Vibro-acoustic Studies of Brake Squeal Noise - UNSWorks

School of Aerospace, Civil and Mechanical Engineering

The University of New South Wales

Australian Defence Force Academy

Vibro-acoustic Studies of

Brake Squeal Noise

Antti Papinniemi

A thesis submitted for the Degree of Doctor of Philosophy

August 2007

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Statement of Originality

I hereby declare that this submission is my own work and to the best of my knowledge

it contains no materials previously published or written by another person, or substantial

proportions of material which have been accepted for the award of any other degree or

diploma at UNSW or any other educational institution, except where due

acknowledgement is made in the thesis. Any contribution made to the research by

others, with whom I have worked at UNSW or elsewhere, is explicitly acknowledged in

the thesis. I also declare that the intellectual content of this thesis is the product of my

own work, except to the extent that assistance from others in the project’s design and

conception or in style, presentation and linguistic expression is acknowledged.

Antti Papinniemi

August 2007

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Copyright Statement

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use in future works (such as articles or books) all or part of this thesis or dissertation. I

also authorise University Microfilms to use the 350 word abstract of my thesis in

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Abstract

Squeal noise has been an on-going concern with automotive brake systems since their

inception. Even after many decades of research no single theory exists that adequately

describes the phenomenon, and no general methods for eliminating squeal noise exist.

Broadly speaking, three primary methods of analysis have been applied to

understanding and eliminating brake squeal: analytical, experimental and numerical.

Analytical models provide some insight into the mechanisms involved when a brake

squeals, but have limitations in applicability to specific brake systems. Experimental

methods provide the backbone of brake squeal investigations, especially in an industrial

environment. However, the core focus of this thesis is to use a large scale finite element

analysis (FEA) model to investigate brake squeal.

Initially the FEA model was developed and the dynamic characteristics were validated

against experimental modal analysis results. A complex eigenvalue analysis was

performed to identify potential squeal modes which appear as unstable system vibration

modes.

Further techniques are described that allow the deeper probing of unstable brake system

modes. Feed-in energy, which is the conversion of friction work into vibrational energy

during the onset of squeal, is used to determine the relative contribution of each brake

pad to the overall system vibration. The distribution of the feed-in energy across the

face of a brake pad is also calculated. Component strain energy distributions are

determined for a brake system as a guide to identifying which components might best be

modified in addressing an unstable system mode. Finally modal participation is

assessed by calculating the Modal Assurance Criterion (MAC) between component free

modes and the component in the assembly during squeal. This allows participating

modes to be visualised and aids in the development of countermeasures.

The majority of the work in this thesis was performed using the commercial FEA code

MSC.Nastran with user defined friction interfaces. An alternative approach using a

contact element formulation available in Abaqus was also implemented and compared

to the MSC.Nastran results. This analysis showed that considerable differences were

noted in the results even though the overall predicted stability correlated relatively well

to observed squeal. Abaqus was also used in a case study into the design of a brake

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rotor in a noisy brake system. The results of this study provided good correlation to

observed squeal and facilitated effective rotor countermeasures to be developed.

Some success was achieved in the main aims of predicting brake squeal and developing

countermeasures. However, while the tools presented do allow a deeper probing of

system behaviour during squeal, their use requires good correlation to observed squeal

on brake system to be established. As such, their use as up-front design tools is still

limited. This shortcoming stems from the complexity of brake squeal itself and the

limitations in modelling the true nature of the non-linearities within a brake system.

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Acknowledgments

I would like to thank Professor Joseph Lai who supervised my studies, and I am grateful

for his direction, patience and encouragement over this long period of time. I would

also like to thank Dr Jiye Zhao for providing his support and advice during the project.

The project was funded jointly by the Australian Research Council and PBR

Automotive under the SPIRT scheme. I also thank the ARC for providing me with

APA(I) and UNSW@ADFA for a completion scholarship.

Many members of staff at the School of Aerospace, Civil and Mechanical Engineering,

UNSW@ADFA have contributed to this project, including Mr Robert Clark, Mrs

Marion Burgess, Mr John Waggener, Dr Andrew Dombek, Dr Alex Tarnopolsky, and

many members of the mechanical and electrical workshops.

I would also like to thank some of my fellow students who I shared many enjoyable

times with, both socially and on an intellectual level, including David Martinez-Munoz,

Jon Couldrick, Orio Kieboom, Jeff Mcguire, Stephen Moore, and many others who

came and went during my studies.

Finally I would like to thank Mum and Dad, and the rest of my family. Hopefully we

finally got there.

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Table of Contents

Statement of Originality iii

Copyright Statement iv

Authenticity Statement iv

Abstract v

Acknowledgments vii

Table of Contents ix

List of Figures xiii

List of Tables xix

Nomenclature xxi

Chapter 1 Introduction 1

1.1 Automotive Disc Brakes 1

1.2 Brake System Description 3

1.3 Characteristics of Brake Squeal 6

1.4 Thesis outline 11

Chapter 2 Literature Review 13

2.1 Introduction 13

2.2 Reviews of Brake Squeal 15

2.3 Analytical Approaches to Brake Squeal 16

2.3.1 Variable µ Analysis 16

2.3.2 Sprag-Slip Analysis 18

2.3.3 Mode Coupling Analysis 20

2.4 Experimental Approaches to Brake Squeal 23

2.5 Numerical Methods for Brake Squeal Analysis 27

2.6 Brake Squeal Noise in Practice 30

2.7 Summary 34

Chapter 3 Experimental Determination of Vibrational and Acoustical

Characteristics of a Brake System 35

3.1 Introduction 35

3.2 Frequency Response Function 36

3.3 Experimental Set-up 37

3.4 Component Testing 39

3.5 Rotor 39

3.5.1 Brake Rotor Mode Shape Descriptions 39

3.5.2 Rotor Test Grid 42

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3.5.3 Rotor Test Results 44

3.5.4 In-Plane Mode Detection 46

3.5.5 In-Plane Measurement 48

3.6 Pad 51

3.7 Caliper Housing 53

3.8 Anchor Bracket 54

3.9 Assembled Brake System 56

3.10 Modal Analysis Summary 62

3.11 Brake System Noise Evaluation 63

Chapter 4 Finite Element Modal Analysis 67

4.1 Introduction 67

4.2 Application of FEA to Dynamical Problems 67

4.3 Modelling Approach 69

4.4 Individual Component models 70

4.4.1 Brake rotor 72

4.4.2 Anchor Bracket 75

4.4.3 Caliper Housing 78

4.4.4 Brake Pad 79

4.5 Mounted Rotor 82

4.6 Assembled Models 84

4.6.1 Component Interfaces 84

4.6.1.1 Multi-point constraints 84

4.6.1.2 Linear springs 86

4.6.1.3 Linear vs. non-linear static analysis 88

4.6.1.4 Spring interface tuning 89

4.6.1.5 Friction interface 91

4.7 Summary 93

Chapter 5 Prediction of Unstable Modes 95

5.1 Introduction 95

5.2 Complex Eigenvalue Analysis 99

5.3 Implementation for a Brake System 100

5.3.1 Generation of a validated FEA model 101

5.3.2 Static analysis 101

5.3.3 Friction Model 101

5.3.4 Implementation with MSC.Nastran 103

5.4 Brake System Analysis 104

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5.5 Summary 112

Chapter 6 Numerical Methods for Assessing Brake Squeal Propensity 113

6.1 Introduction 113

6.2 Strain energy 114

6.2.1 Viscous Work 116

6.3 Feed-in Energy 118

6.3.1 Feed-in Energy vs. Viscous Work 122

6.4 Modal Participation with Modal Assurance Criterion 123

6.5 Example 4DOF System 124

6.6 Analysis of Numerical Model 132

6.4.1 Model Description and Unstable Modes 132

6.4.2 Feed-in Energy for a Numerical Model 133

6.4.3 Strain Energy for a Numerical Model 137

6.6.4 Modal Participation for a Numerical Model 139

6.6.5 Example Unstable Mode Investigation 142

6.7 Summary 147

Chapter 7 Parametric Study 149


7.2 Parameters Under Investigation 151

7.3 Baseline System 151

7.3.1 Complex Eigenvalues 151

7.3.2 Baseline Strain Energy Distributions 152

7.3.3 Baseline Feed-in Energy 154

7.3.4 Baseline Component MAC Modal Participation 154

7.4 Material Properties Sensitivities 161

7.5 Contact Distribution Sensitivities 168

7.6 Damping Shims 172

7.7 Summary 174

Chapter 8 Comparison of Contact Modelling Methods 177


8.2 Contact Elements 177

8.2.1 Tied Contact 178

8.2.2 Deformable-Deformable Contact 180

8.2.3 Non-linear Static Analysis 182

8.2.4 Contact Set-up and Solution Steps 183

8.3 Material Properties and Load Cases 185

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8.4 Analysis Results 187

8.4.1 Abaqus Results for varying Pressure 187

8.4.2 Abaqus vs Nastran Stability 188

8.4.3 Abaqus vs Nastran MAC 189

8.5 Summary 200

Chapter 9 Applications to Rotor Design in an Industrial Environment 203


9.2 Brake system Under Investigation 203

9.3 Noise Evaluation 205

9.4 Mode Description 208

9.5 Stability Prediction 210

9.6 Rotor Modification 212

9.7 Summary 218

Chapter 10 Conclusions 219

10.1 Conclusions 219

10.2 Recommendations for Future Work 222

References 225

Publications Arising From This Thesis 233

Journal Papers 233

Conference Papers 233

Reports 234

Appendix A: Measurement Grids 235

Appendix B: Free Rotor Mode Shapes 239

Appendix C: Example Nastran Input Deck 241

Appendix D: bdfread Source Code 247

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List of Figures

Figure 1.1 Schematic of a siding disc brake caliper. Note that the anchor bracket is

not included for clarity 2

Figure 1.2 Ford AU series II rear brake assembly CAD model 4

Figure 1.3 Ford AU II rear brake assembly components. Clockwise for the top left;

pads, piston, guide pins, caliper housing, and bracket and disc rotor 4

Figure 1.4 Ford AU II rear brake assembly as installed on the vehicle 5

Figure 1.5 Spectrum of a brake squeal event 7

Figure 1.6 Comparison of the same section of two nominally identical tests on the

noise dynamometer. The green dots represent noise occurrences during a single stop 10

