Numerical analysis of automotive disc brake squeal: a reviearahim/ouyang_review.pdf · numerical analysis work on disc brake squeal. A recent comprehensive review paper (Kinkaid et

Int. J. Vehicle Noise and Vibration, Vol. 1, Nos. 3/4, 2005 207

Copyright © 2005 Inderscience Enterprises Ltd.

Numerical analysis of automotive disc brake squeal: a review

Huajiang Ouyang* Department of Engineering, University of Liverpool, Brownlow Street, Liverpool L69 3GH, UK Fax: 0044 151 79 44848 E-mail: [email protected] *Corresponding author

Wayne Nack Technical Center, General Motor Corporation, Warren, Michigan, USA

Yongbin Yuan Technical Center, TRW Automotive, Livonia, Michigan, USA

Frank Chen Advanced Engineering Center, Ford Motor Company, Dearborn, Michigan, USA

Abstract: This paper reviews numerical methods and analysis procedures used in the study of automotive disc brake squeal. It covers two major approaches used in the automotive industry, the complex eigenvalue analysis and the transient analysis. The advantages and limitations of each approach are examined. This review can help analysts to choose right methods and make decisions on new areas of method development. It points out some outstanding issues in modelling and analysis of disc brake squeal and proposes new research topics. It is found that the complex eigenvalue analysis is still the approach favoured by the automotive industry and the transient analysis is gaining increasing popularity.

Keywords: automotive disc brake; friction; contact; vibration; noise; squeal; modelling; numerical methods; the finite element method; complex eigenvalue analysis; transient analysis.

Reference to this paper should be made as follows: Ouyang, H., Nack, W., Yuan, Y. and Chen, F. (2005) ‘Numerical analysis of automotive disc brake squeal: a review’, Int. J. Vehicle Noise and Vibration, Vol. 1, Nos. 3/4, pp.207–231.

Biographical notes: Huajiang Ouyang is a Senior Lecturer at the Department of Engineering, University of Liverpool, UK. He was awarded a BE in Engineering Mechanics in 1982, an ME in Solid Mechanics in 1985, and a PhD in Structural Engineering in 1989, from Dalian University of Technology. He has published over 70 conference and journal papers. His major research interests are structural dynamics and computational mechanics.

208 H. Ouyang, W. Nack, Y. Yuan and F. Chen

Wayne V. Nack has published papers in brake squeal and moan as well as in vehicle structural dynamics and optimisation. He received his BS in Engineering at Rose Hulman Institute, and his PhD in Theoretical and Applied Mechanics at the University of Illinois in Computational Mechanics. Wayne has 30 years of experience in developing new methods for the automotive industry and has published papers in the above areas.

Yongbin Yuan is Chief Engineer of TRW Automotive, Shanghai. He was a Senior Manager and Fellow of TRW Automotive at Livonia, MI, before his present post. He has a PhD in Material Science. He has published a number of papers on disc brake noise and friction.

Frank Chen is a Technical Specialist at the Ford Motor Company. He has been working in noise and vibration area for more than ten years. He has a PhD in Mechanical System Engineering and has published a number of technical papers on disc brake noise.

1 Introduction

Automotive disc brake squeal has been a challenging issue for many engineers and researchers due to its immense complexity. Much progress and insight have been gained in recent years and brakes have become quieter. However, squeal still occurs frequently and therefore much still needs to be understood and done. CAE simulation and analysis methods play an important role in understanding brake squeal mechanisms. It can also be used to interpret test results, prepare for upfront DoE (design of experiment), simulate structural modifications and explore innovative ideas. The methodology is still under development. Some methods have matured and others are being refined. It is thought that the time has come to review the established methods to date.

There is a wealth of literature on automotive disc brake squeal. Reviews (North, 1976; Crolla and Lang, 1991; Nishiwaki, 1990; Yang and Gibson, 1997; Kinkaid et al., 2003) conducted in the last 30 years provide a comprehensive source of information. However, there has not been any paper that reviews the area of numerical analysis work on disc brake squeal. A recent comprehensive review paper (Kinkaid et al., 2003) has only lightly touched on finite element models of disc brakes. Therefore, this paper serves to fill this apparent gap and complement the existing literature.

According to the mechanism of generation, brake noise can be classified into three types. The first type is called creep-groan, which is caused by the stick-slip motion between the friction material and the rotor surface (Abdelhamid, 1995; Brecht et al., 1997). Creep-groan occurs at near-zero vehicle speed. The second type of noise is often called hot judder or rumble, which is caused by periodic features on the rotor surface that result in cyclic brake torques (Abdelhamid, 1997; Swartzfager and Seingo, 1998; Kubota et al., 1998). The salient feature of this type of noise is that its frequency is a multiple of the rotor speed of rotation. The third type of noise is characterised by:

Numerical analysis of automotive disc brake squeal: a review 209

• the absence of apparent sticking at the rotor/pad sliding interface • in general, one dominant high frequency fairly independent of the rotor speed • occurrence of in-plane and/or out-of-plane flexible rotor modes.

This type of noise is usually called squeal. Noise with a dominant frequency over 1 kHz (or above the first out-of-plane rotor frequency) generally belongs to this category. These are meant as a description of disc brake squeal rather than a definition, for there is no generally accepted precise definition (Kinkaid et al., 2003).

Squeal has been the primary subject of past studies on brake noise and is the focus of this review.

Investigation into brake squeal has been conducted by various experimental and analytic methods. Experimental methods, for all their advantages, are expensive mainly due to hardware cost and long turnaround time for design iterations. Frequently discoveries made on a particular type of brakes or on a particular type of vehicles are not transferable to other types of brakes or vehicles. Product development is frequently carried out on a trial-and-error basis. There is also a limitation on the feasibility of the hardware implementation of ideas. A stability margin is usually not found experimentally. Unfortunately, this produces designs that could be only marginally stable.

Analytical or numerical modelling, on the other hand, can simulate different structures, material compositions and operating conditions of a disc brake or of different brakes or of even different vehicles, when used rightly. With these methods, noise improvement measures can be examined conceptually before a prototype is made and tested. Theoretical results can also provide guidance to an experimental set-up and help interpret experimental findings. The authors’ intention is to offer a review of the state-of-art of CAE simulation and analysis for disc brake squeal. The authors believe that theoretical methods and experimental methods are equally important. Both are needed to capture the underlying physics so that eventually a commercial code would be developed to complement and partly replace expensive and time-consuming experimental study. These methods will remain an indispensable tool for understanding brake squeal and for achieving improvements on noise performance in the foreseeable future.

Simulation and analysis methods may be divided into two big categories: complex eigenvalue analysis in the frequency domain and transient analysis in the time domain. The transient results can be converted to the frequency domain by an FFT. Brake models with a large number of degrees-of-freedom (DoFs) or with infinite number of DoFs (continuous media) are of interest to the authors. Models composed of a small number of degrees-of-freedom (lumped-parameter models) will not be covered in this review. The papers being reviewed here have all been published in the public domain (except Yuan, 1997). These exclude internal reports and private communications. This review could not be all-inclusive even though the best possible effort has been attempted.

Brake squeal is the result of friction-induced vibration, which was reviewed comprehensively by Oden and Martins (1985), and also by Ibrahim (1994). Friction-generated sound and noise in general was reviewed recently by Akay (2002).

This review paper consists of five major sections: Introduction, Complex Eigenvalue Analysis, Transient Analysis, Outstanding Issues and Conclusions. There is also an appendix for notations used in the paper after references.


2 Complex eigenvalue analysis

The complex eigenvalue approach refers to what results are sought, not on how the structural model is constructed. The model may be derived by analytical methods, the assumed-modes method and the finite element method. There must be a means (known as a squeal mechanism) of incorporating friction as a sort of nonconservative force in the model. Squeal mechanisms were reviewed by North (1976), Crolla and Lang (1991), Nishiwaki (1990), Yang and Gibson (1997) and Kinkaid et al. (2003).

2.1 Analytical methods

Most early works on instability analysis were performed with hand-derived equations for mass-spring models that were used to represent real structures. Insight gained from such simple models in the early days of research greatly enhanced the understanding of the mechanisms of friction-induced noise and vibration. The analyses for more complicated mass-spring models by Jarvis and Mills (1963–1964), Earles and Lee (1976), North (1976), Millner (1978), and many others have revealed that even when the friction coefficient is constant, the model can be unstable if the friction force couples two degrees-of-freedom together. A large-scale finite element analysis of the stability of the linearised brake system also confirms that instability arises when two modes coalesce under the influence of friction.

2.2 The assumed-modes method

Brake components are continuous media of infinite number of DoFs. Because these components are of complicated geometrical shape, the finite element method is most appropriate. Among the brake components, the rotor is the only component that has rather regular geometry. A solid rotor has a number of cyclic symmetry that equals the number of mounting holes in the top-hat section. A vented rotor is also quite close to having many degrees of cyclic symmetry. Both solid rotors and vented rotors possess pairs of very close frequencies (an axially symmetric rotor possesses pairs of identical frequencies).

