brake squeal virtual design.pdf

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2003-01-1625

Virtual Design of Brake Squeal

Chih-Hung Jerry ChungWilliam Steed

Jianrong Dong

MTS SYSTEMS

Bong Soo KimGeun Soo RyuHyundai Motor Co

Copyright 2003 SAE International

INTRODUCTION

Disk Brake Squeal noise is a problem that continues toconfront automobile manufacturers. Customer complaintsresult in significant warranty costs yearly. Furthermore,customer dissatisfaction can cause a loss of futurebusiness. Physically, squeal noise occurs when thefriction coupling between the rotor and pad creates adynamic instability. This leads to vibration of thestructure, which radiates a high frequency noise in the 1-15 kHz range.

Many analytical approaches have been proposed in theliterature to evaluate the Brake Squeal Dynamics and the

most popular approach is the Complex Eigenvalueapproach[3]-[6]. Root Locus analysis is a furtherapplication of the Complex Eigenvalue approach, whichcan trace the unstable modes back to meaningfulstructure mode pairs. While the Root Locus methodprovides a way to identify potential critical modes, itsuffers from several drawbacks, such as long solutiontime and lack of resolution to provide the detail modeinteraction among complex roots.

Chung, et. al. [1] presented a new analysis approach bytransferring the brake system equation of motion fromtransient domain to modal domain. The modal domain

transformation significantly reduces the complexity of theComplex Eigenvalue analysis and provides mechanismof mode coupling phenomenon. The presentedapproach has been successfully applied to solveAutomotive Brake Squeal problem [1] and Motorcyclebrake squeal problem [2]. The goal of this paper is toapply the Modal Domain analysis approach to minimizedesign iterations using FE models and provides a newmethod to calculate Complex Eigenvalues to reducesolution time.

VIRTUAL DESIGN STUDY CONCEPT

The title of the paper is called Virtual Design of BrakeSqueal. Industry usually refers the Virtual Design asusing models to perform design studies before reallychanging the hardware. The Virtual Design concept inthis paper is slightly different. Since building a BrakeSystem model and performing Complex Eigenvaluesolution is time consuming, it is still not efficient if we donot know what kinds of design changes are effective bu

just randomly modifies the model for design studiesTherefore, authors propose a method that uses thebaseline model to perform sensitivity study and use thecalculated sensitivity data to determine optimumcombinations to improve squeals noise. The wholeoptimization process does not involve re-solving the

model and can establish modal frequency tuning targetsfor brake structure. With the modal target, engineers canthen focus on how to modify the structure FE model toachieve the modal tuning target to suppress BrakeSqueal problem. Authors named the whole optimizationprocess as Virtual Design Studies of Brake Squeal.

The common Brake Squeal analysis process is shown inFigure 1. Before the model can be used for designstudies, the system model needs to be correlated withtest data. During system model correlation stage, manyiterations are necessary so as the Complex EigenvalueSolutions. After the system model is correlated, the

current Complex Eigenvalues are used as a criterion todetermine the brake system instability but does noprovide how to change the structure to improve thesystem unstable modes. Engineers then modify themodel with no clear design direction and re-run theComplex Eigenvalue solves until stable system ComplexEigenvalues are found. During this design studyprocess, a lot of models need to be built and manyComplex Eigenvalue Solutions are involved. Building aBrake System model and solve Complex Eigenvaluesare very time consuming so it will take a long time tocomplete a whole development cycle.


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NASTRAN

Complex

Eigenvalue Solve

correlationTest Data

Brake FE Model

FE Model of

Design Study

NASTRAN

Complex

Eigenvalue Solve

Stable

Complex

NASTRAN

Complex

Eigenvalue Solve


Brake FE Model

FE Model of

Design Study

NASTRAN

Complex

Eigenvalue Solve

Stable

Complex

Eigenvalues

NASTRAN

Complex

Eigenvalue Solve


Brake FE Model

FE Model of

Design Study

NASTRAN

Complex

Eigenvalue Solve

Stable

Complex

NASTRAN

Complex

Eigenvalue Solve


Brake FE Model

FE Model of

Design Study

NASTRAN

Complex

Eigenvalue Solve

Stable

Complex

Eigenvalues

Figure 1 Common Brake Squeal Design Study Process.