Figure 1.7 Example noise sensitivities to brake pressure and IBT 11

Figure 2.1 (a) Number of brake squeal papers according to Thomson ISI Web of

Science on 4 August 2006, (b) Number of brake squeal papers as a percentage of the

papers presented at the SAE Annual Brake Colloquium 14

Figure 2.2 A single degree-of-freedom featuring a block rubbing on a moving

conveyor 17

Figure 2.3 Negative slope µ-v characteristic 17

Figure 2.4 (a) Single strut rubbing against surface, (b) sprag-slip system 19

Figure 2.5 Two degree-of-freedom system analysed by Hoffman, et al 21

Figure 2.6 Example shim material 31

Figure 2.7 Example slotted backplate shim designed to shift the centre of contact

pressure on the piston / pad interface 32

Figure 2.8 Example of pad chamfers 32

Figure 3.1: Schematic of the experimental set-up 38

Figure 3.2 Cross section of a solid drum-in-hat (DIH) rear rotor 40

Figure 3.3 Bending mode descriptions 41

Figure 3.4 In-plane mode descriptions 42

Figure 3.5: Brake rotor experimental grid (384 points). Point numbering is omitted

for clarity 43

Figure 3.6: Cross sectional view of brake rotor showing location of grid points. Each

line radiating from the centre of rotor contains 8 points. Excitation was applied at

point 6 in the z direction 44

Figure 3.7: Spatially averaged rotor FRF traces 45

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Figure 3.8 (a) Out-of-plane vibration, (b)In-plane vibration. Note that the

longitudinal compression is accompanied by expansion in the lateral direction and

longitudinal expansion is accompanied by contraction 47

Figure 3.9 Rotor in an in-plane vibration mode. The top hat provides unsymmetrical

support to the disc itself and may cause out of plane deformation 48

Figure 3.10 In-plane test configuration 49

Figure 3.11 Mode shape for the first circumferential in-plane mode (7840 Hz). The

y-axis represents the normalised displacement from the mean position around the

circumference. The value in this case was actually accelerance, but, once normalised,

it is equivalent to displacement 51

Figure 3.12 (a) Brake pad, (b) Brake pad experimental grid geometry (25 points).

The direction of excitation and measurement in the out-of-plane direction 52

Figure 3.13 (a) Caliper housing. Also seen are the slide pins which were not analysed

as part of these experiments. (b) Caliper housing experimental grid geometry (65

points) 53

Figure 3.14 (a) Anchor bracket, (b) Anchor Bracket experimental grid geometry (36

points) 54

Figure 3.15 Spatially averaged FRF data. (a) Pad, (b) caliper housing, and (c) anchor

bracket 56

Figure 3.16 Assembled brake system test configuration 57

Figure 3.17 Comparison of spatially averaged FRF plots. (a) free rotor, (b) mounted

rotor 59

Figure 3.17 Comparison of spatially averaged FRF plots. (c) assembled no pressure,

(d) assembled 20 bar 60

Figure 3.18 Comparison of the damping factors for the free rotor and 3 assembled

conditions 62

Figure 3.19 Baseline noise performance of the Ford Falcon AUII rear brake system 66

Figure 4.1 Flow chart for generating a validated FEA assembly 70

Figure 4.2 8-node CHEXA element showing grid points 1 through 8 71

Figure 4.3 Rotor mesh generation by revolving a cross sectional plane of shell

elements about the rotor rotation axis 72

Figure 4.4 (a) Rotor mesh of 4916 8-node brick elements. (b) Zoomed in detail of the

pad interface region of the rotor 72

Figure 4.5 Comparison between experimental and FEA predicted driving point FRF

for the free rotor with 0.2% structural damping applied to the FEA model 75

Figure 4.6 Final mesh for the anchor bracket featuring 1434 8-node brick elements 76

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Figure 4.7 Comparison of experimental and FEA predicted driving point FRF for the

free anchor bracket 77

Figure 4.8 Caliper mesh consisting of 1648 8-node brick elements 78

Figure 4.9 Driving point FRF comparison between experimental and FEA results for

the free caliper 79

Figure 4.10 (a) Outer pad, (b) inner pad. Both pads feature 430 8-node brick elements 80

Figure 4.11 Driving point FRF comparison between the brake pad experimental and

FEA results. The FEA model had isotropic lining material properties and 1%

structural damping added 82

Figure 4.12 (a) The bolt holes on the rotor were fully constrained in all DOFs. (b)

Close up detail of the constraints shown as a wireframe 83

Figure 4.13 Comparison of driving point FRF for the mounted rotor below 10 kHz 83

Figure 4.14: Implementation of an MPC for creating a sliding connection. The

annular ring on the piston is connected to a central node, as is an annular ring from the

caliper. The central nodes are directly coupled in all DOFs 85

Figure 4.15: Schematic diagram of nodes on adjacent components connected with

linear springs. Note: the gap between the components is illustrative only, and the

nodes are coincident within the FEA model 86

Figure 4.16 Component interface connection schematic 88

Figure 4.17 Overlays of the Nastran static solution and Abaqus non-linear solution

contact areas. The areas in red represent the footprint of the pad from the Abaqus

solution. The blue dots represent active nodes from the Nastran solution 91

Figure 4.18: Basic friction force diagram 92

Figure 5.1 Single degree-of-freedom system with viscous damping 96

Figure 5.2: Response of an unstable SDOF system for various levels of damping 98

Figure 5.3 Location of a an eigenvalue on the complex plane. Its position provides

the level of damping as well as the frequency 100

Figure 5.4 Coincident node mesh at the pad / rotor interface. Note that it is shown

with a gap for clarity 102

Figure 5.5 The assembled FEA model 105

Figure 5.6 108 eigenvalues extracted from the baseline brake system plotted on the

complex plane. The 7 unstable mode pairs appear as symmetric pair about the

imaginary axis 106

Figure 5.7 Damping vs. frequency for the base brake system analysis 106

Figure 5.8 Mode 27 at 3322 Hz. Guide pins are not displayed 109


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Figure 5.14 Mode 105 at 11860Hz. Guide pins are not displayed 112

Figure 6.1 Schematic of approach to reduce brake squeal propensity 114

Figure 6.2 Undamped single degree-of-freedom system of mass m and spring

stiffness k 114

Figure 6.3. Two degree-of-freedom system 115

Figure 6.4 Viscous damped SDOF system 117

Figure 6.5 A simple 2DOF system with sliding friction 119

Figure 6.6 Phase plot of y vs. x displacement for the 2DOF system in Figure 6.4 with

0 < (θy - θx) < 90° 120

Figure 6.7 Phase plots for the system from Figure 6.4. (a) 0 < (θy - θx) < 90°, (b) 90°

< (θy - θx) < 180°, (c) (θy - θx) = 90°, (d) -90° < (θy - θx) < 0, (e) (θy - θx) = 0, (f) (θy -

θx) = 180° 122

Figure 6.8 4DOF system with sliding friction 124

Figure 6.9 (a) A spring element undergoing a displacement of ∆y at one end, and (b)

displacement decomposed into the x’-y’ coordinate frame. Note that the infinitesimal

displacement does not change the angle θ. 125

Figure 6.10 Contact stiffness and forces at the friction interface 128

Figure 6.11 Change in eigenvalues of modes 104 and 105 due to an increase in the

coefficient of friction 133

Figure 6.12 Feed-in energy across the pad surfaces for mode 27. (a) inner pad, (b)

outer pad 135

Figure 6.13 Feed-in energy across the pad surfaces for mode 105. (a) inner pad, (b)

outer pad 136

Figure 6.14 Strain energy distribution for unstable modes of the baseline system with

µ = 0.5 138

Figure 6.15 Strain energy distribution for (a) mode 104, (b) mode 105. The average

strain energy distribution for 108 modes the base system (µ = 0) is also shown in each

chart 139

Figure 6.16 Modal assurance criterion for the unstable mode 27 at 3322 Hz. (a)

Rotor, (b) anchor, (c) caliper, (d) inner pad and (e) outer pad. In each case, only more

significant modes are shown 141

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Figure 6.17 Modal assurance criterion for the unstable mode 105 at 11860 Hz. (a)

Rotor, (b) anchor, (c) caliper, (d) inner pad and (e) outer pad. Only more significant

modes are shown 142

Figure 6.18 Mode shapes of free pad for 6383 Hz, 7536 Hz and 12533 Hz 144

Figure 6.19 Deformed mode shape of free caliper at 11063 Hz 145

Figure 6.20 2nd

Tangential in-plane rotor mode shape of free rotor at 11838 Hz 145

Figure 6.21 2nd

order radial in-plane rotor mode at 2944 Hz 147

Figure 6.22 Caliper housing mode shape at 2763 Hz 147

Figure 7.1 Baseline system, unstable modes with µ varied from 0.3 to 0.6 152

Figure 7.2 Strain energy distribution for the 7 unstable modes of the baseline system

with µ = 0.5 153

Figure 7.3 Mode 27 3322 Hz MAC values 155







Figure 7.10 Negative damping levels of system modes for different rotor modulus

levels, µ = 0.5 163

Figure 7.11 Negative damping levels of system modes for different anchor modulus

levels, µ = 0.5 165

Figure 7.12: Negative damping levels of system modes for reduced caliper modulus, µ

