ratio.



In some contact sliding systems, if c and k are not the dominant values determining b thenb � Amv2=Fl. For this case increasing Fl or decreasing A, m, v could suppress the vibrations.

Next we give the closed form approximation solutions for Equation 4.53. Assume the coeffi-cient of restitution being approximated to unity or the contact stiffness approaching infinity. This isreasonable for many applications, like the head–disk impact in hard disk drive system usuallyassuming r¼ 1. Far from resonance solution, near-primary resonance solution, superharmonic andsubharmonic resonance solutions are systematically derived as follows:

1. Free vibration (under unit initial condition)

z ¼ v0

2v1cosv1t þ 1� v0

2v1

� v1

v2cos (v2t þ b0)

þ 1p

X1n¼1

sin na0½cos(v1 þ nv0)t þ cos(v1 � nv0)t�n

� v1

pv2

X1n¼1

sin na0 cos½(v2 þ nv0)t þ b0� þ cos½(v2 � nv0)t þ b0�n

(4:55)

in which

v0 ¼ 2v1v2=(v1 þ v2), a0 ¼ pv0=(2v1) (4:56)

b0 ¼ p(1� v2=v1)=2

2. Far from resonance solutions

z¼ F(v22�v2

1)

pm(v21�v2)(v2

2�v2)þ F(v2

1þv22� 2v2)

2m(v21�v2)(v2

2�v2)sinvt

þ 2F(v21�v2

2)

pm(v21�v2)(v2

2�v2)

X1n¼1

cos2nvt(2nþ 1)(2n� 1)

þ 12½A1 sinv1tþA2 sin(v2tþb0)� þ

A1

p

X1n¼1,3,5

cos(v1� nv)t� cos(v1þ nv)t

n

�A2

p

X1n¼1,3,5

cos½(v2� nv)tþb0� � cos½(v2þ nv)tþb0�n

(4:57)

in which b0 ¼ �pv2=v, A1 ¼ Fv=½mv1(v2 � v2

1)�, A2 ¼ �Fv=½mv2(v2 � v2

2)�

3. Near-primary resonance solutions

z ¼ 2F

(v22 � v2

1)m(2Dþ 1)þ 3F

4v2m(2Dþ 1)2

� �sinvt � pF

2v2m(2Dþ 1)cosvt

þ F

pm(2Dþ 1)v2

X1n¼1,3,5

cos(n� 1)vt � cos(nþ 1)vtn

� F(vt � p)

pm(2Dþ 1)v2

X1n¼1,3,5

sin(n� 1)vt þ sin(nþ 1)vtn (4:58)

where D ¼ (v21 � v2)=(v2

2 � v21)



4. Superharmonic resonance solutions

z ¼ F(2n2v2 � v2 � v21)

m(n2 � 1)2v4sinvt þ 2F(v2

2 � v21)

pm(n2 � 1)2(v21 � n2v2)v2

cosvt

� 2F(v22 � v2

1)

pm(n2 � 1)2(v21 � n2v2)v2

cos nvt, n ¼ 2, 3, 4, . . . (4:59)

5. Subharmonic resonance solutionsFor the case of v2 � v1, it can be derived that the approximate solution is in the

proximity of subharmonic resonant frequency

z¼ 4lFp2mv1(l2�1)(v� lv1)

X1n¼1,3,5

cos½(nþ1)vt=l��cos½(n�1)vt=l�n

, l¼2, 4, 6,. . . (4:60)

4.2.3.2 Frequency Response

The curve in Figure 4.18 exhibits a series of resonant peaks including the one-half and one-third ordersuperharmonic resonances, second-, third-, and fourth-order subharmonic resonances, as well as theprimary resonance. There exist unstable regions in between the resonant peaks.

The excitations causing the mass vibrating motion may come from the roughness, waviness, andrunout of the surface. Its response exhibits a series of peaks including the primary, superharmonic,and subharmonic resonances. The surface excitation can be directly treated as external excitation,but the effective amplitude of the excitation should be replaced by A(v) which is dependent on theamplitude and frequency of surface topography.

For a simplified situation, it is attributed to be a typical case with unsymmetrical stiffness underexternal excitation. The system is then simply characterized by only two parameters—the ratio ofthe stiffness and the linear damping ratio of the system. For these types of systems, chaoticvibrations are characterized by an irregular or ragged waveform such as illustrated by Ehrich[49]. Although there may be recurrent patterns in the waveform, they are not precisely alike, andthey repeat at irregular intervals; so the motion is truly nonperiodic. A remarkable response behaviorassociated with chaotic vibration is the cascade of period-doubling bifurcations or tree-like structurein peak amplitude response curve that may take place in the transition from simple periodic responseto chaotic response. A remarkable property of chaotic vibrations is evident in the Poincaré section ofthe motion, shown typically in Figure 4.20.

Vel

ocity

Displacement

FIGURE 4.20 Poincaré section: subcritical chaotic transition between the first and second successive super-harmonic orders (k1=k2¼ 0.005, j¼ 0.05, v=vn¼ 0.56).



In vibro-impact system, there is a similar phenomenon corresponding to sticking, which isreferred to as dwell, for which the oscillator could appear to be resting after innumerable impacts,then the motion excitation will drive it to motion again. During a dwell, the displacement andvelocity are both constrained; thus the collapse reduces the dimension by half.

We noted that in the previous friction vibration system stick–slip could be modeled in discon-tinuous velocity space, with a collapsed phase space. During a stick, there is a collapse in thedimension of state space in which the velocity is constrained during a stick.

4.2.3.3 Friction System with Variable Normal Force and Vibro-Impact

We discuss further the friction system with variable normal force, and here we no longer limit theamplitude of dynamic normal force as less than the static normal load. This allows the possibilityof the case of loss of contact. To illustrate this case, we use the results given by Feeny and Moon[50,51]:

€xþ 2j _xþ xþ m( _x) f (x) ¼ A cos rt (4:61)

in which m( _x)¼m sgn( _x). For _x 6¼ 0, �1 � m( _x) � 1; for _x¼ 0, m(0) can be any value between1 and �1. Consider the simplified case of no damping j¼ 0 and simple friction law ms¼mk¼ 1.Assume the normal load to be linearly dependent on the sliding displacement; the normal force isgiven by

f (x) ¼ 1þ ax,0,

�x > �1=ax < �1=a

(4:62)

in which f(x)¼ 1 gives constant load. This model allows for the loss of contact state. Equations 4.61and 4.62 can be cast into a piecewise linear equation.

For contact condition

€xþ (1þ a)x ¼ �1þ kuo cos rt for _x > 0, x > �1=a

€xþ (1� a)x ¼ 1þ kuo cos rt for _x < 0, x > �1=a (4:63)

For noncontact condition

€xþ x ¼ kuo cos rt for x < �1=a (4:64)

This piecewise-linear system has both periodic and chaotic solutions. The principal motion couldhave one or two periods, or higher period subharmonics. This system exhibits stick–slip chaos on abranched manifold, which has a single-bumped one-dimensional map.

Figure 4.21 shows a typical phase portrait of a chaotic solution. The stick takes place at thetrajectory cusps in the funnel structure. The system has three state variables: displacement x,velocity _x, and time t.

Since the equation of motion has a discontinuity at _x¼ 0, the plane in space (x, _x,t) defined by_x¼ 0 is a natural place to make a Poincaré section. In this Poincaré section, the bifurcation diagramis shown in Figure 4.22. It includes the trajectories which bounce off the underside ( _x < 0) ofthe (x,t) plane. Some trajectories meet the (x,t) plane from below, stick, and then return to below theplane. They have periodic windows.

Such stick–slip causes a collapse in phase space. For a three-dimensional system, this leadsto one-dimensional map dynamics. This phenomenon occurs approximately for smooth friction laws.



4.3 VIBRATIONS OF MULTI-DEGREE-OF-FREEDOM SYSTEMSWITH FRICTION

4.3.1 NEGATIVE DAMPING INSTABILITY DUE TO NEGATIVE SLOPE

OF FRICTION–VELOCITY CURVE

In single-degree-of-freedom systems with velocity-dependent friction, we had presented that thefundamental frequency of response could be lower than the undamped natural frequency of thesystem. The magnitude and the proximity of the fundamental frequency to the natural frequencydepend on the friction properties. For multi-degree-of-freedom systems with velocity-dependentfriction, there is a similar feature. The fundamental frequency of vibration could be lower and closerto any one of the natural frequencies of the system without friction. The value of the fundamentalfrequency and its proximity to the original natural frequency depend on the dynamical character of

−1

−1

−0.5

0

0.5

1

1.5

2

−0.5 0 0.5Displacement

Vel

ocity

1 1.5

FIGURE 4.21 Phase portrait of a chaotic solution (r¼ 1.25, A¼ 1.9, a¼ 1.5). (Reprinted with permissionfrom Feeny, B., Dynamics with Friction: Modeling, Analysis and Experiment, edited by A. Guran, F. Pfeiffer,and K. Popp, World Scientific Publishing, New Jersey, 31, 1996. Copyright World Scientific Publishing, 1996.)

1.2

0

0.2

0.4

0.6

0.8

1

1.2

1.3 1.4 1.5

Amplitude of input

Am

plitu

de o

f out

put

1.6 1.7 1.8

FIGURE 4.22 Abifurcation diagram shows period doubling to be the route to chaos. (Reprinted with permissionfrom Feeny, B., Dynamics with Friction: Modeling, Analysis and Experiment, edited by A. Guran, F. Pfeiffer,and K. Popp, World Scientific Publishing, New Jersey, 31, 1996. Copyright World Scientific Publishing, 1996.)



the system and the friction properties. This matching process between the fundamental and thenatural frequency of the system is sometimes called ‘‘mode lock-in.’’

Next we discuss multi-degree-of-freedom systems with friction law of velocity-dependence. Westart by analyzing a simplified two-degree-of-freedom model as shown in Figure 4.23. The systemparameters are represented by m1, k1, c1, m2, k2, c2, Ff ¼FNm(vr). The friction law is assumed to beas follows:

m(vr) ¼ ms sgn(vr)� avr (4:65)

where vr¼ v0 þ _x2 � _x1. This system was solved by Shin et al. [52]. In the next, a linear analysis isused to determine the stability of the system equilibrium points. The procedure is as briefed inChapter 2. The features of the points determine the existence or nonexistence of a limit cycle;thereby the linear analysis can be used to deal with stick–slip motion approximately. The results ofnumerical nonlinear analysis are also presented to describe the limit cycle features in phase space.

The equations of motion can be written as

m1€x1 þ c1 _x1 þ k1x1 � FNa( _x1 � _x2) ¼ FN(m� av0)

m2€x2 þ c2 _x2 þ k2x2 � FNa( _x2 � _x1) ¼ �FN(m� av0)(4:66)

For stability analysis, we ignore the static effect term on the right-hand side of the equation. We alsoignore the velocity reversal situation. The characteristic equation becomes

detl2 þ c11lþ k11 c12l

c21l l2 þ c22lþ k22

� �¼ 0 (4:67)

in which c11¼ (c1 � FNa)=m1, c22¼ (c2 � FNa)=m2, c12¼FN=m1, c21¼FN=m2, k11¼ k1=m1, andk22¼ k2=m2.

The above equation can be written in the form of fourth-order polynomial

l4 þ a1l3 þ a2l

2 þ a3lþ a4 ¼ 0 (4:68)

x, V x1

x2

k1

k2

c2

c1

m1

m2

FIGURE 4.23 Two-degree-of-freedom model with friction.



in which

a1 ¼ c11 þ c22 ¼ m2(c1 � FNa)þ m1(c2 � FNa)½ �=m1m2

a2 ¼ c11c12 � c12c21 þ k11 þ k22 ¼ (c1 � FNa)(c2 � FNa1)� (FNa)2 þ m1k2 þ m2k1

� �=m1m2

a3 ¼ k11c22 þ k22c11 ¼ k1(c2 � FNa)þ k2(c1 � FNa)½ �=m1m2

a4 ¼ k11k22 ¼ (k1k2)=m1m2

By using Routh criterion, we have the following condition for instability:

a1 < 0, or a2 < 0, or a3 < 0, or a4 < 0, or a1a2 � a3 < 0

or a1a2a3 � a21a4 � a23 < 0(4:69)

Detailed analysis illustrates that when m1 � m2 and k1 � k2, smaller values of FNa=c1 could makethe system unstable. It implies that when the natural frequencies of two subsystems (m1, k1, c1 andm2, k2, c2, respectively) are same, the system will be less stable for the given conditions. Increasingdamping in either of the two subsystems results in a beneficial effect to stability.

For a stable condition, FNa should never be greater than c1 or c2. Generally, to maximize systemstability, the natural frequencies of two subsystems should be well separated and well damped.

A noticeable stability criterion for the case of sufficient natural frequencies’ separation or largeratio of natural frequencies (like >2) is given by

min(c1, c2) > FNa (4:70)

where min(c1,c2) denotes the minimum value. It is noted that c2=c1 is never smaller than twiceFNa=c1 while maintaining its stability. Thus, it is guaranteed that the system is always stableprovided that

min(c1, c2) > 2FNa (4:71)

The criteria imply that no matter how much damping is added to one of the subsystems, the systemcan still become unstable unless an appropriate level of damping is added to the other side to makeEquation 4.71 hold.

The linear analysis is able to predict the existence of instability. For some cases, the resultingsize of limit cycle may be very small and hence the generated instability could be trivial and isnegligible. Nonlinear analysis can quantify the size of a limit cycle and thus provide the informationof the level of the instability. In the next analysis, we still use the linear function of friction law, butwe consider a general case with negative relative velocity being allowed; thus there could be adiscontinuity at zero relative velocity. This can cause highly nonlinear behavior and produce stick–slip motion. When analyzing this type of nonlinear system, the difficulty arises from the discon-tinuity in the friction force. The motion of the system is governed by the static friction force in thestick motion and by a velocity-dependent friction force in the slip motion. For the stick mode, thestatic friction force is limited by the maximum state friction force, i.e., jFsj � msFN and is balancedwith the reaction forces acting on the masses. Considering the relative motion between the twomasses the static friction force can be written as

Fs ¼ k1x1 þ c1 _x1 � k2x2 � c2 _x2 (4:72)

and the frictional force can be described by

Ff ¼min( Fsj j, msFN)sgn(Fs), vr ¼ 0

m(vr)FN sgn(vr), vr 6¼ 0

�(4:73)



For numerical analysis, the friction force is switched at zero relative velocity appropriately accord-ing to the type of motion, and a small region of the relative velocity is defined as vr < « in which«� v0. The equations of motion for the system are given by

m1€x1 þ c1 _x1 þ k1x1 ¼ Ff (vr)� Ff (v0)

m2€x2 þ c2 _x2 þ k2x2 ¼ �½Ff (vr)� Ff (v0)�(4:74)

in which Ff(v0)¼FN(ms � av0) is introduced to compensate for the offset. For the nonlinearnumerical analysis, the simplest case with m1¼m2, c1¼ c2, k1¼ k2 is considered here. For thiscase, the resulting motions of both masses are the same because the same friction force acts uponboth the masses. Moreover, the dynamics are similar to that of the single-degree-of-freedomsystems described before, and stick–slip limit cycle motion is dominant provided that dampingis sufficiently small. As a numerical example, set m1¼m2¼ k1¼ k2¼ 1, c1¼ c2¼ 0.01, and thestatic friction coefficient is set to ms¼ 0.6. The stick–slip motions for various values of frictionparameter a are shown in Figure 4.24a, in which only mass 1 motion is shown. The mass 2 motionis almost identical. From Figure 4.24a, it can be seen that a steady limit cycle occurs if a is verysmall. This could be because the dynamic friction coefficient is similar to the static frictioncoefficient in this case. As a is increased, however, a stick–slip limit cycle occurs and the sizeof limit cycle increases. This shows that instability will probably become worse when the negativeslope of the friction–velocity curve increases. A similar feature is observed for the case of normalforce FN. The size of limit cycle increases with the increasing values of FN. Finally, the effect ofthe velocity v0 is shown in Figure 4.24b, which shows that the size of limit cycle increases withthe increasing values of v0. The above results further verify the importance of the term FNadescribed in the previous section. Also, it is shown that the input velocity v0 affects the size of thelimit cycle.

It is also found that the size of the limit cycles decreases as the damping of both masses 1 and 2increases simultaneously finally ending up as fixed points when the damping parameters aresufficiently large. However, if the damping in the system is increased only on one side of theinterface, for example in mass 1, the size of limit cycle corresponding to mass 1 decreases, whereasthe limit cycle of mass 2 increases. It is also found that the system does not go to a fixed point nomatter how much damping is added.

1

(a) (b)

0.5

x 2 x 2

0a = 0.0005

n0=0.6n0=1.2

n0=2

n0=3

a = 0.005a = 0.015

a = 0.025

−0.5

−1

−1.5

2

1

0

−1

−2

−3−3 −2 −1 0 1 2−2 −1.5 −1 −0.5 0 0.5 1

FIGURE 4.24 Limit cycle motions for various values of parameters: (a) motions of mass 1 for various valuesof a; (b) motions of the pad for various values of v0. (Courtesy of Shin, K., Breman, M.J., and Oh, J.-E.)



The analysis shows that when the natural frequencies of two subsystems are in close proximitythen a system is more likely to be unstable. It has also been found that the amount and distribution ofdamping in the system is a key factor. The damping of masses 1 and 2 are of equal importance in theprevention of instability. The results have shown that the increasing damping of either mass 1 ormass 2 alone can potentially have detrimental effects on the system stability. In Section 4.3.5, amore detailed numerical case will be presented.

4.3.2 INTERNAL COMBINATION RESONANCE DUE TO VELOCITY-DEPENDENT FRICTION

AND VARIABLE NORMAL FORCE

Next we discuss the effect of normal motion on the vibrations of system with velocity-dependentfriction. For this case, the combined effect of normal motion and velocity-dependent frictionintroduces a new type of friction instability. Consider a system with velocity-dependent friction lawsimilar to Equation 4.65, and the system is excited by a periodic rough surface input. Such a typicalkind of system was solved by Berger et al. [53]. For this kind of system, the coupling of normaloscillation and friction causes a parametric excitation to the equations of motion in transversaldirection. This results in locally unstable oscillations called internal combination resonance. Thecombination of velocity-dependent friction and a harmonically varying normal force is shown toproduce large-amplitude oscillations, in some cases leading to stick–slip responses. In the nextanalysis, after a modal change of variables, first-order averaging is used to convert system equationto a set of autonomous equations of motion. Then the eigenvalue analysis of the averaged equationsis used to give stability predictions for the steady sliding position.

Due to the coupling of the oscillating normal force and the velocity-dependent friction, aparametric damping term arises in the equations of motion in transversal direction. The averagingmethod is used to determine the combination resonances of the stability due to parametric excitation.A stability criterion for the combination resonance is a function of system damping, friction charac-teristics, normal force variations, and frequency. The relationship between forcing frequency andsystem natural frequencies is critical for the criterion. It is found that the unstable oscillations for thecombination resonance are related to the magnitude of the slope of friction–velocity curve and theratio of the exciting frequency to natural frequencies. This is different from the negative damping-typeinstability discussed above, which depends on the negative slope of friction–velocity curve.

Consider a three mass system having a primary mass and two subsystems connected to itindividually. The primary mass has only a transversal degree of freedom. Each mass in the subsystemhas both a normal and a transversal degree of freedom. Each subsystem is connected with the primarymass through a spring and damper in transversal direction. Apart from that, each subsystem hascontact in normal direction with a moving surface which has a periodic profile. The system equationof motion in transversal direction has three degree of freedom and can be represented as

mp 0 0

0 m1 0

0 0 m2

264

375

€xp

€x1

€x2

8><>:

9>=>;þ

cp 0 0

0 c1 0

0 0 c2

264

375

_xp

_x1

_x2

8><>:

9>=>;þ

kp þ k1 þ k2 �k1 �k2

�k1 k1 0

�k2 0 k2

264

375

xp

x1

x2

8><>:

9>=>; ¼

0

ff 1

ff 2

8><>:

9>=>;

(4:75)

in which mp is the primary mass, and m1, m2 are the first and second masses of the two subsystems,respectively. For simplification, we assume that the normal motion of subsystems due to the surfaceroughness profile remains sinusoidal and the steady-state normal motion is given by

yi(t) ¼ Yi cos(vit � fi), i ¼ 1, 2 (4:76)



Then the friction force for the first and second subsystems is given by

ffi ¼ mi(vri) FNi � miYiv2i cos(vit � fi)

� �, i ¼ 1, 2 (4:77)

Here we do not consider the situation of sgn(vri) cross zero and a continuous sliding is assumed tooccur for small oscillations about the steady sliding position. The friction law is assumed to be inlinear form

mi(vri) ¼ m0i � aivri ¼ m0i þ si _xi, i ¼ 1, 2 (4:78)

For convenience, it is assumed that the same friction law applies to both subsystems, but thecontacting velocities of the subsystems are independent. Using the linear friction law, the frictionforce becomes

ffi ¼ Di þ di þ bi cos(vit � fi)þ gi cos(vit � fi) _xi (4:79)

in which Di¼m0FNi, di¼ siFNi, bi¼�Yivi2m0, gi¼�Yivi

2siTo solve Equation 4.75, using modal transformation {x}¼ [F]{z}, the system equations of

motion can be transformed to

I½ � €hf g þ j½ � � �(t)½ � _hf g þ V2n

� �hf g ¼ �bf g (4:80)

in which

hf g ¼ zf g � V2n

� ��1 �D �

, �D � ¼ F½ �T 0,D1,D2f gT (4:81)

�(t)½ � ¼ F½ �T0 0 00 g1 cos(vit � f1) 00 0 g2 cos(v2t � f2)

24

35 F½ � (4:82)

§½ � ¼ F½ �T C½ � þ0 0 00 d1 00 0 d2

24

35

8<:

9=; F½ � (4:83)

f�bg ¼ ½F�T ½0 b1 cos(v1t � f1)b2 cos(v2t � f2)�T

If the slope of the friction curve si is assumed to be of the same order as the damping, it is able toscale the following parameters:

I½ � €hf g þ « �§½ � � ��(t)½ � _hf g þ V2n

� �hf g ¼ �bf g (4:84)

The equation can be solved by using the average method to obtain the resultant averaged equation.Assume

hi(t) ¼ Ai cos(Vit) þ Bi sin(Vit)þX2j¼1

Fij cos(v1t � f1) (4:85)

where Fij ¼Fjibj

V2i � v2

j

We can get a set of four first-order differential equations about _Ai, _Bi, i¼ 1, 2. Consider thespecial case when Vi Vj � vk; this is the so-called near internal combination resonance. Assumevk¼Vi þ Vj þ «s; the averaged equation is given by



_A1

_B1

_A2

_B2

8>>>><>>>>:

9>>>>=>>>>;

¼ «

§112

�s�g12V2C

4(vk �V2)

g12V2S

4(vk �V2)

s§112

g12V2S

4(vk �V2)

g12V2C

4(vk �V2)

�g12(vk �V2)C

4V2

g12(vk �V2)S

4V2

§222

0

�g12(vk �V2)S

4V2

g12(vk �V2)C

4V20

§222

26666666666664

37777777777775

A1

B1

A2

B2

8>>>><>>>>:

9>>>>=>>>>;

(4:86)

in whichC¼ cosfk, S¼�sinfk

The eigenvalues of the coefficient matrix in averaged equation can be obtained

l1,2,3,4 ¼ 14

�(§11 þ §22) 2is ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(§11 � §22)

2 þ �212 � 4s2 þ 4is(§22 � §11)

q� �(4:87)

To keep the system stable, the eigenvalues should have negative real parts. This gives the followingrelationship:

§11 > 0, §22 > 0, �12j j < �c12j j (4:88)

in which the critical value of modal parametric excitation is

�c12 ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffi§11§22

p§11 þ §22

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(§11 þ §22)

2 þ 4s2

q(4:89)

For this case, it is not enough for the system to remain stable with periodic solution, by only keepingthe effective modal damping positive. Increasing the system damping makes the effective dampinglarger. Increasing the static part of the normal force plays two different roles depending on the signof the friction curve slope. For the positive slope, increasing the static normal force has a stabilizingeffect, while for the negative slope, increasing the static normal force has a destabilizing effect.Larger slope magnitudes produce larger parametric excitation amplitudes, but depending on the signof the slope, they produce either more or less friction-related damping.

The internal combination resonance may arise in the multi-degree-of-freedom system, wheretime-varying normal forces and velocity-dependent friction coexist. It is a type of parametricresonance instability. The unstable oscillation is also friction induced because it originates fromthe coupling of the oscillatory normal force and the velocity-dependent friction coefficient. Thisparametrically excited instability does not exist for the case of either constant normal force orconstant friction. The coupling of velocity-dependent friction with time-varying normal force cancause locally unstable oscillations.

4.3.3 MODE-COUPLING INSTABILITY DUE TO PROXIMITY OF MODES OF SUBSYSTEMS

WITH CONSTANT FRICTION

The presence of sufficient negative damping due to velocity-dependent friction leads to the oscillationwith fundamental frequency close to anaturalmodeof the system, or themode lock-in. The coexistenceof velocity-dependent friction and variable normal force could cause local mode instability.

We next discuss another type of friction-induced instability called mode-coupling. This type ofinstability is characterized by the proximity of two natural modes of sliding subsystems, whichmerge when they interact under sufficient friction.



Consider a two-degree-of-freedom system shown in Figure 4.25. This model has sliding surfacewith constant velocity, and is in contact with a normal force against a mass m through a spring k3.The mass is connected with two linear springs k1 and k2. Assume the friction is characterized by aconstant friction coefficient m0. The equations of motion for the mass are

m 00 m

� �€x€y

� þ k11 k12

k21 k22

� �xy

� ¼ f

N

� (4:90)

where m, kij (i,j¼ 1,2), f, and N are the mass coefficient, stiffness coefficient, friction force, andcontact force, respectively. The coefficients of stiffness matrix, friction force, and contact force aregiven by

k11 ¼ k1 cos2 aþ k2 cos

2 b, k12 ¼ k1 sina cosa� k2 sinb cosb

k21 ¼ k12, k22 ¼ k1 sin2 aþ k2 sin

2 b(4:91)

f ¼ m0k3(yþ y0), N ¼ �k3(yþ y0)

By ignoring the static term, we have

m 00 m

� �€x€y

� þ k11 k12 � m0k3

k21 k22 þ k3

� �xy

� ¼ 0

0

� (4:92)

This system has a nonsymmetric stiffness matrix, which corresponds to a complex eigenvalueproblem. Before analyzing the complex eigenvalue problem and the instability of the system, weconsider a numerical example to discuss some related basic features. Assume a¼ 308, b¼ 1508,m¼ 1, k1¼ 1, k2¼ 3, k3¼ 8, and assume that the system has constant coefficient of friction m0 inthe range of 0–2. Figure 4.26 shows the calculated system complex eigenvalues as a functionof coefficient of friction m0. We can see that the system’s two resonant modes merge when thecoefficient of friction reaches 1.4. When the coefficient of friction continues to increase, thesystem starts to have positive real eigenvalue that yields mathematical solution with infiniteamplitude, which suggests that the system turns to instability. This is the mode-coupling typeinstability.

Next we investigate the effect of system parameters on the instability. In Equation 4.92, whenm0k3¼ k12, the transversal motion is uncoupled completely from the normal motion. The equationhas the following fundamental solution: x1¼ x10 cos v0t, y1¼ y10t sin v0t. The first corresponds to a

m x

y

k1

k3

k2

Slippage

a b

FIGURE 4.25 Two-degree-of-freedom system with modal coupling due to friction.



transversal vibration of constant amplitude. The second is a normal vibration whose amplitudeincreases linearly with time. For this case, the transversal motion is completely independent ofnormal oscillation, but the transversal motion works as an external force to the normal motion atresonance. The instability occurs gently with linear growth of amplitude.

When k12 is slightly higher than m0k3, the system has two natural frequencies which are veryclose. This causes a kind of beating type vibration. In contrast to the beating in a conservative two-degree-of-freedom system, the total beating vibration energy is not conserved here. The motionconsists of a periodic energy exchange between transversal and normal motion, and the beatingfrequency is half of the frequency difference of the natural frequencies. The beating phenomenonhere is the existence of phase shifts between transversal and normal motion.

When m0k3 is closer to k12, the beating frequency decreases continuously. When m0k3¼ k12, thebeating tends to be infinite. In the case of m0k3¼ k12, the coupling term in the x-direction vanishes,whereas the coupling term in the y-direction remains the same. In this case, the x-vibrationcontinuously supplies energy to the y-vibration, but the y-vibration does not return energy to thex-vibration. When m0k3 is larger than k12, the system turns unstable and vibration solutions increaseexponentially in time.

We considered only the change of friction coefficient above. Changes in contact stiffness, k3,also have a strong effect on the features of system instability. Figure 4.27 shows the complexeigenvalue of the system as functions of friction coefficient for k3¼ 7 and 9, respectively. Thisillustrates that the contact stiffness has strong effects on the modal coupling. It is noted that changesin contact stiffness could change the threshold of friction coefficient corresponding to the onset ofmodal coupling. The changes in contact stiffness actually change the separation of the natural modesof the system and the fundamental frequency of the coupled mode.

In many engineering applications, the contact stiffness is load-dependent and load is usuallytime-dependent subject to operation. This is the partial reason why the fundamental frequency of theinstability is time-dependent, and the oscillations exhibit nonstationary features. If contact stiffness

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 22

4

6

8

10

COF(a)

(b)

Ima

gin

ary

par

t

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−4

−2

0

2

4

COF

Re

al p

art

FIGURE 4.26 Complex eigenvalue of the system mode as functions of friction coefficient: (a) imaginary part;(b) real part.



is transitional and allows mode-coupling to occur, we could have an instantaneous mode-coupling.The fundamental frequency corresponding to this transient coupled mode is neither equal to theoriginal natural modes of subsystems nor to the coupled modes under constant or averaged load. Inthis sense, the dynamic contact and friction can change the inherent spectral features of thedynamics system.

The instability described is analogous to the binary flutter instability from aeroelasticity. Thesetypes of systems mathematically have motion equation with nonsymmetric matrices, which can bedecomposed into a symmetric and a skew-symmetric part. Sometimes systems with so-callednonconservative displacement-dependent forces or follower forces yield similar system equations.Let us illustrate this by using a two-degree-of-freedom model. As shown in Figure 4.28, assume arigid beam of mass M, thickness 2h, moment of inertia J, length L, and total stiffness k. It has twodegrees of freedom x and y. The friction forces F1 and F2 acting on the beam are

F1 ¼ m(k1yþ N), F2 ¼ m(�k1yþ N) (4:93)

these forces are follower forces and N is the static preload. The equations of motion are of the form

M €X þ K €X ¼ 0 (4:94)

in which

X ¼ yu

� , M ¼ M 0

0 J

� �, K ¼ kt þ k1 2mN

�2mhk1 kr þ k1L2=3

� �

The stiffnesses kt, kr are connected to the beam, k2¼ k1L2=3.

0

(a)

(b)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

COF

Ima

gin

ary

par

t

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−4

−2

0

2

4

COF

Rea

l par

t

FIGURE 4.27 Complex eigenvalue of the system as functions of the friction of coefficient for k3¼ 7 (thin solidline) and k3¼ 9 (thick dotted line): (a) imaginary part; (b) real part.



The equation is stable at equilibrium. The eigenvalues of the equation also consist of complexconjugate purely imaginary pairs. As the parameter of the system varies, the pairs could get coupledand the equilibrium becomes unstable. This is the so-called flutter instability and it is due to theskew-symmetric component of the stiffness matrix. The criterion for instability of the equilibriumwas established as

(kt þ k1)J � (kr þ k1L2=3)M

� �2=MJ � 16m2hk1N (4:95)

4.3.4 COMPLEX MODAL ANALYSIS

We discuss further the complex modal analysis, which has been used as an analytical tool fordealing with the system with mode-coupling instability [56–65]. Consider a multi-degree-of-freedomsystem consisting of two subsystems 1 and 2. They are connected by a contact sliding interface withcoefficient of friction m. The equation of motion of the system can be expressed as

M½ � €uf g þ C½ � _uf g þ K½ � uf g ¼ Ff

�(4:96)

where [M], [C], and [K] are the mass, viscous damping, and stiffness matrices for the nonfrictionsystem, respectively, {u} is the displacement vector, and {Ff} is the friction force vectorbetween the two subsystems. The friction system consists of an interface with contact and connectedin the normal direction, but not in the tangential direction. The tangential friction is modeled asa force

Ff

� ¼ m Nstaticf g þ Ndynamic

�� (4:97)

where {Nstatic}, {Ndynamic} are the static and dynamic normal forces, respectively. For a solution tothe eigenvalue problem, the static force is removed from the equation of motion. The dynamicnormal force is caused by the vibration of subsystems 1 and 2, and is represented by

Ndynamic

� ¼ Ks uN1f g þ uN2f gð Þ (4:98)

y

θkt

kr

k1k2

L F1

F2

2h

FIGURE 4.28 Two-degree-of-freedom model with binary instability.



where{uN1}, {uN2} denote the displacements of the interface in normal directionKs is the local contact stiffness

Hence the system equation of motion becomes

M½ � €uf g þ C½ � _uf g þ K½ � uf g ¼ �mKs Kf

� �uf g (4:99)

The matrix [Kf] is the effective stiffness due to friction in interface. It is a nonsymmetric matrix thatcouples the relative normal displacement with the tangential force. If the system has no damping andfriction, it can be expressed as

M½ � €uf g þ K½ � uf g ¼ 0f g (4:100)

The modal domain transformation can be obtained from the equation as

fug ¼ ½c�fgg

where[c] is the modal matrix of system equation{g} is the modal coordinate vector

With the modal transformation, the following relationship can be established:

c½ �T M½ � c½ � ¼ I½ � (4:101)

c½ �T K½ � c½ � ¼

v21 0 � � � 0

0 v22

..

.

..

. . ..

00 � � � 0 v2

n

266664

377775 (4:102)

where vi is the ith natural frequency of system equation (see Equation 4.100). From (4.101–4.102)and (4.99), the complex eigenvalue equation can be expressed as

s2

1 0 � � � 0

0 1 ...

..

. . ..

0

0 � � � 0 1

2666664

3777775þ s

2jv1 0 � � � 0

0 2jv2...

..

. . ..

0

0 � � � 0 2jvn

2666664

3777775þ

v21 0 � � � 0

0 v22

..

.

..

. . ..

0

0 � � � 0 v2n

26666664

37777775þmKs �f

� �8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

gf g ¼ 0f g

(4:103)

In the earlier sections, we discussed the friction-involved single-degree-of-freedom systems, wherethere are periodic limit-cycle oscillation solutions. Similarly, there are limit-cycle oscillationsolutions in multi-degree-of-freedom systems as well. The linear complex mode analysis can beused to predict the onset of the limit cycle. Although it cannot predict the magnitude and thecharacteristics of the limit cycle, it helps in providing stability margin or threshold.

Usually, the stable propensity can be evaluated by using a two-dimensional plot of complexeigenvalues called root locus plot or stability chart, in which the ordinate represents the frequencyand the abscissa represents the real part of complex eigenvalue. The real part of complex eigenvalueis sometimes used to evaluate the stability propensity. Any point on the left side of the stability chart



is considered as a stable mode as the vibration magnitude corresponding to the point will decay withtime. Any point on the right side of the chart represents an unstable mode as the vibration magnitudeof the point will grow as time progresses.

For the complex modes solved from Equation 4.103, the complex roots can be plotted on a rootlocus diagram. Roots in the right half plane have damping that produces growing exponentialsolutions that are unstable. Roots in the left half plane have stable damping. As shown inFigure 4.29, assume two typical roots are initially at B1 and B2. Subject to change, parameterslike the coefficient of friction in the system allow the roots to come together at A. Then theparameter is further increased until the roots split apart. One root remains in the left half planewhile the other goes to the right half plane. Moreover, the roots that lie along the curve B*1 � A� B*2are symmetrically located near point A. This symmetry is retained for a finite parameterchange. It means that the two modes B*1 and B*2 vibrate at the same frequency. Hence they arecoupled. These coupled motions exist within one system mode.

For this type of solution from system equation 4.103, at least one of the roots lies in the righthalf of the complex plane. The occurrence of a stable limit cycle due to system parameter change isthe so-called Hopf bifurcation. During this process, two kinds of motion modes, one having stableand the other unstable motions couple inside a complex mode. These motions exchange energy backand forth until energy is released and the system state changed. The noncrossing of the eigenvalue islinked to the symmetric properties of the eigenvalue functions. This is the self-excited oscillation.The excitation is from motion and interface friction and a local change in interface parameter doesnot necessarily cause local or small effects on a global level stability for the system.

Next we consider how to measure so as to evaluate the level of coupling for different systems.To this aim, the system complex eigenvalue is expressed as

�f

� � c½ �T Kf

� �c½ � ¼ Ks f cTr½ �, cNr½ �f g (4:104)

Obviously, the complex eigenvalue is dependent on friction coefficient. The relevant equation canbe solved directly for different values of friction to establish the relationship. However, it isinstructive to develop an analytical expression that relates these two parameters. An approximationis obtained by a perturbation analysis with respect to the coefficient of friction to the second order.The complex eigenvalue of Equation 4.104 can be expanded by the second-order series as

s2i � �v2i � m �f

� �iiþm2

Xnl¼1l6¼i

�f

� �il�f

� �li

v2l � v2

i

� � (4:105)

From the calculation of the real model in the fundamental frequency range of coupled mode, itusually has vi

2 � m[Lf]ii. Hence, the diagonal terms of [Lf]ii can be neglected from the equation.

Re

ImB1

B2

B1B2

UnstableStable A

**

FIGURE 4.29 Bifurcation roots.



From Equation 4.105, clearly the coupling strength between any two modes depends on thedistance between the modes and the product of the cross-coupling terms [Lf]il[Lf]li. Therefore, thecoupling strength can be defined as

CSil ¼�f

� �il�f

� �li

v2l � v2

i

� � (4:106)

The coupling strength provides a measure of how fast the eigenvalues will move when the coefficientof friction increases from zero, and it can be used to quantify the interaction between modes.

Equation 4.105 is a closed form solution that relates the complex eigenvalues with friction. Thefirst- and second-order terms in (4.105) are the effect of friction. Equation 4.105 shows the effectof friction on the complex eigenvalues of the system. It gives an indication of instability, whichcomes from the second-order term. It involves the dot products of relative normal and tangentialdisplacements between two modes. A pair of modes that has a high dot product will either have astrong tendency to converge or diverge depending on the sign function. Since it is known thatinstability is the result of frequency convergence of two modes, those pairs that have high second-order terms may have the potential to become unstable.

It is noted that the first-order term in Equation 4.105 is typically negligible and has no effect oninstability. Moreover, a stability index can be used for the estimation of instability propensity.Assume the complex conjugate roots are l¼s iv. The corresponding characteristic equation is

(l�(s þ iv))(l�(s � iv)) ¼ 0 (4:107)

It can be simplified to the following:

l2 � 2slþ s2 þ v2 ¼ 0

The modal damping ratio is defined as

§ ¼ �sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 þ v2

p (4:108)

The modal damping ratio gives an index for a stability margin estimation. The friction involved invibration systems could not only have nonsymmetric stiffness matrix as illustrated above, it couldalso have mass coupling as shown in Figure 4.30.

In this model subsystem 1 is represented by a three-degree-of-freedom system at the top, andsubsystem 2 is modeled as a two-degree-of-freedom system at the bottom. The force P representsthe applied force.

Subsystem 2 travels horizontally at a constant speed V. The relative motion between subsystems1 and 2 leads to friction forces between them. Let the interaction normal force be N; the frictionforce is then mN, where m is the friction coefficient. The system motion equation is given by

M11

M12

M12

M2

M2

26666664

37777775

€y11€y12€x1€y2€x2

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

þ

K1y �K1y

�K1y K1y

K1x

K2y

K2x

26666664

37777775

y11y12x1

y2x2

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

¼

P

N

mN

�N

�mN

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

(4:109)

On the right-hand side of the above equations, except for the applied force P, all the other terms areinternal forces that are to be determined by the response of the system. Defining the equilibrium



state of the system as quasi-static equilibrium when the subsystem 2 moves at a constant velocity Vand does not vibrate, we are able to derive the equilibrium state from the following equations:

K1y �K1y

�K1y K1y

K1x

K2y

K2x

266664

377775

y110y120x10y20x20

8>>>><>>>>:

9>>>>=>>>>;

¼

PN0

mN0

�N0

�mN0

8>>>><>>>>:

9>>>>=>>>>;

(4:110)

Let

y11y12x1y2x2

8>>><>>>:

9>>>=>>>;

¼

Y11Y12X1Y2X2

8>>><>>>:

9>>>=>>>;

þ

y110y120x10y20x20

8>>><>>>:

9>>>=>>>;

and N(t) ¼ N0 þ ~N(t)

We are now able to obtain the dynamic equations of the system about its equilibrium state asfollows:

M11

M12

M12

M2

M2

2666664

3777775

€Y11

€Y12

€X1

€Y2

€X2

8>>>>><>>>>>:

9>>>>>=>>>>>;

þ

K1y �K1y

�K1y K1y

K1x

K2y

K2x

2666664

3777775

Y11Y12X1

Y2X2

8>>>>><>>>>>:

9>>>>>=>>>>>;

¼

0

N

m(N0 þ N)� mVN0

�N

�m(N0 þ N)þ mVN0

8>>>>><>>>>>:

9>>>>>=>>>>>;

(4:111)

m11

m12

m2

k1x

k1y

k2x

k2y

P

m

FIGURE 4.30 System with nonsymmetric mass matrix.



From the second equation above, we express N as follows:

N ¼ M22 €Y12 � K2yY11 þ K2yY12 (4:112)

On the other hand, the linear friction law is applied

m ¼ mv � a( _X2 � _X1) (4:113)

Then, we have the following:

m(N0 þ N)� mvN0 ¼ mv � a( _X2 � _X1)� �

(N0 þM1 €Y2 � K1yY11 þ K1yY12)� mvN0

¼ mvM1 €Y2 � aN0( _X2 � _X1)� mvK1y(Y11 � Y2)þ O(X2,Y11, . . .) (4:114)

Substituting Equations 4.112 and 4.114 into Equation 4.111, the linearized equations of the systemcan be derived. We assume that the sliding contact interface does not have separation; we thus havethe constraint condition Y2¼ Y12. Finally, we obtain the motion equations of the system

M11

M12 þM2

�mVM12 M12

mVM12 M2

26664

37775

€Y11

€Y12

€X1

€X2

8>>><>>>:

9>>>=>>>;

þ �aN0 aN0

aN0 �aN0

26664

37775

_Y11

_Y12

_X1

_X2

8>>><>>>:

9>>>=>>>;

K1y �K1y

�K1y K1y þ K2y

mVK1y �mVK1y K1x

�mVK1y mVK1y K2x

26664

37775

Y11Y12X1

X2

8>>><>>>:

9>>>=>>>;

¼

0

0

0

0

8>>><>>>:

9>>>=>>>;

(4:115)

The friction coupling makes both the mass and the stiffness matrices nonsymmetric, which causesthe eigenvalue solutions to be complex. Using standard complex eigenvalue analysis algorithm,one is able to obtain the complex eigenvalues. Usually, double shift QR or QZ algorithms couldhave advantages for this type of problem. Based on obtained complex eigenvalues, the stability ofthe system can be further evaluated if any of the eigenvalues have negative damping characteristics.Figure 4.31 shows a typical 3D plot of eigenvalue solution.

11,400

11,300

11,200

11,100 0

0.2

0.4

0.60

200

400

600

Real part ofcomplex

eigenvalue

Coefficient offriction

Frequency(Hz)

FIGURE 4.31 3D plot of eigenvalue. (Courtesy of Flint, J. and Hulten, J.)



We next discuss the effect of damping. The stable propensity of the system could be increasedby properly adding damping to some part of the system.

In terms of stability chart representation, for a typical multi-degree-of-freedom system,Figure 4.32 depicts the effect of damping on system instability. This is actually a real example ofan insulator in a brake system modeled by finite element method. In the baseline model, no dampingwas assigned to any component. When damping was included in the insulator (viscoelastic layer),many of the real parts of the complex eigenvalue moved toward the negative end of the abscissa;thus some of the unstable modes became more stable.

It was shown in the previous section that a friction system could become unstable due to modalcoupling with high friction coefficient. Next we discuss the effect of nonconstant friction coefficient.Consider linear friction law to be m(vr)¼ms � a1vr. Equation 4.96 can be transferred to

�M½ � €uf g þ �C½ � _uf g þ �K½ � uf g ¼ 0f g (4:116)

It is noted that in motion equation, the damping matrix [ �C] is incorporated with the dynamic frictioncoefficient portion and the normal contact forces of the system. Both [ �K] and [ �M] matrices areasymmetrical due to friction interaction.

Through a transformation, Equation 4.116 can be written as

B½ � _wf g þ A½ � wf g ¼ 0f g (4:117)

in which

B½ � ¼ M½ � 00 I½ �

� �, A½ � ¼ C½ � K½ �

� I½ � 0

� �, wf g ¼ u

_u

�

Then we can get a generalized, first-order eigenvalue problem

(Aþ lB) wf g ¼ 0 (4:118)

wherel is the complex eigenvaluew is the right eigenvector

The complex eigenvalue analysis shows that a negative slope of friction–velocity could renderthe system more unstable compared with constant friction. Figure 4.33 shows the effect of negative

0

2,000

4,000

6,000

Fre

quen

cy (H

z) 8,000

10,000

12,000

−2,000 −1,500 −1,000 −500 500 1,000 1,500 2,000

Real part of complex eigenvalue

No damping

With damping

0

FIGURE 4.32 Effect of damping.



slope friction–velocity on eigenvalues. This is also based on an example of brake system modeledby finite element method. It shows that with the presence of negative slope friction, the real part ofthe eigenvalue moves toward the right side of the abscissa, thereby tending to be more unstable.

4.3.5 NONLINEAR NUMERICAL ANALYSIS

4.3.5.1 Numerical Approaches

The linear stability analysis shown above, despite being approximate, is more economical than the fulltransient analysis using numerical approaches. However, for a deeper understanding, we have to usenumerical transient analysis by taking into account the nonlinearity of material and friction law.Figure 4.34 describes the general methodology used for the analysis of stability of frictional systems,which was suggested by Tworzydlo et al. [66].

In the following we discuss two problems on how to deal with nondiscontinuous friction lawand how to enhance calculation accuracy and reduce numerical errors in transient dynamic analysis.

For many cases, the modeling of vibration systems with friction yields a set of differentialequations with discontinuity. The standard methods to solve discontinuous differential equationsconsist of the smoothing method (normalization method) and the switch method. The smoothingmethod replaces the discontinuous system by a smooth adjoint system. The smoothingmethod yields asystem of ordinary but stiff differential equations and consequently leads to large computational times.

Consider the friction law, m(vr)¼m0 sgn(vr). The transition at zero point is not smooth if staticfriction is not zero. It can be approximated by

m(vr) ¼ (2m0=p) tan�1(«vr) (4:119)

in which a trigonometric function is used to smooth the transition. Sometimes, this form could even bea more realistic representation of a physical friction. « is a steepness parameter. Increasing thesteepness parameter improves the approximation, especially for vr close to zero. Even for trulynondiscontinuous curves, the friction curve will almost always be the same for large values of thesteepness parameter such as «¼ 106. However, a steep slope arises at vr¼ 0 and is given by�(2=p)«m0. This in the end causes a stiff differential equation, which is numerically costly to integrate.

The problems in the smoothing method led to the development of models which switch betweendifferent sets of equations by using alternate friction models for positive and negative velocity,respectively. The switch model starts from an initial state with a set of differential equations.After each time step the state vector is inspected on a possible discontinuous transition within this

0

2,000

4,000

6,000

8,000

10,000

12,000

−500 −300 −100 100 300 500


Fre

quen

cy (H

z)

Constant friction

Negative friction slope

FIGURE 4.33 Effect of negative slope friction vs. velocity on eigenvalues.



time step. If it meets an event, the integration process is paused, and an iteration procedure is used tofind the accurate switching point. With the switching point, a new integration process is started witha modified set of differential equations and initial conditions. An advantage of the switch model isthat it is able to incorporate a variety of friction laws discussed in the previous section, such as thetime-dependent static friction.

We next discuss the algorithms used for transient dynamic analysis shown in Figure 4.34. Theglobal equation of motion of the friction system can be written as

M€X(t)þ C _X(t)þKX(t) ¼ F(t) (4:120)

where M, C, and K are, respectively, the mass, damping, and stiffness matrices of the system. Theforcing term F(t) includes external force, contact force, and friction force. X(t) is the unknowndisplacement vector and a dot denotes differentiation with respect to time t. Conventionally, theRunge–Kutta family and linear multistep method, such as Newmark method, are used to solvethis kind of problem. It has been demonstrated that the linear multistep method is more efficientfor this type of problem. After balancing many requirements for simulating Equation 4.120,

Interface response– Surface profile– Asperity-based model– Statistical homogenization technique

Interface model– Contact– Friction

Dynamic modelof the system

– Finite element– Rigid bodies– Contact surfaces

Computational model

– Newton’s methodNonlinear analysis

– Complex eigenvalues

Linearized dynamic analysis

– Newmark method

Transient dynamic analysis

– Kinetic friction– Stick–slip motion– Frictional vibrations

– Dynamic stability– Self-excited modes

Steady-sliding

FIGURE 4.34 A flowchart for the general approach to modeling of friction-induced vibrations. (Courtesy ofTworzydlo, W.W., Hamzeh, O.N., Zaton, W., and Judek, T.J.)



e.g., low calculation cost, unconditional stability, avoidance of nondissipation in all frequencies,Newmark method with parameters d> 1=2, a> 1=4 has been suggested and applied widely.The Newmark method is expressed as

_XtþDt ¼ _Xt þ ½(1� d)€Xt þ d€XtþDt�Dt (4:121a)

XtþDt ¼ Xt þ _XtDt þ ½(1=2� a� d)€Xt þ a€XtþDt�Dt2 (4:121b)

The method has the following features:

1. Unconditional stability when applied to linear problems2. No more than one set of implicit equations to be solved at each step3. Controllable algorithmic dissipation in the higher modes

However, it is only first-order accurate and therefore is improper to integrate some equations ofstructural dynamics from the viewpoint of methodology. For instance, it was once alerted that a falseresult could be obtained due to the truncation error of the method for contact recording dynamics.There are many developments to extend the low-order methods to higher-order, unconditionallystable methods [67–74].

We next introduce the Sheng–Liu (SL) method [67,68], which uses the Newmark method withd¼ 1=2, a¼ 1=4 as underline algorithm, and uses extrapolation techniques to introduce the neces-sary dissipation and simultaneously improve the order of accuracy of the basic algorithm. Uncon-ditionally stable, higher-order accurate algorithm family is derived by choosing proper extrapolationparameters. The third- and fourth-order accurate algorithms are given as follows.

To generally implement the SL method, an sth-order accurate approximate amplification matrixcan be constructed as follows:

As(Dt) ¼Xs�1

i¼0

aiA1(biDt) (4:122)

Given the initial vector U0, vector U1 at the end of one time step can be rewritten as

U1 ¼ As(Dt)U0 ¼Xs�1

i¼0

aiA1(biDt)U0 ¼Xs�1

i¼0

aiU(biDt) (4:123)

where U(biDt)¼A1(biDt)U0

Based on Equation 4.123, the computational procedures can be briefly summarized as

(1) With the given initial value X0, _X0, €X0, according to following Newmark method (d¼ 1=2,a¼ 1=4) evaluate X(ti), _X(ti), €X(ti), at ti¼biDt for i¼ 0, 1, . . . , s � 1.

_XtþDt ¼ _Xt þ ½(1� d)€Xt þ d€XtþDt�Dt

XtþDt ¼ Xt þ _XtDt þ ½(1=2� a� d)€Xt þ a€XtþDt�Dt2 (4:124)

M€XtþDt þ C _XtþDt þKXtþDt ¼ FtþDt

Note that when b0¼ 0, no evaluation is required.



(2) X1, _X1, €X1 at the end of the time step is then obtained by summing up the results ofeach evaluated X(biDt) in step (1) and the corresponding weighting factor ai for i¼ 0,1, . . . , s � 1.

X1 ¼Xs�1

i¼0

aiX(biDt)

_X1 ¼Xs�1

i¼0

ai_X(biDt)

€X1 ¼Xs�1

i¼0

ai€X(biDt)

(4:125)

(3) X1, _X1, €X1 obtained in the current time step becomes the initial value X0, _X0, €X0 for thenext time step.

(4) Steps (1) to (3) are repeated until the whole time span of interest is covered. The partitionparameter bi and the weighting factor ai are determined by the following equations:

�ai ¼ 1 for s ¼ 0

�aibi ¼ 1 for s ¼ 1

�aib2i ¼ 1 for s ¼ 2

�aib3i ¼ 2=3 for s ¼ 3

�aib4i ¼ 1=3 for s ¼ 4

�aib5i ¼ 2=15 for s ¼ 5

(4:126)

For instance, if the first four equations in Equation 4.126 are satisfied, the algorithm will beimproved to third-order accurate. ai and bi can be solved in terms of b2 as

a0 ¼ 3b23 � 9b2

2 þ 8b2 � 2

b2(2� 3b2)(6b2 � 3b22 � 2)

, b0 ¼ 0

a1 ¼ (3� 3b2)(6b2 � 3b22 � 3)

(2� 3b2)(6b2 � 3b22 � 2)

, b1¼ 2� 3b2

3� 3b2

a2 ¼ 1

b2(6b2 � 3b22 � 2)

(4:127)

A nearly optimized unconditionally stable algorithm can be achieved by choosing anarbitrary b2 close to (3þ ffiffiffi

3p

)=3.

4.3.5.2 Numerical Example

Numerical analyses can provide results explaining the underlying mechanism of different types ofsystem responses and revealing the nonlinear nature of the friction vibration system. We nextpresent a comprehensive numerical example to further elaborate the solution and some features ofthe friction vibration system. This case was comprehensively investigated by Bengisu and Akay[75], next we present their analysis and recaculate their example. In addition to the mode lock-insingle mode, this example also illustrates that the response of a linear multi-degree-of-freedomsystem with friction may constitute several independent fundamental frequencies and their harmon-ics. Depending on the system parameters and the friction, the fundamental frequencies of the



response may relate to each other, exhibiting the harmonics of the lowest one and the synchroniza-tion of the fundamental frequencies. The fundamental frequencies may also appear to be totallynonsynchronizing, each fundamental frequency having its own family of harmonics.

Following [75], we consider a multi-degree-of-freedom system defined by Equation 4.96, withfriction law given by

Ff ¼ m(vr)FN ¼ FN sgn(vr)(1� e�bv vrj j) 1þ ( fr � 1)e�av vrj j� �(4:128)

where fr, av, and bv are constants. After the normalization of eigenvectors of linear homogeneoussystem with respect to mass, the equation of motion can be expressed under its modal coordinates.

For simplification, we rewrite Equation 4.96 in the following tight form; we use the nondimen-sional time t¼v1t

�½ �2 u00f g þ �½ � �½ � u0f g þ I½ � uf g ¼ F½ �T Ff

�(4:129)

�½ � ¼

1 0 � � � 0

0 v2=v1...

..

. . ..

00 � � � 0 vn=v1

26664

37775 �½ � ¼

2j1 0 � � � 0

0 2j2...

..

. . ..

00 � � � 0 2jn

266664

377775

The force vector is given by

Ff

� ¼F01 sgn(vr1) 1� e�bv1 vr1j j� �

1þ ( fr1 � 1)e�av1 vr1j j� �...

F0n sgn(vrn) 1� e�bvn vrnj j� �1þ ( frn � 1)e�avn vrnj j� �

8>><>>:

9>>=>>; (4:130)

in which F0i¼FNi=m1g, bvi¼bvig=v1, avi¼avig=v1, vri ¼ Vriv1=g� X0i*

X0i*(t) ¼ fi1v

2i x

0i1(t)=v

21 þ � � � þ finv

2i x

0in(t)=v

21

� �Let {y(t)}¼ {u0(t)} then the equation can be written as

0 L2

L2 C

� �y0(t)u0(t)

� ¼ L2 0

0 I

� �yu

� þ 0

P

� (4:131)

in which damping matrix is C½ � ¼ �� and force matrix is [P]¼ [F]T{Ff}.Let us first consider the linear stability analysis. It is conducted around the equilibrium point of

the system. The basic procedure has been described before. The roots of its characteristic equationare solved. If the roots have negative real parts, the nonlinear system is asymptotically stable. If anypair of the roots has positive real parts, then the nonlinear system is not asymptotically stable; it haseither a bounded oscillatory response or an unbounded response. If the real part of any pair of theroots is zero while all the others have negative real parts, the stability of the system cannot bedetermined by the procedure.

The equation can be linearized by the following approach. At first, the equilibrium points of thesystem are determined by the following equation:

L2 00 I

� �yu

� ¼ 0

P

� (4:132)



The solution of this set of equations yields

y(t) ¼ 0, u(t) ¼ Pjy¼0

Then the linearized equation at the equilibrium point of a nonlinear system is given by

y0(t)u0(t)

� ¼ J(ye,ue)

y(t)x(t)

� (4:133)

in which J(ye,ue) is the Jacobian matrix evaluated at the equilibrium point. The linearized equationof the motion is

0 L2

L2 C

� �y0(t)u0(t)

� ¼ L2 0

0 �I

� �yu

� þ 0 0

G 0

� �yu

� (4:134)

in which G is the friction-related matrix

� ¼ ffgTi ffgi�2Sf

where {f}i is the ith row of the eigenvector matrix F and

Sf ¼ @Ff

@X0*

��y¼0

¼ F0 a0( fr � 1)(1� e�b0V*)e�a0V*� b0e�b0V*(1þ ( fr � 1)e�a0V*)

h i(4:135)

Here Sf is the slope of the friction at the equilibrium point. In the general case, there are frictionforces acting on the lumped mass

G ¼ ff gT1 ff g1�2@Ff 1

@X01*

��y¼0

þ � � � þ ff gTn ff gn�2 @Ffn

@X0n*

��y¼0

(4:136)

The linearized equation can be rearranged as

0 L2

L2 C� G

� �y0(t)u0(t)

� ¼ L2 0

0 �I

� �yu

� (4:137)

The friction participates in the equation through a damping form. The system behavior can beestimated from the sign of [C] � [G]. [G] is the coupling matrix by which all the modes of theoriginal linear system are coupled. As its dominant parameter is Sf, the stability of the linearizedsystem can be investigated by using Sf as an independent variable. On the other hand, Sf is a functionof the friction law determined by multiple parameters; thus a change of each of the five parametersin friction law will change Sf thereby changing the stability behavior.

We consider next a numerical example; a three-degree-of-freedom system is shown inFigure 4.35. The system parameters are given by m1¼m2¼m, m3¼ 2m, k1¼ k2¼ k, k3¼ 2k,with mode matrix solved as

�½ � ¼1

3:545:43

24

35, F½ � ¼

�0:269 0:878 0:395�0:501 0:223 �0:863�0:582 �0:299 0:268

24

35, �½ � ¼ 2

0:030:01

0:02

24

35



In the solution, the root locus diagram of the linearized equation is solved first for stability analysis.When Sf < 0.1, all the roots in the stable region and system have steady sliding. When Sf is increasedbeyond 0.1, a pair of roots move to the right-hand side of the complex plane and the system exhibitsunstable and sustained oscillations. When Sf is further increased beyond 0.7, the second pair of rootsmoves to the right-hand side of the complex plane. Finally, When Sf is further increased beyond 1.7,the third pair of roots moves to the right-hand side of the complex plane.

It is noted that when Sf is increased to 6.6, one of the pairs of roots moves back to the left-handside of the complex plane. This is a subcritical or inverse Hopf bifurcation. Thereby an independentfrequency is removed and there are only two independent frequencies left in the response; furtherincreasing Sf will remove one more frequency and leave only one frequency in the response.

Therefore, the big picture of phase space is that a limit cycle corresponds to a pair of roots on theright-hand side of the complex plane, a closed trajectory on the surface of a three-dimensional toruswhen two pairs of roots are on the right-hand side, and a closed trajectory on the surface of a four-dimensional torus when three pairs of roots are on the right-hand side.

For this example, the first bifurcation corresponds to the first unstable root or the secondeigenvalue l¼ 3.54. Figure 4.36 shows this phase plot which was presented in [75] and here isrecaculated plot. The response spectral of the nonlinear system is shown in Figure 4.37, withfundamental frequency 3.45 just below the second natural frequency. The vibrations are consideredto be locked in the second mode of the linear system.

When Sf is further increased, the fundamental frequency is further reduced away from naturalfrequency and it is accompanied with an increase of the harmonic component in the response, whichallows the response to be more close to pure periodic.

By increasing the normal contact force, a second bifurcation is attained. This is roughlyassociated with the crossover of the second pair of roots to the right-hand side of the complex

m1 m2

k2

c2

k1

c1

m3

k3

c3

L

V

FIGURE 4.35 Three-degree-of-freedom system with friction.

2

0

−2

−40 2

(a) (c) (b)

4 6 8Displacement

Vel

ocity

FIGURE 4.36 Projection of the phase-space trajectories onto the velocity–displacement plane of the drivingpoint response for different sets of parameters: V¼ 1, a0¼ 1.1, Fr¼ 2.3; (a) one bifurcation, b0¼ 3.33,Fn0¼ 2.8; (b) two bifurcations, b0¼ 3.33, Fn0¼ 15.5; (c) three bifurcations, b0¼ 3.33, Fn0¼ 10.0. (Reused withpermission from Bengisu, M.T. and Akay, A., J. Sound and Vib., 171, 4, 557, 1994. Copyright Elsevier, 1994.)



plane. The phase plot is shown in Figure 4.36. The response of the nonlinear system is acombination of two periodic functions with synchronized fundamental frequencies. The spectrumof the response is shown in Figure 4.38. The value of the fundamental frequency is 0.921 just belowthe eigenvalue l1¼ 1 of the second unstable pair of roots of the corresponding linear system. Nowthe system vibration is locked in the first mode of the linear system.

The second fundamental frequency is not obvious in the response as it is synchronized with thefirst fundamental frequency and looks like one of the harmonics. When Sf is increased beyond 1.7,the system response has all of the fundamental frequencies, and they get synchronized. Theprojection of the corresponding phase–space trajectory onto the velocity–displacement planeof driving point exhibits four loops as in Figure 4.36. The spectrum of the response is shown inFigure 4.39.

In the case of single-degree-of-freedom systems, there is only one bifurcation point thatseparates the asymptotically stable state from the state of limit cycle, and this corresponds to thecrossover of the roots to the right-hand side of the complex plane. For the multi-degree-of-freedomsystem, the similar behavior holds essentially, but it is roughly for second or higher pairs crossingover which may need enough crossover distance away from the left-hand side so as to allow thesystem response to change substantially or bifurcate.

When each bifurcation manifests itself as an independent fundamental frequency, a newperiodic function is added to the response of the system. Moreover, each fundamental frequencyhas its harmonics family. Particularly, the system could exhibit periodic response with onefundamental frequency and its harmonics, due to locked-in one of natural frequencies. There

Frequency

Vel

ocity

0 8 16 24 32 40

0.5

0.4

0.3

0.2

0.1

0

FIGURE 4.37 Vibration–velocity spectrum of the driving point with one fundamental frequency and itsharmonics in the response corresponding to case (a) in Figure 4.36. (Reused with permission from Bengisu,M.T. and Akay, A., J. Sound and Vib., 171, 4, 557, 1994. Copyright Elsevier, 1994.)

0.5

0.4

0.3

0.2

0.1

00 8 16 24 32 40

Frequency

Vel

ocity

FIGURE 4.38 Vibration–velocity spectrum of the driving point with two fundamental frequencies and theirharmonics in the response corresponding to case (b) in Figure 4.36. (Reused with permission from Bengisu,M.T. and Akay, A., J. Sound and Vib., 171, 4, 557, 1994. Copyright Elsevier, 1994.)



could be several independent but synchronized periodic functions subject to the friction features andthe eigenvector feature of the linear system.

Moreover, when the normal motion experiences severe oscillations, loss of contact could occur,which could lead to strong vibration of system without correlation with the natural frequencies ofthe system in the absence of friction. Furthermore, as slip passes through zero, the direction of thefriction force experiences a sharp change and this induces complex vibrations as well.

4.4 VIBRATIONS AND SOUND OF CONTINUUM SYSTEMS WITH FRICTION

4.4.1 LONGITUDINAL VIBRATIONS OF ROD WITH VELOCITY-DEPENDENT FRICTION

In this section, we discuss the vibrations of continuum systems with friction. Basically, theprocedure to deal with friction continuum systems consists of two steps. The first one is to changea discrete continuum system into a multi-degree-of-freedom system, and the second one is to solvethe multi-degree-of-freedom system with friction. There is no fundamental difficulty in dealing withfriction continuum systems with friction using the analytical and numerical approaches presented inthe previous sections of this chapter and in Chapter 2.

We next present several cases of continuum systems to illustrate this procedure. At first weconsider a system of moving rod with friction [76]. Consider a flexible rod traveling at a constantspeed V between two fixed guides that are separated by a distance L. Assume a friction force appliedat a fixed point x0. Assume the longitudinal deformation of the rod is represented as u(x,t); theequation of motion can be written as

rA v2@2u

@x2þ 2v

@2u

@x@tþ r

@2u

@t2

� � EA

@2u

@x2¼ �m(vr)FN(t)d(x� x0) (4:138)

where r, E, and A represent the density, Young’s modulus, and the cross-sectional area of the rod,respectively. d is the Dirac function. FN(t) represents the normal components of the contact force atfixed point x0. m(vr) represents the coefficient of friction at the contact point. It is noted that the firsttwo terms in the equation represent the effect of Centripetal and Carioles acceleration components,respectively, vr¼ _u � V. The following friction law is used:

m(vr) ¼ ms sgn(vr)� a1vr þ a3v3r (4:139)

0

Vel

ocity

0

0.1

0.2

0.3

0.4

0.5

8 16

Frequency

24 32 40

FIGURE 4.39 Velocity spectrum of the driving point with three fundamental frequencies and their harmonicsin the response corresponding to case (c) in Figure 4.36. (Reused with permission from Bengisu, M.T. andAkay, A., J. Sound and Vib., 171, 4, 557, 1994. Copyright Elsevier, 1994.)



