~~~'~;~'~~;~~~~;.~ programs In 5trUcturai 7:~~:f~::~;,::,,~:~~'~r~~~1!11 '~; Dyn;t.mics, 4th edition features a ti ~ an 'introduction to the dynamtc analysis of s'U'Uctu;es

Method a new addition to the chaptet on Random Vibration' ~ "~i~nse of strtu:tuni~. c.... . ,',

modeled as 11 multi degrea-of.freedom 5},st.em,5ubje

Structural Dynamics Theory and Computation Fourth Edition

. I

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Structural Dynamics Theory and Computation Fourth Edition

Mario Paz Speed Scientific School University of Louisville Louisville, KY

m CHAPMAN & HALL

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Copyright 1997 by Chapm.n & Hall

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2 3 4 5 6 7 B 9 10 XXX 01 00 99 98 97

Library of Congress Ca!aloging~in.l?'ublication Data

Paz.. Mario. Struct~ dynamics: theory and .;::ompulation I by Mario Paz. ~ 4th

CONTENTS

PREFACE TO THE FOURTH EDITION I xv PREFACE TO THE FIRST EDITION I xxi

PART I STRUCTURES MODELED AS A SINGLE-DEGREE-OF-FREEDOM SYSTEM 1

UN;)I\MPE;) SINGLE-DEGREE-OF-FREEDOM SYSTEM 3 U Degrees of Freedom I 3 U Undamped System I 5 L3 Springs in Parallel or in Series I 6 1.4 Newton's Law of Motion i 8 LS Free Body Diagram I 9 1.6 D' Alembert'S Plinciple I 10 1.7 Solution of the Differential Equation of Motion I II 1.8 Frequency and Period I 13 1.9 Amplitude of Motion I 15 1.!O Undamped Single-Degree-of-Freedom Systems Using COSMOS I 20 U I Summary I 22

Problems i 23

viii Contents

2

3

4

DAMPED SINGLE.DEGREEOFFREEDOM SYSTEM 3 J

2.1 Viscous Damping i 31 2.2 Equation of Motion i 32 2.3 Critically Damped System I 33 2.4 Overdamped System / 34 2.5 Underdamped System / 35 2.6 Logarithmic Decrement I 37 2.7 Summary / 43

Problems i 44

RESPONSE OF ONEDEGREEOFFREEDOM SYSTElY, TO HARMONIC LOADING 47

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

:3.10

3.1l

Undamped System: Harmonic Excitation / 47 Damped System: Hannonic Excitation / 50 Evaluation of Damping at Resonance I 58 Bandwidth Method (HalfPower) to Evaluate Damping, / 59 Energy Dissipated by Viscous Damping I 61 Equivalent Viscous Damping I 63 Response to Suppon Motion / 66 Force Transmitted to the Foundation I 76 Seismic InstrUments I 79 Response of OneDegree-ofFreedom System to Harmonic Loading Using COSMOS i 81 Summary I 88 Problems I 92

RESPONSE TO GE':-lERAL DYNAMIC LOADING 96

4.1 4.2

4.3

4.4 4.5 4.6 4.7 4.8

Impulsive Loading and Duhamel's Integral I 96 Numerical Evaluation of Duhamel's Integral-Undamped System i 105 Numerical Evaluation of Duhamel's Integral-Damped System I 109 Response by Direct Integration I 11 0 Program 2-Response by Direct Integration I 116 Program 3-Response to Impulsive Excitation f 119 Response to General DynamiC Loading Using COSMOS I 124 Summary I 131

5

6

7

Conten:s ix

FOURIER ANALYSIS AND RESPO':-lSE IN THE FREQUENCY DOMAIN 139

S.I Fourier Analys" I 139 5.2 Response to a Loading Represented by Fourier Series I 140 5.3 Fourier Coefficients for Piecewise Linear Functions / 143 5.4 Exponential Form of Fourier Series I 144 5.5 Discrete Fourier Analysis I ]45 5.6 Fast Fourier Transform I 148 5.7 Program 4-Response in the Frequency Domain i 150 5.8 Summary I 156

Problems I 156

GENERALlZED COORDINATES AND RA YLEtGH'S METHOD 162

6.1 6.2

6.3

6.4 6.5 6.6 6.7 6.8 6.9 6.10

Principle of Virtonl Work I 162 GeneraLlzed Slngle~Degree-of-Freedom System-Rigid Body I 164 Generalized Single-Degree-of-Freedom System-Distlibu~ed Elasticity I ,67 Shear Fo,ces and Bending Moments I 172 Generalized Equation of Motion for a Multistory Building / 177 Shape Function I 180 Rayleigh's Method i 185 Improved Rayleigh's Method! 192 Shear Walls I 195 Summary I 199 Problems I 200

NONLINEAR STRUCTURAL RESPONSE 205

7.1 7.2 73 7,4 7.5 7.6 7.7

7.8 7.9 7.10

Nonlinear Single DegreeofFreedom Model I 206 Integrmion of [he Nonlinear Equation of Motion I 208 Constant Acceleration MeLood I 208 Linear Acceleration Step-byStep Method I 2l! The Newmark Beta Method I 2 14 Elastoplasric Behavior I 215 Algorithm for the Stepby-Step Solution for Elastoplastic Single-Degree-of-Freedom System I 217 Program S-Response for ElastopJastic Behavioa, System I 221 Nonlinear Stn.lClUra; Response Using COSMOS I 224 Summary I 228 Problems I 229

x Contents

s RESPONSE SPECTRA 233

8.1 Construction of Response Spectrum I 233 8.2 Response Spectrum for Support Excitation! 237 8.3 Tripartite Response Spectra I 238 8.4 Response Spectra for Elastic Design I 241 8.5 Influence of Local Soil Conditions! 245 8.6 Response Spectra for Inelastic Systems I 247 8.7 Response Spectra for Inelastic Design ! 250 8.8 Program 6-Seismic Response Spectra .I 257 8.9 Response Spectra Using COSMOS I 260 8,10 Summary I 265

Problems I 266

PART II STRUCTURES MODELED AS SHEAR BUILDINGS 271

9 THE MULTISTORY SHEAR BUILDING 271

9.1 Stiffness Equations for the Shear Building J 272 9.2 P-Ll Effect on a Plane Shear Building! 275 9.3 Flexibility Equations for the Shear Buildip.g J 278 9,4 Relationship Between Stiffness and Flexibility Mame""s J 280 9.5 Program 7-Modeling Structures as Shear Buildings I 281 9.6 Summa,), I 283

Problems ! 283

10 FREE VIBRATION OF A SHEAR BL1LDING 287

II

10.1 Natural Frequencies and Noonal Modes! 287 10,2 Orthogonality Property of Ute Noonal Modes I 294 10.3 Rayleigh's Quotient J 298 lOA Program 8-Natural Frequencies and Normal Modes I 300 10.5 Free Vibration of a Shear Building Using COSMOS! 301 lO.6 Summary I 304

Problems J 305

FORCED MOTION OF SHEAR BUILDING 310

ILl Modal Superposition Ylethod i 310 11.2 Response of a Shear Building to Base Ylotlon I 317

12

Contents Xl

11,4 Ha:monic Forced Excitation I 326 11.5 Program lO~Harmonic Response I 331 1 1.6 Combining Maximum Values of Modal Response I 334 11.7 Forced Motion of a Shear Building Using COSMOS .I 335 11.8 Summary J 346

Prob;ems ! 348

DAMPED MOTION OF SHEAR BUILDIt-;GS 352

12.1 12,2 123 12.4 12,5

Equations for Damped Shear Building I 353 Uncoupled Damped Equations I 354 Conditions for Damping Uncoupling J 355 Program ll~-Absolute Damping From Damping Ratios I 362 Summary I 364 Problems I 364

13 REDUCTION OF DYNAMIC MATRICES 366

13.1 13.2 133 ]3.4 13.5 13.6

Sta:ic Condensation ! 367 Static Condensation Applied to Dynamic Problems J 370 Dynamic Condensation J 380 Modified Dynamic Condensation! 387 Program 12-Reduction of the Dynamic Problem! 391 Summary I 393 Problems I 393

PART III STRUCTURES MODELED AS DISCRETE

14

MUL TIDEGREE,OF-FREEDOM SYSTEMS 397

DYNAMIC ANALYSIS OF BEAMS 399

14.1 14.2 14.3 14.4 14.5 14.6 14.7 14,8 14.9 14.10 14.1 I

Static Properties fnr a Beam Segment J 400 System Stiffness Matrix J 405 Inertial Properties-.Lumped Mass ! 408 Inel1iai Properties-Consistent Mass I 41O Damping Properties ! 414 External Loads ! 414 Geometric Stiffness ! 416 Equations of Motion I 420 Element Forces at Nodal Coordinates I 427 Program 13----Modeling Structures as Beams J 430 Dynamic Analysis of Beams Using COSYlOS I 433

XII Contents

15

14.12 Summary I 437 Problems I 438

DYNAMIC ANALYSIS OF PLA;-!E FRAMES 442

15.1 15,2 15.3 15.4 15.5 15,6

Eleme"' Sliffness Matrix for Axial Effects I 443 Element Mass Motrlx for Axial Effects I 444 Coordinate Transformation I 449 Prooram 14-Modeiinc:r Structures as Plane Frames! 458 ~ ~ Dynamic Analysis of Frumes Using COSMOS I 460 Summary / 465 Problems I 466

16 DYNAMIC ANALYSIS OF GRIDS 469

17

18

16.1 16,2 16.3

16,4 16.5 16.6 16.7 16,8 16.9

Local and Global Coordinate Systems f 470 Torsional Effects I 471 Stiffness :vIatrix for a Grid Element I 472 Consistent Mass Matrix for a Grid Element I 473 Lumped Mass Matrix for a Grid Element f 473 Transformation of Coordinates I 474 Program IS-Modeling Structures as Grid Frames I 480 Dynamic Analysis of Grids Using COSMOS f 483 Summary I 487 Problems f 488

17.1 Element Stiffness Matrix f 49l 17.2 Element Mass :vIatrix f 493 17.3 Element Damping :vIatrix I 494 17.4 Transformation of Coordinates f 494 17.5 Diffetenrial Equation of Motion f 503 17.6 Dynamic Response I 504 17,7 Ptogtam 16-Modeling Structures as Space Frames I 504 17 ,8 Dynamic Response of Three-Dimensional Frames Using

COSMOS f 507 17.9 Summary I 510

Problems I 5\0

DYNAMIC ANALYSIS OF TRUSSES 511

18,1 Stiffness and Mass Matrices for ttie Plane Truss f 512 I .:;'11

19

18.3 18.4 18.5 18.6 187 18.8

Conten:s xlii

Program 17-Modeling Structures as Plane Trusses f 520 Stiffness and Mass Matrices for Space Trusses I 522 Equation of Motion for Space Trusses I 525 Program l8-Modeling Str'Jctures as Space Trusses I 526 Dynamic Analysis of Tmsses Using COSMOS f 528 Summary I 536 Problems I 536

DYI'\AM!C ANALYSIS OF STRUCTURES USING THE Fll'\ITE ELEMEI'\T METHOD 538

19, I Plane Elasticity Problems f 539 19, J.l Triangular Plate Element for Plane EJastlcitv

Problems I 540 -19, L2 Library of Plane ElosticilY Elements

(2D Eleoents) I 552 19.2 Plute Bending I 555

j9.2.1 Rectangular Finite EleT:1ent for Plate Bending I 556 19.2.2 COSMOS Library of Plate and Shell Elements f 565

19.3 Summary I 573 Problems I 575

20 T1ME HISTORY RESPO;-!SE OF :VIULTiDEGREE-OFFREEDOM SYSTEMS 577

20,1 202 20.3

20A 205 20,6 20.7 20.8 20.9 20.10 20.11