= 0.5 166

Figure 7.13: Negative damping levels of system modes for changes in friction material

modulus, µ = 0. 166

Figure 7.14: Structural damping 167

Figure 7.15: Backplate damping 168

Figure 7.16 Cross section of a modified puck with landing and trailing chamfers 169

Figure 7.17 Slotted shim which removes pressure from one end of the piston, helping

to alter the contact pressure at the friction interface 169

Figure 7.18: Shims applied to the assembly 170

Figure 7.19: Contact distribution 171

Figure 7:20 Negative damping levels of system modes for changes damping shim µ =

0.5 174

Figure 8.1 Master/slave contact in 2 dimensions 179

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Figure 8.2 Abaqus complex eigenvalue results at 4 different pressure levels 187

Figure 8.3 Comparison of complex eigenvalue results of Abaqus and Nastran 188










Figure 9.1 Brake system under investigation in this chapter 204

Figure 9.2 Layout of the brake rotor to identify main features 204

Figure 9.3 Noise dynamometer results for problem brake system with the baseline

rotor design 207

Figure 9.4 FRF in the out-of-plane direction 209

Figure 9.5 FRF in the tangential in-plane direction 209

Figure 9.6 Simplified FE model used to perform the analysis 210

Figure 9.7 Unstable modes in the region of the 2nd tangential in-plane mode. (a)

modal frequencies, (b) negative damping level 211

Figure 9.8 (a) Mode shape of unstable 9458Hz mode, (b) Rotor mode shape for 9473

Hz 213

Figure 9.9 Proposed rotor modifications. (a) sides of hat replaced with conical

section, (b) three stiffeners added to the swan neck 214

Figure 9.10 2nd in-plane modes of the modified rotors as a function of µ. (a) new hat,

(b) stiffened rotor 215

Figure 9.11 New hat design rotor noise dynamometer results (a) SPL vs Frequency,

(b) cumulative occurrence 216

Figure 9.12 3 Stiffeners design rotor noise dynamometer results (a) SPL vs

Frequency, (b) cumulative occurrence 217

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List of Tables

Table 1.1 Specification of the Ford Falcon AU II brake system 5

Table 1.2 Main categories of brake noise 6

Table 2.1 Layer specification of typical damping shim material 31

Table 3.1 Experimental equipment for modal measurements 37

Table 3.2 Coordinates for the rotor grid 43

Table 3.3 Rotor modal parameters. Log magnitude is with a reference value of 1

m/Ns2. Mode shapes refer to nodal diameters and circumferences of bending modes

except where prefix is added. TH indicates deformation in predominantly in top hat

region, RI is radial in-plane and CI is circumferential in-plane 46

Table 3.4 Modal parameters of the brake pad 53

Table 3.5: Caliper housing modal parameters 54

Table 3.6 Anchor bracket modal Parameters 55

Table 3.7 Modal parameters for the free rotor and assembled conditions 61

Table 3.8 Parameters for determining the SPL from the test chamber during a noise

test 64

Table 4.1 Final material properties for the brake rotor FEA model 73

Table 4.2 Comparison of the experimental and FEA modal analysis of the free brake

rotor 74

Table 4.3 Final material properties for the anchor bracket FEA model 76

Table 4.4 Comparison of the experimental and FEA modal frequencies of the free

anchor bracket below 10 kHz 77

Table 4.5 Final material properties for the caliper FEA model 78


caliper housing 79

Table 4.7 Example composition of a friction material 80

Table 4.8 Summary of material properties for the brake pad 81

Table 4.9 Comparison of the experimental and FEA modal frequencies of the brake

pad 81

Table 4.10: Interface tuning parameters for baseline model 90

Table 5.1 Summary of unstable modes for the baseline brake system 107

Table 6.1 Parameters for example 4DOF system 129

Table 6.2 Complex eigenvalues from example analysis 130

Table 6.3 Mode shapes from example analysis 130

Table 6.4 Mode three data from example 4DOF analysis 131

Table 6.5 Mode three data from example analysis 131

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Table 6.6 Summary of unstable modes up to 12 kHz for the assembled brake system,

µ = 0.5 132

Table 6.7 Summary of feed-in energy for the unstable modes of the assembled brake

system, µ = 0.5 134

Table 6.8 Distribution of strain energy for the unstable modes of the assembled brake

system, µ = 0.5 . The average strain energy distribution for all modes 1st 108 modes

from the base (µ = 0) system is included 138

Table 7.1 Summary of parameters under investigation 150

Table 7.2 Summary of unstable modes for baseline system with µ = 0.5 152

Table 7.3 Summary of most active components for each unstable mode 153

Table 7.4 Baseline feed-in energy values for the 7 seven unstable modes 154

Table 7.5 Simplified shim structure used in the FEA study 172

Table 8.1 Contact interfaces in the Abaqus brake assembly model 184

Table 8.2 Material properties for the assembled FEA model 186

Table 8.3 Summary of test, Nastran and Abaqus results 189

Table 9.1 Brake system specification 205

Table 9.2 PBR TS640 test procedure summary 206

Table 9.3 Cumulative Noise occurrence acceptability for TS640. Each of the warm

and cold sections are calculated and assessed separately 206

Table A.1: Coordinates for the rotor grid 235

Table A.2: Pad grid coordinates. Measurement direction is in the local z direction for

all points 235

Table A.3: Caliper housing grid coordinates and measurement direction 236

Table A.4: Anchor bracket grid coordinates and measurement direction 237

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Nomenclature

C Damping matrix

c Damping

CI Circumferential in-plane

DIH Drum-in-hat (type of rotor)

DOF/DOFs Degree-of-freedom / Degrees-of-freedom

E Elastic modulus

FEA Finite element analysis

FFT Fast Fourier Transform

FRF Frequency response function

IBT Initial brake temperature

K Stiffness matrix

Kf Friction stiffness matrix

k Spring stiffness

M Mass matrix

m Mass

MAC Modal assurance criterion

MPC Multi-point constraint

ND Nodal diameter

NVH Noise vibration and harshness

ODS Operating deflection shape

Ui Strain energy of ith

element

u Displacement vector

RI Radial in-plane

SPC Single-point constraint

λ Eigenvalue

µ Coefficient of friction

ν Poisson’s ratio

ρ Mass density

ω Circular frequency, undamped natural frequency

ωd Damped natural frequency

ζ Damping ratio

1

Chapter 1

Introduction

Noise is an unwanted by-product of many mechanical processes including frictional

processes. Automotive brake systems, which utilise a friction interface to convert

kinetic energy into heat, are also prone to producing noise.

Much work has been done since early last century to address brake noise issues, yet the

problems still persist today. Indeed, with the continued improvement in vehicle NVH

(Noise, Vibration and Harshness) and subsequent reduction in interior noise levels,

brake noise now attracts more effort from automotive manufacturers than at any time in

the past. Of the types of NVH issues that can be present during braking, brake squeal

has received the most attention in both academic and industrial research and

development.

This thesis is focused on the development of numerical methods for investigating disc

brake squeal. The aim of these methods is to provide a deeper understanding of the

mechanisms involved in disc brake squeal and to provide practical tools for addressing

brake noise and countermeasure development.

1.1 Automotive Disc Brakes

A variety of brake systems have been used since the inception of the motor car, but in

principle they are all similar. Work is done by a friction contact interface which

converts the vehicle’s kinetic energy into heat. This facilitates control of the vehicle

speed and is fundamental for safe motor vehicle operation.

As the demands for braking performance have continued to become greater, the modern

disc brake has grown in popularity. Disc brakes started to become widely used for

passenger car front brake systems in the 1970s, and now almost all cars feature front

and rear disc brakes.

A schematic of a disc brake assembly is shown in Figure 1.1. A large flat circular plate

disc, referred to as the brake rotor, is mounted to the wheel axle and rotates with the

wheel around its axis. The brake operates as follows. When hydraulic pressure is

2

applied to the piston, the inner pad is forced against the brake rotor. The caliper

housing itself “floats”, i.e. is free to slide back and forth in the direction of the wheel

axis, and moves in the opposite direction to the piston. Fingers on the caliper force the

outer pad into contact with the other side of the rotor clamping it between the pads. The

caliper assembly is constrained by the anchor bracket from moving about the wheel

axis; hence a braking torque is generated. Note that the anchor bracket has been

removed from the schematic for clarity.

Figure 1.1 Schematic of a siding disc brake caliper. Note that the anchor

bracket is not included for clarity.

Variations to this arrangement exist, some including fixed calipers with pistons that act

on both the inner and outer pads directly, but the sliding caliper is used on the majority

of automotive brake systems.

Fundamental to the operation of the brake of course is that there is some friction force

that acts when the brake pads are forced into contact with the brake rotor. For the

purposes of brake system modelling, the classical Coulomb friction model is usually

used since it is mathematically convenient, yet is applicable in a wide variety of

practical situations. Mathematically the friction force can be expressed as

Hydraulic

fluid

Piston

Caliper

housing

Disc rotor

Rotation axis

Inner pad Outer pad

Seal

3

uNF µ−= (1.1)

where µ the coefficient of friction, N is the normal force at the friction interface and u is

the unit vector in slip direction. The friction force acts in a direction opposite to travel

as denoted by the negative sign, and independent of the slip velocity or the area of

contact.

For a brake system, the retarding torque for a sliding caliper is calculated by the

following equation

pistoneff pArT 2= (1.2)

where reff is the effective radius of pressure application about the wheel axis, taken

usually as the piston centreline, p is the hydraulic line pressure and Apiston is the piston

area.

The equivalent moment of inertia about its rotational axis that a brake is subjected to

during braking is given by

2

rollwheel rmI = (1.3)

where mwheel is the effective mass loading on the wheel during braking and rroll is the

rolling radius of the tyre. mwheel is governed by the weight distribution of the vehicle

and a specific rate of deceleration; thus the brake system balance is optimised for one

specific condition.

