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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 usedin the study of automotive disc brake squeal. It covers two major approachesused in the automotive industry, the complex eigenvalue analysis and thetransient analysis. The advantages and limitations of each approach areexamined. This review can help analysts to choose right methods and makedecisions on new areas of method development. It points out some outstandingissues in modelling and analysis of disc brake squeal and proposes newresearch topics. It is found that the complex eigenvalue analysis is still theapproach favoured by the automotive industry and the transient analysis isgaining increasing popularity.

Keywords: automotive disc brake; friction; contact; vibration; noise; squeal;modelling; numerical methods; the finite element method; complex eigenvalueanalysis; 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 brakesqueal: 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 Departmentof Engineering, University of Liverpool, UK. He was awarded a BE inEngineering Mechanics in 1982, an ME in Solid Mechanics in 1985, and a PhDin Structural Engineering in 1989, from Dalian University of Technology. Hehas published over 70 conference and journal papers. His major researchinterests are structural dynamics and computational mechanics.



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 nottransferable 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 believethat 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 reviewcould 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.



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

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 themechanisms 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 coupledtogether 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 thicknessusing 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 analyticalsolution (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 relativedisplacements 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 anegative 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 0

0 0

0 0

0 0

mk mk f uk k n v

mk mk f u

k k n 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 K f 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 k c is infinitely large, the contact conditions are exactly applied. However,

in the actual numerical analysis, k c 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 isfriendlier 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 k c 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 K f .

Nishiwaki (1993) also put forward a general formulation to accommodate more than onesqueal 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 tosqueal 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 anonrotational 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 couldmodel 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 frequenciesfound 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 wouldreveal which system modes of vibration were unstable. He pointed out that

“knowing the unstable modes facilitates several courses of action; modalfrequencies could be moved by altering components or damping could beadded 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 sidehalf 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 indicatedthat 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 elementanalysis 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 finiteelement assemblages the element damping parameters, in particular thedamping 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 greatdifficulties 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 andnonstationary, 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




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 andDr. 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.

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Appendix

C Finite element damping matrix.

c The speed of sound.

1 2, 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.

ouyang review

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