The friction has a negative slope over the speed range 0 < vr < vm; vm ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffia1=3a3

pcorresponds

to the speed at which m0(vr)¼ 0. By using dimensional parameters, the equation is cast in thesymbolic form

Mutt þ Gut þ Ku ¼ «N(u,ut) (4:140)

in which M, G, K, N, and « are, respectively, the mass, gyroscope, stiffness, friction operators, andsmall factors. They are given as follows:

M ¼ I, G ¼ 2v@=@x, K ¼ �(1� v2)@2=@x2, N ¼ �d(x� x0)½ms sgn(vr)� a1vr þ a3v3r �

First of all, the continuum system is discretized, and the natural modes of the original linear systemvn Cn are obtained. The nonlinear response with friction is obtained by using average method. Thenonlinear response of the system is assumed as

u(x; a,u) ¼ u(0)(x; a,u)þ � � � þ u(1)(x; a,u) (4:141)

The amplitude and total phase evolution are given by

_a ¼ «P1(a)þ � � � , _u ¼ Q0 þ «Q1(a)þ � � � (4:142)

in which the following solution represents free vibration of the original linear system in a specifiedmode with frequency Q0¼vn and mode shape Cn

u(0)(x; a,u) ¼ 12aC(x)eiu þ 1

2aC(x)eiu

� �*(4:143)

in which �ð Þ* denotes the complex conjugate. The differential equation that governs the evolution ofthe amplitude and phase of the nth mode are derived as

_a ¼ «m1 1� (v=vm)2

� �sin2(npx0)a 1� (a=a1)2

� �(4:144)

_u ¼ np(1� v2)þ (1=2)a1«av 1� (v=vm)2

� �sin(npx0) 1� (a=a1)2

� �in which the characteristic amplitude is

a1 ¼ 1np

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2(v2m � v2)

v2 þ (1� v2) sin2(npx0)

s(4:145)

This is the first-order approximation with _a¼ «P1(a), _u ¼ Q0 þ «Q1(a).The above is the solution of the self-excited vibration of the system. Over the range 0< vr< vm,

there exists limit cycle. The small amplitude motions grow and large amplitude motions decay. Bothapproach the stable limit cycle with amplitude a1. This amplitude is independent of initialconditions, but is dependent on speed, friction law, and the friction location.

4.4.2 BEAM TRANSVERSE VIBRATIONS DUE TO BOWING EFFECTS

We next consider the vibration of a clamp-free beam under rubbing excitation at free end in thetransverse direction. The equation of motion is given by

EI@4w

@x4þ rA

@2w

@t2¼ m(vr)FN(t)d(x� x0) (4:146)



where r, E, I, and A represent the density, Young’s modulus, cross-sectional moment of inertia, andcross-sectional area of the beam, respectively. Rubbing the free end of the cantilever beam with abow can produce mode lock-in. Depending on the application of the bow, the spectra of beamtransversal vibrations display the first, second, or even the third natural frequency. It may have morethan one natural frequency appearing simultaneously. Mostly, the oscillation of the beam locks intoa particular mode and oscillates in a rather stable manner. Because bow impedance in the directionof excitation differs significantly from that of the beam, the dynamics of the two do not criticallyinteract, and the beam actually responds at one of its natural frequencies with correspondingharmonics. Figure 4.40 shows the transfer function of the beam at the free end and the radiationspectrum of the beam under bowing, from which we can see that a variety of mode lock-in isobtained by applying friction to a clamp-free beam using a violin bow.

�20

0

20

40

60

Frequency (Hz)0 400 800 1200 1600

0 400 800 1200 1600 0 400 800 1200 1600

0 400 800 1200 1600 0 400 800 1200 1600

0 20 40 60 80 100

Am

plit

ud

e (d

B)

1

23

4 5

(a)

0

20

40

60

80

Frequency (Hz)

Am

plit

ud

e (d

B)

1st mode

(b)

Am

plit

ude

(d

B) 4th mode

0

20

40

60

80

Frequency (Hz)

0

20

40

60

80

Frequency (Hz)

Am

plit

ud

e (d

B)

5th mode

(e) (f)

60

80

Am

plitu

de (

dB

)

2nd mode

0

20

40

60

80

Frequency (Hz)

Am

plit

ud

e (d

B)

3rd mode

0

20

40

Frequency (Hz)(c) (d)

FIGURE 4.40 The mode lock-in obtained by applying friction to a clamp-free beam using a violin bow.(The transfer function of the beam and the radiation spectrum with 1st–5th order mode lock-in.) (Reused withpermission from Akay, A., J. Acoust. Soc. Am., 111, 4, 1525, 2002. Copyright 2002, Acoustical Society ofAmerica.)



4.4.3 BEAM TRANSVERSE VIBRATIONS DUE TO AXIAL STICK–SLIP EXCITATION

We next present an analysis of the vibration of a random excited beam with an axial stick–slip endcondition. The results were derived by Bigelow and Miles [77]. Consider a beam clamped at twoends with friction support at one end x¼ 1; the beam is subject to a base acceleration €w0(t)of Gaussian white noise. Assume the transverse deflection and axial displacement of the beam arew(x,t) and u(x,t), respectively. Assume the resonance in the axial direction is outside the frequencyrange; the axial inertia is thus neglected in axial equation. The clamping force is FN, and thecoefficient of friction is assumed to be a constant m. Under the random excitation, the transversalvibration could lead to the axial motion. Because of the friction constraints at one end, the systemcould have two states in axial motion: if the axial force is smaller than mFN, the end sticks; if theaxial force increases up to mFN, the slip occurs and the force remains the same. Assume the frictionforce supplied by support is ff (t), which is always equal to the axial force due to transversal motion

ff (t) ¼ (EA=l)u(l,t)þ 12

ðl0

@w

@x

� 2

dx (4:147)

The motion equation of the system in transversal direction is given by

EI@4w

@x4� ff (t)

@2w

@x2þ c

@w

@tþ rA

@2w

@t2¼ �rA€w0 (4:148)

Based on the Galerkin method, assume the motion is approximated by the lowest resonant modevector and eigenvalues, f, v0, w(x,t)¼a(t)f(x). Then the motion equation can be transformed to

€aþ v20 þ (EA=l) u(l,t)þ (a2=2)

ðl0

@f

@x

� 2dx

24

35aþ 2v0j _a ¼ b€w0(t) for stiction (4:149)

in which g¼ 36=(rAl2) and

(EA=l) u(l,t)þ (a2=2)ðl0

@f

@x

� 2dx

24

35

¼ mFNsgn u(l,t)þ (a2=2)ðl0

@f

@x

� 2dx

24

35€aþ v2

0

� mFNsgn u(l,t)þ (a2=2)ðl0

@f

@x

� 2dx

24

35gaþ 2v0j _a ¼ b€w0(t) for slip (4:150)

The analytical model can be used to quantify the likelihood of the slipping in the friction support asa function of the beam parameters. The time history obtained by using the numerical approach isshown in Figure 4.41. The numerical analysis uses white noise excitation of power spectral densityof 0.01 g2=Hz for four given friction values ff¼mFN, 4=3mFN, 5=3mFN, and 2mFN. Figure 4.41shows that as the friction force increases, the time before the first slip event also increases. Anincreased friction force causes the slipping events to be less frequent. The position of the beam endeventually reaches a roughly steady-state condition; the larger the friction value, the larger thesteady-state displacement. Furthermore, the time history of the axial force, the probability densitiesof the transverse displacement, and the rate of slipping can be quantified.



4.4.4 BEAM TRANSVERSE VIBRATIONS WITH MODAL COUPLING DUE TO

GEOMETRY CONSTRAINTS

In the last section we have discussed the two-degree-of-freedom model with binary instability. Theforces that change their directions as the geometry changes are called follower forces. The frictionforces applied on continuum system can act as follower forces, which are nonconservative and canalso result in instabilities. For instance, when friction forces act on the surface of continuum system,they are always parallel to the surface and a sliding structure segment could result in a transversecomponent of the friction force which serves as follower force. Moreover, for this situation, thesliding structure segment also leads to a moment due to the friction action through geometry distance.These types of combined effects of geometry constraint and friction result in nonsymmetric term instiffness matrix, thereby giving rise to dynamic instability. Next we use beam structure to illustrate themodal coupling type instability due to friction and geometry constraints [62–64].

Consider a segment of moving beam in sliding contact with elastic foundations as shown inFigure 4.42. A normal preload pressure FN is applied and the foundation stiffness per length, k,

–0.0020 40 80 120

–0.0015

–0.001

–0.0005

0

0.0005

–0.002

–0.0015

–0.001

–0.0005

0

0.0005

–0.002

–0.0015

–0.001

–0.0005

0

0.0005

–0.002

–0.0015

–0.001

–0.0005

0

0.0005

(a) (b)

(c) (d)

Time (s)

0 40 80 120 0 40 80 120

0 40 80 120

Dis

plac

emen

t of b

eam

end

FIGURE 4.41 The time history of displacement of the beam held in a stick–slip fixture for four given frictionforces: (a) mFN¼ 60; (b) mFN¼ 80; (c) mFN¼ 100; (d) mFN¼ 120. (Courtesy of Bigelow, S.P. and Miles, R.N.)

V

l

FIGURE 4.42 A moving beam model in contact with elastic foundation.



creates a reaction force to the transverse displacement of beam w(x,t). A friction-related forcep¼mkhw and a follower force 2mFN @w=@x are created due to the translation and rotation motion ofthe segment under friction, respectively, in which m is the coefficient of friction between beam andfoundation and h is the height of the beam. Figure 4.43 shows the differential element of movingbeam on elastic foundation.

To derive the equation of motion of the beam, we reconsider the moment equilibrium equationof differential element in Equation 2.150 of Chapter 2, but here we need to include the distributedmoment p effects due to foundation action

@M

@xdxþ p dx� Q dx� @Q

@x(dx)2 � q

(dx)2

2¼ 0 (4:151)

Neglecting the higher-order terms containing (dx)2, we obtain

Q ¼ @M

@xþ p (4:152)

The equation of motion in the vertical direction in Equation 2.152 still holds the following form:

@Q

@xdxþ q dx ¼ � dx

@2w

@t2(4:153)

By combining Equations 4.151 and 4.153 and ignoring the (dx)2 term we get

EI@2w

@x2þ @p

@x� qþ �

@2w

@t2¼ 0 (4:154)

in which q is external load per unit length due to foundation effect. The external bending momentdue to the friction force is p¼mkwh. The external force due to the foundation stiffness and thefollower force is

q ¼ �2kw� 2mFN@w

@x(4:155)

Then the final motion equation becomes

EI@4w

@x4þ mkh

@w

@xþ 2mFN

@w

@xþ 2kwþ �

@2w

@t2¼ 0 (4:156)

p dx

dx

dxx

w

M

U

q dx

∂QQ +

Q

A

∂x

dx∂M

M +∂x

FIGURE 4.43 Differential element of moving beam on elastic foundation.



From Equation 4.156 it can be readily seen that the two terms, 2mFN @w=@x and mkh @w=@x, bothcouple lateral displacement and rotation motion. The following force term 2mFN @w=@x creates atransverse force due to rotation @w=@x and friction force mFN. The elastic deformation termmkh @w=@x creates some bending moment, due to displacement w, friction-related force mkhw,and geometry effect. These friction coupling terms couple translation and rotation, and thus causenonsymmetric terms in the stiffness of the system equation. Thereby they could result in instabilityof binary flutter type, if there are two natural frequencies of the beam in near proximity under theeffect of sufficient coefficient of friction.

With the Galerkin approach, Equation 4.156 can be discretized and is attributed to as a complexmode problem. The discrete equations of motion are then solved in terms of eigenvalues andeigenvectors.

It is noted that for a refined and balanced treatment, Timoshenko beam theory is needed whichaccounts for the effects of transverse shear deformation and rotary inertia. This could be critical inaddressing high frequency problems for thick beam.

It is noted that there exists similar modal coupling due to geometry constraint for the vibrationsof plate under the effect of friction; moreover, the Mindlin plate theory should be employed whichaccounts for the effects of transverse shear deformation and rotary inertia for thick plate.

4.4.5 WEAK CONTACT AND STRONG CONTACT: RANDOM INTERACTIONS OF FRICTION

4.4.5.1 Weak Contact and Strong Contact

Depending on the load level and system stiffness situation, the interaction between two subsystemswith frictional sliding could be a weak contact or a strong contact. The weak contact is generally forlight load cases where mode lock-in and mode-coupling do not take place. The weak contactinteraction can be treated as a forced vibration. This includes typically the rubbing vibrations andsound purely due to asperity interaction and the resulting surface waves. The strong contact involvesmode lock-in and=or mode-coupling. It leads to self-excited vibrations with the magnitude ofvibration and sound controlled and sustained by the rubbing interaction of the interface.

4.4.5.2 Random Interactions of Friction

Wenext discuss the vibrations under random interactions inweak contact systems.This is also valid forstrong contact systems. In vibration theory, when we consider the transversal and longitudinalvibrations of beam, we usually assume that both have no coupling. In this sense, rubbing along thelength of a smooth beamwith friction force only produces longitudinal vibrations. The friction force, inthis case, travels along abeamor rod and causes longitudinalwaves. The equation ofmotion is given by

�E@2u

@x2þ r

@2u

@t2¼ 1

Am(vr)FNd(x� Vt) (4:157)

in which coefficient of friction is assumed as m(vr)¼ms� avr, and vr¼ _u�V, FN is constant normalforce.

However, when rubbing a rough beam along its length with a rough surface, it actually producesboth longitudinal and transversal vibrations due to the surface irregularity effect. Unlike the case ofa smooth beam, the interface force is no longer a constant; it actually has dynamic components inboth tangential and normal directions, and therefore generates both longitudinal and bending wavesin the beam simultaneously. The interface force developed due to the roughness effect has bothdynamic friction force component and dynamic normal force component. Despite the application ofa constant normal force, the interface force is a time series with random and=or impulsive features.In this case, Equation 4.157 still holds for longitudinal vibrations, but the normal force FN isreplaced by FN(s,t) which is actually affected by the roughness or corrugations of the beam,characterized by s. Under this situation, the transverse vibration is characterized by



EI@4w

@x4þ rA

@2w

@t2¼ FN(s,t)d(x� Vt) (4:158)

In this situation, even if the apparent normal load is given as constant, the variation of the roughnessand the formed deformation could cause an excitation in the transversal direction through defor-mation. To illustrate this, we consider the following simplified model:

m€xþ c1 _xþ k1x ¼ m(vr)FNd(x� Vt)

m€yþ c2( _y� _z)þ k2(y� z) ¼ FNd(x� Vt) (4:159)

m(vr) ¼ ms � avr ¼ ms � a _xþ aV , z ¼ z(s,vr)

in which z is the macrodisplacement in the normal direction caused by the microroughness profileand it could also be a function of slip velocity vr.

These types of interactions due to surface imperfections are very common in applications,particularly noticeable for weak contact interface which will be described in the following. This typeof sliding contact usually gives rise to either random excitation, or periodic impacts or single impactto structure, and results in a variety of vibrations of structure, including resonant modes. This will befurther developed in Chapter 5, where the caused ultrasonic resonant frequency is applied todetermine contact status. These types of interactions are also compatible with other kinds ofinstabilities, such as in a sliding rubber; this interaction is associated with stick–slip and waves ofdetachment (Schallamach wave).

Rubbing the surface of a nonrotation disk under a free–free boundary condition was demon-strated to be able to excite a variety of natural modes of the brake system components [78].

The test was performed by using a rubber-like material to rub the disk surface and thenrecording power spectra both in the in-plane and the out-of-plane directions. The rubbing processis shown to be able to induce the out-of-plane vibration at all the major out-of-plane resonantfrequencies under certain rubbing conditions, and may also just generate some dominant frequenciesusing different rubbing conditions. The rubbing–friction process can also create an in-planevibration either at all the major in-plane resonant frequencies or just at selective frequenciesunder different conditions. The vibration amplitudes, peaks, and frequencies can vary subject torubbing speed, pressure, etc. In general, the variation of the distribution of the normal and tangentialforce can induce vibrations at the entire majority of the out-of-plane and the in-plane resonantfrequencies. However, the dominant frequency induced by the friction process can be right at thefrequency where the in-plane resonant is aligned with the out-of-plane resonant of the rotor.

In the above process both weak and strong contact could take place, even though the modalcoupling is unlikely due to the larger difference of impedance between rubber-like material and disk.In general, the response of the system to friction can be of the types of either forced vibrations or self-excited vibrations. Instabilities in the presence of friction usually develop through one or two majormechanisms: negative slope of friction–velocity curve and friction–geometry constraints. Typically,the former has a trend to lead to mode lock-in, the latter has a propensity to lead to mode-coupling.

To address the friction–geometry constraints–related instability problems, both subsystemsshould be taken into account. Next we consider a beam–disk coupled system with friction betweenthe free-end of a cantilever beam and a rotating disk. Conventionally there are many investigationson the friction coupling between rotating disk and beam-type suspension structures [79–86].Consider a cantilever beam that makes an angle w with the normal to the surface of a disk, andthe beam contacts the rotating disk at end L. After neglecting the effects of shear and rotatory inertiaof the beam, the bending and longitudinal vibrations of the beam are governed by

EI@4w

@x4þ rA

@2w

@t2þ C

@w

@t¼ FN(t)½m(vr) coswþ sinw�d(x� L) (4:160)



EI@2u

@x2� rA

@2u

@t2� C0 @u

@t¼ FN(t)½�m(vr) sinwþ cosw�d(x� L) (4:161)

where r,E, I, A are the beam properties; C and C 0 are damping constants; m(vr) is coefficient offriction; FN(t) represents the normal force at the contact point; and vr is disk velocity. Transversalvibrations of the disk excited at a point (r0,u0) on its flat surface are described by

Dr4hþ rdhd@2h

@t2þ C00 @h

@t¼ �FN(t)d(r � r0)d(u� u0) (4:162a)

The circumferential vibration equation of disk is

EI

d4@3ur@u3

� @2uu@u2

� � EA

d2@2uu@u2

þ @ur@u

� þ rA

@2uu@t2

¼ FN(t)m(vr) � d(r � r0)d(u� u0) (4:162b)

where D ¼ Edh3d=12(1� v2) represents the bending rigidity of the disk, and hd and rd are itsthickness and density, respectively. C00 represents an equivalent damping proportional to velocity.The two primary sources of nonlinearities in the equations given above come from the coefficient offriction expressions and the contact forces. Another equation for radial vibration has the same formas Equation 2.199.

For this coupled system, the friction force acting tangentially on the disk surface excites thein-plane waves of the disk and friction at its tip develops simultaneously with the bending waves ofthe beam. The contact force at the interface develops a dynamic component normal to the surface ofthe disk and readily excites the disk bending vibrations. If the disk has its in-plane and bendingvibration frequencies in close proximity, mode-coupling instability develops with sufficient coeffi-cient of friction. As an illustrative example, Figure 4.44 shows the development of mode-coupling

40

70

60

50

40

30

20

10

50

30

10

−10

−40

−20

0

20

40

0 0.5 1 1.5

Frequency (kHz)

(a) (b)

(c) (d)

Am

plitu

de (

dB)

2 2.5 30 0.5 1 1.5 2 2.5 3

0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3

20

0

−20

−40

FIGURE 4.44 Development of mode lock-in between a beam and a disk: (a) transfer function of disk; (b)transfer function of beam; (c) transfer function of beam and disk in stationary contact; (d) response duringsliding contact. (Reused with permission from Akay, A., J. Acoust. Soc. Am., 111, 4, 1525, 2002. Copyright2002, Acoustical Society of America.)



between the beam and the disk [1], in which figures (a) and (b) show the transfer function of diskand beam, respectively. Figure (c) shows the transfer function of coupled beam and disk instationary contact. There is a split of frequency that corresponds to the (0,2) mode of the diskwhen in contact with the beam. Figure (d) shows the coupled response during sliding contact. Thespectrum of response exhibits a mode-coupling at the higher of the split frequencies.

4.4.6 SPRAG–SLIP EXCITED VIBRATIONS

We next discuss a unique instability feature encountered in some friction vibration systems. The termsprag–slip refers to the friction force that can increase significantly above its normal level due to thegeometry constraint of the interface; then at some level the frictional force returns to its original stateonly to repeat the cycle due to the elastic and plastic deformation of the contact interface.

To illustrate this point, consider the system shown in Figure 4.45; a hinged beam is in slidingcontact at an acute angle with a moving plate. Assume Ff¼mFN and consider the equilibrium of thesystem; the following equation is derived:

FN ¼ L

1� m tan u, Ff ¼ mL

1� m tan u(4:163)

Based on this expression, if u approaches tan�1 (1=m), then Ff approaches infinite. This critical caseis what is termed as spragging.

If the structures considered here are rigid and spragging occurs, the motion could be halted.However, if we consider the system with real elastic and=or local plastic interface, the flexibility ofthese components allows a release from the spragging situation by slipping, once they have beensufficiently deformed by the large normal and friction forces at the contact interface. After thespragging has been relieved, the tight contact situation rebuilds, the contact force grows again, andthe next run sprag–slip takes place. In this manner, spragging in an elastic system can cause a sprag–slip limit cycle similar to stick–slip. This is known as geometrically induced or kinematic constraintinstability, which occurs even though the coefficient of friction is constant.

Notice that the variation of friction force in the above limit cycle theory is achieved by varyingthe normal force, instead of the varying coefficient of friction. The sprag–slip is likely to happen insome practical systems. However, since it is difficult to use a theoretical approach to quantify thecomplex interactions associated with sprag–slip in many systems, the applicability of sprag–slip toquantitatively address practical problems has been limited.

V

Rigid rodpivoted at O

Moving plate

O

L

Ff

FN

q

FIGURE 4.45 Schematic of the beam model for spragging.



On the other hand, sprag–slip theory does provide some good insight into themechanismof certaintypes of nonstationary vibrations and sound induced by friction. Sometimes it can be used to highlightthe physical phenomena that occur in real systems where other mechanisms are invalid. In thefollowing we use an example to elaborate on how to treat the sprag–slip in a system, as shown in [87].

At first, let us look at some conventional experiments summarized in [87]. Historically, therewere lots of results of pin-on-disk experiments. A variety of friction-induced vibration and instabil-ity could occur depending on the value of the normal load and system conditions. Some typicalcases include the linear or nonlinear increase of the frictional force with the normal load and theconstancy or increase of the coefficient of friction with the normal load; all of these are characterizedby small amplitude random vibrations of the pin in the tangential, normal, and torsional degrees offreedom, which fall in the weak contact regime. Some other typical cases include the intermittentvariation of the friction force, the mean friction force at a sufficiently high value associated with atemporary burst of self-excited vibrations, and the fall of the friction force to a low value. Othercases include the self-excited vibration associated with the mean friction force at a very low value,which is accompanied by high amplitude periodic self-excited oscillations. These cases are charac-terized by strong contact where the normal load results in an unstable limit cycle. The source of suchlimit cycles is the nonlinearity due to both the nonlinear contact forces and the coupling between thedegrees of freedom. These experimental results do not cover the effect of the sprag–slip phenom-enon. Sprag–slip could result in oscillations or squeal sounds that occur at numerous frequenciesassociated with nonlinearity phenomena. There are sudden jumps in frequency in a single squealassociated with simultaneous changes in the friction coefficient. In [87], the experiments on a pin–disk model characterize the interfacial forces between the pin and the rotating disk, in which themeasured normal and friction forces were essentially random processes. But the normal and frictionforces have different features when the disk reverses rotation. The dependence of the root-mean-square of the friction force on the relative velocity for clockwise rotation was differentfrom that obtained for counterclockwise rotation. When the disk rotates clockwise, the sprag–slipphenomenon was observed. It is the structural misalignments and surface irregularities that result inthe sprag–slip occurrence in the pin–disk, despite the apparent angle of attack between the frictionelement and the disk surface being set at 908. When the disk rotates clockwise, the actual angle ofattack between the friction element and the disk surface is less than 908. On the other hand, if therotation is counterclockwise the angle of attack becomes greater than 908. Obviously if the attackingangle is less than 908 the mechanism of sprag–slip is created due to the generation of a strongrestraining force between the disk and the friction element. The friction element may experiencesevere friction-induced vibrations at an attack angle less than 908. The surface asperities only lead toan additive small random component to the normal load.

Figure 4.46 shows time history records of normal and friction forces for disk speed of 3 rpmclockwise. The kinematic constraint takes place with clockwise rotation due to misalignment. Thenormal force is set at a constant of 55 N, but the real time history indicates irregular fluctuations in awide range. Particularly, occasional noncontact could take place when the friction element losescontact with the disk surface. As the constraint force increases due to asperity on the disk surface,the normal and the friction force increase. Figure 4.47 shows the probability density function of thefriction force, which is essentially non-Gaussian.

When the disk rotation is reversed to counterclockwise, the angle of attack becomes greater than908. The constraint force is not significant and the interfacial forces experience high-frequencyfluctuations over those disk zones with surface asperity. The contact forces exhibit slight randomfluctuations.

The coefficient of friction–velocity curves for clockwise and counterclockwise disk rotations areshown in Figure 4.48. The friction–velocity curve for the clockwise case has a higher negative slopeat low values of relative speed than that of the counterclockwise case.

In addition to the experimental observation, [87] also presents an analytical stochastic model tocorrelate with the experimental results. The analytical model consists of a rigid beam with its free



0

140

105

70

35

0

40

10

−20

−50

−8020

Time (s)

Fric

tion

forc

e (N

)N

orm

al fo

rce

(N)

40 60

FIGURE 4.46 Time history records of normal and friction forces for disk speed of 3 rpm clockwise. (Reused withpermission from Qiao, S.L. and Ibrahim, R.A., J. Sound and Vib., 223, 1, 115, 1999. Copyright 1999, Elsevier.)

0.05

0.04

0.03

0.02

0.01

−90 −60 −30 0

Friction force

Pdf

(fr

ictio

n fo

rce)

30

FIGURE 4.47 Probability density function of the friction force: measured curve (solid line); Gaussian curve(dotted line).

0

0.28

0.56

0.84

1.12

1.4

Disk velocity (m/s)

0 0.009 0.018 0.027 0.036 0.045

CO

F

FIGURE 4.48 Coefficient of friction–velocity curves for clockwise disk speed (dot line) and counterclockwisedisk speed (solid line).



end facing the sliding disk while the other end is pivoted at O as shown in Figure 4.45. Theexception is that the model allows a torsional spring of stiffness ku to attach the beam. The endof the rigid beam in contact with the disk is subjected to random normal force N(t) and tangentialforce F(t). For a small angle u, the governing equation of motion of the beam can be written as

€uþ 2jv0 _uþ v20 � a(t)

� �u ¼ �b(t) (4:164)

wherev0 ¼

ffiffiffiffiffiffiffiffiffiffiku=I0

pa(t)¼ LN(t)=I0b(t)¼ LF(t)=I0L is the length of the beamI0 is the moment of inertia of the beam about the point Ou is the rotational angle of the beam measured from the static equilibrium position u0

Both clockwise and counterclockwise frictions are modeled using similar friction law but withdifferent parameters

mc(vr) ¼ mcs sgn(vr)� ac1vr þ ac3v3cr for clockwise (4:165a)

m(vr) ¼ ms sgn(vr)� a1vr þ a3v3r for counterclockwise (4:165b)

The friction and normal force have the following relationship:

b(t) ¼ m(vr)a(t) (4:166)

Note that Equations 4.164 through 4.166 are nonlinear stochastic differential equations. Thenonlinearity is due to the nonlinear friction and the random parametric excitation of the normalforce. The interfacial forces are random, nonstationary, and essentially non-Gaussian processes asshown in Figure 4.47. These types of systems can be solved using the stochastic averagingmethod. The amplitude extreme for both clockwise and counterclockwise disk speed cases canbe solved in terms of disk velocity and friction spectral density level. The amplitude extreme ofthe response probability density for the clockwise case was more complicated than for thecounterclockwise case due to the sprag–slip phenomenon. The friction statistical parameters forthe clockwise case are significantly different from those of the counterclockwise case. The resultsof both cases approach each other as the disk speed increases.

4.4.7 RUBBING SOUND WAVE UNDER WEAK CONTACT

Next we discuss the properties of stress wave propagation or solid sound in elastic continuumgenerated by rubbing two objects under weak contact. As the sound is generated due to the rubbingbetween two contacting rough surfaces of the objects, it is necessary to account for the role playedby the surface roughness. As the roughness and interface interaction is considered to be random, thekey effort is to establish the spectral formula between the response and the random roughness input.The established spectral formula shows that the natural mode of vibration or the tonal features of thesound can be generated due to the finiteness of the rubbing surfaces. It is shown that the vibrationsolutions or sound field can be expressed in terms of a series of natural modes, leading to tonalfeatures. In addition, the analysis shows that with increasing rubbing speed, more and more high-frequency modes can be excited and the frequency band keeps getting broader [88–90].

Consider the problem of two solids rubbing together. The schematic is shown in Figure 4.49.One of the surfaces is moving with a constant velocity along the positive y axis, while the other is



assumed to be at rest. For simplicity, we assume that the two objects are identical. For differentsizes, the effective rubbing surface will be equal to that of the smaller one. The shape of the objectsis cubic with thickness L, width W, and height H along the x, y, and z axes, respectively.

When rubbing occurs, a shear displacement field will be generated at the surfaces and appear asthe shear waves. Such shear waves can leak out through defects or radiate out at the boundaries, andtransform into the air sound.

The governing equation for the shear waves inside the elastic continuum is

@2

@t2� c2sr2

� u(x, y, z, t) ¼ 0 (4:167)

where cs is the shear speed of the objects. The boundary conditions are

@u

@z

��z¼0,H

¼ 0, ujy¼0,W¼ 0 (4:168)

@u

@x

��x¼L

¼ 0, G@u

@x

��x¼0

¼ F(y, z, t) (4:169)

whereG is the shear modulus of the materialF(y,z,t) is the surface shear stress

The shear stress can be separated into two parts: the mean and the variation

F(y, z, t) ¼ F(y, z, t)h i þ DF(y, z, t) (4:170)

z

y

H

W

Lx

v¡

FIGURE 4.49 Schematic of rubbing surfaces of two solids.



whereh�i denotes the ensemble averagehF(y, z, t)i is independent of y, z

The averaged stress can only excite the uninteresting zero-mode, and will thus be ignored. ByFourier transformation

u(x, y, z, t) ¼ðe�ivt u(x, y, z,v) dv (4:171)

Equation 4.167 becomes

(r2 þ k2s ) u(x, y, z,v) ¼ 0 (4:172)

in which ks¼v=cs. The general solution to Equation 4.172 with Equations 4.168 and 4.169 is

u(x, y, z,v) ¼X1

n¼0,m¼0

Amn cos kmn(x� L)½ � sin(kmy) cos(knz) (4:173)

in which

kn ¼ np=H, km ¼ mp=W , kmn ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2s � k2m � k2n

q(4:174)

and m, n are positive integers. The coefficient Amn is determined from

X1n,m

kmnAmn sin (kmnL) sin(kmy) cos(knz) ¼ F(y, z,v)=G (4:175)

Amn ¼ 4HW(1þ dn0)(1þ dm0)

1kmn sin (kmnL)G

ðH0

dz

ðW0

F(y, z,v) sin(kmy) cos(knz) dy (4:176)

To obtain the solution, we need to have the correlation function of the fluctuation part of the stress atthe surface, hDF(y, z, t)DF(y0, z0, t0)i. The stress fluctuation is caused by the roughness of thecontacting surfaces. If we assume that the roughness is homogeneous and completely random,i.e., spatially uncorrelated at different points when the system is at rest, we have

DF(y, z, t)DF(y0, z0, t0)h i ¼ Sd(z� z0)d y� y0 � V(t � t0)½ � (4:177)

where S is a strength factor. We finally get

u(x, y, z, t)u*(x, y, z, t)h i ¼ S

G2

ð Xn

cos2 (knz) Qn(x, y,v)j j2 dv (4:178)

in which

Qn(x, y,v) ¼Xm

2W sin(kmnL)

cos(kmn(x� L)) sin(kmy)km

(v=V)2 � k2m(�1)meiWv=V � 1h i

(4:179)

Therefore the frequency spectrum of the sound intensity field generated by rubbing is

P(v) ¼ S

G2

Xcos2(knz) Qn(x, y,v)j j2 (4:180)



A few general features can be observed from the spectral formula. First, the strength factor Scontrols the overall sound intensity level. It depends on a few parameters, such as the frictioncoefficients and the mechanical properties of the surfaces. Second, due to the factor sin(kmnL) in thedenominator of Qn(x,y,v) in Equation 4.179, the resonance could take place when sin(kmnL)¼ 0,which causes the phenomenon of tonal sound. This also shows that the thickness mainly defines theresonance feature. Next, the rubbing speed V enters Qn(x,y,v) in the form of v=V in the denomin-ator. For a fixed frequency, the decrease of moving speed will reduce the strength of Qn(x,y,v).Therefore with decreasing speed, high-frequency components tend to decay. Furthermore, we have

kmn ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(v=cs)

2 � (mp=W)2 � (np=H)2q

(4:181)

Then the cutoff frequencies are determined by

(v=cs)2 � (mp=W)2 � (np=H)2 � 0 (4:182)

The frequency spectra for a given system with two different moving speeds are plotted in Figure 4.50.The units are arbitrary. Here it is clearly shown that there is indeed a cutoff frequency at about 200for the lower speed curve. Also the discussed resonance feature does appear for the lower speedcase. For the higher moving speed, the spectrum is broader, but the resonance tends to be weaker.

In Chapter 5, we will further explore the application of using rubbing sound to detect interfacephenomena. In many applications, it is required to empirically decide the relationship between thesound and interface parameters. One of the applications is to quantify the relationship betweenindustrial noise and surface roughness.

Machines with moving parts inevitably produce noise when in operation. Noise can begenerated as a result of the occurrence of mechanical resonances due to force excitation, or a resultof the instability due to self-excitation at particular speeds. In weak contact, a contribution to thenoise spectrum can arise as a direct result of surface roughness. Sometimes surface roughness couldbe noticeable sources of noise; for instance, noise generated by the elastic deformation and releaseof form and waviness features during rolling. This type of noise can be generated in the elastohy-drodynamic lubrication case, where at high speeds it contributes to the noise spectrum. The shocknoise could be caused by the elastic deformation and release of asperities within the Hertzian contactzone. This form of noise could arise in both rolling and sliding contacts. This type of surface-generated noise may affect a broad range of frequencies. There are some efforts made to identify theinterface parameters by analyzing the noise, by which the surface contact sliding–generated noise

00 100 200 300 400 500

0.6

1.2

1.8

2.4

3

Frequency

Fre

quen

cy s

pect

rum

V

2V

FIGURE 4.50 Frequency spectra for two rubbing speeds.



may be valuable as a form of ‘‘noninvasive’’ surface monitoring. For instance, it has been shownthat interface-generated noise can be used to detect wear and to diagnose potential failure in avariety of interface contacts.

For stylus sliding on material surface, it was found that the filtered noise signal within a certainspectrum bandwidth contains a specified frequency at which the amplitude is maximum. Thisfrequency, called the dominant frequency, was found to be a material constant independent ofsurface roughness and contact load. The dominant frequency of a given material is proportional tothe sonic speed of that material. Experiments show that the sound pressure level of noise depends onsurface material, roughness, sliding velocity, and spring load. As a typical case, consider a springwhose end has a tungsten carbide tip rubbing against steel, brass, or aluminum material; therelationship between the generated sound pressure levels (SPL) and average roughness Rn wasfound to be SPL ¼ (Bþ CF)R1=n

n where F is the contact force and B, C, and n are experimentalparameters [2]. This indicates that SPL can be used as an alternative means of quantifying theaverage surface roughness at a given location on the surface. It is found that SPL has a peak value ata given frequency depending upon the material under investigation. The variation in surfaceroughness and contact load generally alters only the magnitude of the maximum SPL, but not thefrequency at which this maximum occurs. For tested samples, different materials have differentdominant frequencies, which vary linearly with the sonic speed, v, as well as the speed of wavepropagation over the surface vR (Rayleigh waves). v ¼ ffiffiffiffiffiffiffiffiffiffiffi

Eg=rp

, vR ¼ 0:9194ffiffiffiffiffiffiffiffiffiffiffiGg=r

pwhere E is

modulus of elasticity, G shear modulus, r specific weight, and g gravitational acceleration.On the other hand, friction-induced acoustic emission has been used to investigate interface

interactions. Acoustic emission (AE) is a technical term for the ultrasonic sound emitted bymaterials or structures when they are subjected to dynamic stress. The types of stresses can bemechanical, thermal, or chemical. This acoustic emission or stress wave propagation usually isultrasonic and is caused by the rapid release of energy from a localized source or sources within amaterial or structure due to events such as crack formation, and the subsequent extension occurringunder an applied stress, generating transient elastic waves that can be detected by proper sensors.AE is the generation of transient elastic waves in solid due to the rapid release of stored energy froma located source. In principle, from the detected AE signals, the locations and activity level of thesources can be determined in real time by reversal analysis, and applied to evaluate the structure andinterface interaction.

AE has been used as material investigation technique that is extremely sensitive and can detectdefects such as a few atom movements and is sensitive to dynamic microscopic process. AE canthus provide the early information on defect or deformation in any material or structure. If theatomic bonds break during an integrity test, the energy released propagates through the materialaccording to the laws of acoustics. AE has been used in industry to detect growing defects instructures, in pressure vessels, welding applications, and corrosion processes, various processmonitoring applications like manufacturing, global or local long-term monitoring of civil engineer-ing structures, and fault detection in rotating elements and reciprocating machines. It has been oneof the critical means to monitor the integrity of a structure or components in service, as anondestructive testing technique. Particularly, AE has been the standard inspection method ofchoice for disk surface screening and quality control in information storage industry. We willfurther elaborate this in Chapter 5.

4.4.8 INSTABILITY OF WAVES UNDER FRICTIONAL SLIDING OF STRONG CONTACT

We next discuss the stability of waves in elastic medium under steady frictional sliding of strongcontact. For convenience we limit our discussion on the stability of waves in periodic linear elasticcontinuum under frictional sliding.

The illustrative model is a one-dimensional elastic system with distributed contacts and periodicboundary conditions, which was worked out by Jung and Feeny [91]. The solutions of the waves



show that finite elastic systems with periodic boundary conditions and a constant coefficient offriction can have unstable steady sliding.

We consider a linear elastic continuum, placed between a moving surface and a frictionlesslinear bearing, representing a one-dimensional, undamped, continuous system in distributed slidingcontact, as shown in Figure 4.51. The friction coefficient is assumed to be constant, and thenonuniform motions, such as stick–slip motion or loss of contact, are not included.

For a homogeneous undamped linear elastic continuum, the equation of motion in axialdirection can be written as

A@sx(x,t)

@xþ f (x,t) ¼ r

@2u

@t2(4:183)

whereA is a cross-sectional area of an elastic continuum with width w and height hr is the mass per unit lengthsx(x,t) is a stress over the cross sectionu(x,t) is an axial displacementf(x,t) is a friction force per unit length

In view of the linear stress–strain relationship, stress can be expressed as sx(x,t)¼E«x(x,t), inwhich E is Young’s modulus. Axial stress could cause an area change of open elastic continuum dueto the Poisson effect. However, the system here is constrained geometrically, and thereby experi-ences a change in contact normal stress. The friction force including the Poisson ratio effect per unitlength is expressed as

f (x,t) ¼ �mwsy(x,t) ¼ �mw s0 þ vsx(x,t)f g (4:184)

where m is the coefficient of friction, sy(x,t) is a contact normal stress, v is the Poisson ratio, and s0

is a preloaded normal stress per unit length, which should always be less than zero due tocompression, so as to create friction stress and maintain contact with the sliding rigid body. Thisdistributed friction force contributes to the axial stresses in the continuum through Equation 4.183.As such, the friction stress is modeled to be proportional to the normal stress. The normal stressconsists of a static load and a variation portion due to the axial stress. This is accompanied byvariations in pressure on the continuum constrained between a sliding surface and a bearing. In viewof the linear strain–displacement relation

«x(x,t) ¼ @u(x,t)=@x (4:185)

V

u(x,t)x

l

Moving plate

Elastic continuum

Frictionless bearing

Rigid

FIGURE 4.51 A schematic diagram for a one-dimensional elastic medium subjected to distributed friction.



The nondimensional equation of motion can be derived as

@2u*@x*2

� a@u*@x*

þ ab ¼ @2u*@t*2

(4:186)

The dimensionless parameters used in Equation 4.186 include a¼mwvl=A¼mvl=h, b¼�s0l=vE,u*¼ u=l, x*¼ x=l, t* ¼ t

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirl 2=AE

p, in which l denotes contact length, and u*, x*, t* are the

dimensionless displacement, coordinate, and time, respectively. For convenience, the notation* will be neglected in the next analysis. The periodic boundary conditions are given by

u(0,t) ¼ u(1,t),du(0,t)dx

¼ du(1,t)dx

(4:187)

For the study of elastic waves, we seek a dynamic solution of the equation of motion with respect toa rigid-body solution of Equation 4.186. To this aim, we let

u(x,t) ¼ ur(t)þ u(x,t) (4:188)

The rigid-body solution is

ur(t) ¼ u0 þ v0t þ «bt2=2 (4:189)

This solution in reality would be physically limited by damping or restraints. The continuum couldbe grounded with a distributed spring, but the above analysis does not introduce the grounding so asto give a direct comparison with the problem of fixed boundary conditions. It is noted thatgrounding the system with springs will affect the details in the stability analysis, but the generalphenomenon is similar. The dynamic equation of motion can be written in self-adjoint form as

@

@xe�ax @u

@x

� ¼ e�ax @

2u

@t2(4:190)

in which system parameter a is a combined factor contributed by constant coefficient of friction andthe Poisson ratio. In view of the periodic boundary conditions (4.187), the solution is assumed tohave the form

u1(x,t) ¼ Real ei2pk(x�ct) �

(4:191)

in which k is a positive number representing the angular frequency of solutions along the x-axis, asthe value 1=k represents the wavelengths along the x-axis. The integer values of k satisfy the periodicboundary conditions. c can be a complex value and plays an important role in dynamic systemstability. In the case of a real value of c, pure waves of constant shape could occur. This implies thatconservative nondispersing waves exist in the elastic medium and the system is in a neutrally stablestate without damping. Moreover, a complex value of c contains information about the character-istics of the waves. Assume c¼R þ Ii; Equation 4.191 can be expressed as

u(x, t) ¼ Real ei2pk(x�Rt)e2pkIt �

(4:192)

A positive R indicates that there is a wave propagating toward the positive direction and a positive Iindicates that there is an unstable wave which increases in amplitude exponentially in time.Moreover, a negative R indicates that there is a wave propagating toward the negative direction



and a negative I indicates that there is a stable wave which decreases in amplitude exponentially intime. Therefore, the imaginary component of the characteristic solution represents the stability of thewave. Substituting Equation 4.192 into Equation 4.191, the characteristic equation of c is obtained

c2 � 1þ a

2pki

� �¼ 0 (4:193)

The imaginary and real parts of the characteristic solution for c are

R ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ (a=2pk)2

p2

s(4:194)

I ¼ a

4pkR¼ a

4pk

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ (a=2pk)2

p� �=2

r (4:195)

If there is no friction, the characteristic solution would have pure real solutions for c and thetraveling waves are pure sinusoidal functions, which retain their wave shapes in time.

When constant friction is included, the characteristic equation yields general complex solutionsfor c. Based on Equation 4.195, unstable waves propagate toward the positive x-axis, as indicated by apositive value of R, since they have positive imaginary components in the characteristic solution. Onthe other hand, any wave that propagates toward the negative x-axis is stable. Figure 4.52 shows theimaginary and real parts of the characteristic solution corresponding to an unstable wave as a functionof parameter for various undetermined frequency factors k. Asa increases, waves traveling toward thepositive x-axis are increasingly unstable in any finite sliding velocity. With respect to the travelingdirection of unstable waves, the direction of the moving rigid body indicates the direction of theunstable waves. It is noted that the instability occurs even if the coefficient of friction is constant.

0.7 1.20

1.18

1.16

1.14

1.12

1.10

1.08

1.06

1.04

1.02

1.00

0.6

0.5

0.4

0.3

0.2

0.1

00 5 10 0 5

aa(b)(a)

Imag

inar

y pa

rt o

f c

Rea

l par

t of c

10

k =1

k =1

k =2

k =2

k =3

k =3

k =10

FIGURE 4.52 The unstable characteristic solutions for the undamped, periodic boundary conditioned model:(a) imaginary part; (b) real part of the characteristic solution vs. a. (Reused with permission from Jung, C.M.and Feeny, B.F., J. Sound and Vib., 252, 5, 945, 2002. Copyright 2002, Elsevier.)



The above analysis shows that finite elastic systems subjected to distributed friction and periodicboundary conditions can also be unstable in the presence of a constant coefficient of friction. Underperiodic boundary conditions, unstable traveling waves in the one-dimensional elastic system werefound to be dependent upon a constant coefficient of friction and the Poisson ratio. In real situations,such unstable waves are expected to lead to nonuniform motions, such as stick–slip oscillations orloss of contact in materials. The characteristic analysis showed that dynamic instability occurs in theform of self-excited, unstable, traveling waves. Low- and high-frequency terms of the travelingwaves are stabilized by adding external and internal damping, respectively.

Many general analyses of the instability of interface waves can be found in [92–97]. Byconsidering the properties of asperities of the contact surface and an infinite beam subjected todistributed friction, which was modeled mathematically as a fourth-order partial differential equa-tion, it has been proved that one-dimensional traveling waves cause system instability. The interfacewaves and the transient response of a three-dimensional sliding interface with friction have beencharacterized. In the problem of the elastic half-spaces sliding with friction, a rich variety ofdynamic behaviors exist due to the nonlinearity of the system as it undergoes sticking and slippingevents. Adams has shown that steady sliding between a dissimilar material pair under constantfriction is unstable with respect to two-dimensional perturbations in the direction of the originalmotion for a wide variety of material pairs and frictional levels [93–96].

For the dynamic response of two flat sliding layered half-spaces with a constant coefficient offriction and a wide range of material combinations, there are friction-induced dynamic instabilities.The magnitude of these dynamic instabilities depends on the shear modulus ratios and on themismatches of the shear wave speeds of the materials. The greater the mismatch in shear wavespeeds, the less likely is stick–slip as opposed to loss of contact. These dynamic instabilities are dueto destabilization of waves at interface, and they can eventually lead to partial separation or toregions of stick–slip. The higher the contact pressure and the slower the sliding speeds, the higherthe likelihood of stick–slip occurrence. The dynamic normal and shearing stresses can be greaterthan the nominal steady-state stresses. The dynamic stresses fluctuate rapidly as they travel with theslip-wave velocity. Thus these dynamic stresses can affect the delamination of surface layers.

The effect of surface waviness on steady sliding has been quantified. It has been shown that thespeed and the friction coefficient can alter the contact area. It was found that stick-and-slip waves(a superposition of slip waves which forms a rectangular wave in the contact pressure and in the slipvelocity) could propagate along the interface and allow for the interface sliding conditions to differfrom the observed sliding conditions. The slip waves (generalized Rayleigh waves) may be generatedby the discovered instability mechanism of steady sliding. It was found that the apparent coefficient offriction could be less than the interface friction coefficient. Furthermore, the apparent frictioncoefficient can decrease with sliding speed even though the interface friction coefficient is constant.Hence the measured coefficient of friction does not necessarily represent the behavior of the slidinginterface. It is found that that slip-wave destabilization provides a physical mechanism, which iscapable of predicting a decrease in the coefficient of friction as a function of sliding speed. Finally,the presence of slip waves may make it possible for two frictional bodies to slide without a resistingshear stress. It is emphasized that many other effects may influence the negative slope of theapparent-coefficient-of-friction vs. velocity curve. It has been shown that the presence of slip wavescan at least partially explain this commonly encountered friction phenomenon.

The influence of body waves on the steady sliding of two elastic half-spaces has beeninvestigated. It was shown that steady sliding is compatible with the formation of a pair of bodywaves (a plane dilatational wave and a plane shear wave) in each body radiated away from thesliding interface. This phenomenon exists if the friction coefficient is greater than a certainminimum value and occurs with a speed-dependent or speed-independent friction law. Each wavepropagates at a different angle such that the trace velocities along the interface are equal. The anglesof wave propagation are determined by the elastic properties, the densities, and by the coefficient offriction. The amplitude of the waves is subject only to the restriction that the perturbations in



interface contact, pressure, and tangential velocity satisfy the inequality constraints for unilateralsliding contact. The possible existence of a slip pulse, formed by a superposition of these waves, hasbeen established. Such a pulse constitutes a propagating slip zone in which the frictional slipcondition is satisfied. The formation of such a slip zone, surrounded by regions of stick, allowsfor the apparent friction coefficient to be less than the interface friction coefficient.

It has also been found that an incident rectangular dilatational wave can allow for relative slidingmotion of two elastic bodies with a ratio of remote shear to normal stress, which is significantly lessthan the coefficient of friction. This reduction in friction is due to periodic stick zones, whichpropagate along the interface. Thus, it is possible that dilatational waves can be generated and usedin order to cause two surfaces to slide with very little (ideally zero) or even negative friction.Furthermore, by changing the angle of the incident waves, the direction of sliding can be reversed.

The nature of this instability is related to the existence of the generalized Rayleigh waves, whichcause changes in normal traction. The amplitude grows until the interface either sticks or opens.

4.5 FRICTION-INDUCED VIBRATIONS AND SOUND IN SOCIAL LIFE,NATURE, SCIENCE, AND ENGINEERING

4.5.1 FRICTION MUSIC SOUND AND FRICTION SOUND SYNTHESIS

4.5.1.1 Music Sound of Bowed String Instruments

Bowed string instruments, such as violin, cello, and er-hu, make use of dry friction to create music[98–103]. Most musical instruments are designed to produce harmonic or near harmonic sounds.They have a clear pitch and can blend with other harmonic sounds in pleasing ways, which isrelatively rare in the natural environment.

Figure 4.53 is a schematic of a violin string played with a bow. The player usually controls bowvelocity, bow position, distance of bow contact point to the bridge, and the bow pressure (or theforce pressing the bow against the string), to generate the desired sound.

As the bow is drawn across a violin string with a uniform motion, the string could vibrate at itsnatural frequency. When the string moves in the same direction as the bow, the slip speed betweenthe string and the bow is low and the frictional force is high in the direction of motion. When thestring moves in the opposite direction to the bow, the velocity is high and the frictional force is lowin the direction resisting the motion of the string. Thus the vibration of the string is amplified.

Figure 4.54 is the schematic of an ideal motion of string at contact point. The angle variations ofthe string at the bridge are illustrated in Figure 4.54a. The string force at the bridge is shown inFigure 4.54b and the typical tone spectrum of the bridge force is depicted in Figure 4.54c. Thesawtooth force leads to response with discrete spectrum.

String

Bow

V

Contact point

FIGURE 4.53 Schematic of a violin string played with a bow.



The typical bowed string results in a sawtooth motion at the bowing point, a short slip phase anda long stick phase. The player must monitor the contact point, the ‘‘bow pressure,’’ and the bowvelocity with high accuracy, especially while playing close to the bridge. For a perfect tonal start thebow force and the bow acceleration must be accurately selected.

The sound of a violin depends on the transfer of vibrations from the bowed string through thebridge to the body of the instrument. The violin body acts as a resonator for the vibration generatedfrom the strings. The coupling of air cavity modes and top and back plate modes produces thecomplex filtering, which contributes strongly to the characteristic timbre of the violin. The violinbody acts as an acoustical amplifier which gives two kinds of amplifications.

Firstly the vibration of the strings results in the vibration of the body walls, since the vibratingstring produces a vibration force on the bridge, which is transmitted via the bridge to the top plateand thereafter to the entire body. Secondly, the resonances of the violin body give extra amplifica-tion at specific frequencies.

The bowing action divides a string into two approximately straight-line sections, on either sideof the ‘‘corner,’’ caused by the bow contact. During bowing, the string moves with the bow until thetension in the string exceeds the friction force, at which point the string separates from the bow untilthey reconnect. The corner travels along the string at the transverse speed of string waves; thisbehavior of the string describes its free vibrations. At each instant, the string maintains the twoapproximately straight-line segments between the corner and the fixed ends. Thus, the completemotion of the string during bowing consists of the classic circulating corner of the Helmholtzmotion, which is the homogeneous solution of the equations of motion, superposed on the particularintegral that describes a stationary corner at bow contact.

The structure of the violin often modifies the pure Helmholtz motion, the transverse motion ofthe string under ideal conditions, through the compliance of pegs at one end and the triangular

Time

(a)

(b)

(c) f

An

f1 f2 f3 …

Time

Str

ing

vibr

atio

nS

trin

g fo

rce

FIGURE 4.54 String motion at contact point: (a) angle variations of string at the bridge; (b) string forces at thebridge; (c) tone spectrum of bridge force.



tailpiece at the other end. As the pegs and tailpiece deform, the tension and length of the stringchange to make the string vibrations quasi-harmonic. In addition to such nonlinearities, the torsionaland longitudinal vibrations of the string, as well as the vibration of the bow, also participate in thetransmission of structure-borne sounds through the bridge, thereby adding to the complexity of theproblem.

In the following, we present an analysis of the friction-induced torsional and transversalvibrations of the string, which was worked out by Leine et al. [103], as one of the illustrativeexamples out of numerous cases of violin sound investigations.

Consider a single violin string with radius r and a bow that moves at a constant velocity V overthe string as schematically shown in Figure 4.55. The friction force between bow and string willinduce the transversal displacement x and rotation angular displacement w to the string.

The equations of motion of string in the slip phase yield

m€xþ kx ¼ Ff (vr), J €wþ ktw ¼ Ff (vr)r2 (4:196)

in which vr ¼ V � _x� _wr. The equations were solved numerically and Figure 4.56a displays thephase portrait for the lateral vibration for d¼ 1 mm. For comparison, the periodic solution by using

V

kkt

r

x

m, J

j

FIGURE 4.55 A model of violin string.

1.5

1

(a) (b)

0.5

−1.5 −1 −0.5 0 0.5 1 1.5

x �10−2 (m) x �10−2 (m)

x (m

/s)

x (m

/s)

2 2.5 3 21.510.50−0.5−5

−4

−3

−2

−1

0

1

2

3

−1

0

−0.5

−1

−1.5

−2

⋅ ⋅

FIGURE 4.56 Phase portraits of the violin string model: (a) d¼ 1 mm; (b) d¼ 0.25 mm. (Reused withpermission from Leine, R.L., Van Campen, R.I., De Kraker, A., and Van Den Steen, L., Nonlinear Dynamics,16, 41, 1998. Copyright 1998, Springer.)



the corresponding single-degree-of-freedom system (only consider transversal displacement x)model is also solved and drawn with a dashed line in the same plot. We can see from the figurethat the single-degree-of-freedom system approximates the two-degree-of-freedom system modelvery well. However, if the diameter of the string is reduced to d¼ 0.25 mm, the qualitative behaviorchanges drastically as shown in Figure 4.56b. The solution of the single-degree-of-freedom systemdiffers significantly from the solution of the two-degree-of-freedom system model.

For the second case of d¼ 0.25 mm, two-degree-of-freedom system model should be used andno periodic solution was found for this case. The obtained solution is possibly chaotic or quasi-periodic, as shown in Figures 4.57 and 4.58 with the time histories and spectra, respectively. Thefrequency spectrum for the second case is quite different from the first one in which the periodicsolution is dominant. The second case has a broadband nature. Moreover, the frequency ratio of thetorsional vibrations relative to the lateral vibrations is proportional to the string diameter

vt

v1¼

ffiffiffiffiffiffiffiffiffikt=J

pffiffiffiffiffiffiffiffiffik=m

p ¼ffiffiffiffiffiffiffiGA

ft

s¼ d

2

ffiffiffiffiffiffiffipG

ft

s

Time (s)(a) (b)

Vel

ocity

(m

/s)

Vel

ocity

(m

/s)

Time (s)

2

2

00 0.010.01 0.020.02

00

−2

−2

FIGURE 4.57 Time histories: (a) d¼ 1 mm; (b) d¼ 0.25 mm. (Reused with permission from Leine, R.L., VanCampen, R.I., De Kraker, A., and Van Den Steen, L., Nonlinear Dynamics, 16, 41, 1998. Copyright 1998,Springer.)

−8

−14

−20

0

(a) (b)

5,000 10,000

Frequency (Hz) Frequency (Hz)

0

Am

plitu

de (

dB)

Am

plitu

de (

dB)

5,000 10,000

−30

−20

−10

FIGURE 4.58 Spectra: (a) d¼ 1 mm; (b) d¼ 0.25 mm. (Reused with permission from Leine, R.L., VanCampen, R.I., De Kraker, A., and Van Den Steen, L., Nonlinear Dynamics, 16, 41, 1998. Copyright 1998,Springer.)



The reason for torsional vibrations having significant influence for d¼ 1 mm is because the torsionalfrequency was much higher than the lateral frequency (vt=v1¼ 35.3). Decreasing the diameter ofthe string to d¼ 0.25 mm leads to the reduction of the frequency ratio (vt=v1¼ 8.8) and therebyresults in the significant effect of the torsional mode.

It is noted that the above model of the bow–string interaction assumes that friction depends onlyon the relative velocity between the bow and the string, in which a velocity-dependent, exponentialform friction law similar to Equation 4.6 is applied.

However, it has recently been found that the actual behavior of friction at the bow point is morecomplicated than a simple friction vs. velocity dependence. The force and sliding velocity aremeasured during stick–slip motion with rosin at the interface, and the results of the plots in thefriction force–velocity plane are shown in Figure 4.59. It can be seen that the classic friction modelsthat consider friction as depending only on the relative velocity between the bow and the string arenot correct. In [100] it has been suggested that friction depends on the variations of temperature inthe interfacial rosin layer. An explanation for this behavior is that during sticking the contact regioncools by heat conduction into the bulk materials behind the contact. This allows the shear strength ofthe interface and the friction coefficient to reach a high value.

Once sliding begins, the contact region is heated due to friction, the rosin layer weakens, and thefriction coefficient falls. Consider the model for this behavior by assuming T as the contacttemperature in the shearing rosin layer; then the coefficient of friction can be expressed as [100,102]

m ¼ Aky(T)

Nsgn(vr) (4:197)

whereN is the normal bow forcevr is the relative speed between bow and stringA is the contact area between bow and stringky(T) is the temperature-dependent shear yield stress

The friction coefficient is plotted in Figure 4.59. This model behaves better than the ‘‘classic’’models. Moreover, further investigations on reconstructing the bowing point friction force confirmthe fact that friction does not depend only on the relative bow string velocity.

0.6

0.4

00 0.05

Velocity (m/s)

Coe

ffici

ent o

f fric

tion

(m)

FIGURE 4.59 Behavior of friction at the bow point. The thick curve represents the classic friction model; thethin line is adapted from [100].



4.5.1.2 Friction Sound Synthesis

The applications of friction acoustics have extended beyond conventional areas like musicalinstruments and machine noise control; this includes the applications in sound synthesis.

Sound effects have been widely used in computer graphics, animation, film, and gamesindustries. Conventionally, the creation of sound effects remains a slow, labor-intensive process,which has been done by talented sound designers. The desire to remedy this situation, by automat-ically synthesizing sound effects based on the physics of interaction, also has a long history. Therehave been many efforts to create certain types of sound effects due to contact and friction interactionby synthesized approaches automatically [104,105].

There are several methods to automatically generate realistic contact sounds by using physicalparameters obtained from a dynamics simulation with contacts and friction, using physicallymotivated sound synthesis algorithms integrated with the simulation. Once the model parametersare defined, the sounds are created automatically. This enables the use of interactive simulation toexperience realistic responsive auditory feedback such as those expected in real life when touching,sliding, or rolling objects.

In these kinds of efforts, the models of the contact and friction interactions are used as a basis forsound synthesis. The modal synthesis technique has been used for the interaction analysis toquantify sounds including continuous sliding contact sounds. The dynamic simulation algorithmfor computing contact forces, including microsimulation techniques, has been used to produce high-resolution ‘‘audio-force’’ from nominal contact forces produced by dynamics simulation at videorates. The contact force sampled at audio rate is referred to as the ‘‘audio-force’’ to distinguish itfrom the ‘‘dynamics-force’’ which is sampled much coarser.

The surface irregularities, object properties, sliding or rolling contacts, and sound are connectedthrough modeling using the fundamentals described in last two chapters. During scraping andsliding the audio-force is generated by combining an effective surface roughness model and aninteraction dynamics model. The audio-force will depend on the combined surface properties of theobjects in contact. A simple and effective model of scraping a surface is the phonograph needlemodel, whereby a needle exactly follows a given surface track. In order to generate the audio-forceat a sampling rate of fs, an effective surface profile at a spatial resolution of Vmax=fs is generated,where Vmax is the maximum contact velocity in the model. This ensures that at the maximum contactsliding speed the surface profile can get sampled at the audio sampling rate. For a homogeneoussurface this is easy to implement with a wave-table of a surface profile which will be ‘‘played back’’at the rate V=Vmax for a contact moving at speed V, where a rate of one represents the audio-force atmaximum speed. This wave-table can be constructed from a mathematical roughness mode. Thewave-table should be oversampled in order to minimize the reduction of quality on slowdown. Forresampling a linear interpolation algorithm can be used. Spurious high-frequency componentsresulting from the discontinuities of the first derivative can be reduced by using a quadraticinterpolation scheme. Such a model is appropriate for scraping a rough surface or for slidinginteraction. For the simplest case, Coulomb friction, the relationship Ff ¼mFN is used. The scrapingaudio-force volume is approximated to be proportional to

ffiffiffiffiffiffiffiffiffiVFN

p, by assuming the acoustic energy is

proportional to the frictional power loss [105].

4.5.2 FRICTION SOUND IN NATURE

4.5.2.1 Creature Friction Sound

Nature offers many friction sounds for creatures such as fish, crustacean, and insects [106]. Theyhave developed through evolution to meet the needs of communication. One of the mechanisms bywhich invertebrates produce sounds is through stridulation which refers to sounds that result fromthe rubbing of two special bodily structures.



The features of stridulation depend on the physical structures of the stridulatory apparatus thatcreatures have. The features of stridulation also depend on how creatures employ them with propermoving part speed, duration, and muscle force. The friction sound signatures of creatures are highlydiversified. For instance, the frequency content of friction sounds in insects range from 1 to 100 kHz.

It is well known that some spiny lobsters are able to create a loud, rasping buzz sound, by whichthey deter predators. To create sound, spiny lobsters waggle one or both of their antennae (horn),causing a flattened projection (the plectrum) on each of the antenna’s spiky base to rub across anoblong lump (the file) located on either side of the animal’s head, near the eye, as shown inFigure 4.60. In the following we introduce some basic observations by Patek [107,108].

The friction comes from microscopic shingles on the otherwise smooth files. Each time thelobster’s plectrum skids on the file, it produces a pulse of sound. As it travels the length of the file,the plectrum generates between 2 and 24 of these pulses, creating the characteristic raspy squeak.The duration of the sounds depends on the length of the file, which varies considerably from genusto genus. Most spiny lobsters have the special files and plectra, whereas a few do not have files andthus make no noise at all. The great advantage of the stick–slip approach associated in this soundcreation is that a soft structure rubbing against another soft structure works just as well right after amolt as it did beforehand.

The spiny lobster’s morphology reveals that the plectrum is made of soft tissue and the file lacksmacroscopic ridges. The pulsed sound, called the ‘‘rasp,’’ is made by rubbing two macroscopicallysmooth surfaces together, which produces sound through frictional interactions between the sur-faces. These frictional interactions are analogous to the frictional stick–slip mechanism in manyother systems. The spiny lobster’s soft tissue plectrum can be considered as a mobile mass–springsystem that moves over the stationary surface of the file. In this case, the plectrum consists of a massconnected between two springs, and the mass–spring system is pulled over the file. As it is pulled,the static friction between the plectrum and the file surface causes the two surfaces to stick relativeto each other. The two plectrum springs eventually extend or compress to a point at which the staticfriction is exceeded and the plectrum slide relative to the file again. The soft, elastic tissue of theplectrum resists compression and tension and probably stores energy during the stick phase andreleases energy during the slip phase. Figure 4.61 illustrates the recorded sound and movementcorrelation. Each pulse of sound is correlated with the movement of the plectrum. The stepped-likemotion of the plectrum over the file is noted.

File

Flap

Plectrum

FileFlap

Plectrum

FIGURE 4.60 Schematic of sound-producing mechanism found in the Caribbean spiny lobster. (Reused withpermission from Sally Ben Susen, Copyright Natural Magazine.)



The stick–slip mechanism is found at an intriguing location compared with most creatureacoustic mechanisms using adjacent rubbing surfaces. The sound-producing apparatus of creaturesis usually limited to one degree of freedom and does not permit translational motion between twoadjacent surfaces. However, the sound-producing spiny lobster generates sound by rubbing theplectra over the files through a translational movement of the proximal antennal joints. This is aninteresting phenomenon. It is believed that in the evolution of creatures, creatures not only devel-oped special morphology and pattern of two translating body surfaces, but also developed transla-tion motion of rubbing parts especially for producing sound. In an unusual structural rearrangement,some kind of spiny lobsters lost an antennal joint articulation, which transformed this joint frommoving with one degree of freedom into a sliding joint with multiple degrees of freedom. With thissliding joint, ‘‘stick–slip’’ sounds are produced by rubbing the base of each antenna against theantennular plate.

In some lobsters, some portion of the rubbing surface has an additional ridge and in the othertype, the rubbing surfaces are smooth. These differences in rubbing surface features suggest avariation in the stick–slip properties of the system. Translational motion permitted by the slidingjoint is necessary for sound production, and the construction of a sliding joint is a key modificationin the origin of this sound-producing mechanism.

4.5.2.2 Nature Sand Sound

4.5.2.2.1 Sound-Producing Sand AvalanchesSound-producing sand grain is one of nature’s more puzzling and least understood physicalphenomenon. Some type of loud and low-frequency sound of sand during avalanching has been

Sou

nd (

V )

(a)

(b)

Ple

ctru

m m

ovem

ent (

V )

Anterior

0 100 200 300

Time (ms)

Posterior

FIGURE 4.61 Sound and movement correlation: (a) sound; (b) stepped-like motion of the plectrum overthe file. (Reused with permission from Patek, S.N., J. Exp. Biol., 205, 2375, 2002. Copyright 2002, TheCompany of Biologists Ltd.)



the subject of desert folklore and legend for centuries. But a satisfactory explanation for any type ofsound-producing mechanisms is still unavailable and has remained a research topic in the scientificcommunity [109–111]. The sound of sand has been considered to be a kind of friction sound. Wenext describe the work by Sholtz et al. [109]. Sand sound occurs naturally in two different types:squeaking and booming sands. Both varieties of sand produce unexpectedly pure sound whensheared; they differ in their frequency range and duration of emission, as well as the environment inwhich they tend to be found.