20.12

Incremental Equations of Motion I 578 The WIlson-8 Method I 579 Algorithm for Step-by-Step Solution of a Linear System Using the Wilson-8 Method f 582 20.31 Initialization f 582 20.3.2 For Each Time Step f 582 Program 19-Response by Step Integration f 587 Newmark Beta Method f 588 Elastoplastic Behavior of Framed Structures f 589 :vIembet Stiffness Matrix I 590 Membet :vIass Matrix I 593 Rotation of Plastic Hinges I 595 Calculation of Member Ductility Ralio ! 596 Time-History Response of Mul(idegree-of-Preedom Syster.rs Using COSMOS f 597 Summary I 602 Probler.rs f 604

xiv Contents

PART IV STRUCTURES MODELED WITH DISTRIBUTED PROPERTIES 607

2! DYNA.\1IC ANALYSIS OF SYSTEMS WlTH DtSTRIBUTED PROPERTIES 609

2 L I Flexural Vibration of Uniform Beams J 610 21.2 Solution of the Equation of MociOf: in Free Vibration J 6 i ! 2 L3 Nftrura~ Frequencies and Mode Shapes for Uniform Beams / 613

21.3.1 Both Ends Simply Supported I 613 21.3.2 Both Ends Free (Free Beam) ! 617 21.3.3 Both Ends Fixed I 6 i 8 2L3.4 One End Fixed and [he other End Free (Cantilever

Beam) I 620 21.3.5 One End Fixed and the other End Simply

S"pponed I 622 21,4 Orthogonallty Condition Be[weer. Normal Modes I 622 2 L5 Forced Vibration of Beams J 624 21.6 Dynamic Stresses in Beams I 630 21.7 Summary I 632

Problems I 633

22 DISCRETIZATION OF CONTINUOUS SYSTEMS 635

22.1 22,2

,,22.3 22.4 22.5

22.6

22.7

22.8

Dynamic Matrix for Flexural Effects I 636 Dynamic Matrix for Axial Effects I 638 Dynamic Matrix for Torsional Effec[S I 641 Beam Flexure Including Axial-Force Effect J 642 Power Series Expansion of ::,e Dynamic Matrix for Flexural Effects I 646 Power Series Expans~on of the Dynamic MatrIX for Axial and for Torsional Effects j 648 Power Series Expansion of the Dynamic Matrix Including the Effect of Axial Forces I 649 Summary I 650

PAHT V RANDOM VIBRATION 651

23 RA,'fDOM VIBRATION 653

23.1 Statlsrical Description of Random Functions I 654 23.2 Probabiliry Dens;ry Function I 657 23.3 The Normal Distribution I 659

f r:.r.;n

Contents xv

Correlation I 662 The Fourier Transform I 666

23.7 Spectral AnalysIS I 668 23.8 Spectral Density FU .. otlon I 672 13.9 Narrow-Band and Wide-Band Random Processes I 675 23.10 Response::o Random ExcI(3[ion: SingleDegree-of~Freedom

System I 679 23.11 Response to R2.ndom Excitation: Mulr!ple-Degree~of-Freedom

SyStem! 685 23.12 Random Vibration Using COSMOS! 696 23 13 S ummay ! 700

PART VI EAHTHQUAKE ENG:NEERING 705

24 t:NIFORM BUILDIC'lC CODE 1994; EQUIVALENT STATIC LATERAL FORCE METHOD 707

24.1 Earlhquake Ground Motion! 708 24,2 Ec;:..tivalenr Seismic Lateral Force J 7:2 24.3 Eanhquake-ResiSlam 8esign Methods I 712 24.4 Static Lateral Force Mcrhod I 713 145 Distribution of Lateral Forces j 718 24.6 Story Shear Force I 718 24.7 Horizontal Torsional :Vloment I 719 24.8 Oveflureing Moment I 720 24.9 Srory 8rift Limitm'on I 720 24.10 P-Delta Effect (P-Ll) I 721 24.11 Diaphragm Design Force I 723 24.12 Program 23 UBC94 Equivalent Static Lateral Force

Method I 732 24.13 Simplified Three Dimensional Earthquake Resistant Design of

B uilcings I 739 24.l3.l Ylodeiing the Building I 739 24.13.2 Transfomla[jon of Stiffness Coefficients j 740 2413.3 Center of Rigidity I 742 24.13.4 Story Eccentricity I 743 24.13.5 Rotational Stiffness! 744 24.13.6 Fundamental Period I 745 24.13.7 Seismic Factors I 745 24. I 3.8 Base Shear Force! 746 24.13.9 Equivalent Lateral Seismic Forces! 746 24.13.10 Overturning Moments I 747

xvi Contents

24.13.10 Story Shear Force / 747 24.13.12 Torsional Moments / 747 24.13.13 Story Drift and Lateral Displacements / 748 24.13.14 Forces and Moments on Structural Elements I 749 24.13.15 Computer Program / 750

24.14 Equivalent Static Lateral Froce Method Using COSMOS / 756 24.15 Summary I 761

25 UN1FORIvl BUlLDl.t'!G CODE 1994: DYNAMIC :vlETHOD 766

25.1 Modal Seismic Response of Buildings / 766 25. Ll Modal Equation and Participation Factor / 767 25.1.2 Modal Shear Force I 768 25. L3 Effective Modal Weight / 770 25.1.4 Modal Lateral Forces / 771 25.1.5 Modal Displacements / 771 25.1.6 Modal Drift I 772 25.1.7 Modal Overturning Moment / 772 25.1.8 Modal Torsional Moment / 772

25.2 Total Design Values I 773 25.3 Provisions of UBC-94: Dynamic Method I 774 25.4 Scaling of Results I 776 25.5 Program 24-UBC 1994 Dynamic Lateral Force Method i 783 25.6 Summary i 787

Problems I 788

APPEI"1DlCES I 789

Appendix I: Answers to Problems in Part I I 79 I

Appendix II: Computer Programs I 80 I Appendix Ill: Organization and their Acronyms / 804

Glossary i 807

Selected Bibliograpby I 815

Index / 819

Diskette Order Fonn I 825

I

Preface to the Fourth Edition

The basic Structure of the three previous editions is maintained in this fourth edition, although numerous revisions and additions have been introduced. A new chapter to serve as an introduction for the dynamic analysis of structures using the Finite Element Method has been incorporated in Part 1lI, Structures Modeled as Discrete Multidegree,of-Freedom Systems. The chapter on Ran-dom Vibration has been extended to include the response of structures modeled as multidegree-of-freedom systems, subjected to several random forces or to a random motion at the base of ~he structure. The concept of damping ~nc1uding the evaluation of equivalent viscous damping is thoroughly discussed. The constant acceleratlon method to determine the response of nonlinear dynamic systems is presented in addition to the Hnear acceleration method presented in past editions, Chapter 8, Response Spectra now includes the development of seismic response spectra with consideration of local soil conditions at the site of the structure. The secondary effect resulting from the lateral displacements of the building, commonly known as the P-!l effect. is explicitly considered through the calculation of the geometric stiffness matrix. Finally, a greater number of iHustrative examples have been incorporated in the various chapters of the Dook using the educational computer programs developed by the author or tbe professional program COSMOS.

xvm Preface to the Fourth Ed:tion

The use of COSMOS for the analysis and solution of structural dynamics problems is introduced in this new edition. The COSMOS program was selected from among the various professional programs available because It has the capabiiity of solving complex problems in structures, as well as in other engin-eering fields such as Heat Transfer, Fluid F:ow, and Electromagnetic Phenom-ena. COSMOS includes routines for Structural Analysis. Static, or Dynamks with linear or nonlinear behavior (materlu! nonUnearlty or large displacements), and can be used most efficiently in the microcomputer. The larger version of COSMOS has the capacity for the analysis of structures modeled up to 64,000 nodes. This fourth edition uses an introductory version thot has a capabiiity limited to 50 nodes or 50 elements. This version is induded in the supplement, STRUCTURAL DYNAMICS USING COSMOS '.

The sets of educational programs in Structural Dynamics and Earthquake Engineering that accompanied the third edition have now been ex.tended and updated. These sets include programs to determine the response in the time or frequency domain using the foB (Fast Fourier Transform) of structures modeled as a single oscillator. Also included is a program to deler:nine the response of an inelastic system with e!astopiastic behavior and a program for the development of seismic response spectral charts. A set of seven computer programs is included for modeling structures as two-dimensional and three-dimensional frames and trusses. Other programs. incorporating modal super-position or a step-by~step time-history solution, are provided for calculation of the responses to forces or motions exciting the structure. In addition. in this fourth edition, a new program is provided to detennine the response of single-or muItidegree-of-freedom systems subjected to random excitations. The com-puter programs for earthquake-resistant deSign have been updated using the latest published seismic codes.

The book IS organized into six parts. Part I deals with st:1lctures modeled as single-degree-of-freedom systems. It introduces basic concepts and presents methods for the solution of such dynamic systems, Part II introduces conccpts and methodology for solving multidegree-of-freedom systems through the use of structures modeled as shear buildings. Part III describes methods for the dynamic analysis of skeletal structures (beams, frames. and trusses) and of continuous structures such as plates and shells modeled as discrete systems with many degrees of freedom, Part IV presents the mathematical solution for some simple structures modeled as systems with distributed properties. thus baving an infinite number of degrees of freedom. Part V introduces tbe reader to the fascinating topic of random vibrations, which is now extended to

multidegree~of-freedom systems, Finally, Part VI presents the current topic of earthquake engineering with applkations for the design of earthquake-resistant

; A cQnVen:erH form lo order this supplemelH is provided in lhe back of the book.