1.2 Brake System Description

This investigation is focused on the analysis of brake squeal. The test brake system is

the rear brake assembly fitted to a Ford Falcon AU series II. This system is typical of

many modern disc brake assemblies featuring a floating type caliper as shown in Figure

1.2.

Figure 1.3 shows the six components that comprise the brake assembly - pads, piston,

pins, caliper housing, anchor bracket and rotor. The rotor is a drum-in-hat (DIH) type

with a friction surface for the integral park brake incorporated on the inner section of

the disc. Figure 1.4 shows the system as installed on the vehicle.

4

Figure 1.2 Ford AU series II rear brake assembly CAD model.

Figure 1.3 Ford AU II rear brake assembly components. Clockwise for the

top left; pads, piston, guide pins, caliper housing, and bracket and disc rotor.

5

Figure 1.4 Ford AU II rear brake assembly as installed on the vehicle.

Table 1.1 Specification of the Ford Falcon AU II brake system.

Vehicle Type Large passenger sedan

Engine 6/V8 – 150/175 kW

Drive Wheels Rear

Rolling Radius 305 mm

Gross vehicle mass 2100 kg

Static Weight Distribution Front 45%, Rear 55 %

Brake Inertia (rear) 32 kgm2

Rotor Diameter (rear) 278 mm

Rotor Mass (rear) 6.4 kg

Effective Radius (rear) 120 mm

Piston Diameter (rear) 40 mm

6

Under braking, considerable weight transfer occurs to the front of a vehicle. In the

interests of vehicle stability and safety, it is common for the rear brake of a vehicle to be

considerably smaller than the front. The specification for this brake system is shown in

Table 1.1.

1.3 Characteristics of Brake Squeal

Many types of brake NVH concerns can be found described in the literature (Lang and

Smales, 1983). These include judder, shudder, graunch, groan, squeal, squeak and wire

brush. Table 1.2 notes the main categories and their distinguishing features.

Table 1.2 Main categories of brake noise

Type Frequency Features Prime

contributors

Low frequency brake

noise: Groan, moan,

shudder, graunch, clunk

<500 Hz Low frequency, broadband

structure-borne noise

Suspension

components,

brake assembly

Low Frequency Squeal 1 – 5 kHz Low frequency, tonal,

airborne noise

Brake

assembly

High Frequency Squeal > 5 kHz High frequency, tonal,

airborne noise

Brake rotor

and pads

Wire Brush > 5 kHz High frequency, multiple

frequencies

Brake rotor

and pads

Of the types of NVH that can be present during braking, brake squeal has received the

most attention in both academic and industrial research and development. Brake squeal

is defined as a tonal resonant vibration of the brakes systems at frequencies greater than

1 kHz. Figure 1.5 displays the sound pressure level (SPL) spectrum of a typical squeal

event as recorded on a noise dynamometer at a distance 0.5m from the brake. The tonal

nature of the sound can be seen clearly.

7

Figure 1.5 Spectrum of a brake squeal event.

Brake squeal is a tonal resonant vibration of the brake system. Critical to this is the pad

to rotor friction interface. During squeal the system enters into an unstable vibration

mode and exhibits self-excited vibration, where some friction work is converted into

vibrational energy, which in turn is radiated by the large flat surfaces of the brake rotor

and perceived as sound. The energy added to the system’s vibrations is called feed-in

energy. At the onset of squeal, the amount of feed-in energy is much greater than what

is being dissipated by sound radiation, damping, and other system non-linearities.

Analytically it can be shown that the system is unstable and exhibits a level of negative

damping, which is a measure of how quickly the vibration amplitude will initially grow.

However, as the system vibration level increases, it soon settles into limit cycle

behaviour where the added energy and dissipative effects are balanced.

One of the biggest contributors to brake squeal is the friction material itself. Obviously

a critical factor is the friction coefficient which can directly contribute to the energy

input to the system’s vibration. But many other aspects of the friction material are

important including its mechanical properties such as compressibility and damping, and

its friction versus slip velocity characteristics. Many friction materials exhibit a

negative slope friction versus speed characteristic given by

0

20

40

60

80

100

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

Frequency (Hz)

SP

L (

dB

[A]

re:

20

µP

a)

8

0<

dv

dµ

(1.4)

where v is the slip velocity at the friction interface.

A wide variety of factors need to be considered not only for NVH issues, but for other

aspects of brake performance and finalising the friction material selection for a

automotive brake system may take 12 months. This certainly makes it very difficult to

predict a priori the propensity of a brake system to squeal.

Often in the design of a brake system priority is given to requirements such as braking

performance, cost and ease of manufacture. The common practice for the different

components of a brake system to be manufactured by different suppliers further

complicates matters. The large number of vehicles produced means that even a low

squeal propensity found during initial testing of a brake system can become a major

warranty concern once a vehicle is in production due to a much larger population size.

Modifications towards the end of development phase will have two potential risks: (1)

leading to production delays and increased costs to both the brake and vehicle

manufacturers and (2) leading to products not fully validated with a potential field

warranty concern.

The most significant complication in applied brake research is the fugitive nature of

brake squeal; that is, brake squeal can sometimes be non-repeatable and transient in

nature. A brake system can have many potential squeal frequencies (unstable vibration

modes), some of which will rarely, if ever, observed as squeal. Each individual

component has its own natural modes under free-free boundary conditions. The number

of modes for a rotor within the human hearing range may be in excess of 50. The modal

frequencies and mode shapes of the rotor, caliper housing, anchor bracket and pad will

change once these parts are installed in-situ. During a brake application, these parts are

dynamically coupled resulting in a series of coupled vibration modes, which may be

different from the component free vibration modes.

A brake system may not always squeal given the “same” apparent conditions. Critical

changes between components, component interfaces and operational conditions may be

imperceptible on a macroscopic scale. The results of two noise dynamometer tests for

components selected to be matched in terms of component natural frequencies, pad

9

compressibility, environmental conditions and test procedure are compared in Figure

1.6. It is clear that the occurrence of noise on one test was far greater in (a) than (b).

Alternatively, small variations in operating temperature, brake pressure, rotor velocity

or coefficient of friction may result in differing squeal propensities or frequencies.

Figure 1.7 show examples of the percentage occurrence of brake squeal obtained with a

brake noise dynamometer testing a typical brake system. It can be seen from Figure 1.7

(a) that there is no simple relationship between the percentage occurrence and frequency

of the brake squeal and the brake pad pressure. Similarly, the influence of initial brake

temperature (IBT) on both the occurrence and frequency of the brake squeal is equally

complicated as shown in Figure 1.7 (b).

Due to the above mentioned difficulties in designing a noise free brake system, efforts

to eliminate brake squeal have largely been empirical, with problematic brake systems

treated in a case by case manner. The success of these empirical fixes depends on the

mechanism that is responsible for causing the squeal problem.

10

0

100

200

300

400

500

1800 1900 2000 2100 2200

Stop No.

Tem

per

atu

re (

oC

)

0

4000

8000

12000

16000

20000

Fre

quen

cy (

Hz)

Initial Temp

Frequency

(a)

0

100

200

300

400

500

1800 1900 2000 2100 2200

Stop No.

Tem

per

atu

re (

oC

)

0

4000

8000

12000

16000

20000

Fre

quen

cy (

Hz)

Initial Temp

Frequency

(b)

Figure 1.6 Comparison of the same section of two nominally identical tests

on the noise dynamometer. The dots represent noise occurrences during a

single stop.

11

(a) Noise occurrences and Frequency versus brake line pressure

(b) Noise occurrence and frequency versus IBT

Figure 1.7 Example noise sensitivities to brake pressure and IBT

1.4 Thesis outline

In this thesis, many of the difficulties in addressing brake squeal are investigated.

Chapter 2 provides a review of the literature on brake squeal research and outlines the

principal models used for describing the onset of squeal noise.

In Chapter 3, the experimental characterisation of the brake system dynamics is

presented. Firstly the modal properties for the individual components of the brake

system are determined using experimental modal analysis. The properties of the

12

complete brake assembly under various boundary conditions then follow. Finally

practical measurements of brake squeal noise on the brake noise dynamometer are

presented.

Chapters 4 through 6 outline the core numerical investigations using the commercial

finite element analysis (FEA) code MSC.Nastran. Chapter 4 outlines how FEA was

applied in normal modes analysis of both the components and the assembly and Chapter

5 describes the complex eigenvalue analysis procedure used to determine system

stability. Chapter 6 introduces three further analysis techniques to allow further

numerical assessment of the squeal propensity of the test brake system; strain energy,

feed-in energy and modal participation.

In Chapter 7, a parametric study is outlined to illustrate the influence of a variety of

system parameters on the squeal propensity of the brake system.

An alternative FEA code, Abaqus, is also applied to analysing the brake system.

Chapter 8 outlines the differences in approach using Nastran and Abaqus, and compares

their performance in predicting the brake squeal propensity of a brake system. An

example of a practical, real world application of Abaqus to improve the design of a

brake rotor is given in Chapter 9.

Chapter 10 provides a summary of the completed work and discusses areas for future

work.

13

Chapter 2

Literature Review

2.1 Introduction

Research into understanding brake squeal has been ongoing over the last 50 years or

more. Initially drum brakes were studied due to their extensive use in early automotive

brake systems. However, disc brake systems have been common place on passenger

vehicles since the 1960s and are used more extensively in modern vehicles. It follows

that research into brake squeal became focused more onto disc brake systems.

Brake noise, vibration and harshness (NVH) continues to be an extremely active area of

research. As shown in Figure 2.1(a), over the last 20 years, there is a significant

increase in the number of papers on brake squeal published in the period 2001-2005 in

journals included in Thomson ISI citation index. Similarly, Figure 2.1(b) shows that the

number of brake squeal papers presented at the SAE Annual Brake Colloquium has

been increasing steadily from 25% of the total number of papers in the period 1997-

1999 to 37% in the period 2003-2005.