There exist two distinct types of sand that are known to produce manifest sound when sheared.The more common one, called ‘‘squeaking’’ or ‘‘whistling’’ sand, produces a short duration(<1=4 s), high-frequency (500–2500 Hz) ‘‘squeak’’ when sheared or compressed. It is fairlycommon in occurrence, and can be found at numerous beaches, lakeshores, and riverbeds aroundthe world. The other, rarer type of sound-producing sand occurs principally in large, isolated dunesdeep in the desert. The loud, low-frequency (typically 50–300 Hz) acoustic output of this ‘‘boom-ing’’ sand occurs upon avalanching. Booming and squeaking sands each show a markedly differentresponse to water exposure. Booming occurs best when the grains are very dry, preferably severalweeks after the last rain. Small amounts of atmospheric humidity similarly create a liquid surfacecoating on the grains; squeaking sand that is visibly moist is not acoustically active either.

The mean grain size (diameter) of most sand is roughly 300 mm. The frequency of emissiongenerated by squeaking sand is thought to vary as the inverse square root of the mean grain size,although mean grain size by itself does not determine the ability of sand to sound. It is unlikely thatbooming frequencies depend similarly on grain size alone, as a fairly wide range of fundamentalfrequencies is often generated in large-scale slumping events. Particle-size distributions of sound-producing grains usually only extend over a narrow range, a condition that is called ‘‘being well-sorted.’’ Also, booming and squeaking grains tend to be both spherical and lack abrupt surfaceasperities. The latter condition is called ‘‘being well-rounded,’’ a term not to be confused here by ouruse of the word ‘‘polished,’’ which we take to mean granular surfaces which are smooth on the 1 mmlength scale. Both types of sand further exhibit unusually high shear strength, and in the case ofsqueaking sand, a decrease in shear strength has been shown to correspond to a decrease in soundingability. Experiments in which spherical glass beads produce sound emissions similar to those ofgenuine squeaking sand when compressed, albeit under somewhat contrived experimental condi-tions, provide further support for the notion that squeaking is caused primarily by frictional effects.The combination of the four grain properties, including size-sorting, sphericity, roundedness, andresistance to shear, is thought to be critical for the onset of squeaking. Booming, on the other hand,is substantially less sensitive to differences in grain shape and sorting, and is likely governed by theunusually smooth and polished surface texture present in all booming grains. Exactly what governseither sounding mechanisms is still an open question.

Most modern theories stress the importance of intergranular friction. It has been suggested thatthe unusually smooth and polished surfaces of booming grains may allow for exaggerated vibrationat the natural resonant frequency of sand. Differences in grain packing have also been considered.Some complete development concludes that both types of sound generation result from nonlinearoscillations of dispersive stress in the layer of moving grains, or shear layer. But this analysis isunable to address some aspects of the booming mechanism. In any case, a critical amount ofbooming sand must shear before any sort of sound is produced.

The relationship between intergranular frictional effects and sound production in booming sandbecomes clear upon actually sensing the tactile vibrations caused by the grains resonating in acoherent manner during a booming event.

It is likely that the cause of the sound generation is closely related to the frictional behaviorbetween the grains during shearing. It was proposed that intergranular frictional effects may createsound in certain types of squeaking sands. It has been concluded that the grains are in general highlyspherical, well-rounded, well-sorted, and unusually smooth, and postulated that sound producingmust result from collectively ‘‘rubbing’’ grains which exhibit these four properties.



4.5.3 FRICTION-INDUCED INSTABILITY AND SOUND IN GEOLOGICAL SCIENCE AND ENGINEERING

4.5.3.1 Stick–Slip and Dynamic Instability in Earthquakes

Earthquakes have long been recognized as resulting from a stick–slip frictional instability [20,21,112–117]. The development of the constitutive law of rock friction illustrates that the manyearthquake phenomena are all attributed to the manifestations of the friction law.

It has been observed that tectonic earthquakes usually occur by sudden slippage along apreexisting fault or plate interface, instead of sudden onset and propagation of a new shear crack.They are inherently a frictional phenomenon, with brittle fracture playing a secondary role in thelengthening of faults and frictional wear. This observation has been widely accepted and earth-quakes have been considered to be the result of a stick–slip frictional instability. Thus, theearthquake is the ‘‘slip,’’ and the ‘‘stick’’ is the interseismic period of elastic strain accumulation.A comprehensive constitutive law for rock friction has been fully developed and many other aspectsof earthquake phenomena have been related to the nature of the friction on faults. Many conven-tional theories like strength, brittleness, and ductility are covered by the related theory of frictionalstability to address earthquake-related phenomena.

It is noted that there is unique friction feature of rock which is different from the standard stick–slip model. Basically, the static friction ms depends on the sliding history. The static friction islogarithmically proportional to the static contact time. The dynamic friction md is dependent on thesliding velocity, V, when measured in the steady-state sliding regime. The dynamic friction is eitherpositively or negatively logarithmically proportional to V, depending on the rock type and certainother parameters like temperature. If it sustains a sudden change in sliding velocity, friction couldevolve to its new steady-state value over a characteristic slip distance L.

The aging of ms and the velocity dependence of md are inherently related behaviors whichresult from the creep of the interface and the resultant increase of real contact area with the increaseof time.

The critical slip distance L is considered to be a memory distance over which the contactpopulation changes. One of the rate and state-variable constitutive laws known as the Dietrich–Ruina law, is expressed as

t ¼ m�s ¼ m0 þ a lnV

V0

� þ b ln

V0u

L

� � �s (4:198)

where t is the shear stress and �s is the effective normal stress which is the applied normal stress minuspore pressure. u is the state variable. a, b are the values of the constitutive parameters and the nominalcoefficient of friction m0 is the steady-state friction at V¼ 0. The parameter V0 is the normalizingconstant, L is the critical slip distance. Figure 4.62 shows the transient and steady-state changes ofcoefficient of friction, m as a function of slip speed V. The initial change in friction in response to astep change in the slip rate is scaled by the parameter a. The subsequent evolution of friction to a newsteady-state level is scaled by the parameter b. The velocity dependence of steady-state friction is scaledby a� b. The value of u evolves toward a new steady-state value according to the following formula:

du

dt¼ 1� uV

L(4:199a)

In the initial phase of the rate increase, there is an increase in friction due to the direct velocity effect.This is followed by an evolutionary effect with a decrease in friction, b. The friction at steady state is

t ¼ m0 þ a� bð Þ ln V

V0

� � �s (4:199b)



The change in steady-state friction during the step change V is

Dms ¼ (a� b) ln(V=V0)

The velocity dependence of steady-state friction is parameterized by a� b. The steady-state velocityweakening (a � b < 0) has been shown to be a necessary condition for unstable slip.

This form of friction does not seem to be very dependent on material; actually it has also beenillustrated to be applicable to some metals, paper, wood, and plastics. The former distinctionbetween ms and md disappears in this model. The base friction m0 has a value nearly independentof rock type and temperature. This model is modified by second-order effects involving a depend-ence on sliding velocity and a state variable u.

Frictional stability depends on two friction parameters, L, and the combined parameter (a � b),defined as the velocity dependence of steady-state friction

a� b ¼ @md

@ ln (V)½ � (4:200)

If a � b � 0, the material is considered to be velocity strengthening, and will always be stableintrinsically. No earthquake can nucleate in this field, and any earthquake propagating into this fieldwill produce a negative stress drop, which will rapidly terminate propagation.

On the other hand, if a � b < 0, the material is considered to be velocity weakening. There is avelocity jump, DV, necessary to destabilize the system as a function of the applied normal stress, �s.If �s is larger than the critical value, the system is unstable and if �s is smaller than the critical value,the system is conditionally stable. Earthquakes may nucleate only in the unstable field, but maypropagate into the conditionally stable field.

At the border of the stability transition there is a narrow region in which self-excited vibrationoccurs. There is a Hopf bifurcation between an unstable regime and a conditionally stable one.Consider a simple spring–mass model as shown in Figure 4.63a, in which there is a fixed stiffness kand a connected mass m, and the friction follows the rate=state-variable friction law. The stability ofthis system depends entirely on �s, t, k, the friction parameters (a � b) and L, and is independent ofbase friction m0.

The bifurcation occurs at a critical value of effective normal stress, �sc, given by

�sc ¼ kL

�(a� b)(4:201)

0.72

0.74

0.76

0.78

0.80

0.82

0 40 80 120 160 200

Displacement (µm)

Coe

ffic

ien

t o

f fr

ictio

na

b

a − b

V = 1 µm /sV = 10 µm /sV = 1 µm /s

FIGURE 4.62 Transient and steady-state changes of coefficient of friction. m as a function of slip speed V.



If s > �sc, the sliding motion of the mass is unstable under quasi-static loading. In the conditionallystable regime, s < �sc, the sliding motion is stable under quasi-static loading but it can becomeunstable under dynamic loading if subjected to a velocity jump exceeding DV. In a narrow region atthe bifurcation, the sliding motion is a self-excited oscillatory motion as shown in the shaded regionin Figure 4.63b. The friction law can also be written in several ways which differ in detail, but thosedetails do not influence the above definitions of the stability states, which control the seismicbehavior of faults.

With this model, the earthquake-producing mechanism can be interpreted as a friction-inducedinstability. Earthquakes can nucleate only in those regions of a fault that fall within the unstableregime. They may propagate indefinitely into the conditionally stable region, provided that theirdynamic stresses continue to create a large enough velocity jump. If earthquakes propagate into astable region, there will be a negative stress drop and large energy release, and the propagation ofthe earthquake will stop quickly.

The critical parameter for stability (a � b) is a material property, which is dependent ontemperature and normal pressure. For instance, the dependence of (a � b) on temperature forgranite is negative at low temperatures and is positive for temperatures above about 3008C. Anotherexample, halite (rock salt), undergoes the same transition at 258C and a pressure of about 70 MPa.These observations indicate that for faults in granite, we should not expect earthquakes to occurbelow a depth at which the temperature is 3008C, and faults in salt should be seismic under almostall conditions.

Faults are not simply frictional contacts of bare rock surfaces, they are usually lined with weardetritus. The shearing of such granular material involves an additional hardening mechanism whichtends to make (a � b) more positive. For such materials, (a � b) is positive when the material ispoorly consolidated, but decreases at elevated pressure and temperature as the material becomeslithified. Therefore faults may also have a stable region near the surface, owing to the presence ofsuch loosely consolidated material.

4.5.3.2 Characterization of Sediment Grain Using Friction Sound

We next present the application of friction sound for ocean bottom survey. Conventional techniquesfor sea sediment grain characterization based on sediment core sampling and lab analysis are unableto determine sediment grain size quantitatively and rapidly with a high spatial resolution. In a fieldexperiment off the Dutch coast by Koomans and de Meijer [118], the level of friction soundgenerated by detector probe towed over the sea floor was used for sediment grain analysis. Theresults indicate that the friction-sound level reveals the characteristics of the seabed. Laboratoryexperiments show a one-to-one relation between sound level and the median grain size of thesediment, and enable to make the calibration for field test data.

kt

m

m

∆V

Unstable

StableOscillations

Unstable Conditionally stable

s

ssc

(a) (b)

FIGURE 4.63 (a) Simple model; (b) stable regime.



The validation experiments show that the technique can give very high-resolution qualitativeresults on sediment composition and has a large potential for the development of a detector for insitu quantitative monitoring of sediment grain size.

Conventionally, the underwater sediment investigation system has been using various sensors tomonitor sediment properties and is towed over the sediment bed. These measurements are ‘‘trans-lated’’ to sediment characteristics as sand=mud ratios via a ‘‘fingerprinting’’ method. The measure-ment system requires contact between the detector and the sediment bed, and it uses a microphone tomonitor this contact. The friction between sensor and sediment will generate noise (friction sound)during towing. The typical friction-sound levels are plotted in Figure 4.64. This figure illustrates thatthere are subareas with enhanced friction sound intensities in the surveyed area. The variations seemto be correlated to grain size, sand, or mud concentrations or variations in morphology.

The relation between sediment bed properties and sound generated by friction has shown thatthe amplitudes of the generated sound are a function of the median grain size of the soil, the packingand water saturation of the soil, and of the penetration rate. Moreover, the frequency distribution ofthe sound is constant and independent of grain size or penetration rate.

The properties of sediment including size, roundness, and sphericity are studied under con-trolled conditions for calibration. The ripple structures are also considered as the parameters thatmay influence the friction sound. The ‘‘bumping’’ of the detector system over the ripple structurescould change the normal sound levels.

The parameters that may affect friction sound include surface parameters and sliding conditions.The surface parameters consist of sediment characteristics and ripple structures. The sedimentcharacteristics include sediment size, sediment type (shells=sand=clays), and sediment shape(rounded vs. angular). The ripple structures include small- and large-scale ripples. The slidingconditions include towing speed and shape=acoustic properties of detector direction of towing overmorphology.

The experiments have shown the correlation between grain size and sound intensity, as plottedin Figure 4.65. The sound level exhibits a spiky behavior, when ripple structures are present. In thesmooth region (between x¼ 190–220 m) and on the ‘‘seaward’’ part of the breaker bar such spikesare hardly present. If the sound level is affected by morphologic variations due to ripple structures,the length scale of the variations in sound level will be related to the length scale of the ripple

620

618

616

614

612

610

608

606

604

N

Ameland

170 172 174 176 178 180 182 184

X-axis (UTM; km)

Sound level (a.u.)

0 to 500500 to 1000

Y-a

xis

(UT

M; k

m)

186 188 190 192 194

FIGURE 4.64 Friction–sound level of the lines towed off the coast of the Dutch Frisian island Ameland.(Courtesy of Koomans, R. and de Meijer, R.J.)



structures. It is observed that small ripple structures (with a typical length and height of 0.5 and0.05 m, respectively) are flattened by the sledge of the detector. Therefore, the influence of mega-ripples (with lengths between 0.5 and 5 m and heights between 0.05 and 0.5 m) is investigated byapplying a Fourier transform to filter the sound level from variations with length scales less than5 m. Figure 4.66 shows the filtered sound pattern in comparison with the grain-size value.Apparently, the small-scale variations superimposed on the sound level do not have an influenceon the large-scale trend of sound level. It seems that the large-scale trend in sound level reflects thegrain-size variations of the bed whilst sound level variations correlated with small-scale changes inmorphology (like ripple structures) are superimposed as scatter on this trend. Whether these small-scale variations are directly the result of bumping on morphologic features or whether they arisefrom the very small-scale grain-size variations due to sediment sorting within ripple structurescannot be concluded from the present experiments.

To determine a relation between grain size and the measured sound levels, the three successivemeasurements are averaged and the friction-sound levels are averaged over the region of each

100

75

50

25Sound

Profile

100 150 200 2500

1

2

3

4

500

d 50

(µm

)

Hei

ght (

m)

Grain size

Position (m)

Sou

nd (

a.u.

)400

300

200

1005

FIGURE 4.65 Friction–sound level (lines) of two repeated measurements and grain size (dots) distribution(upper plot) along the profile (lower plot) in the LWF, Germany. (Courtesy of Koomans, R. and de Meijer, R.J.)

500

5

4

3

2

1

0

400

300

200

100

250200150100

Profile

Sound

100

75

50

25

Position (m)

Sou

nd (

a.u.

)

d 50

(µm

)

Hei

ght (

m)

Grain size

FIGURE 4.66 Filtered sound pattern in comparison with the grain size value. (Courtesy of Koomans, R. andde Meijer, R.J.)



grain-size patch. These results are presented in Figure 4.67. The uncertainties in the sound intensityrepresent the standard deviation of the sound intensity over the specific patch. The results fromFigure 4.67 show that there is a positive relation between grain size and sound intensity by a linearcurve of the function, I¼ ad þ b, in which a and b are constants and d is the grain size. The resultsof these fits are also given in Figure 4.67. These results show that the sound intensity variessimilarly with grain size for the two velocities, but the curves are transposed for a higher towingvelocity: for a doubling in towing velocity, the sound intensities are on average 20% higher. Theseresults show that the friction sound intensities depend linearly on the grain size. The relation isindependent of towing velocity, but the magnitude of the sound levels is, at low speeds, determinedby the towing velocity.

4.5.4 FRICTION-INDUCED VIBRATIONS IN MEMS DEVICES AND EQUIPMENT

4.5.4.1 Ultrasonic Motor

In Chapter 2, we described the properties of the Rayleigh wave: the wave propagates backwardsclose to the surface, whereas it travels forward downwards the surface. This causes an ellipticalmotion for surface particle in the continuum. This principle has been used to design physical devicesto possess the function of moving component on its surface with micro- and nanolevel magnitude.For instance, ultrasonic motors are surface acoustic wave devices that are widely utilized incommunication instrument signal processing circuits for filtering, in cameras for autofocus, and inmechtronics for precise positioning. The driving force of the ultrasonic motor is elastic wave motionof the surface acoustic wave device, in which through the frictional force between rotor and statorsurface, mechanical output force in unidirection is converted from vibration [119].

The ultrasonic motor utilizes Rayleigh wave for friction driving. The high power density of theRayleigh wave at the friction surface results in high output force and high speed motion, whosemagnitude could be from several to tens of Newtons and up to several meters per second. The utilizationof nanometer-level small amplitude ofRayleighwave enables themotor to perform several or less than 1nm stepping motions. The schematic of a typical ultrasonic motor is shown in Figure 4.68.

The surface particles of the stator move in elliptical motion when the Rayleigh wave propagates.The rotating surface particles contact the slider surface at its wave crests. The particle motion, then,

Velocity 0.2 m/sVelocity 0.1 m/sVelocity 0.2 m/s

Grain size (µm)

Velocity 0.1 m/s

450400350350250200150100504

5

6

7

8

9

Sou

nd in

tens

ity

FIGURE 4.67 Correlation between sound level and grain size of the patches. (Courtesy of Koomans, R. andde Meijer, R.J.)



drives the slider to move linearly through frictional force. For instance, in a typical device [120], theRayleigh wave amplitude on the stator was 8 nm and the wavelength was 40 mm. With theutilization of high frequency, 100 MHz, the electrical power is transformed to elastic wave motionby piezoelectric effect of the substrate in the stator transducer. The Rayleigh wave is excited at aninterdigital transducer and propagates beneath the stator transmitting driving force through thefrictional force. The stator is pressed to the slider to obtain large thrust. For motor operation,traveling wave is required. By traveling wave propagation, the surface particles of the devicemove in elliptical motion as illustrated in Figure 4.68. Between the crests of the wave and thepreloaded stator friction force is at work. The peak vibration velocity of particles at the crest has ahorizontal component. Therefore, linearly driving forces in the stator act in the opposite direction ofthe wave propagation.

Since the amplitude of the elliptical motion is in the order of tens of nanometers, the contactcondition of the stator is very critical. The elastic deformation of the rotor and the stator surface isalso in the order of nanometers. The contact point diameter is from several microns to tens ofmicrons. The conversion from the vibration to the frictional force utilizes a frictional layer installedbetween the stator and the rotor.

There are many types of ultrasonic motors such as traveling wave mode, standing wave mode,hybrid transducer, and multivibration mode. The traveling and standing wave modes are used mostcommonly, the former is the space (or continuous) contact mode using the mechanism of thetraveling wave, the latter is the temporal (or intermittent) contact mode using the mechanismof the standing wave. In the space contact, the contact region on the stator varies as the travelingwave of the stator moves. In the temporal contact, the contact region on the stator is fixed. Amongstthe ultrasonic motors mentioned above, the traveling wave piezoelectric ultrasonic motor hasbecome a viable choice for its long life span and high stability in use. Generally, all the vibratorsare metal and piezoelectric ceramic composites, in which elliptical motion is generated by thesuperposition of the two resonant vibrations excited by two ultrasonic transducers or two groups ofpiezoelectric elements. We next describe some more details about traveling wave ultrasonic motor.

Consider the linear elastic medium under the excitation of piezoelectric elements. Allow thevoltages applied to sections A and B have a phase difference of 908. The vibration response on thesurface of section A can be represented as standing waves

wA(x,t) ¼ wm sin2pl

x sin(vt � f1) (4:202)

l

Preload

Friction driving

Slider

Amplitudewm

Rayleigh

Wavelength

Stator

Elliptical motion of particles

wave

FIGURE 4.68 Schematic of a contact model of ultrasonic motor with stator and slider (rotor).



The vibration response on the surface of section B can be represented as

wB(x,t) ¼ wm cos2pl

x cos(vt � f1) (4:203)

The z directional displacement of the stator can be represented as the combining effects

wz(x,t) ¼ wA(x,t)þ wB(x,t) ¼ wmz cos�vt � 2p

lx� f1

�(4:204)

It suggests that the vibration in the neutral plane of the stator takes the form of the traveling wave.The phase difference between sections A and B results in difference in the rotational direction of therotor. The tangential velocity of a particle on the upper surface of the stator is given by

Vs(x) ¼ �hm@2wz(x,t)@x @t

(4:205)

in which hm is the distance between the neural plane and the upper surface of the stator. SubstitutingEquation 4.204 into Equation 4.205 yields

Vs(x) ¼ �vwmhm2pl

cos�vt � 2p

lx� f1

�(4:206)

which differs from the speed of the traveling waves. By appropriate choice of the modes of vibrationand system geometry, the ultrasonic motors could rotate at varied speeds.

4.5.4.2 Stick–Slip in Contact Mode AFM

We next present the stick–slip phenomenon associated with the application of atomic-force micro-scope (AFM). AFM has many practical applications for measuring surface samples on an atomicscale. It makes use of a sharp tip to scan over a surface with feedback mechanism that enables thepiezoelectric scanner to maintain the tip at a constant force or height above the sample surface. It canbe used to measure surface roughness with a nominal 5 nm lateral and 0.01 nm vertical resolutionson all types of samples. A modified AFM can measure the normal and lateral (friction) force. One ofthe most common ways that AFM can be used to measure surfaces is contact mode, in which the tipscans the sample surface in close contact. The deflection of the cantilever connecting tip ismaintained through feedback interaction of the piezoelectric element. In contact mode AFM,frictional forces between tip and sample determine the properties of the dynamics of the system.On the atomic scale a periodic ‘‘stick–slip’’ motion of the tip could occur for many cases.

When an AFM tip makes contact with a surface and scans on it with the velocity vscan, themotion equation of the tip can be modeled as follows. As the cantilever is twisted due to friction, itcauses a displacement x(t) of the tip apex. The mechanical model of the system is similar to anoscillator with friction. The sample to be measured can be replaced by a surface moving with thevelocity vscan. The cantilever and tip can be replaced by an equivalent single-degree-of-freedomsystem with the mass m, torsional spring constant k, and damping c. This approximation limits theuse of the model only to applications where the higher-order, natural modes of the cantilever do notcontribute to the response. The equation can be represented by Equation 4.31, except that theexternal force is zero and vr¼ vtip � vscan. As such, the established macroscopic friction dynamicstheories in previous sections still hold for the nanoscopic applications.

In the sliding state, the tip of the AFM is supposed to continuously slide over the specimen. Butif there is a fall in the stick–slip regime, the tip could oscillate. For high-resolution surface imaging



purpose, the slip mode is favorable and needs to attain. For interfacial friction investigations,different friction laws of velocity dependence should be properly applied to correlate with experi-mental results and obtain insights.

We describe an application in thin film lubricant investigation by Meurk [121]. Consider themultimolecular layer lubricant, with transition from continuum to molecular behavior of confinedliquid films of thickness down to a few molecular layers, in addition to the transition of film fromliquid-like properties to solid-like properties depending on the film thickness; another change is theadvent of a yield point constituting the transition from steady state to stick–slip at decreased velocitieswith a similarity to macroscopic behavior. Both the phase transition of the interfacial material and theelastic response up to the yield point could be followed by stick–slip. To correlate with the AFMexperimentally observed stick–slip with amplitude of friction in the level of tens of nano-Newtons, theinterfacial lubricant material is modeled as a solid-like material in confinement and is assumed toexhibit ductile solid undergoing strain.When themolecular layers are sufficiently sheared, a nonlinearshear stress–shear rate dependency with static and dynamic stress parameters results in an overshootin the shear stress. This critical shear stress or yield point is analogous to the static friction. This wascharacterized by two relaxation times with one corresponding to the buildup of the yield point and theother to the subsequent slow decay leading to kinetic friction. The following viscous yield relaxationmodel was developed to quantify continuum rheological behavior [122]:

Ff ¼ tA ¼ kvrt þ A1 arctant

td

� þ Fs � Fk

tst e A2(1�t2=t2s )½ � (4:207)

in which vr is the sliding velocity; Fs, Fk are, respectively, the static and kinetic frictions; A1, A2 areconstants; and td, ts are the relaxation times responsible for the buildup and decay of the yield point,respectively. This model is incorporated into the AFM stick–slip model for solution. The solutionwas used to fit the AFM measured stick–slip data for 2-hydroxy stearic acid (2-HAS). 2-HAS underAFM scanning exhibits viscoelastic behavior with a yield point as the static–dynamic frictiontransition. At low velocity and higher loads, steady motion was replaced by microscopic stick–slip. The viscoelastic behavior of 2-HAS initiation of sliding friction to a macroscopic rheologicalmodel based on two relaxation times is responsible for the phase transition from a glassy, solid-likematerial to a more liquid-like state. Above a critical load or below a critical velocity, a discontinuoustransition to microscopic stick–slip occurs. This gives some deeper understanding of a nanorheo-logical behavior in terms of shear-melting and shear-freezing. The rate and state friction modelscould be used for a more refined treatment.

4.5.5 FRICTION-INDUCED VIBRATIONS AND NOISE IN VEHICLES

4.5.5.1 Friction-Induced Noise and Vibrations in Clutches

A variety of clutches have been extensively used in passenger cars where clutches are used toengage different gears in automatic transmissions. Clutch operation or engagement is usuallyattained through friction effects between the driving and driven ends, in which self-excited vibra-tions could be triggered for some systems with improper friction properties and system combina-tions [124–129]. Low-frequency torsional vibrations of power train, called shudder (judder),occasionally occur due to clutch friction during acceleration at low speed. Shudder could exist indry clutches in a manual transmission or in wet clutches in an automatic transmission. Squeal noisecould even occur in dry clutch applications.

The slippage of torque converter clutch can excite a low-frequency torsional response of wholepower train due to the property of velocity-dependent friction. The shudder severity varies withclutch surface characteristics, temperature, lubrication additives, and vehicle operating conditions.



It has been found that some combinations of clutch design and transmission fluid can preventnegative damping associated with velocity-dependent friction. On the other hand, some attenua-tion of problem severity can be obtained with increased damping in the clutch or the torqueconverter damper.

Some vibration and noise problems associated with dry friction clutch are attributed to mode-coupling type self-excited vibrations. During the engagement of dry friction clutch in manualtransmission, a strong squealing noise is produced [124]. For this case, in near full engagementthe pressure plate suddenly starts vibrating with a frequency close to the first natural frequency ofthe rotational subsystem. This problem exhibits typical signs of a dynamic instability associatedwith a constant friction coefficient. It was observed that the instability of the rigid body wobblingmode is controlled by the friction forces and this mode is also affected by the first bending mode ofthe pressure plate. A stiffer plate could lead to a design with a reduced tendency to squeal. It hasbeen demonstrated that there exists potential mode-coupling between the pressure plate–wobblingmode and the first elastic deformation mode of the pressure plate. The stability threshold depends onthe friction coefficient, the pressure plate geometry, and structural stiffness. Two conditions wereobserved to generate the transient squeal noise. First, the engine speed is within the range1500–2500 rpm and the slip speed is relatively high. It may assume a constant coefficient offriction m corresponding to the threshold of noise. This allows noise not to follow the mechanismof classical stick–slip. Second, the clutch pedal motion is in the condition that the clutch is close tofull engagement. Measurements of noise spectra exhibit a dominant frequency fw around 450 Hz.This frequency has been correlated to the wobbling rigid-body motion of the pressure plate (bothout-of-plane rotations) from dynamometer tests and the simple math model. The pressure plate andthe disk cushion are assumed to be the main components that control the squeal phenomenon. Theanalysis of the nonlinear characteristic of the cushion demonstrates that mathematical modelingyields an asymmetric stiffness matrix characterized by the coefficient of friction and the stiffnessratio. The results of complex eigenvalue analysis are in agreement with experimental observations.It is seen that the instability of the rigid body wobbling mode is controlled by the frictionforces. This mode, however, may also be affected by the first bending mode of the pressure plate.Therefore, a stiffer plate could lead to a design with a reduced tendency to squeal.

On the other hand, most of wet clutch–related vibrations are attributed to the velocity-dependentproperties of friction [125–129]. A problem area in most wet clutches working at low velocity is theoccurrence of shudder associated with the negative slope of friction–velocity property of lubricatedinterface. The antishudder performance of automatic transmissions has been of primary interest innew designs featuring lock-up and continuous slip torque converter clutches. Generally stick–slipoccurs at low velocities when the static coefficient of friction is higher than the dynamic coefficientof friction. Friction-induced instability takes place at higher sliding velocities, due to a negativeslope of the friction vs. velocity curve.

The influences of the velocity-dependent friction on the antishudder performance of wetclutches have been documented. Friction–velocity curve with a positive slope is usually consideredto be an advantage, but neither a necessary nor a sufficient condition to guarantee stability. It wasconcluded that factors such as level of friction, engine speed, and engine torque–speed slope alsoaffect the dynamic behavior of the system. It was found that fluid-related factors such as level offriction, friction–velocity characteristics, and friction–pressure characteristics should be optimizedin order to improve the overall performance of clutch. Figure 4.69 shows a typical coefficient offriction of automotive transmission fluid (ATF) with and without additives, which is a key factor ininfluencing system vibrations. Ohtani et al. defined two values m1=m50 and m100=m300 as anindex for the evaluation of the friction property, where m1 is the coefficient of friction at 1 rpm andso on [126]. In this case the first ratio is the antishudder performance at low sliding velocities, andthe second is the antishudder performance at higher sliding velocities. Values of these ratios largerthan 1 may correspond to observed vibrations in vehicles. There are several other factors used asfriction indicators of the potential for frictionally induced vibrations. Other extensions include the



ratio m1=m20, where m1 and m20 are the coefficients of friction at sliding velocities of 1 and20 cm=s, respectively.

The coefficient of friction of clutch is strongly temperature dependent. This influence oftemperature on friction is because of the fact that temperature affects both fluid viscosity as wellas the formation of tribolayers. The surface active additives in the transmission fluid present at thesliding interfaces in the clutch have a significant effect on friction. The rate of generation of thetribolayer is influenced by the temperature-dependent surface activity. In addition, the surfacetemperature will also determine which types of additives will dominate in the tribolayer. It hasbeen shown that static and dynamic coefficients of friction decrease with increasing temperature.Moreover, the change rates of static and dynamic frictions could be different due to increasedadditive or temperature change. This rate of change differed between static and dynamic frictionsand may therefore alter the antishudder properties. Thus a fluid with good antishudder properties athigh temperatures may have very poor properties at low temperatures as seen in Figure 4.70.

For the torque converter clutches in drive shaft system, the friction induced instability is due tothe combined effect by negative slope of friction–velocity curve and the varying normal load, asdiscussed in Section 4.3.2.

4.5.5.2 Friction-Induced Gear Noise

Sliding friction between meshing teeth is one of the primary excitations for noise and vibration ingeared systems [130–134]. Figure 4.71 shows a typical coefficient of friction of gear tooth for

0

0.05

0.1

0.15

0.2

0.25

0.3

Velocity (m/s)

0 0.01 0.02 0.03 0.04 0.05

CO

F

Base ATF

Base ATF + additives

FIGURE 4.69 Coefficient of friction of AFT.

00 0.05 0.1 0.15 0.2 0.25 0.3

0.04

0.08

0.12

0.16

0.2

Velocity (m/s)

CO

F

20�C

−20�C

80�C

FIGURE 4.70 Friction performance change under different temperatures.



different loads under the same rpm. It suggests that the measured maximum friction coefficientsare in the range of 0.05–0.06 subject to load. Friction coefficients increase at low sliding speeds.The reversal of sliding which occurs at the pitch-point does not cause a discontinuity in the frictioncoefficient, which shows a smooth transition as the friction force reverses direction.

The following friction law for gear was derived by Kelley and Lemanski [133] for mixedlubrication conditions

m ¼ C11

1� C2Rqlog10

C3w

h0VsV2R

�rp þ rg

�2" #

(4:208)

whereC1, C2, and C3 are empirical coefficientsw is the distributed load per unit length of contactVR is the rolling velocity at surfaceVs is the sliding velocityh0 is the lubricant dynamic viscosityr is the radius of curvature at the point of contactRq is the surface roughness

In this expression, m is a complex nonlinear function of load distribution, surface kinematics,and tribological factors. Figure 4.72 shows friction force vs. roll angle in terms of different models.Due to its empirical nature, such an expression is valid only for a certain range of parameters.

The sliding friction has been recognized as a significant source of noise and vibration in gearmeshing. Houser et al. [132] found that the dynamic friction force could be the same order ofmagnitude as the forces normal to the tooth profile for certain cases. Friction forces play a pivotalrole in load transmitted to bearing and housing in gear meshing, particularly at high torque and lowspeed. Friction has a predominant effect at higher harmonics of meshing frequency. Both oil viscosityand surface roughness had a large influence on the vibro-acoustic behavior. Some of the experimentalresults in [132] are shown in Figure 4.73. The third- and fourth-order mesh harmonics in noise spectraare plotted in Figure 4.73a and b, for two kinds of lubricants at 1000 rpm under different loads.The third- and fourth-order mesh harmonics in noise spectra are plotted in Figure 4.73c and d, as

0.08

0.06

0.04

0.02

10 15 20Roll angle (�)

CO

F

25 30

0

−0.02

−0.04

−0.06

−0.08

FIGURE 4.71 Coefficient of friction of the gear teeth: rpm¼ 800; solid line: full load; dotted line: 80% load.



–300

–200

–100

0

100

200

300

Roll angle (�)

10 14 18 22

Fric

tion

forc

e (N

)

FIGURE 4.72 Friction force vs. roll angle in terms of different models. (Solid line: Equation 4.208; Dash line:Coulomb law)

300

0 400 800 1200 1600 2000 0 400 800 1200 1600 2000

20 40 60 80 100 0 20 40 60 80 100

40

50

60

70

80

Torque (Nm)

Am

plitu

de (

dB)

30

40

50

60

70

80

Torque (Nm)

Am

plitu

de (

dB)

(a) (b)

30

40

50

60

70

80

Speed (rpm)

Am

plitu

de (

dB)

30

40

50

60

70

80

Speed (rpm)

Am

plitu

de (

dB)

(c) (d)

FIGURE 4.73 The third and fourth mesh harmonic peaks of sound pressure level in gear box with twolubricants, as a function of torque at speed 1000 rpm (a,b); as a function of rpm at torque 102 Nm (c,d).(Courtesy of Houser, D.R.)



a function of speed. The difference in measured sound spectra for the two lubricants is attributed to thelower values of friction coefficient. A distinct reduction in noise level is achieved with the highviscosity lubricant. Particularly, a good correlation is observed between vibration level in the slidingdirection and the overall sound pressure levels.

4.5.5.3 Piston Stick–Slip Noise in Engine

The piston motion within the cylinder of some engines could experience stick–slip oscillations asthe piston reverses direction in operation. The noise, called diesel sounding knock, has beenreported to be associated with the stick–slip motion of piston caused by the abrupt change offriction coefficient from static to dynamic when the engine is in idling speed. The excitation actingon the piston in turn excites the crankshaft at its resonant frequency via the connecting rods. Someof the identified noise frequencies are approximately 1100 Hz and are thought to be fundamental tothe crankshaft [135]. The diesel sounding knock usually occurs for new vehicles and disappears formileages above 10,000 miles. The reduced sound level with mileage is a result of a friction forcechange between the piston=rings and cylinder, due to the surface finish change from the originalmanufacturing condition, which is consistent with the stick–slip theory. To remedy this, severalfriction modifiers were added to the engine lubricant and their effects were evaluated. Some of themodifiers showed significant noise reduction. The noise could be reduced by 4 dB with the additionof some kind of additive which has a 3% volume concentration. However, subsequent cold startswith this modifier failed to meet the cold start requirement. Another modifier (complex aminephosphate salt) with 1% concentration gives 3.5 dB noise reductions without compromising on thecold start requirement. The friction modifier is only required in the initial factory fill lubricantbecause the noise would not be audible at higher mileage.

The diesel sounding knock at idling speed was determined to be a second-order phenomenon. Itdecreases rapidly with accumulated mileage; a decrease of 4 dB in the first 80 miles and from 9 dBto a minimum level at 2000 miles. Optimizing piston parameters and adding friction modifiers to thelubricant were the most practical short-term solutions and provided 10 dB noise reductions. This isconsistent with the piston stick–slip theory.

4.5.5.4 Friction-Induced Wiper Blade Noise

As automobiles become increasingly quieter, wiper operational noise that occurs in rubber–glassinterface of wiper blades becomes more perceptible and thereby a quality concern [136,137].

Wiper noise is classified into three groups, squeal noise, chattering, and reversal noise. Squealnoise or squeaky noise is a high-frequency vibration of about 1000 Hz, and is easily generatedbefore and after the wiper reverses direction. Chattering or beep noise is a low-frequency vibrationof 100 Hz or less. Reversal noise is an impulsive sound with a frequency of 500 Hz when the wiperreverses direction.

The wiper operation velocity may change drastically before and after the wiper reversesdirection. As the velocity reduces the friction load increases. Therefore, in an area where the relativevelocity between the blade rubber and glass decreases immediately before and after the wiperreverses direction, the friction force increases, and this causes an increase friction coefficient vs.velocity decrease. Thus, a self-excited vibration is generated in such areas.

The squeal noise generates in such areas where there is a steep negative slope of friction–velocity curve, and this area is close to the reverse point (zero speed). The quietness design wasevaluated by using a span of speed which yields squeal noise. Several countermeasures, such as theapplication of friction reduction coating for rubber surface treatment, increasing of material damp-ing, and optimization of geometry, were tried to improve the noise performance. The surfacemodification had the most favorable effect. Figure 4.74 shows the coefficient of friction as afunction of velocity for blade rubber with and without coating. It was found that the velocitythreshold within which the squeal noise occurs was reduced significantly by the coating treatment.



4.5.5.5 Friction Noises of Tire–Road Interaction and Wheel–Rail Interaction

In transportations, both tire–road interaction in road vehicle and wheel–rail interaction in rail vehiclecause noise [138–140]. Part of road vehicle rubber tire noise is attributed to the tire vibrations associatedwith the adhesion and friction created in the contact patch between the tire and the road surface.

When a tire rolls on the road, it flattens in the contact patch. The changing radial deflectionproduces tangential forces between the tire and the road. These forces are resisted by friction and tirestiffness. The residual forces are dissipated by slip of the tread material over the road surface.

Friction between the tread and the surface can be divided into hysteresis and adhesion compon-ents. The adhesion component has its origins at a molecular level and is governed to a large extent bythe small-scale roughness characteristics or microtexture of the road surface. During relative slidingbetween the tire and the road surface, the adhesion bonds that have been formed between them beginto rupture and break apart, so that contact is effectively lost and the elastomer is free to slip across theroad surface. Contact may be regained as these residual forces are dissipated. The hysteresis force isdue to a bulk phenomenon which also acts at the sliding patch. In the contact zone tread rubber drapesaround asperities in the road surface and the pressure distribution about each asperity is roughlysymmetrical in the case of no slip. When slip occurs, tread rubber tends to accumulate at the leadingedges of these surface irregularities and begins to break contact on the downward slope of the surfaceprofile. This causes an asymmetric pressure distribution and a net force which opposes the slidingmotion. At high speeds this mechanism is largely responsible for the tread element regaining contactwith the road surface. The hysteresis component of tire surface friction is largely controlled by thesurface macrotexture which comprises texture wavelengths corresponding to the size of the aggregateused in the surface material. It is not the slippage of the tread elements alone that cause vibrationalexcitation of the tire. It is the combination of the slip of the tread elements in the contact patch and thehysteresis effect due to the deformation of the tread that causes a ‘‘slip–stick’’ process in the contactpatch, which causes the vibrational excitation of the tire. Tire vibration and noise generated by thismechanism have been related to the slip velocity of the tread elements. The highest velocities tend tobe found to the rear of the contact patch and may contribute to block ‘‘snap out’’ effects.

The rail–wheel interaction in rail vehicle could create horrible noise in some situations when arail vehicle negotiates a narrow or tight curve. The high-pitched, tonal squeal noise with usuallyrecorded frequencies of 500–1000 Hz is called curve shrieking or curve squeal.

This noise is the sound radiated from individual wheels that are excited by the dry frictionforces. The friction is associated with a sliding of the wheels against the rail. It is generally accepted

00 20 40 60 80 100

0.1

0.2

0.3

0.4

Torque (Nm)

CO

F

FIGURE 4.74 Coefficient of friction as a function of velocity for blades. (Solid line: With Coating; Dash line:Without Coating.)



that ‘‘wheel crabbing’’ is responsible for the generation of curve squeal. Wheel crabbing usuallyhappens when a truck with two (or more) rigid parallel axles negotiates a curve, where its wheelscannot align themselves tangentially to the rail; instead, the wheels on the front axle tend to run outof the curve, and those on the trailing axle tend to run into the curve. Thus, the speed of anindividual wheel has two components: the rolling speed, which is tangential to the wheel, and thecrabbing speed, which is perpendicular to the wheel and depends on the angle between the wheeland rail. This crabbing motion induces a friction force, which is normal to the plane of the wheel.This normal force excites out-of-plane or bending oscillations of the wheel. When bendingoscillations are excited, the wheel turns to a very efficient radiator of sound, and this is one reasonfor the high intensity of curve squeal.

Curve squeal has been verified to arise from lateral crabbing of the wheels across the rail head.This induces a lateral friction force acting at the contact of each wheel with the rail. The stick–sliphas been considered to be one of the root causes, in which the lateral modes of the wheel and rail areactuated by the system yaw angle–lateral slip velocity. This phenomenon can be modeled byconsidering a round disk, with several out-of-plane modes, excited at one point along the edge bya dry-friction force which is dependent on the disk velocity.

There are several approaches that can be used to combat shrieking, which include the reductionsof the yaw angle, coefficient of friction, and excited vibrations as well as the application ofautomatic device to spray liquid on the rails before rail vehicle starts to negotiate the narrow curve.

4.5.5.6 Squeak and Rattle in Automobiles

The interior noise of a passenger car consists of many elements. A fundamental element is related tothe irritating noises of squeak and rattle (S&R) comprised of buzzes, squeaks, ticks, rattles, etc. dueto the friction and=or impact interaction of rubbing or loose interfaces of part pairs. These partsinclude instrument panel, dashboard, door closing, windshield and backlight headers, roof=doorweather strips, body closure, underbody exterior, and seats, just to name a few. They are a primaryindicator for customers of the quality of the vehicle [141–167].

Squeaks are friction-induced noises caused by relative motion resulting from a stick–slipphenomenon between interfacing surfaces. The intermittent stick–slip burst cycle usually occursdue to suspension inputs. However, the slip transition from stick is usually associated with therelease of energy in a very short time. The resultant impulsive motion could produce a structuralresponse that is likely to cause audible squeaks in a wide frequency range from 100 to 15,000 Hzsubject to the affected structures and constraints. The amplitude and frequency of the squeak dependon a complex of factors such as material constituents, coefficient of fiction, normal load and loadhistory, sliding velocity, inertia and thermal effects, age and wear characteristics, temperature andhumidity conditions, etc.

Rattles are impact-induced noises that occur when there is a relative motion between compon-ents with a short-duration loss of contact. It is generally caused by loose or overly flexible elementsunder forced excitation. Rattle impacts are caused when surfaces close to each other moveperpendicular to each other due to insufficient attachments or insufficient structural strength, forcingrepeated separation and reestablishment of contact. The frequency range of audible rattles is from100 Hz to above 10 kHz, sometimes with strong contribution around 4 kHz, a frequency region alsovery significant in human hearing. The major exciting inputs are suspension vibrations induced bythe road surface, but there can be other contributions by other inputs such as power train vibrations.On the other hand, the slip phase of the stick–slip event is always associated with energy quickrelease; thereby an impulsive velocity response is attained for many cases. This also increases thelikelihood of the additional impact and rattle. Figure 4.75 shows the typical stick–slip and thevelocity time history of an interior material structure undergoing friction test.

Modern technical progress in automobiles tends to reduce the general level of sound producedby the major sound sources such as the power train, wind, road, and tire. Therefore, customers



increasingly perceive S&R as a direct indicator of vehicle build quality and durability. The high-profile nature of S&R has led manufacturers to formulate numerous specifications for S&Rperformance of assemblies and subcomponents.

Over the last two decades, numerous efforts have been made by the automotive community totest, analyze, and prevent S&R. Numerous S&R events have been reported and addressed experi-mentally and phenomenologically. The predominant approach to prevent S&R is screen testing andverification testing. CAE application has been limited to structural analysis aspect in this area. Thetrial-and-error approach has been used. A simple model for squeak and rattle is briefed as follows.The basic dynamics features and signatures can be derived using the theory presented in precedingsections. The schematic of squeak model is shown in Figure 4.76a. Assume mass m, spring constantk, displacement of mass x, vn ¼


p, motion excitation u(t)¼ u0 cosvt, v(t)¼ v0 sinvt,

v0¼�vu0, r¼v=vn, damping constant c, normal force FN, and relative velocity vr¼ _x þ vo sinvt.The friction force Ff (vr)¼m(vr)FN follows the nonlinear friction law represented by

m(vr) ¼ ms sgn(vr)þ a1vr þ a3v3r (4:209)

The equation of motion is given by

m(€xþ v2u0 cosvt)þ c( _xþ vu0 sinvt)þ k(x� u0 cosvt)þ Ff (vr) ¼ 0 (4:210)

0.6

0.5

0.4

0.3

0.2

0.1

00 0.2 0.4 0.6 0.8

Time (s)

Velocity

COF

CO

F / V

eloc

ity (

cm/s

)

1 1.2 1.4 1.6

FIGURE 4.75 Typical stick–slip and the velocity time history of moving parts.

(a) (b)

u(t)u(t)

m

kk

mm

x(t )

x(t )

d

FIGURE 4.76 Schematic of squeak and rattle models: (a) friction-induced oscillator; (b) impact-inducedoscillator.



For small-amplitude excitation, displacements in some ranges are stable equilibrium positions. Thesystem motion could exhibit periodic stick–slip or chaotic stick–slip. For larger values of theexcitation amplitude, the motion exhibits complex properties.

The schematic of rattle model is shown in Figure 4.76b. Assume mass m, spring constant k,displacement of mass x, motion excitation u(t)¼ u0 cosvt, vn ¼


p, r¼v=vn, damping con-

stant c, spacing d, contact stiffness K. The equation of motion is given by

m(€xþ v2u0 cosvt)þ c( _xþ vu0 sinvt)þ k(x� u0 cosvt) ¼ 0, x < dþ u0 cosvt (4:211)

m(€xþ v2u0 cosvt)þ c( _x� vu0 sinvt)þ (k þ K)(x� u0 cosvt) ¼ 0, x � dþ u0 cosvt

(4:212)

For this kind of system, periodic motions, resonance, bifurcations, and chaos have been proven toexist depending on different system parameter combinations.

Based on the modeling, we can observe that S&R control approaches include reducing thebase motion as well as the relative motion, enhancing the structure stiffness and damping,optimizing the friction pairs to have a favorable friction law, and properly controlling the geometrywith tight tollerance of gapes. Many previous researches have given many practical guidelines.Relativemotions are due to structural deficiencies such as insufficient stiffness, excessive input forces,or poor modal alignment. A stiff or improperly designed suspension causes large input forces,thereby generating larger base motion. Insufficient global, local, and attachment stiffness cancontribute heavily to S&R issues. Not only the stiffness, but also the modal alignment, shouldbe well controlled together with the other major body=engine=seat modes so as to avoid highresponse levels.

Degradation and wear of subsystems and interfaces due to aging result in progressiveloosening over the life of the vehicle. Temperature and humidity conditions not only causedimensional variations in components, but also heavily change the friction properties, especiallyfor elastomer components. Depending on their compatibility (chemical compositions, surfaceproperties, environmental conditions, etc.), interfaces can produce various squeak noises whileimpacts can produce rattles. Improper geometry conditions will likely lead to S&R events.Inappropriate separation of the two surfaces will likely lead to S&R as well. The proper controlof tolerances in both design and=or production can improve the situation. Tribology testing hasbeen widely conducted to screen materials in order to help choose compatible material pairs. Thegeneral procedure of tribology testing had been discussed in preceding sections. After the frictionlaws for various materials have been quantified, the next step is to rank them in terms offriction performance, durability, and reliability. The simple material-selection criterion in termsof friction is the friction peak-valley, namely, the static friction minus the dynamic friction, or therelative value of the slope of the friction–velocity curve. Consideration is also given to this indexover a broad range of environmental and operating conditions or those conditions consideredcrucial. The smaller the static friction minus dynamic friction value, or the smaller the slope offriction vs. velocity, the less the likelihood of friction pairs generating noise. It is noted that thestatic friction could be time-dependent and could be heavily dependent on humidity. System andsubsystem structural testing has been conducted as a standard procedure for many years.Structural integrity validation, manufacturing tolerance control, and early design S&R consider-ation have helped to reduce the S&R. Good structural integrity implies adequate static anddynamic stiffness both globally and locally. The stiffness requirement is generally tested againstthe constraints of cost and weight. Modal analysis of a vehicle structure or component is helpfulto quantify S&R. The optimization criteria are to isolate modal frequencies and minimize energyinteraction and resonance. Good dynamic stiffness requires adequate modal alignment and mode-shape continuity.



Squeak and rattle evaluation are often performed in conjunction with durability testing. For thesound testing aspect, sound pressure testing has been used to record noise, and time-frequencyanalysis is usually conducted to identify the spectrum signature or S&R events. Since the S&R fallinto the nonstationary noise category, the nonstationary Zwicker loudness is a straightforwardchoice for sound quality evaluation. Practice has proven that it is a good tool for objectivequantification of S&R noise. Figure 4.77 shows a typical established relationship between loudnessand the coefficient of friction peak-valley (difference of static and kinetic friction coefficient).

Currently many S&R estimation and prevention efforts are conducted by subjective evaluation.Typically, judgment of the relative loudness is difficult to undertake up to five levels. There areseveral topics concerning human perception of S&R. It has been known that loudness depends onthe frequency, with the most sensitive region between 1000 and 5000 Hz, peaking at about 4000 Hz.Loudness is affected by bandwidth. Sounds with a large bandwidth are perceived as louder thanthose with a narrow bandwidth, even if they have the same energy. For sounds shorter than about asecond, loudness increases with duration: for each halving of duration, the intensity of a noisewhose burst is shorter than 200 ms must be increased by about 3 dB in order to maintain loudness.However, many of these types of loudness dependence were established under simple conditionsusing pure tones.

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158. Cerrato-Jay, G., Gabiniewicz, J., Gatt, J., and Pickering, D.J., Automatic detection of buzz, squeak andrattle events, SAE 2001-01-1479, 2001.

159. Hunt, K., Rediers, B., Brines, R., McCormick, R., Leist, T., and Artale, T., Towards a standard formaterial friction pair testing to reduce automotive squeaks, SAE 2001-01-1547, 2001.

160. Kuo, E.Y., Up-front body structural designs for squeak and rattle prevention, SAE 2003-01-1523, 2003.161. Yang, R., Experimental investigation of a friction induced squeak between head restraint fabric and glass,

SAE Paper 951269, 1995.162. Hurd, Y., Combining accelerated laboratory durability with squeak and rattle evaluation, SAE Paper

911051, 1991.163. Ueda, U., Detection of transient noise of car interior using non-stationary signal analysis, SAE Paper

980589, 1998.164. Malinow, M.S. and Guttal, R., Rattle noise prediction of a seat belt sensor using numerical methods, SAE

2001-01-1551.165. Lee, P., Rediers, B., Hunt, K., and Brines, R., Squeak studies on material pair compatibility, SAE

2001-01-1546.166. Stucklschwaiger, W., de Mendonca, A., and Alves dos Santos, M., The creation of a car interior noise

quality index for the evaluation of rattle phenomena, SAE Paper 972018, 1997.167. Shaw, H. and Borowski, H., A CAE methodology for reducing rattle in structural components, SAE Paper

972057, 1997.



5 Friction-Induced Vibrationsand Acoustic Emission andApplications in Hard DiskDrive System

5.1 INTRODUCTION

In this chapter, we present the friction-induced vibrations and acoustic emission of slider in harddisk drive system, as well as their critical applications in the hard disk data storage industry. Themechanical spacing between the slider and the disk in computer hard disk drive has to be reduced toless than 5 nm in order to achieve an area density of 1 Tbit=in.2. At such a low flying height (FH),intermittent slider–disk contact and friction are unavoidable.

Currently, the peak-to-valley roughness of commercial available disk is below 5 nm and thesurfaces are usually well controlled with designated finishes. This also provides a desirable case tounderstand the interface interaction in the micro- and nanoscale.

Contact sliding, whether a continuous or intermittent one, is usually a nonstationary process.Continuous sliding contact causes random excitation to slider. Noncontinuous sliding contact likethe gliding of flying slider on spinning disk creates intermittent excitation to slider. The strength ofcontact decides the type of waves and oscillations that develop during sliding.

In a lightly loaded friction pair, like a flying slider in making a near contact with a disk, weakcontacts generate light impact as asperities come into contact continuously or intermittently. Thisleads to dynamic wave motion and vibration response in both parts of the friction pair, and theresponses mainly depend on the respective frequency response function, almost independent of theother part of the friction pair. The sliding contact can be considered as an excitation of time series,which results in the linear forced vibrations of system [1–41].

For conventional laser or mechanically textured disks which have regularly corrugated surfaces,slider sliding over disk under light normal loads produces regular impulsive contact forces; thus itdevelops response at the ‘‘textured frequency’’ with components in both tangential and normaldirections to the interface. Experiments show that contact forces produce response with fundamentalfrequencies being a combination of textured and natural frequencies of each component.

We do not use a clear definition on interface interaction to distinguish contact or impact asusually done in some related publications. For convenience, we call the interaction between theflying slider and a single isolated asperity or bump on disk to be impact, as it is usually associatedwith a large response of slider. We also call the interaction between the slider and the disk to becontact when the slider makes continuous or intermittent contact with the disk in which the forcescan be considered as quasi-static in the application. The impact could be very weak under lowvelocity or in the case of low bump height. The contact may be a superposition of a series ofseparate weak impacts.


249

Friction-induced acoustic emission (AE) could be considered as the stress wave or solid soundemanating from the regions of localized deformation of a solid structure; the source of theseemissions is closely associated with the dislocation movement accompanying deformation and theinitiation and extension of cracks in a structure under stress.

Over the last two decades, the AE detection technique has been used by the hard disk driveindustry as an essential tool for disk product screening and quality control [42–54]. In this chapter,we discuss the friction-induced vibrations and the AE in hard disk drive systems, and then presentthe techniques of friction vibration reduction and AE detection. Finally, we present the disk surfacescreening technique using the AE technique for mass production.

AE sensor is generally a piezoelectric element, which transforms the particle motion producedby an elastic wave into an electrical signal. The property and atomic structure of a piezoelectricmaterial are such that an applied force (pressure) will cause a charge (voltage) to be developedacross its ends. The intensity of the force applied is directly proportional to the amount of chargeproduced. Most of the sensors employ a plate of piezoelectric ceramic or crystal and respond tovelocity or acceleration normal to the face of a sensor. This is usually to couple piezoelectrictransducer on the surface of the structure under test, by means of a fluid couplant and is secured withadhesive bond to form AE sensor.

The frequency responses of AE sensors are usually calibrated against standard using compres-sion waves at normal incidence. Typically, AE sensors can collect data over the entire range of AEthat occurs in most materials, approximately 50 kHz to 3 MHz.

AE signals can be very weak; therefore a quality preamp is usually required in order to collectdata and is filtered to remove any extraneous noise and further processed by suitable electronicequipment. The detected waveform can then be subjected to a series of analysis techniques whichcan be used to detect, locate, and identify defects.

Mechanical deformation and fracture are the primary sources of AE. The elastic waves travelthrough the structure to a sensor. The AE pulse and the associated parameters are usually employedas characteristic AE parameters, which include emission count, the number of times that the AEsignal amplitude exceeds a preset threshold during any selected portion of a test, the emission rate orcount rate, the time rate at which emission counts occur, the AE event energy, the total elasticenergy released by an emission event, as well as various spectrum expressions.

Based on prior understanding of the slider system dynamics, appropriate band-pass filtershave been used to eliminate the unwanted signals and extract meaningful characteristic param-eters. The waveform-based analysis of the signal has been applied widely to replace the conven-tional statistical analysis to see visually those frequencies that are detected. The time–frequencyanalysis and the wavelet analysis allow the possibility of unwanted signal attenuation throughcomputer processing.

For friction-induced AE, two types of AE signals are frequently encountered. They are burst-type and continuous-type. Burst-type emissions arise from distinct events of elastic energy release,such as impact or crack-like fracture delamination. The typical case is slider–bump response, whichhas a sharp rise, followed by an exponential decay. This usually accompanies resonance responses,whose ringing pattern arises from various resonances in a structure and the resonances of the sensor.Continuous-type emissions are produced by many overlapping events and observed from plastic andelastic deformation. Normally, ultrasonic frequencies of 30 kHz to 2.5 MHz are used for AE systemto detect slider–disk contact event, both burst-type and continuous-type. System noise includingairborne noise could interfere with AE measurements at lower frequency, while signal attenuationmakes the higher frequency range difficult to use.

In slider–disk interface, the fundamental frequencies of friction-induced AE in slider body oftencorrespond to natural frequencies of slider; therefore a mechanical impact model has been used toquantify the event. The impact involves the repeated interaction of one imperfect surface overanother; both are characterized by micro- and macroroughness and waviness in the surface. Theunstable interactions may cause time series impact, which leads to either single impact, or a series of



continuous or intermittent impacts, or even stick–slip interaction acting at slider. These impacts can,in principle, excite natural mode of the slider, which is then detected by an AE sensor on slider.

The interface of slider–disk is usually under weak contact conditions; the friction pair is not acoupled system and therefore the system can be treated as a linear response system.

5.2 CONTACT AND FRICTION-INDUCED VIBRATIONS OF SLIDERIN MAGNETIC HARD DISK DRIVE

5.2.1 SLIDER–DISK INTERFACE

The recording density of hard disk drive has increased more than a million times since it was firstintroduced half century ago. There are several key factors in supporting the hard disk industry inachieving its annual increment of 60% recording density, among which one of the most importanttechniques is the substantial reduction of the slider–disk spacing.

Figure 5.1 is the schematic of a slider–disk interface. The signal read–write function of thesystem is mechanically implemented by a slider having a magnetic read–write head at its rear tip,flying on a spinning magnetic disk. The slider is supported by a flexure and suspension arm; the railsof the slider are deigned such that it generates air-bearing force when the disk is spinning at properspeed. The special patterned surface of the slider rails is named as air-bearing surface (ABS).

In the current state-of-the-art design, the slider–disk spacing is required to be about 5 nm. Insuch a low spacing, high-speed head–disk interaction is inevitable and pseudo-contact designs arenecessary. Therefore, friction-induced vibrations and stability of slider become a critical issue.

5.2.2 NANOMETER SPACING INTERFACE AND BUDGETS

Conventionally, sliders have been designed to fly on spinning disk in operation. Over the last tenyears, the head–disk spacing, or the FH of the slider, has entered into the levels of disk surfaceroughness and ABS roughness. In this region, slider is likely to make contact with the disk via theroughness of the two surfaces. Some slider designs, such as the tripad slider and the negativepressure tripad slider, have the properties of making proximity contact or near contact with the diskin operation. The further developments are leading to the contact recording. The schematic of theoptional interfaces is shown in Figure 5.2. The schematic of the possible forces acting on sliderduring near-contact is shown in Figure 5.3, in which Fa is the aerodynamic drag force, Fb is the air-bearing force, Fc is the contact force, Ff is the friction force, Fv is the viscous shear force, Fx is thex-directional suspension force, and Fz is the z-directional suspension force.

We next discuss the near-contact slider–disk interface to illustrate the contact and frictiondynamics of near-contact slider with nanometer spacing. The schematic of the near-contact interface

Suspension

Slider

Read–write element

Flexure

Disk

FIGURE 5.1 Schematic of head–disk interface.


Friction-Induced Vibrations and Acoustic Emission and Applications 251

is shown in Figure 5.4. The disk surface profile is characterized by disk microwaviness androughness, which are part of the factors influencing the slider flying. The slider surface profile ischaracterized by head roughness. When the slider flies on the spinning disk, its FH actually hasunavoidable fluctuation. Therefore, when a slider flies on a disk without contacting the disk, itsminimum FH is the summarization of the disk microwaviness, head and disk roughness, and theslider fluctuation magnitude. This minimum FH is defined as take-off height (TOH), above which,slider can fly on spinning disk without contact. Moreover, for all kinds of disk products, a thin layerof lubricant of nanometer level is deployed on the top of disk for durability purpose, which furthercomplicates the interface phenomena due to the lubricant effects.

5.2.3 SLIDER–DISK CONTACT MODEL AND INTERACTIONS

In current products, slider flies at spacing distance much less than the mean-free-path of airmolecules (60 nm at 1 atm). As the mechanical spacing is driven down toward 5 nm, successivebarriers are encountered in tribology. There are several issues that need to be addressed: thelubricant thickness is comparable to the air gap, therefore liquid meniscus effect could occur; vander Waals and electrostatic force could interact with air-pressure force which affects the interactionamong the slider and the lubricant as well as the disk. The short-range forces have been demon-strated to have critical effects on slider contact dynamics.

The disk roughness and waviness have great influences on the slider’s FH and near-contactfeatures including glide avalanche, due to their great effect on the friction force.

To employ the roughness contact mechanics described in Chapter 3 to deal with the sliderinterface dynamics, more details need to be explored. The characteristics of the roughness contact

Asperity

(a) (b) (c)

FIGURE 5.2 Schematic of various slider–disk interfaces: (a) flying slider; (b) near-contact slider; (c) contactslider.

Fa

Fz

Fx

Ff

FcFb

Fv

Disk

Slider

FIGURE 5.3 Schematic of forces acting on a near-contact slider.



mechanics between disk surface and ABS are widely studied. To incorporate these short-rangeforces into contact and friction dynamics analysis, the following roughness contact model [55–58]was developed to quantify the contact force, which is associated with the roughness and theair-bearing forces, and is shown to be efficient and effective.

Air-Bearing Contact Model. Considering a near-contact interface, the schematic of the roughnessand slider–disk spacing is illustrated in Figure 5.5. The balanced state of the slider can be obtainedby solving the coupled model containing the air-bearing force, the contact force, the friction force,and the loading force.

The surface roughness of ABS or disk can be depicted by the modified Gaussian distribution

g(z) ¼ hffiffiffiffiffiffi2p

psexp � (z� z0)

2

2s2

� �

zmin � z � zmax

h ¼ðzmax

zmin

1ffiffiffiffiffiffi2p

psexp � (z� z0)

2

2s2

� �dz

8<:

9=;

�1

(5:1)

Head roughnessHead fluctuationDisk roughnesMicrowaviness

TOH budget

Disk

Slider

FIGURE 5.4 Take-off height budget.

h max

h l (>

0)y

z z = 0 plane

y = 0 plane

h 0

Roughness valley plane

Roughness valley plane

z min

z min

hl (< 0) (contact depth)

FIGURE 5.5 Schematic of roughness and clearance of slider–disk interface.



For a possible acting point on the ABS and a corresponding point on the disk surface, the liftingforce acting on the two points can be described by

Ppoint ¼ðhmax

d

Fa«a dhþð0

hmin

Fc«c dh

hmax ¼ h0 � zdmin � zsmin

hmin ¼ h0 � zdmax � zsmax

(5:2)

where h0 is the clearance between the mean roughness plane of disk and the mean roughness planeof ABS, and Fa, «a are the air-bearing force and the probability at ABS–disk clearance h, and Fc, «care the contact force and the probability at ABS–disk contact depth h. For convenience, theroughness height is assumed to be zero on the mean plane. The subscripts d and s refer to thedisk and slider, and the low integral interval d is the clearance below which the air molecule cannotpass through. Extending the point-to-point acting force to the whole ABS region, the total liftingforce Ptotal and its force center X0 and Y0 can be calculated as

Ptotal ¼ðð

ABSPpoint dx dy

X0 ¼ðð

ABSPpoint x dx dy=Ptotal

Y0 ¼ðð

ABSPpoint y dx dy=Ptotal

(5:3)

For a clearance h0 between the two mean planes, if the local clearance hl, which may vary accordingto the probability, is greater than d, then the efficient clearance he can be calculated by thearithmetically averaged film clearance and the harmonically averaged clearance

hme ¼ hhm þ (1� h)hm, m ¼ 1, 2, 3

h3 ¼ 1=h�3, h2 ¼ 1=h�2, h ¼ h�2=h�3(5:4)

The coefficient h in the above equation is related to the ABS roughness and the disk surfaceroughness. The arithmetically averaged film clearance and the harmonically averaged film clearancecan be obtained as

hm ¼ðzmaxd

�zmind

gd(z)

ðzmaxs

�zmins

hml ~cegs(y) dy dz=Ce

(m ¼ 1, 2, 3)

hm ¼ Ce=

ðzmaxd

�zmind

gd(z)

ðzmaxs

�zmins

h�ml ~cegs(y) dy dz

(5:5)

(m ¼ 2, 3)

Here gd and gs are the roughness distributions of the disk surface and the ABS, respectively.We denote y and z as the local ABS roughness height and the disk surface roughness height,



respectively, and for convenience, we define that y and z are positive when the points are above themean planes and toward the outside of the surface. The local clearance hl and the coefficient Ce canbe expressed as

hl ¼ h0 � y� z

Ce ¼ðzmaxd

�zmind

gd(z)

ðzmaxs

�zmins

~cegs(y) dy dz

~ce ¼ 1; hl > d

~ce ¼ 0; hl � d

(5:6)

When calculating the air-bearing force, the obtained force should be discounted by the bearingcoefficient Ce due to the reduction of actual air-bearing region. Therefore, the air-bearing part can begiven as

Pair ¼ðð

ABS( pCe � 1) dx dy

Mx-air ¼ðð

ABS( pCe � 1)x dx dy

My-air ¼ðð

ABS( pCe � 1)y dx dy

(5:7)

where Pair, Mx-air, and My-air are the air-bearing force and the air-bearing torques along pitch angleand roll angle, respectively, and p is the air pressure of ABS, and it satisfies the Boltzmann equationdescribed as follows:

r(~qphe3rp� 6m~Vphe) ¼ 0 (5:8)

The efficient clearance he is determined by Equation 5.4. The contact pressure pc as a function ofclearance h for the elastic contact model can be described as follows:

pc ¼ b(1� h=a)m for h < a0 for h �a

�(5:9)

wherea is the maximum asperity heightb and m are the material-dependent proportionality constants1� h=a is the strain of the material

In the power law model, the contact pressure is proportional to the m power order of contactstress, and is usually assumed as 1.5 � m � 2.5.