The proposed Virtual Design Study process is shown inFigure 2. Due to the new Modal Domain Analysisapproach, Complex Eigenvalues is no longer obtained byusing NASTRAN Complex Eigenvalue Solver. TheComplex Eigenvalues can be calculated using MATLABMatrix Eigenvalue Solution with the inputs of Normalmode frequencies and mode shapes of pad-rotor contactDegrees of Freedoms of non-friction Brake Systemmodel. The detail discussion of formulation will bediscussed in the following section. The new ComplexEigenvalue evaluation process reduces significantsolution time especially trying to establish detailEigenvalue trace shown in Figure 3. The new method

only needs to solve system normal modes once and useMATLAB to calculate all Eigenvalues. Comparing to thecurrent approach which needs to run ComplexEigenvalue solve for every data point, the time saving for

just generate a Complex Eigenvalue trace plot istremendous.

The new method uses the natural frequency trace suchas Figure 3 to identify the critical modes that convergeand create unstable system Complex Eigenvalues.Since the new analysis method can provide closed-formEigenvalue solution, the sensitivity of critical mode traceto the change of coefficient of friction can be estimated

easily without resolving the system model. Then thesensitivity values are used to determine how muchfrequency shift of critical modes or mode couplingstrength reduction to avoid unstable mode at operationcoefficient of friction. Engineer can analyze the criticamodes and determine effective modifications can reachtheir frequency target. Hence, the brake squeal designstudies are no longer trial-and-error iterations but withspecific design directions. Once the critical modes

achieve target frequencies, Complex Eigenvalues will becompared to baseline results to verify its result.

NASTRAN

Normal Mode

Solution


Brake FE Model

Critical mode pair

identification and

sensitivity

calculation

Stable

Complex

Eigenvalues

MATLAB

Eigenvalue

Solution

NASTRAN

Normal Mode

Solution

MATLAB

Eigenvalue

Solution

Determine Critical

Mode Targets

frequency and

coupling Strength

Change FE Model

to meet Normal

Mode Target Freq

NASTRAN

Normal Mode

Solution


Brake FE Model

Critical mode pair

identification and

sensitivity

calculation

Stable

Complex

Eigenvalues

MATLAB

Eigenvalue

Solution

NASTRAN

Normal Mode

Solution

MATLAB

Eigenvalue

Solution

Determine Critical

Mode Targets

frequency and

coupling Strength

Change FE Model

to meet Normal

Mode Target Freq

Figure 2 Proposed Virtual Design Process


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Figure 3. Imaginary part of complex Eigenvalue of a Brake System.

MODAL DOMAIN FORMULATION

The brake system equations of motion can be expressedas

[ ]{ } [ ]{ } [ ]{ } fFuKuCuM =++ (1)

where, M , C, and K are the mass, viscous damping,and stiffness matrices respectively for the non-friction

system, u is the displacement vector of the system, fF

is the friction force applied to the system. A dot over thedisplacement denotes differentiation with respect to time.The non-friction system is defined as the brake system,which has its pad and rotor in contact but the frictionforce is zero. The friction force is modeled as

{ } dynamicstaticf NNF += (2)

where and N are the coefficient of friction and

normal force respectively. For the Eigenvalue problem,the static force, which is due to the pressure applied bythe piston, is removed from the equation of motion. Thedynamic normal force is caused by the vibration of thepad and rotor and is represented by

{ }( )padNrotorNsdynamic uuKN ,, = (3)

where padNu , denotes the displacement of pad in normal

direction. Hence the system equation of motionbecomes

[ ]{ } [ ]{ } [ ]{ } { }uKKuKuCuM fs=++ (4)

The brake system with no damping and friction can beexpressed as

[ ]{ } [ ]{ } { }0=+ uKuM (5)

The modal domain transformation can be obtained foequation (5),

{ } [ ]{ }=u (6)

where [ ] is the modal matrix of system equation (5)With the modal transformation, the following relationship(7) can be established.