Unfortunately, the large body of research into brake squeal has failed to provide a

complete understanding of, or the ability to predict its occurrence. Certainly the

understanding of certain aspects of brake squeal has improved, and specific modelling

techniques have been shown to offer good correlation to observed squeal in some cases,

but no reliable procedure for eliminating brake squeal altogether has been developed. It

appears likely that a practical, generally applicable cure for all brake squeal will never

come to fruition.

This inability to fully characterise and control squeal should not be viewed as a failure

on the part of researchers. It is partly due to the shear complexity of the mechanisms

that cause brake squeal at both the micro and macroscopic levels, and partly due to the

transient and elusive nature of brake squeal that often limit the direct probing of a

squealing brake.

14

0

10

20

30

40

50

1981-

1985

1986-

1990

1991-

1995

1996-

2000

2001-

2005

Year

(a)

0%

5%

10%

15%

20%

25%

30%

35%

40%

1997-1999 2000-2002 2003-2005

Year

(b)

Figure 2.1 (a) Number of brake squeal papers according to Thomson ISI

Web of Science on 4 August 2006, (b) Number of brake squeal papers as a

percentage of the papers presented at the SAE Annual Brake Colloquium.

The competitive nature of the automotive industry, which limits the amount of

cooperative research that is published in the open literature, has also been an

impediment, although some of these hurdles are being reduced. While component

suppliers and manufacturers may still not always openly disclose the nature of their

research, much progress has been made within European and US working groups on

brake noise in openly discussing some of the key areas of concern and introducing some

standardised testing procedures. This allows more opportunity for researchers to be in

open communication within each other.

This chapter will endeavour to provide an overview on the published research into brake

squeal noise. Firstly the previous reviews of brake squeal will be covered before

15

moving into the main approaches used by previous researchers. These can be broadly

classed as:

1. Analytical

2. Experimental

3. Numerical

2.2 Reviews of Brake Squeal

Several reviews of brake squeal have appeared in the literature. North (1976) provided

coverage of the basic theories used to analyse brake squeal up until that time which

included highly simplified geometry based models, such as cantilevers and pin on disc

models. Use of numerical analysis such as the finite element method (FEM) was not

yet practical at that time so the analytical models were simple with only a limited

number of degrees-of-freedom (DOFs) that didn’t directly relate to any specific brake

geometry.

Yang and Gibson (1997) conducted a review of the modern aspects of brake squeal

research. In particular, they noted the limitation of using large multi-DOF FEM models

for analysis lies in the difficulty in modelling the friction interface. They expressed

hope for the future, but noted that experimental methods had been more productive in

addressing brake noise concerns.

Arguably the most comprehensive review of brake squeal has only recently been

published. Kinkaid et al (2002) examined most aspects of the research that has been

conducted over the years including not only the approaches used by previous

researchers in tackling brake squeal, but also provided a background of modern brake

system operation. While they felt that a truly useful general theory of brake squeal may

not become reality, or may be too complicated to be practical, they did note that there is

considerable scope for improvements in the current modelling methods of brake squeal.

A four part series covering many aspects of automotive brake squeal was published by

the SAE in 2003. Part I covered the mechanisms thought to be the root of brake squeal

(Chen et al, 2003a), Part II looked into simulation and analysis (Ouyang et al, 2003),

Part III tackled testing and evaluation (Chen et al, 2003b), and Part IV covered brake

squeal reduction and prevention (Chen et al, 2003c).

16

2.3 Analytical Approaches to Brake Squeal

Much of the early work into brake squeal can be described as analytical. The theories

used to describe the phenomena hinged upon relatively simple models with a small

number of DOFs that were tractable for hand calculation. The goal was then, as it is

now, to find the source of the instability within a brake system during squeal.

The models that are reviewed in this section were used as possible models to describe

the mechanism of brake squeal. In practice, the squeal may be due to one or more

mechanisms, or not at all. It is difficult to confirm with certainty that they are the exact

physical squeal mechanism in the brake system under investigation.

2.3.1 Variable µµµµ Analysis

Consider the block of mass m on a sliding conveyor with a velocity v as shown in

Figure 2.2. If the coefficient of friction µ is independent of the sliding velocity, and the

sliding velocity is greater than the vibration velocity, then the equation of motion for

vibration is

0=++ kxxcxm &&& (2.1)

where c is the damping coefficient and k is the spring stiffness. The coordinate x is a

measure of the displacement of the mass from the equilibrium position, which is

determined from the values of µ, m, k and the acceleration due to gravity g. If

disturbed, the system undergoes stable damped oscillation for all positive values of m

and k, and when c lies between 0 and km2 .

17

Figure 2.2 A single degree-of-freedom featuring a block rubbing on a

moving conveyor.

However, if the coefficient of friction was not constant but a function of velocity as

shown in Figure 2.3, then the potential arises for the system to become unstable.

Figure 2.3 Negative slope µ-v characteristic.

The coefficient of friction decreases as velocity increases giving

ss vv αµµ −= 0)( (2.2)

where µ0 is the coefficient of static friction, α is the negative slope of the friction curve,

and vs is the sliding velocity given by xv &− .

m

k

c

x

v

µ

v

µ0

α

1

18

Incorporating the negative µ - v into from equation (2.2) into equation (2.1) yields

0)( =+−+ kxxmgcxm &&& α (2.3)

where g is the acceleration due to gravity and other symbols as described earlier. It can

be seen that it is possible for the damping within the system to take a negative value, if

mg

c>α (2.4)

This gives rise to the instability and the system can undergo self-excited oscillation.

As cited by Kinkaid et al (2002), this line of thinking prompted early researchers into

brake squeal, such as Mills (1938), and Fosberry and Holubecki (Fosberry and

Holubecki, 1959, Fosbery and Holubecki, 1961), to suggest that the variation in the

friction coefficient with sliding velocity was the root cause of the instability driving

squeal.

The negative µ - v characteristic has since fallen out of favour and it has been shown to

be of little or no importance in brake noise generation other than for very low speed,

low frequency noise such as creep-groan (Lang and Smales, 1983, Eriksson and

Jacobson, 2001).

2.3.2 Sprag-Slip Analysis

Squeal has been shown to occur in brake systems where the coefficient of kinetic

friction is constant. The earliest analysis performed to explain this was by Spurr (1961)

in his investigations into railway, drum and disc brakes in the early 1960s. Spurr

proposed an early sprag-slip model that describes a geometric coupling hypothesis.

Consider a strut inclined at an angle θ to a sliding surface as shown in figure 2.4(a). The

magnitude of the friction force is given by

θµ

µ

tan1−=

LF (2.5)

where µ is the coefficient of friction and L is the load. It can be seen that the friction

force will approach infinity as µ approaches cot θ. When µ = cot θ the strut ‘sprags’ or

19

locks and the surface can move no further. Spurr’s sprag-slip model consisted of a

double cantilever as shown in figure 2.4(b). Here the arm O′P is inclined at an angle θ′

to a moving surface. The arm will rotate about an elastic pivot O′ as P moves under the

influence of the friction force F once the spragging angle has been reached. Eventually

the moment opposing the rotation about O′ becomes so large that O″P replaces O′P, and

the inclination angle is reduced to θ″. The elastic energy stored in O′can now be

released and the O′P swings in the opposite direction to the moving surface. The cycle

can now recommence resulting in oscillatory behaviour.

(a) (b)

Figure 2.4 (a) Single strut rubbing against surface, (b) sprag-slip system

Others extended this idea in an attempt to model a brake system more completely.

Jarvis and Mills (1963) used a cantilever rubbing against a rotating disc, and Earles and

Soar (1971) used a pin-disc model.

Millner (1978) modelled the disc, pad and caliper as a 6-degree of freedom, lumped

parameter model and found good agreement between predicted and observed squeal.

Complex eigenvalue analysis was used to determine which configurations would be

unstable, and tried to apply as much physical relevance to his model as possible.

Parameters investigated included the coefficient of pad friction, Young’s modulus of

pad material, and the mass and stiffness of caliper. Squeal propensity was found to

increase steeply with the coefficient of friction, but squeal would not occur below a cut

off value of 0.28. He found that for a constant friction value, the occurrence of squeal

and squeal frequency depends on the stiffness of pad material (Young’s modulus).

20

Caliper mass and stiffness also displayed distinct narrow regions where squeal

propensity was high.

The terms “geometrically induced instability” and “kinematic constraint instability”

were introduced by Earles and Chambers (1987). These terms and sprag-slip are often

applied interchangeably to the type of behaviour described in these models.

Murakami et al (1984) applied a combination of negative µ-v and sprag-slip in the

models they created and attributed a portion of blame to both phenomena. The µ-v

characteristic was reasoned to provide an energy source for the squeal, and the

geometric instability provides the pathway for squeal generation.

Nishiwaki (1991) argued that the source for all brake noise was the same; variation in

friction forces through a cycle. This can be seen with either negative µ-v friction

behaviour or the variation in normal forces in sprag-slip type models with constant µ.

The common conclusions of kinematic constraint stability models are that brake squeal

can be caused by geometrically induced instabilities that do not require variations in the

coefficient of friction. The variations in frictional forces during the cycle can be

geometrically driven.

2.3.3 Mode Coupling Analysis

Another type of instability that has been described in relation to brake systems is due to

mode coupling. Other names used in the literature include binary flutter and non-

conservative displacement dependant forces. Again, the variation through the cycle of

frictional forces drives the self-excited vibration, but in this case the resulting motions

form when two adjacent vibration modes coalesce.

