chapter 4 evaluation of disc brake...

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CHAPTER 4

EVALUATION OF DISC BRAKE SQUEAL

4.1 INTRODUCTION

Though much work has been done on the disc brake squeal, it

requires continuous refinement for the prediction accuracy of finite element

models to provide engineers appropriate tools to design quiet brakes. There

are two main categories of numerical approaches (Ouyang et al 2005) that are

used to study this problem:

1. Complex eigenvalue analysis (CEA)

2. Dynamic transient analysis (DTA)

Although complex eigenvalue analysis has been the more popular

approach used for stability studies of brake squeal, dynamic transient analysis

is gradually gaining popularity. In contrast to complex eigenvalue analysis

which is capable of evaluating only the stability of a system, dynamic

transient analysis is capable of considering squeal as vibration problem in the

time domain, from which dominant frequency components associated with

high levels of noise can be determined through a Fourier transformation.

Comparison between CEA and DTA is given in detail in reference (Mahajan

et al 1999, Ouyang et al 2005).

In the following section, both experimental squeal test and dynamic

transient analysis are performed for validating the findings of the complex

eigenvalue analysis. Three improved approaches are considered in order to

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increase the prediction accuracy of CEA results. The correlation between the

two FE methods in terms of predicted unstable frequencies are discussed.

4.2 EXPERIMENTAL SQUEAL TEST

This section describes the use of a brake test rig to investigate the

squeal phenomena of the disc brake assembly, as shown in Figure 4.1. The

purpose of this experimental work is measuring the brake squeal at different

applied pressure and disc rotating speed in order to verify the FE model

results, which will be used for further analysis of the brake squeal. The

experimental work will be presented in three main parts. The first part

describes the brake test rig set-up. The second part explains the squeal testing

methodology. The third part contains the test results.

4.2.1 Test Rig Set-up

The brake test rig can be divided into three main groups: the

driving unit, the braking unit and the measurement facilities. The detail of

each unit is given below:

4.2.1.1 Driving unit

Since the brake squeal occurred at low speeds below 30 km/h and

at low brake pressures below 2 MPa, no large power was required in the

drive. The driving unit consists of a 3.7 kW dc motor that has a maximum

speed of 1500 rpm with a variable speed controller to rotate the driving shaft

at different rotating speeds. Disc brake assembly is connected directly with

the motor through a mild steel coupling and driving shaft, which fixed on the

base of the test rig by two bearing located in between motor and the brake

assembly to ensure the disc is align horizontally and vertically. The brake test

rig used for this experiment included a steering knuckle and wheel hub as a

part of suspensions corner assembly that attaches with the disc brake

assembly.

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4.2.1.2 Braking unit

The braking unit is used to apply the required pressure to disc brake

system. The braking torque is applied by hydraulic pressure pump with

braking line fluid which is connected through hydraulic control valve and

pressure gauge. The pressure gauge is used to measure brake line pressure to

estimate the braking force applied to the brake pads.

4.2.1.3 Measurement facilities

The test rig is equipped with available instruments to measure the

required parameters during squeal event. In order to measure braking torque,

a digital watt meter type C.A 8210 is used to measure the power consumption

through the braking process. A digital tachometer type HTM-590 is used to

read the number of rotations of the disc brake. 1/4" Electret-Measuring

Microphone M 360 Class 1 DIN EN 60 651 was used with frequency range

from 20 Hz to 20 kHz and free-field sound pressure level from 35 dB to130

dB. A dynamic data logger (DEWE-201) is used to monitor sound pressure

level and data from the accelerometer located at the brake pad.

Figure 4.1 Layout of the brake test rig

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4.2.2 Squeal Testing Methodology and Results

Before conducting squeal tests, bedding-in process is performed

(according to SAE J2521 test) for two hours at a low pressure (0.7 MPa) and

low disc speed (50 rpm). The purpose of the bedding-in process is to deposit

an even layer of friction material, or transfer layer, on the rubbing surface of

the rotor disc. The operation of the test rig involves starting the motor and

reduces its speed manually by controlling variable speed drive until it reaches

the desired value. A series of squeal tests are carried out at different rotational

speed and different pressure. The speed of the drive axle is measured through

tachometer. The brake line pressure can be applied by hydraulic pump and

controlled by hydraulic valve to a certain value, which can be displayed in

pressure gauge. The brake torque is determined by measuring the power

consumption through the braking process using a digital wattmeter and rotor

speed is measured using a digital tachometer. Then, coefficient of friction is

determined using Equation 4.1.