For a certain clearance h0, when the asperity of the slider and the disk is in contact, the contactpressure of the slider and the disk is the same. Therefore, we have

bdDmd

d ¼ bsDms

s

Ddad þ Dsas ¼ zþ y� h0(5:10)

and

ad ¼ zmaxd � zmind

as ¼ zmaxs � zmins



wherea refers to the peak-valley value of the disk or the sliderD is the strain

Thus, the contact force can be calculated by the following expression:

Pc ¼ðð

contact

ðzmaxd

h0�zmaxs

gd(z)

ðzmaxs

h0�zmaxd

pcgs(y) dy dz dS (5:11)

where pc is the contact pressure at clearance h0. So pc can be expressed as

pc ¼ bdDdmd ¼ bsDs

ms when zþ y > h00 when zþ y � h0

�(5:12)

In the simulation, we only need to consider the region in which the ABS and the disk surface arepossibly in contact. The friction force can be represented as Ff¼mPc.

When FH of a slider decreases to the sub-5 nm region, the short-range forces, including theintermolecular force, the electrostatic force, and the lubricant meniscus force, start to have signifi-cant influences on the stability and reliability of the interface. Therefore, their effects need to beincorporated into the contact analysis [24,59–76] via the above model.

5.2.3.1 Intermolecular Force

The intermolecular force-induced instabilities have been widely studied. They are considered as oneof the main reasons for the take-off and touch-down hystereses, which are one of the most famousphenomena in near-contact air-bearing slider dynamics.

When the surfaces of slider and disk get close enough, intermolecular forces or van derWaals forcestake effective action. At the beginning, it is an attraction force, and its strength increases with decreasingdistance until a maximum value is reached as described in Section 3.4; then it decreases with furtherdistance reduction. When the distance is reduced further, the force becomes repulsive and increasesdramatically. The potential energy between the two atoms is a function of their distance. For slidersflyingwith a few nanometers spacing, the repulsive potential term can be ignored, and when the spacingapproaches about 0.3 nm, the attraction potential and the repulsive potential in the equation are equal.

Equation 3.39 is the intermolecular force between each volume of material of the slider and thedisk. The attractive and repulsive portions of the force have different acting ranges. The attractivevan der Waals force has a much longer acting range than the repulsive portion. The two solidsurfaces first experience the attraction force when the distance between them is less than about10 nm. The strength of the attractive force increases with the reduction of the spacing until thespacing becomes so small that the short-range repulsive force becomes active.

It is noticed that when the clearance approaches zero, the intermolecular pressure appears as aninfinite repulsive form. It means that the two surfaces can never be touched. Actually, when theasperities are close enough, the molecules of the asperities tend to rearrange their positions (contactdeformation) instead of being pressed together to get infinite molecular repulsive forces. This is dueto the reason that the contact stiffness is much less than the molecular repulsive stiffness. The HLSZmodel is especially useful to deal with this case, as the model extends the calculation of theintermolecular force to the various FH regions without limit.

5.2.3.2 Electrostatic Force

The electrostatic force is one kind of short-range surface force and is also considered as theinstability factor of the interface and its effects are intensively studied. It has also been consideredas one of the reasons for causing the take-off and touch-down hystereses. It is believed to relate to



the tribocharging of the interface during the take-off period, and the lubricant pickup mayincrease the tribocharging and delay the takeoff of the slider.

The surface forces become significant in the slider–disk interactions for a head–disk spacing ofless than 10 nm. For 3–10 nm range, this is especially true for the electrostatic force as it is arelatively long range force compared with the van der Waals force which is significant at a spacingabout 2 nm or below. By applying a controlled voltage on the interface, the FH of the slider isdemonstrated to be modulated. For instance, 1 V potential could lead to 1 nm FH change. Theeffects of electrostatic forces can be even larger when FH approaches 5 nm, where electrostatic forcecould be a major factor for FH drop.

The electrical charge at HDI has been generally considered to arise from two major processes:transfer of electron through contact potential (surface potential difference) and transfer of materialthrough wear-induced tribo-emissions. It was reported that the contact potential at a typical HDI is atabout 0.3 V with the head positively charged, but ranged between �1 and þ1 V depending on theexperimental setup.

During sliding contact with a stainless steel or glass stylus, a lubricant like PTFE substrate coulddevelop a negative charge due to tribocharging.

The negative contact charging mechanisms of polymers have been widely investigated and wellunderstood. The basic concepts center on bringing electron-free energies into equilibrium, andhaving simultaneously available charge carriers and unoccupied states.

On the other hand, efforts have been made to make use of electrostatic force to reduce FH andstabilize interface by applying proper DC voltage across the interface. The applied electrostaticforce can easily be calculated by modeling the slider–disk as a capacitor with one surface at an angleequal to the pitch of the slider. For instance, the magnitude of the force could be around 10 mN withthe applied voltage of about 5 V for a typical slider–disk interface.

5.2.3.3 Meniscus Forces

The interaction between the lubricant on disk and the near-contact slider has been a critical topic.The lubricant meniscus force has also been considered as the main reason of the hysteresis. Onenoticeable phenomenon associated with lubricant is the ‘‘washboard effect.’’ When a slider fliesover a disk covered with a layer of lubricant, the lubricant exhibits a periodic thickness modulationin the downtrack direction, and the modulation frequency is correlated to the air-bearing sliderfrequency. However, the modeling of the dynamic meniscus force of lubricant has been a greatchallenge due to the lack of experiment support for correlation.

In the near-contact region, free lubricant present at the interface can disrupt the air bearing beforethe limit for van derWaals forces is reached, which take strong effective action at critical values of lessthan 5 nm separation. A theoretical viewpoint and conjectures from experimental results suggest theexistence of lubricant bridging as predicted. Lubricant moguls, related to surface topography, weredescribed and lubricant ripples, related to slider roll mode frequencies, have also been reported.

When enough lubricant was made available at the interface, a stick–slip type of sustainedoscillation was observed. The excess lubricant ended up on the disk or on the slider deposited end.The meniscus principle described in Chapter 3 will be elaborated in the following, for furtheranalysis of near-contact slider. Assume that the meniscus is formed when the distance between acontact pad and an asperity summit is smaller than the lubricant thickness h, and consider thatthe form of the meniscus is always in ‘‘toe-dipping regime’’ with meniscus surrounding only oneasperity. The contact angle of the meniscus is always 08. The meniscus force acts as a tensileforce on the formed asperity, and the asperities are not independent of each other with respect to themeniscus force. Under the above assumptions, when the meniscus is formed, the meniscus force Fm

acting on the asperity summit and the contact pad is given by

Fm ¼ 4pRg (5:13)



whereR is the radius of curvature of the asperity summitg is the surface tension of the lubricant

This equation is considered to be the maximum value of the meniscus force in the toe-dippingregime. The present meniscus model can be applied only to the thin lubricant case in which thelubricant thickness is less than several nanometers.

The strong effects of meniscus in the region of sub-10 nm have been verified by experiments,including the take-off and touch-down hysteresis effects. A typical experimental result of hysteresis inslider–disk interface with lubricant mediated is plotted in Figure 5.6 [74–76]. As the separation ddecreases, the pad of slider is drawn to the disk surface by themeniscus force. In addition, the absolutevalue of the negative contact force when the pad separates from the disk surface is larger than thatwhen the pad comes into the disk surface. This is because the meniscus force acts as a tensile force, sothat the number of menisci when the pad separates from the disk surface is larger than that when thepad comes into the disk surface. The effect of this meniscus contributes to the hysteresis on the FH ofthe slider. In Figure 5.6, when FH is changed from 8 to 0 nm, i.e., the touch-down process, and whenFH is changed from 0 to 8 nm, i.e., the take-off process, are indicated by black and gray lines,respectively. The slider is at first drawn to the disk surface by the meniscus force at 5 nm in the touch-down process, and it separates from the disk surface at 6 nm in the take-off process. The difference ofthe touch-down FH and the take-off FH is more than 3 nm when the lubricant thickness hl is 2 nm.These hystereses phenomena of the slider FH have also been observed in other experiments.

5.2.4 NONLINEAR DYNAMICS NEAR-CONTACT AIR-BEARING SLIDER

AND THE FRICTION-INDUCED VIBRATIONS

This section introduces the nonlinear dynamics of air-bearing slider and its friction-inducedvibrations in near-contact regime.

As the basis, the dynamics of air-bearing slider is discussed first. The stiffness coefficients of airbearing are nonlinear, and they are also affected by the contact force which leads to larger naturalfrequencies. Moreover, slider in near-contact regime exhibits nonlinear dynamic properties of vibro-impact when the state of loss of contact occurs. Both of the properties render the system to exhibitnonlinear spectral features. Resonances with fold and fractional frequencies of the primary naturalfrequency as well as primary resonance were usually observed. A thorough understanding of bothslider air-bearing nonlinear dynamics and vibro-impact dynamics of contact slider is crucial to thesuccessful design and implementation of near-contact recording interface [77–101].

10

0

2

4

6

8

0 2 4 6 8

Nominal flying height (nm)

Mea

n va

lue

of s

paci

ng (

nm)

FIGURE 5.6 Hysteresis due to meniscus force during slider touch-down and take-off processes.



Finally, the self-excited vibration due to friction is described, and this is associated withsub-5 nm region where different short-range forces take strong effect.

5.2.4.1 Nonlinearity of Air Bearing

Analysis reveals that the air-bearing slider in near-contact region exhibits nonlinear dynamicfeatures due to nonlinear equivalent stiffness. Approximate analysis also shows that the naturalfrequency of near-contact air-bearing slider depends on the contact. The system response couldexhibit primary, subharmonic, and superharmonic resonances.

Consider the rigid motion of an air-bearing slider described in Equation 5.8 to obtain thedynamic behavior of slider fluctuation around the steady state; perturbation method has been usedto obtain linear stiffness coefficients kij and damping coefficients cij of air bearing [6]

k1j þ ivc1j ¼ �ðGj dA

k2j þ ivc2j ¼ �ðGj(xg � x) dA

k3j þ ivc3j ¼ �ðGj( yg � y) dA ( j ¼ 1, 2, 3)

(5:14)

The subscripts 1, 2, and 3 refer to vertical displacement, pitch and roll angles, respectively. Gj is thecomplex functions of spatial variables x and y.

The dynamic coefficients given in Equation 5.14 can be assumed as the parameters of first-orderapproximation to Equation 5.8. For simplification, we ignore the roll mode; thus the motionequation of slider can be expressed as

m€zþ Fs ¼ fz(t)þðð

(p� pa) dA ¼ fz(t)þ Qz(z, u, _z, _u)

Iu€uþMsu ¼ fu(t)þðð

(p� pa)(xg � x) dA ¼ fu(t)þ Qu(z, u, _z, _u)

(5:15)

in which z, u are slider’s vertical vibration displacement at slider’s center and slider’s pitch vibra-tion from their steady flying condition. Develop the air-bearing lifts Qz and Qu to be powerseries from the slider steady flying condition

m€z ¼ fz(t)þX1l,k¼0

a1lkzluk þ

X1l,k¼0

a2lk _zluk þ

X1l,k¼0

a3lkzl _uk þ

X1l,k¼0

a4lk _zl _uk

¼ fz(t)þ k11zþ k12uþ c11 _zþ c12 _uþ O(z2, . . . )

Iu€u ¼ fu(t)þX1l,k¼0

b1lkzluk þ

X1l,k¼0

b2lk _zluk þ

X1l,k¼0

b3lkzl _uk þ

X1l,k¼0

b4lk _zl _uk

¼ fu(t)þ k21zþ k22uþ c21 _zþ c22 _uþ O(z2, . . . )

(5:16)

in which ailk, bilk, (i¼ 1, 2, 3, 4) are the coefficients of the developed power series with linear terms:k11¼ a110, k12¼ a101, k21¼ b110, k22¼ b101, c11¼ a210, c12¼ a301, c21¼ b210, c22¼ b301. By remain-ing the second-order terms, a higher order approximation is expressed as

m€z ¼ fz(t)þ k110zþ k120uþ c110 _zþ c120 _uþ k111z2 þ k121u

2 þ c111 _z2 þ c121 _u

2 þ � � �¼ fz(t)þ k11(z,u)zþ k12(z,u)uþ c11(z,u)_zþ c12(z,u) _u

Iu€u ¼ fu(t)þ k210zþ k220uþ c210 _zþ c220 _uþ k211z2 þ k221u

2 þ c211 _z2 þ c221 _u

2 þ � � �¼ fu(t)þ k21(z,u)zþ k22(z,u)uþ c21(z,u)_zþ c22(z,u) _u

(5:17)



The system can be assumed as a small damping case as the modal damping for most of the airbearing is smaller than 5%. As such the velocity relevant terms can be neglected for nonlinearanalysis. In order to obtain an approximation to the nonlinear dynamic coefficients kij, cij, weconsider the case that slider vibrates from steady state to a transient (zq,uq) (q¼ 1, 2, . . . , N) with thetransient inertia force and moments (Fq,Mq) (q¼ 1, 2, . . . , N). We assume that by applying an equalstatic force and moments (Fq,Mq), q¼ 1, 2, . . . to the slider, it will move from steady state to (zq,uq),q¼ 1, 2, . . . . Then we are able to calculate the corresponding kij(zq,uq), cij(zq,uq) by using Equations5.15 through 5.17 with the loads. The obtained kij(zq,uq), cij(zq,uq) can be used as estimation to kij,cij. With enough database kij(zq,uq), cij(zq,uq), the parameters for kij, cij can be identified by usingnonlinear least-square method. For simplification, kij(zq,uq), cij(zq,uq) can be further reduced to asingle variable function by remaining a dominant variable or removing one variable by average. Onthe whole, it gives an approximation measure for the nonlinearity. Figure 5.7 shows the nonlinearstiffness of air bearing (k11) for three different near-contact sliders.

5.2.4.2 Contact Vibrations

We next discuss the contact vibration of air-bearing slider by taking the case of loss of contact intoaccount. For convenience, assume that the contact force is dependent on vibrating displacement zcand velocity _zc at slider trailing edge. It can be developed as Taylor series at the static equilibriumcontact point

Fc(zc,_zc) ¼ Fc(0,0)þ @Fc(0,0)@zc

zc þ @Fc(0,0)@ _zc

_zc þ @2Fc(0,0)@z2c

z2c þ@2Fc(0,0)

@ _z2c_z2c þ � � � (5:18)

If the first two terms are taken into account, it mathematically represents a model with the contactforce as linear spring force, and @F(zc, _zc)=@zc¼ kp is the linear stiffness coefficient. Under constantcontact force, assume the mean asperity peak of disk has a static compression zg; then it can beapproximated as kp¼Fc=zg. For time domain analysis, the contact force can be treated moresophisticatedly by using the GW model or a more advanced one as described above. Different

Flying height (nm)

2 3 4 5 6 7 8 9A

ir-be

arin

g st

iffne

ss (

M N

/m)

0

1

2

3

4

5

6

7

Slider A

Slider B

Slider C

(a) (b) (c)

FIGURE 5.7 Nonlinear stiffness of air bearings of three types of sliders.



contact models yield different information on the contact force, duration of contact, and FH.However, regarding the global spectrum characteristics of the slider vibration, different modelsdraw out consistent results.

Consider an air-bearing slider in near contact with a spinning disk with constant contact force. Ifa proper disturbance is applied, a small vibration could take place around its steady state, and bothcontact and loss of contact could occur. For simplification, the higher order terms in Equation 5.17,which characterize nonlinearity of air bearing, are ignored. Then the motion equation including bothcontact and loss of contact is given by

m€zv þ k11zv þ k12uv þ kp(zv þ buv) ¼ 0

I €uv þ k21zv þ k22uv þ kp(zv þ buv)b ¼ 0for zpv < zg, zpv ¼ zv þ buv (5:19a)

m€zv þ k11zv þ k12uv þ kpzg ¼ 0

I €uv þ k21zv þ k22uv þ kpzgb ¼ 0for zpv > zg (5:19b)

in which zv and uv are, respectively, the slider vertical and pitch vibration motions relative to itssteady flying state, and zpv is the transient vertical vibration of slider trailing edge.

Generally, Equation 5.19 represents a highly nonlinear system. First of all, consider the simplestcase, zpv < zg, in which the slider never loses the partial contact with the disk in vibrations. Then thesystem is linear and the squares of the natural frequency are derived as

v221,22 ¼ (k011=mþ k022=I)=2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(k011=mþ k022=I)

2 � 4(k011k022=(mI)� k012k

021=(mI))

q=2 (5:20)

in which k011 ¼ k11 þ kp, k012 ¼ k12 þ bkp, k021 ¼ k21 þ bkp, k022 ¼ k22 þ b2kp: Furthermore, whenkp¼ 0, it deteriorates to the natural frequency of the air bearing. Equation 5.20 illustrates the effectof contact on the natural frequency of air bearing.

For the case zpv > zg, the slider would experience both contact and loss of contact states andtherefore exhibits a nonlinear feature.

Consider the cases in which the slider is excited by disturbance. The disturbance could be thedisk waviness or roughness, the excitations of suspension and actuator system. However, fordifferent excitations except for friction, the slider vibration equations can be derived as thefollowing identical equations by different transformations:

m€zv þ k11zv þ k12uv þ kp(zv þ buv) ¼ F sinvt

I €uv þ k21zv þ k22uv þ kp(zv þ buv)b ¼ M sinvtfor zpv < zg (5:21a)

m€zv þ k11zv þ k12uv þ kpzg ¼ F sinvt

I €uv þ k21zv þ k22uv þ kpzgb ¼ M sinvtfor zpv > zg (5:21b)

where F and M are the effective force and moment disturbance applied on slider. It is noted thatEquation 5.21a,b is symmetric in terms of pitch mode and vertical mode. Consider the case in whichk11 is very large; vertical and pitch mode motions can be considered decoupled as they have similarfeatures. Then Equation 5.21a,b can be approximated by

m€zv þ k11zv þ kpzv ¼ F sinvt for zv < zg (5:22a)

m€zv þ k11zv þ kpzg ¼ F sinvt for zv > zg (5:22b)

The vibro-impact properties of equation described by Equation 5.22 have been investigated inChapter 4.



The nonlinear properties of air bearing and vibro-impact of near-contact slider have beenexperimentally investigated. In the following, a typical case is illustrated. The sliders used in thestudy were 50% nanotripad sliders made from Al2O3–TiC. The FH of slider at trailing edge wasmeasured using an optical FH tester (Phase Matrics). The protective layer and lubricant layer areformed on the experimental glass disk media, with a carbon protective layer thickness of 5 nm and alubricant thickness of 1.2 nm. The disk surface roughness was measured by atomic force micro-scope (AFM) with Ra¼ 0.68 nm. It is observed that the direct slider–disk interaction makes theslider vibrate severely on the disk surface. Figure 5.8 shows the spectrum of the slider FH atdifferent spindle speeds. The slider–disk contact occurs when the spindle speed is 3000 rpm or less,which was also confirmed by the recorded AE signal. The air-bearing natural frequency (112 kHz)is excited by such a contact interaction. When the spindle speed is further reduced, the har-monic frequencies (224 and 336 kHz) and the other frequency (160 � 172.2 kHz) can also beobserved, though its amplitude is relatively small. An interesting observation could be made

200.08

70 120 170 220 270 320 370

52.4 kHz

75.8 kHz

85.4 kHz

86.2 kHz

59.0 kHz

77.2 kHz52.4 kHz

112.0 kHz

112 kHz

336 kHz172.2 kHz

224 kHz

160 kHz 224 kHz

112 kHz

Am

plitu

de (

norm

aliz

ed a

rbitr

ary

unit)

8 krpm

6 krpm

4 krpm

3 krpm

2.8 krpm

2 krpm

Frequency (kHz)

(a)

(b)

(c)

(d)

(e)

(f)

FIGURE 5.8 Spectrum of slider–disk spacing at different spindle speed.



from Figure 5.8; the spectrum becomes rather smooth at spindle speed between 4800 and 3500 rpm.The spectrum on the lower range becomes rather salient at spindle speed from 6000 to 8000 rpm,which represents the other fundamental frequency.

5.2.4.3 Friction-Induced Vibrations with Effect of Intermolecular Forces

We next discuss the effect of friction forces on near-contact slider dynamics. In sub-10 nm range,when near-contact slider makes substantial contact with the disk, the friction effect due to strongeffect of short-range surface forces could be critical to slider vibrations.

By incorporating intermolecular forces into motion equation of air-bearing slider, the solutionexperiences some dramatical changes. It has been found that at ultra low fly heights the stiffnesscould decrease as the fly height decreases and becomes negative below a critical fly height value[59–62]. Negative stiffness means that the bearing is unable to maintain a mechanical spacingbetween the slider and the disk, which implies that contact will occur between the slider and thedisk. To maintain interface stability, the total stiffness in the vertical, pitch, and roll directionsmust be positive. If all three eigenvalues of the stiffness matrix are positive, then the system isstable. But if one or more of the eigenvalues of the stiffness matrix are negative, then the systemis unstable. The complexity of the effect of intermolecular force effect is illustrated by Gupta andBogy [61,62] using the so-called fly height diagram shown in Figure 5.9. The points on curve 1 givethe steady-state fly height vs. disk rpm without considering the effect of intermolecular forces. Allthe points on this curve have positive stiffness values and hence are stable. Curves 2 and 3 plot thevariations of fly heights vs. disk rpm taking into consideration the effect of intermolecular forces.From Figure 5.9 we observe multiple equilibrium points for disk rotations per minute between 1900and 4300. The equilibrium points on curve 2 have positive stiffness values and hence are stableequilibrium points. However, the points on curve 3 have negative stiffness values and hence areunstable equilibrium points.

From the fly height diagrams we observe that as the disk rpm decreases from 12,000 to as low as1,900, a stable fly height is given by curve 2. Below 1,900 rpm corresponding to a fly height of3.4 nm the slider becomes unstable and contact occurs between the slider and the disk. This valueof disk rpm gives the touch-down rpm. If the disk rpm increases from 1,900, the slider remainsunstable until a disk rpm of 4300. This is because at a disk rpm of 1,900, the slider is in contact withthe disk. This means that the slider is initially disturbed from its stable equilibrium position curve2 and hence it will oscillate between the multiple equilibriums that exist until a disk rpm of 4300.For rotations per minute above 4,300 corresponding to a fly height of 5.05 nm there is only one

00

2

4

6

8

10

3

3

2

1

6Disk speed (krpm)

Fly

ing

heig

ht (

nm)

9 12

FIGURE 5.9 Fly height diagram.



equilibrium point which is stable and hence the stability of the slider is restored at 4300. This valueof rpm gives the take-off rpm. For this case, the stable fly height for the system is 5.05 nm,above which the system will always converge to a stable equilibrium if it is perturbed from its steadystate due to external disturbances such as air flow, disk roughness, etc. This fly height is called the‘‘desired fly height.’’ The result suggests that the rpm range of the curve 2 in the fly height diagramgives an estimate of the hysteresis observed in touch-down–take-off experiments. It also suggeststhat the larger the range of the unstable region on curve 2, the more the hysteresis will be observedin the touch-down–take-off experiments.

Due to the attractive nature of the intermolecular forces, there could be a reduction in fly heightvalue. These forces also increase the magnitude of the pitch and roll angles slightly. At low diskrotations per minute there exists three equilibrium points, one of which corresponds to contact at theslider–disk interface. Among the two equilibrium points that can exist at low disk rotations perminute, one is stable and the other one is unstable. If the slider is perturbed from its stableequilibrium position in the bi-equilibrium region, it will undergo large excursions.

A higher Hamaker constant for the interface layers was found to result in an increasedmagnitude of the intermolecular forces and hence a more unstable system. The desired fly heightalso increases with the increase in Hamaker constant, which in turn limits the areal density that canbe achieved. The analysis concludes that higher pitch angles stabilize the system, and also suggeststhat increasing the preload reduces the desired fly height value, as such, higher disk rpm is needed toattain this fly height.

The above analysis with intermolecular force is based on the assumption of the smooth surfaceof the slider pad. It has a limit that when spacing approaches zero, there is infinite repulsive term ofthe incorporated intermolecular force. Thus, the corresponding result is invalid.

A refined treatment is to use the HLSZ model to incorporate the roughness effect. Comparedwith the intermolecular pressure with smooth surfaces, the results based on HLSZ model showgreater attractive pressure at the FH higher than 0.7 nm approximately, but much smaller between0.26 and 0.7 nm approximately. A negative stiffness region exists when the minimum FH is below1.2 nm [58].

5.2.4.4 Friction-Induced Vibrations with the Effect of Meniscus Force

Under certain conditions, the slider bouncing vibrations in a near-contact regime could be self-excited vibrations due to friction effect instead of forced vibrations due to the irregularity of the disksurface as illustrated previously. Sheng [96] discussed a simple case where self-excited vibrationscould occur if the initial growth rate of friction force is larger than half of the product of suspensionhorizontal stiffness and the sliding speed projection in the direction perpendicular to suspension. Wenext present a more detailed example given by Ono et al. [101]. Consider a 2-DOF analytical modelof air-bearing slider shown in Figure 5.10. The slider is assumed to be a rectangular with length a,height b, mass M, and pitch moment of inertia J at the mass center. The slider suspension system isrepresented as normal spring stiffness k, angular spring stiffness ku, and damping coefficients c andcu, respectively. The front and rear air-bearing effects are represented by two lumped linear springs(kf0,kr0) and dampers (cf0,cr0) located at the bearing pressure centers. This linear model is dedicatedto study the instability of friction interaction, as such; the nonlinear quantity of air-bearing force andother effects are ignored. In the first-order analysis of nonlinear vibrations, it is commonly knownthat an effective linear stiffness averaged over one period motion can approximately be used forthe calculation of the natural frequency. The distance from the center of the mass of the slider to thefront and rear pressure centers and the head gap point are denoted by df, dr, and dh, respectively. Thespacing at the head gap is denoted by zp. The disk is assumed to have no microwaviness andthe severe bouncing vibration is considered due to a self-excited vibration from friction force.Assume that the contact between the slider and the disk takes place only in the rear air-bearing padand that the concentrated normal contact force Fc and the friction force mFcr act on the head gap



point, where Fcr is the real contact force including adhesion force. Because the bouncing vibration iscaused by the adhesion force and friction force in the interface between the air-bearing pad and thedisk surface, it is important to properly evaluate these forces in the asperity contact with a thininterlayer of lubricant.

Many researchers considered that the adhesion force between a slider and a running disk iscaused not by a meniscus force but by an intermolecular force or by an electrostatic force, becauseof difficulty of the formation of a meniscus under the high-speed sliding and vibrating condition; onthe other hand, some experiments show that the meniscus force at each contacting asperity mainlycontributes to the adhesion force between the contacting slider and the disk. It was experimentallyshown that the adhesion force in a macroscopic meniscus can be valid in a nanomeniscus where thelubricant layer has a thickness of a few nanometers and the meniscus radius is less than 1 nm.Therefore, it is able to consider that the adhesion force at one asperity contact with interlayer oflubricant is approximated by Equation 5.13 even in dynamic contact conditions. Ono et al. [74–76]analyzed contact force characteristics of a rough contacting surface including bulk deformation andthe adhesion force (Equation 5.13) of the lubricant layer at each asperity and obtained the touch-down and take-off hystereses with respect to decreasing and increasing processes of nominal FH byusing a simple model. On the other hand, even the intermolecular adhesion force between a rigidsphere and a rigid plane can be expressed by Equation 5.13 except that R is the sphere radius and gis the surface energy of mating materials. For the contacting rough surface covered with a thinlubricant layer, both the intermolecular attractive force of lubricant and the adhesion force betweenan asperity and a flat with an interlayer of lubricant are given by Equation 5.13, where g is thesurface energy of the lubricant.

In Figure 5.10, consider that the slider is in a static equilibrium condition at the nominal FH atthe head gap contact point and pitch angle u0 when the air-bearing force, suspension load, andnormal contact force without meniscus force balance each other. If the vertical displacement of thecenter of mass and the angular displacement in the counterclockwise direction from the equilibriumstate are denoted by zg and u, the equation of motion of the slider is written in the form

M 00 J

� �€zg€u

� �þ c11 c12

c21 c22

� �_zg_u

� �þ k11 k12

k21 k22

� �zgu

� �¼ Fc(zp)�F0N(FH)

� 12bmFcr(zp)� dh(Fc(zp)�F0N(FH))

� �(5:23)

zp kr 0cr 0

drdf

dh

Fc

mFcr

z

cf 0

x

b

M, j

a

q

G

O

ccq

kq k

kf 0

FIGURE 5.10 2-DOF slider model for analysis.



where F0N(¼NF0) is the normal contact force neglecting any interfacial force of the surface andlubricant at the nominal FH, and m is the coefficient of friction. The nominal FH is defined as theseparation d between the head gap point and themean height of disk surface in a state of equilibrium. k11,k12, k21, k22 are given by

k11 ¼ kf0 þ kr0 þ k, k12 ¼ kf0df þ kr0dr ¼ k21, k22 ¼ kf0d2f þ kr0d

2r þ ku (5:24)

On the other hand, the elastic contact force at the same contact area becomes larger because themuch larger dynamic spacing zp at the head gap point is given by zp¼ zg � dku þ FH.

Figure 5.11 is the calculated response of the above model. The parametric study shows thatthese bouncing vibrations and the hysteresis phenomenon are obtained only when both the adhesionforce and frictional coefficient are taken into account. The slider bouncing vibrations in a near-contact regime could be a self-excited vibration due to friction effect, instead of a forced vibrationdue to the irregularity of the disk surface.

The bouncing vibration of a slider in near-contact and contact regimes could also be a self-exited vibration caused by the combination effect of an adhesion force and a friction force. Theslider bouncing vibration usually shows a periodical steady waveform in the time domain, but itusually has many higher order components and sometimes has fractional frequency components inthe frequency spectrum.

The fundamental frequency of the unstable vibration is closely related to the front air-bearingstiffness. The self-excited vibration has such a mode that the leading side of the slider swingsgreatly. In order to realize a stable flying slider without bouncing vibration and FH hysteresisin near-contact regime, the approaches include to increase both the front and rear air-bearingstiffnesses, and to reduce adhesion force and frictional coefficient.

5.2.4.5 Experimental Observations

Extensive experimental studies have been conducted to investigate near-contact slider dynamics inthe last decade. Figure 5.12 shows the schematic of a typical experimental setup to investigate

0

5

10

15

20

0 0.025 0.05 0.075 0.1 0 0.025 0.05 0.075 0.1Time (ms)

Dis

plac

emen

t (n

m)

0

5

10

15

20

Time (ms)

Dis

plac

emen

t (n

m)

0 200 400 600 800 1000 0 200 400 600 800 10001.00E − 08

1.00E − 06

1.00E − 04

1.00E − 02

1.00E + 00

1.00E − 08

1.00E − 06

1.00E − 04

1.00E − 02

1.00E + 00

Frequency (kHz)

Vel

ocity

(m

m/s

)

(a)Frequency (kHz)

Vel

ocity

(m

m/s

)

(b)

FIGURE 5.11 Calculated results of displacement time history and velocity frequency spectrum at the headgap position of a slider (N¼ 5.0, m¼ 3.0, kf0¼ 53 105 N=m, kf0¼ 33 105 N=m): (a) nominal flyingheight¼ 5.8 nm; (b) nominal flying height¼ 9.5 nm.



slider–disk interface. The velocity of the vertical motion at the trailing edge of the slider wasmeasured by ultra-precision Laser Doppler Vibrometer (LDV), which usually has a bandwidthranging from 10 kHz to 10 MHz. The contact event is usually monitored using AE sensor mountedon the base of suspension. AE sensor usually has a bandwidth ranging from 20 kHz to 2.5 MHz. Allof the signals were collected by using signal analyzer.

The LDV spot was placed on either the outer trailing edge or the center trailing edge of theslider. The AE and the LDV signals were acquired and triggered simultaneously and were usuallyband-pass filtered between 20 kHz and 2 MHz. The roughness of the disks was measured with anAFM and with optical profilometry. In the following illustrative example, the roughness has acenterline average roughness value of 0.3 nm. Disks with the following three values of lubricantfilm thickness were used: 1.5, 2.5, and 3.5 nm. The lubricant used was Z-Dol. The sliders availablefor this study were subambient pressure picosliders. Experimental results illustrated that for thedifferent cases, there are different kinds of nonlinear oscillations existing in the system: the slider–disk contact-induced nonlinearity and the air-bearing inherent nonlinearity, as well as friction-induced self-excited vibrations. It was shown that the contact bouncing vibration and air-bearingoscillation as well as friction-induced instability render the system to exhibit complex nonlinearfeatures, including primary, superharmonic, and subharmonic resonances, as well as parametricresonance.

Bump response test is usually applied for contact validation as illustrated in the followingcase. The sliders used in the study were 30% picosliders made from Al2O3–TiC. The disks used inthe experiment were supersmooth disks with 1.5 nm carbon overcoat and 1.5 nm lubricant, androughness Ra¼ 0.4 nm. The sliders were designed to have 8 nm FH at the trailing edge. The sliderFH design conditions were met at a test radius of 40 mm on the spinning disk of 3600 rpm. Theoutput velocity signal of the LDV was fed into a high-pass filter with a cutoff frequency of 20kHz. The resonant modes of the slider air bearing are in the range of 30–200 kHz from itsspecification. The major natural frequencies of the slider body are in the range of 1–2 MHzby using FEM analysis. The velocity or displacement output was fed into a band-pass filter withthe frequency range from 2 to 300 kHz for the air-bearing analysis. Alternatively, the band-passfilter was removed to analyze the high-frequency signal for slider body resonance due to slider–disk contact. To validate the contact test, an experiment was conducted by allowing the 8 nm FHslider to fly across a bump with 10 nm height and 10 mm radius on disk. Figure 5.13 is the profileof a semisphere bump with 10 nm height fabricated by using laser process on the supersmoothdisk. A cubic bump fabricated by using etching techniques was also used for the experimentalverification. The flying slider is likely to make contact with the bump at properly low diskspinning speed in the experiment.

HP filter Digital scope

Signalconditioner

LP filter PC

Digital scope

BP filter

Preamplifier

AE

Slider

LDV

Fric

tion

Air bearing

Disk

FIGURE 5.12 Schematic of a typical test setup.



Figure 5.14 is the LDV measured slider–bump response and time–frequency spectrum. Fromthe figure we can see that both air-bearing resonant modes (below 500 kHz) and slider bodyresonant modes (between 1 and 2 MHz) were excited during the bump excitation. It indicates thatthe slider made contact with the bump.

nm

0 10.0 20.0

Spectrum

−10.

00

10.0

µM

FIGURE 5.13 AFM (atomic force microscope) measured profile of the bump with 10 nm height ondisk. (Reprinted with permission from Zhang, J. and Sheng, G., IEEE Trans. Mag., 39, 5, 2003. IEEECopyright 2003.)

0 0.5 1 1.5

�10−3

−2

−1

0

1

2

3

Time (s)

V

0 1 2 3

�106

10−4

10−2

100

102

Power spectral density

Frequency (Hz)

A

LDV response history

Time (s)

F (

Hz)

0 1 2 3 4 5 6 7 8 9

�10−4

0

0.5

1

1.5

2

2.5�106

Time – frequency spectrum

FIGURE 5.14 LDV measured response, time–frequency spectrum, and spectrum peak hold of slider responseto a bump.



Figure 5.15a is the phase plot of the slider–bump response, which indicates that the slider–diskcontact event is characterized by ‘‘sharp corner’’ and the rest of the oscillations of air bearing on thetrajectory are characterized by ‘‘smooth corner.’’ Multiple short consecutive contacts could bedistinguished in time–frequency spectrum with specific slider body natural frequency. Phase plotscould be used to identify the slider–disk contact vibration and air-bearing nonlinear oscillation; theformer is characterized by sharp corner when the noncontact phase transfers into contact phase, andthe latter is characterized by smooth corner due to the stiffness–softening feature of air bearing. Forcomparison, Figure 5.15b is the phase plot of a near-contact response of a slider that makescontinuous contact with the disk, which indicates that the head–disk contact is characterized bysharp corner and the induced air-bearing oscillation is characterized by smooth corner.

The near-contact process of a flying slider can be realized by gradually reducing the rpm ofspinning disk until the slider touch-down on the disk. Figure 5.16 shows power spectral density ofLDV signal of slider trailing part at several disk velocities. During flying (8, 6, 4 m=s), we can seethe air-bearing resonance frequencies of f1 and f2 are around 100 and 200 kHz. Particularly, f2increases with the decrease of the velocities; this is because of the nonlinearity of the air-bearingstiffness. When the speed is reduced to 2 m=s, the slider starts to make contact with the disk, which

−25 −20 −15 −10 −5 0 5 10 15−40

−30

−20

−10

0

(a) (b)

10

20

30

40

50

60

Displacement (nm)

Ve

loci

ty (

mm

/s)

Phase plot Phase plot

−15 −10 −5 0 5 10 15−15

−10

−5

0

5

10

15

Displacement (nm)

Ve

loci

ty (

mm

/s)

ContactNoncontact

Contact Noncontact

FIGURE 5.15 Phase plots of slider response: (a) flying slider response to a bump impact; (b) proximity contactslider response.

0

20

40

60

80

100

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

f1 f2

ft

fb2 m/s

4 m/s

8 m/s

6 m/s

Frequency (MHz)

Po

wer

spe

ctra

l den

sity

(20

dB/d

iv.)

FIGURE 5.16 Power spectral density of proximity contact slider.



is indicated by the detected slider body resonant frequencies in the spectrum. The torsional andbending frequencies of this kind of slider body are ft¼ 1.3 and fb¼ 1.7 MHz, respectively.

We next describe another experiment of near-contact slider conducted by Ono et al. [75].Contrary to the above example, in this example the FH reduction is attained by gradually reducingthe pressure of chamber where the slider–disk system is installed. The tested slider was a commer-cially available picoslider with ABS shown in Figure 5.17; it is applied with 4200 rpm, at the radialposition of 30 mm. The slider was flying without intermittent contact with the disk at about 15 nmheight at atmospheric pressure, with resonance frequency between 100 and 200 kHz. However, asthe chamber pressure is gradually decreased, the FH decreases so that the slider comes in contactwith the disk surface and exhibits bouncing vibration. The contact between the slider and the disksurface is monitored by means of an AE sensor.

When the slider is perfectly flying, the high-frequency vibration of the slider cannot be observedand the slider tracks the disk surface runout. When the ambient pressure is reduced to 0.4 atm, theslider begins to contact the disk surface and displays bouncing vibration in touchdown. Then,the chamber pressure is gradually increased, but the slider still continues the bouncing vibrationuntil the ambient pressure is increased to 0.56 atm for takeoff. This pressure difference between thetouchdown and the takeoff corresponds to a difference of 2–3 nm of a nominal FH.

Figure 5.18 shows the typical time history of the slider displacement and frequency spectrum ofthe slider velocity at the trailing edge when bouncing vibration happens. The displacement wasobtained by numerical integration of the LDV measured velocity signal.

Figure 5.18a through c shows the three cases: just after the touch-down point, an intermediatepoint between the touch-down and take-off points, and just before the take-off point, respectively.

From the displacement waveform in Figure 5.18a which represents the case of just after thetouchdown, it is noted that the slider exhibits severe random vibration whose maximum heightsometimes increases to more than 200 nm. From the waveforms in Figure 5.18b and c, we note thatthe vibration at the trailing edge becomes quasi-periodic and whose total amplitude is about 50 nm.From the frequency spectrum of Figure 5.18b and c, we can see that the vibration waveformcontains many frequency components but that the dominant fundamental frequency is about100 kHz. This corresponds to the lower resonance frequency of the slider.

These experimental results suggest that the bouncing vibration is a self-excited vibration of thelower natural frequency of the slider and air-bearing system. The destabilizing effect is generated

1.25 (mm)

Leading edge Trailing edge

Rear air-bearingFront air-bearing

Pad

Shallow step

Deep step

1.0

(mm

)

FIGURE 5.17 Air-bearing surface of slider used for the experiment.



from the interfacial force at contact between the air-bearing pad and the disk surface. Due to theconstraint of contact, the waveform contains many higher order components and sideband com-ponents. Because of the nonlinear effect of the contact force, the bouncing vibration shows ahysteresis phenomenon of the different touch-down and take-off points when decreasing andincreasing the nominal FH. This indicates that the slider may exhibit bouncing vibration bydisturbances even if the slider is stably flying over the disk surface at a nominal FH in this hysteresisrange. It is obvious that a flying slider cannot be used properly without eliminating this self-excitedbouncing vibration and hysteresis range in near-contact regime.

Apart from the lubricant effect on the hysteresis phenomena associated with take-off and touch-down processes in near-contact dynamics, the roughness of disk also influences the frictionhysteresis transition processes when a near-contact slider takes off or touches down on disk.Figure 5.19 shows the frictions as functions of FH for a slider touching –down on a smooth and arough disk, respectively. When a near-contact slider flies over a rough disk, friction force increasesgradually with decreasing FH; the increase in the amplitude of the slider vibration is within a fewnanometers. When the slider flies over a smooth disk, the friction and vibration show a steepincrease upon contact; the vibration could increase to an amplitude of 10 nm. Moreover, the thickerthe lubricant, the larger the vibration amplitudes. This is also attributed to the effects of intermo-lecular and meniscus forces. The surface topography of supersmooth disk (Ra is in the vicinity of

(a)

Time (ms)0

0 500 1000

0.05 0.1

Time (ms)0 0.05 0.1

Frequency (Hz)

0 500 1000Frequency (Hz)

Frequency (Hz)

Vel

ocity

(m

m/s

)D

ispl

acem

ent (

nm

)

Vel

ocity

(m

m/s

)D

ispl

acem

ent (

nm

)

(c)

(b)

Time (ms)

Vel

ocity

(m

m/s

)D

ispl

acem

ent (

nm)

00 0.05 0.1

0

100

1

0.01500 1000

100

200

0

100

1

0.01

100

200

0

100

1

0.01

100

200

FIGURE 5.18 Experimental results of bouncing vibrations of a slider when the flying height is decreased andincreased by changing ambient pressure: (a) in touch-down point (0.4 atm); (b) during touch-down and take-offpoint (0.5 atm); (c) in take-off point (0.56 atm). (Reprinted from Ono, K. and Ohara, S., ASME J. Tribol., 365,127, 2005. Copyright 2005, ASME.)



1–2 Å) could rapidly approach molecular dimension, where short-range force takes strong action.Sharp friction increase or step-like increase of friction is likely accompanied by the rapid onset ofadhesion. This can be attributed to the rapid increase of larger real area of contact, which leads to theformation of increased number of microjunctions at the onset of contact.

Figure 5.20 shows the AE as functions of time in typical contact start–stop operation fordifferent disks with various roughnesses. The contact start–stop operation of disk consists of threephases, namely, disk rpm ramp-up, steady spinning with constant rpm, and ramp-down to stop asshown in Figure 5.20a. From Figure 5.20b we can see that the effect of disk roughness on AE isconsistent with the effect of disk roughness on friction shown in Figure 5.19. For rough disks, thetransition is a gradual process; for very smooth disks, the onset of AE is a sharp or step-rise process.

Apart from the micromeniscus effect, there are certain studies that reported the effects ofelectrostatic force on FH under certain conditions.

5.3 MODELING OF FRICTION-INDUCED ACOUSTIC EMISSION

Over the last two decades, AE method has been one of the most sensitive and reliable approaches todetect and quantify slider–disk contact and certify disk surface mechanical asperities. AE sensorand PZT element have been applied to detect AE signal in slider–disk contact. AE sensor isusually stand-alone and mounted on the base of the suspension assembly, and PZT sensor is usuallyswitched between the slider and the suspension or glued on slider.

Basically, both PZT and AE sensors operate on the same principle, and use similar materialswith piezoelectric properties (they usually are both PZT-PbZrmTinO3). The level of ‘‘sensitivity’’depends on the physical crystal dimension, mounting, and material type. Since the mountinglocation of the PZT sensor is closer to the source of vibration, the output signals at certain frequencyranges can be larger than in an AE sensor. The mounting of the PZT sensor on the slider may not bepossible or practical for many cases; the use of an AE sensor further up the mounting fixture maybe the only option for these cases. With proper mounting and conditioning electronics, an AE sensorcan produce signals with the same level of signal to noise as the PZT. For high-frequency detection(higher than 1 MHz), an AE sensor may be better suited due to a variety of sensor choices andmounting flexibility. If the requirement involves an actual slider with read–write data head, then AEis the way to go. However, for disk certification in mass production, the PZT has been used throughspecial fabrications. Near-contact slider usually generate signals with distinct frequencies in shorttime intervals which are usually nonstationary. Figure 5.21 is the typical AE sensor output of sliderresponse to bump impact, which is very consistent with the results measured by using high precisionLDV. Therefore, it is a very high efficient approach to detect and investigate slider dynamics andslider–disk interaction. It has been demonstrated that AE can be used to characterize different

0

5

10

15

0 5 10 15

Textured disk

Flying height (nm)

Fri

ctio

n (m

N)

Smoother disk

FIGURE 5.19 Friction as a function of flying height for smooth and rough disks.



elements of a slider take-off profile in both the time and frequency domains, but is subject to othermechanical noise such as the dimple separation slip on suspension. We next discuss the modeling ofAE of a flying slider when it makes contact with the disk.

5.3.1 EMPIRICAL MODEL

The AE approach has been widely used in the tribology community to detect and monitor interfaceinteraction to correlate friction and wear change. However, the rms value of AE signal, after filteringout the lower components, has also been used to estimate the contact forces [102–134].

The following relationship has been used for the estimation of the energy ofAE signal per unit time:

E=t / V2 (5:25)

wheret is the timeV is the output of the AE signal

0

2

4

6

Time (s)

0 3 6 9

krpm

Time (s)0 2 4 6 8 10

Aco

ustic

em

issi

on (

a.u.

)

16 nm

13 nm

8 nm

20 nm

5 nm

(b)

(a)

2

1

0/2

1

0/2

1

0/2

0/2

1

0

1

FIGURE 5.20 (a) rpm profile as time function in a start–stop operation; (b) AE as a function of time in contactstart–stop operation for different disks with various roughnesses.



Diei [103] performed pin-on-disk sliding test and proposed the following model to quantify therelationship between the root-mean-square (rms) voltage of the AE signal Vrms and the work of thefriction force per unit time:

Vrms ¼ (kArtV)m=2 (5:26)

wherek and m are the constantst is the shear strength of the interfaceAr is the real area of contactV is the sliding velocity

Kita et al. [42] were the first to apply AE for the interface investigation of hard disk drive.Ganapathi et al. [118] proposed the following model for AE signal Vrms as a function of diskvelocity and contact force:

Vrms ¼ kFV2 (5:27)

where k is the constant. Khurshudov and Talke [102] summarized some experimental formulationsof AE signal, and indicated that it is difficult to give a meaningful physical interpretation about thedependence of Vrms on velocity and contact force. Based on some experiments for specific interfaceof ceramic ball sliding on disk, they fitted the following linear relationship:

Vrms ¼ kFV (5:28)

It is simply interpreted that the AE signal is related to the energy dissipated at the interface perunit time.

Slider responsedue to bump

impact

Slider response due tointeractions (solid–solid or solid–

lubricant)

Disk speed 3 m/s

Disk speed 5 m/s

Disk speed 10 m/s

200160120Time (s)Bump

Sen

sor

outp

ut (

mV

)100

0

100

100

0

100

100

0

−100

80400

FIGURE 5.21 Typical AE sensor response on an interaction between the slider and the bump.



There are many efforts dedicated to calibrate AE model parameters k using low rpm data toextrapolate contact force in high rpm region. It is noted that for near-contact slider the contact forceserves as an excitation force for slider vibration. The locations, on which the contact force is applied,are different in the low- and high-speed regions. In low-speed regions, the contact force is adistributed force, and in the high-speed regions, the contact force is a concentrated force actingon the trailing edge of the slider. Even if the sum of the distributed force is equal to the concentratedforce, the response of the slider will be much different in the two states. Therefore, the application ofthe constant k obtained from the low-speed state to predict the contact force in the high-speed stateneeds further modeling treatment. As the AE method is very convenient and has no obvious sideeffects on the system, it has been used to qualitatively evaluate the contact force. The above modelsonly consider the static part or rms of the force by using the AE method. In the following, wedescribe the SL (Sheng–Liu) model [131,132], which improves the above empirical models bygiving suitable functional forms and incorporating the entire interface parameter.

5.3.2 SL MODEL

An AE signal contains lot of information from AE sources and the propagation paths. The detectedsignal from a sensor is generally given by convolving the sourced function, the impulse response of theslider–suspension system to the source, and the impulse response of the sensor. In principle, the sourcefunction can be recovered by deconvolutionwhenwideband sensor is employed, and thefirst motions ofsignal are discernible. However, mathematically, there is some difficulty to handle illness matrixproblem, and in practice, these efforts are often controversial because they usually lack a valid analyticaljustification of the signal features used to sort the experimental signals into different types of sources.

Some AE source-identification experiments have been carried out with various specimengeometries and sensor locations such that signals are obtained from the direct wave propagationanalysis. By comparing relative experimental bulk wave amplitudes in different directions with thecalculated results for a series of possible sources, the experimental sources were identified in a moresatisfying fashion. Even if the direct method is not completely successful, these results couldprovide insight into the most relevant results to input.

The interactions in friction interfaces could be either burst-like impact which is associated withisolated asperity impact and system resonance, or continuous interaction that is of random form oftime series as shown in Figure 5.21, or both.

There has been a need to give theoretical model of slider–disk interaction and AE sensingprocess for interpreting interface phenomena and estimating unknown parameters. In the following,the AE sensor system is modeled as a linear system excited by deterministic and random interfaceexcitations. The time history form output is approximated by a conventional closed form, whereasrms form output of the system is extracted as a function of velocity, contact force, surfacetopographic, mechanical parameters, lubricant thickness, and system transfer functions.

An AE is a stress wave due to quick release of the strain energy from materials under stress. Thestress could come from either deformation or fracture. The deformation may be induced by theimpact or contact process, whereas the fracture may be induced by the crack initiation andpropagation. Part of this energy, released from the material in the form of elastic stress waves,can be detected by an AE sensor.

Consider a linear mechanical system, which consists of a near-contact slider attached to asuspension. An AE sensor is mounted on the base of the suspension. The near-contact slider isflying on a spinning disk. Assuming that the slider is impacted by an asperity on the sliding disksurface and the resultant stress is s(x,y,z,t), the impact force can be represented as f(t)¼ Ð

As dA andthe output of the sensor can be represented by

V(t) ¼ðt0h(t � t)f (t) dt (5:29)



in which h(t) is the system transfer function. The rms output of the transducer is

Vrms ¼ 12p

ð1�1

H(v)j j2Sf (v) dv� �1=2

(5:30)

in which Sf(v) is the power spectral function of impact force. H(v)¼H3(v)H2(v)H1(v) is thefrequency domain system transfer function including slider body transfer function H1(v), suspen-sion transfer function H2(v), and sensor transfer function H3(v). The band-limited rms output of thesensor can be written as

Vrms ¼ 12p

ðv2

v1

H(v)j j2Sf (v) dv� �1=2

(5:31)

in which v1, v2 are the cutoff frequencies of system filters.To model and estimate impact forces in detail is usually difficult in practice. However, for the

simplest case in which single bump impacts with slider, the elastic impact force can be approxi-mated by

f (t) ¼ (5 mV=3)3=5f4E ffiffir

p=½3(1� v2)�g2=5 sin (pt=ti) (5:32)

in which m and r are the equivalent mass and the radius of the bump, respectively, V is the diskspeed, ti is the impact duration period, and E and n are Young’s modulus and Poisson’s ratio,respectively.

The power spectral density of the output signal for the system can be represented as

Sv(v) ¼ H(v)j j2Sf (v) (5:33)

The frequency spectral distribution of AE signal is dependent on the relevant resonance mode of theentire system and the impact force. For the impact with enough short duration, Sf (v) is a constant inthe interested frequency range; then Sv(v) is fully dependent on the system transfer function. If thesystem resonance modes of sensor and suspension are known beforehand, their effects on the Sv(v)can be separated; then the rest peaks on the curve of Sv(v) are fully dependent on peaks of sliderbody transfer functions H1(v) which reflect the resonance mode of the slider body. The slider bodyresonance frequencies usually have high frequencies (usually >300 kHz for 70% slider, >650 kHzfor 50% slider, and >1200 kHz for 30% slider, with 100% slider denoting slider with 2 by 2 mmplane size). They are much higher than the dominant peaks of suspension (including gimbal andfixture), transfer function H2(v) (usually <20 kHz), and the air-bearing resonance frequency(usually <150 kHz) which usually contribute greatly to the interface excitation f(t). The peaks ofH1(v) can be readily triggered by the slider–disk impact, and thereby can be used as a sensitivemeasure to monitor slider–disk impact by means of filter technique. For commercially available AEsensors, we can choose ones with flat response curves in the range from 0.5 to 2 MHz. Figure 5.22shows the torsional and bending mode shapes of commercial 50% tripad slider by finite elementmodeling; the corresponding natural frequencies are 650 and 870 kHz, respectively. The resonantfrequencies of slider body or the peaks of H1(v) are known beforehand, and then based on thedetected signal these peaks are verified and used as criteria to judge the slider–disk impact or tofurther identify contact force.

Theoretically, impact force could be back calculated in terms of the above equations as long asAE signal is measured. However, due to the inherent ill-conditioned nature of this technique,accurate results are seldom obtained, particularly for small form factor sliders.



In the next subsection, we further develop the model to address the AE of slider under generalsliding contact interaction using approximate energy average method. The impact case will befurther discussed in the next section using modal analysis approach.

For AE of a near-contact slider, in addition to individual or periodic impact due to large asperity orbump collision, major input in near-contact process comes from interface continuous or intermittentcontact, friction and wear process, which are governed by the contact deformation associated withsurface geometric irregularities and the materials removal associated with the micro- and macrocrackprocesses. Typically, measured waviness wavelength and roughness wavelength of disk may be aslow as 20 and 1 mm, respectively. Assuming the velocity of slider is about 20 m=s, the interferencefrequency of the waviness ranges from low frequency to 1 MHz. Also assuming the velocity of slideris about 4 m=s, the excitation frequency of the roughness interaction ranges from low frequency to4 MHz. Apart from slider–asperity contact, the uncertainty of the contact point or region, friction andwear between the slider and the disk, lubricant transferring, debris pickup and break-off, as well asmechanical excitation such as runout and disk flutter all contribute to the detected AE signal.

It is convenient to characterize the dynamic responses of an elastic system under random excitationby using the energymethod. For structural dynamic systems, the input mechanical power is equal to theoutput power adding dissipated power. Generally, most of friction power is transferred to thermalpower, whereas stress wave is linkedwith surface deformation power and surface crack removal power.In view of the relevant input power to the contact slider consisting of removal power _Er and deformationpower _Ed, the AE rms yielded by near-contact slider can be represented as

Vrms ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig _Er þ b _Ed

q(5:34)

where g, b are the characteristic parameters depending on transfer functions. We next derive theinterface removal power and elastic deformation power, respectively.

Archard’s assumption can give an approximation of the removal power. When two asperitiescome into contact each time to form a junction, there is a constant probability k that an adhesivefragment will be formed. The schematic of the removal of a hemispherical wear particle by adhesiveforce at the juncture is plotted in Figure 5.23a. In terms of Equation 3.81, each fragment can beassumed to be an adhesive fragment and is given by dv¼ kF dL=(3H), in which F is the contactforce, dL is the sliding distance, and H is the hardness of material. Since the energy required toremove a volume dv of material is Er¼H dv, the removal power can be represented as

_Er ¼ dEr=dt ¼ H dv=dt ¼ d(kFL=3)=dt ¼ kFV=3 (5:35)

As the coefficient k is the probability that any junction leads to the formation of a transferred debris,the coefficient depends on the ratio of lubricated and readily penetrated areas, and is dominated by

Mode 1624.0 kHz

Mode 2824.5 kHz

(a) (b)

FIGURE 5.22 Finite element modeling of tripad slider: (a) torsional mode; (b) bending mode.



the readily penetrated areas, k¼akd þ (1 � a)kl � akd, where a is the fraction coefficient of thereadily penetrated area and kd is the probability coefficient for readily penetrated area, which can beapproximated by three times of dry wear coefficient. The readily penetrated area can be defined asan area with lubricant thickness thinner than hc. Therefore, for normal distributed lubricantthickness, the readily penetrated area fraction can be estimated as

a ¼ðhc

�1½exp(�(h� h0)

2=2s 2l )=

ffiffiffiffiffiffi2p

ps1� dh (5:36)

in which h0 is the mean lube thickness and sl is its variance.On the other hand, the deformation energy can be readily modeled in terms of Hertz theory of

elastic contact. The schematic of the compressing and restoring process of an asperity on slider by asliding disk is plotted in Figure 5.23b. Assume P is the load on a single asperity, R is the radius ofthe peak asperity, and E* is the composite Young’s modulus of elasticity. In terms of Equations 3.8through 3.10, the contact size is given by a¼ [3PR=(4E*)]1=3 and the compression is given byd¼ a2=R. The elastic deformation energy of contacting asperity can be represented as

e ¼ðP dd ¼ 8

15E*R1=2d5=2 (5:37)

In view of the loading transient of the compressing process being dt¼ a/V, the corresponding powercan be represented as

_e ¼ de=dt ¼ 815

34

� �4=3½P 4=3V �=½(E*)1=3R2=3� (5:38)

and the total power can be represented as

_Ed ¼ hA

ð_e�(z) dz (5:39)

whereh is the asperity densityA is the real contact area of sliderF(z) is the probability distribution of the asperity height z

The contact force of the interface is

F ¼ hA

ðP�(z) dz (5:40)

FFragment

V

2a

P

V d

Adhesion

(a) (b)

FIGURE 5.23 (a) Schematic of the removal of hemispherical wear particle by adhesive force at the junction;(b) Schematic of the deformation process of an asperity compressed by sliding contact.



For simplification, the peak-height distribution can be assumed as a normal distribution function. Assumethe limit case where the asperity has the same shape and the same height, the power can be derived as

_Ed ¼ 815

34

� �4=3

½F4=3V �=½(E*)1=3R2=3(hA)1=3� (5:41)

It has been experimentally found that plastic deformation and fracture process play significant rolesin contributing AE energy. As the deformation in slider–disk interface has been found with elasticcharacteristics, the AE energy could be mainly attributed to the removal power due to microfracture.

The take-off process of flying slider in conventional contact start–stop operation is a typicaltransition process from boundary lubrication to aerodynamic lubrication process as the Stribeckdiagram illustrates. In this process, the friction can be measured as a function of velocity and thetransition can be characterized by means of proper indicators relevant to the velocity. The contactforce can be reasonably estimated by dividing friction force with constant friction coefficient. Fortypical flying slider, the slider contact force during take-off operation can be approximated by

F ¼ F0e�kt(V=Vt)

2(5:42)

where F0 is the static contact force and Vt is the take-off velocity. kt is the parameter determining thecontact force at takeoff, and it can be determined by air-bearing simulation incorporated withcontact analysis. Consider a typical operation with a constant acceleration L, the AE model canbe approximated as

_E ¼ kFV=3 ¼ (k�F0=3)te�kt(�t=Vt)

2(5:43)

For near-contact slider, the change of its contact force in the take-off process is more complicateddue to its complicated flying altitude variation. It could maintain gliding or contact state, even whenthe disk speed has been increased above its take-off speed, as the FH increment caused by increasein velocity could be balanced by its drop due to the simultaneous increase of pitch angle. Inapplication, different modifications can be further incorporated into the AE model. For instance,the energy given out by stress wave and detected by the AE transducer can be affected by specifiedsuspension, and the energy stored in the suspension spring can be estimated as [130]

E ¼ 2k3RF4E

� �2=3

2� F

10Rk

� �(5:44)

in which k is the spring constant of suspension. Moreover, another model by Baranov et al. [133]can be considered as a special case of the AE model described above under the condition of pureelastic deformation.

In the following, we describe some experimental results of AE to correlate the modeling. Thesliders used include 50%Al2O3–TiC TPC flying sliders and contact sliders which comprise of contactpads glued to 850 type suspensions. The contact pads are cubic pads with a dimension of0.53 0.453 0.38 mm, which are cut from commercially available 50% Al2O3–TiC slider. Thedisks are standard 95 mm, carbon coated, and PFPE lubricated. All the testing are conducted at radius25 mm on a friction tester (CSS tester) and the AE sensors used include three models from PhysicalAcoustic Corp (Micro-100D and Pico-models) and from Vallen System Corp (Pico-Z model).

A constant speed drag test is conducted by using TPC slider with 300 rpm. The AE time historyis recorded by using AE sensor with 100 kHz high-pass and 1 MHz low-pass filters. The spectralanalysis results of seven AE signal samples obtained at different time denoted by A0–A6, respect-ively, are illustrated in Figure 5.24. It can be seen that there exist two peaks at 360 and 700 kHz,



respectively. The former is the resonance frequency of the sensor and the latter the torsionalresonance frequency of the slider. The presence of the torsional frequency indicates that the slidermakes contact with the disk at the leading or trailing edges of the slider rails. It is noted that for thetwo cases, A0 and A5, the torsional resonance peaks are very low and almost undetectable. Thissuggests that for impact investigation the narrowband resonance monitoring of 50% slider, refinedfilters, and average technique should be used as described in the next section.

The contact sliders are used to conduct the measurements. The AE sensing system is used with a1.2 MHz upper cutoff frequency filter. As the contact sliders have basic resonance frequencieshigher than 2 MHz, the effect of their resonance on the detected AE signal could be ignored.Figure 5.25a illustrates the AE rms as the function of contact force under the same velocity but withdifferent filters F1–F3, denoting three different filters which have the same upper cutoff frequencyof 1.2 MHz but have different lower cutoff frequencies of 0, 100, and 500 kHz, respectively.Figure 5.25b illustrates the measured AE rms as the function of disk rpm under the same load butwith different filters F1–F3. In the next experiment, the disks have similar roughness (Ra) about6 nm but have different lubricant thicknesses of 0, 1.7, 2.6, and 3.8 nm. The sensing system employs100 kHz high-pass and 1 MHz low-pass filters. The measurements are conducted at 200 (V1) and400 (V2) rpm, respectively. The effect of lubricant thickness of disk on the AE rms is shown inFigure 5.26a. Finally, the flying sliders are used for the measurements. Figure 5.26b shows the AE

0.016

0.012

0.008

0.004

MHz

A0A1

A2A3

A4A5

A6

0

0.19

0.25

0.31

0.38

0.44

0.50

0.57

0.63

0.69

0.76

0.82

0.88

0.95

FIGURE 5.24 Spectrum of AE signal for seven samples at different times. (Reused with permission fromSheng, G. and Liu, B., Tribol. Lett., 6, 233, 1999. Copyright 1999, Springer.)

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2 2.5 3 3.5

Grams

AE

rm

s (V

)

F1

F2

F3

0

0.1

0.2

0.3

0.4

0 1000 2000 3000 4000

rpm

AE

rm

s (V

)

F1F2F3

(a) (b)

FIGURE 5.25 (a) Measured AE rms as the function of contact forces under different filters; (b) measured AErms as the function of velocity. (Reused with permission from Sheng, G. and Liu, B., Tribol. Lett., 6, 233,1999. Copyright 1999, Springer.)



rms vs. disk rpm under different filters for TPC slider. From Figure 5.26b it can be seen that theAE rms level comes down to the original noise level as the rpm increases to above 1500 rpm, nomatter what kind of filters are used.

5.4 IDENTIFICATION OF INTERFACE CONTACT AND FORCE USINGACOUSTIC EMISSION SIGNAL

5.4.1 CONTACT LOCATIONS

Based on the detection of the presence of slide body resonance, AE signal can be used not only todetect the presence of slider–disk impact, but also to identify the detailed dynamic characteristicsassociated with contact. When a slider glides or slides on disk making impact or continuous contactwith bumps or asperities on disk, AE signal can be recorded continuously and is Fourier transformedinto the frequency domain. Then the contact event can be identified by verifying the presence ofslider body resonance. Moreover, the acoustic energy associated with the bending and torsionalmode of slider vibrations as shown in Figure 5.22 can be estimated and separated approximately,which give more information about the attitude of slider motion.

Before using this approach, the resonant frequencies and mode shapes of slider were predeter-mined by using FEM. The relative energy of the torsional and bending mode frequencies in AEsignal can be estimated as a function of velocity. When the power spectrum of AE signal iscalculated, the following equation was suggested by Knigge et al. [140]:

E ¼Xk¼N�1

k¼0

X(k)j j2 (5:45)

then the energy corresponding to the torsional and bending mode can be calculated by defining awindow around the corresponding frequency such that

Et ¼Xk¼t2

k¼t1

X(k)j j2 (5:46)

Eb ¼Xk¼b2

k¼b1

X(k)j j2 (5:47)

where t1, t2 and b1, b2 are the lower and upper limits of the window around the torsional and bendingmode frequency, respectively.

0

0.2

0.4

0.6

0.8

1

1.2

Lubricant thickness (nm)

AE

rm

s (V

)

V1

V2

(a)

0 1 2 3 40

0.005

0.01

0.015

0.02

0.025

rpm

0 500 1000 1500 2000

AE

rm

s (V

)

F1F2F3

(b)

FIGURE 5.26 (a) Measured AE rms as the function of lubricant thickness under different velocities; (b)measured AE rms vs. rpm under different filters for TPC slider. (Reused with permission from Sheng, G. andLiu, B., Tribol. Lett., 6, 233, 1999. Copyright 1999, Springer.)



The change in energy in the AE signal corresponding to the torsional and bending modes ofvibration slider is found to be associated with the change in the contact location between the sliderand the disk from the slider rails to the trailing edge. This reflects the transition from sliding toflying. Figure 5.27 shows the separated AE energy for subambient tripad slider as a functionof velocity.

From Figure 5.27 we can see that at velocity below 4 m=s, the AE energy in the torsional modeis much larger than the AE energy in the bending mode. This suggests that at low velocity mostimpacts occur on the side rails of the subambient pressure slider. At speed above 4 m=s, AE energyhas a drop in the order of magnitude, and then the bending mode AE energy becomes larger than theAE energy of the torsional mode, which means that the tripad region of the slider makes contactwith the disk rather than the side rails. It is noted that a small amount of AE energy from bendingmode vibrations is present even at speeds exceeding 10 m=s; it is apparent that the subambientpressure tripad slider is a true proximity recording slider.

Figure 5.28 shows the separated AE energy for tripad slider as a function of velocity for both thetorsional and the bending mode. It can be seen that the AE energy in the torsional mode is muchsmaller than the AE energy in the bending mode. The AE energy in the torsional mode increasesinitially from 1 to 1.5 m=s. Since the bending mode AE energy is much larger than the AE energy ofthe torsional mode, it is apparent that the slider makes contact mainly with the rear tripad. The AEenergy associated with individual slider body resonance gives an index indicating the location ofslider–disk contacts during takeoff and the slider attitude.