Preface to the Fourth Edilion xix

buildings following the provisions of the Cnifonn Building Code In use in the United States. There is a detailed presentation of the seismic analysis of buildings modeled as three-dimensional structures with two independent hori~ zontal motions and one rotational motion about a vertical axis for each story of the building. A computer program for the implementation of this simplified method for seismic analys~s of buildings is jncluded in the set of educational programs.

Scientific knowledge may be presented from a general all-encompassing theory from which particular or simple situations are obtained by introducing restricting conditions. Alternativeiy, the presentation may begin by considering particular or simple situations that are progressively extended. The author has adopted this latter approach in which the presentation begins with particular or simple cases that are extended to more general and complex situations. Funhennore, the author believes that a combination of knowledge of applied mathematics, theory of structures. and the use of computer programs is needed today for the succe$.').fu! profes;;;iollal pfflctice of engineering. To provide the reader with such a combination of knowledge has been the primary objective of this book. The reader is encouraged to inform the author on the extent to which this objective has been fulfined.

Many of my students. colieagues, and practidng professionals have sugges-ted improvements, identified typographical errorS. and recommended additional topics for inclusion. Ali these suggestions were carefully considered and have been included in this fourth edition whenever possible.

I was fortunate to have received valuable assistance and insight from many individuals to whom I wish to express my appreciation. I am grateful to Jeffrey S. Janover, a consulting engineer from New Jersey, who shared his expertise in the implementation of professional computer programs for the solution of complex engineering probtems. I appreciate the discussions and comments offered by my colleagues Drs. Michael A. Cassaro and Julius Wong who helped me in refining my exposition. I am also grateful to my friend Dr, Farzad Naeim who has coIlabornted with me on Seismic Response Spe

xx Preface to the Fourth Edllion

in the book; and Cleryl Hoskins who most carefully checked the solution of the problems for some chapters of the book

A special acknowledgement of gratitude is extended to Dr. Edwin A. Tuttle, emeritus professor of educatlon, who provided many suggestions that helped to improve the clarity of my presentation. I aiso wish to express my sincere gratitude to my friend Jack Bension for his professional help in editing the revised sections of the book. My thanks also go to Ms. Debbie Jones for her competent typing sieills in the revisions.

To those people whom I recognized in the prefaces to the previous editions for their help, I again express my wholehearted appreciation. To my wife Jean a special thanks for carefully checking the structure of the book and for most graciously aUowing me time to prepare this new edition, particularlY during sev.eral "working vacations." As with the third edition, this volume is dedicated to the everlasting memory of my parents.

March, 1997

Preface to the First Edition

Natural phenomena and human activities impose forces of time~dependent variability on structures as simple as a concrete beam or a steel pile, or as complex as a multistory building or a nuclear power plant constructed from different materiais. Analysis and design of such Stn.lctures subjected to dy-nawic loads involve consideration of time-dependent inertial forces. The res is-lance to displacemenl exhibited by a struclure may include forces which are functions of the displacement and the velocity. As a consequence, the govern-ing equations of motion of the dynamic system are generaUy nonlinear partial differential equations which are extremely difficult to solve in mathematical terms. Nevertheless. recent developments in the field of structural dynamks enable such analYSls and design to be accomplished in a practical and efficient manner. This work is facilitated through the use of simplifying assumptions and mathematical models, and of matrix methods and modem computarionai techniques,

In the process of teaching courses on the SUbject of structural dynamics, the author came to the realization thai there was a definite need for a text which would be suitable for the advanced undergr'J.duar:e or the beginning graduate engineering stJdent being introduced to this subject. The author is familiar with tJ}e existence of several excellent lexts of an advanced nature but gen-

xxi

xxli Preface to the First Edmon

eraUy these texts are. in his view, beyond the expected comprehension of the student Consequently, it was his principal aim in writir.g this book to incorpor-ate modem methods of analysis and lechniques adaptable to computer program-ming in a manner as clear and easy as the subject permits. He felt that computer programs should be induded in the book in order to assist the student in the application of modem methods associated with computer usage. In addition, the author hopes that thIs text will serve the practicing engineer for purposes of self-study and as a reference source.

In writing this text. the author also had in mind the use of the book as a possible source for research topics in structurdl dynamics for students working toward an advanced degree in engineering who are required [0 write a thesis, At Speed Scientific School, University of Louisville, most engineering students complete a fifth year of study with n thesis requirement leading to n Master in Engineering degree. The author'S experience as a thesis ndvisor Jeads him to believe that this book may weB serve the students in their senrch and selection of topics in subjects cunently under investigation in structural dynamics.

Should the text fulfill the expectations of the nuthor in some measure, par~ ticuiariy the elucidation of this subject, he will [hen feel rewarded for his efforts in the preparation and development of the material in this book.

MARIO PAZ December, 1979

PART I Structures Modeled as a Single-Degree-of-Freedom System

1 Undamped Single Degree-of-Freedom System

It is not always possible to obtain rigorous mathematical solutions for engin-eering problems, In bct, analYlical solutions can be obtaine

4 Slructures Modeled as a Single"Degree-ol-Freedom System

Fjrl

~F~t=~ , ,.,

~ pm llwnlllHl 1ill1lj 1

'"

lfJ-' Ft.-_-==~---;

n Fig. 1.1 Examples of Structures modeled as one-degree~ofMfreedom systems,

infinite number of degrees of freedom, Nevertheless. the process of idealization or selection of an appropriate mathematical model permits the reduction in the number of degrees of freedom to a discrete number and in some cases to just a single degree of freedom. Figure I.! shows some examples of structures that may be represented for dynamic analysis as one-degree-offreedom systems, that is. structures modeled as systems with a single displacement coordlnate. These one-degree~of~freedom systems may be described conveniently by the mathematical mode} shown in Fig. 1,2 which has the foHowing elements: (1) a mass element m representing the mass and inertial characteristic of the structure; (2) a spring element k representing the eiastic restOing force and potential energy storage of the stnlcture~ (3) a damping element c representing the frictional characteristics and energy losses of the structure; and (4) an excitation force F(t) representing the external forces acting on the structural system. The force F(t) is written this way to indicate that it is a function of time. In adopting the mathematical model shown in Fig. 1.2, it is assumed that each element in the system represents a Single property; that is, the mass m represents only the property of inertia and not elasticity or energy diSSIpation, whereas [he spring k represents exclusively elasticity and not inertia or energy dissipation. FinaIly, the damper c only dissipates energy. The reader certainly realizes thar such "pure" elements do not exist 1n our physical world and that mathematical models are only conceptual idealizations of real structures. As such, mathematical models may provide complete and accurate knowledge of the behavior of the model itself, but only limited or approximate information

Fig. 1.2 Mathematical mode! for one-degree-Qf-freedom systems,

Undamped Sjng!e~Oegree"of~Freedom System 5

on the behavior of the real physical system. Nevertheless, from a practical point of view, the information acquired from the analysis of the mathematicai model may very well be sufficient for an adequate understanding of the dynamic behavior of the physical system, including design and safety requirements.

1.2 UNDAMPED SYSTEM \Ve start our study of structural dynamics with the analysis of a fundamental and simpIe system, the one~degree-of-freedom system in which we disregard or "neglect" frictional forces or damping. In addition, we consider the system, during its motion or Vibration, to be free from external actions or forces, Under these conditIons, the system is in motion governed only by the influence of the so-caBed inith.l conditions, that is, the given displacement and velocity at time I = 0 when the study of the system is lnitiated. This undamped, one-degree~of-freedom system is often referred to as the simple undamped oscil-lator, It is usually represented as shown in Fig, 1.3(0) or Fig, L3(b) or nny similar arrangements. These two figures represent mathematical models that are dynamically equivalent It is only a matter of personal preference to adopt one or the other. In these models the mass m is restrained by the spring k and is limited to rectilinear motion along one coordinate axis.

:Ine mechanical characteristic of a spring is described by the relation be-tween the magnitude of the force Fs applied to Its free end and the resulting end displacement y, as shown graphically in Fig. 1 A for three different springs.

~ >v

tj~J/ fa) Ib)

Fig. 1.3 Alternate representations of mathematical models for one~degreeMof~freedom systems,

,

1 '.

~~---------------, Fig, 1.4 Force displacement relation. (a) Hard spring, (b) Linear spring. (c) Soft spring.

6 Structures Modeled as a Sing;e-Degree~of-Ffeedom System'

The curve labeled (a) in Fjg, 1.4 represents the behavior of a "hard spring," in which the force required to produce a given displacement becomes increas-ingly greater as the spring is defonned. The second spring (b) is designated a linear spring because the deformation is directly proportional to the force and the graphicaJ representation of its characteristic is a straight 1ine. The constant of proportionality between the force and displacement (slope of line (b)] of a linear spring is referred to as the spn'ng constalll, usually designated by the lelter k" Consequently, we may write the foUowing relatlon between force and displacement for a linear spring.

F,=ky (l.l)

A spring with characteristics shown by curve (c) in Fjg. 1.4 is known as a "soft spring," For such a spring the incremental force required to produce additional deformation decreases as the spring deformation increases. Undoubtedly, the reader is aware from his previous exposure to mathemat-ical modeling of physical systems that the linear spring is tne simplest type to manage analytically. It should not come as a surprise [0 learn that most of !be technical literature on structural dynamics deals wHh mode's using linear springs_ In other words, either because the elastic characterisrics of the structural system are, in fact, essentially linear, or simply because of analytical expediency, it is usually assumed that the force-deformation pmperties of the system are linear. In support of [his practice, it should be noted that in many cases the displacements produced in the. structure by the action of external forces or disturbances are small in magnitude (Zone E in Fig. 1.4), thus rendering lbe linear approximation close to tbe actual structural behavior.

1.3 SPRINGS IN PARALLEL OR IN SERIES Sometimes it 1s necessary to determine the equivalent spring constant for a system in which two or more springs are arranged in parallel as shown in Fig. 1.5(.) or in series as in Fig. L5(b).

Fig. 1.5 Combination of springs. (3) Springs in parallel. (b) Springs in series.

Undamped Slngle.-Degree-orFreedom System 7

For two springs in parallel the total force requJred to produce a relative displacement of their ends of one unit is equal to the sum of their spring constants. This total (orce is by definition the equivalent spring constant k( and (s given by

(12) In general for n springs 1n paralic.!