Chan et al. (1994) considered a brake rotor as a thin, annular plate and thus expressed the transverse vibration of the rotor as a linear sum of analytic component modes. This approach has been followed by Mottershead and coworkers (Mottershead et al., 1997; Ouyang et al., 1998). Hulten and Flint (1999), and Flint (2000) used the assumed modes method for their beam model of the rotor. In the work of Chowdhary et al. (2001), individual brake components (disc and pads both modelled as thin plates) were modelled and solved separately for their modal characteristics. Then, these were coupled together at the contact interface and the equations of motion were derived through the Lagrangian approach. Chakraborty et al. (2002) also used thin plate model for the disc. They introduced (cubic) nonlinear spring for the pads. von Wagner et al. (2003) further demonstrated that the frequency of the limit-cycle vibration of the disc was very close to that of the linear unstable vibration obtained through a complex eigenvalue analysis.

Wauver and Heilig (2003) considered both friction and heating at the contact points between the pads (lumped masses with springs) and the disc (a rotating ring). Like Chakraborty et al. (2002) and von Wagner et al. (2003), Galerkin’s approach was used to discretise the equations of motion.


Recognising the importance of the in-plane modes of the disc, Tzou et al. (1998) derived the analytical solution of 3-D elastodynamics for a cylinder of arbitrary thickness using the Ritz method. They concluded “in-plane loading of the disc, with concomitant resonance of an in-plane mode, is expected to be one mechanism for the generation of out-of-plane vibration”. Tseng and Wickert (1998) obtained the analytic solution of the in-plane stresses of a rotating disc and used it in the equation of motion of the disc. The in-plane shear stresses due to the friction were considered as follower forces (distributed load acting on a sector of the disc), instead of friction itself being the follower force. Gyroscopic and centripetal effects were included so that the work can apply to discs rotating at higher speeds.

Discs (rotors in the automotive industry) are basic mechanical elements that are widely used in engineering industry, such as wood saws, computer disks, and turbine discs and so on. Disc vibration was reviewed by Mottershead (1998).

Since the assumed-modes method normally applies to a model where an analytical solution (closed-form or approximate) may be found, for example, a beam or a plate model for the disc, it is not as widely used as the finite element method. It deserves a place when testing a squeal mechanism, such as in Hulten and Flint (1999), or when incorporating some special features, such as moving loads (Ouyang and Mottershead, 2000), where a complicated system model should be avoided initially.

2.3 The finite element method

The complex eigenvalue approach linearises the brake squeal solution at static steady sliding states. It is a good approximation if it is linearised properly in the vicinity of an equilibrium point like the steady sliding position. This procedure is referred to as a linearised stability analysis and it is discussed by Nack (1995, 2000). Currently, most production work in industry uses this method. Nonlinear transient solutions in multibody and crash codes have also emerged as an alternative.

Liles’s (1989) paper with the help of SDRC (Structural Dynamics Research Corporation) appears to be the first published paper to present a large-scale FE analysis of a real disc brake. He built a finite element model for each brake component using solid elements. Crucially, he also carried out modal testing on these components and validated his component finite element models using experimental results. Then, the less certain connections between individual components were located by ‘using engineering intuition and an iterative testing procedure’. Friction was incorporated into the model as a geometric coupling. A complex eigenvalue formulation was derived for the system. The real parts of the eigenvalues dictate the stability (or lack of it) of the system. If the real part of an eigenvalue is positive, the corresponding imaginary part was thought to be a possible squeal frequency. He constructed the friction stiffness matrix using relative displacements between mating surfaces. However, many theoretical details were not given. This lack of details is common in many papers.

Nack (1995, 2000) presented a detailed method on how to construct the friction stiffness and clearly stated the meaning of using eigenvalue analysis, that is, to determine the necessary condition for a system to become unstable and grow into a state of limit-cycles. From sound signals of squealing events, he also observed that the amplitude exponentially increased at the beginning (positive real part at work here) and then somewhat stabilised (limit-cycle at work now), as shown in Figure 1. This phenomenon can also be found in a simulation: the motion becomes divergent for a linear model but


grows up to a certain magnitude for the corresponding nonlinear model, as demonstrated by Thompson and Stewart (2002) for a different nonlinear system. If a mode has a negative real part in the eigenvalue, then the motion will not have a chance to grow into a limit-cycle and thus cause sustained noise. Engineering design is often more concerned with if a brake may squeal and less with how loud the brake may squeal. Therefore, for engineering design at present, a complex eigenvalue analysis offers a pragmatic approach. Limit-cycle vibration induced by dry friction was studied earlier by D’Souza and Dweib (1990).

Figure 1 An experimentally measured squeal signal

The most common way of incorporating friction into a finite element model is through a geometric coupling. This can be illustrated by a spring that links a pair of nodes on the surface of the rotor and the surface of the pad in contact, as shown in Figure 2.

Figure 2 Contact spring with friction loading

Reprinted with permission from SAE 2003-01-0684 © 2003 SAE International.

The element stiffness matrix for this friction-loaded spring is (Nack, 1995; Ripin, 1995)

c c1 1

c c1 1

c c2 2

c c2 2

0 00 00 00 0

mk mkf uk kn vmk mkf uk kn v

− − = − −

which is asymmetric. The equation of motion for such a system is (Nack, 1995; Murakami et al., 1984; Okamura and Nishiwaki, 1988; Yuan, 1996)

f( ) 0+ + + =Mx Cx K K x


where Kf is made of the element stiffness matrices of the friction-loaded springs. Some assumptions made in this formulation are also discussed by Nack (1995, 2000).

Liles and Nack’s method requires connection of coincident nodes on the interface with massless springs. This is to enforce the condition that the friction interfaces do not separate in the normal contact direction during squeal. Mathematically, the spring connection is equivalent to applying a constraint with the penalty method. When the spring stiffness kc is infinitely large, the contact conditions are exactly applied. However, in the actual numerical analysis, kc has to be finite and thus leads to concerns on accuracy. Otherwise, the eigenvalue solver will become numerically unstable because of the singularly large terms in the matrix.

Yuan’s (1996) paper was written to tackle this issue. The paper presented a general formulation for contact problems involving sliding friction. Constraints for contact in the normal direction are mathematically imposed. For the same problem, the formulation in Yuan (1996) results in a matrix smaller than Liles’s spring method and is friendlier to eigenvalue solvers because the stiffness matrix does not contain very large terms. Besides, the friction model incorporated in the formulation is a linear function of relative sliding speed and normal contact pressure. This formulation was implemented in NASTRAN and was used to conduct a parametric analysis for a small brake model. The results presented by Yuan (1997) revealed that when kc was in the order of 109, results obtained with the formulation in Yuan (1997) and with Liles and Nack’s method are the same. This general formulation is applicable to any types of elements used at the rotor/pads interface, not just limited to contact springs.

North first attributed disc brake squeal to flutter instability (North, 1976). All the FE analyses by the complex eigenvalue approach indicate that when two modes coalesce under the influence of friction, the system becomes unstable. This phenomenon was noted by a number of researchers (Nack, 1995; Okamura and Nishiwaki, 1988; Tworzydlo et al., 1997; Kung et al., 2000; Chan, 1995). The mode coalescence (or mode coupling) in large structures is in essence the same phenomenon as the coupling of two degrees-of-freedom by friction mentioned in the analytic methods earlier. Linear aeroelastic flutter occurs due to mode coalescence too, but the loading comes from a different source, airflow.

There are other ways of incorporating friction into a system model, for example, as a follower force (North, 1976; Chan et al., 1994; Mottershead et al., 1997; Ouyang et al., 1998), or as a friction couple (Hulten and Flint, 1999). Yuan (1995) incorporated the negative friction-velocity gradient, geometric coupling and the follower force into one finite element model and compared one against the other. In recognition that massless springs might not be suitable for representing rotor/pads contact at high squeal frequencies, Yuan (1996) proposed a new way of constructing Kf. Nishiwaki (1993) also put forward a general formulation to accommodate more than one squeal mechanism.

Nishiwaki et al. (1989) studied the effects of slits cut into the rotor. They found that squeal frequencies often occurred in the vicinity of the natural frequencies of the rotor and they devised a method for eliminating squeal by modifying the frequencies and modes of the rotor. It is noted here that Fieldhouse and Newcomb (1996) found from their experiments a squeal frequency that was not close to any frequencies of the rotor.

Kido et al. (1996) studied the connection between the squeal propensity and the ratio of the eigenvalues of the rotor and the caliper. They showed in the finite element results and experimental results that a higher ratio increased the tendency to squeal.


Matsushima et al. (1998) concluded that squeal occurred within a region of piston line pressures and this region widened with increasing friction coefficient.