0

0.2

0.4

0.6

0.8

1

0 3.5 7 10.5 14Velocity (m/s)

AE

(no

rmal

ized

)

FIGURE 5.27 AE energy for subambient tripad slider as a function of velocity. (Dashed line: Total; Thin line:Torsional; Thick line: Bending.)

0

0.2

0.4

0.6

0.8

1

Velocity (m/s)

AE

(no

rmal

ized

)

0 3.5 7 10.5 14

FIGURE 5.28 AE energy for tripad slider as a function of velocity. (Dashed line: Total; Thin line: Bending;Thick line: Torsional.)



Contact dynamics of near-contact slider with the interactions of air bearing, slider bodyresonance, and lubricant effects have been investigated using high bandwidth LDV and highbandwidth AE sensor. It has been observed that the role of a lubricant is dominant in the velocityrange near the slider take-off velocity (glide avalanche point). In general, increased stiction andfriction is observed with increased lubricant thickness. It is experimentally observed that increasedtorsional and bending mode vibrations occur with increased lubricant thickness. The slider air-bearing modes are excited more strongly at lower speeds and on disks with thinner lubricant films.Multiple air-bearing harmonics are found to occur predominantly on thin lubricant films. Thestandard deviations of LDV and AE signal increase with increased lubricant thickness. This canbe attributed to vertical stick–slip behavior leading to ‘‘bouncing’’ vibrations. The increased ‘‘take-off’’ velocity on thick lubricant films can be explained by an uneven distribution of the lubricantfilm over the disk surface. The uneven distribution causes the slider to interfere with the lubricant athigher FH. Generally, thick lubricant films are not bonded as strongly to the disk as thin lubricantfilms. This in turn leads to increased lubricant accumulation on the ABS of the slider and thereby toincreased slider torsional and bending mode vibrations.

For thick lubricant films, additional peaks are observed near the bending mode frequency in theLDV spectrum. These sidebands are also observed in the AE spectrum for all values of lubricantthickness. The amplitude of the sidebands is proportional to the lubricant thickness. Thus, it isjustifiable to conjecture that the sidebands are related to slider–lubricant interactions.

Apart from the effect of lubricant, the air-bearing oscillations also have an effect on the detectedresonance of slider body. To illustrate the interactions, we describe the experimental results given byMcMillan and Talke [141], in which the sliding contact of air-bearing slider near its glide avalanchepoint is studied for disks with lubricant thickness of 1.5, 2.5, and 3.5 nm, respectively. Slider air-bearing modes, slider body torsional, and bending mode vibrations are investigated as a function ofdisk velocity and lubricant thickness. Nonlinear features of interactions among air bearing, sliderbody, and lubricant are observed, which leads to additional peaks in the spectrum near the bendingmode frequency and the variation of the peak value. In Figure 5.29, an AE spectrum is shown as afunction of disk velocity for a disk with a thick lubricant film (3.5 nm). The spectrum of the AEsignal contains a number of additional peaks in the range from 200 kHz to 1 MHz. They are salient

Frequency (MHz)

1.0E10

1.0E05

1

Ave

rage

d P

SD

Roll

Pitch

Torsional

4.6 m/s

4.4 m/s

4.2 m/s

4.1 m/s

3.8 m/s

S1

S2

S3S4

Bending

0 0.4 0.8 1.2 1.6 2

FIGURE 5.29 Spectrum of AE signal with thick lubricant (3.5 nm) at various disk velocities. (From McMillan,T.C., Talke, F.E., and Harrison, J.C., IEEE Trans. Mag., 34, 4, 1998. With permission from IEEE, Copyright1998, IEEE.)



at the speed that is near the transition from contact to flying. These peaks appear to be the result ofdispersion, reflection, and interference of stress waves, compared to the LDV signal.

The sidebands, S1, S2, S3, and S4, were present in both LDV and AE spectrum. In AE spectrum,S1 is barely detectable, the magnitude of S2 is much larger, S3 and S4 can be detected but theiramplitudes are very small. It is noted that the amplitude ratio between S2 and the bending mode isnearly constant for all disk velocities. This was also found for the 2.5 and 1.5 nm thick lubricant films.However, the amplitude ratio between S2 and the bending mode is a function of lubricant thickness.

It is observed that the amplitude of S2 at 1.56 MHz is a function of the lubricant thickness. Thus,the amplitude ratio between S2 and the bending mode could possibly be used, after propercalibration, to determine lubricant thickness using AE measurements.

It is also observed that the sideband frequencies S2 and S3 are separated from the bendingfrequency by approximately 120 kHz, while the sidebands S1 and S4 are separated by more than200 kHz. These frequency differences are of the same order as the roll and pitch frequency,respectively, and it appears justifiable to suggest that the sidebands are related to the roll andpitch frequency of the air bearing.

Near the transition from contact to flying, the air bearing is almost fully developed and contactsbetween the slider and the disk are related to the roll and pitch motion of the slider. Furthermore, asthe disk velocity increases, the pitch of the slider increases and the contact area between the sliderand the disk is shifted toward the trailing edge. This causes the excitation of bending mode ofvibrations of the slider which is modulated mainly by the pitch frequency, i.e., the pitch motionof the air bearing excites the bending mode frequency of the slider at 1.68 MHz. When the sliderbegins to fly, the amplitude of the pitch and roll motion becomes much smaller and torsional andbending mode vibrations disappear.

Since the sideband occurs only near the avalanche point on disk with thick lubricant films, it isassumed that the nonlinear effects are due to the thickness change in the lubricant films from 1.5 to3.5 nm. This assumption seems justifiable since the lubricant film is changed from a more ‘‘solid-like’’ behavior to a more ‘‘fluid-like’’ behavior as the lubricant thickness increases. When the sliderhits the disk ‘‘hard,’’ the lubricant becomes thin and solid-like. However, for ‘‘soft’’ impacts, thelubricant film remains thick and more fluid-like. Thus, the stiffness of the lubricant film is a functionof contact force.

With the above observations, the dynamics of the slider are affected by the stiffness that isaffected by the air-bearing forces related to pitch and roll; in [141] a simple model is proposed tointerpret the nonlinear modulation of the slider bending mode by air-bearing mode. This model isbased on the assumption that contact stiffness is a function of contact force:

€x(t)þ 2jv _x(t)þ v2x(t) 1þ bf (t)½ � ¼ f (t)=m (5:48)

The stiffness of the system is nonlinear due to the term bf(t). v is the natural frequency of bendingmode. The input force function f(t) is the summarization of the pitch and roll contribution of air-bearing slider and is concerned with lubricant effect, i.e.,

f (t) ¼ A sin(2pfrollt)þ B sin(2pfpitcht)

The velocity spectrum of the numerical solution of Equation 5.48 is shown in Figure 5.30. Thestiffness parameter is assumed as b¼ 0.1. It shows that the amplitude of the sidebands is stronglyrelated to the stiffness change due to air bearing and the effect of lubricant.

5.4.1.1 Force Identification

Contacts between sliders and disks are unavoidable during normal operation in almost all currentdisk drives, and they can result in undesirable head crash, wear failure, or disturbance of theread-back signal. Apart from being extensively used to detect contact events, AE method has also



been used to estimate contact force. The suitable calibration method for the AE contact forcemeasurement has been explored. Conventionally, two methods for obtaining a calibration of the AEchannel include the direct calibration method and the system identification by using FEM modelingas well as LDV measurements.

Quantitatively measuring contact forces is much more difficult than detecting whether thecontact occurs. The position of the contact force action is usually variable and unknown. The contactforce is an impact type force whose root-mean-square is much smaller than the air-bearing force, butthe peak amplitude is relatively large. The contact force estimation has been based on slider resonantringing resulting from solid contact between the slider and the disk. Measurement and calibrationare very difficult for such small parts with a wide-frequency range. The LDV is usually wellcalibrated, accurate in a wide-frequency range, and thus suitable for noncontact measurement.

The force identification using AE signal usually consists of two steps: (i) modeling the system andobtaining the frequency response functions of the system, and (ii) measuring the responses andcalculating the forces. Matsumoto et al. [43] and Briggs et al. [128] used this method to identify thecontact forces. They adopted a different approaches in the first step, and a similar approach in thesecond. Matsumoto et al. [43] used a direct calibration method (ball drop method and breaking pencillead) to obtain the FRFs of a 70% slider. But it is difficult to use this method in nano (50%) and pico(30%) sliders. Matsumoto et al. [43] measured the responses of four sensors (one on each corner of theslider), as shown in Figure 5.31, by applying a known input force (breaking pencil load or dropping aball) at the ABS; a pulse or step excitation of the system with a known input force is generated. Thesignal from anAE sensor on the suspension is then the convolution of the slider=suspension systemwiththe input force function. From the known input (force) and themeasured output (AE signal), the transferfunction of the system can be determined. Thereafter, any unknown impact force can be obtained fromthis transfer function and the measuredAE output signal. For most applications, the input force functionis a delta function, i.e., all the frequencies of the system are excited with the same intensity.

The step calibration of an AE sensor using the pencil brake method is currently a NIST standard.It was found that the contact force estimate via a transfer function is a function of sensor location.By placing the AE or PZT sensor as close as possible to the impact point, the accuracy of thetransfer function can be significantly improved.

The second method is the system identification method combining finite element analysis resultsand experimental data to model the system [128]. The accuracy of the calculated modal frequencies

Frequency (MHz)

0 0.4 0.8 1.2 1.6 2

Pitch

RollBending

S1

S2 S3

S4

1.0E−6

1.0E−10

1.0E−14

Ave

rage

d P

SD

FIGURE 5.30 Simulation result. (From McMillan, T.C., Talke, F.E., and Harrison, J.C., IEEE Trans. Mag.,34, 4, 1998. With permission from IEEE, Copyright 1998, IEEE.)



and the lack of the damping parameters limit the application of this method. A small error in the modalfrequencies can result in a large error in the identified forces. Briggs et al. [128] also suggested that themagnitude of vibration of the modes could be used like a ‘‘fingerprint’’ to estimate the force locationalong the slider rails. The magnitude of the impact can be determined from the overall vibration level.

The interaction force waveform can be calculated from the measured responses with theobtained transfer function. For instance, the following are two typical types of identified impactforces between the slider and the disk: one has an amplitude of about 20 mN and an interaction timeof about 5 ms; the other has an amplitude of about 4 mN and an interaction time of about 22 ms.

MEMS technology was also once applied in the design and realization of an impact sensorarray. The array was integrated in a silicon slider that can be used to measure slider–diskinteractions. The impact sensor array consists of thin film piezoelectric sensor elements that areplaced close to the ABS. The local deformations due to the impact force are measured, which resultsin a high sensitivity and wide measuring bandwidth.

To illustrate the calibration details, we describe the recent improvement of the two methods forAE calibration by Knigge and Talke [104] in the following sections.

5.4.1.2 Method of Direct Calibration

As discussed above, contact is indicated by the occurrence of torsional and bending mode vibrationsof the slider body. These vibrations occur at approximately 700 and 900 kHz, respectively, fornanosliders, and at 1.3 and 1.6 MHz, respectively, for picosliders.

For the direct calibration of the PZT and AE sensor, the ball drop method was used togetherwith Hertz’ theory and finite element simulation. Small steel and glass balls were dropped onto theslider rail at the trailing edge. To ensure that subsequent ball drops occurred at the same locationon the slider, the spheres were dropped through a chute onto the slider rail. Two different balldrop situations were tested. In the first case, the nanoglide slider was separated from thesuspension=gimbal and glued onto a metal block. For the second test, the two-rail glide sliderwas attached on the suspension=gimbal. The impact force and the impact time were varied bychoosing different balls and different drop heights. Good impact repeatability was obtained in boththe time and frequency domains. Only one impact is observed for the first case but multipleconsecutive contacts and vibrations were observed in the second test. The wavelet–time–frequencyanalysis of the second test indicated that the frequencies at around 530 kHz correspond to aslider torsion and PZT tilting motion, and frequencies at around 930 kHz correspond to a secondtorsion=PZT glue elongation. Using Hertz’ theory and finite element analysis, the contact force andthe calibration constant for the PZT glide sensor is calculated to be 90 N m=V with the preamp set at40 dB gain. For the AE sensor, the calibration is less accurate. This is because the stress waves thatare excited at the slider rail are strongly attenuated when traveling through the slider–gimbal

Piezoelectric sensor

Disk

Elastic wave(acoustic emission)

Projection

Slider

FIGURE 5.31 Schematic of PZT sensors on slider.



interface and through the suspension before they reach the AE sensor. Due to the attenuation of thestress waves, small contacts cannot be detected with the AE sensor. The calibration constant wasfound to strongly depend on how the AE sensor was mounted. With the preamp set to 40 dB, atypical calibration constant was around 100 mN=V for the AE sensor.

5.4.1.3 Method of System Identification

For the calibration based on system identification, a high bandwidth LDV was used. The transferfunction was established from the harmonic response derived at different vibration modes andfrequencies. A finite element method based on transient response simulation of impact was used toestimate the velocity and stress response of the slider. The contact forces were found to be in therange of 5–25 mN for nanosliders and 2–10 mN for picosliders.

To determine the contact force, the slider is considered to be a multi-degree-of-freedom linearsystem that is excited by a harmonic impact force leading to bending and torsional mode vibrations.The velocity and displacement of these vibrations can be measured directly with a high bandwidthLDV. By using system identification and finite element modeling, the measured natural frequencies,mode shapes, and modal masses are obtained. These values are used in a second-order harmonicoscillator differential equation and the transfer function is estimated by using modal summation. Thevelocity response v(t) at an arbitrary point in a structure is related to the impact force F(t) by the timeconvolution. In the frequency domain, the convolution is given by multiplication, i.e.,

V(v) ¼ F(v)H(v) or F(v) ¼ V(v)

H(v)(5:49)

Assuming a damped harmonic oscillator, we have the multi-degree-of-freedom system model forthe system. The transfer function or frequency response function (FRF) is then

a(v) ¼ U(v)

F(v)¼ 1

(K � v2M)þ ivC(5:50)

wherea(v) is called the receptance of the systemv is the frequency

The receptance a(v) is obtained if the measured response parameter is the displacement u(t).In general, both the theoretical and experimental approaches can be used to modal model. In the

modal domain, the receptance matrix can also be written as

a(v) ¼ �½ � v2r � v2

� �1F½ �T (5:51)

whereF is the modal shapevr is the resonance frequency

Making use of the orthogonality properties of the eigenvectors F, and by applying massnormalization, we can write Equation 5.51 as

ajk(v) ¼XNr¼1

rAjk

v2r � v2 þ ihrv

2r

(5:52)

whererAjk is the modal constant of resonance rhr is the damping or loss factor of the resonance peak

The loss factor can be estimated via the logarithmic decrement method or by the width of theresonance region



�r ¼ v2a � v2

b

v2r

¼ Dv

v(5:53)

The modal constant of each mode is

Ar ¼ aj jv2r �r (5:54)

After the receptance matrix is calculated from Equation 5.51, the impact force F(v) can bedetermined from Equation 5.50 using

F(v) ¼ a(v)½ ��1U(v) (5:55)

where U(v) is the displacement response. Since the receptance matrix a(v) is often rank deficient, itis better to determine F(v) using the pseudo-inverse approach [104]

F(v) ¼ a(v)½ �T a(v)½ ��1 �

a(v)½ �TU(v) (5:56)

If one estimates contact forces using the above equations, errors on the order from 50% to 200% areobserved. To avoid such large errors, a more practical method for estimating the contact force is atrial and error approach. In this case, the force function F(v) is first estimated and then its value isadjusted iteratively until the solution of U(v) approaches the measured response.

After the input force function is estimated, the LDV signal is used to calibrate the AE sensorsince the AE sensor shows a significantly higher contact sensitivity than the LDV. However, the AEsignal is subject to dispersion and reflection and has therefore additional frequency componentscompared to the LDV signal. Therefore, calibration of the AE sensor is useful if one compares aband-pass filtered signal from both LDV and AE.

For a nano subambient pressure slider sliding on the disk at 2 m=s, the slider receptance FRFand the modal fit are shown in Figure 5.32. The slider air-bearing system is fitted with six distinctmodes. The first two modes at 158 and 282 kHz correspond to air-bearing frequencies whereasthe slider body resonance flexural modes are at 694 kHz (first torsion), 851 kHz (first bending),1134 kHz (second bending), and 1434 kHz (second torsion). The slider mode shapes and the mode

Torsional

Bending

Second bending

Second torsional

Raw data

Fit

Frequency (MHz)0 0.4 0.8 1.2 1.6 2

−90

−110

−130

150

Air-bearing modes

Fre

quen

cy r

espo

nse

func

tion

(dB

)

FIGURE 5.32 Frequency response function of nanoslider and modal fit. (From Knigge, B. and Talke, F.E.,IEEE Trans. Mag., 37, 900, 2001. With permission from IEEE, Copyright 2001, IEEE.)



frequencies were verified using finite element analysis. Figure 5.33 shows the estimated forcefunction using the pseudo-inverse technique.

A comparison of the standard deviation between LDV and AE response is a linear relation. TheLDV was used to establish a transfer function and to calibrate the AE signal. The transfer function isbased on modal summation of linear harmonic oscillators. From LDV measurements, the contactforce for a ‘‘dragging’’ contact was found to be between 5 and 25 mN.

5.5 DISK SURFACE SCREENING USING ACOUSTIC EMISSION TECHNIQUEFOR MASS PRODUCTION

5.5.1 GLIDE TESTING FOR DISK SURFACE FINISH CERTIFICATION

In the mass production of magnetic disk, AE method has been widely used to screen disk todetermine glide height and decide the take-off height (TOH) value [113–121,135–144]. Glide heightis the height from centerline of disk surface above which flying slider is unable to meet any asperityon disk. It is usually decided by using a flying glide slider to scan the entire disk surface withgradual decrease in FH. The glide slider has PZT sensor which outputs AE burst signal wheneverthe flying makes contact with asperity. TOH is the height from centerline of disk surface belowwhich flying slider will make continuous contact with the disk. The mechanical glide test is veryimportant among the means for disk surface certification since a mechanical defect could developand rapidly degrade the interface to the point of failure.

To this aim, the PZT crystal piece is usually integrated on the sliders, either switched in betweenthe slider and the suspension or glued on a corner of the slider. The ‘‘glide head’’ is usuallycommercially available glider test slider with PZT element. The disk products used for disk driveare usually under glide test certification, which employs a glide head to scan the entire disk surfacewith specific glide height. The output of the AE signal can be mapped on a disk surface withdifferent tracks and sectors. It is usually divided as 231 tracks and 1024 sectors. Using the recordedAE rms output in each partitioned sector, a 3D plot of the AE rms representing slide–disk contactevents can be obtained, as shown in Figure 5.34. With this information, the defected disk can beidentified for further treatment or removed from production line.

16

12

8

For

ce (

mN

)

4

00 20 40 60

Time (s)80 100

FIGURE 5.33 Estimated force function using pseudo-inverse technique. The contact force ranges from 8 to20 mN. (From Knigge, B. and Talke, F.E., IEEE Trans. Mag., 37, 900, 2001. With permission from IEEE,Copyright 2001, IEEE.)



The goal of the glide test is to certify disk by measuring slider–disk impact for a given FH. Thisis done by measuring various AE from flying slider during the scanning of the spinning disk. Thechallenge is to isolate the impact type of AE from other type of AE signals, such as the air-bearingvibration-induced AE output. This is usually done by conditioning electronics like band-pass filterto allow only the frequencies of the slider bending mode(s) to pass.

Glide heads need to be ‘‘dialed-in’’ to a particular FH via changing the relative disk velocity. Fornegative pressure slider designs, changingvelocitiesmay result in changingFH in the opposite directionthan expected. Lower velocity may result in a higher FH (decreased subambient pressure) and highervelocity often means lower FH (increased subambient pressure). Changing velocity also meanschanging the energy level of a particular impact. Slower velocity may mean a larger impact area forglide heads but at reduced impact speed. Tomaintain the same impact velocity and to reliably avalanchethe negative pressure heads, a popular method is elevation avalanche. The air pressure is changed tosimulate various elevations. The higher the elevation, the lower the air pressure and nominal FH.

The main requirement for increasing data density is the lowering of the head–disk spacing. Formagnetic disk makers, the term flying height often translates to ‘‘glide.’’ This is the pass and fail testto determine whether the slider impacted the disk at a particular FH, otherwise known as ‘‘glideheight.’’ The glide height specification is usually chosen by subtracting the nominal FH from thethree-sigma of the FH variations. Disk drive makers often will further lower the glide height for thepurpose of guard banding against process variations. The glide test itself involves flying a genericslider (of positive pressure design) equipped with a PZT sensor mounted in close proximity to theslider to maximize the detection sensitivity. In most cases, the PZT sensor is mounted on the slideritself. Since one hit fails a disk, a ‘‘burnish’’ step precedes the glide test to maximize the yield. Thisburnish step is typically done by dragging a special nonflying head (waffle), equipped with sharpedges to ‘‘cut down’’ any potential protrusions.

Proper filtering is the most difficult portion to implement in all electronics for AE detection.Typically, a sharp (8-pole) programmable band-pass filter is employed. The programmable range isbetween 50 kHz and 2.5 MHz. With an 8-bit range of programmability for each side, a total of 4096combinations are available. Due to the nature of the test, the data acquisition (digitizer) packagemust have high resolution (12 bit minimum) and high speed to cover at least Nyquist (this means4 Mega-samples=sec minimum), and have deep memory to acquire a full revolution of data atlow speed.

The setup is equipped with a high performance spinstand that has a top speed of 20,000 rpm anda low runout. The controller can be programmed. The spinstand is also equipped with a pressurechamber to simulate 5,000 to 45,000 ft.

The analysis program could cover band-pass, band-stop, high-pass, low-pass, rms, and fre-quency analysis tools such as FFT, time–frequency analysis, and wavelet.

(a) (b)

FIGURE 5.34 3D plot of the PZT output of glide head after it scanned the disk surface. (a) Disk withoutdefects; (b) disk with defects.



5.5.2 NANOMETER CLEARANCE CERTIFICATION AND CALIBRATION

Apart from the glide height characterization, AE approach is also used to characterize the disk TOH.TOH has been used as one of the most important technical specifications of the disk for the entiredisk drive design.

Conventionally, TOH is decided by using AE rms signal at contact as indicator. From the kneeor bend point on the AE signal vs. rpm curve, the take-off velocity can be decided. Then from flyingheight–flying velocity calibration curve, the FH can be determined.

Another option is to make use of asperity counting of AE sensor output as an indicator to decideTOH. The TOH measurement based on counting the number of asperities, as a function of FH, isshown to allow easier correlation with a peak mean roughness statistic, and more importantly, toavoid errors. Figure 5.35 shows the output of PZT sensor as a function of disk velocity by using rmsand hit number. For conventional disks, take-off velocity measurements using rms as indicatorcould be affected by isolated defects. It tends to give out higher value, and accordingly gives outhigher TOH evaluation. Depending on individual defected disk, TOH using rms could be overesti-mated up to 2 nm. This phenomenon actually occurs in many of the TOH tests despite the use of theglide slider. This issue can be solved by offsetting rms threshold in advance or preferably, usinghigh-frequency (HF) asperity counting method.

The asperity counting approach can provide more detailed information about slider–diskinteraction. By partitioning disk surface into sectors, the output of AE at each sector can be mappedas the 3D hit profile, 2D contour, and hit distribution. Figure 5.36 is a typical hit profile (top), hitcontour (middle), and hit distribution (bottom) of a disk without salient defects. Figure 5.37 is the hitprofile (top), hit contour (middle), and hit distribution (bottom) of a disk with salient defects.

Another critical problem is the accurate calibration of fly height of the test sliders used for glideand TOH test.

How to accurately calibrate the slider FH has been one of the critical problems to enabletechnology to attain sub-10 nm FH. Conventionally, optical FH tester is used to measure the sliderFH, but this method becomes unreliable as FH approaches sub-10 nm. Another method is standardbump disk method, which reduces disk speed to allow a flying slider to touch down on a bump withknown height. A curve of different bump heights vs. the slider–bump touch-down speed can beobtained for the slider. The typical specified bump with known height is illustrated in Figure 5.13.

This method has been considered to be efficient and reliable. However, it is difficult to estimatethe accuracy of the method due to the complicated slider–bump response process. It is not clear whatthe minimum slider–bump interference is for the contact event being detectable. This interferenceconstitutes one of the major measurement errors of this method.

Basically, when a slider flies over a bump, the slider reacts slightly as the bump travels under thefront taper and then experiences larger oscillation as the bump encounters the more sensitive lower

0

0.2

0.4

0.6

0.8

1

300 340 380 420Velocity (IPS)

Nor

mal

ized

hit

num

ber/

norm

aliz

ed r

ms

Normalized rms

Normalized hit number(1 = 100% hits)

FIGURE 5.35 Output of sensor as a function of disk velocity by using rms and hit number.



film thickness portion of the air bearing. The slider trailing edge or pad may or may not contact thebump subject to the air-bearing oscillation and the bump height. There exists a minimum bumpheight above which the slider will contact the bump when the slider flies over it at a given FH. Theslider size, air-bearing stiffness, minimum FH, and bump size determine the minimum bump height.The contact onset condition is critical and it needs to estimate how many interferences will beenough to cause the contact. Some simulation results by Hu and Bogy [21–23] show that the airbearings of a nanoslider and a picoslider with 15 nm FH could be disturbed by even passing a bumpwith 10 nm height and 25 mm width, the FH drop could be 3 and 1 nm, respectively. Theirsimulation also shows that a slider with 12 nm FH will not contact the bump as long as the bump

050 100

150

010

20300

0.5

1

Sectors

Hit profile (V )

Tracks

20 40 60 80 100 120

10

20

s ec to rs

Hit contour

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

Hit distributionLog(

hit n

umbe

rs)

Voltages (V )

Tra

cks

FIGURE 5.36 Hit profile (top), hit contour (middle), and hit distribution (bottom) of a normal disk.

0 50 100 150

01020

300

0.5

1

Sectors

Hit profile (V )

Tracks

20 40 60 80 100 120

10

20

se ct ors

Tra

cks

Hit contour

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

Log(

hit n

umbe

rs)

Hit distribution Voltages (V )

FIGURE 5.37 Hit profile (top), hit contour (middle), and hit distribution (bottom) of a defect disk.



height is below 14 nm, whereas another slider with 14 nm FH will not contact the bump unless thebump height reaches up to 21 nm. This complicates the application of slider FH calibration by usingresponse to bump with known height, which requires the occurrence of slider–bump contact withlowest slider–bump interference. Moreover, it is desired that the air-bearing oscillation is nottriggered before the flying slider contacts the bump; otherwise, the detected transient FH is thedisturbed FH instead of the normal FH.

This problem could be bypassed by using small bump size, which is unlikely to affect the airbearing. The bump sizes are limited by the fabrication availability and application durability. Thereis a critical size of bump below which slider air bearing will not be disturbed if actual slider–bumpcontact does not take place when slider flies over it.

To illustrate the complicated dynamic behavior of slider–bump interaction through experiments,two different sliders, type A and type B with different ABS designs, were employed to conductbump avalanche tests on a PM MSA900 test stand. The signal of PZT sensor, which is sandwichedbetween the slider and the suspension, was applied to identify the slider air-bearing resonance andslider body resonance. By comparing the onset of signals from these two sources, impropermeasurement results from the air-bearing oscillation of slider can be eliminated. The LDV measuredslider–bump response and analysis are also used to correlate the results.

The resonant modes of the slider air bearing are in the range of 30–200 kHz. The PZT elementswere fabricated to have the same length and width as that of the slider and were sandwiched betweenslider and suspension tabs. The major resonant modes are in the range of 1–2 MHz. To capture andidentify the air-bearing resonance modes and slider body resonance modes, two measurementchannels were used. One is the narrowband (NB) channel with band-pass filter 1–2 MHz. Theother is the wideband (WB) channel that consists of two band-pass filters including 25–200 kHz aswell as 1–2 MHz.

To determine slider–bump touch-down velocity through test, the central rail of slider needspositioning over the designated bump track by using actuator and load–unload mechanism on thetester. Starting from a high velocity, where slider flies over disk bump track without contacting it,the velocity is then decreased in discrete steps. At each velocity step the PZT peak signal afteramplifier and filter channels is measured and averaged. The speed corresponding to the rise of PZTsignal just above noise level or the knee point is the touch-down speed. The scanning procedure isused to eliminate other possible errors. The multiple track scan is conducted by a step-by-stepmoving of the slider trail completely across a bump at a proper velocity that allows some slider–bump interference.

Figure 5.38 shows a typical measured output of PZT signal due to slider–bump interferencewhen the flying slider A is scanned across the bump track. Based on the measured bump profiles,proper slider and track are selected to avoid errors caused by slider=bump surface asperity and roll

1.100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.12 1.14 1.16 1.18

Radius (in.)

RM

S g

lide

sign

al (

V )

1.20 1.22 1.24

FIGURE 5.38 Output of PZT signal vs. disk radius when flying glide slider scans across the bump track.



effects. The bump touch-down test is then conducted by positioning the slider on the proper bumptrack. Wideband signal of PZT has been suggested to detect slider–asperity interaction. The wide-band signal could include the air-bearing resonance, sensor resonance, as well as slider bodyresonance. However, the wideband signal detection has an advantage of higher sensitivity. Tocalibrate FH by using slider–bump response, we need to accurately detect the onset of slider–bumpcontact, and need to make sure that slider air bearing and FH are not disturbed before the onset ofslider–bump contact. Both WB and NB outputs of PZT are measured to monitor the slider–bumpcontact and air-bearing disturbance. Figure 5.39 shows the typical measured results for the twosliders. For slider A, Figure 5.39a shows that the WB output occurs earlier than NB when the speedis reduced. It suggests that at very high speed the slider is unable to make contact with the bump butthe air bearing is disturbed when passing the bump. By further reducing the speed, the WB outputbecomes larger, and at the same time, the NB output becomes detectable, which suggests that theslider starts to make contact with the bump. For this case, it is difficult to determine whetherthe slider–bump contact and the air-bearing disturbance take place at the same time. If the onsetof the slider–bump contact is later than the air-bearing disturbance, then the height at which theslider contacts the bump is the modulated height instead of the normal FH. Figure 5.39b shows themeasured results for slider B, which shows that the WB and NB outputs of PZT become detectableat the same time when the speed is reduced. It suggests that the air-bearing and the slider bodyresonance are likely to be triggered at the same time, and that the bump alone does not disturb the airbearing without contact. Only in this case, the knee point is the true touch-down velocity. As the bumpused is the same, the difference of the two sliders’ reaction to the bump comes from their air-bearingsurface and air-bearing stiffness difference. Slider B has a wider trailing rail which supplies larger airbearing. The above approach is used to enhance calibration accuracy for sub-10 nm FHmeasurementof slider by using consistent responses induced by bump. The approach is implemented by comparingthe coincidence of air-bearing resonance and slider resonance, which is able to eliminate themeasurement error caused by early response of air bearing to bump. It was found that with 10 mmwidth bump, the air-bearing resonance of a tested slider was disturbed even though the slider did notactually contact the bump. However, the air bearing of another slider with different ABS design doesnot react with the bump until the slider makes actual contact with the bump. Obviously, for the firstcase, the detected slider body resonance is not reliable to be used directly to calibrate the slider FH,as the detected contact height is likely the disturbed FH of the slider instead of its normal FH. Forthe second slider, the implication of the measured results is that the FH is unlikely disturbed beforeslider–bump contact, and therefore it matches with the bump height when the slider starts to contactthe bump. The LDV measurement and time–spectrum analysis for a commercial slider also alert theinconsistent onsets of air-bearing resonance and slider body resonance.

NB

WB

(a)

Amplified PZT Output of Slider A for10 nm Bump Spindown

0

0.2

0.4

0.6

0.8

Velocity (m/s)

0.00 5.00 10.00 15.00

Am

plitu

de (

V ) NB

WB

Amplified PZT Output of Slider B for 10 nm Bump Spindown

0

0.5

1

1.5

2

2.5

Velocity (m/s)

Am

plitu

de (

V ) NB

WB

(b)

5.00 7.00 9.00 11.00 13.00 15.00

FIGURE 5.39 Narrowband and wideband output of PZT signal vs. different velocities for slider flying overa bump.



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138. Suzuki, S., Prediction of glide avalanche based on composite roughness, Inf. Stor. Proc. Syst., ISPS-4,23, 1998.

139. Yamamoto, T., Yokohata, T., and Kasamatsu, Y., Stiction free slider for lightly textured disks, IEEETrans. Mag., 34, 4, 1753, 1998.

140. Khurshudov, A.G. and Talke, F.E., A study of subambient pressure tri-pad sliders using acousticemission, ASME J. Tribol., 120, 54, 1998.

141. Knigge, B., Talke, F.E., Baumgart, P., and Harrison, J., Slider disk contact dynamics of glide heads andproximity recording sliders using joint time–frequency analysis and scalogram techniques, ASME Trib-Vol. 9, Proc. Symp. Interface Technology Towards 100 Gbit=in.2, 23, 1999.

142. McMillan, T.C., Talke, F.E., and Harrison, J.C., Identification of slider=disk contacts using the energy ofthe acoustic emission signal, IEEE Trans. Mag., 34, 4, 1998.

143. Zhang, J. and Sheng, G., Calibration of sub-10nm flying height using the bump response, IEEE Trans.Mag., 39, 5, 331, 2003.

144. Truong, T., Glide, the forgotten tribology focus: Time and frequency domain analysis of negativepressure slider elevation avalanche glide signals, Tribol. Inf. Stor. Devices TISD’99, Santa ClaraUniversity, USA, 1999.



6 Friction-Induced Noisein Power TransmissionBelt Systems

6.1 INTRODUCTION

Because of the vast breadth and depth of the relevant dynamics and tribology, understanding powertransmission belt dynamics and stability fully has been a classic yet critical topic for the mechanicaland automotive engineering community.

Early studies on belts were mostly confined to obtaining the coefficient of friction and naturalfrequencies and modes by simple analysis. For the dynamics aspect, analytical models for axiallymoving systems have been extensively studied in the last 50 years. Sack [1], Miranker [2], Mote andWu [3–5], and Ulsoy et al. [6–8] were among the pioneers in this research area. The followingresearchers have extended the research to all the other aspects [8–62]. Belt drive dynamics belong tothe class of hybrid continuous=discrete dynamics systems, and all the dynamic issues discussedabove fall in the low-frequency range. For instance, automotive multirib V-belt drive system is oneof the typical kinds of belt drive systems, which has been widely used in automobiles. Such kindof system exhibits complex dynamic behavior, including rotational vibrations of pulleys with thebelt spans serving as coupling springs, transverse and torsional as well as lateral vibrations invarious belt spans. Over the last decade, a tremendous progress has been made to understand andimprove belt drive dynamic properties of this type. The related issues have been successfullyaddressed by the industry and academics with specific softwares [59–61] and standard commercialsoftwares such as SimDrive [62]. For these kinds of efforts, the general solution of the governingnonlinear equations provides the coupled longitudinal and transverse response of the translating beltdrive system. Typical simulation outputs include pulley hub loads, belt impact dynamic forces, andbelt slip rates at the pulleys.

On the other hand, belt tribology as the critical area concerning power transmission capacity hasbeen studied by numerous researchers. The pioneers include Swift [63], Gough [64], Grosch [65],Childs et al. [66–71], Gerbert and Gerbert [72–77], and others [78–96].

However, the friction and dynamics interaction, instability, and noise problems have rarely beenaddressed until the last decade [97–110]. The studies of V-ribbed belt squeal have not receivedenough attention until recently [97–108]. In real application, the belt drive system failures have beenestimated and judged by the loss of power transmission capability (overslip), wear, and noise.Connell and Rorrer [97] investigated the friction-induced vibrations in V-ribbed belt applications.Connell and Meckstroth [98] studied the influence of pulley profile on automotive accessory drivenoise. Meckstroth [99] presented a dynamic test procedure under varied conditions to quantify thevariations in load carrying capability of different belts and pulleys. Meckstroth et al. [100]investigated the accessory drive belt pulley entry friction and belt misalignment noise. SAEStandard J2432 [101] gives out the standard method for belt effective COF testing and the methodfor misalignment noise testing. Moon and Wickert [17] investigated the noise of misaligned V-beltdrives. Dalgarno et al. [102] investigated the tangential slippage and squeal of V-ribbed belts with


301

experiments and finite element analyses, and attributed the noise from the excitation of thefundamental vibration of belts. Sheng et al. [103] proposed a model and experimental investigationof belt slip and misalignment noise in automotive accessory belt drive systems. Okura and Tatsumi[104] analyzed the squeal radiated from serpentine belt for accessories drive system. In thesevibration and instability analyses, the squeal other than the load carrying capability loss has beenemphasized. Sheng et al. [105–108] characterized the fundamental properties of belt friction-induced noise and instability by extensive experiments, including dry and wet conditions, tangentialand radial type slip, and they proposed several models addressing different instabilities and noise.

A number of previous publications have documented working progress on noise analysis fromacademics to industries. It has been considered that belt noises are due to belt self-excited vibrationsfrom belt friction stick–slip or velocity-dependent friction (negative slope of friction vs. slip), modallock-in or modal coupling, or a combination of some of them.

Conventionally, due to lack of fundamental understanding, the belt noise prevention has been asubjectively driven trial and error process. The only way to eliminate these kinds of ‘‘patch and fix’’solutions is to acquire a fundamental understanding of the intricate interdependency between all thecomponents of the drive system. For this we have to take into account the effects and theinteractions of materials, belt construction, drive system dynamics, components and integration,boundary conditions at the friction interface, manufacturing and assembly tolerances, wear, non-linearity, elasticity, temperature and humidity variations, operation conditions, etc. The basic featureis that a small change in one of the items listed above does not necessarily cause minor changes inthe interface friction and noise. In other words, the friction and noise are controlled by many factors.

Further development of the work leads to a strong requirement on the analysis and prediction ofthe instability and noise using a computer analysis–based engineering process. Following a compre-hensive review, this chapter describes the noise generation mechanisms of ABDS. The proposedconcepts and approaches can be applied to the noise treatment of multiple V-ribbed belts and V-beltsas well as the slip noise treatment of flat belts in all kinds of belt power transmission systems.

Many power transmission belts are friction driven. Friction driven belts rely on coefficient offriction to transmit the power and require tension to maintain the coefficient of friction. Automotiveaccessory drives have limited space available in engine compartments. By the late 1980s, most ofautomotive drives were driven by a single V-ribbed belt (K section) and were tensioned bytensioner. Figure 6.1a shows an automotive engine and its accessory belt drive system at the front

ALTIDR

P/S

W/P

A/CCRK

TEN

(a) (b)

�0.0 x +

0.0�Y

FIGURE 6.1 Schematic layout of a typical accessory belt drive system. (a) Automotive accessory belt drivesystem. (b) Schematic of accessory belt drive system.



end of the engine. Figure 6.1b shows a schematic of accessory belt drive system in which thecrankshaft pulley (CRK) rotates clockwise and transmits power to accessories through a single belt.The belt connects all the accessory pulleys including a tensioner (TEN) pulley, an idler (IDR)pulley, an air conditioning compressor (A=C) pulley, a power steering pump (P=S) pulley, analternator (ALT) pulley, and a water pump (W=P) pulley.

Figure 6.2 shows the schematic of a typical V-ribbed belt structure and the belt–pulley interface.The main belt constituents are polyester cords as the tension member, a cushion elastomercompound which envelops the cords, and a rib elastomer compound which forms the belt ribs,and which may be stiffened through the use of short fiber reinforcement in the axial direction. Thetension member provides the V-ribbed belt with a high tensile strength and carries up to 95% of thetorque loading during the operation of the V-ribbed belt drive system. The rubber layer providesthe V-ribbed belt with frictional and absorbing shock properties and transmits the torque betweenthe pulley and the tension member. The textile top cover is located on the top surface of theV-ribbed belt to protect the belt during the drive system operation. Compared with the V-belt, theflexibility of the V-ribbed belts arises from the small section depth, and the power transmissioncapacity from the V-shaped ribs which develop traction.

Another type of belt used in automobiles for engine combustion timing control is timing belt, akind of synchronous belt, which is a toothed belt. Its teeth fit in the matching grooves of toothedpulleys (sprockets) as shown in Figure 6.3a. The schematic of synchronous belt structure is shownin Figure 6.3b. Most synchronous belts have a wear-resistant tooth-facing material to protect thetooth surfaces. This material presents a low coefficient of friction to decrease the friction losseswhen the belt teeth enter and leave the sprocket teeth. In this chapter we focus on the discussion ofV-ribbed belt instability and noise, we also touch on the friction noise of timing belt.

Rib

(a) (b)

Cord

Interface

Pulley

Belt

FIGURE 6.2 Schematic of (a) V-ribbed belt and (b) belt–pulley interface.

Synchronous belt

Sprocket

Facing fabric

Cord

(a) (b)

Elastomer compound

FIGURE 6.3 Schematic of (a) synchronous belt transmission system and (b) synchronous belt structure.


Friction-Induced Noise in Power Transmission Belt Systems 303

In automotive V-ribbed belt applications, noise has substituted the conventional issues of lifespan or slippage to be the top failure mode for automotive belt products. The low belt tension ortension variation application could result in belt longitudinal slippage, and accordingly, leads to beltslip noise. The unavoidable mounting misalignments of pulleys lead to belt radial slip, which couldcause another type of noise—misalignment noise. Both types of friction-induced noises usuallyhave high frequency and are quite perceptible compared with the conventional drive system noise.This is likely to give the customer a very strong adverse impact on the quality perception.

From a practical viewpoint of noise, appropriate limitations to system design, manufacturingand assembly tolerance could be applied, e.g., a permissible belt misalignment maximum of 18, anda permissible belt slip maximum of 3% (peak value). However, in practice, some drives with slipmaximum peaking up to 30% can be observed when running under unfavorable conditions.

Figure 6.4 is the typical graph of the belt load carry capability vs. slippage measured on a drybelt system. The belt slip s is defined as s¼Dv=vr where vr is the speed of the driver pulley and Dvis the difference in speed between the driver and driven pulleys. Assume there is no slip between beltand driver pulley; the slip rate at the driven pulley is defined as the relative velocity betweenthe belt and the pulley, divided by the pulley velocity, expressed as a percentage.

In the test the torque ranges from 10 to 20 Nm, the tight side tension on the driven=driverpulleys ranges from 10 to 45 kgf=rib, and the belt speed is 1000 rpm.

In addition to tangential slip between the belt and the pulley, there exists radial friction slipbetween the wedge wall of V-ribbed belts and the inside groove walls of the pulley when the beltruns on misaligned pulleys. Misalignment noise is due to parallel or angular pulley misalignment-induced belt radial slip.

Slip and misalignment noises are due to some slippages in longitudinal and radial directions,respectively. Both are friction-induced noises. The slippage depends on the grip between the pulleywalls and the belt to transmit torque.

Since belt drives employ friction to transmit motion and torque, their load capability isdetermined by the level at which longitudinal slippage begins between the belt and its groovedpulley. In the most general case, relative motion at the belt=pulley interface comprises twocomponents: one along the pulley’s circumference (termed longitudinal or tangential slippage)and the other along its radius (radial slippage). The former limits the drive’s torque capacity, withslippage occurring ‘‘rest on the smaller diameter pulley,’’ if any, as the overload condition isreached. Compared with the ‘‘micro-slip,’’ such macroscopic tangential slippage contributes toaccelerated wear, fatigue, the production of friction-induced vibration, and squeal noise. Macro-scopic radial slippage also contributes to the belt’s vibration and noise. Other imperfections such asthe pulley’s and belt’s geometry variations, manufacturing variations, aging, and unequal load-sharing characteristics could also interact with the interface and contribute to the slippage and noise.

Power transmission capability

010 20 30 40 50

5

10

15

20

Load (kgf/rib)

Slip

rat

e (%

)

Tested

Designed

QQ

FIGURE 6.4 The measured load carrying capability of a belt.



This chapter deals with belt friction noise and focuses on V-ribbed belt friction noise. The beltfriction is presented in Section 6.2. Belt vibrations are presented in Section 6.3. Misalignment beltnoise analysis is given in Section 6.4. In Sections 6.5 and 6.6, the belt slip noise under dry andwet situations is presented, respectively. Finally, the friction noise in timing belt is discussed inSection 6.7.

6.2 BELT FRICTION

6.2.1 LAWS OF V-BELT FRICTION AND BELT EFFECTIVE FRICTION

In order to estimate belt system torque carrying capacity, the friction must be known, not only with anew belt and pulley, but with aged belts and pulleys and over a broad range of environmental andapplication conditions as well. Conventionally, the coefficient of friction has been used to charac-terize friction, and it is usually measured with stationary, semi-stationary, or dynamic test equip-ments. Such equipments are usually unable to provide a very repeatable, unique coefficient offriction for the entire belt length, over the full range of applied and environmental conditions, aswell as for varied automotive applications.

The coefficients of friction of belts have been calculated using classical Euler belt equation.While this equation may not capture all the subtleties of belt slip condition, it provides a gooddynamic average characterization of a belt–pulley limit torque situation. It has been used as SAEstandard and by the entire automotive industry. This equation applies only at a slip condition. Thisequation implies that the coefficient of friction is not a function of pulley diameter, speed, or numberof ribs. When a V-ribbed belt drive system operates at a high angular speed v, the centrifugal forceFr on an arc portion of the V-ribbed belt is known as rRv2, where r is the density of the belt and Ris the radius of the arc portion of the belt. The general relationship is given by

Th � rAV2 ¼ (Tl � rAV2)emb

sin (u=2) þ I _v=R (6:1)

whererA is the length specific mass of beltV is the circumferential belt speedTh is the tight side tensionTl is the low side tensionI is the inertia of pulleyb is the belt wrap angle (radians)m is the coefficient of friction of the belt to pulley, excluding the V wedge angle effectu is the V wedge angle

Since this is for one rib of a V-ribbed belt, multiplying the tension by the number of ribs in theleft and right terms will not change the equation; thus the above equation is independent of thenumber of ribs, and the total tension of the belt can be used in this equation. If the wedging effect ofthe V angle is included in the coefficient of friction, the equation becomes

me ¼1bln

Th � rAV2 � I _v=R

Tl � rAV2

� �(6:2)

in which me¼m=sin(u=2) represents the effective dynamic friction coefficient on the contactsurfaces between the V-ribbed belt and the pulley. It is a larger number than the dynamic frictioncoefficient as it includes the wedging effect. An average or effective coefficient of friction canbe obtained between belt and pulley, but this value is often subject to wide variations. Suchinconsistency is due to an absence of uniform sliding velocity and cleanliness, inconsistent surface



topography, and a complex distribution of interface pressure between belt and pulley, particularlythe system dynamics effects. In industrial engineering design, the coefficient of friction has beenassumed to be constant for a given belt, which is usually characterized by a standard method usingstandard conditions and pulley. The actually measured coefficient of friction is highly dependent onvelocity or slip [70,74,75], and also on other factors such as pressure. A particularly interestingfeature is the negative slope of coefficient of friction vs. slip, as this specific friction feature causesnegative damping in system dynamics; therefore it is likely to result in instability or self-excitedvibrations. Based on the existing research, the negative slope of friction exists for most of the casesof wet belt systems, whereas it only appears for some dry belt systems, particularly at high slip rateregion. The wet belt interface usually forms unsteady mixed lubrication and some boundarylubrication. For wet belt applications, the full fluid lubrication is unlikely to take place due to therelatively low velocity, low viscosity of water, and the open interface. The water lubrication takesplace in the mixed or boundary regimes, so that the frictional resistance is developed due to acombination of fluid film shear over the entire contact and boundary film and solid friction atlocalized asperity contacts. In mixed lubrication, the load is carried primarily by the fluid with solid-to-solid interaction also contributing significantly to the friction. In boundary lubrication, the load iscarried primarily by the solid asperities with the fluid shear still contributing significantly to thefriction. Furthermore, under typical application conditions, coefficient of friction is a function of thesurface velocity, curvature, and normal contact load on the mating surfaces. A unified principle thatcan represent the interface under all conditions has eluded most researchers and many alternativeformulations have to be carefully applied based on different application regimes.

6.2.2 BELT DYNAMIC FRICTION TESTING

V-rib-based serpentine belt drive systems have greatly improved reliability and maintenancerequirement compared to multiple V-belt drives and are widely applied in automobiles. Theautomotive industry conventionally used a simple coefficient of friction test stand to quantifycoefficient of friction. It uses a belt rubbing on stationary pulley as shown in Figure 6.5a. Thelow tension is controlled with a weight, and the high tension is measured with a force transducer.The torque is calculated based on the high and low tensions and the pulley pitch diameter.

By ignoring the effects of dynamic inertia, Equation 6.2 can be simplified as the Euler equationme¼ ln(Th=Tl)=b. An alternative method is to use stationary belt to run against a rotating pulley asshown in Figure 6.5b.

To simulate the friction of real accessory system shown in Figure 6.1, more sophisticated testershave been developed. Currently, the SAE dynamic friction tester are widely applied in the USautomotive industry. It allows the study of the belt–pulley interface under much more realisticconditions. This tester can test K section V-ribbed belts ranging from 3 to 8 ribs in width. Both thegrooved side and the flat back part of the belt can be studied. The principles involved here can be

V

Pulley

Belt

(a) (b)Weight

Load sensor

T2

T1

Pulley

Belt

Weight

Load sensor

T2

T1

qq

w

FIGURE 6.5 Schematic of simple test stands for belt COFmeasurement. (a) Stationary pulley. (b) Stationary belt.



applied to any other size V or flat belt. The photography of the test stand is shown in Figure 6.6 withthe belt grooved side setup. The driven test pulley is connected to a torque transducer and anelectromagnetic dynamometer. Tension is maintained on the slack side of the driven pulley by theuse of weights and cables. A force transducer is used to monitor the tension in the slack side at onepulley. Idler pulleys with encoders are used to measure the belt speed. The test stand can be quicklyreconfigured to measure the coefficient of friction on the back part of the belt. The coefficient offriction is calculated using the above classical Euler belt equation. While this equation may notcapture all the subtleties of the belt slip condition, it provides a good dynamic average character-ization of a belt–pulley limit torque situation. This equation applies only at a slip condition andimplies that the coefficient of friction is not a function of pulley diameter, speed, or number of ribs.However, the coefficient of friction is not a constant subject to many variations. For instance,according to Meckstroth [99], some new belts (less than 3000 miles in the field) tend to behavesomewhat inconsistently, probably due to minor instabilities in the chemical and cure state of therubber. In many applications, coefficient of friction is a function of many parameters like speed,number of ribs, etc.

6.2.3 BELT FRICTION FEATURES

In wide applications, belt coefficient of friction generally increases with the increase of slip velocityat some range, and then decreases with the further increase of slip velocity, as shown in Figure 6.7.There are many exceptions where coefficient of friction increases monotonously with the increaseof slip velocity, and others where coefficient of friction decreases monotonously with the increase ofslip velocity.

The sensitivity of typical engineering rubber coefficient of friction to load and temperature isshown in Figure 6.8 [87]. The coefficient of friction generally decreases with the increase of normalload. There are some exceptions, as when the mechanism of friction changes from typical adhesivefriction sliding to plowing friction. The coefficient of friction generally decreases with the increaseof temperature at low temperature range, but in very high temperature range it could increase withthe increase of temperature due to softening=adhesion increase (hardness reduction). In general,the maximum coefficient of friction correlates with glass transition temperature, Tg. For example,in the rubber applications of tires, the coefficient of friction of some rubbers goes through amaximum value at about �108C or �208C on ice. Basically, coefficient of friction is affected bymany conflicting factors (damping [tan d], modulus, hardness, surface parameters, etc.) when

FIGURE 6.6 Photography of SAE dynamic belt COF test stand.



temperature changes. A general interpretation of temperature effect can be given by considering therelationship between coefficient of friction and damping and modulus, and the relationship betweendamping=modulus and temperature (note that Tg corresponds to the maximum of damping).Roughly, there is a relationship between friction and the values of damping and modulus for rubber,F¼K(G0)1=3 tan d, where F is the friction force, G0 is the dynamic shear modulus, tan d isthe dynamic damping or dissipation factor, and K is a constant. The typical COF range of rubberis 0.3–2.5, but it could be as high as 4 near glass transition temperature. Shear modulus usuallyincreases with the temperature decrease. On the other hand, with the decrease of temperature,damping decreases gradually, then increases rapidly and attains the maximum at glass transitiontemperature, and decreases again. The combined effect of damping and modulus causes the COFcharacteristics of rubber. However, the particular results for belts are complicated and subject todetailed compounding, particularly the surface features.

For aged belts, the belt surface could form a boundary layer and transformer film, and=or getenough belt fiber exposed. For this situation, the boundary layer could dominate the coefficient offriction instead of the rubber substrate. Then the formed surface film could be less sensitive totemperature compared with the original surface.

SAE tester offers several specific procedures to characterize the coefficient of friction of belt. Itenables to quantify the COF sensitivity to other parameters including wrap change, torque variation,and various environmental conditions. They are categorically, the variable wrap test, the variabletorque test, and the variable environment test. Variable wrap test determines the point of tractionlimit for a belt and pulley by unwrapping the test pulley under a constant torque load until gross slip

Slip rate (%)

CO

F

00

0.5

1

1.5

2

2.5

3

20 40 60 80 100

FIGURE 6.7 Coefficient of friction of dry belts (Wayne Phillips) as function of slip rate (%).

1.5

1.25

1

0.75

0.5

0.25

CO

F

00 0.2 0.4 0.6

Pressure (MPa)

0.8 10

0.2

0.4

0.6

0.8

1

CO

F

Temperature (�C)(b)(a)0 10 20 30 40 50

FIGURE 6.8 Coefficient of friction of some belts as function of (a) pressure and (b) temperature.



occurs. Variable torque test holds the belt wrap and ramps up the torque until slip occurs. Variableenvironmental test is conducted by allowing to preset the test pulley wrap angle, speed, and reactiontorque; then the pulley is subjected to a water flow rate of constant value usually between 0 and 1L=min. This procedure has been used primarily with a wet condition of 300 mL of water flow permin. The purpose of this run is to characterize the belt–pulley systems that develop low coefficientsof friction immediately upon exposure to water, so as to distinguish good and poor performers ofbelts in terms of resistance to a drop in coefficient of friction.

Belt to pulley wet friction and dry friction are different. Figure 6.9 shows the wet friction as afunction of percent slip. The test runs show that the coefficient of friction drops significantly withwetness of the belt friction surface, each driven test pulley was run at 1600 rpm, and water wasapplied at the middle of the grooved area of the belt at a rate of 1 L=min. It was applied on the beltwithin a few inches of the entrance to the driven test pulley. Note that there is a significant variationof wet coefficient of friction among belts, suggesting that there is an opportunity to improve the wetcoefficient of friction by modifying the belt compound or cross-sectional geometry.

Some sources [99] affecting belt friction variation include belt manufacturing process andchemistry control; environmental factors including belt conditioning in the field by heat, cold,moisture, excessive slip, and thermal cycling; belt storage factors like storing of belts under differenttemperature and humidity conditions to abrasive or bad-shaped pulleys; and geometric factors likethe degradation of the belt profile, which can be due to the manufacturing process related to setupand tool wear. These sources of product variability suggest that any tolerances set on the coefficientof friction of a belt–pulley interface should be large, so as not to compromise on the performance ofpower transmission.

Belt to pulley frictions are also affected by pulley angle. Figure 6.10 shows the effect of pulleyV angle, given 408 nominal dry belts. The same belt was used for the different V angle pulleys testedin each group. For dry pulleys and belts, the smaller V angles result in slightly lower coefficients offriction as would be predicted by classic theory.

Belt to pulley friction is also affected by the number of ribs, both on the grooved and backparts. Figure 6.11 shows the approximate independence of coefficient of friction on belt width(number of ribs). Figure 6.11 shows that using the same initial 8K belt and cutting down first to a 6Kand then to a 4K results in minimal change in friction. Note that the 8K belt actually demonstratesthe lowest friction of the group, although the run-to-run variation is in the same range as the ribnumber variation. The reason the 8K is not significantly higher is probably because the 8K wouldrun the coolest, therefore not incurring a heat-related increase in coefficient of friction, a pheno-menon not addressed directly by the Euler equation [99].

Slip rate (%)

CO

F

0

0

0.3

0.6

0.9

1.2

1.5

20 40 60 80 100

FIGURE 6.9 Coefficient of friction of wet belts (Wayne Phillips) as function of slip rate (%).



6.3 BELT VIBRATIONS

6.3.1 BELT VIBRATIONS ANALYSIS

There are numerous research papers dealing with linear and nonlinear transverse and longitudinalbelt vibration responses. The cases investigated include the effects of initial tension, transportvelocity, bending rigidity, support flexibility, large displacement, belt and pulley imperfections, etc.

Generally, during its operation, a power transmission belt can undergo transverse (vertical),longitudinal (line), torsional (twist), as well as lateral motions or their combinations as schematicallyshown in Figure 6.12. A power transmission belt is a flexible element subject to initial tensionmoving in the axial direction. Transverse motion of belt can be modeled using a moving string, amoving beam, or even a moving band model.

Depending on the combinations of bending rigidity EI, initial tension T0, and span length L ofbelt considered, the simple string model can be used to predict the belt dynamics for many cases.

Slip rate (%)

CO

F

0

0

0.5

1

1.5

2

2.5

3

2 4 6 8 10

FIGURE 6.10 Coefficient of friction as function of slip rate (%) for dry belts (Wayne Phillips) with differentV angle (thin line: 308; dashed line: 358; thick line: 378).

Slip rate (%)

CO

F

0

0

0.5

1

1.5

2

2.5

3

2 4 6 8 10

FIGURE 6.11 Coefficient of friction as function of slip rate (%) for dry belts (Wayne Phillips) with different ribs(thin line: 4 ribs; dashed line: 6 ribs; thick line: 8 ribs).



In other situations, the effect of finite bending rigidity must be included. When the belt is stationary,the equation of motion for a Bernoulli–Euler beam subject to an initial tension T0 is

EIWxxxx � T0Wxx � rAWtt ¼ 0 (6:3)

whererA is the mass per unit lengthW is the transverse displacementt is the timex is the spatial coordinate

Consider free vibrations; the displacement function can be written as

W(x,t) ¼ A sin(npx=L) exp(ivt) (6:4)

It satisfies the simply supported boundary conditions

W(0,t) ¼ W(L,t) ¼ Wxx(0,t) ¼ Wxx(L,t) (6:5)

Substituting Equation 6.4 in Equation 6.3 gives the natural frequencies of the beam under initialtension as

v2n ¼ v2

sn þ v2bn (6:6)

wheren is the mode numbervsn¼ (np=L)(T0=rA)

1=2 is the nth natural frequency of a stringvbn¼ (np=L)2(EI=rA)1=2 is the nth natural frequency of a simply supported beam

Basically, string theory can accurately predict the relationship between natural frequencies andtension, except for short span belt. When the bending rigidity of the belt is negligible, a movingstring model can be used. On the other hand, for short span belt and high-frequency problems,bending rigidity and even the shear deformation and rotational inertia need to be taken into account.

(a) (b)

(c) (d)

C

C

FIGURE 6.12 Schematic of vibrations of belt span: (a) vertical vibration; (b) longitudinal vibration; (c) twistvibration; (d) lateral vibration.



There exist very rich nonlinear dynamics phenomena in belt drive systems. If the pulley is takeninto account, the linear system vibration coupling exists between the pulley rotation and thetransverse span deflections. Models that treat the belt as a string and neglect the belt bendingstiffness cannot explain this coupling phenomenon.

There exists instability of an axially moving string supported by a discrete or distributed elasticfoundation, which corresponds to the critical speeds. The elastically supported string shows uniquestability behavior that is considerably different from unsupported axially moving string. In particu-lar, any elastic foundation (discrete or distributed) leads to multiple critical speeds and a singleregion of divergence instability above the first critical speed, whereas the unsupported string has onecritical speed and is stable at all supercritical speeds. Additionally, the elastically supported stringcritical speeds are bounded above, and the maximum critical speed is the upper bound of thedivergent speed region.

There exists the vibration localization phenomenon in multiple span axially moving beams.Depending on the magnitude of the disorder and the internal coupling of the structure, its modes offree vibration may become spatially localized, such that vibrational energy is contained to aparticular geometric region instead of being distributed throughout the structure. This phenomenonis referred to as normal mode localization. Researches illustrate that normal mode localizationoccurs for both stationary and translating disordered two span beams, especially for small interspancoupling. The occurrence of localization is characterized by a detected peak that is much greater inone span than in the other. In the stationary disordered case, localization becomes more pronouncedas the span axial tension increases. In the axially moving disordered case, the transport speed has asignificant influence on localization and, generally speaking, localization becomes stronger withincreasing speed.

There exists the coupling of longitudinal and transverse (or lateral) vibrations. Belt longitudinalvibration could trigger complicated parametrical transverse or lateral vibrations of belt, and beltresonance occurs because of modal coupling of the longitudinal and transverse (or lateral) modevibrations.

The equation governing a belt span longitudinal vibration can be approximated by

EAUxx � rAUtt � cUt ¼ 0 (6:7)

whereEA is the belt axial stiffnessU is the belt longitudinal displacement

The longitudinal natural frequencies are

vLn ¼ (np=L)(E=r)1=2 (6:8)

Particularly, if belt system has torque variation–induced longitudinal dynamic load, the system islikely to have longitudinal vibrations. This longitudinal vibration leads to a variable tension, whichcan be represented as

T(t) ¼Xni¼1

Ti cosvit (6:9)

By substituting Equation 6.9 into Equation 6.3, we get a parametric transverse vibration equation

EIWxxxx �Xn1

Ti cosvit þ T0

!Wxx � rAWtt ¼ 0 (6:10)



This equation characterizes a parameter-excited system. The transverse resonance due to the parameterexcitation could exhibit many resonant forms, such as fundamental resonance, harmonic resonance,combination resonance, etc. The parametric resonance could occur whenever the natural frequency oftransverse vibration equals to n times the frequency of longitudinal vibration. If any of the longitudinalvibration modes coincides with one of the belt transverse vibration modes, system vibration couplesstrongly and the modal lock resonance will occur; there exists similar coupling between longitudinaland lateral vibrations. The transverse vibrations unavoidably cause noise radiation, but this is the drivesystem noise, which is different from the slip-related friction-induced noise.

6.3.2 BELT VIBRATIONS MEASUREMENTS

Figure 6.13 shows the schematic of measurement methods of accessory belt movement by usinghigh-speed camera and laser sensor proposed by Fujii et al. [57]. The observation of belt behaviorusing high-speed cameras and laser sensors reveals the existence of the four vibrational modesshown in Figure 6.12. For instance, Figure 6.14 shows the schematic of captured lateral vibration atcross-section C–C in Figure 6.12d, in which the arrows show movement on both sides of the belt.This fluctuation is belt movement in the thrust direction, and also indicates movement of the beltback surface to the outside due to torsion. Belt movement in the thrust direction could be as great as30 mm, which is equivalent to twice the pulley width. The experiments show that the accessory beltlateral vibration resonance occurs when the accessory belt’s lateral natural frequency equals half ofthe frequency of belt longitudinal fluctuation in the belt tension direction. The longitudinal fluctu-ation is caused by crankshaft torsional vibration. A correlation exists between belt lateral vibrationand belt length fluctuation in the belt tension direction, with the belt running off the pulley when thethreshold value is exceeded. Vibration in the lateral direction occurs due to the excitation of naturalvibrational mode of lateral vibration. This holds for the relationship between the transverse vibrationand belt length fluctuation.

Driven pulley

Driving pulley

Belt

High-speed camera

High-speed camera

Laser sensor

FIGURE 6.13 Schematic of typical setup for belt vibration measurement.

A

AB

B

C C

FIGURE 6.14 Schematic of lateral vibration of belt span at cross-section C–C in Figure 6.12, arrows indicatingmovement of both sides of the belt.



Most of the above vibrations measured are correlated with drive system noise directly orindirectly. However, they are not directly correlated with the friction-induced noise. It is notedthat V-ribbed belts have cross-section type vibrational modes as shown in the Figure 6.15, which aremore likely to correlate with friction-induced noise.

6.4 MISALIGNMENT BELT NOISE

In this section, the experimental results of misaligned belt chirp noise are presented. For conveni-ence, the instability generation mechanism was described using existing formulation for V-beltdeveloped by Moon and Wickert [17]. The belt misalignment excitation due to radial slip andfriction actually results in a boundary excitation to the influenced belt span, which leads to forcedtransverse vibration of the belt span (lateral and torsional as well). On the other hand, radial slip andfriction generate direct excitation to the cross-sectional modes of the belt. The dominant componentsof misalignment noise (ticking, chirp, or squeal) can be attributed to the radial friction-inducedmode coupling and=or mode lock-in of belt cross-section vibrational modes.

As a belt enters a pulley with misalignment, it is moving in a slightly different direction than thepulley. This slight difference in the direction of motion, or misalignment, can excite instability andnoise referred to as misalignment noise. The difference in directions of the belt and pulley isresolved by forcing the belt inward radially and sideways as shown in Figure 6.16. However, theforces on the belt will then act to move the belt outward radially until it is forced to slip back asshown in Figure 6.17. This struggle for equilibrium between the belt tension and friction forcesresults in continuous climb–jump events, exhibiting sawtooth type motion at some locations of thebelt where the belt leaves the pulley. The climb–jump event behaves like ‘‘stick–slip’’ in the radialdirection. The sawtooth type boundary motion relative to the belt span forms a boundary excitation.The boundary motion frequency is proportional to the speed of the drive. The faster the drive rotatesthe higher the boundary motion frequency. This type of boundary motion due to belt misalignmentcreates rich vibrations of belt spans. However, basically the misalignment-induced belt spanvibrations are not the dominant component of the misalignment noise, whereas the associatedhigh-frequency cross-sectional resonant mode constitutes the dominant component of chirp noise.

3 rib belt

4 rib belt

6 rib belt

FIGURE 6.15 FEA predicted cross-section vibrational mode shapes of 3, 4, and 6 rib belts (the 4th, 5th, andthe 8th modes, respectively).



The misalignment noise analysis has shown the dominant chirp or squeal noise corresponds to aparticular cross-sectional resonant mode of the belt structure as indicated by modal finite elementanalysis; this suggests that pulley–belt interface friction-induced rib cross-section vibrations, insteadof the other type of belt span vibrations, are the root causes of this noise.