,

K, = 2: k, (1.3) ;"'1

For two springs assembled in series as s.hown in Fig_ 1.5(b), the force P produces the relative displacements in (he springs

LlYI

and

Lj., =~ 1. k2

Then. (he total displacement y of the free end of the spring assembly is equal to y = LiYI -;. Ll)'2. or substituting LiYI and LlY2,

y (1.4)

Consequently, the force necessary 1.0 produce one unit dis.placement {equival-ent spring constant) is given by

k=P , y

Substituting y from this last relation lnto eq, (1.4), we may conveniently express the reciprocal value of :.he equivalent spring constant as

1 k,

+_. k, (1.5)

In general for n springs in series the equivalent spring constant m.ay be oblained from

k, k, ( 1.6)

8 Struc\ures Modeled as a Slng\e~Oegree-of-Freedom System

1,4 NEWTON'S LAW OF MOTION We continue now with the study of the simple oscillator depicted in Fig. L3. The objective is to describe its motion. that is, to predict the displaceme~t or velocity of the mass In at any time f. for a given set of initial conditions at time f = O. The analytical relation between ~he displacement. y, and time, (, is given by Newton's Second Law of Motion, which in mod~rn notation may be , expressed as

F=ma (1.7) where F is the resultant force acting on a particle of mass m and a is its resultant tlcce1eration_ The reader should recognize rhat eq. (1.7) is a vector relation and as such it can be written in equivalent fonn in terms of irs components along the coordinate axes .x, y, and z, namely

(1.8a) (I,8b) (L8c)

The acceleration 1S defined as the second derivative of the position vector with respect to time; it foHows that eqs. (1.8) are indeed differentiar equations. The reader should also be reminded that these equations as stated by Newton are directly applicable only to bodies idealized as panides, that is, bodies that possess mass but no volume, However. as is proved in elementary mechanics, Newton's Law of Motion is also directly applicable to bodies of finite dimen-sions undergoing translatory motion.

For plane motion of a rigid body that is symmetric with respect to the reference piane of motion (x-y plane), Newton's Law of Motion yields the following equations:

L F, = m(ad, IFy mead)' IMr; = fcCt

(1.93) (L9b) (J ,9c)

In the above equations (aG);~ and (aG)" are the acceleration components, along the x and v axes, of the center of mass G of the body; Ct is the angular accelerati{}J;; lr; is the mass moment of inertia of the body with respect to an axis through G, the center of mass; and 'kMc; is the sum of the moments of all the forces acting on the body with respect to an axis through G, perpendicular to the x-y plane, Equations (1.9) are certainly a}so applicable to the motion of a rigid body in pure rotation about a fixed axis. For this particular type of plane motion, altematively, eq. (L9c) may be replaced by

( 1.9d)

Undamped ~jngle'Oegree'of"Freedom Sys!em 9

in which the mass moment of inertia 10 and the moment of the forces Mo are determined with respeet to the fixed axis of rotation. The general motioo of a rigid body is described by two vector equations, one expressing the relation between tbe forces and the acceleration of the mass center and anorher relatioo-the moments of the forces and the angular motion of' the body. This Ias~ equation expressed in its scalar components is rather complicated, but seldom needed in structural dynamics.

1.5 FREE BODY DIAGRAM At this point, it is advisable to follow a method conducive to an organized and systematic analysis In the solution of dynamics problems, The first and probably the most important practice to follow in any dynamic analysis IS to draw a free body diagram of the system, prior to writing a mathematical description of the system.

The free body diagram (FBD). as the student may recall, is a sketch of the body isolated from all other bodies, in which all the forces external to the body are shown, For the case at hand, Fig, 1,6(b) depicts the FBD of the mass In of the oscillator, displaced in the positive direction with reference to coor~ dinate y, aqd acted upon by the spring force F~ ky (assuming a Jjnear spI1ng). The weight of the bod) mg and the nonnal reaction N of tbe sUp'Porting surface are also shown for compieteness, though these forces, acting in the vertical direction, do not enter into the equation of motion written for the y direction. The appHcation of Newton's Law of Motion gives

-.. Icy = mji (LlO) where the spring force acting in the negative direction has a minus sign, and where the acceleration has been indicated by y. Ir. this notation, double over-dots denote the second derivative with respect to time and obviously a single overdot denotes the first derivative with respect to time. that is, the velocity.

,.) (0) 'c)

Fig. 1.6 Alternate free body diagrams: (a) Singie degree-of-freedom system. {b) Show~ jng only external forces. (c) Showing external and inertial forces,

Sj;uclu,es Modeled as a SlngleDegreeo!Freedom System

1.6 O'ALEMBERT'S PRINCIPLE An alternative approach to obtain eq. (LlO) is to make use of D'Alembert's Principle which states that a system may be sei in a state of dynamic equili-brium by adding to the external forces a fictitious force thai is commonly known as the inertial force.

Figure L6(c) shows the FED with inclusion of the inert:al force my. This force is equal (0 the mass multiplied by the acceleration, and should always be directed negatively with respect to the corresponding coordinate. The ap-plication of D'Alembert's Prineiple allows us to use equations of equilibrium in obtainJng the equation of motion. For example, in Fig. 1.6(c), the summation of forces in the y direclion gives directly

mji + J .. :y = 0 (LlI)

which obviously is equivalent to eg. (1.10). The use of D' Alembert's Principle in this case appears to be triviaL This

will not be the case for a more complex problem, in which the application of D'Alembert's Principle, in conjunction with the Principle of Virtual Work, constitutes a powerful tool of analysis. As wUi be explained later, the Principle of Virtual Work is directly applicable [0 any system in equHibriuffi. ft follows then that this principle may also be applied to the solution of dynamic prob-lems, provided that D' Alembert' s Principle is used to establish lhe dynamic equilibrium of the system.

Exam ple 1.1. Show that the same differential equat:on is obtained for a spring-supported body moving verlically as for the same body vibrating along a horizontal axis, as shown in Figs. 1.7(0) and 1.7(b).

SOlution: The FEDs for these two representations of the simple oscillator are shown in 1.7(c) and 1.7(e), where the inertial forces are inchIded. Equating to zero the sum of the forces in Fig. L7(c), we obtain

m'j+ i..:y;;; 0 (a)

When the body in Fig. 1.7(d) is in the static equilibrium posit;on, lhe spring is stretched Yo units and exerts a force kyo = W upward on the body, where W is the weight of the body. 'When {he body is displaced a distance y downward from this position of equilibrium the magnitude of the spring force is given by F. =. k(yo + y) or W + ky, since kyo = W. Using this result and applying it to the body in Fig. L7(e), we obtain from Newton's Second Law of Motion

- (W+Ay) + W=my (b)

Undamped Single~Deg(eeot-Freedom System 11

,

(0)

w

'

12 structures Modeled as a Sing!e-Oegree-of-Freedom System

or

y=B sin W! (Ll3) where A and B are constantS depending on the initia~ion of the motion whlle w is a quantity denoting a physical characteristic of the systert: as it wiE be shown next. The substitution of eq. (1.12) into eq. (1.1 j) gives

(- In,,} + k) A cos w!;;;:; 0 (1.14) If this equation is to be satisfied at any time, the factor in parentheses must be equal to z.ero or

(1.15)

The reader should verify that eq. (1. 13) is also a solution of the differential equation (LlI), with OJ also satisfying eg. (US).

The positive roO! of eq. (LlS), (1.160)

is known as the natural frequency of the system for reasons that will soon be apparent.

Equation (1.16a) may be expressed in tem.s of the static displacement resulting from the weight W = mg. The substitution into eq. (1. i6) of In = Wlg results in

(kg w= i-y W

Hence

'8 w=;-

V YSl (LJ6b)

where Y$! = W Ik is the static displacement due to the weight W. Since eilher eq, (Ll2) or eq. (Ll3) is a solution of eq, (l.ll), and since

this differentiaI equation 1$ linear, the superposition of these two solutions, indicated by eg, (1.l7) below, is also a solution. FurthemlOre, eq. (Ll?), having two constants 0: integration, A and B, is, in fact, the general solution for this second~order differential equation,

y=A cos wt-B SIn wI (1.17) The expression for velocity, ):, is found simply by differentiating eq. (1, 17) with respect to time; that is,

y = Aw sin wi + Bw cos wt (US)

Undamped Single .. Degreeof-Freedom Sysiem 13

Next, we should determine the constants of integration ,4, and B. These constants are determined from known values for the motion of the system which almost invariably are the displacement Yo and the velocity Va at the jojtiation of the motion. that is, at time t = O. Tbese two conditions are referred to as initial conditionJ, and the problem of solving the differential equation for the lnltial conditions is caUed an initial value problem.

After substituting. for l = 0, y ~ Yo, and }' = Uo into eqs. (Ll?) and (!.IS) we find that

Yo=A

Vo= Bw

(LJ9a) (l.!9b)

Finally, the substitution of A and B from eqs. (1.l9) into eq. (Ll?) gives

Vo . y=yocos wt+-sm wt w

( 1.20)

which is the expression of the displacement y of the simple oscillator as a function of the time variable 1; thus we have accomplished our objective of describing the motion of the simple undamped oscillator mOdeling structures with a single degree of freedom.

1.8 FREQUENCY AND PERIOD An examination of eq. (1.20) shows that the motion described by this equation is harmonic and, therefore, periodic; that is, it can be expressed by a sine or cosine function of the same frequency w. The period may easily be found since the functions sine and cosine both have a period of 211', The period T of the motion is determined from

or

wT= 211'

T= 21T W

(1.21 )

The period is usually expressed in seconds per cycle or simply in seconds, whh the tacit understanding,that it is '

:"".'

14 Structures Modeled as a Single-Oegreeo{Freedom System

l~l;n-ll M~T 10.691~lit\, 114 in.

Fig. 1.8 System for Example 1.2_

Tne natural frequency f is usually expressed in hertz or cycles per second (cps). Because the quantity w d!ffers from the natural frequency f only by the constant factor, 2'lT, w also is sometimes referred to as the natural frequency. To distinguish between these two expressjons for natural frequency, w may be called the circular or an8ular natural frequency. Most often, the distinction j5 understood from the context or from the units. The natural frequency f is measured in cps (IS indicated. while the circular frequency w should be given in radians per second (radfsec).

Example 1.2. Detennine the natural frequency of the system shown in Fig. 1.8 consisting of a weight of W = 50,7 lb attached to a horizontal cami-lever beam through the coil spring ka, The cantilever beam has a thickness t ~ in, a width b = 1 in modulus of eiasticity E= 30 X 106 psi, and a length I. = ! 2.5 in. The coil spring has a stiffness, k, = 10.69 (Ib lin).

Solution: The deflection L1 at the free end of a uniform car-tilever beam acted UpOrl by a static force P at the free end is given by

PI.' 3EI

Tne corresponding spring constant k, is then

P 3EI k!=-;:;;o-)-

j I.