Guan and Jiang (1998) constructed finite element models of individual disc brake components and then built a reduced multibody system model. They calculated the amplitude coefficients (modal participation factors) for a squealing mode and experimented with adjustment of some frequencies of the component modes. In so doing, the influential component modes on the squeal may be found. Kung et al. (2000) also computed the modal participation factors of component modes. The information would be useful in finding ways of suppressing certain unwanted modes and thus reducing the likelihood of a particular squeal frequency. The above investigations revealed that there was more than one dominant component mode at squealing and all the major components of the disc brake could make comparable contributions. Zhang et al. (2003) focused on one particular squeal frequency and established the contribution to the squeal mode from individual component modes. The above investigations of the component contribution to squeal modes suggest that those analytical models that have ignored some brake components or have modelled some brake components as lumped masses would not predict squeal events well.

Shi et al. (2001) performed structural design optimisation for suppressing squeal. Forty-six 1-gram masses were evenly distributed on the back of both pads. They found that the total mass after optimisation increased by 245 grams. All the roots (complex eigenvalues) were shifted into the left half of the root locus plane and this was impossible to achieve by trial-and-error runs.

Matsuzaki and Izumihara (1993) were the early researchers who realised that some high squeal frequencies were the result of the in-plane vibration of the rotor. Their finite element results were reported to be in good agreement with experimental results. Due to lack of details on the FE model, it is unclear how friction was represented in their work. Nevertheless, this was a groundbreaking piece of research.

The presence of the in-plane vibration at squeal led to the new classification of squeal into two subclasses by Dunlap et al. (1999): the low frequency squeal in the range of 1 kHz up to the frequency of the mode whose deformation in the rotor is of its first in-plane pattern, in which range the bending vibration of the rotor dominates; and the high frequency squeal, which is above the frequency of the first in-plane rotor mode. Chen et al. (2000, 2002) used laser Doppler to measure the rotor’s in-plane modes and provided more cases in which the coupling of the in-plane rotor modes and out-of-plane rotor modes may be responsible for the generation of high frequency squeal, especially at the frequencies in the three zones of the first, second and third pairs of in-plane and out-of-plane rotor modes.

The role of the in-plane motion of the rotor in generating squeal was gradually recognised by the brake community. Baba et al. (2001) designed what they called a nonrotational hat (by creating an asymmetry in the hat). Their experimental results and finite element results found a favourable configuration of the modified hat structure. Another very interesting piece of work was presented by Bae and Wickert (2000). They built a very detailed finite element model of the top-hat rotor (alone) and discovered that certain modes appearing on the annulus part of the rotor could be suppressed by modifying the hat part of the rotor.

In a very recent study, Yang and Afeneh (2004) found that the lineup between the in-plane circumferential frequencies and the out-of-plane diametric frequencies was not correlated with vehicle squeal. This observation seemed to contradict the above findings


or the hypothesis that squeal was due to the coupling of the in-plane and the out-of-plane modes.

Feld and Fehr (1995) published a paper on large-scale FE analysis of squeal in aircraft disc brakes.

The latest version of ABAQUS (version 6.4) provides a new complex eigensolver that is particularly useful for the complex eigenvalue analysis of the disc brake squeal problem. This new capability was created in collaboration with industrial partners and has produced very useful results (Kung et al., 2003; Bajer et al., 2003). It uses direct contact coupling at the disc/pads interface described by Yuan (1996) and there is no need to introduce contact springs at the interface. Predicted complex eigenvalues for a typical disc brake using ABAQUS are shown in Figure 3.

Figure 3 Predicted unstable complex eigenvalues using ABAQUS 6.4

Source: Dr. Shih-Wei Kung of Delphi Corporation kindly provided this figure

In many applications involving discs, the (relative) rotational speed of the discs is very high. It has been a tradition to model the force acting on these discs as moving loads since the pioneering work of Mote (1970) on a stationary disc subjected to a point-wise moving load and of Iwan and Moeller (1976) on a disc spinning past a point-wise stationary load. It should be mentioned that friction is absent in these models. Ono et al. (1991) introduced friction as a follower force to their model of a spinning floppy disc. Chan et al. (1994) put in friction as a follower force to their model of a stationary brake disc.

Historically, in the study of brake squeal, the moving load nature of the problem normally has been ignored in the assumption that the low rotor speed range in which squeal tends to occur does not warrant this complication. Chan et al. (1994) introduced the element of moving loads into the modelling of brake discs. Ouyang et al. formally proposed to treat the disc brake vibration and squeal as a moving load problem (Ouyang and Mottershead, 2000; Ouyang et al., 2003b) and put forward an analytical-numerical combined method (Ouyang et al., 2000). The numerical simulation of a real brake indicated that as the rotational speed varies, some initially stable modes become unstable while some stable modes may become even more stable (Ouyang et al., 2003a). In Ouyang and Mottershead (2000) and Ouyang et al. (2000, 2003a, 2003b), the disc brake was conceptually divided into two parts: the rotating rotor and the nonrotating components. The former was solved by an analytic method while the latter by the finite element method. Predicted complex eigenvalues for one typical disc brake including the moving loads at one disc speed are shown in Figure 4.


Figure 4 Noise indices vs. unstable frequencies from a moving-load simulation

Wauver and Heilig (2003) also considered moving loads in their simplified analytical model of a disc brake. Talbot et al. (2004) introduced a spectral method which could model moving boundary conditions.

The notion of a moving load is to consider that a force moves to the circumferential position of θ + Ωt from its initial circumferential position of θ after time t elapses. As such, moving loads act at different locations at different times, which can bring about resonance even if the force itself is constant. Problems like this belong to a subject called (self-excited) parametric vibration. One recent paper (Ouyang et al., 2004) showed good agreement of predicted unstable frequencies with experimental squeal frequencies in the low squeal frequency range, using the moving load approach. Whether it is crucial to bring in the moving load concept into the study of disc brake squeal needs further investigations.

3 Transient analysis

For solutions close to steady sliding states, the linear complex eigenvalue approach may be sufficient for a stability analysis. A nonlinear transient solution can analyse the effect of time-dependent loads as well as perform a stability analysis. The solutions are appropriate for situations that are far from the equilibrium position or when the effects of nonlinearities are present. It is possible to run the nonlinear transient process from the initial state to steady state. This would simulate the operating conditions. If large amplitude limit-cycles form, then the associated frequencies are thought to represent squeal frequencies (Tworzydlo et al., 1997). The time signal of an experimentally measured squeal noise is displayed in Figure 1. The frequency content can be extracted from the time series by applying an FFT procedure. If nonlinearity is present and not weak, the only way to analyse properly the dynamic behaviour of such systems is to use a transient analysis. In the time-domain integration, both explicit (Nagy et al., 1994) and implicit (Chargin et al., 1997) methods may be used. If system parameters are time-dependent (but not periodic), a transient analysis is necessary. A dynamic system with time-periodic coefficients can be solved approximately using the Floquet theory, or perturbation methods when there is at least one small system parameter (Nayfeh and Mook, 1979).

Transient analyses of simple, lumped parameter models of disc brakes have been performed in the past. Normally, sophisticated friction laws were used. In fact, that was


the main reason to adopt the transient analysis method in the past. Either the finite element method or the assumed-modes method may be used in a transient analysis.

Nagy et al. (1994) pioneered the transient analysis of disc brakes using the finite element method. The nonlinearity of the friction at the rotor/pads contact was considered. They found that the stability of the system was mainly influenced by the friction coupling the rotor and the pads. Hu and Nagy (1997) developed the method of Nagy et al. (1994) and tackled large finite element models of disc brakes. Dynamic contact was considered by incorporating a penalty function to model the contact surface in the virtual work statement. Another merit of this work was the use of pressure-dependent friction coefficient obtained from dynamometer tests. To speed up the explicit numerical integration and yet maintain accuracy, they used a one-point-integration solid element with stiffness hourglass control that they developed with Belytschko. The numerical time-domain information was converted to power spectral density data through FFT. Several predicted frequencies of high sound level compared well with squeal frequencies found from experiments. Chargin et al. (1997) also performed a transient analysis for a simple finite element model of a brake. They used implicit integration with tangent matrices of the steady-state solution. This was the point to start the complex eigenvalue analysis. Again, the contact constraint was imposed by the Lagrange multiplier. A stiff equation solver was necessary to solve the resultant differential equations.

Hu et al. (1999) investigated the effects of friction characteristics and structural modifications (such as pad chamfer or slot). An optimal design was found using Taguchi method. Mahajan et al. (1999) used the same method of Nagy et al. (1994) and the procedure of Hu et al. (1999) and compared the transient analysis, complex eigenvalue analysis and normal mode analysis and found that they were useful at different stages of design.

Following Nagy and Hu’s approach, Chern et al. (2002) performed a nonlinear transient analysis of a disc brake including rotor, pads, two pistons, sliding pins, and anchor bracket using LS/DYNA with constant dynamic friction coefficient. Figures 5 and 6 show the squeal frequency and the corresponding operational deflection shape of the rotor, respectively.

Figure 5 Velocity frequency domain response at rotor with squeal frequency of 7.16 kHz



Figure 6 Operational deflection shape of a rotor at squeal of 7.16 kHz


Van der Auweraer et al. (2002) concentrated on the determination of the contact area in a dynamic process in the transient analysis. Rook et al. (2001) used the ADAMS multibody code with NASTRAN super elements. Travis (1995) carried out a nonlinear transient analysis of aircraft brake squeal.