A detailed analysis of a two DOF system was performed by Hoffman et al ( 2002) in an

effort to provide physical insight into the mode coupling instability. Consider the

system shown in Figure 2.5. Linear springs k1 and k2 are inclined at angles to the

normal (x) and tangential (y) directions respectively, coupling them. A further linear

spring k3 represents the contact stiffness between the mass and the sliding surface, and

coulomb friction coefficient µ acts tangentially.

21

Figure 2.5 Two degree-of-freedom system analysed by Hoffman, et al.

The equations of motion can be expressed as the matrix equation

=

−+

N

F

F

F

y

x

kk

kkk

y

x

m

m

2221

31211

0

0 µ

&&

&& (2.6)

where

k11 = k1cos2α1 + k2cos

2α2,

k12 = k21 = k1sinα1cosα1 + k2sinα2cosα2,

k22 = k1sin2α1 + k2sin

2α2 + k3,

and x&& and y&& are the 2nd

time derivatives of the displacements x and y respectively.

The friction coupling term ∆ = k3µ. appears as an off diagonal term in the system

stiffness matrix, hence it is unsymmetric. With suitably chosen values of m, α1, α2, k1,

k2, and k3, the system possesses two, possibly complex, eigenvalues

[ ]2

1

2,1 ∆−±±= bas (2.7)

where a and b are real numbers.

The value of ∆ is of critical importance to the resulting eigenvalues and eigenvectors. If

∆ < b then two normal, undamped modes occur with distinct frequencies. As ∆

approaches b the modal frequencies converge. At ∆ = b the modes coalesce and form a

stable and an unstable mode pair. The motion of the modes is no longer normal, that is,

k3

k2 k1

m

α1

α2

FN FF

y

x

22

they no longer move perfectly in or out of phase with each other, but now a phase

relationship between the DOFs allows the mass to take on a loop in the x-y plane.

This type of behaviour has been analysed on a variety of levels, ranging from simplified

models of a limited number of degrees-of-freedom in the vein of the sprag-slip models,

through to large scale models using FEA. In systems of much larger DOFs, the mode

coupling can occur between many mode pairs, and a variety of stable/unstable modes

may exist. These are characterised by the same shift away from normal mode behaviour

at the point of coalescence, and the formation of complex modes where non 0° or 180°

phase relationships exist between DOFs.

North (1976) presented an eight-degree of freedom model that included a bar

representing the disc. The model attempted to mimic real aspects of a brake system

more than earlier simplified models, and displayed binary flutter type behaviour.

Work has been done on examining the influence of geometry and symmetry on mode

coupling using relatively simple models. Mottershead and Chan (Mottershead and

Chan, 1992, Mottershead and Chan, 1995) have published several works on the

behaviour of repeated modes or “doublets” in symmetric structures like brake rotors, as

have Lang and co-workers (Lang and Newcomb, 1990, Lang et al, 1993). It should be

noted that much of this work is shifting away from simplified analytical models toward

larger scale FEA models.

Further more recent analysis of simple models have been conducted by Brooks et al

(1993) and El Butch and Ibrahim (1999). Both studies included efforts to optimise

piston location to reduce squeal propensity.

Since these closed form theoretical approaches discussed in previous sections cannot

adequately model the complex interactions between components found in practical

brake systems their applicability has been limited. However, they do provide some

good insight into the mechanisms of brake squeal by probing the fundamental physical

phenomena that occur when a brake system squeals.

Analytical models are not confined to history with the advent of more sophisticated

modelling tools such as FEA. Much work continues in trying to understand the

mechanisms of brake squeal using relatively simple analytical models. An example of

this approach in a practical setting is work performed by Denou and Nishiwaki (2001)

23

who used simple models to provide guidance on the general direction a design should

take before conducting more sophisticated and detailed analysis.

2.4 Experimental Approaches to Brake Squeal

Experimental approaches to brake squeal analysis revolve about understanding the

characteristics of the brake system during a squeal event. This includes evaluating the

modal properties of the brake system, both at a component level and as an assembly,

investigating the nature of the friction processes and interactions within the system, and

also determining the sound radiation of the characteristics of the brake system. A good

understanding of the characteristics is required in support of other brake squeal

analyses, especially in the case of validating a large scale FEA model.

Experimental modal analysis techniques are well suited to determine the modal

properties of a brake; that is, the modal frequencies, damping and mode shapes. A good

example of the characterisation of a brake system as part of a larger research project

was described by Richmond et al (1999). Firstly the components were analysed

individually and their modal properties determined before moving on to the system as

an assembly. Traditional accelerometer based measurements are suitable for component

level analysis, although the advent of laser vibrometry greatly increases the speed at

which this can be achieved.

Experimental modal analysis is of critical importance in understanding the behaviour of

the disc brake rotor itself. Note only is it required to establish the modal properties of

the brake rotor, but the complexity and modes found with a brake rotor, and their

interactions have been found to be key in many studies on high frequency brake squeal,

ie, squeal above 5 kHz (Matsuzaki and Izumihara, 1993, Dunlap et al, 1999, Chen et al

2000, Chen et al 2002).

Brake rotors possess several different types of vibration modes, but can be broadly

classified as in-plane and out-of-plane. In-plane modes came in several types, but the

squeal frequency of a brake system often correlates strongly with the tangential (also

called circumferential or longitudinal) in-plane vibration modes of the rotor (Matsuzaki

and Izumihara, 1993, Dunlap et al, 1999, Chen et al 2000, Chen et al 2002).

Matsuzaki and Izumihara (1993) showed that the squeal noise in their study correlated

with a rotor tangential in-plane noise rather than rotor bending modes. Modifying the

24

brake rotor by cutting slits into the brake rotor friction faces to modify the behaviour of

the in-plane modes and reduced the occurrence of squeal.

Dunlap et al (1999) examined a variety of brake squeal problems, low and high

frequency, but again the high frequency problem was found to correlate strongly with

the existence of tangential in-plane rotor modes.

Chen et al (2000) investigated the relationship between in-plane modes and out-of-plane

modes that were close in frequency. It was reasoned that energy from in-plane modes

could be efficiently transferred to out-of-plane modes if they occurred at the same

frequency driving the squeal. A further investigation provided an in-depth study on 8

different rotor designs to establish the frequency relationship between the in-plane and

bending modes of a rotor (Chen et al, 2002). They concluded that coupling of rotor

tangential in-plane modes and rotor bending modes were the primary cause of high

frequency brake squeal. The squeal frequency would occur at the rotor in-plane

frequency, but the mode shape would correspond to the coupled out-of-plane mode.

The design guideline they provided stipulated that the rotor tangential in-plane modes

should lie within the central 1/3 frequency band of adjacent dominant out-of-plane

modes.

A further study by Chen et al (2004) again focused strongly on the relationship between

in-plane and out-of-plane modes. A key influence other than the rotor design was the

brake pad. Friction forces tend to excite the in-plane modes, and pad bending can excite

out-of-plane modes. Pad chamfering, which reduces the pad foot print on the pad at the

rotor / pad interface, can help alleviate high frequency in-plane related noise. Altering

pad resonant frequencies can also help.

While high frequency brake squeal highlights the importance of the brake rotor modes,

low frequency brake squeal in range of 1 to 4 kHz has been found to depend more

strongly on the caliper components in addition to the rotor.

Baba et al (1995) studied low frequency squeal and found it particularly sensitive to the

characteristics of the caliper mounting bracket. Dunlap et al (1999) showed the

importance of both rotor and caliper modes in the appearance of “modal locking.”

Ishihara et al (1996) concentrated on rotor behaviour, highlighting that any of the major

components are important in low frequency squeal.

25

Control of rotor resonant frequencies pose a concern for both low and high frequency

brake occurrences. Grey cast iron, which is the usual material for brake rotors, is

somewhat unique in that it can change significantly in modulus and therefore modal

frequencies which are proportional to the square root of the modulus. Other brake

components, with the exception of the brake pads, are not made of material that can

vary significantly in modulus or density, so their compositions are not the subject of

specific studies.

Several studies have been done on the nature of a brake rotor’s sensitivity to carbon

content, or more usually carbon equivalent (CE). Malosh (1998) provides an analysis

on the relationship between the elastic modulus E of grey cast iron and the carbon

equivalent as

CEGPaE 1.543.310)( −= (2.8)

for the range of 8.43.3 ≤≤ CE , the range for typical grey cast iron. The carbon

equivalent is given as

3

%%

SiCCE += (2.9)

where %C and %Si are the percentage compositions by weight of carbon and silicon

respectively.

Chatterley and Macnaughton (1999) do not provide an explicit relationship between CE

and modulus, but merely discuss similar trends on the basis of a survey of typical

European rotor materials, their compositions and applications. Also, the equation they

gave for CE takes the form of a carbon equivalent liquidus (CEL) taking into account

the addition of phosphorous also

2

%

4

%%

PSiCCEL ++= (2.10)

To assist in understanding the behaviour of a brake system, it is of great importance to

be able to visualise dynamical deformations, or operating deflection shapes (ODS),

during a squeal event. Taking measurements of a squealing brake is difficult since the

brake rotor itself rotates, so that it is not possible to use conventional measurements, for

example with accelerometers. Further complications to the measurement process also

include the transient and fugitive nature of squeal, which ideally requires an

26

instantaneous measurement, and a large number of components deforming in different

scales, which places demands on the resolution requirements.

A number of optical techniques have been developed in an attempt to overcome these

difficulties. The main type that has achieved practical success is Double Pulse

Holographic Interferometry (DPHI). DPHI provides excellent temporal and spatial

resolution, but is somewhat specialised and suitable mostly for laboratory research.