where I -?7.btJ (for rectangular section). Now, the cantilever and the coil spring of this system are connected as springs in series. Consequently, the equivalent spring constant as given from eq. (1.5) is

",:i;

Undamped Single~Oegree,of-Freedom System 15

Substituting corresponding numericai values, we obtain

1 I] \1' 1 . 4 1= - X I X - = (JIl) 12 \ 4 J 3 X 30 X 10'

k, = (12.5)' X 768 = 60 !blin

and

I 1 =-+---

60 ;0.69 k, = 9.07 lb/in

The natural frequency for this system is then given by eq. (1.16a) as

w=Jkt:!m (m Vligandg=386 in/sec2) w = /"9ih X 386/50,7 w = 831 rad/sec

or using eq. (1.22) 1.32 cps

1.9 AMPLITUDE OF MOTION

(Ans.)

Let us now examine in more detail eq. (1,20), the solution de.scribing the free vibratory motion of the undamped osciHator. A simple trigonometric trans-formatJon may show us (hat we can rewrite this equation in the equivalent foons, namely

or

where

and

y""'-C sin(wl+ a)

y = C cos (",1- fl)

c = j y~ + (uo}w)J. y,

ran a

uo}w tan fl=--

y

(123)

(1.24)

(1.25)

(1.26)

( 1.27)

16 Structures Modeled as a Single-Degree-ofFreedom System

Fig. 1.9 Definition of angle Ck'.

The simplest way to obtain eq. (1.23) or eq. (1.24) is: to :nultip;y and divide eg, (1.20) by the factor C defined in 0'1. (L25) and to defme a (or (3) by eq, (1.26) [or 0'1, (1.27)J, Thus

I Yo Y = C\CCOS wi +

With the assistance of Fig. L9, we recognize that

and

sin ex = Yo . C

uo!w cos 0: =

The substitution of eqs, (1,29) and (1.30) into eg, (1.28) gives y :;;;: C (sin a cos wI + cos 0' sin wt)

(1.28)

(L29)

(1.30)

(1.31)

The expression within the parentheses of eq. (1.3 J) is identical to sin (wt + a), which yields eq. (1.23), Similarly, the reader should verify, without difficulty, the fo:m of solution given by eq, (1.24).

The value of C in eq_ (Ll3) [or eq, (L24)] is referred to as the ampliLOde of motion and the angle 0' (or f3) as the phase The solution for the motioo of the simple oscillator is shown graphically in Fig, 1. [0.

y

Fig. 1,10 Undamped free-vibralion response.

I I I I

~

I

Undamped Sing!l-Degree-of~Freedom System 17

Example 1.3. Consider the frame shown in Fig. 1. 11 (a). ThIS is a rigid steel frame to which a horizontal dynamic force is applied at the upper level. As part of the overall structurai design it is required to determine the natural frequency of the frame. Two assumptions are made: (1) the masses of the columns and walls are negJigible; and (2) the horizontal members are suffi-ciently rigid to prevent rotation at the tops of the columns. These assumptions are not mandatory for the solution of the problem, but they serve to simplify the analysis. Under lhese conditions, the frame may be modeled by the spring-mass system shown in Fig. 1. 11 (b).

Solution: The parameters of this model may be computed as follows:

W=200 X 25 = 5000 lb

1 82.5 in4

E = 30 X 10'psi

k

k 1O,185Ibli"

12 X 30 X 10' X 165 (15x 12)'

Note: A unit displflcement oftlte top of afixed column requires aforce equal 1o 12EUI},

Therefore, the Mruralfrequency from eqs, (U6) and (1.22) is

= _1_1 10,185 X 386 tv 21T~ 5000

4,46 cps (Ans_)

r-'

lj t:: m FII) 11///////5J~ .L we x 24 -~ -----L '" 15' I tbl

Fig. 1.11 One-degree~of~freedom frar:le and corresponding mathematical model for Example 1.3.

18 Structures Modeled as a Single-DegfeeofFreedom System

k

,,' ~b)

Fig. 1.12 (a) Water lower tank of Example 1.4; (b) Mathemutical modeL

Example 1.4. The elevated water tower tank with capacity for 5000 gal-lons of water shown in Fig. 1.12(a) has a natural period in laterai vibration of 1.0 sec when empty_ When the tank is fuiI of water. its period lengthens to 2.2 sec. Determine the lateral stiffness k of the tower and [he weight \.11 of the tank. Neglect the mass of the s~lpporting columns (one gallon of water weighs approximately 8.34 Ib)

Solution: In its iatem! motion, the water tower is modeled by [he simple oscillator shown in Fig. L12(b) in which k is the lateral stiffness of tile tower and m is the vibrating mass of the tank.

fa) Natural frequency We (tank empty):

(b) Natural frequency Wy: (tank full of water) Weight of water W",:

w, ~ 5000 x 8.34 = 41,700 Ib

_ 2r. _ 21T _ I--kg-Wt- ---.-- )-_ .. _--

T, 2.2 \ 1'1 + 41.700

(a)

(b)

Squaring eqs. (a) and (b) and dividing correspondingly the left and right sides of these equations, results in

(2.2)' W + 41,700 (1.0)' =. W

U,'ldamped Sing~eDegTee-of~Freedom System

, y

w r

F k ~

...........-

t:::k-~-J ' ' ~ M, ",ml,}"""

'Ir ~a)

Fig. 1.13 (a) Fru:ne of Example 1.5; (b) Mathematical model.

and solving for W

Subs~ituling in10 eq. (a), VI

and

1'1= 10,860 ib

10,860 10 and g = 386 inlsec', yields

211' 1,0

I k386 I-~\ 10,860

k= lila Ibiin

19

(Ans.)

(Ans.) Example 1.5. The steel frame shown in Fig l.13(a) is fixed at the base

and has a rigid top that weighs 1000 lb. Experimentally) it has been found that its naturai period in iateral vibration, is equal to 1/10 of a second. It IS required to shorten or lengthen its period by 20% by adding weight or strengthening the columns. Determine needed additional weight or additional stiffness (ne-glect the weight of the COlumns).

Solurion: The frame is modeled by the spring-mass system shown in Fig, 1.13(b). Its stiffness is calculated from

as

217 0.:

~L 1000

2r. _ [T -Y-V In

k = 10,228 In lin

:11 :~ tl !: r , ,

20 Structures MOdeled as a Sing!e-Degfee-of-Freedom System

(a) Lengthen the period lO T, = L2 X 0.10 = 0.12 sec by adding weight LlW:

2 Ti" ! 10,228 x 386 w = -0-. '-2 = Y -;-;, OcoOO"'+-:-iLlC::CW';-

Solve for LlW:

LlW=440 lb (Ans.)

(b) Shorten the peliod to T, = O.S X 0 1 = O.oS sec by strengthening columns in Llk:

co =2..". = i -,(_10~.2_2_8-,.~.,. "M,....:..)(",3",8",,-6) 0.08 Y 1000

Solve for M:

Llk= 5753 Ib/in

1.10 UNDAMPED SINGLE-DEGREE-OF-FR!::EDOM SYSTEMS USING COSMOS

(Ans.)

The foi;owing example is presented to illustrate the use of the program COSMOS in the analysis of structures modeled as single degree-of-freedom systems. Detailed explanations for the llse of COSMOS including numerous examples with data preparation and results are presented in the supplement STRUC-TURAL DYNAMICS USING COSMOS .

Example 1.6. An instrument of mass m = 0.026 (lb . secL/in) is mOl;.nted on isolation springs of total spring constant k = 29.30 (lb/in). Model the system as an undamped single-degree-of-freedom system (Fig. 1.14) and de;-tennine its natural frequer:cy.

Fig. 1.14 Mathematical model for Example 1.6,

A tonVenlelll foffi'l ror or>Jering ~his supplement is provided :in the lacs! page of this volume,

undamp'ect Sing\e-Degree-of-Freedom System 21

Solution: The analysis is performed using a single spring elemem with one concentrated mass element. The foBowing com vn:w_PAR > VJ EW VIEW, 0, 0, 1,

(2) Define the XY plane at Z ~ 0: GEOMETRY > GRID > PLANE PL>.NE, Z, 0, 1

(3) Establish a grid with two diVIsions in the X and Y directions, then use the scale command;

GEOHE'l'RY > GlUDON GRIDON, C, 0, 1, 1., 2, 2, 2

DISPLAY > DISP_PAR > SCALE SCALE, 0

(4) Generate a curve from 1, 0, O. to I, 1, 0: GEOMETRY > CiJRVES > CRPCORD CRPCORD, , \.I, 0, :, ~, C-

(5) Define element group using the SPRING element formulation with two nodes:

PROPSETS > EGROUP EGROUP, 1, SPRING, 0, 2, 1, 0, 0, 0, 0

(6) Define real constant for spling element: k = 29.3 (ib/in): PROPSETS > RCONST ReOM'ST, 1, I, }, 1, 29.3

(7) Generate one spring element along curve 1: MESHING > PAR..b.Y" ... J,!:ESH > }CCR

~CR, L 1 f L 2, 1,. 1

(8) Define element group 2 using the MASS element fonnulation: PROPSETS > EGROUP EGHOUP, 2. MASS, 0, 0, 0, 0, 0, C,

(9) Define real constant for mass element: m = 0.026 (lb . sec2 /in): PROPSETS > RCONS1'

~CON$'l', 2, 2, 1, 7, 0, 0.026, 0, 0, 0, 0, Q

22 Structures Modeled as a Single-Deg.ee-ol-Free:dom Syslem

(10) Gene,rate one mass element at point 2: MESHING > PARhlCMESH > !'CPT l"CFT .. 2, 2, ~

(11) Merge nodes; MESH ING > NODES. NMERG NMEHGE, 1, 3. 1, O.aOOL O. 0, 0

(12) App!y constraints in an degrees of freedom at node 1. and an degrees of freedom exce.pt UY at node 2:

LOADS-Be > STRUCTURAL > ;)ISPLHKT'S > DPT DP1', 1, l\L, 0, 1, 1 VPT, 2, UX, 0, 2, 1, UZ, RX, RY. RZ

(13) Set the options for the frequency analysis to extract one frequency using the Subspace Iteration Method with a maximum of 16 iterations, and run the frequency analysis:

A,NAI.''iSIS > FREQ!BlJCK > A_FREQUENCY A-FPQUCNCY, 1, 5, 16, 0, 0, 0, 0, IE-OS, 0, 1-06, 0, 0, 0, ~ ANALYSIS > FREQ/BUCK > R_FREQUENCY

~FREQUENCY

(14) List the natural frequency of the system: RESULTS > LIST > FREQLIST

F:~E:QLIST

FREQUENCY# 1

1.11 SUMMARY

FREQUK'CY (2.AD / SEC)

3,35597E+Ol

FREQUENCY (CYCLES/SEC) 5.34278:+:):)

Several '::>as1c concepts were inLroduced in this chapter.