6.4.1 EXPERIMENTAL OBSERVATION OF V-BELT RADIAL MOTION DUE TO MISALIGNMENT

Figure 6.17 shows the schematic of the radial motion of V-ribbed belt due to misalignment.The fully seated V-ribbed belt rides in a multigrooved pulley as shown in Figure 6.17a. Ideally,the belt rib enters the pulley making simultaneously contact with each side of the V-groove as thebelt sits into place. Whenever there is misalignment of pulleys as shown in Figure 6.16, thebelt enters the pulley and makes initial contact with only one side of the V-groove as shown inFigure 6.17b. The belt subsequently slides into the fully seated position guided by the action ofthe V-ribs on the groove face. This sliding motion, in which the belt rib follows a sliding path downone side of the pulley groove, is referred to as radial sliding. The radial sliding develops whenever abelt span entering the pulley is not coincidental with the pulley’s plane of rotation, and theangle between belt span and pulley plane is referred to as the drive misalignment angle as shownin Figure 6.16.

Driven+

Driver+

Misalignment angle

l

V

W

U

FIGURE 6.16 Schematic of misalignment of pulleys.

(a) (b)

FIGURE 6.17 Schematic of the radial motion of belt due to misalignment. (a) Fully seated belt. (b) One sidecontacted belt.



To further elaborate the radial motion, we use the example of V-belt radial motion due tomisalignment [17]. Figure 6.18 shows the measured radial displacement of V-belt underlateral misalignment. This signal is obtained by eliminating the motions from other resources,such as the pulley’s eccentricity which is essentially independent of misalignment; the transversemotion of the belt span associated with the sawtooth forms the boundary excitation. As ischaracteristic of stick–slip behavior, the belt gradually rose, and then rapidly settled into the pulley’sgroove; the process then repeated itself periodically. This component of the belt’s motion has apeak-to-peak amplitude of approximately 150 mm, and one complete stick–slip cycle occurred asthe pulley rotated through roughly 18. Periodic radial oscillation and slippage of the belt in thismanner result from frictional stick–slip response at the belt–pulley interface. The belt experiences aprescribed sawtooth-like motion on its boundary. Periodic slippage of the belt occurs in the presenceof misalignment, even without torque overload, and it contributes to the belt’s high-frequencyresponse. Measurements correlate the belt’s radial oscillation and slippage with periodic bursts inthe near-field sound pressure. Figure 6.19 shows the synchronized measurements of displacementand sound over a single cycle of stick–slip excitation. In the measurement, a noncontact opticdisplacement transducer is used when the belt moves onto the pulley, and a microphone is used to

Rotation angle (�)

Bou

ndar

y di

spla

cem

ent (

mm

)

0.0−0.05

0.00

0.10

0.15

0.20

1.5 3.0

FIGURE 6.18 Measured radial displacement of V-belt under misalignment. (Reprinted with permission fromMoon, J. and Wickert, J.A., J. Sound and Vib., 225, 527, 1999. Copyright 1999, Elsevier.)

Dis

plac

eme

nt (

mm

)

Sou

nd p

ress

ure

(Pa)

Time (s)

Rotation angle (�)

0 0.5

0

1

0.05 0.1−0.5

0

0.5

−0.15

0

(a) (b)

0.15

FIGURE 6.19 Synchronized measurements of (a) displacement and (b) sound over a single cycle of stick–slipexcitation. (Reprinted with permission from Moon, J. and Wickert, J.A., J. Sound and Vib., 225, 527, 1999.Copyright 1999, Elsevier.)



correlate local boundary motion to the sound produced as the belt slips. The displacement wasrecorded along the pulley’s radial direction, which is equivalent to the direction of the belt’stransverse vibration. Measured displacement and sound pressure signals were synchronized.

As is common in phenomena involving friction–vibration interaction, stick–slip motion isoften accompanied by the generation of noise. With the belt operating at a transport speed of2.5 m=s, a sequence of sound bursts was recorded. One burst occurred for essentially each degreeof the pulley’s rotation. The sequence of sound bursts and the corresponding slip motion of thebelt were not observed when the pulleys were aligned within the precision of the apparatus.Measurements of radial displacement and sound pressure are synchronized in Figure 6.19 over asingle cycle of response as the pulley was rotated slowly essentially under no load. The motion ofthe belt within the groove is depicted by the three insets so as to highlight the asymmetry of belt–pulley contact that occurs in the presence of misalignment. The misalignment generates biasedcontact against one side of the groove. At the onset of rotation, normal and frictional forcesdeveloped between the belt and the pulley. With an incremental increase of the rotation angle,the belt was elevated higher within the groove (as in the second loading stage), and it wassupported on the left face of the groove under higher normal and friction forces. In the thirdloading stage, the belt reached its maximum radial elevation, d. The corresponding rotation anglerelative to the first loading stage is defined as the critical angle, a. Under any further rotation ofthe pulley, the belt rapidly slipped back to its fully seated position, and this sequence repeateditself continuously. The synchronized measurements of Figure 6.19 demonstrate the radial slip-page of the belt correlated with the sound burst, and so the problem is actually attributed to befriction and misalignment-induced vibration.

Figure 6.20 shows the predicted values of the critical angle and elevation rise as a function of thepulley’s radius. Figure 6.21 shows predicted values of the critical angle and elevation rise as a functionof the belt’s pre-tension for an application r¼ 4.98, R¼ 54 mm, and v¼ 2.5 m=s. The formulationgiven by Moon and Wickert [17] will be introduced in Section 6.4.4.

6.4.2 CHIRP AND SQUEAL OF AUTOMOTIVE V-RIBBED BELT

The noise burst due to belt misalignment is usually termed as ‘‘chirp’’ noise. In automotive accessorybelt drive system, belt chirp noise has been recognized as one of the major failure modes affectingproducts’ competitiveness. Similar to V-belt, the noise of misaligned V-ribbed belt is associated with

0

0.2

0.4

0.6

0.8

4 24 44 64 84

Pulley radius (mm)

Ele

vatio

n (m

m)

FIGURE 6.20 Predicted values of the critical angle and elevation rise as a function of the pulley’s radius:r¼ 4.98 and v¼ 2.5 m=s.



radial stick–slip-induced vibrations. The squeal noise, which takes place at some belt misalignmentconditions, is a continuous chirp noise and shares the same features as individual chirp. The friction-induced mode coupling or mode lock-in instability for radial rib slippage can be considered as thedirect causes of chirp. Chirp noise control is often achieved by very tight pulley alignment; forinstance, a conventional guideline could be about 0.38 maximum belt entry angle into each groovedpulley. Occasionally belts will chirp at pulleys where the system alignment is this good or better.

The automotive industry has been using a misalignment noise tester shown in Figure 6.22 tocharacterize belt alignment noise. The driven pulley is allowed to have controlled axle displacement,which gives specified misalignment angle to the belt. The test rpm is usually controlled close to theidle speed of the engine. Figure 6.23 shows typical misalignment noise waveforms and time–frequency spectra for a belt under different misalignment angles.

Figure 6.24 shows the comparison of noise spectra of a belt with 08 and 38 misaligned pulleys.This indicates the remarkably larger magnitude of high-frequency components in chirp noise

0.14

0.16

0.18

0.2

75 100 125 150

Tension (N)

Ele

vatio

n (m

m)

FIGURE 6.21 Predicted values of the critical angle and elevation rise as functions of the belt’s pre-tension foran application: r¼ 4.98, R¼ 54 mm, and v¼ 2.5 m=s.

FIGURE 6.22 Photography of belt misaligment noise tester. (Courtesy of Gates Corporation.)



spectrum. The dominant spectral property in misaligned belt chirp is characterized by harmonicoscillation. It is noted that the radial stick–slip complicated system vibrations. The radial stick–slipof misaligned belt is a particular boundary excitation associated with the misalignment and it justleads to simple forced vibrations. Experiments show that the stick–slip frequency increases withincreasing rotational speed. Basically, span vibration induces noise and is a portion of the entirenoise, whereas the high-frequency ‘‘harmonic oscillation’’ or high-frequency component of stick–slip burst constitutes the dominant component of chirp noise.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04�10

�5

0

5

10

Time (s)

Pre

ssur

e (P

a)

Time (s)

Fre

quen

cy (

Hz)

0.005 0.01 0.015 0.02 0.025 0.03 0.0350

1

2

x 104

�50

0

50

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04�10

0

10

20

Time (s)

Pre

ssur

e (P

a)

Time (s)

Fre

quen

cy (

Hz)

0.005 0.01 0.015 0.02 0.025 0.03 0.0350

1

2

x 104

�50

0

50

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04�40

�20

0

20

40

Time (s)

Pre

ssur

e (P

a)

Time (s)

Fre

quen

cy (

Hz)

0.005 0.01 0.015 0.02 0.025 0.03 0.0350

1

2

x 104

�20

0

20

40

60

80

(b)(a)

(c)

FIGURE 6.23 Misalignment noise waveform and time–frequency spectrum for different misalignments:(a) 0.58 misalignment; (b) 1.58 misalignment; (c) 2.58 misalignment. (Courtesy of Gates Corporation.)

130

Sou

nd p

ress

ure

leve

l (dB

)

50

70

90

110 0�3�

10

Frequency (kHz)

FIGURE 6.24 Comparison of noise spectra of 08 and 38 misaligned pulleys. (Courtesy of Gates Corporation.)



Finally, as shown in Figure 6.24, the harmonic oscillation has multiple harmonics; along with the5 kHz principal harmonic, there are also 10 and 15 kHz harmonic components. This suggests that theoccurrence of the harmonic oscillation allows the system to exhibit strong nonlinear features. Thesystem has larger magnitude oscillations associated with the occurrence of harmonic oscillations.

Figure 6.25 shows the noise waveforms for different rpm speeds of 30, 300, 700, and 1000. Asshown in Figure 6.25, at very low sliding speeds (i.e., low rotational speeds) the noise was presentedprimarily as individual stick–slip noise burst after initiation of the radial slip event.

As the rotational speed increases the stick–slip frequency increases (period between chirpsdecreases). Finally, the dwell time between stick–slip events is reduced to zero and the noise burstbecomes a continuous or intermittent noise squeal. The fundamental frequency of stick–slip eventnearly disappears in proper high rpm. At this point it appears that the belt no longer attains a true‘‘stick’’ condition (sliding velocity¼ 0); instead, it continues to slide in a continuous mannerexhibiting strong contact. Further increasing the speed does not change the feature of this oscilla-tion. However, the fundamental frequency of chirp noise remains by and large unchanged, andexhibits a continuous and sharp squeal.

Figure 6.26 shows more details of the power spectra of the time waveforms in Figure 6.25.Figure 6.26 shows that from the stick–slip burst (or chirp) to the continuous squeal (or harmonicoscillation) the dominant spectrum peak experiences little change in frequency as speed is increasedalthough the spectral definition narrows considerably. This suggests that the chirp noise and squealsare consistent with the harmonic oscillation. In other words, the individual stick–slip burst or chirpis a separated harmonic oscillation or squeal. The frequency of stick–slip itself is dependent on rpmand much lower than that of harmonic oscillation. The chirp and squeal have similar frequencyspectrum signature of harmonic oscillation that is not affected by the change of speed. AsFigures 6.25 and 6.26 show, the burst or harmonic oscillation component is the dominant noise.The maximum stick–slip frequency when speed increases is still much lower than the harmonicoscillation frequency. This suggests that there is no fundamental relationship between the stick–slipevent and the harmonic oscillation; only slip event, either burst or continuous type, correlates with

Nor

mal

ized

sou

nd p

ress

ure

Time (ms)

1

1

1

1

0 5 10 15

30 rp

300 rp

700 rp

1000 rp

30 rpm

300 rpm

700 rpm

1000 rpm

FIGURE 6.25 Misalignment noise waveform of 30, 300, 700, and 1000 rpm speeds. (Courtesy of GatesCorporation.)



the harmonic oscillation (chirp or squeal). Figure 6.27 shows the stick–slip frequency and harmonicoscillation (chirp or squeal) frequency as a function of rpm.

6.4.3 INFLUENCE OF RIB AND PULLEY PROFILE

Some test results about the effect of pulley diameter indicate that smaller pulleys are less susceptibleto misalignment noise. Smaller pulleys also show an improved ability to tolerate greater beltmisaligned angles prior to the belt skipping pulley grooves. With larger pulleys, a misalignedbelt has the potential for a greater length of belt to undergo radial sliding. Correspondingly, thefriction force associated with this sliding is also likely to be greater.

1

2

3

4

90

70

50

30

0.11

10

1

2

3

4

1

Sou

nd p

ress

ure

leve

l (dB

)

Frequency (kHz)

1000rpm

0

FIGURE 6.26 Noise spectra of 30, 300, 700, and 1000 rpm speeds. (Courtesy of Gates Corporation.)

7000

6000

5000

4000

3000

2000

1000

250 500 750 1000 1250 1500 1750Shaft speed (rpm)

Stic

k–sl

ip fr

eque

ncy

(Hz)

0

FIGURE 6.27 Stick–slip frequency (solid line) and harmonic oscillation frequency (horizontal broken line).(Courtesy of Gates Corporation.)



As the friction force opposite the radial seating of the belt increases, the likelihood for bothnoise and diminished belt stability increases proportionally. The testing results suggest that theopportunities to achieve substantial improvements in misaligned angle sensitivity, and thus mis-alignment noise, by modifying the standard K section pulley profile may be rather limited. In orderto verify the effect of changing belt mass on the resonant characteristics of the belt, the ribs weretruncated first by 30% and then by 50% of the original rib height by grinding the end of the ribs inbetween tests. The resulting spectra are shown in Figure 6.28. As the rib height is reduced both thestick–slip and the harmonic oscillation frequencies shift to higher levels. A 30% reduction in ribheight (25% reduction in the effective mass) causes the harmonic oscillation frequency to increaseby 18% to 5.2 kHz. A total reduction of 50% of rib height (35% mass reduction) causes thesame harmonic oscillation peak frequency to increase by 28% over the full rib height to 5.7 kHz.These changes in oscillation frequency correspond very closely to what is predicted by the simplenatural frequency relation of frequency equaling to root of stiffness above mass. Moreover, there is areduction in the average magnitude of the harmonic oscillation peaks. This can be linked with thedecreased area of interface contact, the reduced radial sliding distance, and the concurrent decreasein the amount of stored energy available for burst.

The effect of pulley profile variations on belt misalignment noise performance was studied.Smaller diameter pulleys exhibit less sensitivity to misalignment noise due to the smaller area ofsliding contact between belt and pulley.

A profile modification which reduces the available ‘‘space’’ in the groove leads to higher radialsliding pressures and subsequently greater tendency toward noise. The pulley profile variationsalone cannot be expected to create substantial improvements in misalignment noise. Figure 6.29shows the two profile modifications with radial slotted and grooved. Figure 6.30 shows thecomparisons of their effect on the band limited noise level. In addition to the above two profilemodifications in Figure 6.29, the results of other modifications are also plotted in Figure 6.30.Basically, various profile changes from K section including opened, tightened, convex, concave,368 V angle, 438 V angle, relieved, and larger tip radius have only about 4 dB variations. However,roughened or ‘‘interrupted’’ surfaces increase noise level substantially. Pulley size and span lengthare more sensitive to ensure quiet operation.

6.4.4 MODELING OF RADIAL STICK–SLIP MOTION OF V-BELT

For elaboration purpose, in the following, the stick–slip motion of misaligned V-belts is formulatedmathematically by using Moon and Wickert’s approach [17]. The analysis characterizes the

1,000

50 dB

60 dB

70 dB

80 dB

90 dB

100 dB

110 dB 100% rib height70% rib height50% rib height

10,000Frequency (Hz)

Sou

nd p

ress

ure

(dB

L rq

: 20

µpa)

FIGURE 6.28 Noise spectra for belts with different rib heights. (Courtesy of Gates Corporation.)



properties of the boundary excitation vibrations induced by belt misalignment. Consider themisaligned belt shown in Figure 6.31; the belt undergoes continuous transition from being straightwithin the span to conforming with the pulley’s circular shape. The instantaneous curvature of thebelt is denoted as r. This transition region spans angle a and is set by the misalignment angle r, theelevation angle g¼ tan�1(d=L), and the radius R. The boundary excitation mechanism of misalignedbelt can be characterized by two quantities: a, the critical rotation angle of the pulley at whichslippage occurs, and d, the belt’s rise in elevation as shown in Figure 6.31. The period of excitationis a=v, where v¼V=R is the pulley’s rotation rate. With these quantities known, the boundarymotion of the belt can be represented in the first approximation by a sawtooth-like displacementprofile, as shown in Figure 6.32.

Radial slotted

Grooved

FIGURE 6.29 Schematic of modified grooves of pulleys.

80.7

68.3

40

45

50

55

60

65

70

75

80

Ban

d lim

ited

soun

d le

vel (

dBL)

(3.6

kH

z ba

ndpa

ss) 66.7 67.9

70.8

Tighte

ned

Opene

d

Groov

ed

Slotte

d

Rough

ened

FIGURE 6.30 Band limited noise level for same belt tested on different pulleys with various modified grooves,each under four kinds of misalignments: 08, 0.88, 1.68, and 2.48. (Courtesy of Gates Corporation.)



The equations of motion for the transverse vibration of misaligned belt span can be repre-sented as

EIWxxxx � (T þ T0)Wxx � rA(Wtt þ 2cWxt þ kc2Wxx)þ b(Wt þ cWx) ¼ 0 (6:11)

whererA is the mass per unit lengthW is the transverse displacementt is the timex is the spatial coordinateT0 is the mean tensionT is the dynamic tensionk is the pulley support system constantb is the belt damping coefficient

The boundary condition due to misalignment-induced slippage is considered as sawtooth-likewave of elevation–slippage function

ev

e2

er a

L

g

r

def

R

FIGURE 6.31 Schematic of misaligned V-belt model.

Elevation, d

TimePeriod,

Fin

e sc

ale

boun

dary

mot

ion

aW

FIGURE 6.32 Idealization of the belt’s boundary motion in terms of the model’s critical angle and elevation rise.



x ¼ 0: W ¼ 0

x ¼ l: W ¼ d 1� 1p

X1i¼1

1isin(iv0t)

" #, v0 ¼ 2pv=a

(6:12)

wherev is the pulley rotating frequencyd is the amplitude of radial slippagea is a derived parameter corresponding to the angle where the belt starts to slip

To solve Equations 6.11 and 6.12, we substitute transverse displacement W with

W ¼ Z þ d 1� 1p

X1i¼1

1isin(iv0t)

" #x

l(6:13)

Then we get following forced vibration equations with homogeneous boundary condition:

(T þ T0)Zxx � rA(Ztt þ 2cZxt þ kc2Zxx)þ b(Zt þ cZx) ¼ f (x,t) (6:14)

x ¼ 0: Z ¼ 0; x ¼ l: Z ¼ 0, f (x,t) ¼ div20

plsin(iv0t)

By ignoring the damping effect, the solution of Equation 6.14 is derived as

Z ¼X1i¼1

d

i

X�1

i¼�1

sin npx

p2nlv2n(1� iv0=vn)

� h1 cos(iv0t þ npvx)þ cos(np) sin (iv0t � npv(1� x))þ h1 cos(iv0t � npv(1� x))½ �f g(6:15)

wherevn¼ np(1 � v2)h1¼ 2v(vn � iv0)=(vniv0)

Then we obtain transverse vibrations

W ¼X1i¼1

d

i

X�1

i¼�1

sin npx


� h1 cos(iv0t þ npvx)þ cos(np) sin(iv0t � npv(1� x))þ h1 cos(iv0t � npv(1� x))½ �f g

þ d 1� 1p

X1i¼1

1isin(iv0t)

" #x

l(6:16)

By using proper assumption and considering the force and moment equilibrium of the effective arcof belt, the slippage amplitude or elevation rise of V-belt within groove during the climbing–slippage process was derived as a function [17]

d ¼ 2EIrL0aT1R2 cos2 r

� R cos r ¼ T1 cos(eva � 1)L� 12n0EIz tan r=LT1((n2 � naþ 1)na=(n2a)� 1=n þ a=2)

(6:17)



where

n ¼ m cos k

sinbþ m sin k cosb0 , n0 ¼ m cos k

cosb� m sink sinb,

g¼ tan�1(d=L), k¼ sin�1(d1=d2), where d1 and d2 are the specific distances determined from thelocation of the belt’s cross-sectional centroid and the wedge angle. EIr and EIz are the flexuralstiffnesses. The studies illustrate the behavior of pretension and pulley radius on the boundaryexcitation mechanism.

The effects of pulley size are experimentally investigated for three radii in [17], which verifiedthe model prediction in Figure 6.20. As the radius of the pulley was increased, both the elevationsand critical angle decreased measurably. For the belt parameter values used in the case study, themodel indicates that beyond a proper radius, d becomes insensitive to R.

Moon and Wickert also shows the relationship between the initial pre-tension, and d by usingexperiment and the model’s predictions as shown in Figure 6.21. d decreases monotonically byabout 25% over the factor-of-two range in pretension that was examined, and those values werecalculated using the procedure discussed above. The larger the tension, the lesser the belt rise withinthe pulley, and the higher the frequency of the stick–slip motion.

This kind of model study in conjunction with lab data offers some possibility to fundamentallylink the coefficient of friction of the belt to the belt chirp noise phenomenon, and allows theprojection of the belt’s general tendency to chirp to be predicted by the measurement of beltcoefficient of friction on a test stand with limited success. This study also provides formulas suitablefor computer coding based on measurable data for determining how much misalignment can beallowed. The data needed are the groove side coefficient of friction of the belt on the pulley, thepulley diameter, the belt tension, and the pulley V angle.

The above analysis is readily applicable to V-ribbed belts in principle. But it is noted that realbelts are very sensitive to many other factors that could dilute the effects indicated by the aboveformula. As such, the experiment-based trend could be more reliable to give design guideline.Figure 6.33 shows the typical noise threshold as functions of coefficient of friction and misalign-ment values for V-ribbed belts with different wedged V angles suggested by Meckstroth et al. [100].

Based on empirical and theoretical analysis, it appears that there is a fundamental inclination ofbelts to ride up the side of the groove for some short entry wrap angle within the belt alignment and

0

0.7

1.4

2.1

2.8

3.5

0 0.2 0.4 0.6 0.8 1

Misalignment angle (°)

CO

F

FIGURE 6.33 Noise threshold as function of coefficient of friction and misalignment for belt with differentV angles (solid thin line: 348; dashed line: 408; solid thick line: 468).



coefficients of friction ranges that are usually used in accessory drive systems. Note that it ispossible for belt to climb one side of the groove with zero misalignment if the coefficient of frictionis high. For a 4000 Hz repeated snap-back chirp phenomenon, the more the misalignment, the moreeasily the belt will ride up one side of the groove. A bigger V angle allows a slightly higherlikelihood for a belt to ride up one side of the groove. Reducing the V angle will reduce the tendencyto chirp, but a change from 408 to 348 would only equal roughly a 0.058 improvement in beltalignment. Belts with very high dry coefficient of friction will be more prone to chirp. Reducing thecoefficient of friction will reduce the tendency to chirp, but typically since we count on at least a drycoefficient of friction of 1.0 for torque reaction, there are limits to this approach.

On the other hand, the model suggests that the root cause of misalignment noise is the belt radialslippage; a straightforward way of reducing this noise is to reduce the slippage amplitude. Frommodified equation we infer that the slippage amplitude decreases as the belt thickness increases. Assuch, higher belt thickness is desired for low misalignment noise. Six categories of belts werecompared for their misalignment noise level in terms of thin and thick types. Each category of beltshas two kinds of thickness, noted as thick (A, B, C, D, E, F) and thin (A1, B1, C1, D1, E1, F1). Forcomparison convenience, misalignment threshold is used to represent the noise performance ofbelts, which is defined as the half value of peak–peak axial offset of test pulley when squeal or chirpnoise at 75–80 dBA level is measured. A smaller value of offset threshold means more sensitivity tomisalignment, or worse noise performance. The tested results are shown in Figure 6.34. FromFigure 6.34 we can see that thick belts have higher offset threshold for all categories; namely, thickbelts have better noise performance for this investigation.

From Equation 6.15 we can see clearly that the radial slippage-induced vibration is a kind offorced vibration, which is excited by stick–slip boundary motion. It is noted that this slippage alsoexcites lateral vibrations as well as torsional vibrations. The belt lateral motion equation isrepresented as

EIVxxxx � (T þ T0)Vxx � rA(Vtt þ 2cVxt þ kc2Vxx)þ b(Vt þ cVx) ¼ 0 (6:18)

whererA is the mass per unit lengthV is the lateral displacementt is the timex is the spatial coordinate

3

2.5

2

1.5

1

0.5

0AA1 BB1 CC1

Thr

esho

ld o

ffset

(m

m)

DD1 EE1 FF1

Thick Thin

FIGURE 6.34 Effect of belt thickness on the alignment threshold offset for noise generation. (Reprinted fromSheng, G., Liu, K., Otremba, J., Pang, J., Qatu, M., and Dukkati, R., Int. J. Vehicle Noise Vib., 1, 1=2, 2004.With permission from Inderscience.)



T0 is the mean tensionT is the dynamic tensionk is the pulley support system constantb is the belt damping coefficient

The lateral excitation boundary condition due to the slippage can be represented as

x ¼ l : V ¼ tgu

2

� �d 1� 1

p

Xi

1isin(iv0t)

" #(6:19)

where V is the lateral displacement of belt. By using a similar substitution as in Equation 6.13, weobtain

V ¼X1i¼1

tg(u=2)di

X�1

i¼�1

sin npx


� h1 cos(iv0t þ npvx)þ cos(np) sin(iv0t � npv(1� x))þ h1 cos(iv0t � npv(1� x))½ �f g

þ tgu

2

� �d 1� 1

p

X1i¼1

1isin(iv0t)

" #x

l(6:20)

Moreover, torsional vibration motion of the belt could be excited by asymmetrical forces due to theone-side effect of the slippage within the pulley groove. This kind of torsional vibration has asimilar spectrum signature as transverse vibration defined by Equation 6.16 or lateral vibrationgiven by Equation 6.20.

The above model is consistent with conventional results of V-ribbed belt misalignment vibra-tions: the radial stick–slip excitation frequency increases with increasing rotational speed; thesystem vibrations exhibit various resonant modes including transverse mode, lateral mode, torsionalmode, and their combinations. The system vibrations exhibit various vibration spectrum signaturesbesides the resonant modes, which are caused by the particular excitation spectrum signature(particular stick–slip pattern).

Basically, pulley misalignment leads to belt boundary excitation or a forced vibration asformulated above. This stick–slip excitation is a saw-like wave and has a dominant period(frequency) and harmonics. This dominant frequency could range from 1 to 2 kHz, it could havemany harmonics with magnitude lower and frequency higher than or lower than 1–2 kHz due toFourier development of saw-like waveform. It could lead to span vibrations, stick–slip vibrations(its frequency is consistent with the dominant excitation frequency 1–2 kHz), and even higherfrequency vibrations. However, all of the measured higher frequency ‘‘harmonic oscillations’’ (withone identical frequency within 4–6 kHz, and their double and triple harmonics) are the uniquedominant parts of the chirp noise. It suggests that the harmonic oscillation does not fall in thevibration regimes defined by the above boundary excitation vibrations.

It was found that the harmonic oscillation frequency range corresponds to a particular resonantmode of the belt structure as verified by modal finite element analysis shown in Figure 6.15. Onlythese mode frequencies are close or identical to the chirp noise frequency. The actual tests showthat the harmonic oscillation frequency varies little with belt width. The only belt vibration modewhich meets this criterion for all belt widths is the one which provides for independent semirigidbody motion of rib. In other words, the chirp noise correlates with the rib cross section modes, orlocal mode. In each case, the modal displacement character and frequency are similar despitedifferences in belt width or rib numbers. The FEA modal analysis indicates that the harmonicoscillation frequency is due to rotational vibration of the individual ribs as viewed from the belt



cross-section. This gives rise to the fact that the frequency is relatively independent of the totallateral width of the belt and, therefore, to a degree independent of the number of ribs. The radialslip excites the rotational modes of the belt around the cord line. Advanced models help to clarifywhy and how only a specific mode is excited severely and the oscillation is well concentrated,whereas all the other modes cannot be excited or their components are trivial. Since the noise-related oscillation is very strong, it likely triggers nonlinear oscillations which have been illustratedby the harmonics; that is why we always observe high-frequency harmonics in the experiments.

6.5 DRY BELT SLIP NOISE

In this section, the experiment and analysis of dry belt slip noise are presented. The belt slip noise isattributed to a longitudinal friction-induced instability due to mode coupling or mode lock-in=velocity-dependent friction of rib natural vibrational modes. The effects of various parameters onnoise are discussed.

6.5.1 DRY BELT SLIP AND FRICTION

For typical ABDS systems shown in Figure 6.1, the desired condition is that there is no slippagebetween pulley and belts among all of the interfaces, from CRK, TEN, IDR, A=C, P=S, ALT toW=P. However, when belt drive angular vibration is beyond the control of the tensioner(tension=damping) or the system has excessive torque load in some pulley hub, belt tangentialslippage takes place on the pulley interface. For a given system, the belt slip property is alsodependent on the running time. Figure 6.35 shows the effect of run time on belt slip. In belt slipprocess, intermittent longitudinal slip noise, ‘‘chirp’’ or ‘‘squeak,’’ could occur due to friction-induced instability subject to operational conditions. These types of belt noises have been observedand reported for many years, but it has been very difficult to establish a relationship between theunique feature of the noise and all kinds of resonances predicted in conventional ways.

6.5.2 DRY BELT FRICTION AND FRICTION NOISE

The longitudinal slip noise associated with a serpentine belt can be categorized as dry and wet slipnoise. Under certain conditions, belt instability can be triggered often resulting in noise. An example

0.6

0.5

0.4

0.3

0.2

0.1

0500 700 900 1100

Total tension (N)

Slip

1300 1500

FIGURE 6.35 Effect of run time on belt slip: diamond, new belts; square, 200 h aged belts; triangle, 400 haged belts. Filled symbols for 7 Nm torque, hollow symbols for 10 Nm torque. (Courtesy of Day, A.J.)



would be an instability resulting from a drop in the coefficient of friction with an increase invelocity. The dynamic instability often results from the frictional interaction of the belt and thepulley. While the conditions that create this instability are understood in terms of modal coupling ormode lock-in, what happens after the instability is less understood. In some cases, strong nonlinearoscillations around some natural frequencies and their harmonics have been identified.

Typical friction–velocity features and slip noise features of V-ribbed belts can be experimentallycharacterized by directly using modified SAE belt friction=slip test rig shown in Figure 6.6. The testrig is used to simulate real belt conditions by circling belts on a driving pulley, driven pulley, andidlers. Belt motion is driven by a driving pulley. For slip noise testing, the driven pulley has resistanttorque controlled with an electric dynamometer. The wrap angle on the driven pulley can beadjusted, and the tension is maintained on the slack side of the driven pulley by the use of weights

COF–slip

0.0

0.4

0.8

1.2

1.6

2.0

2.4

0 1 2 3 4 5 6

Slip speed (m/s)

CO

FC

OF

80

90

100

110

120

130

140

COF

SPL (dB)

COF–slip

Slip speed (m/s)

2.4

2.0

1.6

1.2

0.8

0.4

0.00 1 2 3 4 5 6

140COF

SPL (dB) 130

120

110

SP

L (d

B)

100

90

80

SP

L (d

B)

FIGURE 6.36 Coefficient of friction and slip noise vs. slip speed for two kinds of dry belts. (Reprinted fromSheng, G., Liu, K., Brown, L., Otremba, J., Pang, J., Qatu, M., and Dukkati, R., Int. J. Vehicle Noise Vib., 2, 4,305, 2006. With permission from Inderscience.)



and cables. During the test, when the resistant torque of driven pulley increases to a high enoughlevel, the belt will slip, and noise will generate when the transmitted torque load reaches a certainlevel. All the testing parameters are recorded by sensors including torque sensors, thermal couple,velocity sensors and microphone, data acquisition system, and computer. Currently, using thelaboratory equipment, research and development engineers are able to create many of the conditionsthat result in belt drive noise. This testing and analysis has verified some previous thought andprovided new insight into this complex phenomenon.

Many belts with a negative slope of friction–velocity generate slip noise. However, thereare many belts with zero or even positive slope of friction–velocity that also generate slip noise.Figure 6.36 shows a typical case.

Dry slip noise is characterized by a high-frequency noise (for instance, around 5000 Hz and itsharmonics). It can be identified by its spectrum as shown in Figure 6.37. The noise usually occurssuddenly when proper slip velocity and=or load is attained. Belt slip occurs when pulley hub loadtorque exceeds the load carrying capacity of the belt and pulley interface. This capacity is the limitof the amount of friction force that can be generated. A combination of the coefficient of friction,angle of wrap, and tension usually decides the threshold for the specific case. The slip noise couldoccur for both positive and negative slopes of friction vs. velocity. Moreover, even for the case ofboth positive slope of friction and noise occuring at certain speed, the coefficient of friction and theload carrying capability have drops at higher slip velocity range.

Conventionally, it has been considered that belt slip squeal noises are associated with belt self-excited vibrations due to belt stick–slip or belt negative slope friction or both. However, manyexperimental results exhibit conflicting observations. Larger friction usually correlates to the higherlikelihood of slip noise generation. Belt wrap angle and belt tension have strong effects on the

10,000

80

60

40

20

0

−20

5,000

0

(a)

(b) Time (s)

Fre

quen

cy (

Hz)

0.01 0.02 0.03 0.04 0.05 0.06 0.07

40

20

0

−20

−400 0.01 0.02 0.03 0.04

Time (s)

Pre

ssur

e (P

a)

0.05 0.06 0.07 0.08

FIGURE 6.37 Dry slip noise (a) waveform and (b) time–frequency spectrum.



threshold of noise generation. The slip noise usually occurs radically and suddenly even when beltrunning condition is gradually tuned. The slip noise is associated with a sudden occurrence of severesystem oscillation with an identical spectrum signature.

To investigate system dynamics and noise features, the following experiments were conductedon the test rig. The accelerations measurement at the hubs of idlers were synchronized with slipnoise measurement. Figure 6.38 shows the spectrum of measured acceleration at idler hub and slipnoise; they are quite consistent.

The results show that the synchronized measurement of acceleration is not sensitive to thedirections. The accelerations are similar, either parallel or perpendicular to the belt motion. Theresults suggest that the slip noise occurs suddenly and has 10–20 dB jump when test rig changestesting conditions (react torque, or slip power, or slippage velocity) gradually; the running condi-tions do not change radically before and after the onset of noise; once the slip noise occurs, thesystem acceleration levels increase up to 3–5 times compared with the acceleration level before thenoise onset, and the spectrum of acceleration is correlated with that of the slip noise very well. Thissuggests that the entire system could exhibit a similar concentrated dynamic characteristic (accel-eration spectrum), in spite of location and direction, which suggests that the system undergoes akind of ‘‘resonance’’ state when the slip noise is triggered. The noise usually has a very clear pattern:for instance, for frequency cluster centered close to 5 kHz and close to its harmonic, 10 kHz, thereare some mode transitions, which merge or diverge subject to time or running conditions. It is notedthat the belt has very rich transverse=torsional=lateral=longitudinal resonant modes that are far belowthe recorded oscillation frequency. But these comparatively low resonant modes were below the slipnoise frequency range. On the other hand, some rib cross-sectional modes have a good correlationwith the slip chirp or squeal modes. Consider the driven pulley–belt as the only slippage interface;the slip of the belt and the interface friction are likely to cause the mode coupling or mode lock-in,through which the belt system instability and noise are triggered. Figure 6.39 shows the fundamentalfrequency of noise for belts with different thicknesses.

2,000 4,000 6,000 8,000

Frequency (Hz)

Frequency (Hz)

Noi

se le

vel (

dBA

)R

elat

ive

acce

lera

tion

(dB

)

10,000 12,000 14,0000

(a)

(b)

2,000 4,000 6,000 8,000 10,000 12,000 14,0000

−40

20

40

60

80

100

120

−60

−80

−100

−120

FIGURE 6.38 Spectrum of (a) measured acceleration at idler hub and (b) measured sound.



In addition to friction, many other factors affect squeal noise. Figure 6.40 shows the slip noisesound pressure level as the function of belt tension for different rpm. Figure 6.41 shows the slipnoise sound pressure level as the function of wrap angle for different rpm. Obviously, theexperimental results show that the higher the tension and the larger the wrap angle, the higherthe likelihood of generating slip noise.

Conventionally, a straightforward way of reducing vibration is to enhance system dampingand tune stiffness. However, this solution path does not necessarily produce favorable results due

Structural vibration vs. thickness

Belt thickness (mm)

Fun

dam

enta

l fre

quen

cy (

Hz)

3.5 4.2

8000

7000

6000

5000

4000

3000

2000

1000

04.8 5.2

FIGURE 6.39 Fundamental frequency of noise of belts with different thickness.

130

120

110

100

90

80

70

4590

180

400 rpm

Belt tension (N)

SP

L (d

B)

600 rpm

800 rpm

FIGURE 6.40 Sound pressure level of slip noise vs. belt tensions under different rpm. (Reprinted from Sheng,G., Liu, K., Brown, L., Otremba, J., Pang, J., Qatu, M., and Dukkati, R., Int. J. Vehicle Noise Vib., 2, 4, 305,2006. With permission from Inderscience.)



to the complicated performance of the system characterized by the following equation, whichgives an approximate relationship between coefficient of friction and Young’s modulus=tan d(damping)

m ¼ ½4b1c(p=E)n þ b2(t0=H)� tan d (6:21)

wherep is interface contact pressureb1 and n are constants for various asperity shapesc is asperity density factorb2 is proportional constantt0 is shear stressH is hardnesstan d is the value characterizing damping

By taking into account Equation 6.21, we see a dilemma in reducing vibration by enhancing beltdamping: If the belt damping is enhanced by compound selection=modifier, its coefficient of frictionis automatically raised and the friction feature is changed. Changing the damping and the frictionrelevant force function in Equation 6.21 simultaneously may or may not lead to a smaller solution. Itis noted that larger coefficient of friction tends to give larger mode coupling. Generally, a systemwith large coefficient of friction releases more energy once slippage occurs or the self-excitedvibration threshold is passed. Finally, it leads to stronger or more audible noise. Figure 6.42 showsthe measured coefficient of friction vs. measured Young’s modulus and tan d (damping) for similarbelt material with different compound for regulating damping. It can be seen that the results areconsistent with Equation 6.21. Table 6.1 shows a correlation between noise and coefficient offriction of belts with different Young’s modulus and damping. The experiment shows that the effectof modulus and damping on noise performance is not distinguishable.

70

2030

40

400 rpm

600 rpm

Wrap angle (�)

SP

L (d

B)

800 rpm80

90

100

110

120

130

FIGURE 6.41 Sound pressure level of slip noise vs. belt wrap angle under different rpm. (Reprinted fromSheng, G., Liu, K., Brown, L., Otremba, J., Pang, J., Qatu, M., and Dukkati, R., Int. J. Vehicle Noise Vib., 2, 4,305, 2006. With permission from Inderscience.)



6.6 WET BELT FRICTION AND FRICTION-INDUCED NOISE IN AUTOMOTIVEACCESSORY BELT DRIVE SYSTEM

In this section, the experiment and analysis of wet belt slip noise are presented. The wet beltslip noise is attributed to the instability associated with negative slope of friction vs. velocity. Thewater-mediated stiction exaggerates the case. The effects of various parameters on wet slip noise arediscussed.

6.6.1 WET FRICTION

The experimental results show that the wet belt interface exhibits an unsteady mixed lubricationhaving negative slope of friction–velocity. Extensive experiments have shown that for certain cases,

2

1.5

1

0.5

0.147

0.09

Tan d

Modulus

COF vs. modulus and damping

0

908

578

407

301

FIGURE 6.42 Measured coefficient of friction vs. measured Young’s modulus and tan d (damping). (Reprintedfrom Sheng, G., Liu, K., Otremba, J., Pang, J., Qatu, M., and Dukkati, R., Int. J. Vehicle Noise Vib., 1, 1=2, 2004.With permission from Inderscience.)

TABLE 6.1Noise and Coefficient of Friction of Belts with Different Young’s Modulus and Damping

Belt ID DuroModulus at

100% Extension Tand MDR COF at 10% Slip Peak COFSlip Noise

dBA

1 80.8 908 0.081 0.43 1.01 105 dB2 76.2 650 0.084 0.77 1.76 112 dB

3 64.9 249 0.147 0.87 1.57 93 dB4 69.5 334 0.126 0.57 1.27 111 dB5 69.8 301 0.124 0.89 1.55 111 dB

6 75.3 543 0.091 0.41 1.02 94 dB7 75.6 578 0.09 0.51 1.06 95 dB

Source: Reprinted from Sheng, G., Liu, K., Otremba, J., Pang, J., Qatu, M., and Dukkati, R., Int. J. Vehicle Noise Vib., 1,

1=2, 2004. With permission from Inderscience.



wet belt creates perceptible noise at low slip speed, which is always associated with the steepnegative slope. The belt wet slip noise is basically attributed to the self-excited belt vibration underthe negative slope of friction–velocity.

Usually, torque overload could lead to belt tangential slippage, and the slippage could beincreased remarkably when the belt is used in wet condition. Existing experiments show that astrong noise, called wet belt slip noise, could occur for some wet belt applications under slippagecondition, which usually occurs suddenly and jumps to a much higher sound pressure level whenthe system running conditions are changed gradually.

Typical friction features of wet belts were experimentally characterized by directly using modi-fied SAE belt friction=slip test rig shown in Figure 6.22. The wet belt is simulated by continuouslyapplying water with a rate of 300 mL=min into the interface. The belt tension is set at 180 N, the testpulley was applied 800 rpm with 458wrap angle. The torque load varies from 0 to 20 Nm. During therunning process, when the resistant torque of driven pulley increases to a high enough level, beltslippage could take place. Figure 6.43 shows the measured coefficients of friction as a function ofinterface slip velocity for two typical belts under wet conditions. Usually between the wet and drycase, there is a remarkable difference in the frictions of belts. For most of tested belts, wet belts exhibitthe following features: (i) the wet belt dynamic friction is significantly smaller than the dry case; and(ii) the dynamic friction always has a negative slope on the friction–velocity curve. Basically, the wetbelt interface always has a negative slope of friction–velocity curve, whereas the normal dry belt

COF–slip1.2

1.0

0.8

0.6

0.4

0.2

0.00

(a)2 4 6

80

90

100

COF

SPL (dB)

110

SP

L (d

B)

CO

F

120

Slip speed (m/s)

COF–slip1.2

1.0

0.8

0.6

0.4

0.2

0.00

(b)

2 4 680

90

100

COF

SPL (dB)

110

SP

L (d

B)

CO

F

120

Slip speed (m/s)

FIGURE 6.43 Measured coefficient of friction and noise as a function of slip velocity for two kinds ofwet belts. (a) Withoug stiction. (b) With stiction. (Reprinted from Sheng, G., Miller, L., Brown, L., Otremba,J., Pang, J., Qatu, M., and Dukkati, R., Int. J. Vehicle Noise Vib., 2, 3, 2006. With permission fromInderscience.)



usually has at least a positive slope region at relatively low slip velocity range. The coefficient offriction of belt could reduce by 200% at high slip velocity when the belt is wet. This essential featureholds for most of tested belts. This shows that water mediation plays a critical role in wet belt interfaceand changes the friction essentially. Figure 6.44 shows the measured coefficient of friction as afunction of slip velocity for belt slip acceleration and deceleration processes. It is interesting that thetwo curves are almost identical at high slip velocity, and have a huge difference at low slip velocity,which shows that wet friction has a strong ‘‘hysteretic’’ feature.

Based on extensive experiments on the friction properties of wet belt–pulley interface, thetribological property of the interface is characterized to be unsteady mixed lubrication with water,which possesses the feature of negative slope of friction with velocity as shown in Figure 6.43. InFigure 6.44, the investigation shows that the friction in the deceleration process exhibits weakerfeatures than that in the acceleration process.

While wet, the way in which force is transferred from the belt to the pulley is changed. Underthe wet condition, the force is transferred through a combination of a fluid and solid interface. In thiscondition, the dynamic coefficient of friction exhibit negative slope trend. Stiction also contributesto wet slip associated instability and noise. Stiction is where a fluid suction force acts in a mannerthat results in a higher static coefficient of friction. When the belt begins to move relative to thepulley, the stiction force is negated, the effective COF drops, and the system becomes unstable.

By comparing the experimental results of Figures 6.43 and 6.44 with the fundamentalprinciples presented in Chapter 3, we conclude wet belt interface exhibits the unsteady mixedlubrication and some boundary lubrication. The lubrication takes place in the mixed or boundaryregimes, such that the frictional resistance is developed due to a combination of fluid shear overentire contact and solid friction at localized asperity contacts. In mixed lubrication, the load iscarried primarily by the fluid with solid-to-solid interaction also contributing significantly to thefriction. In boundary lubrication, the load is carried primarily by the solid asperities with the fluidshear still contributing significantly to the friction. Conventionally, the boundary lubrication isimportant because of its role in playing stick–slip. However, for both wet belt and wet rubbercompound interfaces, the measured static frictions are only limitedly higher than dynamic

COF–slip

CO

F

COF

SPL (dB)

1.2

1.0

0.8

0.6

0.4

0.2

0.00 1 2 3

Slip speed (m/s)

SP

L (d

B)

4 5 680

90

100

110

120

140

160

FIGURE 6.44 Measured coefficient of friction and noise of wet belt as a function of velocity, during slipacceleration and slip deceleration process. (Reprinted from Sheng, G., Miller, L., Brown, L., Otremba, J., Pang,J., Qatu, M., and Dukkati, R., Int. J. Vehicle Noise Vib., 2, 3, 2006. With permission from Inderscience.)



frictions; thus the stick–slip may not be a concern. However, the boundary lubrication determinesthe highest friction that accordingly dominates the initial magnitude of negative slope of friction.On the other hand, the mixed lubrication determines the overall trend of negative slope of friction–velocity curve. In mixed lubrication regime, the friction is a function of velocity because thephysical process of shear in the junction changes with velocity. Details of the friction–velocitycurve depend on the degree of boundary lubrication and the details of mixed lubrication. In Figure6.43, case (a) arises when water provides a little or no boundary lubrication. When boundarylubrication is more effective, the case (b) takes place, where the friction remains a higher constantvalue until a higher velocity at which mixed lubrication begins to play a dominant role, and thefriction starts to drop.

Water film stiction could be another reason why wet belt friction at low slip velocity is largerthan the dynamic friction at higher velocity. The liquid involved interface at zero or very small sliprate has a higher suction force called ‘‘stiction’’ due to film junction, which is partially due to themeniscus force. This extra interface suction force could enhance the friction at low slip rate. Theextra adhesion force caused by water mediation can be estimated using the formulation presented inChapter 3.

6.6.2 WET BELT SLIP NOISE

The wet belt slip noise could occur for some kinds of belts under special conditions. It usuallyaccompanies comparatively larger negative slope at low slip speed or wet start-up where slip speedis very small. To investigate wet slip noise, the sound pressures are recorded by a microphonesystem attached to the SAE belt friction=slip test rig shown in Figure 6.16. During the belt runningprocess, when the resistant torque of driven pulley is increased to a high enough level, wet beltslippage and noise could be triggered.

The measured friction and sound pressure levels of slip noise vs. belt slip velocity are alsoplotted in Figures 6.43 and 6.44. Wet belt slip noise occurs only at relative low slip velocity. Onlysome wet belts with relatively larger negative slope of friction–velocity correspond to the noiseonset. Both slip acceleration and deceleration processes could trigger wet noise. Slip noise usuallyoccurs at proper condition combinations when test rig changes testing conditions (both resistanttorque and slippage velocity). Wet slip noise always occurs rapidly with a sound pressure level jump

120.00

80.00

40.00

0.000.00 4,000.00 8,000.00

Frequency (Hz)

SP

L (d

BA

)

12,000.00

10,000

−50

0

50

5,000

0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

0.01 0.02 0.03 0.04

Time (s)

Time (s)

Pre

ssur

e (P

a)

0.05 0.06 0.07

80

60

40

20

0

−20

Fre

quen

cy (

Hz)

(a)

(b)

(c)

FIGURE 6.45 (a) Wet belt slip noise spectrum; (b) waveform; and (c) and time–frequency spectrum. (Courtesyof Gates Corporation.)



of up to 10 dB, and then dies down, despite the gradual changing of running conditions. It is shownthat the noise is directly associated with the negative slope of the friction–velocity curve. In Figure6.44, the friction in the deceleration process exhibits weaker features than in the accelerationprocess. However, both processes could generate noises. Figure 6.45 shows the wet belt slipnoise spectrum, waveform, and time–frequency spectrum. The wet belt noise has a lower frequency(about 400–2000 Hz and its harmonics) than dry slip noise. Another distinguishing feature is thatas the fluid is eliminated from the system and returns to the dry state the wet noise vanishes (seeFigure 6.46).

6.7 TIMING BELT FRICTION-INDUCED NOISE

6.7.1 TIMING BELT FRICTION

Timing belt is one kind of synchronous belt which is widely used for automotive engines asdriving parts of valve train mechanism. Timing belts are positive drive elements which transmitpower between shafts through the action of molded teeth in a performed sprocket. The belts arecomposite in nature, with a tensile cord to carry the extensional loads on the belt, an elastomercompound forming the backing of the belt and the bulk of the belt teeth, and a polyamide facing

Time history

0 0.02 0.04 0.06 0.08Time (s)

Time–frequency spectrum FFT Spectrum

Frequency (Hz)

1

0.5

0

−0.5

0.04

0.02

015,000

10,0005,000

0 0

0.05

0.25

0.2

0.15

0.1

0.05

00 5,000 10,000

Frequency (Hz)

Time (s)0

0

5,000

10,000

Time–frequency spectrum

Fre

quen

cy

Nor

mal

ized

sou

nd p

ress

ure

Am

plitu

de

0.02 0.04 0.06

15,000

Time (s)

Am

plitu

de

0.1

−1

FIGURE 6.46 Wet belt noise waveform and spectrum. (Courtesy of Gates Corporation.)



fabric covering the belt face in contact with the sprocket, to reinforce the belt teeth and protect thebelt wear. It has many general applications with the largest market being for automotive camshaftdrive belts.

The friction force acting on belt groove between two belt teeth was found to relate to thedeflection of the belt groove in tangential direction by Karolev and Gold [93]. The friction forceis equal or smaller than the maximum friction force derived from Euler’s formula. Underinitial tension, wrap angle u is equal to the pitch angle a. But in a drive transmitting power theactual wrap angle is somewhat smaller. Figure 6.47 shows a sprocket and belt tooth loadsand friction forces at a fixed tooth position in the course of one pulley revolution. During themotion, the slip motion of belt relative to sprocket is unavoidable; this could cause friction-induced noise.

6.7.2 FRICTION-INDUCED NOISE

Timing belt noise is low in level but it is an unpleasant noise. With improvement in quietness ofvehicles, it is necessary to reduce the timing belt noise for higher quality of vehicles. Therehave been many studies on timing belt noise with belt unit testing machines and engines. Tokoro,Nakamura, and Sugiura [110] demonstrated that the noise consists mainly of the low-frequencycomponent occurring at the meshing order and the high-frequency component at about 5 kHz.There are some studies on intermittent noise above 5 kHz. The high-frequency noise occurs once pera pitch of meshing and over a wide range of engine speeds. It was found that the high-frequencynoise occurs mainly from the friction at the beginning and end of meshing between the belt andsprocket.

It was identified that the portion of high-frequency noise is generated by the discontinuousslips between belt and sprocket. Improved meshing smoothness can reduce this type of noise.

6

4

2

0

1 9 17

SBIIQBIIRBII

Fu

25 33

Step number k

Pul

ley

toot

h lo

ad S

BII

(N)

Bel

t too

th lo

ad Q

BII

(N)

Fric

tion

forc

e R

BII

(N)

41 49 57 65

−60

−40

−20

0

Tot

al c

ircum

fere

ntia

l loa

d F

u (N

)

20

40

60

−2

−4

−6

FIGURE 6.47 Pulley and belt tooth loads and friction forces at a fixed tooth position during one pulleyrevolution. (From Karolev, N.A. and Gold, P.W., Mech. Machine Theory, 30, 4, 553–565, 1995. Withpermission from Elsevier, Copyright 1995, Elsevier.)



The fundamental frequency of friction noise is very high and is as high as 10 kHz for polyurethanebelt and above 4000 Hz for rubber belt, and changes its value when the speed, tension, width,and length of the belt change. It depends mainly on belt materials and geometry. The noise orderspectrum of polyurethane belt in Figure 6.48 shows that the predominant frequency of the noiseis in the range of 5–8 kHz and its level is high, which is independent of rpm. The measuredtransverse vibration of belt using laser velocimeter and displacement transducer from the back faceof the belt cannot detect any high-frequency component above 5 kHz as shown in Figure 6.49.This indicates that the high-frequency noise has nothing to do with the transverse vibrations [110].

Meshing noise

Frequency (kHz)

Eng

ine

spee

d (r

pm)

0

1 k

2 k

3 k

4 k

2 4 6 8 10

High-frequency noise

FIGURE 6.48 Order spectrum of timing belt noise.

Meshing order

Frequency (kHz)

0

1 k

2 k

Eng

ine

spee

d (r

pm)

3 k

4 k

2 4 6 8 10

FIGURE 6.49 Order spectrum of transverse vibration.



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7 Friction-Induced Vibrationsand Noise in VehicleBrake System

7.1 BRAKE STRUCTURE, MATERIALS, VIBRATION, AND NOISE

7.1.1 BRAKE STRUCTURE AND MATERIALS

Brake systems have been widely used in automobiles, motorcycles, rail vehicles, and aircrafts. Inbrake system, friction is both a principal performance factor and a potential cause of undesirablenoise and vibrations. The structures and principles of varied brake systems in different vehicles areanalogous and similar. The friction-induced vibration and noise have been one of the mostchallenging problems in brake industry [1–36].

Friction-induced vibrations and noise impact on the reliability and quality of brake systems inmany ways. For instance, the brake squeal has been the number one challenging issue in automotivebrake system, as it has been percepted to be equal to the quality of products by customers. It wasestimated that the noise and vibration cost approximately 1 billion=year in warranty work in Detroitalone. In aircraft braking systems, friction-induced torque oscillations can lead to excessivelyhigh loads in the landing gear and brake structure. This results in passenger discomfort and=orcomponent failure and thereby warranty claims.

There are two major types of brakes, the disk brake and the drum brake. Disk brakes have beenwidely used in passenger vehicles, while drum brakes are predominantly used in trucks, buses andsome rare applications of some passenger vehicles. We focus our discussion on the disk brakes inthis chapter.

As illustrated in Figure 7.1, a disk brake system consists of a rotor that can rotate about thewheel axis. The caliper assembly in brake system is mounted to chassis. The caliper housing canslide on the bracket. The pads can also slide on the anchor bracket and the piston can slide inside thecaliper housing. When hydraulic pressure is supplied, the piston and caliper press the pads to contactagainst the rotor thus generating braking torque through friction.

The brake pad assemblies consist of a friction material which is mounted to a steel backingplate. The mounting can be achieved by using rivets, using adhesives or integrally moldingthe friction material to the backing plate. The two brake pad assemblies in a single disk brake areusually distinguished as inboard and outboard. These assemblies are assembled to the caliper orthe caliper-mounting bracket in various ways. It is common to use squeal prevention measureslike pad insulators (or shims) and chamfers.

The rotors usually made of grey cast iron have dominated the market. In the last couple of years,other materials like SiC-reinforced aluminum, carbon–SiC composites, and sintered carbon havestarted to emerge.

Most of the pad friction materials are organic materials that are based on a metal fiber reinforcedorganic matrix. Other types of friction materials include metallic, semimetallic, and carbon based.Organic pads are a compound of a number of different materials (could be up to 25 kinds) with the


347

functional groups of binder for adhesion, structural materials for strength enhancing, fillers forregulatingcost andvolume,and frictional additives for frictioncontrol.Thebinder ormatrix isnormallysome kind of phenolic resin. The fibers include nonasbestos organic (NAO) or resin bonded metallic.

The structural materials include metal-, carbon-, glass-, and=or kevlar fibers that are added to thelinings to give mechanical stability. Fillers are principally used to make the material less expensiveand=or to improve the main material. The frictional additives are added to control the coefficient offriction (COF) or to change the type of wear. Lubricants such as graphite are used to decrease thecoefficient of friction. Abrasive particles are used to increase the coefficient of friction or to increasethe brake disk wear with the purpose to remove iron oxides from the disk surface, thus providingmore consistent braking properties.

7.1.2 FRICTION-INDUCED VIBRATION AND NOISE IN BRAKE SYSTEM

Almost all of the power dissipated during abrake stopoperation is in the formof heat; usually less than athousandth part is radiated as sound. However, even this small portion can cause intolerable noise.

In braking, the rotor and the pads experience normal contact force and tangential friction forcesat their interface. These forces could interact and develop complicated vibrations. Even assumingfriction force is constant, the tangential forces acting on the surfaces of a rotor can developin-plane vibrations and the normal forces can develop out-of-plane vibrations. A combination of in-plane vibrations and out-of-plane vibrations can change the contact area thereby changing thepresumed constant behavior of the friction force. Oscillatory normal force components alwaysaccompany the real oscillatory friction force, thus yielding bending waves and transverse vibrationsin the rotor and pads.

Pad vibrations can easily travel to the brake caliper and beyond, and cause caliper and otherchassis subsystem resonances. Over the years, disk brake vibration and noise have been givenvarious names in terms of the pattern of the sound generated such as moan, groan, judder, squeak,squeal, and wire brush.

A simple definition is to classify the brake noise to be low-frequency noise and squeal,depending on whether the noise frequency is lower or higher than about 1 kHz. The low-frequency

CaliperPiston

Pads

Rotor

FIGURE 7.1 Schematic of an automotive disk brake system.



noise mainly consists of groan and judder. This category of noise is caused by friction materialexcitation, and the energy is transmitted as a vibrational response through the brake corner andcouples with other chassis components.

Squeal is an annoying, usually close to a pure-tone, high-pitched noise. The squeal includeslow-frequency squeal and high-frequency squeal. Low-frequency squeal is generally classified as thesqueal having a narrow frequency bandwidth in the frequency range above 1000 Hz yet below the firstcircumferential mode of the rotor. The high-frequency brake squeal is defined as noise which containsthe first circumferential mode of the rotor, which is produced by friction-induced excitation, usuallyinvolves in coupled mode resonances with squeal occurring at frequencies above 5 kHz.

A typical groan could have a spectrum ranging from 10 to 50 Hz, their harmonics extending to 500Hz. It occurs at low speeds under moderate braking conditions. A groan is caused by stick–slip whichexcites the low-frequency resonances brake system. In particular, resonances of the rigid-rotation modeof the caliper and the local suspension develop and radiate sound without the participation of rotor.

Judder is due to rotational speed dependent, continuous displacement pulsations between therotor and pads, which could magnify itself as a low-frequency vibration whose frequencies areinteger multiples of the rotational speed of the wheel. It can transmit to the chassis and steering andexhibit to be vibration and harshness rather than noise. The conditions that lead to judder generallyresult from modulation and nonuniform friction force between the rotor and pads. The nonunifor-mity results from circumferential disk thickness variation (DTV) and variation in surface finish.

A low-frequency squeal usually contains first four nodal diameters vibrational mode of rotor,which has nodal spacing larger than the pad length, and can also involve the resonance of certainmode of the caliper bracket. Whereas a high-frequency squeal usually contains certain circumfer-ential mode of rotor. The mode lock in some natural resonant mode due to velocity dependentfriction, or the mode coupling of two close different modes, could be the typical mechanism ofthe squeal. The low-frequency noise is sometimes more likely attributed to the friction speeddependence (due to the difference between static and kinetic friction coefficients and=or negativeslope of the friction coefficient with respect to sliding speed).

In general, squeal could be due to the effect of mode lock-in or mode coupling or itscombinations with the effects of negative friction slope, and even sprag–slip, nonsmooth effect(friction reversal involved), parametric resonance, bouncing and impulsive rubbing effect offriction. Squeal is usually high pitched with nearly fixed frequency, which could be close to oneof the natural frequencies of a component, or in between of two natural frequencies of one or morecomponents (coupling), or even their harmonics.

Squeal could exhibit nonstationary feature due to the nonstationary of instantaneous friction andcontact effect and give rise to limit cycle due to friction motion excitation. Squeal usually has afundamental frequency, which can be assumed to be the vibrational mode of the instantaneously formedsystem through friction=contact coupling. It is an instantaneousmode and usually named as squeal mode.

To provide a precise explanation to specific squeal phenomena, we may need to use certainabove mechanisms. Which mechanism is dominant depends on the components, system=interface,operational conditions, environmental conditions, aging of the brake, and some stochastic factors.

7.1.3 GROAN AND JUDDER

In brake community, friction-induced low-frequency vibration and noise are detailed by judder,groan, and moan, which correspond to be around 10, 100, and 1000 Hz, respectively.

Groan denotes unpleasant high intensity, low-frequency vibration (20–200 Hz), and noisegenerated by the brake systems at very low speed, either accelerating or decelerating, with morefrequently decelerating. Some brake run-in burnishing is usually necessary before groan will occur.The noise could produce through the entire stop process in the most severe cases. Usually, it isproduced from the middle to the end of the braking event. The caused motion of the brake corner isrigid-body twisting of the caliper housing, caliper bracket, and knuckle.


Friction-Induced Vibrations and Noise in Vehicle Brake System 349

While groan is rarely a symptom of any brakemalfunction, it poses a serious customer satisfactionproblem which often results in costly warranty claims. The phenomenon may exhibit when only alight pressure is applied by the driver on the brake and some driving forces are acting on the vehicle. Itoccurs when there is simultaneous application of torque to the wheel and gradual release=increase ofbrake pressure, eventually the wheel torque load breaks the friction between the pad and rotor causingslippage and energy release. If the torque load is not large enough to maintain slippage, sustainedstick–slip vibrations can occur which transmits a low-frequency noise to the vehicle interior which isthe percepted groan. In some cases, the noise levels caused by groan can be quite objectionable.

The root cause of the groan is the stick–slip of the brake pads at the rotor surface due to frictioncoefficient–velocity dependence, or static coefficient of friction higher than the dynamic value. Theeffective brake friction material is likely to exhibit these types of characteristics.

The groan of the low-frequency vibration involves rigid-body oscillations of the full assembly,which includes brake pads, caliper, and knuckle through which the system is mount on chassis, it isamplified and emitted by the chassis components (axle and=or suspension).

Sometimes, the above described groan is specified as ‘‘Creep groan,’’ so as to distinguish withanother ‘‘dynamic groan,’’ which occurs in high temperature stop, caused by forced vibration relatedto rotor Vane-pitch. Dynamic groan is order based and rarely occurs.

The spectral content of the groan vibration shows response at multiples of a certain fundamentalfrequency. Bettella, Harrison, and Sharp found that the stick–slip vibration typically occurs at thefirst, second, or third order of the fundamental frequency [37]. Figure 7.2 shows a typical case. Thiscondition is common in automatic transmission vehicles since wheel torque is present at idle, but itcan also occur with manual transmission vehicles that are stopped on an incline.

Another type of low-frequency brake noise is called moan due to stick–slip that is similar togroan in principle. It is typically a low-frequency pure-tone noise of several hundred Hertz. It can beconsidered as an ultra low-frequency ‘‘squeal’’ because it is similar to squeal in many ways. It couldbe either a system instability issue or resonance of certain low-frequency mode of brake assembly.

Brake groan=moan can be very difficult for a brake supplier to both detect and design, because itis often dependent upon factors outside direct control of the brake supplier, such as axle andsuspension configuration. A successful investigation of this problem requires a system-level analy-sis, including elements of the suspension as well as the foundation brake components. For instance,one of the effective countermeasures of moan is called ‘‘moan brace,’’ meaning stiffening the axlerotational degree of freedom.

Since it is the friction material characteristics that make the system prone to a stick–slipbehavior, most engineering efforts up to date have been concentrating on the characteristics betweenthe friction material and the contacting surfaces and its effect on creep–groan vibration.Reducing the magnitude of the negative slope of the friction curve can significantly reduce the

50

40

30

20

10

0

−10

−200 60 120

Frequency (Hz)

Pow

er s

pect

rum

(dB

)

180 240 300

FIGURE 7.2 Power spectral density function. Vertical acceleration measured on front (solid line) and rearcaliper (dotted line) during creep–groan tonal phase.



overall vibration levels. System mode frequencies usually do not shift, although the frequency ofstick–slip vibration can jump to a different order.

Brake judder (roughness or shudder) can be roughly categorized into three major types. The firstis the disk thickness variation (DTV) type in which aggressiveness=wear of brake friction materialgenerates unveven wear on the rotor surface. The second is the corrosion-type brake judder in whichtorque fluctuates due to corrosion-induced variation and surface damage. The third is the hot juddertype in which torque fluctuates due to disk rotor distortion caused by high temperature (braketemperature >2008C). Other types include green roughness, new brake wet roughness, water soakedhigh-speed judder, etc. Brake judder is the vehicle vibration, jerkiness, or pulsation that is felt by thedriver and=or passenger, which is usually sensed through steering wheel (shake or nibble), floor-board or dash panel, and=or car seat brake pedal during stop process. It is amplified by the resonancemodes of the vehicle body (vibration transmission system) in relation to hydraulic pressure or torquefluctuations. It is created by repetitive variation in torque output from the brake, typically in the5–60 Hz range frequency which varies with wheel speed typically 1 to 2 pulsations per revolutionbut can reach up to 10. Although all types have been observed to cause torque variation, the mostserious case could be caused by brake pressure fluctuation mainly due to DTV.

The uneven rotor wear caused by varying pad contact while driving at highway speed is themain factor influencing on creating brake judder. There can be selective wear on either one side ofthe rotor or both sides of the rotor. Typically, DTV in the range of 15–20 mm can create torquevariation in the range of 50 Nm. This normally is sufficient to create vehicle judder.

DTV is usually caused by uneven wear or combined effect with corrosion of the disk rotor, whichprones to yield a vibration called cold judder. In brake systems, frictional heat is not uniformlydistributed due to many reasons like nonuniform contact, imperfections in geometry as well asthermal expansion. Moreover, the thermoelastic distortion due to frictional heating could furtheraffect the contact pressure distribution and likely lead to thermoelastic instability by which the contactload is concentrated in one or more small regions on the brake disk surface. These local contactregions reach very high temperatures and the passage of these hot spots moving under the brakepads can cause low-frequency vibration called brake hot judder. The nonuniform contact can alsobe generated by disk transverse runout. This type of judder occurs at higher orders of wheel rotation(6–12 have been typically noticed). The mechanism of this type of judder is considered to be thevariation of the friction coefficient around the rotor. This variation is caused by chemical changesoccurring to the friction couple when intense heat is generated at the friction interface. Visualobservation of hot or blue spots around the rotor is common. Even after cooling one can notice thiseffect by the color of the rotor surface. The order of vibration is related directly to the number of hotspots. The engineering development efforts for judder reduction include the control of characteristicparameters of the friction material and the improvement of brake assembly and vehicle sensitivity.