DPHI has been applied with considerable success since the 1970s. High energy pulses

are generated by a powerful laser such as a ruby laser and illuminate the vibrating object

at two points during a vibration cycle. Each pulse is split into a direct reference beam

falling directly onto a holographic plate and an object beam reflected from the vibrating

object. The difference in path length of the object beam for the two pulses due to the

deformation causes an interference pattern on the holographic plate. The fringe pattern

represents contours of displacement during the cycle. Felske et al (1978) successfully

visualised a squealing disc rotor and caliper components. Time average holography was

also used to visualise an artificially excited brake system. DPHI has further been used

by a number of researchers for visualising the brake systems including Murukami et al

(1984), Nishiwaki et al (1989), and Ichiba and Nagasawa (1993). Fieldhouse and

colleagues have also used DPHI on many disc and drum brake squeal problems

(Fieldhouse and Newcomb, 1991, Fieldhouse and Newcomb, 1996, Fieldhouse and

Rennison, 1998, Fieldhouse and Beveridge, 2000, Fieldhouse and Beveridge, 2001,

Talbot and Fieldhouse, 2001).

While pulse laser systems have improved in their user friendliness and flexibility over

the last several decades, the main features of DPHI applied to brake systems have not

changed a great deal. The main limitations revolve about trying to implement a

practical system that could be used in an industrial environment in a “turn-key” fashion.

Scanning Doppler Laser Vibrometer (DLV) systems offer considerable practical

advantages over DPHI, and now practical, easy to use systems have been employed by

several research groups. McDaniel et al (1999) used a scanning laser vibrometer to

measure the ODS of a statically loaded and artificially excited brake system. These

results were then used to assist the study of sound radiated by the system. Richmond et

al (1999) and Chen et al (2002) also made measurements on a static system in this

manner.

27

Scanning laser systems utilising three heads to resolve 3-D motions are only just

beginning to be employed in industrial research (Polytec, 2007). The future of these

systems is bright since they allow the visualisation of in-plane motions is addition to

out-of-plane. The 3-D systems have been developed specifically to allow investigation

of squealing disc brake systems.

The techniques described in this section have provided some good and useful tools for

investigations into brake vibration characteristics. In particular, accelerometer and laser

based modal property and ODS measurements can be resolved with great accuracy for

static systems. However, the shortfall is in measurements of a system during squeal.

DPHI systems are difficult to implement in practice and scanning laser systems struggle

to provide temporal and spatial resolution for transient events. Hence, these tools need

to be further developed to be useful in the industry. The experimental techniques used in

industry are described in Section 3.11.

The focus of this section has been on understanding the vibration characteristics of the

brake system. These characteristics alone do not determine if a squeal concern will

arise. Understanding the system’s sound radiation provides the link between the

vibration behaviour and a squeal concern. For example, sound radiation efficiency

using the two-microphone technique, or acoustic holography with a microphone array

can be used to investigate the acoustic sound field around the brake system as a result of

system vibration.

2.5 Numerical Methods for Brake Squeal Analysis

Numerical analysis of brake squeal can be considered a more sophisticated extension of

the analytical models discussed earlier. However, by using powerful computers and

finite element analysis (FEA), it is possible to model individual components as

continuous structures and incorporate their elastic properties into a dynamical

description of the system.

FEA has been used to determine the modal frequencies and mode shapes of individual

components. This is also required before a larger assembly analysis can be performed

to validate the individual component modes against experimental measurements.

Examples of this type of work were the work done by Richmond et al (1999) on the

components of an assembly and Saka and Wada (2003) who undertook a study to

28

provide a systematic method for naming mounting bracket modes in terms of a ring and

a bent ring. Kumemura et al (2001) analysed low frequency behaviour of a brake

anchor bracket / suspension knuckle assembly and increased stiffness of the bracket to

improve a low frequency squeal concern.

The prediction of system stability is one area where there has been a large amount of

research using FEA. Complex eigenvalue analysis is the main solution technique that

has been used in assessing system stability. Complex eigenvalue analysis is a large scale

extension of the mode coupling type instability discussed in section 2.3.3. By far the

majority of the research that has been performed has utilised the commercial FEA code

Nastran. Nastran is favoured because it has well established eigenvalue solution

schemes and until recently was the only commercial code to perform the analysis. The

analysis proceeds by solving a static step to determine the static position of the brake

assembly components. The system is then linearised about this base position and a

complex eigenvalue extraction is performed to determine the stability of the system

modes based on the appearance of negatively damped modes.

The first major published work was a study performed by Liles at GM (Liles, 1989).

Liles manually inserted springs and friction elements to model the friction coupling at

the rotor/pad interface, thereby creating an unsymmetrical system stiffness matrix; a

necessary, but not sufficient, condition to produce coupled modes and system instability

(Liles, 1989). Liles went on to study a variety of brake system parameters including

coefficient of friction, pad geometry, caliper and rotor stiffness, and system damping.

This type of analysis was further championed by Nack, also at GM, using much the

same analysis methods as Liles. Nack and Joshi (1995) studied low frequency brake

noise and observed the coupling of system modes at the onset of instability. A further

study focusing on brake squeal was later presented (Nack, 1999). The main limitation

expressed in these studies was the limited modelling of variable friction coefficient, and

future work suggestion was to move toward time domain analysis.

Complex eigenvalue analysis to assess stability forms the first step in many studies and

further analyses have been applied to probe the behaviour of the brake system and to

develop countermeasures. Kung et al (2001a) used the Modal Assurance Criterion

(MAC) to identify modal participation of component modes in unstable modes extracted

from a complex eigenvalue analysis. A parametric study was conducted to observe the

29

influence of varying friction coefficient and brake rotor modulus. A further study was

conducted that involved shifting the frequencies of two system modes further apart in

frequency to reduce their tendency to form a coupled unstable system mode (Kung et al,

2000b).

Park et al (2001) used MAC to assess the modal participation in the unstable system

mode which was the focus of their study, but also used component strain energy to

further identify which component would be best modified.

Zhang et al (2003) also used both MAC and strain energy to aid identifying potential

countermeasures, but also included a number of alternative measures of component

participation. The 1-norm and infinity-norm were calculated and compared as

indicators of component contributions along with strain energy. The norms offered

similar information as to which components contributed the most. Strain energy was

found to offer quite different conclusions due to the significant differences in size and

material properties of each of the components. They suggested that each of the

indicators should be used in a complimentary manner.

Another stream of complex eigenvalue analysis began in 2003 when HKS Abaqus

provided an appropriate eigenvalue solver. The main feature of Abaqus is advance non-

linear and contact solving capabilities in the non-linear static parts of an analysis. Bajer

et al (2003) described the basic formulation of this scheme, and Kung et al (2003)

investigated the instability related to various in-plane rotor modes.

The major advantage of Abaqus over the previous analysis using Nastran is the

seamless integration of non-linear static analysis and dynamic analysis without the

requirement for manual inserted friction coupling terms. The contact element

formulation removes the need for coincident node meshing at the component interfaces.

Abaqus also provides the ability to utilise some aspects of a velocity dependent friction

coefficient.

Guan and Huang (2003) presented the method of feed-in energy analysis for a brake

system. This is a method that calculates the frictional work converted into vibration

energy by considering the relative motions of a friction contact pair at the friction

interface. If the feed-in energy is positive then system vibrational energy is increasing

and the system is unstable, and they showed that feed-in energy is an alternative to

considering the real part of eigenvalue as an indicator of stability.

30

Another form of stability prediction that has been used much less widely is non-linear

transient analysis. This type of analysis follows the evolution of displacements in the

time domain and an unstable state is identified by divergence away from equilibrium. A

Fast Fourier Transform (FFT) analysis is performed to identify frequencies associated

with the noise. Hu and Nagy (1997) performed a non-linear transient analysis using LS-

Dyna and achieved good correlation of measured noise performance. The key benefits

of the analysis as compared to complex eigenvalue analysis that they identified were:

the ability to implement more sophisticated friction behaviour, it was not necessary to

determine system static state prior to analysis, and finally the variation in component

contact during vibration is captured. Chargin et al (1997) also presented a study using

non-linear transient analysis. They did not include a direct correlation to a specific

brake system, but concentrated on a detailed description of their modelling procedure.

Chern at al (2002) have also performed non-linear transient analysis using LS-Dyna.

The main drawback of the technique is the demand on computer resources. The size of

time increments, which is a function of element mesh size and propagation velocity of

vibration waves, requires an enormous number of solutions to be preformed.

The methods covered in this section involve large scale FEA models to investigate the

vibration behaviour of a brake system and components. However, such analysis is not

capable of predicting whether a particular vibration mode will cause an audible squeal

concern. The boundary element method (BEM) can be applied to determine the sound

radiation efficiency of the system, and thus link the vibration characteristics to the

radiated sound.

2.6 Brake Squeal Noise in Practice

The methods for addressing brake squeal noise during development of a production

brake system remain largely empirical. Counter-measures are developed during final

development of the system typically by trial and error in conjunction with noise

dynamometer and on-vehicle testing. This section gives a feel for the types of counter-

measures that are applied to many automotive brake systems in production today.

The addition of shims to the back of pad backplates is used on most brake systems

today, usually in the form of a multilayer form. Two common types of shim are in wide

spread use. The first type utilises several layers of steel shim with grease applied

between the layers. The second type features a multi-layer design of alternating rubber

31

and steel, bonded together to the back of the backplate. The rubber forms a constrained

layer and induces significant damping through shear of the rubber layer. An example

material is shown in Figure 2.6 and material properties in Table 2.1.

Table 2.1 Layer specification of typical damping shim material

Layer (from piston side) Thickness (mm) Rust prevention 0.02

Steel 0.40

Nitrile rubber 0.24

Adhesive 0.12

Total 0.78 Additional removable protective paper layer 0.15

Figure 2.6 Example shim material

The function of shims operates on many different levels (Flint and Hald, 2003). Shims

can provide additional damping for the brake system, shims can decouple the caliper /

pad interface, and shims can alter the pressure distribution between components by

geometry. Figure 2.7 is an example of a shim that has been designed to shift the centre

of contact pressure of the piston on the back of the pad.