PERIOD {SECONDS)

:.87:68-:)1

(1) The mathematical model of a structure is an idealized representation for its analysis.

{2) The number of degrees of freedom of a system is equal to the number of independent coordinates necessary to describe its ,Position.

(3) The free body diagram (FBD) for dynamic equilibrium (to allow application of D' A~embert's Principle) is a diagram of the system isolated from all orher bodies. showing an the external forces on the system. including the inertial force.

(4) The stiffness or spring constant of a iinear system is the force neces-sary to produce a unir displacement

Undamped Single-Oegree-of-Freedom System 23

(5) The differential equalion of the undamped simpie oscHlator in free motion is

and its general solution is

)' = A cos fJJJ + B sin mt where A and Bare conslants of integration determined from initial condiiions:

A; Yo B =v(jiw w=Jkim is the r.atural frequency in rad/sec

w f = ... -... ~ is the nalural frequency in cps 2.". 1

T = 7 is the natural period in seconds (6) The equation of motion may be written in the alternate fonns:

y = C sin (UJI + a) or

)' """ C eos((ul - f3) where

c

and

tan a

llnfJ

PROBLEMS L 1 Determine the natural period [or the system in Fig. P U. Assume Lhllt the, bellm

and springs supporting the weight Ware massless. 1.2 The foHowing numerical values are given in Problem 1.1; L = 100 in, E!

lOlI(lb_inl), W=3000 Ib, and k 2000!biin. If the weight W has an initial

24

1.3

1.4

Struclures Modeled as a Single-Degrce-Ql-Freedom SysLer

EI T , Y

Fig. Pl.1.

displacement of Yo'= 1.0 in and an initial velocity OJ'=' 20 in/sec, cetermlne the displacement and the veloctry 1 sec later. Determjne the natural frequency for norlzontaf motion of the sreel frame in Fjg. P1.3. Assume the horizontal girder to be Infinitely rigid and neglect the mass of the columns.

Fig, Pl.3.

Calculate the natural frequency in the horizontal mode of the steel frame in Fig. PL4 for the following cases: (a) me horiz,ontal member is assumed:o be infmitely rigid; (v) the horizontal member is flexible and made of steel~WJ:O X 33.

W"" 25 Kips

15' W10 x 33

I _L

1---'5' Fig. PlA.

,,j; ..

Undampeij Sjngle~Degree-Qj-Freedom System 25

1.S Detennine the natural frequency of the fixed beam in Fig. PLS carrying a concen-trated weight W at its center. Neglect the mass of the beam.

~ ~. .. fl =I I ml../2 -1.12 ~ Y

Fig. P1.5.

1.6 The numerical values for Problem 1,5 are given as; L = 120 in, El = 109 (Ib . in2), and W 5000 lb. If [he initial displacement and (he initial velociry of the weight are, respectiveiy, Yo = 0.5 in and Vo = 15 in/sec, determine the displacement, vel~ odty, and acceleration of W when time (= 2 sec,

1.1 Consider the simple pendulum of weight W illustrated in Fig. Pl.?, A simple pendulum is a particle or concentrated weight that oscillates in a vertical arc and is supported by a weightless cord. The only forces acting are those of gravity and cord tension (I.e., frictional resistance is neglected). If the cord length is L, determine the motion ir the maximum 9sdllation angle B is small and the lnllial displacement and' velocity are 80 and 80 respectively.

1.8

1.9

~ I\~ I \ I I \

:-' ~ ; ; W

Fig. Pl.7.

A diver standing at the end of a. diving board that cantilevers 2 ft osciIlates at a frequency 2 cps. Determine the flexural rigidity EI of the diving board. The wejght of the dIver is 180 lb. (Neglect the mass of the diving board). A bullet weigh1ng 0.2 Ib is fired at a. speed of 100 ft/sec into a wooden block weighing 50!b and supported by a spring of stiffness 300 IbJin (Fig, PL9). Determine the displacement y{t) and velocity v(t) of the block after r sec,

;: II il 'I 11

"

26 Structures Modeled as a Single-Degrae-o!Freedom System

Fig. P1.9.

1.10 An c;evator weighing 500 Ib is suspended from" spring having a stiffness of 600 IbJin. A weight of 300 Ib is suspended through a cable to the elevator as shown schematically in Fig. PLIO, Determine the equation of motion of the elevator if the ca:::'Je of the suspended weight suddenly breaks.

=p ,

Fig. F1.l0.

1.11 Write the differential equation of motion fo~ lhe inverted pendulum shown in Fig, P Lll and determine its natur 1,13 may be expressed as

i--~ / = fn y 1 - W~,

where 10 is the natura! frequency calculated neglecting the effect of gravity and W

28 Struc1ures Modeled as a Slngle,Degreeo/-Freedom System

w

k L_-.l 2 I

I

~k JA,;. ,,, (bi

!---'-t-"r [WJi b--l

1.16

~ Ie) lei

Fig. P1.15.

A system (see Fig. Pl.16) 15 modeled by two freely v;brating masses m; and!n2 interconnected by a spring having a constant k. Detennine for this system the differential equation of mo!ion for the relative displacement u = Y2 - Yl between the two masses. Also determine the corresponding natural frequency of the system.

~.- k I nl, ~.------ '1J'l

/=#/d')"'W/A/d/,Pflil.awV~$pp1bh'fliIP Fig. P1.16.

1.17 Calculate ~he natura! frequency for the vibration of the mass In shown in Fig. PL17. Member AE 15 rigid with a hinge at C and a supporting spring of stiffness k at D~ (Problem contributed by Professors Vladimir N. Alekhio and A!eksey A Antipin of t::,e Urals State Technical University, Russia.)

1.18

1.19

Undamped Single-Degreeol-Freedom Syslem 29

Rigid Bearo

~Hinge { ~ 0 1: 8 C

I I 4 a a I a Fig. P1.17,

Determine the natural freqt:e::cy of v;bration in t,1.e vertical direction for t.ie rigid foundation (Fig. PLlS) transmitting a uniformly distributed pressure on Ihe soil having a resuliant force Q = 2COO kN, The area of the foot of the foundation is A = 10 mJ The coefficient of elastic compression of the soil is k "'" 25,000 kN 1m3 (Problem contributed by Professors Vladimir N, Alekhin and Aleksey A Antipin of the Urals Siate Technicai University, Russia.)

Fig. PL18.

Calculate the natural frequency of free vibration of the chimney on elastic foundation (Fig. PL 19), pennitting the rotation of {he structure as a rigid body about the horizontal axis x-x. The total weight of the structure is W with its center of gravity at a height h from the base of the foundation. The mass moment of inertia of the structure about the axis x-x is 1 and the rotational stIffness of soil is k (resisting moment of the soil per unit rotation). (Problem contributed by Professors Vladimir N. Alekhin and AIeksey A. Antipin of the Urals State Technical University, Russia.)

30 Struct:1r'eS Modeled as a Single-Degree-of-Freedorn System

n ~T./Jw r- I :

b

~--- . +j

Fig. Pl.19.

r, "

iJ " ..

~ .~ , "

2 Damped Single Degree-of-Freedom System

We have seen in the preceding chapter that the simple oscillator under idealized conditions of no damping, once excited, will oscillate indefinitely with a constant amplitude at its natural frequency_ Experience indicates, how~ ever, that it is not possihie to have a device that vibrates under lhese ideal conditions. Forces designatec as frictional or dampIng forces are always pres~ ent in any physical system undergoing motion. These forces dissipate energy; more precisely, the unavoidable presence of these frictional forces constitutes a mechanism through wh:ch the mechanical energy of the system, kinetic or potential energy> is transformed to other forms of energy such as heaL The, mechanism of this energy transformation or CissJpation is qujle compiex and is not completely :mderstood a! this time. In order to account for these dissipative forces in the una:ysis of dynamic systems, it is necessary to make some assumptions about these forces, on the basis of experience.

2.1 VISCOUS DAMPING In considering damping forces ;n the dynamic analysis of strJctures. it is usually assumed (hal these forces are proportionaJ to the magnitude of the

32 Structures Modeled as a Single-Deg:rse-or-Freedom System

velocity> and opposite to the direction of motion. This type of damping is known as viscous damping; it is the type of damping force that could be developed in a body restramed in its motion by a surrounding viscous fluid.

There are situations in which the assumption of viscous damping is realistic and in which the dissipative mechanism is approximately viscous. Neverthe-less, the assumption of viscous damping is often made regardless of the actual dissipative characteristics of the system, The primary reason for such wide use of this method is that it leads to a relatively simple mathematical analysis.

2.2 EQUATION OF MOTION Let us assume that we have modeled a str"JclUral system as a simple oscillator with viscous damping, as shown in Fig. 2.1(a). In this figure, In and k are, respectively. the mass and spring constant of the osci!!ator and c is the viscous damping coefficIent We proceed. as in the case of the undamped oscillator, to draw the free body diagram (FED) and apply Newton's Law to obtain the differential equation of motion, Figure 2.1(b) shows the FED of the camped oscillator in which the inertial force my is also shown, so that we can use D' Alernbert's Principle. The summation of forces in the y direction gives the differential equation of motion,

my+cy+ky 0 (2,1)

The reader may verify that a trial solution y ;;;; A sin wt or y = B cos t:J! will not satisfy eg. (2.l). However, the exponential function yo:: Cef'i coes satisfy this equation. Substitution of this function into eg. (2.1) results in the equation

mCp',,' + cCpe" + kCe" = 0

(,j

(0) A,,', : I ~~" , "'---Ic~j Fig. 2.1 (a) Viscous damped OSCj1Jl'HOf. (b) Free body diagram.