2

B

p eff

T

P A R (4.1)

where; TB is braking torque, P is applied pressure, Ap is area of piston and Reff

is the effective radius (radius between centre of the rotor and centre of piston).

Sound pressure level (SPL) is measured using the microphone

which is suspended by its cable about 500 mm from the disc brake assembly.

Simultaneously vibrations were sensed by an accelerometer fixed on the outer

pad back plate. The microphone output signal was fed to a Fast Fourier

Transform (FFT) analyzer and the SPL spectrum was calculated using

(DEWEsoft). The recorded data is plotted as sound pressure level (dB) against

frequency (Hz). SPL value exceeds 70 dB is considered as a squeal. The

results of the rig testing are illustrated in Figure 4.2. It is found that

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experimental squeal frequencies are dominant at 1438 Hz, 2370 Hz, 7442 Hz

and 8557 Hz. Figure 4.3 shows an unstable state leading to a limit cycle, with

strong vibration of the brake system and generate squeal. In particular case for

brake-line pressure of 0.7 MPa and rotational speed of 5 rad/s, it is also found

that there are four squeal frequencies at the same values, as shown in Figure

4.4. This particular case will be used for further study to correct FE results.

Figure 4.2 Sound pressure level versus squeal frequency

-6

-4

-2

0

2

4

6

0 0.2 0.4 0.6 0.8 1

Time (sec)

Ac

ce

lera

tio

n (

g)

Figure 4.3 Brake squeal limit-cycle vibration

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0

20

40

60

80

100

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Frequency (Hz)

Sou

nd

Pre

ssu

re L

eve

l (d

B)

Figure 4.4 SPL of brake squeal at pressure 0.7 MPa and speed 5 rad/sec

4.3 COMPLEX EIGENVALUE ANALYSIS

Recently, the complex eigenvalue analysis, which reveals the

squeal phenomenon as mode coupling, has become an accepted approach for

investigating the stability of brake system. Complex eigenvalues result from

the frictional coupling of brake components due to the off-diagonal terms that

arise in the stiffness matrix of the system causing it to be asymmetrical. The

positive real parts of the complex eigenvalues indicate the degree of

instability of the disc brake assembly and are thought to indicate the

likelihood of squeal occurrence. Generally, disc brake components have been

modeled for their connections by so-called friction springs to generate friction

coupling terms. To solve the CEA the coupling mechanisms between the

contact bodies were modeled by linear spring elements in order that the

stiffness matrix to become asymmetric (Liles 1989, Lee et al 1998,

Papinniemi et al 2008, Dai and Lim 2008).

In recent years, researchers have suggested an alternative method

associated with the direct connection of pads and rotor and the elimination of

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these "imaginary" springs. The interface pressure distribution is thought to be

important method 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. Blaschke et al (2000) proposed the direct connection of

the rotor with the pads, therefore eliminating the "imaginary springs" which

could result in errors when CEA is performed within a high frequency region.

A more detailed description of the CEA is given in the following sections.

4.3.1 Theory of Complex Eigenvalue Analysis

Complex eigenvalue analysis was first employed on lumped

parameter models (Ibrahim, 1994, Kinkaid et al 2003) and subsequently

advances in computer technology have enabled researchers to make analytical

models of great complexity, with the use of three-dimensional finite element

models (Lee et al 1998, Bajer et al 2003, Abu Bakar and Ouyang 2006, Liu et

al 2007, Trichés et al 2008, Dai and Lim 2008).