[ ] [ ][ ] [ ] [ ] [ ][ ]

==

2

2

2

2

1

00

0

0

00

,

n

TTKIM

(7)

wherej is the j-th natural frequency of system

equation (5). From (7) and (4), the Complex Eigenvalueof a non-damping Brake System with friction can beexpressed as

[ ] { } { }0

00

00

00

2

2

2

2

1

2 =

+

+

fs

n

Ks

(8)

The system Complex Eigenvalue is expressed as sand

the dynamic friction term becomes

[ ] [ ] [ ][ ]Dcontactdirectionnormal

T

DOFcontactdirectionforcefriction

ffK

(9)

The inputs of system equation (8) are natural frequenciesof non-friction system, the mode shape of non-frictionsystem at contact degrees of freedoms and the value ocoefficient of friction. Therefore, establishing root locusplot only needs to solve the system model once and thenuse numerical tool such as Matlab to calcuate complex

eigenvalues with different values.

VIRTUAL DESIGN APPLICATION

A brake system model is shown in Figure 4.


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Figure 4 Brake Squeal System model.

The first process of virtual design is system model

analysis. The test data and the model ComplexEigenvalue correlation are shown in Figure 5.

Figure 5 The figure above shows the Brake Dyno test data and squeal

around 3.2 KHz; the red dot represents a squeal event. The lower plot

shows the root locus plot of the system and unstable Complex

Eigenvalues at 3.3 kHz. Note that the straight line of the lower plot

represents the system reference damping value.

From Figure 3, there are two critical modes at 3150 Hzand 3200 Hz converging and create an unstableComplex Eigenvalue at about 3200 Hz. The middlemode at 3175 Hz is close to two critical modes but itdoes not have strong interaction with these two critica

modes and it remains at same frequency as increases.

In order to fix a Brake Squeal problem whose coefficien

of friction 0.5, as shown in Figure 5(b), we eithe

reduce the convergence speed of two critical modes orfurther separate these two modes. However, thequestions are how much mode separation is necessaryor how much convergence speed reduction is necessaryto fix the problem.

MODE CONVERGENCE SPEED

To quantify the critical mode convergence speed, we cantake advantage of the symbolic form of the systemequation of motion (8) and calculate the closed formsolution of

( )[ ] [ ]

jifijf

ij

d

sd *speedeconvergenc

2

2

(10)

Note that the convergence speed is mainly controlled by

the mode shapes of the system because f is madefrom the mode shapes of rotor-pad contact degrees ofreedom.

Virtual Design Approach

It is time consuming to generate a Brake System mode

and it is also time consuming to make a designmodification of component to re-construct a modifiedbrake system model. In todays industry environmentthe product development cycle is cut down significantlyHence, the old trial-and-error method is no longesuitable. It also applies to the Brake Squeal problemsolving, brake industry definitely needs a quick turn-around re-design method.

The proposed Virtual Design method can definitely cudown the trial-and-error time and provides a clear designmodification target for fixing Brake Squeal. The virtuadesign concept is based on equation (8) and (10)

Equation (8) shows that the system ComplexEigenvalues are functions of non-friction natura

frequencyi and mode shapes , and equation (10

shows that the convergence speed is controlled by the

product ofjifijf * . There are two ways to

perform the virtual design study.

The first virtual design is to evaluate the improvement oseparation critical modes. For example, moving thecritical mode of Figure 3 from 3150 Hz to 3100 Hz suchthat the mode convergence may happen at highercoefficient of friction to create a more stable system. The


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way to perform this virtual design study is reassign the

value of Hzi 3150= to Hzi 3100= of equation (8)

and recalculate the system Complex Eigenvalues. Thereason we can make this change is based on minorchanges of the structure, such as removing somematerial, may shift natural frequency and make minorinfluence on its mode shape. Therefore, we can useEquation (8) to calculate the virtual design ComplexEigenvalues. Since all information need for Equation (8)

are already existed, we do not need to re-solve thesystem model so it does not increase any calculationtime. The virtual design result compared to the baselineresult is shown in Figure 6. The critical separation by 1.5% is an efficient way to suppress the system instability.