7.1.4 LOW-FREQUENCY SQUEAL

The low-frequency squeal can be associated with modal lock-in or modal coupling of brake cornercomponents due to frictional excitation. Many cases are due to the coupling of two or more modesof various structures including caliper, pad, and rotor.

A typical case on low-frequency squeal noise was attributed as the modal coupling of the caliperand rotor. The friction variation between pads and rotor induces the rotational vibration. Therotational vibration of caliper leads to longitudinal resonance of caliper, which results in pressurevariation of pads. The pressure variation could further increase friction vibration between pads androtor. Thus, a feedback mechanism is established and the caliper system exhibits self-excitation andmagnifies itself leading to squeal noise. Thus, separating the related mode of rotor and caliperlongitudinal mode can avoid the coupling and the vibrations. This leads to the solution of decouplingof caliper and rotor modes by changing the rotor material from gray cast iron to ‘‘damped’’ iron.

The low-frequency squeal could be pad induced, caliper-bracket induced, or both. The padinduced denotes the vibrations are triggered by out-of-plane motions of the pad ends (pad-end



flutter), which excite the out-of-plane vibration modes of the rotor. A typical case is the pad-induced out-of-plane squeal (4–11 kHz), which is aligned with the out-of-plane modes of the rotor(15 and 16 in. brake corners with cast iron rotors). Another typical case could be the caliper-bracketinduced squeal (2–6.5 kHz) for some floating caliper design.

The low-frequency squeal could occur during low brake deceleration at about 5 ft=s2, vehiclespeeds of 5 to 10 mile=h, and initial pad temperatures of 308F to 408F. Low-frequency squealcould occur under special conditions such as very cold and=or high humidity environment, and theoccurrence rate is less than the high-frequency squeal. This type of squeal could occur in the morningbecause of the overnight environmental conditions, and it is called morning sickness=squeal. Coldsqueal tends to appear at low speed under low temperature.

7.1.5 HIGH-FREQUENCY SQUEAL

A high-frequency squeal typically involves the higher order disk modes, with 5 to 10 nodaldiameters. The squeal frequencies usually range between 5 and 15 kHz. Nodal spacing betweenthe excited modes is comparable to or less than the length of brake pads. For instance, a typicalhigh-frequency squeal takes place in 30 revolutions per minute, 1000 kPa line pressure, 1508C bulkrotor temperature. The system can squeal at a number of distinct frequencies.

An illustrative example given by Baba et al. [38] is shown in Figure 7.3, the out-of-plane modeof typical rotor disk exists throughout the audible frequency range, for instance, the 2 nodal diameter

FIGURE 7.3 Rotor disk frequency response function of out-of-plane and in-plane modes and their modalshapes close to squeal mode (6900 Hz). (Reprinted from Baba, H., Wada, T., and Takgi, T., SAE 2001-0103158, 2001. With permission from SAE.)



mode near 1 kHz and the 12 nodal diameter mode near 15 kHz. On the other hand, the modaldensity of in-plane modes is much lower. The first three circumferential (tangential) modes of therotor disk consist of 1, 2, and 3 nodal diameters. A circumferential mode of a disk can be viewed asa compression wave in the disk circumference. From Figure 7.3, the rotor disk frequency responsefunction data illustrates the strong relationship between the resonance frequency of the in-plane modesand the frequency of squeal. There is an in-plane primary eigenvalue at 7200 Hz that was close to thefrequency of 6900 Hz at which the brake squeal mainly occurred. The squealing frequencies areusually slightly lower than the natural frequencies of the stationary rotor. Increased coefficient offriction between the rotor and pads of friction material increases the propensity for squeal.

In most cases of high-frequency squeal, the squeal frequency relates to an in-plane resonancefrequency of the rotor disk, not an out-of-plane resonance. The frequencies at which high-frequencynoise occurs are related to the in-plane modes of the rotor disk. Unlike the pure in-plane modeof disk described inChapter 2, the disk–hat shape of rotor renders its in-planemode to have out-of-planecomponents, thus it can radiate noise. Sometimes, the cross coupling of an in-plane and out-of-planemodes of the rotor or an in-planemode of rotor and another mode of the brake system could occur.

Some existing work illustrated the squeal’s good correlation with the in-plane circumferentialmodes despite not locked precisely in its frequency, irrelevant with the out-of-plane diametric modes,and despite the out-of-plane modes usually providing more efficient way for sound radiation. It isattributed to the velocity dependence of friction triggering the in-plane circumferential modes, whichleads to the out-of-plane vibration and sound radiation through normal force coupling. In some othercases, the direct in-plane and out-of-plane mode coupling could occur and lead to squeal.

Basically, high-frequency squeal is a result of the excitation of the in-plane modes of the rotor.The basic types of tangential rotor modes are compression whose primary controlling parameter is thedisk diameter. For the family of rotors used in automotive disk brakes (15 and 16 in. brake cornerswith cast iron rotors), the first three in-plane modes are in the range of 6–7, 9–11, and 14–16 kHz,respectively.

For instance, a high-frequency brake squeal was characterized as an intermittent squeal thatoccurs once or twice per revolution of the brake rotor for short time duration. The squeal eventstended to occur at different physical locations on the brake rotor between tests, and the loudness foreach event varied with changes in brake operating conditions. The amplitude of vibration ofsquealing brakes is on the order of microns. The vibration of a squealing spinning rotor could bea standing wave or a traveling wave. For instance, a 13 kHz brake squeal was characterized as anintermittent squeal that occurs once or twice per revolution of the brake rotor for short time duration.The squeal events tend to occur at different physical locations on the brake rotor between tests.These squeal characteristics are depicted in the acceleration time history shown in Figure 7.4 [39].

Acc

eler

atio

n (m

/s2 )

500

250

0 0.1 0.2Time (s)

0.3 0.4 0.5

−250

−500

0

FIGURE 7.4 Intermittent squeal events with amplitude fluctuations over time.



Mode lock-in of in-plane mode, or mode coupling between in-plane and out-of-plane, or simplyrubbing excitation is the inherent characteristic of the rotor for a brake to generate high-frequencysqueal. However, this alone is not sufficient. It needs the right friction, pressure, and temperature tomake them lock-in, couple, or direct exciting to trigger squeal. Proper combination of geometric=structural=material parameters and assembly configurations as well as operational conditions giverise to proper threshold for the brake to generate squeal.

So far there is no determined approach to determine the squeal threshold to evaluate the mode ofthe components of a disk brake system which can be used to infer a squeal propensity. The time-dependent nature of friction properties, manufacturing tolerances with respect to flatness andparallelism of rotor surfaces, and variability of material properties, all of them render the quantita-tive prediction of squeal as elusive. A recommended approach to deal with squeal is to look at theentire brake corner as an interactive system and address all these interactions instead of sticking ononly one component, sole factor, or identical mechanism.

Random impulsive excitation combined with self-excited system yielding a nonstationaryoscillation or instantaneous mode solution constitutes the basic characteristics of brake noise.Generally, the rotor is the noise radiator. The driving factor is the magnitude=slope of the frictionforce under specific operation and environmental conditions. The overall dynamic characteristicsof the caliper=pad=rotor system determine the squeal instability sensitivity. For low-frequency noise(groan and judder), axle assembly subsystem as well as brake assembly may need to take intoaccount.

7.2 NUMERICAL APPROACHES AND ANALYSIS OF BRAKE NOISE

Analytical and numerical approaches have been used to simulate different structure design options,material compositions, and operational conditions for brake system design. With these approaches,the design and prototype can be optimized to reduce the likelihood of vibration and noise occur-rence. They also help set up experiments and help interpret experimental results, allowing topinpoint the NVH failure root cause. Both analytical=numerical approach and experimentalapproach are indispensable tools for understanding and improving squeal issues in brake.

Analytical and numerical methods can be classified as three categories: real mode analysis,complex eigenvalue analysis in frequency domain, and transient analysis in time domain. The threemethods have been used to model and analyze brake noise mechanisms in automotive industry.

7.2.1 REAL MODAL ANALYSIS

Real modal analysis can be implemented readily using widely available tools for structural dynamicanalysis. Real mode method analyzes the natural frequencies and modes of rotor, pads, caliper, andanchor bracket components. The obtained natural frequencies of the components are then checkedfor alignment and=or coupling to other components. If the components had close natural frequen-cies, the propensity of instability or squeal through coupling would be considered.

It can provide a quick approach in investigating the vibrations of components. The noncrossingof the eigenvalue is considered to be linked to the symmetric properties of the eigenvalue functions.By using the basic concept of modal lock-in or coupling, it simply assumes that noise could betriggered either when a natural mode is locked-in through velocity dependent friction, or majorcomponents have close modal frequencies, or when certain rotor tangential and normal vibrationfrequencies are aligned with each other.

7.2.2 COMPLEX EIGENVALUE ANALYSIS

To take into account of more parameter details in noise analysis, like incorporating the effectof friction or damping, complex eigenvalue analysis and transient analysis have to be used [40–61].



In the complex eigenvalue method, friction effects between the pads and rotor are treated as theexcitation mechanism. In the dynamic governing equation of mass, damping, and structuralstiffness, even the negative damping forces caused by friction can be introduced to evaluate theinstability of brake system. Based on the analysis as illustrated in Chapters 2 and 4, a complexeigenvalue problem is usually attained for a system containing contact friction interface. Thetreatment of complex eigenvalue problem yields a linear stability analysis, which is a goodapproximation in the vicinity of an equilibrium point during steady contact sliding.

Despite the results of complex eigenvalue analysis can be used to specify unstable modestheoretically, however, it is not feasible to directly use the results of complex eigenvalue analysisto predict squeal occurrence. Actually, there are many unstable modes for a brake system, whichare only the potential modes for squeal. Each component in brake system has its own natural modes.Even for a typical rotor, the number of modes may be up to 80 below 20 kHz. Theoretically,many of them can be locked-in as instability subjected to velocity dependence friction features.Moreover, the natural modes of the rotor, caliper, anchor, and pad could dynamically coupletogether resulting in a series of coupled vibration modes, which are different from the free vibrationmodes of individual component. The inclusion of the friction coupling forces results in the stiffnessmatrix containing unsymmetric coupling terms. This unsymmetric coupling term could be theroot cause of the brake squeal. Complex eigenvalue analysis usually uncovers a lot of unstablemode, but only very small number or even just one corresponds to real squeal which is subject tovery specified operational condition and environmental condition. This is partially because a minorchange in the interface does not necessarily result in insignificant effects on a global vibration andnoise level for the system. The small variations in operating conditions and environmental condi-tions can lead to significant changes in coefficient of friction, thereby cause differing squealpropensities or frequencies. The brake squeal usually exhibits a scholastic nature which is usuallynonrepeatable. A brake may not always remain squeal for a given condition that is apparentlyunchanged. On the other hand, the squeal always exhibits sustainable and limited oscillationamplitudes, which suggests the occurrence of a limit cycle due to system parameter change isHopf bifurcation. The excitation is fundamentally from motion and friction. The instability couldbe triggered by proper friction or friction properties, and the limit cycle oscillations could bemaintained for a short duration. Overall, it exhibits the combined feature of motion of limit cycleand scholastics as well as nonstationary.

Usually, finite element models are used to modal component with complicated geometry and arevalidated through component modal testing. The complex eigenvalue analysis is used for brakenoise analysis and reduction by identifying and reducing unstable modes. The dynamometer andvehicle tests are used to diagnose and validate squeal occurrence. The complex mode shapes arecapable of depicting the noise generating mechanisms, and the complex eigenvalues are able todepict the noise frequencies.

Real part of the complex eigenvalue, lR, has been used as squeal factor in brake complexeigenvalue analysis [62–72]. From Equation 4.105, the second-order term is the summation of crossmode coupling between the ith mode and other modes. The complex eigenvalue trace is usuallycontrolled by cross mode coupling. Hence, a basic characteristic of brake squeal instability is that itis mainly due to cross mode coupling. Generally, a few strong coupling mode pairs give the largestcontribution to the summation. Brake squeal happens when the strong coupling mode pairs are closeenough in frequency to provide a chance for the two modes to merge and become an unstablecomplex eigenvalue. Thus, identification of mode pairs with high coupling could be an indicationof the possibility for instability. There are two requirements for the cross coupling term to have ahigh value:

(1) Similarity of relative normal and tangential shapeEquation 4.104 shows that the coupling value is large if {�N,r} and {�T,r} have similar

shapes for mode i and l. This means that either the normal force for mode i corresponds



with a relative tangential displacement for mode l and=or the normal force for modelcorresponds with a relative tangential displacement for mode i. The coupling is highestwhen both conditions are true, but coupling can be high even when only one is true. Herethe normal force shape from mode i is similar to the relative tangential displacement shapeof mode l. Thus, another characteristic of brake squeal instability is that the strong couplingmodes have similar normal and tangential mode shapes at the pad–rotor interface.

(2) High normal force shapeIn addition to mode shape similarity it is also believed that the shape for the normal

force itself must be high. This is partly evident from Equation 4.105, and also followsfrom the belief that out-of-plane bending modes of the rotor are the source for theradiated squeal noise. Logically, bending modes of the rotor tend to compress the softerlining and create high normal force. This leads to the assertion that a high normal forceis required.

Generally, the complex eigenvalue results from numerical analysis need experimental valid-ation. As an illustrative example, Figure 7.5 shows the comparison of a tested operational defor-mation shape of rotor and the corresponding mode shape by modeling, which was given by Shawet al. [70]. Figure 7.6 is the tested squeal frequency and the root locus diagram from complexeigenvalue analysis. The complex eigenvalue analysis could provide more than one unstablecoupling modes. But in experiments there are much smaller modes or even one squeal mode.For instance, as shown in Figure 7.6a, the laboratory testing shows squeal frequencies between 3500and 3800 Hz. The analysis showed unstable frequencies at 3712 Hz and approximately 3300 Hz as

ODS FEM Modeling

FIGURE 7.5 Rotor mode shape correlation between test and modeling. (Reprinted with permission fromShaw, P.S., Riehle, M.A., and Kung, S.W., SAE 2003-01-1619, 2003. SAE Copyright 2003.)

12010080

SP

L (d

BA

)

Fre

quen

cy (

kHz)

6040200

0 2 4Frequency (kHz)

(a) (b)

6 8−20 −10 0

Real part of eigenvalue10 20

4

3

2

1

0

FIGURE 7.6 (a) Laboratory testing results and (b) analysis results.



illustrated in Figure 7.6b. Only 3712 Hz correlates with the squeal frequencies observed duringtesting. The instability at 3300 Hz is an overprediction analysis. This is very common in complexeigenvalue analysis. It is the overprediction like this that prevents complex eigenvalue analysis frombeing a true predictive tool. Complex eigenvalue analysis is best used as an analysis tool after theoverpredictions are clarified by test data.

Another limit of complex eigenvalue approach is that the predictions of unstable modes aresensitive to slight changes to the system model. Since small shifts (less than 3%) in modalfrequencies can make the difference in the occurrence of instability, it is easily possible to missthe instability prediction. This sensitivity to condition is also partially the reason why in practicesqueal noise does not always occur with every brake application. A further drawback of the methodis that it does not provide any insight into the instability mechanism.

7.2.3 TRANSIENT ANALYSIS

The complex eigenvalue analysis could give sufficient estimation to linear instability. However, inorder to capture complicated interface parameters in which coefficient of friction is parameterdependent, time dependent, or being of random feature, a nonlinear transient analysis in timedomain is needed. For instance, the transient analysis of finite element models is needed for diskbrake squeal analysis so as to feature a coefficient of friction depending on velocity, interfacetemperature field, interface pressure field, etc. Transient analysis can deal with the system where anexperimentally fitted law of contact force or stiffness between the pads and the rotor is involved.

The transient analysis has the promise of simulating the brake event as real dynamics. However,it is quite difficult to relate detailed design iterations with the result of transient analysis. Thesolution time is also prohibitively long for a comprehensive modeling. The transient analysis hasbeen conducted by incorporating a nonlinear frictional contact mode into finite element model. Asan illustrative example given by Hamzeh et al. [71], the constitutive model of the interface, extendedversion of Oden–Martins law given by Equations 4.23 through 4.27, is applied to deal with anaircraft brake. In the investigations, the statistical information of the surface and the materialproperties of the bulk material are used to define the response of the surface to normal and tangentialloading. A representative asperity is defined as having the tip curvature radius of Ra¼ 5 mm. For thisasperity, elasto-plastic solution is obtained using an analytical solution to a spherical contactproblem. The calculated response of a single asperity is then used in conjunction with a statisticalhomogenization procedure to produce constitutive characteristics of the interface. This includesnormal pressure versus approach, real contact area versus approach, and friction force versusapproach. The approach means the coming together of two surfaces from the point of initial contactof the tips of the tallest asperities. Two of these curves, namely normal pressure and frictionalresistance, are shown in Figure 7.7.

Figure 7.8 shows the phase plots of the normal and tangential direction motions of a selectednode, which lies on the face of the pad=stator in contact with the spinning surface directlyunderneath the inner edge of a piston friction.

It is noted that both oscillations are excited and grown in this case. The strength of instabilitycan be quantitatively compared for different types of instabilities induced either by mode lock due tofriction–velocity dependence or modal coupling.

Due to nonlinear effects in this problem, the oscillations do not grow infinitely, but developinto a limit cycle oscillation. In transient analysis, the time dependence of friction force betweendisk and pads can be readily incorporated into system equation to investigate the generation of low-and high-frequency squeal [73,74]. In a study by Beloiu and Ibrahim [73] both deterministic andrandom friction coefficient are considered. Figure 7.9 shows a typical friction coefficient–velocityrelationship with random component included. The numerical simulation with random friction helpsto reveal a modulated intermittent response of both disk and pad. It is found that the frequencycomponent corresponding to circumferential vibrations of the disk is always dominant in the case of



0

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Penetration (micron)

N/m

2 c

Pressure

Friction

FIGURE 7.7 Normal pressure and frictional resistance as a function of approach.

Vx

(mm

/s)

150

25

−25−0.0025 0.00250

00

−150−0.01 0

Ux (mm) Uz (mm)

0.01

FIGURE 7.8 Phase plots of (a) tangential and (b) normal motions of selected nodes during unstable oscillationsdue to dynamic coupling. (Courtesy of Tworzydlo, W.W.)

Dimensionless relative speed

CO

F

0.6

�0.6

−6000 0 6000

0

FIGURE 7.9 Friction coefficient–velocity relationship. (Reprinted with permission from Beloiu, D.M. andIbrahim, I.M., SAE 2004-01-0826, 2004. SAE Copyright 2004.)



deterministic friction. However, random friction creates impulsive forces that excite the transversemode and has little effect on circumferential vibrations. Figure 7.10 shows a typical time–frequencyspectrum of circumferential disk motion with asymmetric mode excited.

The results show that contact forces create modulated intermittent motions of both disk andpads. The intermittent motion consists of high-frequency oscillations in addition to a low-frequencycomponent. The frequency content is changing with time due to the time dependency of contactforces as the relative velocity was changing with time. The time length of high=low-frequencyalternating motions is found to depend on the system parameters such as damping, brake force, anddisk speed. Intermittent character of the response is also maintained in the case of random friction.

The time evolution of high-frequency component is shown in Figure 7.11. It can be observedthat oscillations with f¼ 6.7 dimensionless frequency are triggered first, and then both magnitudeand frequency increase with time to the dominant peak of f ¼ 8.2.

The time length of low- and high-frequency alternating zones is very much influenced by thebrake force, damping, disk speed, and nonlinear characteristic of friction coefficient. The unstablemode could remain at some fixed frequency component over the entire time domain. The frequencyswitching could occur for some unstable mode. The operational deflection shapes (ODS) of rotorduring the squeal event can be extracted from numerical time-domain data. Moreover, many detailslike the areas with largest displacements on the pads during the squeal event could be depicted.

7.2.4 INTERACTIVE ANALYSIS

Generally, one of the major difficulties in modeling brake noise is to bring different scale problemstogether. Effects on very small scales in length and time (i.e., microscopic contact phenomena andhigh-frequency vibrations) interact in critical ways with effects on larger scales (such as motions anddynamics of large brake substructures). Other major difficulty in modeling brake noise is to properlypredict the invasive process like friction and wear over the life of the brake and quantify frictionwhich is a multiple function with many parameters. Moreover, there is difficulty in modeling high-frequency natural mode of component due to its high sensitivity to minor structure and propertyuncertainty. The large-scale motion (braking operation) could lead to the substantial changes ofinterface in the small scale (surface), but the resultant small-scale change does not necessarily have

frequency

Mag

nitu

de (

dB)

3000

1412

108

6Time 4

2

0 35004000

Frequency4500

5000

0

−50

−100

−150

−200

FIGURE 7.10 Time–frequency spectrum of circumferential disk motion: asymmetric mode. (Reprinted withpermission from Beloiu, D.M. and Ibrahim, I.M., SAE 2004-01-0826, 2004. SAE Copyright 2004.)



substantial effect on the large-scale quantities like frictions, vibrations, and squeal in the next operation.It is this unpredictable change in small scale that makes the accurate modeling and prediction elusive.

The modeling and analysis of brake vibration and noise is a challenging multidisciplinaryproblem in physics from nano-, micro- to macrolevels, involving contact mechanics, nonlineardynamics, acoustics, and tribology. The problems are generally characterized by progressiveinterface due to environmental and operational effects, the time-varying dynamic property, thespatial and timescales, wide frequency range from tens of Hz to 20 kHz.

The modeling and analysis of groan and judder is a challenging engineering problem involvinginterface friction properties, component structural vibrations, and system dynamics including bothbrake system and suspensions. A number of theories have been formulated to explain the mechan-isms of brake squeal, and numerous studies have tried with varied success to apply them to thedynamics of disk brakes. However, none of these models have attempted to include the effects at allscales mentioned above. This has led to models which capture some features of brake squeal welland ignore many others. Experimental studies also tend to have limited applicability with theirresults only pertaining to one brake configuration or to one automobile type.

The experiments have been the most reliable methods for investigating brake squeal. In fact,experiments have been so far considered to be the sole means for verifying any solution to brakesqueal. It has been argued that the finite element method=lumped parameter models are as yetincapable of accurately modeling the complicated boundary conditions and friction mechanismsfound in brake systems, and that the finite element method=lumped parameter models used to modelsquealing brakes simplify the problem to a degree that many of the important parameters areneglected. Nevertheless, the finite element method has become an indispensable tool for modelingdisk brake systems and providing new insights into the problem of brake squeal.

Dimensionless frequency

Wav

efor

mM

agni

tude

Mag

nitu

deM

agni

tude

2X 10−3

10−6

10−5

10−4

3550

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10

3600 3650 3700 3750

1

0

−1

FIGURE 7.11 Magnification of time history for disk circumferential motion and FFT for different timesections. (Reprinted with permission from Beloiu, D.M. and Ibrahim, I.M., SAE 2004-01-0826, 2004. SAECopyright 2004.)



There are many possible mechanisms leading to the noise and vibrations and there is no singlemechanism which can comprehensively capture all of them and put into a unified model. Given thenumber of factors involved in and the wide range of design options, a unified sole model intended tocapture all may be too complex to be useful.

Therefore, the multilayer modeling and interactive analysis incorporating with testing could bean appropriate approach to achieve the analysis and diagnosis goal [75–82]. These might include theinfluence of the variable speed and pressures, the inclusion of thermal and wear effects, and theincorporation of more complex friction models and constitutive models for the pads of frictionmaterial, to ensure that all the important factors are covered. By the interactive analysis, theirinteractions are uncovered and the root cause is identified.

. Multilayer modeling to cover all the possible mechanisms. Using finite element modeling for real modal analysis. Using finite element modeling for complex eigenvalue analysis. Using continuummodel or lumped parameter model to quantify complicated phenomena

under special excitation like sprag–slip, nonsmooth effect, impact effects, etc.. Using proper friction law that is quantified in parameter space to quantify contact and

friction. Define entire operational condition, environmental condition, and extrapolate the aging

condition. Define uncertainty from structure and boundary condition

. Interactive analysis. Conduct comprehensive testing, establish the database of vibration, noise, and tribology

parameters versus environmental and operational conditions and time. Correlate multilayer analysis with testing results. Eliminate the overpredictions and refine the analysis where the correct correlation and

prediction have been attained. Update the analysis and test by interactive procedure

7.3 TEST TECHNIQUE

The conventional approaches dealing with brake noise issues are largely based on trial and errorapproach using experiments. The experiments mainly focus on determining noise by using brakedynamometers testing in lab controlled conditions and on-road vehicle testing. One of the majorexperiments in the lab is to evaluate the system vibration modes or ODS so as to correlate with thesqueal mode and track the rootcause of noise. In addition to the measurement of noise and vibration,experimental techniques are also extended to determine pressure and temperature as well astribology parameters. Recently, industry put on more requirements on noise test, in addition tothe sound pressure level and the occurrence (propensity, frequencies, conditions, etc.), sound qualityanalysis is also required for specific cases.

7.3.1 MODAL TEST AND OPERATIONAL DEFLECTION SHAPE TEST

As the rotating disk does not allow any physical contact with the rotor during its operation, the useof accelerometers is difficult and very time-consuming. Noncontact optical techniques have beenwidely used for these applications [4,83]. Basically, it can provide a very high temporal resolution,but it is still challenging for nonsteady vibrational problems. Recently, the applications have movedfrom one-dimensional deformation determination to the three-dimensional vibration behavior,especially the analysis of both the in-plane and out-of-plane components of the vibration.

Conventionally, experimental modal analysis has been the robust approach to determine systemvibrational modes. But the conventional modal analysis approach needs to determine both system



responses and excitations. To evaluate excitation input is quite difficult for many operationalsystems. Over the last decades, ODS method has been widely developed and applied in noncontactmeasurement. ODS is a technique where modal parameters are estimated solely from response datawithout knowing the input loading force. The ODS estimation approaches are divided into twogroups, frequency-domain decomposition and stochastic subspace identification. The former is anextension of the classical frequency-domain approach, which is based on signal processing usingFourier transformation and the modal transformation. The stochastic subspace identification is anapproach using time-domain techniques based on state-space formulations incorporating with ortho-gonal projection technique, and the modal decomposition is implemented in state-space. ODS is anexperimental method which can be used to determine the deflection shapes of a structure at thefrequency of specific vibration. It is a method that gives the deflection shape of structure underoperation state and themeasured vibration is a superposition of several eigenmodes. The mode shapesdetermined by modal analysis are usually insufficient to describe the deflection of a structure. This isparticularly significant in some cases like the structure vibrating in coupled modes, the structuresubject to unknown excitation forces, or the system being substantially nonlinear.Measurement of theODS usually requires steady-state conditions, and the system does not need to be linear.

Three noncontact optical techniques have been used to measure the ODS of a rotating brakerotor. They are double holographic interferometry, electronic speckle pattern interferometry, andlaser Doppler vibrometry. The double-pulsed laser holographic interferometry has been successfullyapplied to squealing brake systems. This has allowed the coupled mode shapes of a complete brakesystem to be determined while it is squealing. A holographic image is produced by triggering a laserat the maximum and minimum amplitude of a vibrating object. The difference in optical path length,caused by the deformed shape of the vibrating object, creates an interference fringe pattern on aholographic plate. The mode shape can then be determined by interpreting the fringe pattern.

The advantage of holographic interferometry is that the mode shapes of a brake rotor can bedecided while it is squealing. The acquired holographic image can be the rotor as well as the pads,anchor bracket, and caliper. The double-pulsed laser electronic speckle pattern interferometry usestwo laser pulses: one laser pulse is used to acquire the initial speckle pattern of a rotor and the otherpulse is utilized to obtain the speckle pattern of the rotor when it is deforming. Two very short laserpulses with a predetermined time separation are used to illuminate the component under investi-gation including rotating brake disk with engaged caliper components. The images at these two laserilluminations are recorded with a special high-speed camera and analyzed in a computer. Thecalculated result is a deformation map of each point of the components which describes themovement of the individual points of the brake between these two laser pulses.

As such, this technique literally freezes two moments of the movement of the brake and gives aninstantaneous view of the overall deformation. Recent developments have introduced the three-dimensional systems, which provide the complete three-dimensional vibration map of the brake[83]. Special algorithms are used in order to separate the rotation component of the brake disk fromthe in-plane vibrations. ESPI has already been extended to full field measuring of brake squeal toobtain complete maps of the vibration of any component of the brake during test. Brake disks,drums, calipers, and pads have been measured with this technique. Figure 7.12 shows the ESPIin-plane mode and out-of-plane mode measured by Krupka, Walz, and Ettemeyer.

Both ESPI and DPHI can be used to obtain three-dimensional displacement fields of the imagedobject; these techniques can isolate various squeal mechanisms. Scanning laser Doppler vibrometeris a scanning technique in which a relatively high spatial resolution deflection data is obtained bypoint scanning [4,36]. Since squeal is a transient event, laser vibrometry records approximately theODS by repeated transient events. The test requires robust=repeated squeal events during thescanning time which usually lasts for several minutes. A trigger system helps trigger the laservibrometer to scan when a squeal event occurs. It is usually difficult to have a robust squeal event setup which can last several minutes since the squeal condition highly depends on temperature andother conditions. It is a handy technique to measure squeal deflection shape.



Figure 7.13 is the amplitude map of the mode obtained by a laser vibrometer. Figure 7.13ashows the out-of-plane response using in-plane excitation at 6.944 kHz with light spring constraintat one point on the rotor in the out-of-plane direction. It can be seen that the in-plane excitation at thein-plane resonant frequency can induce the out-of-plane vibration of whose shape is very closeto the mode shape at 7.008 kHz. When the rotor is in a brake assembly, the resonant frequency ofthe in-plane mode is shifted lower to 6.720 kHz, as shown in Figure 7.13b where the squeal noisewas detected.

7.3.2 DYNAMOMETER TEST AND VEHICLE TEST

The brake noise performance is affected not only by the structure design, material selections, andvehicle hardware design, but also significantly by driver behavior, the vehicle usage, the state ofadjustment of the brake hardware, and the overall environment in which the vehicle is driven. As such,different level tests to brake are required to quantify the system and to address noise and vibrationissues. Dynamometers and vehicle tests are two kinds of tests used to test brake systems.

Dynamometers are categorized as inertia and chassis types, respectively. The inertial dyna-mometers test the braking system under inertia simulation conditions. This could be mechanical

FIGURE 7.12 Out-of-plane and in-plane deformation of the rotating disk bottom (counterclockwise rotation).

(a) (b)

FIGURE 7.13 (a) Transverse response at 6.944 kHz under in-plane excitation and (b) in-plane mode of rotor at6.720 kHz. (Courtesy of Chen, F.)



inertia simulation via inertia disks or electrical inertia simulation using an electric motor. Chassisdynamometers test the whole vehicle and like inertial dynamometers they could perform inertiasimulation usually of electric type. In addition to test hardware setups, operational conditions, andbrake system variations which could result in significant discrepancy from vehicle testing, the setup ofan inertia brake dynamometer test does not include the suspension system by which the brake isconnected to and it will not capture the influence from the suspension system that may contribute tolow-frequency noise and vibrations.

Inertial dynamometers use one or more shaft-mounted weights to store a given amount of energywhich is dissipated by the brake materials during testing. Such machines vary in size from laboratory-scale, sub-size units to huge, full-size units that can test aircraft, and heavy truck brake components.It usually requires full-size inertial dynamometers to reflect three main characteristics: effectiveness,fade, and recovery. Effectiveness measures the efficiency of braking under different line pressures.Fade refers to the ability to decelerate quickly time after time (multiple applications) without the needto exert unduly high line pressure. Recovery involves multiple stops at lesser rates of accelerationunder a maximum allowed line pressure of certain value.

The inertia disks on dynamometers must be varied to represent different vehicles and differentvehicle ballast conditions. Such disks have the inherent limitation that they can only representdistinct values of inertia. The exact inertia should be approximated correctly. There is a physicallower limit which cannot be violated. The inertia simulation can be realized by using torquefeedback to the motor controller to maintain the proper amount of torque to match the torque thatis present in a given braking situation. The inertia simulation can also be realized by using speedcontrol.

Dynamometer testing was performed to quantitatively characterize the squeal event observedduring vehicle testing and to gain insight into component behavior. A scanning laser vibrometerwas usually used to define the out-of-plane displacement during a squeal event. This test data wasused to validate and update the analysis model. Even though component level correlationwas performed prior to constructing the system model, it is important that the system responsecorrelates with observed test data. However, correlating to the approximate squeal frequency doesnot necessarily imply that the proper mode has been predicted by the analysis model. To ensure thatthe unstable mode from the analysis is in fact the mode associated with the squeal event, the analysisresults were compared with the ODS data obtained during a squeal event by using laser vibrometer.Brake industry has widely applied brake test dynamometers, which are constructed to be able torepresent most aspects of the operation of the vehicle braking process.

Typically, dynamometer has automated control, which is able to perform programmed matrixtest procedure [84–92]. It can conduct drag- and stop-type tests. It consists of appropriate fixtures,microphone and sensor system, data acquisition system, and infrared sensors or thermal couple, withenvironmental control humidity from 5% to 95% RH, temperature from �308C to 608C. Forinstance, to simulate the ‘‘morning sickness,’’ the high humidity and low temperature near freezingor below are usually used. Figure 7.14 shows the schematic of pressure, speed, and sound recordedduring one typical brake stop [86].

In Figure 7.14, if the number of registered squeals, filled circles, is divided by the total numberof sound measurements, circles, for one stop, the squeal index for that stop is obtained. In the aboveillustrated case, the squeal index is defined as 2=7¼ 0.29 [86]. The squeal patterns are usually verycomplicated, different companies have used different indexes.

The modern test dynamometer has become a sophisticated test platform for identifying thepropensity of a brake to generate squeal and diagnose the squeal problem [93–95].

Compared to vehicle tests, dyno measurements provide an easier-to-handle and a more cost-effective approach to determine brake noise characteristics. Conventionally, the dynamometer testprocedure (like AK Noise developed by European brake community) can be designed to follow thelines of the drag mode test procedures. It uses controlled fixed speeds during braking which enablesa systematic search for squeal conditions by covering varying test parameters such as temperature and



pressure. This procedure known as drag mode gives a broad view of the noise events occurring at thetested brake system. However, it tends to overestimate the noise rating found in vehicle testing.

The second procedure is to set up duplicate vehicle driving conditions leading to ‘‘road load’’ or‘‘in-stop brake’’ noise test procedures. In-stop modules can focus on testing the noise performanceunder conditions related to road tests. On the other hand, they can also be set up in a way less relatedto road test by systematically increasing or decreasing pressures and temperatures. Using this testmethod, it is less likely to miss noise events.

In the United States, the more common approach is to use road load or in-stop mode proceduresemphasizing the replication of vehicle road test. A global noise and vibration matrix which combinesdrag mode and in-stop mode tests has been incorporated into the standard of SAE J2521, which hasseveral sections. The main test sections comprise more than one stop. For the proposed noise andvibration matrix, the test rig must be capable of speed ranges upto 100 km=h and of a changeover inthe direction of rotation, pressure bar control 0–51. It requires flywheels or inertia simulation. Duringthe tests, hydraulic pressure, temperature, and speed have to be measured and controlled. In additionto the noise transducers, information on torque is required to determine friction coefficients. In the firstsection, the break-in of burnish at different pressures is running with 30 stops, at the second sectionconditioning operations for the lining are performed by 32 stops. The third section is used todetermine friction coefficient values by performing 6 stops.

Section 4 that comprises 266 stops is the main part of the noise test. It is the drag module inwhich the brake is applied for different temperatures and pressures at constant speed. During a stopthe pressure is increased and decreased following a ramp. After it, section 5 is applied to warm upwith 24 stops, then the section 6 of backward=forward step is run with 50 stops. In the back-ward=forward module, every drag operation of the matrix is performed first backwards andimmediately afterwards in forward direction to cover situations like parking maneuvers. So faronly drag tests, i.e., constant speed operations, have been performed in this section. Then section 7with 108 stops using deceleration module, a typical US in-stop noise module with a starting velocityof 50 km=h at various pressures and temperatures is incorporated. By including this into theprocedure, a combination of the ‘‘worst case’’ scenario given by the drag stops and the more vehicletest related information given by the in-stop tests is achieved. Then section 8 is completed by a stepwhich gives the current friction characterization by performing 6 stops.

Thewhole sections4–8will be repeated three times,whichadds two setsof sections9–13and14–18.Finally, the procedure consists of a temperature fade module. This enables investigating the noiseperformanceof thebrake systemafter thebrakehasbeenheated in the temperature fadesection. It consists

Sound measurements

Bra

ke li

ne p

ress

ure

(bar

)

Registeredsqueals

rps

0

0

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4

6

8

10

12 4

3

2

1

010 20

Time (s)

30 40

Pressure

Rot

atio

nal s

peed

(rp

s)

FIGURE 7.14 Schematic of sound, pressure, and speed profiles during one stop test. (Courtesy of Eriksson, M.and Jacobson, S.)



of section19, the temperature fademodule; section20, recoverymodule; section21,dragmodule; section22, intermediate conditioning and warm-up; section 23, backward=forward; section 24, decelerationmodule; and section 25, giving the current friction characterization by performing 6 stops.

Figure 7.15 shows a typical noise and vibration matrix test results [94]. The standard resultincludes a graphical summary with all the key sections. Plots for temperature history versus peak

70

50

00 500 1,000 1,500

Step no.

13.9

22.0

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cy (

kHz)

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16Temperature (�C)

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L (d

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empe

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e %

of n

oisy

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ps

Forward

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History

10,000 12,000 14,000 16,000

80

90

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FIGURE 7.15 NVH matrix test results. (Reprinted from Blaschke, P. and Rumold, W., SAE 1999-01-3405,1999. With permission from SAE.)



frequency and the sound level are shown for all stops above the threshold level, for which SAEJ2521 standard threshold is set as 70 dBA. Test data and layout allow assessment of the followingelements: brake and release speed, average deceleration, average and maximum torque, average andmaximum pressure, initial and final rotor temperature, average effectiveness or BEF, peak soundlevel, peak frequency, above threshold or not.

Dynamometer test is not a 100% reflective of vehicle drive condition. Vehicle road test has beenused as final verification test of brake noise performance. For instance, the road test of Los AngelesCity Traffic (LACT) has been widely used in the United States. The specific road test done in LosAngeles City under varied urban severe conditions has been proved to be effective to generatesignificant noise for susceptible vehicle brakes.

LACT testing has been utilized by automotive manufactures and brake component manufac-turers for many years as an indication of how a vehicle=brake system performs in ‘‘real world’’conditions from a noise and wear standpoint. The LACT offers a very good variety of trafficconditions and different climates in one driving pattern which gives a large variety of braketemperatures and pressure applied. This test provided ‘‘real world’’ noise and wear results duringa 25 day (5000 mile) run. Although LACT covers a wide variety of driving conditions, it does notcover all normal driving patterns (for instance, Asian and European driving).

To repeat these conditions in dynamometer test is difficult. The dynamometer tests usually donot correlate with road test well. Actually, even SAE J2521 does not attempt to provide a substitutefor LACT test. There are many differences in components and conditions used in the two tests.Dynamometer tests are used as rapid screening process and refinement as well as failure diagnosistests. In practice, SAE J2521 has been proved to be good tool for screening for the propensity ofbrake generating noise.

7.3.3 NOISE EVALUATION

In LACT test, brake squeal can be identified and classified subjectively by drivers or objectivelyrecorded by using microphones and accelerometers. The subjective rating is a numeric valuebetween 1 and 10, which is supposed to be recorded by driver to make notes of noise occurrence.Rating 10 is given when no noise is observed, while 1 is given when loud objectionable noise ispresent in all brake operation. The rest categories fall in between 1 and 10. It provides person’sperspective of the noise performance.

The objective evaluation is done by using sensors and data acquisition system. Figure 7.16 istypical power spectra of microphone signals recorded at vehicle corners (left-rear, right-rear). Brakesqueal has a distinctive peak with possible harmonics displayed in its FFT spectrum.

It is usually required to correlate noise evaluation between objective evaluation and subjectiveevaluation, and the metrics in addition to peak spectral density need to be considered. The subjec-tive noise index (SNI) derived from psychoacoustics consideration is usually used

SNI ¼

Prating 1 to 10

n �WSR

N

wheren is the total number of noisy stopsWSR is the noise weighting factorN is the total number of stops

The subjective evaluation can be a good representative of final end-customer’s perception, but itcannot give accurate information to investigate the true nature of noise. In-vehicle objective



evaluation based on sensors and data acquisition system has been used to capture accurate noiseinformation. Some objective noise index (ONI) or metric is also developed to quantify the overallbrake system noise performance. The ONI is defined as

ONI ¼Plevels

n �WSPL

N

where WSPL is the noise weighting factor.In advanced applications, because of the nonstationary pattern of friction-induced sound, more

noise metrics are needed so as to quantify the complex characteristic of the human ear and humanpsychology. A basic index, that is the commonly accepted curves of equal loudness contours, isshown in Figure 2.28. The unit of curves and phons was arbitrarily chosen so that at 1 kHz,the loudness in phons is the same as the sound pressure level in dB. The most basic metric is theA-weighting, which approximates the 40-phon line. This simple method does have limitations. Itwas derived from the 40-phon line so it is valid only for low-level sounds (40–60 dB) and for singlepure tones. However, it is not a good measure of loudness of complex sounds consisting of multipletones or broadband noise which is very common for friction-induced noise. The equal loudnesscontours also do not reflect the subjective judgment of relative loudness. In practice, we are notconcerned with tones but broadband signals. Stevens introduced a method for calculating a loudnessindex. This is known as the Mark VI method, which is the basis of the American Standard ANSIS3.4. This standard provides a quantitative measurement of the overall loudness, as well as thecontribution of each octave band. Errors can occur if applied to different types of line spectra or tospectra having sharp maxima that are separated by more than one octave. The Mark VI method wasalso designed for noise that is steady rather than intermittent. Zwicker extended the above methodwith a hearing based method for noise measurement. This method calculates specific loudness andthen total loudness. While similar to Mark VI method it accounts for the upward spread of maskingand can be used with complex sounds with broadband and=or with strong line spectra. His method isthe basis of ISO 532B.

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(a)

(b)

FIGURE 7.16 Power spectra of microphone signals at vehicle corners: (a) left-rear and (b) right-rear.



The equivalent loudness for squeals can be read from the graph or by the following equation:

Y ¼ aþ bX þ cX2

whereY¼ equivalent loudness in phonsX¼ SPL of pure tone, ref 20 m Pa, a, b, and c are functions of frequency f

This calculation assumes that the squeal was a pure tone at a constant level throughout the stop.However, a brake may squeal for only part of the stop either at the end or start and may inducemultiple squeal frequencies, all of which contribute to the overall loudness level. The Zwickermethod of calculating the specific loudness at each frequency band and then the total loudness canbe used to deal with this situation. No single power law relates sound pressure in dB to thepsychoacoustics loudness. Possible relationships occupy a region, determined by spectral shapes.

The spectra are calculated on intervals as short as 40 ms on many devices; it is possible toconvert the objective evaluation from device to subjective evaluation by human perception whichwill react to noise longer than about 200 ms. The basic approach is for a jury to listen to the noiseand then to subjectively rate it against a predefined scale. Many efforts have been made to usemodels to convert the SPL from noise dynamometer tests and chassis dynamometer tests to thesubjective 1–10 rating normally used in vehicle testing.

7.3.4 TRIBOLOGY TESTING

Typical brake noise dynamometer test provides the following data: noise frequency, noise soundpressure level, noise occurrence percentage, and operation conditions including speed, temperature,pressure, and torque [84–92]. The friction coefficient and its variation with temperature and speedare also reported. Dynamometer testing usually provides a convenient means of direct measurementof brake torque. Friction coefficient can be computed from the measurement of the brake torque T,brake pressure P, the total piston area A, and effective radius R of a disk brake; the coefficient offriction is computed by using the formula, m¼ T/(2RPA).

The information helps engineers in diagnosing and determining the important influencingfactors in the overall noise performance of the brake system. The friction coefficient usuallyvaries with pressure, temperature, and speeding in a complicated pattern, which will be furtherelaborated in the next section. In engineering, understanding friction law helps to understand theinteractions of influence factors. A tribological investigation into the interface details gives deeperunderstanding of the friction natures. The final qualification test for brake materials involvesextensive on-vehicle tests with full-sized components. A number of materials tests (friction test,wear test, compression test, hardness, thermal conductivity measurements, etc.) are employedduring the development of brake materials and additives. In order to maximize the considerationsof the possible influences of braking systems into laboratory test, the test conditions that simulatedriving conditions are introduced. To reduce preliminary material qualification costs and to facilitateresearch, a variety of laboratory-scale test machines have been developed. These range frommassive, inertial dynamometers with electronic controls and sensors to small, rub-shoe machinesthat can sit on a bench-top.

Off-vehicle brake material test methods range from simple drag tests at constant speed andcontact pressure to complex, multistage qualification tests involving programmed changes incontact pressure, speed, temperature, and repetitive contacts that simulate vehicle braking events(e.g., SAE J1652).

The basic elements of friction test include a means to apply a force, use of conformal contact,and a means to measure frictional torque. Some tests involve constant speed, but others involvedeceleration. Use of multiple load applications is common, as is temperature measurement.



The friction assessment and screening test (FAST) machine was introduced by Ford MotorCompany as a quality assurance test. It uses small block specimens, about 6.35 mm square, draggingon the circumference of a test ring at constant torque (actuator pressure is adjusted to maintaintorque) for 90 min. Some investigators have attempted to use this to evaluate new materials forvehicles, but it was only intended as a quality assurance tool for brake materials, not as adevelopment tool. According to some reports, the FAST machine does not replicate road conditionswell enough to be used in brake materials research and development.

The Chase Machine is used to perform SAE test J661a and was involved in the development ofthreshold. It consists of a rotating drum with a 25.4 mm square pad of friction material loadedagainst the inner diameter of the drum (279.4 mm) by an air pressure system. Friction and wear datacan be obtained. The wear is usually reported in terms of weight loss of the pad and thickness lossfor the drum. Depending on the types of brake phenomena under investigation (e.g., noise,vibration, friction level, thermoelastic effects, etc.) good correlations may or may not exist betweensub-scale and full-scale tests, like inertial dynamometer tests. For certain studies, however, sub-scalemachines constitute a cost-effective tool for brake materials developers. For example, when theamount of experimental materials is limited or when the fabrication cost for full-scale prototypesis prohibitively expensive, sub-scale tests can provide enough screening information to down-select the most promising candidates. A number of tribological instruments including SEM,optical profilometry and nanoindentation, light optical microscopy, and macrophotography havebeen used to characterize the surface of brake rotor and pad, so as to attain insights into the tribologynature.

Normally, brake pads are composites of friction pads which are made of compound materialsconsisting of approximately 10 different types of materials. These include organic and inorganicreinforcement fibers, friction modifiers, and fillers, which are solidified with a binder such as phenolresin. One characteristic of friction materials is that they are commonly used under high-temperatureconditions that exceed the temperature at the time of manufacture. For example, the temperatureduring braking may exceed 5008C. Additionally, friction materials must provide a service lifecapable of withstanding frequent braking during vehicle operation over several tens of thousandsof kilometers. Pad materials have very different properties. The soft components, such as resins andsolid lubricants, have a hardness of around 200 MPa, while the abrasive particles and fibers insome cases may have a hardness of up to 20 GPa. The components show a correspondingly widespectra of wear resistances. These differences result in a complex contact situation. According toEriksson et al., unevenly distributed wear and compaction of wear debris result in a surfacecharacterized by flat plateaus, rising above the rest of the surface [84–90]. The plateaus can bespotted visually as shiny spots scattered over the pad surface. A scanning electron microscope(SEM) or a profilometer investigation shows that the plateaus are of varying sizes, ranging from 50to 500 mm in diameter and a few microns high. They can be defined as the areas of the pad showingsigns of sliding contact with the disk. Typically, these signs involve a relatively flat surface withshallow grooves in the sliding direction. The number of plateaus on one pad is typically on the orderof 105 and their total area is 10%–30% of the nominal area of the pad. Figure 7.17 showsa typical topographic appearance of contact plateaus on a brake pad, which is acquired by using awhite light interferometry.

As in all other sliding contact situations, the real contact area transfers the friction forces. Due tothe topography of the brake pads, the real contact area is confined within the contact plateaus. Thesize and composition of the plateaus thus have a crucial influence on the friction behavior of the pad.

The presented contact situation is unique for the given material combination, with a coarseinhomogeneous composite sliding against a solid metal disk. If the brake pad is replaced by a pieceof solid cast iron with the same geometry, a completely different situation would occur. The numberof contact areas would be much fewer, due to the higher stiffness of the iron. Lining materials aregenerally more compliant than solid metals and thus, the contact areas will be more evenly scatteredover the surface. If a plateau experiences a high load, the low modulus of the matrix will help to



unload it. The load will be transferred to the neighboring plateaus, resulting in a more evenlydistributed load.

Figure 7.18 shows a measured surface profile of the friction couple. It includes a padincluding two contact plateaus with the contact plateaus measured parallel and perpendicular

FIGURE 7.17 Typical topographic appearance of contact plateaus on a brake pad. Standard pad 3D imaged bywhite light interferometry. (Reprinted with permission from Eriksson, M., Bergman, F., and Jacobson, S.,Wear, 232, 163, 1999. Copyright 1999, Elsevier.)

6(a) Pad

(b) Contact plateau

(c) Disc

Ra = 2 µm

0 0.1 0.2 0.3Lateral position (mm)

Parallel, Ra = 0.2 µm

Parallel, Ra = 0.1 µm

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ght (

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FIGURE 7.18 Surface profiles of the friction couple (optical profilometry): (a) pad: including two contactplateaus, (b) contact plateau: measured parallel and perpendicular to the sliding direction, and (c) disk:measured parallel and perpendicular to the sliding direction. (Courtesy of Eriksson, M. and Jacobson, S.)



to the sliding direction. The disk surface is measured parallel and perpendicular to the slidingdirection.

The overall measurement of pad material surface results in a typical Ra of 2 mm as shown inFigure 7.18. In the profile, two contact plateaus can be spotted as 0.1 mm wide and relatively flatareas. Generally, the surfaces of two well run-in mating parts in sliding contact exhibit matchingprofiles. This is not valid for the brake pad and disk counterparts as shown in Figure 7.18. Onthe other hand, the profiles of the plateaus match with those of the disk shown in Figure 7.18.Furthermore, the uniform sliding of the plateaus on the disk yields a surface with differentroughness in different directions. Parallel to the sliding direction, the two surfaces in contact arevery smooth, whereas in perpendicular direction, the surface is more than twice as rough. Theplateau is slightly rougher than the disk in both directions.

7.4 FEATURES ASSOCIATED WITH BRAKE NOISE

This section shows a number of factors and their trend that affect friction-induced vibration andnoise in brake system. Since the friction-induced vibration and noise in brake system are socomplicated problems concerning with material properties, structural features, interface parameters,operational conditions, and environmental conditions, it is necessary to give an entire descriptionof the basic features of the affecting factors. However, this aim is beyond the limits, as the scopeis too wide and the actual system has too many variations. Moreover, these affecting factorsinfluence the system properties through an interactive way in terms of tribology, structural dynam-ics, and acoustics. This prevents from using them to interpolate noise and vibration consequences.Nevertheless, we try to present some of them to give some insight into this topic.

7.4.1 INTERFACE AGING EFFECT

The break-in for a new brake disk is usually accompanied by a process where friction coefficientincreases slowly, and sometimes squeal could occur when coefficient of friction reaches a threshold.The coefficient of friction and the accumulated number of squeals for a typical break-in test areshown in Figure 7.19.

1

0.8

0.6

0.4

0.2

00 100 200

Coe

ffici

ent o

f fric

tion

Num

ber

of r

egis

tere

d sq

ueal

s

300

Number of brakings

400 500 600

250

200

150

100

50

0

FIGURE 7.19 Coefficient of friction and registered squeal in a break-in test. (Reprinted with permission fromBergman, F., Eriksson, M., and Jacobson, S., Wear, 225–229, 621, 1999. Copyright 1999, Elsevier.)



The macroscopic friction coefficient increase can be attributed to the increase of the real contactarea. As such, one way to suppress squeal is to allow the real contact area kept smaller than a smoothdisk and the friction force thereby stays lower.

Eriksson et al. [84–90] did a series of tests elaborating the process. Figure 7.20 shows a gradualchange of the SEM-micrographs of disk surface and corresponding friction and squeal generationcurves during the break-in test. The SEM-micrographs of disk surface were taken after 42, 210, 420,and 588 brakings. It illustrates the gradual increase in the fraction of smooth area on the brake disksurface which is part of the area not occupied by the pits. The surface area increases as the pits areworn away, more rapidly at the earlier stages of the test and considerably slower at the end. After 1260brakings, the pits still cover approximately 13% of the surface. The surface profiles obtained from thereplicas revealed a maximum pit depth of 30–40 mm after 378 brakings and roughly 20 mm after 966brakings. The average diameter to depth ratio was generally estimated to 5 mm, i.e., the pits were fivetimes as wide as they were deep. During the initial period, the pits exhibited diameters from 10 to 100mm. EDX analysis showed that this compacted wear debris contained typical pad elements. Assmaller pits were filled to a higher degree than larger ones, the share of filled pits increased as the diskwas worn. In the test, the friction coefficient was initially low (0.3) but quickly increased to 0.4 duringthe first 300 brakings. Even though the friction then started to stabilize, it was still 25% below that of anonprepared disk. As the tests continued, the disks became smoother with the friction graduallyincreasing. When the average friction level exceeded 0.4, squeal started to be generated morefrequently. Brake squeal will not be generated if the coefficient of friction is kept below some criticallevel. For the tested pad–disk combination, this level was found to be 0.4. Also a small increase in thecoefficient of friction in the close vicinity of the critical level causes a dramatic increase in squealgeneration, indicating some threshold behavior of the excitation mechanism.

Figure 7.21 illustrates the development of the coefficient of friction and the estimated nominalfraction of contact area of the disk during a sequence of brakings against a shot blasted disk.

A transfer film composed principally of friction material forms on the brake rotor side of theinterface, and friction occurs between this film and the pad. The types of wears possible to occur atthe interface include adhesive wear, abrasive wear, and compound tribochemical reactions inducedby these wear mechanisms. Temperature has strong effect on friction and wear. The operating

420th Braking

Friction

Squeal generation

Number of brakings

210th Braking 42nd Braking 588th Braking 100 µm 100 µm 100 µm 100 µm

FIGURE 7.20 Gradual change of the SEM-micrographs of disk surface and corresponding friction and squealgeneration curves during a test. SEM-micrographs were taken after 42, 210, 420, and 588 individual brakings,respectively. (Reprinted with permission fromBergman, F., Eriksson,M., and Jacobson, S.,Wear, 225–229, 621,1999. Copyright 1999, Elsevier.)



temperature range is usually from 08C to 5008C. At the upper extremes of this range, the wear rate ofthe friction material increases exponentially.

Other relevant factors affecting interface friction and wear include corrosion and contaminations(dusk or chemical liquids). Figure 7.22a indicates the wear results of rotor disk in a dynamometerexperimental simulation with=without applying a corrosion condition on the disk. Figure 7.22bshows the rotor disk DTV results for the rotor disk with=without applying a corrosion conditionon the disk. It indicates that both disk wear and DTV growth are closely linked to disk corrosionwhich could be the result of long parking. DTV growth and disk wear, with periodic corrosion,are significantly faster and greater in magnitude. DTV is also strongly dependent on the initialon-vehicle mounted rotor runout.

7.4.2 FRICTION–VELOCITY DEPENDENCE AND HYSTERESIS

The friction coefficient itself is a function of pressure, speed, temperature, humidity, real contactarea, surface parameters, contamination, etc. Friction coefficient may vary from acceleration ordeceleration, from forward or backward stops, moreover, many aging factors discussed above makegage repeatability and reproducibility very challenging.

As an example of the friction–velocity dependence, the experimental results of Ahmed et al.[95] are described as follows: Figure 7.23 shows the effect of vehicle speed on the coefficient offriction at constant rotor surface temperature with different line pressure ranging from 10 to 40 bar.These groups of figures indicate that the coefficient of friction increases with the increase of therotor surface temperature till certain limit and then decreases with the increase of the rotor surfacetemperature. The maximum value of coefficient of friction reached is 0.51 at vehicle speed 20 km=hand rotor surface temperature 1008C, while the minimum coefficient of friction is 0.24 at vehiclespeed 80 km=h and rotor surface temperature 1008C. It is realized that at the beginning of thebraking moment, the line pressure tends to be active with nearly 0.15 s lag till it reaches themaximum constant value of line pressure.

Moreover, the friction–velocity dependence could exhibit the hysteresis feature as shown inFigure 7.24 [84].

Number of test sequences

Disk contact area

Coefficientof friction

4 9 16 23 301000.6

0.5

0.4

0.3

0.2

0.1

00 300 600 900 1200

Number of brakings

Nom

inal

dis

k co

ntac

t are

a (%

)

Coe

ffici

ent o

f fric

tion 80

60

40

20

0

FIGURE 7.21 Development of the coefficient of friction and nominal fraction of contact area of the diskduring a sequence of braking against a shot blasted disk. (Reprinted with permission from Eriksson, M.,Bergman, F., and Jacobson, S., Wear, 252, 26, 2002. Copyright 2002, Elsevier.)



7.4.3 FRICTION–PRESSURE DEPENDENCE AND HYSTERESIS

Friction not only exhibits velocity dependence, but also has strong pressure dependence. Thepressure dependence usually has negative slope nature [84–90]. Moreover, there could be stronghysteresis features associating with the pressure dependence. The coefficient of friction versus

0

6

12

18

24

30

0

(a)

(b)

3,000 6,000 9,000 12,000 15,000

Mileage (mile)

0 3,000 6,000 9,000 12,000 15,000

Mileage (mile)

Wea

r (µ

m) With corrosion

Without corrosion

0

6

12

18

24

30

Dis

k th

hick

ness

var

iatio

n (µ

m)

With corrosion

Without corrosion

FIGURE 7.22 (a) Disk wear and (b) disk thickness variation as functions of mileages.

0.6

P = 10 bar

T = 50 degree C

P = 30 bar P = 30 bar

P = 10 bar

P = 40 bar

P = 40 bar

T = 150 degree C

0.5

0.4

0.3

0.2

0.1

0

CO

F

20 40 60 80 1000

(a) (b)Equivalent speed (km/hr)

0.6

0.5

0.4

0.3

0.2

0.1

0

CO

F

20 40 60 80 1000

Equivalent speed (km/hr)

FIGURE 7.23 Effect of vehicle speed on the coefficient of friction at rotor surface temperatures: (a) 508C and(b) 1508C.



brake pressure history constitutes the hysteresis behavior of the friction coefficient as shown inFigure 7.25a for two different pads. For one set of pads, the friction coefficient is clearly lowerwhen the brake pressure is increasing than when it is decreasing. The hysteresis loop exhibitsanticlockwise direction instead of clockwise occurred in speed dependence. However, this hyster-esis behavior is very dependent on the friction characteristics of the pad material. The results from apad of different formulation, exhibiting very little friction hysteresis, are shown in Figure 7.25b.

7.4.4 TEMPERATURE DEPENDENCE

Many materials will in general be softer at higher temperatures. So modulus could decrease forincreased temperature. On the other hand, the coefficient of friction could also be temperaturedependent. In many cases, squeal often appears after heating the brake.

Figure 7.26 shows the effect of rotor surface temperature on the coefficient of friction atconstant speed with different line pressure ranging from 10 to 40 bar [95]. These groups of figuresindicate that the coefficient of friction could increase with an increase of the rotor surface tempera-ture in proper range, and then decrease with a further increase of the rotor surface temperature. Themaximum value of coefficient of friction reached is 0.55. The energy dissipated during brakingfrom speed equivalent to 20 km=h to stop fully is 10 kJ, braking time is 3 s, and the decceleration

0.7

0.65

0.6

0.55

0.5

CO

F0.45

0.4

0.35

0.30 1 2

Rotational speed (rps)

3 4

FIGURE 7.24 Effect of vehicle speed on the coefficient of friction.

0.6

0.55

0.5

0.45

0.4

0.35

0.30 5 10 15

Pressure (bar)

stop

start

stop

start

CO

F

0.6

0.55

0.5

0.45

0.4

0.35

0.3

CO

F

20 25 0 5 10 15

Pressure (bar)

20 25 30

Type A Type B

(a) (b)

FIGURE 7.25 Hysteresis in the coefficient of friction with continuously changing normal load (brake-linepressure). Braking at 2 rps constant speed: (a) hysteresis effects involving higher coefficient of friction fordecreasing than for increasing normal load. Standard pad to the Volvo 850 TX4005. (b) Less obvious hysteresisdisplayed by a noncommercial pad formulation MD631D.



is 1 m=s2. While the energy dissipated during braking from speed equivalent to 80 km=h to stop fullyis 173 kJ, braking time is 7 s, and the maximum deceleration is 8 m=s2.

Figure 7.27 shows the effect of rotor surface temperature on the coefficient of friction atconstant line pressure with different speed ranging from 20 to 80 km=h. These groups of figuresindicate that the coefficient of friction increases with the increase of rotor surface temperature tillcertain limit and then decrease with the increase of rotor surface temperature. The maximum valueof coefficient of friction reached is 0.55 at rotor surface temperature 1008C, while the minimumcoefficient of friction is 0.23 at rotor surface temperature 508C.

When the temperature is further reduced to cross 08, it tends to create higher friction coefficientand negative velocity slope. This phenomenon can be correlated with the so-called ‘‘cold noise’’which is likely to occur in cold and wet conditions.

In addition to friction coefficient, other material parameters like damping and modulus are alsotemperature dependent.

7.4.5 HUMIDITY EFFECT

Exposure to environmental humidity has big impact on interface tribological performance and thenoise generation; the effect could last for short period like several stops. When temperature is lowand humidity is high, for some pad materials it is found that the variation of friction coefficient ishigh at=after a first couple of runs. The factors that influence this noise event range from pad stiction

0

0.1

0.2

0.3

0.4

0.5

0.6

CO

F

P = 10 bar

P = 30 bar

P = 40 bar

V = 20 km/hr

0

0.1

0.2

0.3

0.4

0.5

0.6

0

Temperature (�C)

CO

F

P = 10 bar

P = 30 bar

P = 40 bar

V = 60 km/hr

100 300200

0(a)

(b)

Temperature (�C)

100 300200

FIGURE 7.26 Effect of rotor surface temperature on the coefficient of friction at speed (a) 20 and (b) 60 km=h.



to the rotor, friction humidity dependence, and rotor corrosion. When rotor is kept engaged withcomposite friction material over night like parking, condensing humidity are likely soaked oninterface to develop meniscus force that causes a significantly higher static friction than cleansurfaces. The NAO materials are susceptible to these effects, which need to quantify the moistureabsorption effect to rank the sensitivity by using water dunking or high humidity soak approaches.

It is noted that many squeal phenomenon are promoted by more than one affecting factors, likethe ‘‘cold noise,’’ where usually low temperature, low speed, high humidity, and low brake pressureinteract to create higher static friction coefficient and negative velocity slope, accordingly theinstability and noise.

7.4.6 MICROTRIBOLOGY PERSPECTIVE

Unlike the situation in hard disk drive community where the study and application of tribology hadbeen under full development, the research of tribology of brake system is quite prohibitive, part ofthe reason is that the brake interface is exposed to ambient conditions and the contact is usuallystrong contact, which is associated with fast surface change under irregular operations. The relatedresearch and investigation under specific conditions are usually difficult to be used for the predictionof another case with certainty, and therefore lack general meaning.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 100 200 300

Temperature (�C)

CO

F

P = 30 bar

V = 20 km/hr

V = 60 km/hr V = 80 km/hr

0.6

0.5

0.4

0.3C

OF

0.2

0.1

00

(a)

(b)

100 200Temperature (�C)

300

V = 20 km/hr

V = 60 km/hr

P=10 bar

V = 80 km/hr

FIGURE 7.27 Effect of rotor surface temperature on the coefficient of friction at linepressure (a) 10and (b) 30bar.



Eriksson et al. [84–90] proposed a mechanism of primary and secondary plateaus growthfor a friction law. In that context, it is believed that there are primary and secondary plateausexisted in pad, and the plateau growth and degradation have a strong influence on the coefficient offriction.

When the secondary plateaus grow due to a load increase, the primary plateaus will retain theirsize and thus the average composition of the material sliding against the disk changes. Conse-quently, the compactly grown areas will carry a larger share of the normal force and thus the frictionconditions are different.

As the composition of the plateaus changes with changed load, the friction force needed toshear the top layer can be expected to change. Further, if the plateaus are adapted to a certainbrake pressure and the pressure is decreased, the coefficient of friction should be expected tochange. The macroscopic effects of these processes can be observed for some brake pads as afriction hysteresis.

During the break-in of an unused pad, a large increase in friction can be observed during the firststops. This behavior is due to the formation of primary plateaus. It was observed that defects in theform of pits in the disk surface considerably lowered the coefficient of friction. The decrease isbelieved to be a result of a continuous degradation of the contact plateaus. Each time a plateau is hitby a surface defect, the secondary plateau will be degraded, and thus the coefficient of friction willbe lowered.

The size of the real contact area between the pad and the disk, and also the composition of theoutermost surface layers within this area, is far from constant but will vary due to changing pressure,changing temperatures, deformation, and wear. Both locally and globally, the contact pressure mayvary on different timescales due to different processes.

Naturally, a change in the braking force will result in a corresponding change of the elasticcompression of the pad. The actuation and variation of the braking force can be quick due to manual‘‘fine tuning’’ during a braking. The associated pressure changes are global over the brake pad (theaverage compression of the pad varies) and on a timescale of 1=10 s. A quick brake pressureincrease thus momentarily results in a corresponding elastic compression of the pad. This compres-sion will result in: (a) More contact plateaus becoming engaged. (b) A redistribution of the loadbetween primary and secondary plateaus within each contact plateau. When the load is increased onone plateau, the mechanically more stable part will carry a larger share of the load. Thus, the averagecomposition of the material transferring the load to the disk will change. (c) An increased load onthe already engaged contact plateaus, which will result in a higher area fraction of real contactwithin these plateaus.

The pad compression, and hence the pressure, may also vary locally over the pad surface.Vibrations cause rapid local pressure variations in the brake system, such as brake squeal. Brakesqueal vibrations are associated with bending and wave motions of the pad and disk. Thesedeformations result in local pressure variations over the contact surfaces, often on a timescale ofmilliseconds or less.