In this research, complex eigenvalues are solved using the subspace

projection method. The governing equation of motion for vibrating system is

0Mx Cx Kx (4.2)

where M is the mass matrix, C is the damping matrix, which can include

friction-induced damping effects as well as material damping contribution and

K is the stiffness matrix, and x is the displacement vector. For friction induced

vibration, the stiffness matrix has specific properties:

Structure FrictionK K K (4.3)

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where KStructure is the structural stiffness matrix, KFriction the asymmetrical

friction induced stiffness matrix and the friction coefficient. This

unsymmetrical stiffness matrix leads to both complex eigenvalues and

complex eigenvectors. The governing equation can be rewritten as

2( ) 0M C K (4.4)

where is the eigenvalue and is the corresponding eigenvector. In order to

solve the complex eigenvalue problem, this system is symmetrized by

ignoring the damping matrix C and the asymmetric contributions to the

stiffness matrix K. Therefore the eigenvalue becomes a pure imaginary

number = i , and the eigenvalue problem now becomes

0)( 2

structureKM (4.5)

Then this symmetric eigenvalue problem is solved to find the

projection subspace. The N eigenvectors obtained from the symmetric

eigenvalue problem are expressed in a matrix as ],........,[ 21 N . Next, the

original matrices are projected onto the subspace of N eigenvectors and given

as follows:

],,.........,[],.........,[ 2121

*

N

T

N MM (4.6)

*

1 2 1 2[ , ,......... ] [ , ,......... ],T

N NC C (4.7)

*

1 2 1 2[ , ,......... ] [ , ,......... ],T

N NK K (4.8)

Now the projected complex eigenproblem becomes

2 * * * *( ) 0M C K (4.9)

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which is solved using QZ method for a generalised unsymmetrical eigenvalue

problem. The eigenvectors of the original system are recovered by:

k

N

k *

21 ],.........,[ (4.10)

wherek

is the approximation of the kth eigenvector of the original system.

The eigenvalues and the eigenvectors of Equation 4.4 may be complex,

consisting of both a real and imaginary part. For under damped systems the

eigenvalues always occur in complex conjugate pairs. For a particular mode

the eigenvalue pair is:

1,2i i ii (4.11)

where i is the real part, and i is the imaginary part for the ith mode. The

motion for each mode can be described in terms of the complex conjugate

eigenvalue and eigenvector:

i i i ij t j t

i i ix A e A e (4.12)

or

i i it j t j t

i ix A e e e (4.13)

Using the exponential cosine identity:

cos2

i ij t j t

i

e et (4.14)

and disregarding scaling, the displacement can be rewritten as a damped

sinusoidal wave:

cosit

i i ix A e t (4.15)

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Thus, i and i are the real part and damped natural frequency describing

damped sinusoidal motion. If the real part is negative, decaying oscillations

typical of a stable system result. A positive damping coefficient, however,

causes the amplitude of oscillations to increase with time. Therefore the

system is not stable when the real part is positive.

In this research, the complex eigenvalue analysis (in ABAQUS) is

utilised to determine instability of the disc brake assembly. As reported by

Huang et al (2007) that the finite element packages (ABAQUS) has become

quite capable of performing a complex eigenvalue analysis. A new method is

used by Kung et al (2003) and Bajer et al (2003) in order to simulate brake

squeal by combining the nonlinear static analysis with complex eigenvalue

analysis. Four main steps that are required to perform the analysis as follows:

1. Nonlinear static analysis for applying brake-line pressure.

2. Nonlinear static analysis to impose rotational speed on the

disc.

3. Normal mode analysis to extract natural frequency of

undamped system.

4. Complex eigenvalue analysis that incorporates the effect of

friction coupling.

4.3.2 Prediction of Squeal Using CEA

In order to predict the squeal occurrence of the disc brake, the CEA

is performed between 1 kHz and 10 kHz at a rotational speed of 5 rad/s,

applied pressure of 0.7 MPa, constant friction coefficient of 0.5, and without

considering effects of damping. The CEA result is plotted on the complex

plane. Figure 4.5 shows that all modes have zero damping except where pairs

of modes have become coupled and formed a stable/unstable pair. These

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results in the eigenvalue with conjugate pairs are symmetrically located about

the Y-axis. In this case, there are five unstable (squeal) frequencies predicted

at 2777 Hz, 7573 Hz, 8530 Hz, 9453 Hz and 9722 Hz.