4 Pros and cons of analysis methods

4.1 Complex eigenvalue analysis

As Liles (1989) noted, the complex roots obtained from an eigenvalue analysis would reveal which system modes of vibration were unstable. He pointed out that

“knowing the unstable modes facilitates several courses of action; modal frequencies could be moved by altering components or damping could be added such that the mode in question becomes stable.”

A complex eigenvalue analysis allows all unstable eigenvalues to be found in one run. While in a transient analysis, many runs have to be executed until a limit-cycle motion is found. So, the complex eigenvalue analysis has a clear advantage over the transient analysis in the amount of computations involved. A complex eigenvalue analysis should always be conducted before a transient analysis is attempted to estimate when an unstable motion may occur.

The limitation of the complex eigenvalue analysis is that it includes the linearised nonlinearities, which are only accurate near steady sliding states. The full effect of nonlinearities away from steady sliding states is not present. Moreover, nonstationary features, such as time-dependent material properties, cannot be included.

Another serious drawback is that while the magnitude of the positive real part of an unstable eigenvalue describes the growth rate of an oscillation, it does not necessarily indicate a higher noise level (Liles, 1989). A stability analysis only indicates the tendency of divergence but not the real magnitude of a motion. In time, the growing motion predicted by a stability analysis will be limited by the nonlinearities in the system and a limit-cycle motion of finite amplitude will result (Crolla and Lang, 1991), which is obtainable only through a transient analysis.


Despite the limitations of the complex eigenvalue approach, disc brake engineers prefer it to the transient analysis approach because they think in terms of modes. To them, unstable modes allow structural modifications to be made to prevent these modes from occurring. Limit-cycle vibration, predicted from a transient analysis, evolves from exponentially growing motion that can be predicted in theory by the positive real parts of the system eigenvalues obtained through complex eigenvalue analysis. These growing motions are unstable vibrations occurring in linear models. Therefore, the large amplitude limit-cycle vibrations (squeal) can be avoided if the linear unstable motions are avoided, as pointed out by a number of people (Nack, 1995; Yuan, 1996). Complex eigenvalue analysis provides a conservative approach (Nack, 2000), for it predicts all unstable modes that may grow into limit-cycle vibration. Which modes indeed grow into limit-cycle vibration and the amplitudes of the resultant limit-cycles, found from a transient analysis, depends on triggering mechanisms. The aim of complex eigenvalue analysis is to attempt to shift those complex eigenvalues that appear in the right-hand side half plane to the left-hand side plane in the root locus diagram (Nack, 2000). D’Souza and Dweib (1990) studied the stability of a linearised system through Routh’s criterion and the limit-cycle vibration from a transient analysis. They compared the results from both approaches and found that the same parameter values led to a non-negative real part of the complex eigenvalues of the linearised system and the onset of limit-cycle oscillation of the nonlinear system. Although the system they studied is a 3-DoF pin-on-disc problem (of dynamic frictional contact), the methodology is valid and has been adopted by a number of researchers working on disc brake squeal.

4.2 Transient analysis

The transient analysis in theory does not need to make some of the important assumptions underlying the complex eigenvalue analysis. These assumptions include a constant contact area between the rotor and the pads, linear friction laws and time-independence of material properties. Moving loads can be handled in a better way in a transient analysis than in a complex eigenvalue analysis because of the time-varying properties introduced by moving loads.

A major shortcoming of the transient analysis is its formidable computing time, and hence slow turnaround time for design iterations. Time-domain solutions require a lot of disk storage and the data are hard to analyse for design changes. To make matters worse, the importance of high frequencies means that the time steps must be very small for explicit integration. The conventional criterion of choosing a time step for explicit integration based on a linear system, for example, used in Hu and Nagy (1997), is

∆t ≤ l/c

To include the contributions from high frequencies, very small ∆t and l must be used. For example, if a frequency of 10 kHz is to have a contribution in the vibration, ∆t should not exceed 10–5.

An implicit integration algorithm damps out the contributions from high frequency modes and allows a much larger time step. But the systems equations must be solved at each step. An unanswered question is what effect the neglected high frequencies in the implicit algorithm has on the overall solution accuracy. Multibody and Crash codes can find simplified nonlinear transient solutions. The linear part of the structure is normally partitioned into substructures to improve efficiency. Generally, the friction degrees of


freedom at the interfaces determine the size of the substructure boundary set, how dense the substructures are and thus the solution time.

It has been recognised by Mahajan et al. (1999), for example, that the analysis of squeal ‘cannot be met by one single method’. These different methods all have a place at different stages of the analysis and are useful for different purposes. It is expected that a variety of analysis methods will coexist and the transient analysis will become more popular in future.

5 Outstanding issues and recommendations

5.1 Squeal mechanisms

There are many ways to induce instability, such as by the effect of follower force, mode coupling, dependence of friction coefficient on contact load, or negative µ – v gradient. In practice, it is important to know which of these is dominant. For example, if the follower force effect is the primary cause of brake squeal, one should somehow increase the bending resistance of the rotor; if mode coupling is dominant, then squeal prevention should be focused on the design of caliper, caliper adapter, rotor or drum. On the other hand, if the negative µ – v gradient is most important, then one should concentrate on developing friction materials less sensitive to speed and possibly add more damping to the system.

So far, most of the published studies assume constant friction. This is most likely for the convenience of analysis or due to the lack of adequate commercial FEA tools. The work reported by Yuan (1996, 1997) indicates the influence of negative µ – v gradient should not be ignored. Kung and his coworkers’ recent results on the complex eigenvalue analysis using the new ABAQUS complex eigensolver indicated that the real parts increased with the negative gradient of the friction-velocity curve (Kung et al., 2003).

5.2 The contact analysis

When a brake is applied, the piston line pressure acts on part of the pad back plate. As a result, the interface pressure distribution at the friction interface is highly nonuniform. Furthermore, surface asperities at the disc/pads interface bring about high localised pressures. Manufacturing variations also produces a variation in stiffness at the rotor/pads interface. The matter is further complicated by the relative sliding between the rotor and the pads, leading to an interface pressure offset to the leading edge (Samie and Sheridan, 1990; Tirovic and Day, 1991). Contact surfaces used in the finite element analysis are discussed in Oden and Martins (1985), Kikuchi and Oden (1988), Laursen (2002), Sextro (2002), Wriggers (2002) and Willner and Gaul (1995).

The interface pressure distribution is thought to be important since the dynamic friction at the rotor/pads interface normally depends on the local pressure. Therefore, a number of researchers have investigated the contact area and the contact pressure of disc brakes.

According to Tirovic and Day (1991), Harding and Wintel (1978) first published results of interface pressure distribution in a brake using a 2-D finite element model. So did Samie and Sheridan (1990). Tirovic and Day (1991), Day et al. (1991) studied the


interface pressure distribution and how it varied with the elasticity of the pads and other parameters. They found that not only the pressure distribution was not uniform, but also a partial gap (loss of contact) existed between the disc and the pads. To overcome this difficulty, soft pads or stiff back plates were suggested to help achieve a more uniform pressure distribution. Ghesquire and Castel (1992) considered the static contact area and used the contact stiffness as a system parameter in their model. Their work also demonstrated the need to include all the modes within the frequency range of the instability (modal proximity alone is not enough to initiate squeal), as noted by Ripin (1995). The piston line pressure was used to specify the contact stiffness of inserted springs at the rotor/pads interface by Ripin (1995). Lee et al. (2003b) studied the interface pressure distribution at different values of the contact spring constant and found the stiffness value above that the interface pressure distribution would not vary. They used ANSYS gap elements at the contact interface.

During vibration, the contact area and the interface pressure distribution of a disc brake vary with time. This is a dynamic contact problem, which poses a difficult modelling and computing issue. While in the complex eigenvalue procedure, the roots are found at the steady sliding position. Normally, it is assumed that the contact area determined by a nonlinear static contact analysis is maintained during vibration (Ouyang and Mottershead, 2000; Ripin, 1995; Lee et al., 2003b; Blaschke et al., 2000).

The nonlinear transient analysis approach can use the fully nonlinear contact conditions (Nagy et al., 1994). However, the dynamic constitutive friction law for a rotating rotor/pads contact is still the subject of on-going research. A transient analysis assuming a constant contact area was carried out by Moirot and Nguyen (2000). Rumold and Swift (2002) recently studied the effects of the finger positions and piston locations of a twin piston disc brake of a medium truck. They concluded that a shift of the fingers or the pistons to the trailing side was beneficial since a more uniform interface pressure distribution was achieved. They used a multibody analysis with flexible superelements for efficiency in the transient analysis of the contact pressure. Multibody analyses were also performed by Lötstedt (1982) and Van der Auweraer et al. (2002).