To achieve the 1.5 % natural frequency separationtarget. We can examine the mode shape and strainenergy of these two critical modes. The first criticalmode of 3150 Hz is shown in Figure 7, and its energy ismainly at caliper. Engineer can examine the modeshape and the high strain energy area to determine themost effective changes to shift the natural frequency to

lower frequency. The second critical mode is shown inFigure 8 and its strain energy is in rotor component.Hence, modifications on rotor may move the frequency tohigher frequency.

Figure 6 The comparison between the baseline system and virtual

design system. The red circle is the baseline system and blue

diamond is the virtual design result.

Figure 7 The first critical mode.

Figure 8 The second critical mode

In a real brake structure, it is a highly integrated structure

and only certain areas are allowed for minor changesWith critical mode natural frequency target, engineerscan spend time to focus feasible and effectivemodifications for design modifications.

The second way to perform the virtual design study is toreduce the mode convergence speed. From equation(8), we know that the convergence speed is proportion to

the product ofjifijf * . If we change the value o

ijf , we can change the convergence speed. Fo

example, if we want to reduce the critical modeconvergence speed of Figure 3 to one-half, we can set

ijfijf = *5.0 (11

The cross mode coupling mechanism is discussed inauthors another paper 2003-01-1624. The reduction omode convergence speed can be obtained by reducingthe vibration level of rotor shell modes or reducing themode shape similarity between coupling modes.

Case 2 shows brake system has squeal problem at 5kHz squeal problem as shown in Figure 9, and thehighest mode convergence happens at mode 67 and 69


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Using the virtual design approach, we found thatincreasing mode separation by 2 % is not effective forthis squeal problem as shown in Figure 10, because thisproblem is mainly due to strong mode coupling. Hence,it is only effective by reducing the convergence. Toreduce the convergence speed by 50%, we set

69,6769,67 *5.0 ff = in equation (8). (12)

The modified system is shown in Figure 11 and theunstable mode is brought into more stable region.

The critical modes of the structure is mainly pad androtor modes as shown in Figure 12a,b. One is rotorbutterfly mode and rotor bending mode. The designmodifications should then focus on how to reduce theirvibration level to fix squeal noise.

Figure 9. A brake system with 5 kHz squeal showing in Root locus

plot.

Figure 10 The virtual design result compared to baseline result.

Figure 11 The virtual design result by reducing coupling strength.

Figure 12a (upper plot) shows the pad critical mode; 12b (lower plot)

shows the rotor bending critical mode.

CONCLUSION

The paper presents a new design study process forbrake squeal development. The Computation efficiencyof the new design process is compared with currencommonly used procedure. Based on the modal domainformulation, the Complex Eigenvalue characteristicequation can be expressed in symbolic form so the modeconverging speed can be derived in closed-form solutionSince the System Complex Eigenvalues are functions ofnatural frequency and friction coupling matrix of the nonfriction system, we can make virtual design changes ocritical natural frequency and friction coupling matrix toevaluate the improvement of virtual designs. After the


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satisfied improvement is obtained in the virtual design,the structure design modification target can bedetermined, engineers can focus on effective designchanges to achieve the squeal suppressing goal.

REFERENCES

1. Chung, Chih-Hung, and Steed, William, andKobayashi, and Nakata, Hiroyuki, A New Analysis

Method For Brake Squeal Part I: Theory for ModalDomain Formulation And Stability Analysis, SAE

paper 2001-01-1600.2. Nakata, H., Kobayashi, K, Kajita, M and Chung C.,

A New Analysis Approach For Motorcycle Brake

Squeal Noise and its Adaptation, SAE paperSETC2001-01-1850.

3. Liles, G., Analysis of Disc Brake Squeal Using FiniteElement Methods SAE paper 891150

4. Chen, L.W. and Ku, D, Stability ofNonconservatively Elastic Systems Using Eigenvalue

Sensitivity, ASME Journal of Vibration andAcoustics, Vol. 116, No. 2.

5. Ghesquiere, H., Brake Squeal Noise Analysis AndPrediction, ImechE paper No. 925060.

6. Lewis, T., Analysis and Control of Brake Noise,

SAE Paper 872240.7. Chung, Chih-Hung, and Donley Mark, Mode

Coupling Phenomenon of Brake Squeal Dynamics,SAE Paper 03-NVC-386.

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