0

2000

4000

6000

8000

10000

-120 -80 -40 0 40 80 120

Real Part

un

sta

ble

Fre

qu

en

cy (H

z)

Figure 4.5 Eigenvalues extracted from the disc brake model plotted on

the complex plane

4.3.3 Influence of Friction Coefficient

Brake squeal is generally defined as a friction induced instability

phenomenon. Since friction is the main cause of instability, a complex

eigenvalue analysis has been undertaken to assess the brake stability with the

variation of friction coefficient . In the past, the friction coefficient of the

brake pad in real vehicle was considered to be typically 0.35. However, brake

system today possess friction coefficient that is 0.5 or higher, which increases

the propensity of squeal.

In this section, the effect of friction coefficient of the pad-rotor

interface is performed. The unstable frequencies for varying from 0.1 to 0.5

are plotted as real parts versus frequency in Figure 4.6 to illustrate how the

instability increases with friction level. With the low friction coefficient all of

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the modes of the system will be stable. As the friction coefficient is increased,

modes can be driven closer to one another in frequency. Figure 4.6, shows

results in the form of the real part as a function of frequency for different

friction coefficients. It was found that with µ equal to 0.2 one unstable

frequency is predicted at higher value. With increasing friction coefficient

values up to 0.5, a numbers of unstable modes are seen to appear. It was

observed that the propensity for squeal increases with higher coefficients of

friction. This is because the higher coefficient of friction causes the variable

frictional forces to be higher resulting in the excitation of greater number of

unstable modes.

0

2000

4000

6000

8000

10000

0 0.1 0.2 0.3 0.4 0.5 0.6

Coefficient of Friction

Un

sta

ble

Fre

qu

en

cy (

Hz)

Figure 4.6 Prediction of unstable frequencies with variation of friction

coefficient from 0.1 to 0.5

4.3.4 Mode-Coupling Mechanism

There are many mechanisms of brake squeal, namely, stick–slip,

sprag–slip and mode coupling contributing to brake squeal. Though

considerable amount of research has been done on the squeal problem to date,

it was observed that none of the previous mechanisms alone can explain all

events related to squeal noise. The stick-slip mechanism has not received

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much attention as a mechanism for squeal phenomenon. Chen et al (2003)

noticed that squeal noise can occur in regions with both positive and negative

friction velocity slopes. Hence, there is no correlation between negative

friction velocity slope and the generation of squeal. In spite of all of this,

mode coupling is generally recognized to be one of the most significant

mechanisms and the complex eigenvalues analysis of the brake system is a

popular numerical tool for squeal instability prediction (Ouyang et al 2005).

In this mechanism, the instability is caused by friction induced excitation, due

to coupled resonance between two adjacent vibration modes of the brake

system.

The aim of this section is to help further understanding of the

significance of mode-coupling and extend insights into the brake squeal

generation. The FE analysis by complex modes indicate that when two modes

close to each other in the frequency range coalesce under the influence of

friction (become coupled), the system becomes unstable. To explain coupled

mode analysis, FE model is done with varying the friction coefficient to find

whether system modes become coupled and unstable. The influence of

friction coefficient on the stability of the mode at 9453 Hz is examined.

Figure 4.7 shows real and imaginary part of the eigenvalue pair as functions

of friction coefficient µ . It is seen that when µ is less than 0.22 there are two

distinct modes at different frequencies. As friction reaches 0.22 some of the

adjacent modal frequencies start to merge towards each other and form a

complex conjugate pair. At this point, imaginary parts of eigenvalues

converge to one value and real parts start to diverge. This result indicates that

the coefficient of friction of 0.22 is a critical value of the squeal at 9453 Hz

which gives rise to an unstable mode.