Regardless of the subsequent modal analysis methods, frictional contact may be enforced in more than one way (Nack, 2000; Yuan, 1996; Van der Auweraer et al., 2002; Blaschke et al., 2000; Moirot and Nguyen, 2000). It is worth mentioning that Moirot and Nguyen (2000) and Moirot et al. (2000) imposed unilateral constraint at the contact points.

Once the static contact area is determined, the connection between the rotor and the pads is known. In Nack’s finite element model (Nack, 1995, 2000), contact springs initially were distributed over the whole apparent contact area in the contact model. Those springs found to be in tension after the static contact analysis were later removed from the modal analysis model. This strategy has been adopted by many other researchers, for example, Lee et al. (2003b). While in Ouyang and Mottershead (2000), a single model was constructed for both the contact analysis and the modal analysis. A layer of thin solid elements was inserted between the rotor and the pads and those elements found to be in tension after the static contact analysis were given very small Young’s modulus in the subsequent modal analysis. Direct contact between the disc and the pads can be imposed in ABAQUS.

It is now a common practice to determine the interface pressure distribution and the contact area at the rotor/pads interface prior to a complex eigenvalue analysis (for example, Nack, 2000; Ouyang et al., 2003a; Lee et al., 2003b; Blaschke et al., 2000;


Lötstedt, 1982; Moirot et al., 2000; Lee et al., 2003c). Much less attention has been paid to other contact interfaces. If the leading or trailing edge abutment is deemed a place for noise performance improvement, the contact there needs to be considered properly as well. The rotating rotor tends to push the back plate of the piston-pad against the trailing side abutment, leaving a small gap or less tight contact at the leading side abutment (for a pad with ears but no horns). The trailing side abutment obviously provides resistance to a pad being compressed against it, but not to a pad moving away from it. As such, it supplies only a one-sided constraint to the motion of the pad normal to the contact plane. This kind of constraint was studied in other circumstances (Lötstedt, 1982). It is not clear at the moment whether this would alter the vibration of the pad to a significant extent.

The interface between the back plate of piston-pad and the piston head is another area of great interest. A shim of various configurations/materials may be attached to the back plate or to the piston head. Lubricants may be applied at the interface as a measure for reducing squeal propensity. Linear elastic springs are not thought to be adequate to model the shim or the presence of anti-squeal product (greases, for example). Therefore, there is a nontrivial modelling issue there. The contact interface between the fingers and the finger-pad back plate may need to be modelled too.

Stochastic analysis is used by Oden and Martins (1985), Tworzydlo et al. (1997), Wriggers (2002) and Tworzydlo et al. (1999a) to evaluate the friction stiffness due to random surface asperities.

5.3 Representation of damping

The complex eigenvalue approach normally finds more unstable frequencies than experiment-established squeal frequencies, when damping is omitted. It is argued in Nack (1995) that a conservative complex eigenvalue approach predicts modes that are picked up in a dynamometer test with pressures and velocities varying slightly from the baseline model. If damping is duly specified, there is a very good reason to believe that the predicted unstable frequencies should be close to squeal frequencies. Unfortunately, damping is very difficult to determine and model, in particular for a model of many DoFs. The notion of viscous damping is a convenient one rather than an accurate one. As Bathe and Wilson (1976) commented

“in practice, it is difficult, if not impossible, to determine for general finite element assemblages the element damping parameters, in particular the damping properties are frequency dependent.”

To incorporate damping into the complex eigenvalue procedure, modes could be found in bands with damping evaluated at the central frequency.

If damping is assumed to be proportional, the damping matrix can be prescribed from two so-called Rayleigh coefficients (Bathe and Wilson, 1976). It seems that modal damping is not a good description of energy-dissipation mechanism for brakes (Shi et al., 2001). When a system consists of materials of distinct damping features such as in disc brakes, proportional damping is not a good model (Bathe and Wilson, 1976). In that case, Bathe and Wilson thought that “it may be reasonable to assign in the construction of the damping matrix different Rayleigh coefficients to different parts of the structure”. Other types of damping analysed in Nashif et al. (1985) are also worth looking at.


Interestingly, Hoffmaan and Gaul (2003) demonstrated that viscous damping could promote mode-coupling instability in the friction-induced vibration of a 2-DoF system.

5.4 Structural modifications

Structural modifications include geometric and material modifications of a disc brake system, removal of a part or the whole of an existing component or more often insertion of a new component. They are widely used as a fix to a squealing brake and are also often simulated numerically. The motivation behind the work is to attempt to prevent certain unwanted modes from occurring or to shift some noisy frequencies. They have had a limited success. The crux lies in the fact that the shift of some existing noisy frequencies can create new noisy ones when a modification is made. Nevertheless, these are effective ways of suppressing unwanted vibration. When combined with optimisation or inverse methods, structural modifications can be a major effective tactic. Work on structural modifications is reported in many papers (for example, Liles, 1989; Nishiwaki et al., 1989; Kido et al., 1996; Guan and Jiang, 1998; Shi et al., 2001; Matsuzaki and Izumihara, 1993; Dunlap et al., 1999; Baba et al., 2001; Bae and Wickert, 2000; Hu et al., 1999; Tirovic and Day, 1991; Day et al., 1991; Ghesquire and Castel, 1992; Rumold and Swift, 2002; Abu Bakar and Ouyang, 2004).

5.5 Friction laws and wear

Kinkaid et al. (2003) observed that sophisticated friction laws developed in tribology have not been adopted by the brake research community. Given a very complicated system like a disc brake, it is natural to use simple friction laws in analysis and simulation. This should remain true in near future. As computing power and capacity grows, it can be expected that sophisticated friction laws (for example, those reviewed by Oden and Martins, 1985) will be used in CAE simulation and analysis. Wear, as a by-product of friction, may be accommodated in refined friction laws. New friction formulations are discussed by Tworzydlo et al. (1997), Sextro (2002), Willner and Gaul (1995) and Tworzydlo et al. (1999b). Persson’s (2000) monograph on sliding friction is also a good source of information.

5.6 Uncertainties and stochastic study

It has been well known that brake components possess a large range of variations in material properties and sizes. The influence of the variations can be clearly seen from measured and simulated frequencies on system and component levels. Chen et al. (2003) studied a number of parameter variations and their sources. The variations present great difficulties in validating a numerical model or a physical design. A robust theoretical model taking into consideration the uncertainties due to material and size variations is a step forward. It is recommended here that theoretical uncertainty models should be developed.

The repeatability of friction tests and squeal tests is notoriously poor. Squeal is a largely elusive phenomenon. Surface roughness of the rotor and the pads is known to affect wear and squeal behaviour and appears random. Wear itself is a random variable. Environmental effects that cannot be accommodated well in friction laws also present a degree of unpredictability. The variability and uncertainty abundant in disc brakes calls


for a stochastic model. Qiao and Ibrahim (1999) found in their experiments that the interfacial friction force between a rotating disc and a pin was non-Gaussian and nonstationary, and its power spectra density covered a wide frequency band. Ibrahim et al. (2000) established a beam model for the friction element with random friction force as an external load. As mentioned before, some researchers have studied the random nature of friction stiffness (Oden and Martins, 1985; Tworzydlo et al., 1997; Willner and Gaul, 1995; Tworzydlo et al., 1999a).

There are two possible approaches: the statistic approach and the deterministic approach. Langley (1999) showed that the various probabilistic and possibilistic methods can be expressed in terms of a common mathematical algorithm. The challenge here is to incorporate the uncertainty model with the structural dynamics model. A large pool of test data is needed. There is a serious issue of computing load. Elishakoff and Ren’s (2003) timely monograph describes how to use the finite element method to deal with structures with large stochastic variations.

Chen et al. (2003) reported a robust design of a disc brake through structural optimisation using the complex eigenvalue approach, considering

• friction coefficient range of 0.2 to 0.75. • major elastic constants having 30% variations. • pad resonant frequency change induced by worn-off using 50% thickness friction

material.

Numerical results indicated that the brake system studied remained stable even at a friction coefficient of 0.75.

5.7 Thermal effects

The change of material properties and deformation due to temperature change has been largely omitted in squeal analysis. However, many published papers on low frequency brake noise are available which consider both mechanical and thermal effects, and even coupled thermo-mechanical effects. The work can be borrowed fairly easily. Again, the complexity and the computing load involved are the main obstacle to any work involving the thermal effects. Temperature variation, like wear, had better be accounted for in a sophisticated friction law.

An important part of the brake to consider temperature influence should be the brake fluid. It is well known that the compressibility of the brake fluid can change drastically with rising temperature as a result of a long period of persistent braking application, which may alter the dynamics of the system. Chen et al. (2003) recently presented test results on the effects of temperature variation on a number of system parameters (for example, friction coefficients, Young’s modulus). The variability in material properties due to temperature variation may be tackled by stochastic methods. Incidentally, Qi et al. (2004) simulated temperature distributions on the surface of a pad from transient thermal analysis

5.8 Computational efficiency

With the complex eigenvalue approach, it is a standard practice to run complex eigenvalues analysis repeatedly for different system parameters in order to locate


favourable parameter values. This is a very time-consuming process even with powerful computers. Lee et al. (2003a) introduced the recently developed ABLE (Adaptive Block Lanczos method for non-Hermitian Eigenvalue problems (Bai et al., 1999) algorithm to the complex eigenvalue analysis of disc brake squeal problem and found that this algorithm allowed this analysis to run even on a PC. Gaul and Wagner (2004) recently put forward a new technique for the sensitivity study of eigenvalues and eigenvectors with respect to parameter variations. This technique was shown to be suitable for the complex eigenvalue analysis of the brake squeal problem.