The mechanisms for contact area variations are the same as in the global processes. A numberof slow mechanisms are assumed to operate. The slow processes are due to different kinds of wearand accumulation of debris, and to temperature variations. They typically appear on timescales ofseconds or more, such as during long, low decelerating brakings or as the accumulated resultof numerous brakings. Slow processes appear both on a microscale, that is on the scale ofindividual contact plateaus or smaller, and on a macroscale that is on a scale involving numerousplateaus.

The slow processes include: (a) formation, growth, and disintegration of contact plateaus;(b) shape adaptation on a microlevel; (c) shape adaptation on a macrolevel; (d) thermally induceddeformation on a macrolevel; (e) thermally induced surface property variations; and (f) ‘‘contami-nation and cleaning’’ processes.



The growth and disintegration of contact plateaus involve agglomeration and compac-tion of pad wear debris around a wear resistant nucleus. When the load on a contact plateau isincreased, the small real contact areas within the plateau will flatten elastically, plastically, and bywear. These processes result in an increased real contact area against the disk. When the load isdecreased, the wear and deformation of the points in real contact will tend to reduce their contactwith the disk.

The disk is continually worn, mainly by the harder components in the pads. This wear willinitially polish the disk surface, making it better adapted to the pad. The individual contact plateauson the pad will correspondingly experience milder contact conditions along the less rough slidingpath. Due to the inhomogeneous structure of the materials, the continuous wear on both the disk andthe pad will not be evenly distributed. However, the mutual adaptation to the shape of the countersurface will result in a wavy surface. On the disk, the waves will form concentric circles. In an idealsteady-state situation, the matching between the two parts is perfect, and each individual contactplateau will experience a smooth ride. However, small misalignments or movements betweenbrakings or due to other changes will result in mismatched surfaces and initially a reduced area ofreal contact.

During moments of high and increasing temperature, the pad surface will be hotterthan the interior and the back plate. This will result in convex bending of the pad and hencean uneven pressure distribution. The pressure reduction on the leading and trailing edges will resultin a corresponding uneven distribution of wear, i.e., the pad will become thinner in the center.

When returning to a lower temperature, the pad will straighten out. Then, the uneven wearduring the bent situation will result in reversal of the uneven pressure distribution. The properties ofany surface depend on the prevailing temperature. When the disk and pad are heated during braking,this will affect both the chemical reactivity on their surfaces, the mechanical properties (thermalsoftening, etc.), the structure of the pad (decomposition of polymer constituents, etc.), the tendenciesfor smearing and sticking of wear debris on both surfaces, etc. Both the composition and thetribological properties of the surfaces are affected.

Unloaded contact plateaus are exposed to different ‘‘contamination’’ processes, includingoxidation, smearing out of wear debris and road dust, etc., which will change their composition.When the sliding contact is continued the plateaus will be subjected to a ‘‘cleaning’’ processinvolving the removal of the less wear resistant surface layers. This cleaning results in an increaseddegree of metallic contact. The corresponding processes occur on the disk surface.

The slow contact surface variation processes are responsible for: the in-stop increase of m duringlong brakings; the m hysteresis reported for brakings under varying pressures; and the m increaseduring the run-in of a new disk or a new pad.

It is believed that friction films form on the contact surfaces of brake rotors and pads of frictionmaterial as a result of the compaction of wear debris removed from these components duringbraking actions. Rhee et al. [91,92] reported that, for a semimetallic friction material and a gray castiron disk brake rotor, the formation of the friction film increases the coefficient of friction, and thatbrake squeal did not occur until the films were well formed and the coefficient of friction k hadreached a steady level. This usually occurred in the temperature range of 1008C–3008C. Highertemperatures destroyed the films, and squeal was eliminated. However, this process often led tosevere thermal damage to both the rotor and the pads.

Despite friction plays a key role in brake squeal, it is not immediately clear how to modelaccurately and incorporate it into predictive models for disk brake squeal. Few analyses existed haveincorporated complex tribological processes in their models. It is still in its infancy.

7.4.7 CONTACT STIFFNESS

Contact stiffness is a critical parameter for brake squeal generation. Contact stiffness estimation canbe conducted by using static pressure measurement using ‘‘finger printing,’’ which offers some



information of the contact state. Figure 7.28 shows the obtained ‘‘finger printing’’ of contactpressure distributions for two new sets of pads with the same specifications=configurations andthe same caliper, rotor, and other brake components under the same pressure action [96]. It can beseen that there are noticeable variations of contact pressure distribution between the two assembliesat pad–rotor interface.

The equipment of compressibility measurement under static pressure is predominantly used forquality control measurements. Detailed contact stiffness can be obtained by using specializedinstruments measuring applied pressure and pad deformation. Figure 7.29 is a typical result fortwo tested pad samples [97].

Generally, it is considered that the smaller contact stiffness is favorable to suppress squeal.From system dynamics perspective, this could be attributed to the trend to form a system withsmaller contact stiffness which is likely to give rise to coupling. On the other hand, this could berelated to the analysis of contact plateaus [84]. The results indicate that pads with many smallcontact plateaus have a larger tendency to generate squeal than pads with a few large plateaus.According to measurements, it is possible to define a squeal threshold in the contact plateau size atapproximately 0.01 mm2. The squeal was generated below this plateau size. If the pad types arestudied individually the results show that, in the silent pressure interval, the size of the contactplateaus increases rapidly with brake pressure. It is believed that this pressure dependence of theplateau size has a greater influence on the squeal generation than the absolute size of the plateaus. Atlow pressures in the squealing interval, the pressure had little effect on the size of the contactplateaus. It is concluded that the size of the contact plateaus affects the generation of brake squeal.Pads with many small plateaus generate more squeal than do pads with few large plateaus. Within

FIGURE 7.28 (a) Contact area of inboard pad 1 under 350 PSI; (b) contact area of inboard pad 2 under350 PSI. (Courtesy of Chen, F.)

00

50

100

150

200

50 100

Pressure (bar)

Def

orm

atio

n (µ

m)

150

Pad 2

Pad 1

200

FIGURE 7.29 Measured pad deformations as a function of applied pressure.



the silent pressure interval, the size of the contact plateaus increased with increasing brake pressurefor both tested pads.

Stiffness has important effects on brake. There is an optimum value of stiffness for eachcomponent that makes the system more stable. Some research shows that the stiffer pad usuallyreduces squeal propensity. Pin-on-disk model shows that as the contact stiffness increases to acertain level, the two modes merged to generate a complex mode; however, when the contactstiffness further increases, the complex mode splits into two modes again. The influence is notmonotonic.

7.4.8 DAMPING AND YOUNG’S MODULUS EFFECT

Increasing pad damping by selecting high damped insulators to reduce high-frequency squeal hasbeen extensively practiced and focused. Suppliers are usually reluctant to modify the frictionmaterials, they prefer to modify the pad geometry and optimize the insulator.

A brake insulator is a sandwich of viscoelastic layer and steel layers with a maximum thicknessof around 1 mm. It is attached to the pad back plate. Its functions include material damping,friction damping as well as isolation. The straight theory behind it is that the adding damp-ing helps reduce forced vibrations. A more modest theory is that damping can help reduce coupling.The trace of the complex eigenvalues reveals the effects of damping and stiffness on instability.The real part of the eigenvalue decreases monotonically as the value of damping is increased, ormoves to the left. With the increase of modulus, some trajectories cross, while others veer awayfrom each other.

On the other hand, practices demonstrated that increasing pad damping could reduce high-frequency squeal. However, caution should be taken at low frequency, where the material dampingcould increase the noise due to the increasing of coupling. The damping effect on low frequencymay not be linear or monotonic at relative low damping level, which is subject to the effects ofcompressibility, temperature characteristic, and surface finish. For this complicated reason, theinsulator screening test has been widely used to verify the optimization.

The damping and modulus are affected by other factors. As shown in Figures 7.30 and 7.31, thematerial properties for the midrange and high-temperature dampers are bonded to a brake shoe andlining for the third bending mode only [98]. The material properties have been identified throughmodal experiments and theoretical modal analysis.

It is noted that the effect of Young’s modulus on coupling characteristic of friction material isnot monotonical. The trace of the eigenvalues reveals some pattern: with the increase of modulus,some trajectories cross, while others veer away from each other.

0

0.01

0.1

50 100 150

Type 2

Type 1

Temperature (�C)

Ela

stic

mod

ulus

(P

a)

200 250

FIGURE 7.30 Equivalent Young’s modulus.



The brake insulator also has a function of changing boundary on contact condition betweencaliper and friction pads. Use of insulator adds compressibility to contact system which changes thedynamics of the brake system.

7.4.9 STRUCTURAL EFFECT

Structural components like rotor, caliper bracket, and pads have respective modal properties, andthey all play important roles in squeal event. The structural modification and optimization have beenflexibly used as means to tweak system vibrations and resolve squeal [99–103].

Next we use rotor as example to discuss the effects of structure design variation on the in-planeand out-of-plane modes. In braking operation, rotor disk usually generates a lot of heat. In order toprotect the wheel bearings from the high temperatures induced in a braking action in the rotor–padinterface, the rotor is designed and manufactured with a top–hat as illustrated in Figure 7.31. The hatsection increases both the surface area for cooling and the length of the thermal conductivity pathfrom interface to bearings. On the other hand, the hat element renders the in-plane and transversemodes of an idealized disk of moderate thickness to evolve into fully three-dimensional modes andleads to significantly coupled motions in the in-plane and transverse directions. This section willbrief the effect of hat element.

The related analytical modal analysis of rotor done by Bae and Wickert [99] illustrates the roleplayed by the hat element’s depth in influencing the three-dimensional vibration of the disk. Thelower vibration modes of disk–hat structures are shown to be characterized by the number of nodalcircles NC and nodal diameters ND present on the disk, as well as the phase relationship betweenthe disk’s transverse and radial displacements due to coupling with the hat element. Such modesmap continuously back to the pure bending and in-plane modes of the disk alone appear in orderedpairs, and can exist at close frequencies. Those characteristics are illustrated with respect tosensitivities in the disk’s thickness and the hat’s depth with a view towards shifting particularnatural frequencies or minimizing transverse disk motion in certain vibration modes. As illustratedin Figure 7.32, the structure is axisymmetric and formed of an isotropic homogeneous elasticmaterial, and all surfaces are specified to be traction-free. Four-dimensional parameters a, b, d,and h represent the disk’s inner and outer diameters, the hat depth, and the common disk and hatwall thickness, respectively. All lengths are converted to nondimensionalized parameters withrespect to b and are denoted by variables having an asterisk superscript, d*¼ d=b, a*¼ a=b,h*¼ h=b. Some typical parameter values fall in the ranges 0.5 < a* < 0.7, 0.05 < d* < 0.2, and

0

0.1

1

50 100 150

Temperature (�C)

Loss

fact

or

200 250

Type 2

Type 1

FIGURE 7.31 Equivalent loss factor (two pads with middle- and high-temperature sustainability).



0.04 < h* < 0.10. First, let us discuss the case of an idealized annular disk without a hat element.The lower frequency vibration modes fall into two distinct classes: the bending B and in-plane IPmodes which are further classified according to the number of nodal circles NC and diameters NDpresent. Figure 7.33 depicts the variation of the nondimensionalized B and IP natural frequencies asfunctions of dimensionless disk thickness h*, defined as the ratio of the disk’s physical thickness hto its outer diameter b. The physical circular frequency v is nondimensionalized as v* with respectto the material’s compression wave speed according to v*¼v

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirb2=4(lþ 2m)

p, where r denotes

the volumetric density, and l and m are Lame constants. The calculation is based on an analysisof the governing equations for three-dimensional vibration of a disk having arbitrarythickness. As expected on the basis of plate theory, the natural frequencies of the bendingmodes increase in a substantially linear manner with h* over the depicted range of thickness.

Disk

Hath

h

d

a

b

FIGURE 7.32 Schematic of a prototypical, free, annular, and axisymmetric disk–hat structure.

0.01

0.1

1

0.01 0.1Nondimensional disk thickness

Non

dim

ensi

onal

freq

uenc

y

(0,2)B

(0,2)IP

FIGURE 7.33 Natural frequency spectrum for the three-dimensional vibration of a free annular disk asa function of its thickness; a*¼ 0.5. Over this range of h*, the frequencies of the bending modes B increasewith thickness, while those of the in-plane modes IP are insensitive to it. (Reprinted with permission fromBae, J.C. and Wickert, J.A., J. Sound and Vib., 235, 1, 117, 2000. Copyright 2000, Elsevier.)



The frequencies of the in-plane modes, on the other hand, are largely insensitive to h* overthis range.

Secondly, the natural frequencies and modes of the disk–hat model were determined throughfinite element analysis and laboratory tests.

Values a*¼ 0.5 and h*¼ 0.04 were used in case studies investigating the variation ofnatural frequencies and modes over hat depths 0 < d* < 0.05. Figure 7.34 depicts changes inthe simulated (solid lines) and measured (points) natural frequencies; at d*¼ 0, the values forthe lower bending and in-plane frequencies of a moderately thick annular disk are recovered. Asdeduced from Figure 7.32, at d¼ 0 in Figure 7.33 the first five modes (0,2)B, (1,0)B, (1,3)B,(1,1)B, and (0,4)B are of the bending class, while the sixth mode (0,2)IP is the fundamentalin-plane mode. Beginning with the bending and in-plane modes of the disk alone at d*¼ 0,the loci are extended in Figure 7.34 for the modes (0,ND) with 2 < ND � 5, and (1,ND) with0 < ND � 3.

As the hat depth grows, the natural frequencies experience continuous transition from those of adisk alone to those of a disk–hat structure. The lower modes of the disk can be classified as beingmembers of either the B or IP classes. This kind of representation is not appropriate for disk–hatstructures to the degree that their modes involve coupled transverse and in-plane displacementcomponents. In the present case, each vibration mode of the disk–hat structure maps continuously inthe limit of vanishing d* to either bending or in-plane mode of the disk alone having designation

1.0

0.8

(1,3)−

(0,5)−

(0,4)−

(0,3)−

(0,2)−

(1,2)−

(1,1)−

(1,0)−

(0,2)+

0.6

Non

dim

ensi

onal

freq

uenc

y (w

∗)

Nondimensional hat depth (d∗ )

0.4

0.2

0.00.0 0.1 0.2 0.3 0.4 0.5

FIGURE 7.34 Natural frequency spectrum for the three-dimensional vibration of a disk–hat structure as afunction of hat depth; h*¼ 0.04 and a*¼ 0.5. Loci for the (0,2) + pair of modes are highlighted. Solid line:model predictions; points: measurements. (Reprinted with permission from Bae, J.C. and Wickert, J.A.,J. Sound and Vib., 235, 1, 117, 2000. Copyright 2000, Elsevier.)



(NC,ND). By convention, those modes that transit to a (lower frequency) bending mode of the diskalone are denoted (NC,ND)�, while those tending towards the companion (higher frequency)in-plane mode are denoted (NC,ND)þ. In addition to identifying the relative frequencies of themode pair, the ‘‘+’’ suffix also differentiates the phase relationship between transversal vibration ofthe disk and radial vibration of the hat element. The natural frequencies for members of the (0,ND)class of modes grow moderately with increasing depth over the range considered in Figure 7.34.Members of the (1,ND)� class have frequencies which do not behave monotonically but aregenerally maximized near d*� 0.1 for the lower modes. A hat depth near that value could beconsidered optimal in the sense that those modes are stiffened.

In addition to hat height, the thickness of hat can also give us the similar design flexibility.Moreover, in vented disk design, the disk is designed with two thinner disks connected by a series ofthin fins (ribs), so as to further improve cooling. A prime number (e.g., 37) of fins is typically usedin this case in order to avoid symmetric modes of vibration in the disk. In high performance andcustomized disk brakes, the rotors are often slotted and=or drilled and are not necessarily composedof cast iron.

Typical ventilated rotor also demonstrates the presence of a pair of modes having the samenumber of nodal diameters and circles on the disk but different frequency and phase coupling withthe hat. It is noted that usually Mindlin plate theory is needed to precisely describe the high-frequency modes of disk so as to capture shear deformation and rotational inertia effects.

7.4.10 ACOUSTIC RADIATION EFFICIENCIES

The acoustic radiation efficiencies of rotor can be estimated by using analytical, boundary elementapproach or semianalytical approaches. The velocities can be obtained by experiments and beincorporated into the calculation.

It is usually assumed that the radiation characteristics of rotating system modes are similar tothose of stationary brake system modes. The natural frequencies and mode shapes of a rotationsystem are actually identical or close or to that of the stationary system under the assumptions thatthe systems are linear and that the rotational frequency of the rotor is much smaller than the naturalfrequencies of either system.

Figure 7.35 shows the sound radiation efficiency spectra for a thick annular disk estimated usingsemianalytical procedure based on structural eigen solutions from finite element analysis andanalytical modal sound radiation solutions by Lee and Singh [104]. The results for a different

Frequency (kHz)

Sou

nd r

adia

tion

effic

ienc

y

FIGURE 7.35 Radiation efficiency spectra of two brake rotor: (��), analytical; (---), computed using BEM(Lee and Singh); (...), based on the method of McDaniel et al.



thick annular disk using semiexperimental procedure given by McDaniel et al. [105] are also plottedin the same figure. For the particular systems the remarkably high radiation efficiency occurred atfrequencies above 1 kHz.

7.4.11 UNCERTAINTY

The manufacturing, assembly, and usage variations are usually needed to take into account, whichhave strong impacts on the noise and vibration performance of individual system. It has been shownthat the maximum variations of pad damping can be higher than 15% compared with the mean valueand more than 30% between maximum and minimum values. The modulus of rotor also hasnoticeable variation, the natural frequencies could have variation higher than 5%. This renders thehigh-frequency mode to have substantial variation, for instance, the 6th nodal diametrical mode at7 kHz has the frequency difference as large as 355 Hz. For the 8ND mode frequency at 10 kHz, thefrequency difference between two rotors can be as large as 500 Hz.

The Monte Carlo approach can be used for quantifying model parameter uncertainty. A cloud ofresponse points can be obtained by making a full run for each random vector change in the model.Usually, several hundreds of run need to generate the cloud. A linearized stochastic solution can beapplied surrounding each response point, so as to reduce the run below 100.

Once contact conditions reach to threshold to allow squeal to occur, the onset is certain andrapid. When the conditions cease to promote squeal, the inhibition is also rapid, due to the formationof underlying instability mechanism. However, the very small changes of the contact conditionsduring a test can lead to a transfer from silent to squealing conditions. These changes are so subtlethat they are not easily measured or observed. The macromechanical conditions, such as stiffness,hardness, and design, do not have to change to transfer the brake from silence to squeal. Thus, thesqueal appears to have a random behavior, so that the onset of squeal under certain nominalconditions cannot be predicted, but described as a probability, only. The only properties changingrapidly and unpredictably enough to give this seemingly random behavior are the surface conditionsand the friction properties. Some uncertainty is difficult to quantify, particularly the interfacephenomena.

7.5 PREVENTION

This section briefs some approaches used to combat the friction-induced vibrations and noise.

7.5.1 FRICTION SCREENING

It is generally believed that the coefficient of friction should be controlled below a threshold forspecific design and to avoid friction–velocity dependence yielding negative damping.

7.5.1.1 Groan

For the case of groan, friction screening is shown to be one of the most effective approaches inpreventing moan by identifying lining materials that exhibit change between static and dynamicfriction coefficients.

Design change based on dynamic characteristics at component level is not sufficient for groanprevention. It is observed that the groan is usually impacted by component design with multi-disciplines, from brake systems, wheel=tire manufacturer, to axle assembly.

On the other hand, the difference Dm between kinetic mk and static ms coefficients of friction canbe used as index for material design to control stick–slip type oscillations. For instance, using thisindex, Jang et al. [106] conducted measurements for varied material formulations to investigatecreep–groan propensity of each friction material. Results showed that zircon (zirconium silicate),



steel wool, and phenolic resin showed a tendency to increase Dm. On the other hand, antimonytrisulfide, cashew particles, and rockwool tended to reduce Dm. Based on the results, as illustrated inFigure 7.36, the initial formulation was modified to reduce Dm. The modified friction materialshowed a smaller Dm value than the initial friction material implying low propensity of creep–groanphenomena during braking. The velocity dependence of the friction coefficient was also examinedusing the initial and the modified friction materials. The modified formulation exhibited a lessnegative m–V relation than the initial formulation.

7.5.1.2 Judder

Brake judder is considered to be mainly caused by DTV due to uneven wear of the rotor. DTV hasbeen used as a substitute characteristic index. It is believed that DTV is mainly caused byintermittent contact between friction lining and rotor during no braking operation. Vehicle sensi-tivity also plays a role in the generation of judder. It has been demonstrated that some correctivecharacteristic can be introduced to act to rectify DTV during braking. This corrective featurehas been demonstrated workable and it requires a trade-off with rotor life. It is attained by asemicorrective material formulation, characterized by a higher friction coefficient for controllingaggressiveness to the rotors and material.

This formulation involves applying a suitably thick transfer film to the friction interface and alsoremoving it by abrasion. The film is controlled neither too thick nor excessively removed. Samplesof brake friction materials were prepared by combining certain selected abrasives, lubricants, andfillers with the other materials. The abrasive, having a small particle size and a round shape, wasfound to be advantageous for transfer film formation at the friction interface. It was also observedthat a thermally stable crystal structure was desirable. An organic lubricant is generally effectiveagainst brake squeal because of its superior damping performance. It plays a large role, however, inthe formation of the transfer film on the rotor especially at low temperatures. With a pad materialhaving a friction coefficient larger than 0.4, it is necessary to consider abrasive wear as the principaltype of wear. Ordinarily, three-dimensional abrasive wear is dominant in the high-temperatureregion. Such wear is thought to be caused by friction that occurs between hard wear particles presentbetween the transfer film on the rotor and the friction pad.

7.5.1.3 Squeals

At present, the reduction of squeal noise for specific brake systems is achievable. Theoreticalanalysis of brake systems is difficult due to the complexity of mechanisms and the lack of an

Zircon

∆m

Graphite Sb2S3

0.4

0.32

0.380.36

0.34

Cashew particle

Rubber powerPhenolic resin

∆m

0.40.370.36

0.35

0.34

0.330.32

FIGURE 7.36 Static ms and kinetic mk coefficients of friction, and Dm¼mk � ms on ternary diagrams enclosedby the confined compositions of (a) organic ingredients and (b) solid lubricants and abrasives.



adequate model for the friction interface that causes brake squeal. However, this should not restrictthe developing and applying simplified models as valuable insight. Understanding obtained bystudying simplified models can assist in the interpretation of experimental results and the develop-ment of improved computational tools.

Due to the above-mentioned difficulties in designing a squeal-free brake system, efforts toeliminate brake squeal have largely been empirical, with problematic brake systems treated in acase-by-case manner. The success of these empirical approaches depends on the mechanism that isresponsible for causing the squeal problem. The most fundamental method of eliminating brakesqueal seems to be to reduce the coefficient of friction of the pad material. However, this obviouslyreduces braking performance and is not a preferable method to employ. Instead the screening andoptimization of insulator, chamfer, and lining shape have been favorably preferred.

Usually, there is a threshold of the friction coefficient below which squeal rarely occurs. Whenthe coefficient of friction is increased above some threshold, for instance 0.3–0.4, the squeal willbecome worse.

The most popular approaches to eliminate low-frequency squeal could be the pad treatment withdamping (insulator), chamfers, and optimizing the length of pad, and the treatment of rotor withasymmetry, stiffness, and damping as well as the control of pad coefficient of friction. Moreover,shape and stiffness change of the anchor bracket are effective in reducing the noise if it is caused bythe local vibration of bracket.

Theoretically, the direct way to totally eliminate high-frequency squeal mode of vibration is toslice the rotor in the radial direction through the friction surface so as to control the circumferentialmodes, which could cause issues with uneven pad wear, rotor cheek warping, and durability. Thepractical approach is to limit the response levels of these modes by increasing rotor stiffness throughincreasing cheek thickness and rotor damping (damped iron castings, rings, etc.), trying to userotor asymmetry (slots, weights, etc.), and choosing pad with a proper lower coefficient of friction,in addition to the screening and optimization of insulator, chamfer, and lining shape which are themore flexible and feasible approaches. It is noted that the friction coefficient and pad stiffness aswell as rotor stiffness have interactive relationship to the high-frequency squeal [107] as illustratedin Figure 7.37.

7.5.2 STIFFNESS OPTIMIZATIONS

Component stiffness usually has important effects on brake squeal generation. It is necessary todesign the brake pads, rotor, and calipers such that their natural frequencies in the audible range are

0.10

0.2

0.4

0.6

0.8

1

1 10Pad stiffness (MN/m)

Stable region

Unstable region1 MN/m

10 MN/m100 MN/m

CO

F

100

Rotor stiffness

FIGURE 7.37 Stability mapping of coefficient of friction versus brake pad stiffness for varying levels of brakerotor stiffness.



as isolated as possible to suppress mode coupling. However, it is difficult to decouple the modesbecause of the closely spaced modes. On the other hand, there could be an optimum value ofstiffness for each component that makes the system more stable. Some research shows that thestiffer pad usually increase squeal propensity. Pin-on-disk model shows that as the contact stiffnessincreases to a certain level, two modes could merge to generate a coupled mode, however, when thecontact stiffness further increases, the complex mode splits into two modes again. The influence isnot monotonic.

One way to regulate the modulus of rotor is to change the ingredients of cast iron. One basicindex quantifying the composition of cast iron is the carbon equivalent (CE). CE of gray cast iron isindicated in the following formula [108]: CE¼% carbon þ (% silicon)=3.

The carbon equivalent is a quantity that is commonly reported in the chemical analysis of graycast iron. The % phosphorus is included with the % silicon. The modulus of elasticity of gray castiron is affected by its chemical composition and microstructure. It is not a constant. This is a uniquecharacteristic that is not found in other ferrous materials such as steel. The carbon and siliconcontent as measured by the carbon equivalent of gray iron have a significant effect on modulus ofelasticity. The relationship between carbon equivalent and modulus of elasticity is almost linear forcertain range of values. It can be approximated by the following equation over the range3:3 � CE � 4:8%: E (GPa)¼ 310.3�54.13CE.

A parameter that can impact the propensity to squeal is rotor stiffness. Roughly, squeal propensityis inversely proportional to rotor stiffness. This can be looked upon as increasing the mechanicalimpedance of the rotor and therefore making it more resistive in responding to input forces.An associated and potentially more significant factor of increased rotor stiffness is the shifting ofnormal response modes and the resulting reduction in modal density. The reduction in modal densitycan inhibit the cross coupling of the circumferential and normal modes of the rotor disk.

Moreover, this solution technique could be used to decouple the caliper and rotor modes. Forinstance, the simplest modification that could be made to avoid low-frequency squeal is to substitutea material change for gray cast iron. Some new iron rotor was tested and demonstrated to be able toeliminate low-frequency squeal. Further investigation into the mechanism that eliminated the noiseindicated that damping did not play a role in the solution. Frequency response measurementscomparing gray iron versus ‘‘damped iron’’ rotors show that the resonance at 2600 Hz was shifteddown in frequency approximately to 400 Hz. Damping values measured at those particularfrequencies are the same. Therefore, the modulus change of the rotor not its damping characteristicwas the key to the noise reduction.

Another similar example is given by Kung et al. [109] who illustrated to shift the naturalfrequency of the rotor by changing the amount of graphite in the cast iron. This material substitutionwas classified as ‘‘high carbon equivalent’’. Figure 7.38 shows real and imaginary parts of thecomplex eigenvalues at 2.5 kHz when Young’s modulus of the rotor varies from 96 to 144 GPa.There are two distinct modes at different frequencies when it is small. When it reaches101 GPa, these two modes couple into a complex conjugate pair. The instability reaches itsmaximum value when it equals 120 GPa and starts to go down as it increases beyond 120 GPa.When it equals 140 GPa, the two modes decouple and become stabilized again. These resultsindicate a range of instability in terms of the rotor stiffness. The high carbon equivalent rotor isoutside this range. By changing the rotor material, the 3ND rotor resonance is also shifted. Thiscauses the decoupling of the modal interaction, which may result from the modal coupling of therotor and the caliper or from that of the rotor and the caliper-bracket subsystem.

7.5.3 DAMPING

The use of viscoelastic material (damping material) on the back of back plate can be effectivewhen there is significant pad bending vibration. Increasing pad damping is likely to reduce high-frequency squeal.



There are typically two kinds of brake squeal reduction mechanisms with respect to noiseinsulators. One is the isolator effect between the shoe plate and the caliper where the insulatoraffects the mechanical impedance between the brake caliper and disk brake shoe. The other one isthe damping effect.

To illustrate the effect of damping, the case discussed by Shi et al. [110] is described as follows:in the baseline model, no damping was assigned to any component as shown in Figure 7.39a. Whendamping was included in the viscoelastic layer, some of the unstable modes become more stable.

If no sliding between the metal and shoe was allowed (using an artificially high Young’smodulus for the viscoelastic layer), the stability pattern is different (Figure 7.39b). It is concludedthat: (1) the insulator provides damping, which makes some unstable modes more stable and (2)viscoelastic layer provides sliding, which has a different stability characteristic from the hypothet-ical nonsliding layer.

Increasing pad damping can reduce high-frequency squeal. Complex mode analysis shows thatincreasing damping in pad or rotor can help to stabilize the system. Using insulator to reduce diskbrake high-frequency squeal has been the common practice in brake community. But at lowfrequency where the damping could increase the noise due to the increase of coupling, there existsa competing of increasing and reducing vibrations and noise. The damping effect on low frequencymay not be linear or monotonic at relative low damping level.

The effectiveness of the brake noise insulator is dependent upon pressure, temperature, andfrequency. This dependence is due to brake noise insulator being constructed of viscoelastic mater-ials, which is typically adhesives and rubber-like compounds. The dynamic material properties ofthese types of materials can vary widely with temperature and frequency. As a result the performanceof brake noise insulators, whether supposedly designed to mismatch mechanical impedance or to addstructural damping, can vary significantly in the exposed operating environment.

As displayed, the level of damping in the assembly can change significantly versus temperature.If damping optimization is applied, the damping in the brake shoe assembly should be optimized forthe temperature range at which squeal occurs.

7.5.4 GEOMETRY OPTIMIZATION

For the purpose of seeking cost-effective solution, suppliers are usually reluctant to modify thefriction materials, they prefer to modify the pad geometry and optimize the insulator.

Changing the coupling between the pad and rotor by modifying the shape of the brake pad hasbeen found effective. Other geometrical modifications that have been successful include modifyingcaliper stiffness, the caliper-mounting bracket, pad attachment method, and rotor geometry.

90–100 2000

2150

2300

2450

2600

2750

2900

–75

–50

–25

0

25

50

75

100

100 110 120

Rotor Young’s modulus (GPa)

Rea

l par

t of e

igen

val

ue

Fre

quen

cy (

Hz)

130 140 150 90

(b)(a)

100 110 120

Rotor Young’s modulus (GPa)

130 140 150

FIGURE 7.38 Complex eigenvalues of the 2.5 kHz mode as functions of rotor Young’s modulus: (a) real partand (b) imaginary part.



Brake squeal is usually sensitive to pad stiffness and a factor that can affect brake pad stiffness ispad geometry. Tighter tolerance of pad compressibility is quite important. Brake pad geometry canalso have effect on the brake pad pressure distribution. Chamfer has been used to decouple the padand rotor mode shape alignment and to decrease the sprag–slip possibility. Significant variation inthe noise was observed for the various configurations.

Figure 7.40 depicts some typical approaches to optimize pad geometry given by DiLisio et al.[111]. End and circumferential chamfers are introduced intently to adjust pad loading. Single,double, and diagonal slots are employed intently to break up mode shapes and shift resonantfrequencies. Deeper rotor undercut is used intently to alter resonant response characteristics ofrotor. The end chamfers are experimentally verified to be very effective. Some geometric modifi-cations like certain slots could have adverse effects, which exacerbated on noise.

Chamfer has been widely used to decouple the pad and rotor mode shape alignment and todecrease sprag–slip possibility. Figure 7.41 shows the stability chart of a brake corner [110], theunstable mode at 7.3 kHz is the rotor–pad mode coupling with the rotor radial and axial directionmotion. By cutting chamfer 20 mm on both leading and trailing edges, it is found that unstable modeat 7.3 kHz becomes stable.

System model can be used to help the optimization of the pressure profile of pad. Nextwe present a case given by Swift [112]. The baseline and optimal pressure profiles for an

–2,0000

(a)

(b)

2,000

4,000

6,000

8,000

Without damping

With damping

Without sliding

With sliding

10,000

12,000

12,000

–1,000 0 1,000 2,000


–1,0000

2,000

4,000

6,000

8,000

10,000

–500 0 500 1,000


Fre

quen

cy (

Hz)

Fre

quen

cy (

Hz)

FIGURE 7.39 (a) Effect of insulator damping (no other damping included) and (b) effect of insulatorviscoelastic layer stiffness.



outboard pad are shown in Figure 7.42. For the baseline design, the leading edge of the lining isto the right of the figure and there is a significant increase in pressure over the leading-half of thepad. This is due to the moment generated by the shear forces acting on the pad lining andthe reaction force (which is offset from the lining) between the shoe steel and anchor. Also, notethe significantly higher pressure present at the outer diameter of the lining, due to the inherentstiffness characteristics of the fist-type caliper. A series of design modifications are implemented bychanging the offset to the trailing edge for the lead piston, trail piston, lead caliper finger, centercaliper finger, and trail caliper finger. These changes result in a pad load standard deviationreduction of 29%, from a baseline value of 0.917 to an optimal value of 0.650. The optimal pressureprofile for the outboard pad is contrasted to the baseline in Figure 7.42. Note the significantreduction in pressure on the leading-half of the lining along with the reduction in radial pressuredisparity.

For high-frequency squeal as well as low-frequency squeal, it may be useful to treat rotor withrotor asymmetry, higher stiffness, or higher damping. Rotor asymmetry treatment like slots, unevenvane spacing, and weights can decouple the repeated roots, making it much more difficult toexcite the rotor. It prevents waves from developing within a rotor that make it unable to radiatesound. Radial slits in a rotor reduce its propensity to radiate sound by preventing the circumferential

(a) (b) (c)

(d) (e) (f)

FIGURE 7.40 Pad and rotor geometry changes for noise countermeasure proposals.

FIGURE 7.41 Effect of chamfer for a brake corner.



waves from fully developing. The transfer functions of the rotor show a reduction of peakamplitudes with increasing slit length. The insertion of a metal piece into a radial slit nearlyeliminates the sharpness of the peaks in the transfer function of the rotor. The use of ring dampersaround the circumference of a rotor can completely eliminate the peaks in the bending-wave transferfunction of a brake rotor.

However, the degree of asymmetry necessary to eliminate the excitation of the rotor couldresult in significant asymmetry from a thermal mass perspective, which is not desirable and greatlylimits its real application. By stiffening a rotor, the number of axial modes in the audible frequencyrange and the magnitude of the response levels can be reduced. Stiffening the rotor in the axialdirection can be achieved by using thicker rotor cheeks, more number of vanes, and wider vane width.

One option is to use the hat portionwith nonrotational shape where the thinner wall and the thickerwall sections are arranged at 908, as shown in Figure 7.43a [38]. The rotor of this shape was verifiedusing dynamometer, and the in-plane brake squeal is suppressed as shown in Figure 7.43b.

–500

1

2

3

4

5

6

7

8

–40 –30 –20 –10 0 10

Contact point horizontal position (mm)

Pre

ssur

e (M

Pa)

20 30 40 50

FIGURE 7.42 Lining contact point horizontal position optimization: baseline and optimal pressure profiles forthe outboard pad (dashed line: baseline; solid line: optimized).

(a) (b)

Conventional rotor

Squ

eal g

ener

atio

n ra

te (

%)

0

20

40

60

80

100

Shaped rotor

FIGURE 7.43 The rotor with nonrotational hat element: (a) the shape and (b) squeal evaluation.



7.5.5 OTHER APPROACHES

Several other novel methods for the suppression of brake squeal were proposed. One such methodthat was recently proposed by Cunefare et al. [113–116] involves using dither to eliminate squeal.This is achieved by placing a piezoelectric transducer in contact with the backing plate of theinboard pad of friction material in a floating caliper disk brake as illustrated in Figure 7.44.The transducer can be conveniently placed in the piston. The piezo-ceramic (PZT) stack wasused as the dither implementation device and was housed inside the brake piston as shown inFigure 7.44.

Figure 7.45a shows the spectrum of sound pressure level during squeal before dither control isapplied.

The prominent spike at 5.6 kHz is the squeal tone. There are clear harmonics at 11.2 and16.8 kHz as well, though these tones are some 20 dB down from the main tone at 5.6 kHz. A weakerthird harmonic is present at 22.4 kHz, some 38 dB down from the fundamental.

Figure 7.45b depicts the spectrum of sound pressure level after the control has been applied. Thecontrol signal amplitude is 153 V rms. Dither has suppressed the 5.6 kHz squeal and its entireharmonics. The only prominent tone remaining is that of the 20 kHz dither control signal. Thisdither signal is ultrasonic so it was not audible.

Control signal wire

Caliper piston

Brake pad

Backing platePZT stack

FIGURE 7.44 PZT actuator inside caliper piston.

(a) (b)

FIGURE 7.45 Sound pressure spectrum from the brake system during stages of 20 kHz control: (a) beforecontrol activation and (b) after full control. (Reprinted with permission from Cunefare, K.A. and Graf, A.J.,J. Sound and Vib., 250, 4, 579, 2002. Copyright 2002, Elsevier.)



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59. Lee, L., Xu, K.Q., Malott, B., Matsuzaki, M., and Lou, G., A systematic approach to brake squealsimulation using MacNeal method, SAE Report 2002-01-2610, 2002.

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66. Fieldhouse, J.D. and Rennison, M., An investigation of low frequency drum brake noise, TechnicalReport 982250, SAE, Warrendale, PA, 1998.

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68. Chowdhary, H.V., Bajaj, A.K., and Krousgill, C.M., An analytical approach to model disc brakesystem for squeal prediction, Proceedings of DETC’01, DETC2001=VIB-21560, ASME, Pittsburgh,PA, 1, 2001.

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73. Beloiu, D.M. and Ibrahim, R.A., Disc–pad interaction related to brake squeal, SAE 2004-01-0826, 2004.74. Ibrahim, R.A., Madhavan, S., Qiao, S.L., and Chang, W.K., Experimental investigation of friction-

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83. Krupka, R., Walz, T., and Ettemeyer, A., New techniques and applications for 3D-brake vibrationanalysis, SAE 2000-01-2772, 2000.

84. Eriksson, M., Bergman, F., and Jacobson, S., Surface characterization of brake pads after running undersilent and squealing conditions, Wear, 232, 163, 1999.

85. Bergman, F., Eriksson, M., and Jacobson, S., Influence of disc topography on generation of brake squeal,Wear, 225, 621, 1999.

86. Eriksson, M., Friction and contact phenomena of disc brakes related to squeal, Comprehensive Summar-ies of Uppsala Dissertations from the Faculty of Science and Technology, 537, 2003.

87. Eriksson, M., Bergman, F., and Jacobson, S., On the nature of tribological contact in automotive brakes,Wear, 252, 1=2, 26, 2002.

88. Bergman, F., Eriksson, M., and Jacobson, S., The effect of reduced contact area on the occurrence of discbrake squeals for an automotive brake pad, Proceedings of the Institution of Mechanical Engineers,D214, D5, 561, 2000.

89. Eriksson, M. and Jacobson, S., Friction behavior and squeal generation of disc brakes at low speeds,Proceedings of the Institution of Mechanical Engineers, D215, D12, 1245, 2001.

90. Eriksson, M., Lundquist, A., and Jacobson, S., A study of the influence of humidity on the friction andsqueal generation of automotive brake pads, Proceedings of the Institution of Mechanical Engineers,D215, D3, 329, 2001.

91. Rhee, S.K., Tsang, P.H.S., and Wang, Y.S., Friction-induced noise and vibration of disc brakes, Wear,133, 39, 1989.

92. Rhee, S.K., Jacko, M.G., and Tsang, P.H.S., The role of friction film in friction, wear, and noise ofautomotive brakes, Technical Report 900004, SAE, Warrendale, PA, 1990.

93. Chen, F., Chen, S.E., and Harwood, P., In-plane mode=friction process & their contribution to disc brakesqueal at high frequency, SAE 2000-01-2773, 2000.

94. Blaschke, P. and Rumold, W., Global NVH matrix for brake noise—A Bosch proposal, SAE 1999-01-3405, 1999.

95. Ahmed, I.L.M., Leung, P.S., and Datta, P.K., Experimental investigations of disc brake friction, SAE2000-01-2778, 2000.

96. Chen, F., Abdelhamid, M.K., Blaschke, P., and Swayze, J., On automotive disc brake squeal, part III:Test and evaluation, SAE 2003-01-1622, 2003.

97. Abendrath, H., Advances=progress in NVH brake test technology, SAE 982241, 1998.98. Mignery, L., Wang, J., and Luo, J., Prediction of damper effects in a brake system model, SAE 2001-01-

3140, 2001.99. Bae, J.C. andWickert, J.A., Free vibration of coupled disk hat structures, J. Sound Vib., 235, 1, 117, 2000.100. Kim, M., Moon, J., and Wickert, J.A., Spatial modulation of repeated vibration modes in rotationally

periodic structures, Trans. ASME J. Vib. Acoust., 122, 62, 2000.101. Tseng, J.-G. and Wickert, J.A., Nonconservative stability of a friction loaded disk, Trans. ASME J. Vib.

Acoust., 120, 4, 922, 1998.102. Chang, J.Y. and Wickert, J.A., Response of modulated doublet modes to traveling wave excitation,

J. Sound Vib., 242, 1, 69, 2001.103. Yang, M., Afaneh, A.-H., and Blaschke, P., A study of disc brake high frequency squeals and disk

in-plane=out-of-plane modes, SAE 2003-01-1621, 2003.104. Lee, H. and Singh, R., Sound radiation from a disk brake rotor using a semi-analytical method, SAE

2003-01-1620, 2003.105. McDaniel, J.G., Moore, J., Chen, S.E., and Clarke, C.L., Acoustic radiation models of brake systems

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106. Jang, H., Lee, J.S., and Fash, J.W., Compositional effects of the brake friction material on creep groanphenomena, Wear, 251, 1477, 2001.

107. Dunlap, K.B., Riehle, M.A., and Longhouse, R.E., An investigative overview of automotive disk brakenoise, SAE 1999-01-0142, 1999.

108. Malosh, J.B., Disk brake noise reduction through metallurgical control of rotor resonance, SAE 982236,1998.

109. Kung, S.W., Dunlap, K.B., and Ballinger, R.S., Complex eigenvalue analysis for reducing low frequencybrake squeal, SAE 2000-01-0444, 2000.



110. Shi, T.S., Dessouki, O., Warzecha, T., and Chang, W.K., and Jayasundera, A., Advances in complexeigenvalue analysis for brake noise, SAE 2001-01-1603, 2001.

111. DiLisio, P., Parisi, R., Rieker, J., and Stringham, W., Brake noise resolution on the 1998 Mercedes-BenzM-Class, SAE 982245, 1998.

112. Swift, R.A., A parametric modeling approach for the preliminary design of automotive disk brakes, SAE2003-01-3330, 2003.

113. Cunefare, K.A. and Graf, A.J., Experimental active control of automotive disk brake rotor squeal usingdither, J. Sound Vib., 250, 4, 579, 2002.

114. Cunefare, K.A. and Graf, A.J., Disk brake rotor squeal suppression using dither control, SAE 2001-01-1605, 2001.

115. Cunefare, K.A. and Rye, R., Investigation of disk brake squeal via sound intensity and laser vibrometry,SAE 2001-01-1604, 2001.

116. Badertscher, J. and Cunefare, K.A., Experimental investigation of dither control on effective brakingtorque, SAE 2003-01-1617, 2003.



IndexA

Accessory belt drive system (ABDS), 2Accessory belt movement, measurement

methods, 313Acoustic emission (AE), 210

calibration method for measuring, 285detection technique for, 250disk roughness effect on, 272disk surface screening using, 289empirical model for, 273–275force identification using, 285friction-induced, 250identification of interface contact using, 281lubricant role in, 283modeling of friction-induced, 272sensing system, 280sensor, 250

step calibration of, 285SL model for, 275stress waves, 250torsional and bending mode frequencies in, 281

Acoustic vibrationfriction-induced, 249, 258preventing methods for, 387

Acoustics, define, 1Active noise control, 27ADCK (Armstrong–Hélouvry, Dupont, Canudas de

Wit, and Karnopp) model, 145; see alsoFriction law

Adhesionliquid-mediated adhesion

capillary pressure forces, 98contact angle, determination, 98Kelvin radius, 99Kinetic meniscus force, measurement, 100Laplace equation, 98liquid condensation in interface, 97–98meniscus in liquid film, 99surface energy, 98viscous force, 101Young’s equation, 99

solid–solid adhesionelectrostatic forces, 93–94free surface energy of adhesion, 96JKR (Johnson–Kendall–Roberts) model, 96–97van der Waals forces, 94–96

Aerodynamic drag force, 251Aging effects, 143

Air bearingharmonics, 283lubricants effect on, 283natural frequency of, 261nonlinear properties of, 260, 262nonlinearity of, 259–260oscillations, 269resonant modes, 268slider

contact vibration of, 260–263nonlinear dynamics of, 256, 258–259

surface-roughness of, 253stiffness coefficients of, 258

American Standard ANSI S3.4, 368Aperiodic motion, 8Asymmetric piecewise linear restoring force, 163Atomic force microscope (AFM)

applications of, 127surface roughness measurement using, 262

Automotive accessory belt drive system, 317Automotive camshaft drive belts, 340Automotive multirib V-belt drive system, 301Automotive transmission fluid (ATF), 233

coefficient of friction, 234

B

Ball drop method, for calibrating AE sensors, 286Band-pass filters, 250, 290, 293Beams, 23

lateral motion of, 33Timoshenko beam theory, 33transverse vibration of, 29

Beating phenomenon, phase shifts, 177Belt chirp noise phenomenon, 326Belt dynamic friction, testing of, 306Belt friction-induced noise, 302Belt impact dynamic forces, 301Belt slip noise, 329Belt system torque, methods for estimating, 305Belt-pulley

interface, 303system, characterization of, 309

Bernoulli-Euler beam, equation of motion for, 311Bilinear hysteretic behavior, 146Binary instability, two-degree-of-freedom

model, 178–179Bouc–Wen model, 146–147; see also Friction law


401

Brake friction materials, 388Brake hot judder, 351Brake insulator, impact of damping and Young’s

modulus effect on, 382Brake judder, 351Brake noise

dynamometer test, 369features associated with, 372insulator, effectiveness of, 391numerical approaches and analysis of, 354

complex eigenvalue, 354–357interactive, 359–361real modal, 354transient, 357–359

performancedynamometer test for evaluating, 363–367humidity effect, 377–378subjective rating of, 367verification test of, 367

random impulsive excitation of, 354tribology test, 369–372

Brake padapplications, 370assemblies, 347geometry, effects of, 392

Brake squeal generationcontact stiffness, 380–382stiffness optimizations and effects on, 389structural effect on, 383–386suppression of, 395

Brake squeal instability, characteristic of, 356Brake system; see also Vehicle brake system

distribution of frictional heat, 351friction-induced vibration and noise in, 348principles of, 347structure and materials, 347theoretical analysis of, 388tribology of, 378types of, 347

C

Certificationglide testing for disk surface finish, 289nanometer clearance and calibration, 291

Chaos, behavior of system, 47, 48Chaotic motion, 164–165Chassis dynamometer tests, 369Chirp noise, procedure for controlling, 318Circular disk, 34

in-plane natural modes of, 39out-of-plane natural modes of, 38

Circumferential disk motion, time-frequencyspectrum of, 359

Complex planar, of roots, 12

Contact start-stop (CSS), 117Contact vibration model, 163Control coefficient of friction (COF), 348Corrosion-type brake judder, 351Coulomb friction, 144–147, 151, 220

force, 12law, 144

Crankshaft pulley (CRK), 303Crankshaft torsional vibration, 313Creep groan, 350

D

Damping material, use of, 390Discrete systems, 23Disk brake

application of, 347shoe, 391

Disk platesin-plane vibrations, 38–39out-of-plane vibrations, 34–38

Disk surface finish certification, glide testingfor, 289–290

Disk thickness variation (DTV), 349, 351, 388Dithering effect

equation, 160friction characteristic, 161

Drum brakes, applications of, 347Dry friction

average shear stress, 102contact and slope contact of asperities, 101energy dissipation during friction, 107–108environmental and operational condition,

effect of, 114–115load and velocity effect, 115temperature dependence, 114

friction coefficient dependence onparameters, 102

friction mechanisms, 101–102adhesion, 102–104deformation slope, 105–107

friction transitions, 110–112major phenomena, determining friction, 101plowing of hard sphere asperity, 107rubber friction, 108–110static friction, hysteresis, time, and displacement

dependence, 112friction-velocity curve, 113microslip of magnetic recording tape belt, 114

Dry slip noise, characteristic of, 331Duffing equation, 40; see also Nonlinear vibration

systemsDynamic friction coefficients, 387Dynamic groan, 350Dynamometers, 364



E

Eigen equations, for eigen frequency, 38Eigenvalues, 18

coefficient matrix in averaged equation, 175damped system, 20–21system mode as functions of friction coefficient,

177, 178undamped system, 18–20

Eigenvectors, 18–21Elastic continuum vibrations, 144Elastic wave, particle motion produced by, 250Electromagnetic dynamometer, 307Electronic speckle pattern interferometry, 362Electrostatic force, 94, 256–257Euler belt equation, use in calculating coefficients

of friction of belts, 305

F

Floating caliper disk brake, 395Flying height (FH), 249, 262, 271

electrostatic force on, 272Forced vibrations, 8

oscillator with friction law II, 154SDOF system with

friction damping, 16–17viscous damping, 13–16

Fourth-order Runge–Kutta method, 54; see alsoNonlinear vibration systems

Free energy for electrostatic interaction, 94Free vibrations, 8

friction oscillator, 152under unit initial condition, 166

Frequency response function (FRF), 287Friction, 1, 85

adhesion, 93behavior at bow point, 219damping, 16dry, 101force, 146, 171law, 143–151three-degree-of-freedom system with, 192two-degree-of-freedom

model with, 170system with modal coupling, 176

wet, 115Friction assessment and screening test (FAST)

machine, 370Friction law, 104, 124, 141, 143Friction-induced acoustic emission, modeling

of, 272–273Friction-induced instability and sound

in geological science and engineeringDietrich–Ruina law, 224

filtered sound, 228friction-induced instability, 226friction–sound level, 227, 228Hopf bifurcation, 225sediment grain using friction sound,

characterization of, 226–229sound level and grain size, correlation, 229stick–slip and dynamic instability in

earthquakes, 224–226transient and steady-state changes, of

coefficient of friction, 225Friction-induced noise, 340–341Friction-induced torque oscillations, 347Friction-induced vibrations, 141, 143

intermolecular forces effect on, 263–264in MEMS devices and equipment, 229

contact model of ultrasonic motor with statorand slider, 230

stick–slip in contact mode AFM, 231–232ultrasonic motor, 229–231viscous yield relaxation model, 232

modeling approach, 187and noise in vehicles

coefficient of friction of gear teeth, 235friction force vs. roll angle, 235friction noises of tire–road interaction

and wheel–rail interaction, 238–239gear noise, 234–237noise and vibrations in clutches, 232–234piston stick–slip noise in engine, 237sound pressure level in gear box, 236squeak and rattle in automobiles, 239–242wiper blade noise, 237–238

Friction-induced vibrations and sound, 1, 2, 215flat belts, friction drive, 2friction music sound and synthesis

bowed string instruments, 215–219friction sound synthesis, 220

friction sound in naturecreature friction sound, 220–222nature sand sound, 222–223sound and movement, correlation, 222sound-producing mechanism, Caribbean spiny

lobster, 221sound-producing sand avalanches, 222–223

in information storage industry and automotiveindustry, 2

V-belts and V-ribbed belts, on coefficientof friction, 2

Friction–pressure dependence and hysteresis,375–376

Friction–velocity dependence and hysteresis, 374Friction–vibration interactions

normal oscillations, effect of, 125–127stick–slip, 122–125


Index 403

coefficient of friction vs. relative airhumidity, 123

coefficient of friction vs. slip velocity, 123dry and wet coefficient of friction vs.

velocity, 122periodic and chaotic, 124stick–slip amplitude, 125

G

Gaussian distribution, 253Glide height characterization, 291Grazing bifurcation, 164Groan

causes of, 387impact on braking system, 349–351as symptom of any brake malfunction, 350

H

Hamaker constant, 95, 96, 264Harmonic oscillation frequencies, 322Helmholtz motion, 216Hertz model, 88, 96, 97; see also AdhesionHertz theory, 286

approximation for, 144elastic contact of, 278

High-frequency (HF) asperity counting method, 43High-frequency cross-sectional resonant mode, 314HLSZ

air-bearing contact model, 253model, for deremining effect of intermolecular

pressure on smooth surface, 264Holographic interferometry, 362Hooke’s law, 23, 28, 80Hysteresis, 164

I

In-plane (IP) free vibrations, 38Intensity level (IL), 69Interface aging effect, 372Intermolecular force, 93–96, 256

friction-induced vibrations with effectof, 263–264

J

Judderbraking system impact on, 349–351causes of, 388types of, 351

L

LACT testing, 367Laser Doppler Vibrometer (LDV)

signal for calibrating AE sensor, 288slider velocity measurement using, 267use in measuring and analyzing slider-bump

response, 293Laser Doppler vibrometry, 362Lateral excitation boundary condition, due to

slippage, 328Lennard–Jones potential, 94Limit cycle

motions, 172for Rayleigh’s equation, 47

Lindstedt–Poincare method, 45; see also Nonlinearvibration systems

Linear forced vibrations of system, 249Linear harmonic oscillators, 289Linear multi-degree-of-freedom system, 17Linear single-degree-of-freedom system, 10Linear stiffness coefficient, 260Linear vibrations, 7

linear multiple-degree-of-freedom(MDoF) system, 17

eigenvalues and eigenvectors, 18–21forced vibration solution, 22–23

linear single-degree-of-freedom (SDoF) system, 10free vibration with viscous damping, 10–12with friction damping, 16–17with viscous damping, 13–16viscous and friction damping, 12–13

vibration of continuous systems, 23Galerkin method, 33–34longitudinal vibration of rods, 27–28torsional vibration of shafts, 28–29transverse vibration of beams, 29–33transverse vibrations of string, 23–26vibrations of disk plates, 34–39wave equation, 27

vibration motion, 7–8complex and sinusoidal periodic

motion, 8–9transient nonperiodic motion, 9

Los Angeles City Traffic (LACT), 367Low-frequency squeal, 351–354

M

Macroscopic bodies, sliding behavior of, 144Macroscopic radial slippage, 304Magnetic hard disk drive

contact and friction-induced vibrations ofslider in, 251

signal read-write function of, 251



Mark VI method, 368Mathieu’s equation, 52; see also Nonlinear vibration

systemsstability chart, 53

MEMS technology, for designing impact sensorarrays, 286

Meniscus forcefriction-induced vibrations effects on, 264–266lubricants, 257

Micro-electromechanical systems (MEMS), 128Microphone signals, power spectra of, 368Molecular dynamic, 143Molecular repulsive forces, 256Moon and Wickert’s approach, for determining

stick-slip motion of misalignedV-belts, 322

Multiple-degree-of-freedom (MDoF),7, 17, 18, 20, 22

explicit and implicit integration schemes, 54linear system, 287

N

Nano- and molecular scales, friction, 127–132dimensionless friction stress, 128Frenkel–Kontorova model, for molecular

dynamics simulation, 130–131meniscus formation, different time steps, 132time and length scales, of friction

problems, 129Narrowband (NB) channel, air-bearing and slider

body resonance modes, 293Narrowband resonance monitoring, 280Near-contact recording interface, 258Newton’s second law, application, 24, 29, 30Noise dynamometer tests, 369Nonasbestos organic (NAO) fibers, 348Noncontact optical techniques, 361Nonlinear dynamic coefficients, 260Nonlinear oscillations, 142, 267Nonlinear system, amplitude-frequency

relations, 44Nonlinear transient analysis, for brake noise, 357Nonlinear vibration systems

multiple-degree-of-freedom systems, 54parametrically excited system, 52–54perturbation method, 40–45

amplitude-frequency relations, 44phase plot, 47–48Rayleigh’s equation, 46–47stability of equilibrium, 48–50

focus, l1,2 complex conjugates, 51node, l1,2 real and same sign, 50saddle point, l1,2 real and opposite

sign, 50

Taylor’s series, 49vortex, l1,2 imaginary, 51

variation of parameter, method of, 45–47Nonstationary=nonperiodic resonance, 54Nonstationary vibrations, 52Normal mode summation method, 23

O

Objective noise index (ONI), 368Oden–Martins law, 357Off-vehicle brake material test methods, 369Operational deflection shape (ODS), 359, 361Oscillator

with friction law II, bifurcation, 154with friction law III, 155

bifurcation maps, 155–156with friction laws II and III, 155

Oscillatory normal force components, 348

P

Pad bending vibration, 390Pad friction materials, 347Pad vibrations, 348Pad wear debris, agglomeration and

compaction of, 380Pad-rotor interface, 381Peak-to-valley roughness, 249Piezo-ceramic (PZT) stack, 395Piezoelectric transducer (PZT), 250, 272Pin-on-disk model, 382Poincaré section, subcritical chaotic transition, 167Poisson’s ratio, 35, 80, 88Power transmission belt drive system

coefficients of friction of, 305dynamics of, 301features of, 307friction driven, 302transverse motion for, 310vibration analysis of, 310

PZT glide sensor, 286PZT sensor, 272

R

Radio waves, 27Random impulsive excitation, of brake noise, 354Random vibrations, 54

arbitrary function input, response of SDOF systemto, 58–61

Dirac delta function, 59equation of motion, 58


Index 405

frequency response method, 60impulse response method, 59

autocorrelation function, 57–58Gaussian probability distribution, 58measurement, 58

cross-correlation function, 63–64Gaussian random process, 55–57joint probability density function, 63power spectral density function, 61–62probability density function, 55–57

probability density curve, 56random input

MDoF systems to, 65–67SDoF system to, 64–65

sample functions, ensemble of, 55Rayleigh waves, 210Resonance solutions, vibrations, 166–167Rods, 23

differential element and displacement, 28longitudinal vibration of, 27, 29

Rotating brake rotor, 362Rotating system modes, radiation

characteristics of, 386Rotor

acoustic radiation efficiencies estimationof, 386–387

bending-wave transfer function of, 394stiffness, 390

S

SAE dynamic friction tester, 306Scanning laser Doppler vibrometer, 362Scanning laser vibrometer, 364Second-order harmonic oscillator, 287Self-excited oscillations, 47Self-excited vibration, 8

closed form solution, 157excitation speed, various range, 157–158

friction oscillator, 153Semimetallic friction material, 380Shafts

polar moment of inertia, of cross-sectional area, 29torsional vibration of, 28

Single-degree-of-freedom (SDOF), 7Slider air bearing, resonant modes of, 293Slider airbearing modes, 283Slider bouncing vibrations, 264, 266Slider roll mode frequencies, 257Slider system dynamics, 250Slider–disk

contact model and interactions, 252–253, 257interface of, 251, 279intermolecular force-induced instabilities

in, 256

Slider–lubricant interactions, effects on acousticemission, 283

Slip and misalignment noises, causes of, 304Slip noise generation, 331Slip-line theory, 105; see also Dry frictionSnap-back chirp phenomenon, 327Snell’s law, 77Solid surfaces, contact

multiple asperity contacts, 88identical asperities, 89–90nominally flat rough surfaces, statistical

analysis, 90–92plastic deformation, 92–93

spherical contact, 87–88Sound fundamentals

airborne sound and measures, 68–69differential element in analysis of, 73

elastic solid medium, stress wave propagationin, 78–83

elongation ratio or strain and shear strain, 79Rayleigh wave, speed, 82–83relationship for rotation angles, 79spherical wave, 81wave propagates in terms velocity, 81

nonstationary sound, noise and spectrum analysisof, 69–71

sound interference, 76–78sound refraction, 77

sound perception and weighting curves, 71–72Fletcher–Munson loudness curve, 71friction-induced sound, 72loudness of sound, 72

sound radiation of structures, 75–76sound wave equation, 72–75

weighting curves, 73Sound power level (SWL), 69Sound pressure levels (SPL), 210Sound waves, 27Sprag–slip motion, 143Spring characteristics, hardening and softening, 41Squeal noise

factors affecting, 333methods for reducing, 388

Static friction, 145Stationary brake system modes, 386Stick–slip

boundary motion, 327induced vibrations, 318motion, 142vibrations, in vehicle braking system, 350

Stochastic subspace identification, 362Stribeck diagram, 279Stribeck effect, 145String diameter, 218, 219String motion, contact point, 216



Subjective noise index (SNI), 367Superharmonic resonances, 259Surface roughness

random rough surfaces, characteristics, 85–86roughness in different scales, 86

surface roughness parameters, 86–87centerline-average, 86height of surface from centerline s, 86–87root mean square roughness, 86

Surface topography, of supersmooth disk, 271

T

Take-off and touch-down hysteresis effects, 258Take-off height (TOH), 252, 289Tangential rotor modes, types of, 353Tensioner (TEN) pulley, 303Thermoelastic distortion, due to frictional

heating, 351Thin film piezoelectric sensor, 286Timing belt friction-induced noise, 339Torque transducer, 307Torsional vibrations, 218–219Trade-off friction law, 144Transient motion, 9Transverse traveling wave, height, 27Transverse vibration

beam, 30string, 23–27

Traveling waves, 27Tribology

geometry scales for molecular, nano-, micro-,and macrolevel, 129

hard disk drive, 100, 128micro-electromechanical systems (MEMS), 128

V

V-belts, stick–slip motion of misaligned, 322V-rib-based serpentine belt drive systems, 306V-ribbed belt

applications, 301drive system, 303, 305friction noise, 305slip noise features of, 330

Van der Pol’s equation, 40, 44, 47; see also Nonlinearvibration systems

Van der Waals forces, 257Van der Waals potential, 94, 95Vehicle brake system

friction-induced vibrations and noiseimpact on, 347

types of, 347Vehicle road test, 367Velocity spectrum, 194

Velocity-dependent friction, 142Vibration localization phenomenon, 312Vibration motion, 7–9Vibration, of uniform slender rod, 27Vibration–velocity spectrum, 193Vibrations and sound, of continuum systems with

friction, 194beam transverse vibrations

axial stick–slip excitation, 197–198bowing effects, 195–196with modal coupling due to geometry

constraints, 198–200instability of waves under frictional sliding of

strong contact, 210–215dynamic equation of motion, in self-adjoint

form, 212equation of motion in axial direction, 211friction force, calculation, 211influence of body waves, 214–215instability of interface waves, 214nondimensional equation of motion, 212periodic boundary conditions, 212rectangular dilatational wave, 215rigid-body solution, 212surface waviness effect of, 214unstable characteristic solutions, 213

longitudinal vibrations, rod with velocity-dependent friction, 194–195

amplitude calculation, 195equation of motion, 194friction law, used, 194–195

rubbing sound wave under weak contact, 206–210averaged stress, 208cutoff frequencies, 209equation, for shear waves inside elastic

continuum, 207frequency spectrum of sound intensity field,

208–209rubbing surfaces of two solids, 206–207shear stress, 207–208stress fluctuation, 208

sprag–slip excited vibrations, 203–206coefficient of friction–velocity curves, for

clockwise disk speed, 204–205friction and normal force, relation, 206friction forces for disk speed of 3 rpm

clockwise, 204–205probability density function of friction force,

204–205spragging, 203–204

weak contact and strong contact, randominteractions of friction, 200

bending and longitudinal vibrations of beam,201–202

circumferential vibration equation of disk, 202


Index 407

mode lock-in between beam and disk, 202–203random interactions of friction, 200–203

equation of motion, 200transverse vibration, characterization,200–201

weak contact and strong contact, 200Vibrations, multi-degree-of-freedom systems with

frictioncomplex modal analysis, 179–186

bifurcation roots, 181coupling strength, 182damping ratio, 182dynamic normal force, 179effect of damping, 185eigenvalue equation and second-order

series, 180, 181equation of motion, 179, 180, 182first-order eigenvalue problem, 185Hopf bifurcation, 181nonsymmetric mass matrix, 182–1833D plot of eigenvalue, 184slope friction vs. velocity, 186tangential friction, 179

dither effect, 159–162internal combination resonance, 173–175

eigenvalues of coefficient matrix, 175in multi-degree-of-freedom system, 175parametric damping, 173parametric excitation, 175three mass system, 173–174

mode-coupling instability, 175–179binary instability, 178, 179effect of system parameters on the instability,

176–177two-degree-of-freedom system, 176

negative damping instability, 169–173limit cycle motions, 172two-degree-of-freedom model with

friction, 170nonlinear numerical analysis, 189–194

numerical approaches, 186–189Vibrations, single-degree-of-freedom systems with

friction, 141closed form solution for, 156–159contact vibrations, 163–164

features, 164–167frequency response, 167–168variable normal force and vibro-impact,

168–169forced vibrations, 153free vibrations, 152friction law, 143–151

artificial neural net, 150Dahl model, 147data fitting–based empirical models, 150elastic regime, formulations, 148LuGre model, 147

neural net friction model, 151neural net training, 151Oden–Martins law, 149Prandlt’s elasto-plastic material model, 147Stribeck features of real systems, 144

profiles of, 151–152SDOF, sliding friction and external

excitation, 151self-excited vibration and forced vibration,

combined, 153–156self-excited vibrations, 152–153variable normal force, 162–163

Vibratory behavior, of dynamic systems, 8Vibro-impacts of system with friction, 163; see also

Vibrations, single-degree-of-freedomsystems with friction

Violin string, 215, 217Viscous friction, 145

W

Wavelet–time–frequency analysis, 286Waves, broad classes, 27Wear debris compaction, 380Wear resistance

spectra, 370tooth-facing material, 303

Wet beltdynamic friction, 336interface, 335slip noise, 338

Wet frictionnegative slope of friction-velocity curve,

120–122dry and wet coefficient of friction vs. velocity,

121–122friction-velocity curves, 121

Stribeck curve, 115–117unsteady liquid-mediated friction, 117–120

film thickness, hydrodynamic lubrication, 119specific film thickness or lambda ratio,

calculation, 119–120static friction coefficient, 119static friction force vs. CSS operation cycles,

117–118Wideband (WB) channel, air-bearing and slider body

resonance modes, 293

Y

Young’s modulus, 35, 88

Z

Zwicker method, for calculating specificloudness, 369



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Friction-Induced Vibration and Sound Principle and Application