Damped SingJe-Oegree-of-Freedom System 33

which, after cance1!aiion of the common factors, reduces to an equation called the characteristic equation: for the system, namely

mp2 + cp + k == 0 (2,2) The roots of this ;:;uadratic equation are

P,= ~--~-... :t It~r-~ P2 2m'~ l2m) m (2,3)

Thus the general solution of eq. (2.1) 1s given by the superposition of the two possible solutions, namely

y(l) = C,e'" + C,e" (2,4) where C\ and C1 are constants of integration to be determined from the initial conditions,

The final fonn of eq. (2.4} depends on the sign of the expression under the radical in eq. (2.3). Three distinct cases may occur: the quantity under the racical may either be zero, positive, or negative. The limiting case in which the quantity under the radical is zero is treated first The damping present in this case is called crilical damping,

2,3 CRITICALLY DAMPED SYSTEM For a system osciUating with critical camping, as defined above, the expression under the radical jn eq" (2.3) is e

34 . Structures Modeled as a SingleDegree-of~Freedo'11 System

Since the two roots are equal, the generai solution given by eq. (2.4) would provide only one independent constant of integration, hence, one independent solution, namely

(2.9)

Another independent solution may be found by using the function

(2.10)

This equation, as the reader may verify, also satisfies. the differential equalion (2.1). The general solution for a critically damped system is. then given by the superposition of these two solurions,

(2.11 )

2.4 OVERDAMPED SYSTEM

In an overdamped system, the damping coefficient is greater than the value for critical damping, namely

(2.12)

Therefore, the expression under the radical of eq. (2.3) is positive; thus the two iools of the characteristic equation are real and distinct, and consequently the solution is given direcdy by eq. (2.4). It should be nOled that, for the overdamped or the critjcally damped system, the resulting motion is not oscil~ latory; the magnitude of the osciHations decays exponentially with time to

~ero. Fjgure 2.2 depicts grapbknJJy the response for the simple oscjHator with c'rilicat damping. The response of the overdamped system is similar to the motion of the critically damped system of Fig. 2.2, but the return toward the neutral position requires more time as the damping is increased,

Fig. 2.2 Free-vibraLion response with critical damping,

Dam;JeQ Single-Degree-of-Freedom System 35

2.5 UNDERDAMPED SYSTEM When the value of the damping coefficient is iess [han the critical value (c < c.:r), which occurs when the expression under the radical is negal~ve, the fOots of the characteristic eq. (2,3) are complex conjugates, so that

c Ik ('C'lT --+i 1---

2m - ~ m 2m, p,

(2.13) p,

where i = J~ is the jmaginary unit For this case, i[ is convenient to :nake use of Eu!er's equations which relate

exponential and t.:-igonometric functions. namely

ei~ = cos x + i sin x (2.14) e -H = cos X - i s;n x

T1H~ substitution of the roots Pi and pz from eq. (2.13) into eq, (2.4) together with the use of eq. (2.14) gives lhe foilowing convenient fonn for the general Solullon of the underdamped system;

Y (l) = e -{,.;.'2>:.-jl (A cos woe + n sin (/Jot) (2.15) where A and B are redefined conStants of integration and WD, the damped frequency of the system, is given by

or

(uv= W

This lost result is obtained after substituting, in eq, tbe undamped natural frequency

and defining the damping ratio of the system as

c

(2.16)

(2.17)

(2.16), the expression for

(2.18)

(2.19)

where the critical damping coefficient c'" JS given by eq. (2.6).

36 structures Modeled as a Singje~Degreeol-Freedom System

Finally, when the initial conditions of dispLacement and veiodty, Yo and VOl are introduced, the constants of integration can be evaluated and substituted ilito eq. (2,15), giving

'~I ' "0 + yogw, ) y(t):;:; e ' Yo cos

38 Structures Modeled as a SingleOegree-of-Freedom System

We note from this equation that, when the cosine factor is unity, the displace-ment is on points of the exponential curve y (I) = Ce ~f"" as shown in Fig. 2A However, these points are near but not equal to [he positions of maximum displacement. The points on the exponential curve api?car slightly to the right of the points of maximum amplitude. For most practical problems, the discrep-ancy is negligible and the displacement curve may be assumed to coincide at the peak amplitude. with the curve y (1) = Ce .. wi so that we may write, for two consecutive peaks, Yl at time '1 and h at To seconds later,

Yl Ce- f "",

and

Dividing these two peak amplitudes and taking the natural logarithm, we obtain

(2.27)

or by substituting TD , the damped period, from eq. (2.24),

8= (2.28)

As we can see, the damping ratio {can be calculated from eg. (2,28) after detennlning experimentally the amplitudes of two successive peaks of the system in free vibration. For small values of the damping ratio, eq. (2.28) can be approximated by

(2.29)

Alternatively, the logarithmic decrement may be cak:l1ated as the ratio of two consecutive peak accelerations, wbich are easier to measure experimentaty than dispJaceme:1ls. In this case, raking the first and the second derivatives in eq. (2.20, we obtain

y(l) Ce 1~[-WCOs(wDI-a)-Wosin(wd-a)] Y(I) = Ce -,~ [[ - {w cos (Wol- (X) (Ua sin (wo! - a)] (- {w)

+ [WWD sin (UJol - a) -~ wb cos ((J)vt ~ a)J)

.... "',, ,"

Damped Sing!e-Degree-ofrFreedom System 39

and at time 1'2 ::;::c 1 J + Tv, corresponding to a period later, whcn again the cosine function is equal to one and the sine function is equal to zero

The ratio of the accelerations at times (I and [1 is then

and laking nalural logarithmic results in the logarithmic decrement in term of accelerations as

which is identical to the expression for the logarithmic decrement given by eq. (2.27) in terms of displacements.

Example 2.1. A vibrating syslem consisting of a weight of W = 10 jb and a spring with stiffness k;;::. 20 lb/in is viscously damped so that the ratio of two consecutive amplitudes is LOO to 0.85. Delennine: (a) the natural fre-quency of the undamped system, (b) the logarithmic decrement, (c) the damp~ lng ratio, (d) the damping coefficient, and (e) the damped natural frequency.

Solution: (a) The undamped natural frequency of lhe system in radians per second is

w= Jklm = )(20 !blin X 386 io/sec')110 lb = 27.78 radlsec

or in cycles per second

w J= 4.42 cps

(b) The logarithmic decremel1t is given by eq. (2.26) as

. y, 100 0= In ~ = In = 0.163 y, 0.85

(e) The damping ratio from eq. (2.29) is approximately equal to

8 0.163 = =--=0.026 2,,-

40 St!uctures Modeled as a Single-Degree-ofFreedorn System

(d) The damping coefficient is obrained from eqs. (2,6) and (2.19) as

0.037 Jb . sec

(e) The natural frequency of the damped system is given by 0

,_ f

42 Structures Modeled as a Single-Degree-oJFreedorr. System

L = 4(\itl

I ~.~.=.~= f 1

(3)

(\

Fig, 2,5 (a) System for Ex:affiplc 2.3; (b) Mathematica! modeL

coefficient c for the system assumed to have 10% of the criticai damping. Neglect the mass of the beam.

So/mion: The spring constant k.b for a uniform simple supported beam is obtained from the deflection 3 resulting from a force P applied at the center of the beam:

Hence,

P 481 kb~5 IT

48 X 10' =---= 7500 Jb/in

40'

The equivalent spring constant is then calculated using eq. (1.5) for two springs in series:

I I =.---+----

2000 7500

kF= 15791blin (Ans)

Dampec Single-Oegree-ol-Freedom System

The equivalent mass is:

The critical damp}ng is calculated from eq. (2.6):

c" (lb'sec\

12792 [~~~~-! \ 10 !

The damping is then calculated from etl (2.19):

Ilb.seC) Cs Ccr=O.10X 12792 12.79,-.-

\ 10

2.7 SUMMARY

43

(Ans)

(Ans)

Real stmcturcs dissipate energy while undergoing vibratory motion. The most common and practical method for considering this dissipation of energy is to assume that it is due [0 viscous camping forces, These forces are assumed to be proportional to the magnitude of the veiocilY but acting in the direction opposite to the motion. The ftletar of proportionality is called the viscous dampiflg coefficient. It is expedient to express tbis coefficient as a fraction of the crifical damping in the system (the damping ratio {= c ICe,)' The critical damping may be defined as the least value of the damping coefficient for which the system will ;'lot oscll1ate whe!': disturbed initially, but it simply will return to the equilibrium position.

The differential equation ef motion fer the free vibration of a damped single degree-or-freedom system is given by

my + C)' + k)l = 0

The analytical expression for the solution of this equation depends on the magnitude of the damping ratio. Three cases are possible: (l) eriticaUy damped system 1), For the underdamped system (1;< J) the solution of the differential equation of motion may be written as

44 StnJClures Modeled as a Srngle~Degree~of-Freedom System

in which

(jJ;;; {ki;;; is the undamped frequency is the damped ff"'.....quency

';= clccr is the damping ratio ~

Crr 2.,! km is the critical damping

and Yo and tlo are, respectively, the jnitial displacement and velocity. A common method of detennining the damping present in a sys~em is to

evaluate experimentally the logartthmic decrement, which is defined as the natural logarithm of the ratio of two consecutive ueaks for dis!)iacement or for acceleration. in free vibration, that is, - .

The damping ratio in structural systems is usually less than 10% of the critical damping (f< 0,]). For such systems, 6e damped frequency is approxi-mately equal to the undamped frequency_

PROBLEMS

2.1 Repeat Problem L2 assuming that the system has 15% of crit:cal damping. 2.2 Repeat Problem 1.6 assuming that the system has 10% of critical damping, 23 The ampiitude of vibration of the system shown jn Fig. P2.3 is observed to

decrease 5% on each consecucive cycle of motion. Determine the damping coefficient c of the system~ k = 200 Ib lin and m = 10 lb sec1l!n.

IT_C k , "'T : . , m i , , I I

v

Fig. 1'2.3.

2.4 It is observed experimentally that the ampIitude of free vibration of a certain structure, modeled as a single degree-of freedom system, decreases from 1 to 0.4 in 10 cycles. What is the percentage of critical damping?

Dar:;ped SingleDegree~ofFreedom System 45

2.5 Show that the displacement for critical and overcritical damped systems with init:al displacement Yo and velocity Vo may be written as

2.6

2.7

2,8

y=e-oduced a relative velocity of 1.0 in/sec in the damping element Find: (a) rhe damping ratio fj, (b) [he damped period TD, (e) the logar-ithmic decrement a, and (d) the ratio between two consecutive amplitudes. In 2.4 it is indicated that the tangent points to the displacement curve correspond to cos (wot- a)"" L Therefore the difference in We( between any twO consecutive tangent points is 211". Show lhat the difference in Wpf between any two consecutive peaks of the curve is- also 211", Show that for an underdamped system in free vibration the logarithmic decre-ment may be written as

1 v' (5: = .. ~ In -'-'-k YiH

where k IS the number of cycles separating two measured peak amplitudes Yi and

2.9 It has been estimated that damping in the system of Problem 1.11 is IO% of the critical va~ue. Determine the damped frequency In of the system and the absolute value of the damping coefficient c.

2.10 A single degree-of-freedom system consj.5ts of a mass with a weight of 386 lb and a spring of stiffness k 3000 !b/in. By testing the system it was found that a force of JOO Ib produces a relative velocity i2 in/sec. Find (a) the damping ratio , (b) the damped frequency of vibration If), (c) logarithmic decrement 8, and (d) the ratio of two consecutive amplitudes.