There are sophisticated friction laws. Dynamic contact can be represented in a better manner. The knowledge of where a brake model can be refined is largely available. But it is naïve to assume that it is in the end down to only the computing load that dictates whether a very sophisticated analysis method and model should be used. Increasing computing power will not replace further research into the physics of friction and contact. Thermal and stress coupling is also a serious, poorly understood issue. Analysts and designers should always be ready to embrace new knowledge and also watch out for new, more efficient methods. Explicit and implicit integrations should be compared to determine which is more efficient as well as more accurate. When using stochastic methods, special efficient computational procedures must be established.

5.9 Pin-on-disc models

There is a large body of papers on the so-called pin-on-disc models. The idea is believed to stem from early research work on computer disk systems, pioneered by Iwan and Moeller (1976) and Bogy and his coworkers (Ono et al., 1991). Some of these works attempt to address disc brake squeal (for example, Tuchinda et al., 2001). They are not a topic of this review. Interested readers may consult the review paper by Mottershead (1998).

6 Conclusions

There are numerous factors that influence the occurrence and level of squeal noise in automotive disc brakes. Ideally, all effects should be included in an analysis or simulation exercise. These should include linear super elements, complex eigenvalues, nonlinear transient stage, contact surfaces with friction (Samie and Sheridan, 1990; Tirovic and Day, 1991; Kikuchi and Oden, 1988; Laursen, 2002; Sextro, 2002), nonlinear friction laws (Oden and Martins, 1985; Tworzydlo et al., 1997; Laursen, 2002; Wriggers, 2002; Tworzydlo et al., 1999b) and the effect of relative rotation between the rotor and the pads. The friction laws could be built in and/or read in by user subroutines. Developers of commercial software packages are starting to put in some useful capabilities. Nonlinear finite element crash codes and multibody codes are being extended for studying brake squeal. Both implicit and explicit integrated codes are needed. At present, there does not exist a complete system. Capabilities such as wear and thermal-mechanical effects are needed as well. Stochastic methods for robust design solutions will need further development. Exciting new development in the theoretical and experimental investigation into automotive disc brake squeal can be expected in near future.


This review also finds that the complex eigenvalue analysis is still the industry-favoured approach and the transient analysis is gaining popularity.

Acknowledgments

The authors are indebted to all their coauthors in the cited references. Professor O.M. O’Reilly kindly allowed the authors to have a preview of his manuscript (Kinkaid et al., 2003), which partly provided the inspiration for writing this review paper. Dr. Y. Chern of Ford Motor Company has supplied two figures (Figures 5 and 6) in this paper. A number of people have sent their papers on request. They include Professor A.K. Bajaj of Purdue University, Professor D.C. Barton of University of Leeds, Professor L Gaul of University of Stuttgart, Dr. S-W. Kung of Delphi Corporation, Dr. M. Nishiwaki of Toyota Motor Corporation, Dr. M. Tirovic of Brunel University and Dr. U. von Wagner of TU-Darmstadt, Dr. M. Yang of Bosch Automotive. Discussions with Professor J.E. Mottershead of University of Liverpool, Dr. A. Wang and Dr. S.E. Chen of Ford Motor Company were very useful. Several colleagues have read different versions of this paper and suggested improvements.

This paper has used a large part of SAE Paper 2003–01–0684 written by the same authors. They are grateful for the permission granted by SAE International.

References Abdelhamid, M.K. (1995) Creep-groan of Disc Brakes, SAE Paper 951282. Abdelhamid, M.K. (1997) Brake Judder Analysis Using Transfer Functions, SAE Paper 973018. Abu Bakar, A.R. and Ouyang, H. (2004) ‘Contact pressure distribution by simulated structural

modifications’, Proceedings of IMechE International Conference on Braking 2004: Vehicle Braking and Chassis Control, Professional Engineering Publishing Ltd, pp. 123-132.

Akay, A. (2002) ‘Acoustics of friction’, J. Acoust. Soc. Am., Vol. 111, No. 4, pp.1525–1548. Baba, H., Wada, T. and Takagi, T. (2001) Study on Reduction of Brake Squeal Caused by

In-plane Vibration of Rotor, SAE Paper 2001-01-3158. Bae, J. and Wickert, J. (2000) ‘Free vibration of coupled disc-hat structures’, J. Sound Vib.,

Vol. 235, pp.117–132. Bai, A., Day, D. and Ye, Q. (1999) ‘ABEL: an adaptive block Lanczos method for

non-Hermitian eigenvalue problems’, SIAM J. Matrix Anal. Appl., Vol. 20, pp.1060–1082. Bajer, A., Belsky, V. and Zeng, L.J. (2003) Combing a Nonlinear Static Analysis and Complex

Eigenvalue Extraction in Brake Squeal Simulation, SAE Paper 2003-01-3349. Bathe, K-J. and Wilson, E.L. (1976) Numerical Methods in Finite Element Analysis,

Prentice-Hall, Inc., Blaschke, P., Tan, M. and Wang, A. (2000) On the Analysis of Brake Squeal Propensity Using

Finite Element Method, SAE Paper 2000-01-2765. Brecht, J., Hoffrichter, W. and Dohle, A. (1997) Mechanisms of Brake Creep-groan,

SAE Paper 973026. Chakraborty, G., Jearsiripongkul, T., von Wagner, U. and Hagedorn, P. (2002) ‘A new model for a

floating caliper disk-brake and active squeal control’, VDI-Report 1736, pp.93–102. Chan, S.N., Mottershead, J.E. and Cartmell, M.P. (1994) ‘Parametric resonances at subcritical

speeds in discs with rotating frictional loads’, Proceedings IMechE, Part C 208, pp.417–425.


Chan, S.N., Mottershead, J.E. and Cartmell, M.P. (1995) ‘Instabilities at subcritical speeds in discs with rotating frictional follower loads’, ASME J. Vib. Acoust., Vol. 117, No. 2, pp.240–242.

Chargin, M.L., Dunne, L.W. and Herting, D.N. (1997) ‘Nonlinear dynamics of brake squeal’, Finite Elements in Analysis and Design, Vol. 28, pp.69–82.

Chen, F., Abdelhamid, M.K., Blaschke, P. and Swayze, J. (2003) On Automotive Disc Brake Squeal Part III: Test and Evaluation, SAE Paper 2003-01-1622.

Chen, F., Chen, S-E. and Harwood, P. (2000) In-plane Mode/Friction Process and Their Contribution to Disc Brake Squeal at High Frequency, SAE Paper 2000-01-2773.

Chen, F., Chern, Y. and Swayze, J. (2002) Modal Coupling and Its Effect on Brake Squeal, SAE Paper 2002-01-0922.

Chen, F., Tong, H., Chen, S-E. and Quaglia, R. (2003) On Automotive Disc Brake Squeal. PART IV: Reduction and Prevention, SAE Paper 2003-01-3345.

Chern, Y., Chen, F. and Swayze, J. (2002) Nonlinear Brake Squeal Analysis, SAE Paper 2002-01-3138.

Chowdhary, H.V., Bajaj, A.K. and Krousgril, C.M. (2001) ‘An analytical approach to model disc brake system for squeal prediction’, Proceedings of 2001 ASME Design Engineering Technical Conference and Computers and Information in Engineering Conference, Paper DETC2001/VIB-21560.

Crolla, D.A. and Lang, A.M. (1991) ‘Brake noise and vibration – the state of the art’, Dawson, D., Taylor, C.M. and Godet, M. (Eds.): in Vehicle Technology, No.18, in Tribology Series, pp.165–174.

D’Souza, A.F. and Dweib, A.H. (1990) ‘Self-excited vibration induced by dry friction, part 2: stability and limit-cycle analysis’, J. Sound Vib., Vol. 137, No. 2, pp.177–190.

Day, A.J., Tirovic, M. and Newcomb, T.P. (1991) ‘Thermal effects and pressure distributions in brakes’, Proceedings IMechE, J. Auto. Eng., Vol. 205, pp.199–205.

Dunlap, K.B., Riehle, M.A. and Longhouse, R.E. (1999) An Investigative Overview of Automotive Disc Brake Noise, SAE Paper 1999-01-0142.

Earles, S.W.E. and Lee, C.K. (1976) ‘Instabilities arising from the frictional interaction of a pin-disc system resulting in noise generation’, ASME Journal of Engineering for Industry, Vol. 98, No. 1, pp.81–88.