92

9430

9440

9450

9460

9470

0 0.1 0.2 0.3 0.4 0.5


Fre

qu

en

cy

(H

z)

(a) Imaginary part

-120

-80

-40

0

40

80

120

0 0.1 0.2 0.3 0.4 0.5


Re

al p

art

(b) Real part

Figure 4.7 Complex eigenvalues of the 9453 Hz mode as functions of µ

4.3.5 Verification of the CEA Results

In order to verify the CEA result, a comparison between the squeal

frequencies of the experimental test and the predicted results is made. It is

observed that three squeal frequencies measured by experiments are close

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with the predicted frequencies obtained in section 4.3.2 at the same operating

conditions. It is observed that the complex eigenvalue analysis does not

indicate experimental squeal frequency at 1438 Hz and predict unstable

frequencies at 9453 Hz and 9722 Hz. Hence improvement of CEA is required

to reduce the difference between numerical and experimental values.

To increase the prediction accuracy and overcome on the

limitations of CEA, Chen (2009) in his recent review stated that considering

negative damping tends to minimize under prediction while introducing

positive damping leads to avoid the probability of over prediction. Abu Bakar

and Ouyang (2008) reported that good agreement between the experimental

tests and numerical results using CEA will be achieved by considering

negative and positive damping. Also (Kung et al 2003, Bajer et al 2003)

suggested that combination of realistic surface of brake pads with negative

damping could improve the prediction accuracy in the complex eigenvalue

analysis. The following section describes the methods of prediction the squeal

frequencies by including the above parameters.

4.3.5.1 Influence of positive damping

The first improvement is carried out by considering the influence of

positive friction damping along with a constant friction coefficient to reduce

over predictions. This damping is caused by the friction forces that stabilise

vibration along the contact surface in direction perpendicular to the slip

direction (Abu Bakar 2005).

It is observed in Figure 4.8, that the complex results are not

symmetrical due to the inclusion of friction damping. Therefore, complex

conjugate pairs are not easily identified. This result is similar with (Kung et al

2004). It is also observed that the values of real parts are reduced and the

numbers of unstable frequencies are also reduced. This is because considering

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positive damping lead to dissipate energy from the system and reduce the

probability of over prediction.

0

2000

4000

6000

8000

10000

-150 -100 -50 0 50 100 150

Real Part

Fre

qu

en

cy

(H

z)

Figure 4.8 Effect of positive damping on predicted results

4.3.5.2 Influence of negative damping

The second analysis is performed by considering the effect of

negative friction-velocity slope. In order to activate this effect, two values of

friction coefficient are considered. The static friction coefficient s=0.65, the

dynamic friction coefficient d =0.5 measured at speed 5 rad/s.

From Figure 4.9, it can be seen that the positive part values of

eigenvalues with a negative friction–velocity slope are generally larger than

those with a constant friction. In addition new unstable frequencies are

generated. The result suggests that the propensity of squeal occurrence

increases in the presence of a negative friction velocity slope. This lead to

minimize under prediction, because of negative friction-velocity slope tends

to add energy to the system.

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It also is observed that the prediction of squeal using CEA, which

incorporate negative and positive damping, shows good correlation with

experimental results. However, the numbers of predicted frequencies are

higher than experimental squeal frequencies, which indicate that still some

over prediction of frequencies occurs.

0

2000

4000

6000

8000

10000

-150 -100 -50 0 50 100 150

Real Part

Fre

qu

en

cy

(H

z)

Figure 4.9 Effect of negative and positive damping on predicted results

4.3.5.3 Influence of real pad surface

The third improvement is carried out with CEA by replacing

perfect pad surface with actual pad surface. The real pad surface roughness is

measured using a portable stylus-type profilometer (Taylor Hobson Surtronic

3+). The profilometer has a microprocessor and a digital scale indicator that is

used to measure surface features. In this study, roughness parameter

considered is surface average height (Ra), which can be measured directly at

any point on the surface. The brake pad is fixed on the support and the

guiding element is translated along its own axis as shown in Figure 4.10. The

stylus is made to move in a direction perpendicular to the direction of the

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brake pad. The surface height of the brake pad is measured by considering the

same node mapping obtained from FE model of the brake pad, as shown in

Figure 4.11 (a). The measured values are used in FE model to update the pad

surface by repositioning the surface height at each node, as shown in Figure

4.11 (b).