2.11 Solve Problem 2.10 when the damping coefficient is c=2Ib, sec fin. 2.12 For each 0: the systems considered in Problem L15, determine the equivalent

spring COnstant k! and (he equivalent damping coefficient CE in the mathematical model shown in Fig. P2.12. Assume that the damping in these systems is equal to lO% of the critical damping.

Fig, P2.12.

I.

46 Structures Modeled as a Sing!e-Degree-of~Freadom System

2.13

2.14

A vibration generator with two weights each of 30 lb with an eccentricity of lOin rotating about vertical axes in apposiLe direclions is mouo!ed on the

48 Structures Modeled as a. Sing!e~Degree-ofFreooom System

f.-y...j )b) >y ... ---,[-... -:-t---'h fo 5ir.;:;r

Fig. 3~1 (a) Undamped osclllator harmonically excited. (b) Free body diagrf'.!:1.

is ~"e frequency o~ the force in radi,ans per second. The differential equation ob.amed by summrng all the forces m the free body diagram of Fig. 3, I (b) is

my + ky = Fo sin ?Ix (3.1 )

The solution of eq. (3.I) can be expressed as

(3.2)

,,:here YC(,I) is the complementary solution satisfying the homogeneous equa-tlOn~ that IS, eq: (3.1) wIth the left-hand side set equal to zero; and Yp(l) is the partlc~lar solu~Jon based on the solution satisfying the nonhomogeneous dif-ferentJal equatIOn (3,1). The complementary solution, v (!) is 2:iven by eg (l.I7) as < , -

y .. (1) =A cos Wl-:- B sin (ut (3.3)

where w [kim" The nature of the forcing function in eq. (3.1) suggests that the partIcular

solution be taken as

yp(t) = Y sin iiJt (3.4)

wher~ Y 15 the peak value of the particular solution. The subsritut10n of eq. (3.4) Into eq. (3.1) followed by canceHation of common factors gives

?naif"'" kY= Fo

Respo-:)Se of One-Degree"of-Freedom System 10 Harmonic Loading 49

or

y= Fo k-m&/

(3.5)

in which r represents the ratio (frequency ratio) of the applied forced frequency to the natural frequency of vibration of the system, that is,

iii r=-

'"

Combining eqs. (3.3) through (3..5) with eq. (3.2) yields

. Folk. y(l) ==.1 cos wt+ B Sin wt+~ S10 col

1 - r

(3.6)

(3.7)

If the initlal conditions at time t:=: 0 are taken as zero (Yo";;:; 0, Of) 0), the constants of integration determined from eq. (3.7) are

rFolk A=O, B=~17

which, upon substitution in eq. (33). gives

Folk. .) y (I) ~ ---, (sm iilt - r sm WI 1 - r~ (3.8)

As we can see from eg. (3.8), the response is given by the superposition of two hannonic tenns of different frequencies. The reSUlting motion is not hannonic; however, b the practlcal case, damping forces wiH always be present in the system ar:d will cause the last tenn, i.e., the free frequency term in eq. (3.8), to vanish eventually. For this reason, this term is said to represent the transient response. The fordng frequency term in eq. (3.8). namely

F,lk y{t) = Si;1 wt

1 (3.9)

is referred to as the steady-state response. It 1s clear from eq. (3,8) that in the case of no damp1ng in the system, the transient will not vanish and the response is then given by eg. (3.8). It can also be seen from eg. (3.8) or eg. (3.9) that when the forcing frequency ~s equal to the natural frequency (r == LO), the amplitude of the motion becomes infinitely large. A system acted

50 Structures Modeled as a SingJe-Oegree-of-Fredom System

upon by an external excitation of frequency cOinciding with the natural fre-quency is said to be at resonance, In this circumstance, the ampljtude will' increase gradualiy to infinite. However, materials that are commonly used jn practice are subjected to strength limitations and in actua; structures failures occur tong before extremely large amplitudes can be auained.

3.2 DAMPED SYSTEM: HARMONIC EXCITATION Now consider the case of the onedegree-of-freedom system in Fig. 3.2{a) vibrating under the jnfluence of viscous damping. The differentia! equation of motion is obtained by equating to zerO the sum of the forces in the free body diagram of Fig. 3.2(b). Hence

my + cy + ky = Fo sin [ol (3.10)

The complete solution of this equation ag;::jn consists of the complementary solution Ye(r) and the panicular solution yp(t). The complementary solution is given for (he underdamped case (c < C,.) by eqs. (2.15) and (2.19) as

Y,,{t) = e-~(A cos (,tJo(+ B sin wni) (3.11 ) The particular solution may be foand by substituting Yr, in this case assumed lo be of the fonn

(3.12)

into eq. (3.10) and equating the coefficients of the sine and cosine functions, Here we follow a more elegant approach using Eu}er's reJat:on, namely

e;:" cos wI + i sin WI

(bl

Fig. 3.2 (a) Damped oscitlator harmonically exciled. (b) Free body diagram.

.i

Response of One--Oegree-ofFreedom System to Harmonic Loading 51

For this purpose, t:le reader should realize that we can write eq. (3,10) as

(3.13)

with the understanding that only the imaginary component of Fo etOJ

52 Structures Modeled as a Sir,gle~Degfee..j)f~f(eedom System

The response to the force in Fo sin WI (the imaginary component of Foe i'''') is then the imaginary component of eq. (3.;6), na:nely,

(3.18)

or

Yp= Y sin (Wi ... Il) (3.19) where

is the amplitude of the steady~s"'te motion. Equations (3.18) and (3.17) may conveniently be wnuen in terms of dimensionless ratios as

(3.20)

and

tan B= (3.21)

where Ys, = Folk is seen to be the static deflection of the spring acted upon by the force F 0; g = c lew the damping ratio; and r iJJl lV, the frequency ratio. The total response is then obtained by summing the complementary solution (tran-sient response) from eq. (3.11) and the particular solution (steady~state re-sponse) from eq. (3.20), that is,

(3.22)

The reader should be warned that the constants of integration A and B must be evaluated from initial conditions using the total response given by eg. (3_22) and not from just the transient component of the response given in eq, (3.11). By examining the transie

54 Structures Modeled as a Single~Degree~of-Freedom Syslern

180

f

b-~ ~.-

! I

'-44 I. '( , !: 2 , I I

/ l'

.// o . ~

~I l ~O I. ! i 0.1 ...-r ,

/:02 i..,..- f-j I / /,

..

! 0 / 0,'1 I V

iiz I ' /, ! , V --r-/' ' r-'

l =-~?,

V / / L'

i if 0 4 / 1

/oi/ Va. 1

t=

IIV V --~f:::-r-.

!

!

0 !

,

..

, , I I W I

I ,

2 FreQuency ratio r '" w/w

i

i

,-

-

,

--1

' ,2:: ::::

4 -

~- -!

-i

i I

i +_ I i

3

Fig. 3.4 Phase angle fj as a function of tbe frequency ratio for various amounls of damping.

Although the dynamic magnification factor evaluated al resonance is close to its maximum value, it ~s not exactly lhe max.imum response for a damped system. However, for moderate amounts of damping, the difference between the approximate value of eq. (3,24) and the exact maximum is negligible.

Example 3~1. A simple beam supports at its center a machine having a weight W 16,000 lb. The beam is made of two standard S8 X 23 sections with a clear span L = 12 ft and total cross~sectiona! moment of inertia 1= 2 X 64,2 = 128A in4. The motor runs at 300 rpm, and its rotor is out of balance: to the extent of W' = 40 lb at a radius of eo = 10 in. What will be the amplitude of the steady-state response if the equivalent viscous damping for the system is assumed 10% of the critical?

SoIUlion: This dynamic system may be modeled by the damped osciHato/ The distributed mass of the beam will be neglected in comparison with the

I 'I

Response of One-Degreo-of-Freedom System 10 Harmonic Loading 55

El

~ L-~----{I

Fig, 3.5 Dlagram [or beam-machine. system of Example 3, I.

large mass of the machine. Figures 3.5 and 3.6 show, respectively, the sche-matic diagram of a beam~machine system and the adapled model. The force at the center of a slmply supported beam necessary to deflect this point one unit (i.e., the stiffness coefflclent) is given by the formuia

k=~= 48X30XIO'X 128,4 =61920 Ib/in L'

The natural frequency of the system (neglecting the mass of the beam) lS

w

the force frequency

rT 161~92() 1_ = I-~-- = 38.65 rad/sec Y III V 16,000/386

300 X 27f W = --.. -~ = 31 A I rad/sec 60

+ 1in i:;,

t , m 1- ... y

~, ~~Td. /##,$#/////.&7/.,w///)..w~

1,1

!m-m'}~; TT 'y

'"

Fig. 3.6 (a) Mathematical model for Example 3.L (b) Free body diagram.

I I I

i i

56 Structures ModeJed as a Single-Oegree-of-Free-dorn System

and the frequency ratio

Referring to Fig, 3,6, let m be the total mass of the motor and In J the unbalanced rotating mass. Tnen, if y is the vertical displacement of the non-rotating mass (m - m ') from the equilibrium position, the displacement YI of m' as shown in Fig. 3.6 is

Yt = Y + eo sin Wi (a)

The equation of motion is then obtained by summing forces along the vertical direction in the free body diagram of Fig. 3.6(b), where the inertial forces of both the nonrotating mass and the unbalanced mass are also showll. This summation yields

(m - m')y + m), + cj + ky = 0 (b)

Substitution of YI obtained from eg. (a) gives

(m-m')Y+m'(Y-eoli.l' sin &;f)-:-cj+ky 0

and with a rearrangement of terms

mj.i+ C); +ky = m'eo{i} sin Wl (c)

This last equation 1S of the same form as the equation of motion (3. I 0) for the damped oscillator excited hannonically by a force of amptjrude

(d)

Substjtuting in this equation the numerical values for this example. we obtain

F,=(40)(1O)(31.41)'!386= 1022lb

The amplitude of the steady-state resulting motion from eqs< (3< 19) and (320) :5 then

y 1022!6L920 J (l -- 0

58 SL'lJclures Modeled as a Single-Degree-ofFreedom System

(b) Then, the maximum shear force in the columns is

3EIY V~, = = 2018 Ib

the maximum bending moment

l.1 \l\lU = V mIlA L = 36,324 Ib . in

and the maximum stress

Mm." 36,324 crm~~ = -- = = 2136 psi lie (Ans.)

in whrch J Ie is the section modulus.

3.3 EVALUATION OF DAMPING AT