Elishakoff, I. and Ren, Y. (2003) Finite Element Methods for Structures with Large Stochastic Variations, Oxford University Press.

Feld, D.J. and Fehr, D.J. (1995) ‘Complex eigenvalue analysis applied to an aircraft brake vibration problem’, Proceedings of 1995 ASME Des. Eng. Conf., DE-Vol. 84-1, Vol. 3, Part A, pp.1135–1142.

Fieldhouse, J.D. and Newcomb, T.P. (1996) ‘Double pulsed holography used to investigate noisy brakes’, Optics and Lasers in Engineering, Vol. 25, pp.455–494.

Flint, J. (2000) ‘Modeling of high frequency disc-brake squeal’, Proceedings IMechE Int. Conf. Auto. Brakes – Braking 2000, Leeds, pp.39–50.

Gaul, L. and Wagner, N. (2004) ‘Eigenpath dynamics of non-conservative mechanical systems such as disc brakes’, Proceedings of the 22nd Int. Modal Analysis Conf. (CD-ROM), Society for Experimental Mechanics.

Ghesquire, H. and Castel, L. (1992) ‘Brake squeal analysis and prediction’, Proceedings IMechE Conf. Autotech 1992, Paper C389/257.

Guan, D. and Jiang, D. (1998) A Study on Disc Brake Squeal Using Finite Element Methods, SAE Paper 980597.

Harding, P.R.J. and Wintel, J.B. (1978) ‘Flexural effects in disc brake pads’, Proceedings IMechE, Vol. 192, pp.1–7.

Hoffmaan, N. and Gaul, L. (2003) ‘Effects of damping on mode-coupling instability in friction induced oscillations’, ZAMM Z. Angew. Math. Mech., Vol. 83, No. 8, pp.524–534.


Hu, Y. and Nagy, L.I. (1997) Brake Squeal Analysis Using Nonlinear Transient Finite Element Method, SAE Paper 971510.

Hu, Y., Mahajan, S. and Zhang, K. (1999) Brake Squeal DOE Using Nonlinear Transient Analysis, SAE Paper 1999-01-1737.

Hulten, J.O. and Flint, J. (1999) An Assumed Modes Approach to Disc Brake Squeal Analysis, SAE Paper 1999-01-1335.

Ibrahim, R.A. (1994) ‘Friction-induced vibration, chatter, squeal, and chaos: part I – mechanics of friction; part II – dynamics and modeling’, Applied Mechanics Review, Vol. 47, pp.209–253.

Ibrahim, R.A., Madhavan, S., Qiao, S.L. and Chang, W.K. (2000) ‘Experimental investigation of friction-induced noise in disc brake systems’, Int. J. Vehicle Des., Vol. 23, Nos. 3/4, pp.218–240.

Iwan, W.D. and Moeller, T.L. (1976) ‘The stability of a spinning disc with a transverse load system’, ASME Journal of Applied Mechanics, Vol. 43, pp.485–490.

Jarvis, R.P. and Mills, B. (1963–1964) ‘Vibration induced by dry friction’, Proc. IMechE, Vol. 178, No. 32, pp.847–866.

Kido, I., Kurahachi, T. and Asai, M. (1996) A Study of Low-frequency Brake Squeal Noise, SAE Paper 960993.

Kikuchi, K. and Oden, J.T. (1988) Contact Problems in Elasticity, SIAM. Kinkaid, N.M., O’Reilly, O.M. and Papadopoulos, P. (2003) ‘Automotive disc brake squeal:

a review’, J. Sound Vib., Vol. 267, No. 1, pp.105–166. Kubota, M., Suenaga, T. and Doi, K. (1998) A Study of the Mechanism Causing High-speed Brake

Judder, SAE Paper 980594. Kung, S., Dunlap, K.B. and Ballinger, R.S. (2000) Complex Eigenvalue Analysis for Reducing Low

Frequency Brake Squeal, SAE Paper 2000-01-0444. Kung, S-W., Stelzer, G., Belsky, V. and Bajer, A. (2003) Brake Squeal Analysis Incorporating

Contact Conditions and Other Nonlinear Effects, SAE Paper 2003-01-3343. Langley, R.S. (1999) ‘A unified approach to probabilistic and possibilistic analysis of uncertain

systems’, ASCE J. Eng. Mech., Vol. 126, No. 11, pp.1163–1172. Laursen, T.A. (2002) Computational Contact and Impact Mechanics: Fundamentals of Modeling

Interfacial Phenomena in Nonlinear Finite Element Analysis, Springer. Lee, L., Lou, G., Xu, K., Matsuzaki, M. and Malott, B. (2003a) Direct Finite Element Analysis on

Disc Brake Squeal Using the ABLE Algorithm, SAE Paper 2003-01-3350. Lee, Y.S., Brooks, P.C., Barton, D.C. and Crolla, D.A. (2003b) ‘A predictive tool to evaluate disc

brake squeal propensity part 1: the model philosophy and the contact problem’, Int. J. Vehicle Des., Vol. 31, No. 3, pp.289–308.

Lee, Y.S., Brooks, P.C., Barton, D.C. and Crolla, D.A. (2003c) ‘A predictive tool to evaluate disc brake squeal propensity part 2: system linearisation and modal analysis, part 3: parametric design studies’, Int. J. Vehicle Des., Vol. 31, No. 3, pp.309–353.

Liles, G.D. (1989) Analysis of Disc Brake Squeal Using Finite Element Methods, SAE Paper 891150.

Lötstedt, P. (1982) ‘Mechanical systems of rigid bodies subjected to unilateral constraints’, SIAM J. Appl. Math., Vol. 42, pp.281–296.

Mahajan, S.K., Hu, Y. and Zhang, K. (1999) Vehicle Disc Brake Squeal Simulation and Experiences, SAE Paper 1999-01-1738.

Matsushima, T., Masumo, H., Ito, S. and Nishiwaki, M. (1998) FE Analysis of Low-frequency Disc Brake Squeal, SAE Paper 982251.

Matsuzaki, M. and Izumihara, T. (1993) Brake Noise Caused by Longitudinal Vibration of the Disc Rotor, SAE Paper 930804.

Millner, N. (1978) An Analysis of Disc Brake Squeal, SAE Paper 780332.


Moirot, F. and Nguyen, Q.S. (2000) ‘Brake squeal: a problem of flutter instability of the steady sliding solution’, Arch. Mech., Vol. 52, pp.645–661.

Moirot, F., Nehme, A. and Nguyen, Q.S. (2000) Numerical Simulation to Detect Low-frequency Squeal Propensity, SAE Paper 2000-01-2767.

Mote, C.D. Jr. (1970) ‘Stability of circular plates subjected to moving loads’, Journal of the Franklin Institute, Vol. 290, pp.329–344.

Mottershead, J.E. (1998) ‘Vibration and friction-induced instability in discs’, Shock Vib. Dig., Vol. 30, pp.14–31.

Mottershead, J.E., Ouyang, H., Cartmell, M.P. and Friswell, M.I. (1997) ‘Parametric resonances in an annular disc, with a rotating system of distributed mass and elasticity; and the effects of friction and damping’, Proceedings of the Royal Society of London A, Vol. 453, No. 1, pp.1–19.

Murakami, H., Tsunada, N. and Kitamura, T. (1984) A Study Concerned with a Mechanism of Disc-Brake Squeal, SAE Paper 841233.

Nack, W.V. (1995) Friction Induced Vibration: Brake Moan, SAE Paper 951095. Nack, W.V. (2000) ‘Brake squeal analysis by the finite element method’, Int. J. Vehicle Des.,

Vol. 23, Nos. 3–4, pp.263–275. Nagy, L.I., Cheng, J. and Hu, Y. (1994) A New Method Development to Predict Squeal

Occurrence, SAE Paper 942258. Nashif, A.D., Jones, D.I.G. and Henderson, J.P. (1985) Vibration Damping, Wiley. Nayfeh, H. and Mook, D.T. (1979) Nonlinear Oscillation, Wiley-Intersciences, New York. Nishiwaki, M. (1993) ‘Generalised theory of brake noise’, Proceedings IMechE, J. Auto. Eng.,

Vol. 207, pp.195–202. Nishiwaki, M., Harada, H., Okamura, H. and Ikeuchi, T. (1989) Study on Disc Brake Squeal,

SAE Paper 890864. Nishiwaki, M.R. (1990) ‘Review of study on brake squeal’, Japan Society of Automobile

Engineering Review, Vol. 11, No. 4, pp.48–54. North, N.R. (1976) ‘Disc brake squeal’, Proceedings of IMechE, C38/76, pp.169–176. Oden, J. and Martins, J. (1985) ‘Model and computational methods for dynamic friction

phenomena’, Computer Methods in Appl. Mech. Eng.,Vol. 52, pp.527–634. Okamura, H. and Nishiwaki, M. (1988) ‘‘Study on brake noise’ (first report, on drum brake

squeal)’, Trans. Japan Soc. Mech. Eng., Series C, Vol. 54, No. 497, pp.166–174. Ono, K., Chen, J.S. and Bogy, D.B. (1991) ‘Stability analysis of the head-disk interface in a

flexible disc drive’, ASME J. Appl. Mech.,Vol. 58, pp.1005–1014. Ouyang, H. and Mottershead, J.E. (2000) ‘Vibration and dynamic instability of moving load

systems’, in Sas, P. and Moen, D. (Eds.): Proceedings of the 25th International Conference on Noise and Vibration Engineering, Katholieke Universiteit of Leuven, Leuven, Belgium, pp.1355–1360.