Complex eigenvalue analysis is conducted after replacing perfect

pad surface with actual pad surface. From simulation results, it is observed

that static contact pressure distribution between the brake pad and the rotor is

changed due to consider surface roughness, as shown in Figure 4.12. It is also

found that contact pressures are distributed symmetrically and the red colour

indicates the highest contact pressure. When the disc starts to rotate the

contact pressure distributions are no longer symmetric and are now shifted

towards the leading edge, as shown in Figure 4.13.

Figure 4.10 Measurement of pad surface through profilometer

97

(a) Perfect pad surface (b) Real pad surface

Figure 4.11 FE model of the brake pad


Figure 4.12 Static contact pressure distribution (v=0 rad/s)

98


Figure 4.13 Dynamic contact pressure distribution (v=5rad/s)

(Top of each diagram is the leading edge)

Thus, the CEA is performed by considering the friction positive

damping, negative damping and real pad surface. From the CEA results, it is

found that there are seven unstable (squeal) frequencies at 1472, 2339, 2773,

5816, 7383, 8706 and 9471 Hz, as shown in Figure 4.14, and their mode

shapes are shown in Figure 4.15. It is also found that the predicted unstable

frequencies are in a good agreement with measured experimental squeal

frequencies for all out-of-plane modes at 1472, 2339, 7383 and 8706 Hz,

while experimental measurement could not record squeal due to in-plane

modes at 2773, 5816 and 9471 Hz. This may be due to the incapability of the

instrument and equipment that are used in the experiments to detect any in-

plane vibrations. Similarly, in the squeal tests which were conducted in

University of Liverpool under a series of operating conditions by James

(2003), squeal frequencies were dominated by out-of plane modes and there

was no in-plane modes captured during squeal events even if they exist,

which was later confirmed numerically by Abu Baker (2005). Therefore, it is

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desirable to use another approach to compare with the CEA results in order to

check prediction of squeal at in-plane vibration that could not be measured at

experimental squeal test. A non-linear transient analysis based on FEA will be

performed in the following section to predict squeal and compare its results

with CEA results.

0

2000

4000

6000

8000

10000

-150 -100 -50 0 50 100 150

Real Part

Fre

qu

en

cy

(H

z)

Figure 4.14 Effect of real pad surface including negative and positive

damping on predicted results

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1 - Unstable frequency at 1472 Hz

(out-of-plane)


(out-of-plane)


(in-plane)


(in-plane)

5- Unstable frequency at 7383 Hz

(out-of-plane)

6- Unstable frequency at 8706 Hz

(out-of-plane)

7- Unstable frequency at 9471 Hz (in-plane)

Figure 4.15 Mode shapes for the baseline model at unstable frequencies

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4.4 DYNAMIC TRANSIENT ANALYSIS

Dynamic transient analysis (DTA) is a non-linear finite element

method used to determine the dynamic response of a structure under the

action of any general time-dependent loads. This method includes real life

operational conditions and parameters. It is usually carried out to provide the

solution to nonlinear dynamics problems when material nonlinearity,

geometric nonlinear effects or changes in boundary conditions occur due to

dynamic events. There are a few published papers based on FE models only

consider the transient behaviors of brake systems because of highly

computationally intensive and require a lot of disc storage.

DTA is performed using the commercial nonlinear finite element

code ABAQUS /Explicit v6-8. The explicit dynamics procedure performs a

large number of small time increments efficiently. An explicit central-

difference time integration rule together with diagonal or “lumped” element

mass matrices is used; each increment is relatively inexpensive (compared to

the direct-integration dynamic analysis procedure available in ABAQUS

/Standard) because there is no solution for a set of simultaneous equations.

The following finite element equation of motion is solved:

(4.16)

(4.17)

102

where is acceleration, is velocity and subscript (i) refers to the increment

number in an explicit dynamics step and refer to mid-

increment values. The central-di erence integration operator is explicit in the

sense that the kinematic state is advanced using known values of and

from the previous increment. The explicit integration rule is quite simple

but by itself does not provide the computational e ciency associated with the

explicit dynamics procedure. The key to the computational e ciency of the

explicit procedure is the use of diagonal element mass matrices because the

accelerations at the beginning of the increment are computed by:

(4.18)

where is the diagonal lumped mass matrix, is the applied load vector,

and is the internal force vector. A lumped mass matrix is used because its

inverse is simple to compute. The explicit procedure requires no iterations and

no tangent stiffness matrix. Since the central difference operator is not self-

starting because of the mid-increment of velocity, the initial values at time t =

0 for velocity and acceleration need to be defined. In this case, both

parameters are set to zero as the disc remains stationary at time t = 0. Explicit

dynamic integration does not need a convergent solution before attempting

the next time step. Each time step is so small that its stability limits is

bounded in terms of the highest eigenvalue ( ) in the system:

(4.19)

In this study, dynamic transient analysis of a full brake model is

carried out in order to make comparison with the complex eigenvalue

analysis. The main aim is to see correlation between the two analysis methods

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in terms of predicted unstable frequencies. By using dynamic transient

analysis, instability in the disc brake system can be found with an initially

divergent vibration-time response. This time domain information can then be

converted to frequency domain information by using FFT technique.

For the transient analysis, the time history of the brake-line pressure

and rotational speed are used for describing operating conditions of the disc

brake model, as shown in Figure 4.16. At first, a brake pressure is applied

gradually until time t1, after which the brake pressure is kept uniform. The

disc starts to rotate at t1 and the speed gradually increases up to time t2, after

which the rotational speed becomes constant.

Figure 4.16 Schematic diagram of transient analysis simulation

procedure (Abu Bakar and Ouyang 2006)

The DTA with the same operational conditions as in the CEA

section is used to predict the squeal occurrence of the disc brake. Both time-

domain results and frequency-domain results are presented. The brake

application time of 0.2 s is used until the initial limit cycle oscillations is

obtained. The period of simulated brake application requires a long

computing time; it takes over four weeks to complete one simulation. The

analysis was accomplished using Pentium Core 2 Duo processor and 4 GB

104

RAM under Windows platform. Figure 4.17 shows the time history of the

vibration of a node at the outer radius on the disc surface in contact with the

finger pad.

From the simulation results, it is observed that the vibration seems

to grow into a state of limit cycles; this agrees with the squeal behaviour

observed during the experimental squeal test. The result is converted from

time domain to frequency domain using the Fast Fourier Transform

technique, as shown in Figure 4.18. It is found that the frequency domain

results show various unstable frequencies (peak values) and seven of them are

close to those predicted in the CEA with peaks values from 1 to 7 represent

frequencies at 1472, 2339, 2773, 5816, 7383, 8706 and 9471 Hz respectively.

It is also observed that DTA indicates more unstable frequencies

than those predicted in the CEA. This finding agrees with previous

observations by Bengisu and Akay (1994) who examined the correlation

between the CEA and time domain using a 3-dof analytical model. They

found that if one or two modes are predicted by the CEA to be unstable, the

same system analysed in the time domain might already show three or more

instabilities which are due to either quasi-periodic or chaotic behaviour.

Similarly, Abu Bakar et al (2007) developed a FE model of drum brake and

found that DTA generates more unstable frequencies than those predicted in

the CEA.

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-3

-2

-1

0

1

2

3

0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2

Time (s)

Ac

ce

lera

tio

n (

g)

Figure 4.17 Vibration response predicted by transient analysis

Figure 4.18 Predicted unstable frequencies by transient analysis

4.5 CONCLUDING REMARKS

In this chapter, the FE model is used to predict brake squeal

through CEA at constant friction coefficient. The predicted results are

compared with the squeal events observed in the experiments. It is observed

106

that the results of the CEA at constant friction coefficient are not in good

agreement with experimental test. Hence, some improvements techniques are

considered in order to increase the prediction accuracy of CEA results. It is

also found that the predicted unstable frequencies are in a good agreement

with measured squeal frequencies for all out-of-plane modes, while squeal

due to in-plane modes could not be obtained by experimental measurement.

Thus, DTA is performed although it is a very time-intensive to

validate unstable (squeal) frequencies specially in-plane modes. It is found

that predicted DTA results are close well to those predicted in the complex

eigenvalue analysis. Hence the CEA results could be used with higher

confidence level to reduce squeal occurrence, as discussed in the following

chapters.

chapter 4 evaluation of disc brake...

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