Ouyang, H., Cao Q., Mottershead J.E. and Treyde, T. (2003b) ‘Vibration and squeal of a disc brake: modelling and experimental results’, IMechE J. Auto. Eng., Vol. 217, No. 10, pp.867–875.

Ouyang, H., Cao, Q., Mottershead, J.E. and Treyde, T. (2004) ‘Predicting disc brake squeal frequencies using two distinct approaches’, Proceedings of the 22nd Int. Modal Analysis Conference (CD-ROM), Society for Experimental Mechanics.

Ouyang, H., Mottershead, J.E. and Li, W. (2003a) ‘A moving-load model for disc-brake stability analysis’, ASME Journal of Vibration and Acoustics, Vol. 125, No. 1, pp.1–6.

Ouyang, H., Mottershead, J.E., Brookfield, D.J., James, S. and Cartmell M.P. (2000) ‘A methodology for the determination of dynamic instabilities in a car disc brake’, Int. J. Vehicle Design, Vol. 23, Nos. 3–4, pp.241–262.


Ouyang, H., Mottershead, J.E., Cartmell, M.P. and Friswell, M.I. (1998) ‘Friction-induced parametric resonances in discs: effect of a negative friction-velocity relationship’, Journal of Sound Vibration, Vol. 209, No. 2, pp.251–264.

Persson, B.N.J. (2000) Sliding Friction: Physical Principles and Applications, 2nd ed., Springer. Qi, H.S., Day A.J., Kuan K.H. and Forsala G.F. (2004) ‘A contribution towards understanding

brake interface temperatures’, Proceedings of Int. Conf. Braking 2004: Vehicle Braking and Chassis Control, Professional Engineering Publishing Ltd., pp.251-260

Qiao, S.L. and Ibrahim, R.A. (1999) ‘Stochastic dynamics of systems with friction-induced vibration’, J. Sound. Vib., Vol. 223, No. 1, pp.115–140.

Ripin, Z.B.M. (1995) Analysis of Disc Brake Squeal Using the Finite Element Method, PhD Thesis, Dept. Mech. Eng., University of Leeds.

Rook, T.E., Enright, J.J., Kumar, S. and Balagangsadhar, R. (2001) Simulation of Aircraft Brake Vibration Using Flexible Multibody and Finite Element Methods to Guide Component Testing, SAE Paper 2001-01-3142.

Rumold, W. and Swift, R.A. (2002) ‘Evaluation of disc brake pad pressure distribution by multibody dynamic analysis’, Proceedings IMechE Int. Conf. on Auto. Brakes – Braking 2002, Leeds, pp.75–83.

Samie, F. and Sheridan, D.C. (1990) Contact Analysis for a Passenger Car Disc Brake, SAE Paper 900005.

Sextro, W. (2002) Dynamical Contact Problems with Friction: Models, Experiments and Applications, Springer.

Shi, T.S., Chang, W.K., Dessouki, O., Jayasundera, A.M. and Warzecha, T. (2001) Advances in Complex Eigenvalue Analysis for Brake Noise, SAE Paper 2001-01-1603.

Swartzfager, D.G. and Seingo, R. (1998) Relationship of the Transfer Film Formed under Low Pressure Drag Conditions and Torque Variation, SAE Paper 982255.

Talbot, C., Fieldhouse, J.D., Crampton, A. and Steel, W.P. (2004) ‘Mathematical modeling of brake noise vibrations using spectral methods’, Proceedings of the 22nd Int. Modal Analysis Conference (CD-ROM), Society for Experimental Mechanics.

Thompson, J.M.T. and Stewart, B. (2002) Nonlinear Dynamics and Chaos, 2nd ed., John Wiley and Sons Ltd.

Tirovic, M. and Day, A.J. (1991) ‘Disc brake interface pressure distribution’, Proceedings IMechE, J. Auto. Eng., Vol. 205, pp.137–146.

Travis, M.H. (1995) ‘Nonlinear transient analysis of aircraft landing gear brake whirl and squeal’, Proceedings of 1995 ASME Des. Eng. Conf., DE-Vol. 84-1, Vol. 3, Part A, pp.1209–1216.

Tseng, J-G. and Wickert, J.A. (1998) ‘Nonconservative stability of a friction loaded disk’, ASME J. Vib. Acoust., Vol. 120, pp.922–929.

Tuchinda, A., Hoffmann, N.P., Ewins, D.J. and Keiper, W. (2001) ‘Mode lock-in characteristics and instability study of the pin-on-disc system’, Proceedings of the 19th Int. Modal Analysis Conf., Vol. 1, pp.71–77.

Tworzydlo, W., Hamzeh, N.O., Chang, H.J. and Fryska, S.T. (1999a) Analysis of Friction Induced Instabilities in a Simplified Aircraft Brake, SAE Paper 1999-01-3404.

Tworzydlo, W., Hamzeh, O.N., Zaton, W. and Judek, T. (1999b) ‘Friction induced oscillations of a pin on disk slider: analytical and experimental studies’, Wear, Vol. 236, pp.9–23.

Tworzydlo, W.W., Zaton, Z., Judek, T. and Loftus, H. (1997) ‘Experimental and numerical studies of friction induced oscillations of a pin on disk apparatus’, 1997 ASME Des. Eng. Tech. Conf., Paper DETC97/VIB-3903.

Tzou, K.I., Wickert, J.A. and Akay, A. (1998) ‘In-plane vibration modes of arbitrary thick disc’, ASME J. Vib. Acoust., Vol. 120, pp.384–391.

Van der Auweraer, H., Hendricx, W., Garesci, F. and Pezzutto, A. (2002) Experimental and Numerical Modelling of Friction Induced Noise in Disc Brakes, SAE Paper 2002-01-1192.


von Wagner, U., Jearsiripongkul, T., Vomstein, T., Chakraborty, G. and Hagedorn, P. (2003) ‘Brake squeal: modeling and experiments’, VDI-Report 1749, pp.173–186.

Wauver, J. and Heilig, J. (2003) ‘Friction-induced instabilities in a disc brake model including heat effects’, Proceedings of the 4th Int. Symposium on Vibrations of Continuous Systems, pp.54–56.

Willner, K. and Gaul, L. (1995) ‘A penalty approach for contact description by fem based on interface physics’, Contact Mechanics II, WIT Press.

Wriggers, P. (2002) Computational Contact Mechanics, Wiley. Yang, M. and Afeneh, A-H. (2004) ‘Investigation of mounted disc brake in-plane and

out-of-plane modes in brake squeal study’, Proceedings of the 22nd Int. Modal Analysis Conference (CD-ROM), Society for Experimental Mechanics.

Yang, S. and Gibson, R.F. (1997) ‘Brake vibration and noise: review, comments and proposals’, Int. J. Mater. Product Tech., Vol. 12, Nos.4–6, pp.496–513.

Yuan, Y. (1995) ‘A study of the effects of negative friction-velocity gradient on brake squeal’, 1995 ASME Des. Eng. Tech. Conf., DE-Vol. 84-1, Vol. 3, Part A, pp.1153–1162.

Yuan, Y. (1996) ‘An eigenvalue analysis approach to brake squeal problem’, Proceeding. of the 29th ISATA Conference Automotive Braking Systems, Florence, Italy.

Yuan, Y. (1997) ‘Brake squeal analysis with FEM: numerical results’, The 15th Annual SAE Brake Colloquium, Phoenix, October (oral presentation only).

Zhang, L., Wang, A., Mayer, M. and Blasche, P. (2003) Component Contribution and Eigenvalue Sensitivity Analysis for Brake Squeal, SAE Paper 2003-01-3346.

Appendix

C Finite element damping matrix. c The speed of sound.

1 2,f f Friction forces at nodes 1 and 2 of a contact spring.

K Finite element stiffness matrix.

ck Spring constant of the contact spring at the rotor/pads interface.

M Finite element mass matrix.

l The characteristic element length in a finite element transient analysis.

1 2,n n Normal forces at nodes 1 and 2 of a contact spring.

1 2,u u Displacements at nodes 1 and 2 of a contact spring in the direction of the friction forces.

v Relative velocity between the rotor and the pads.

1 2,v v Displacements at nodes 1 and 2 of a contact spring in the direction of the normal forces.

x Finite element displacement vector. µ Dry friction coefficient.

t∆ Time step length in a transient analysis.

Documents

Numerical analysis of automotive disc brake squeal: a reviearahim/ouyang_review.pdf · numerical analysis work on disc brake squeal. A recent comprehensive review paper (Kinkaid et