Another common strategy for the controlling brake noise is modification of the surface

area of the brake pads at the critical rotor / pad interface, as shown in Figure 2.8. This

is achieved most commonly by inserting chamfers onto the surface of the brake pad.

The final form chamfers take before production begins is often through continuous

refinement and testing.

Piston

side

32

Figure 2.7 Example slotted backplate shim designed to shift the centre of

contact pressure on the piston / pad interface.

Figure 2.8 Example of pad chamfers

Investigation of noise concerns at specific frequencies requires investigation of the

dynamical properties of the components themselves. Care needs to be taken since

properties under free-free boundary conditions may change when the component is

constrained in a brake assembly. However, the free modes near a particular squeal

frequency can be used as a guide when considering structural modifications.

Operational deflection shape (ODS) measurement on a squealing brake can be used to

understand what component modes are active. This leads in the direction of structural

modifications of caliper components or the rotor.

33

Unfortunately the bulk of structural modification implemented in industry proceeds on a

trail and error basis. Changes revolve around changing mass and stiffness

characteristics of components and validation occurs on the brake noise dynamometer or

in vehicle testing.

Most of the structural components within a brake system are made of metal alloys

including steel, aluminium, ductile cast iron and grey cast iron. The elastic moduli and

mass densities of these alloys are usually relatively insensitive to variation in chemical

composition. Hence there is not usually much tuning of material composition in

developing countermeasures. The exception is grey cast iron, which has an elastic

modulus that can vary greatly due to changes in carbon content and other alloying

elements. This, in turn, leads to major changes in the modal frequencies of brake rotors,

and material properties can tuned to address specific noise concerns (Malosh, 1998).

Another key area in brake noise development relates to the compressibility of the brake

pad, or more specifically, the compressibility of the friction material. Brake pads

consist of two main sections; a steel backplate and the friction material which is

moulded to the backplate. The friction material is a composite structure with potentially

dozen of constituents, the exact composition being proprietary information and known

only to the friction supplier. Further, it is not uncommon for an underlayer to be

applied between the friction material and the backplate itself, which can add beneficial

thermal and mechanical properties. The sum of all these components is a pad that is

relatively compressible, with a typical pad showing approximately 0.1mm compression

under high brake pressures.

The static compressibility of the pad can have a strong influence on the noise

performance of a brake system. It is often seen that increased static compressibility

tends to reduce noise occurrences. However, increasing static compressibility can have

a negative impact on other aspects of brake performance such as pedal feel and drag.

To further add to this complexity, friction materials may exhibit behaviour that is a

function of frequency, both in terms of stiffness and damping. For example, the

dynamic compressibility of a friction material may be significantly different from the

static compressibility. While static compressibility measurements are common in

industry, dynamic compressibility, stiffness and damping are not easily quantified and

remain as gaps in the characterisation of friction material behaviour.

34

2.7 Summary

The key methods of brake squeal investigations; analytical, experimental and numerical

have been presented. The analytical work has provided some basic models to help

understand the phenomena within brake systems. However, they have limited

applicability to any specific brake systems and practical brake squeal investigations.

Experimental work continues to be the leading form of brake development, particularly

in industry. While the complex behaviours within the system are only gradually being

revealed as new technologies allow deeper probing into the system, much successful

development work the done on the basis of observation and modification. The

application of optical visualisation techniques such as DPHI and scanning DLV have

allowed more detailed probing into brake system behaviour. Unfortunately DPHI is

difficult to apply in practice and scanning lasers systems have limitations in resolution

for transient events.

The numerical modelling of brake systems has been an area of significant growth over

the last few decades. Complex eigenvalue analysis is in widespread use and no doubt

time domain analysis will follow with the increase in computational power and

reductions in cost. Considerable gaps exist in the numerical techniques. Modelling of

contact interfaces, particularly the key pad / rotor interface, as well as correlation to true

material behaviour across the whole frequency range are still areas undergoing

development.

The focus of this thesis is in the area of numerical modelling to complement industrial

testing and development of brake systems. Complex eigenvalue analysis is the chosen

method of stability analysis due to the lower computational requirements. The areas of

deeper probing into a system behaviour following a complex eigenvalue analysis will be

studied in detail. Feed-in energy analysis as proposed by Guan and Huang (2003) has

not been taken up in widespread use. There is potential to apply it to investigate pad /

rotor interaction and to contrast it to complex eigenvalue prediction. Strain energy has

used widely to identify active components, but meaningful normalisation needs to be

investigated. Finally, addition of MAC calculations to these allows a suite of tools for

understanding system behaviour at system level. Identifying their relevance and

applying them to a large scale FEA model in a complementary manner is a key aim of

this thesis.

35

Chapter 3

Experimental Determination Of

Vibrational and Acoustical Characteristics

of a Brake System

3.1 Introduction

In general, noise problems stem from the dynamic characteristics of any system. One or

more vibration modes are excited and some of the energy is dissipated by radiating

sound. This is the case when a brake squeals. The brake system enters into an unstable

vibration mode and the brake rotor acts as a loudspeaker radiating most of the squeal

sound. Hence, it is of fundamental importance to gain an understanding of the vibration

characteristics of the brake system.

Experimental modal analysis is a technique used to determine the vibration

characteristics of a system. Measurements are made and analysed to evaluate the

vibration modes of the system. Each vibration mode consists of three modal parameters

or properties which together provide a description of the system’s dynamic

characteristics

1. Modal frequency – resonant frequency of oscillation for the mode, usually

expressed in cycles per second (cps) or Hertz (Hz).

2. Modal damping – a measure of the dissipation of vibration energy or the rate at

which the vibration decays, usually expressed as a percentage on the critical

damping; the maximum damping level which will permit free vibration.

3. Mode shape – deformation or displacement pattern of the structure for the

mode, described by complex-valued displacements

Experimental modal analysis can be a very effective analysis tool. Modifying the modal

parameters of a system can control many types of noise problems. Modal testing is also

vital in the development of numerical models such as when using the finite element

method (FEM). The modal analysis results will be required to validate FEM models,

36

and thus ensure that the models accurately reflect the physical system. Investigation of

the modal properties of the brake system with experimental modal analysis forms the

bulk of this present chapter and provides the basis for the developing the FEA models in

subsequent chapters.

3.2 Frequency Response Function

The first step in modal analysis is to obtain frequency response function (FRF)

measurements for the structure of interest. A FRF is a frequency domain function

describing the relationship between two points on a structure, an excitation point and a

response point. In general, the FRF is complex-valued and contains magnitude and

phase information between the points. However, if the damping is zero or very small

then the mode will be “normal” where the phase relationship between points is zero or ±

180°. A sinusoidal input will cause an output of the same frequency scaled by the FRF

magnitude and shifted by the FRF phase. When the excitation and response

measurements are made at a coincident location it is called the driving point FRF;

otherwise it is called a transfer FRF.

Modal analysis assumes linear system behavior. A feature of a transfer FRFs that

follows is that it makes no difference which is the excitation point and which is the

response point. This property, known as reciprocity, makes it possible to move either

the driving point or response point to obtain a sufficient set of measurements for a

structure.

A FRF may contain any number of peaks, each corresponding to a vibration mode.

Within a small range of frequency around a modal peak, the FRF is dominated by a

single vibration mode. Curve-fitting is a mathematical process used to find an analytic

function to describe each mode. The variables solved for this function are the modal

parameters.

FRFs can take three forms – compliance, mobility and accelerance. A compliance FRF

denotes displacement output per unit input force, mobility is velocity output per unit

input force and accelerance is acceleration per unit input force. For accelerometer based

measurements it is acceleration that is usually measured, but mobility or compliance can

be obtained by integrating the acceleration measurement. In this investigation the FRFs

37

were all in terms of accelerance. A comprehensive description of experimental modal

analysis can be found in Modal testing: theory and practice [Ewins, 1984].

3.3 Experimental Set-up

A schematic of the experimental set-up can be seen in Figure 3.1. FRF measurements

were recorded on a fast fourier transform (FFT) analyser before being transferred to a

computer for analysis. Table 3.1 contains a listing of the equipment used for the

measurements. Shaker excitation was used for all of the testing with the exception of the

specific experiments to identify rotor in-plane modes as discussed in section 3.5.5. A

small roving accelerometer that was affixed with bee’s wax was used for the

measurements except for the driving point FRF which was obtained through the

impedance head.

The B&K type 2032 FFT analyser divides the measurement spectrum into 800 lines.

Baseband random noise excitation was used between 0 and 6.4 kHz, and zoom random

noise excitation between 6.4 and 12.8 kHz. This results in a frequency resolution of 8

Hz for 800 lines in each frequency band, which was sufficiently high to allow the FRFs

to be recorded with acceptable accuracy. The FRFs were obtained from 50 averages

using a Hanning window with 50% overlap.

Table 3.1 Experimental equipment for modal measurements.

Item Manufacturer and Type

FFT analyser B&K 2032

Accelerometer B&K 4374

Impedance head B&K 8001

Charge amplifier B&K 2635

Shaker B&K 4810

Power amplifier B&K 2706

Calibrator B&K 4294

Impact hammer B&K 8202

Computer NEC Versa 4050C

Analysis of the experimental data was conducted with STAR Modal v 5.23, a

commercial modal analysis software package. The FRFs were spatially averaged to aid

the identification of modal peaks. Modal peaks can be identified from individual FRFs,

but they can usually be identified more reliably from a spatially averaged spectrum.

Cursor bands were set up around the modal peaks in the averaged FRF data. A cursor

38

band defines a small region of the frequency spectrum where curve-fitting will be

attempted.

Five curve-fitting methods are available in STAR Modal - Coincident, Quadrature,

Peak, Polynomial and Global [Star SytemUser’s Guide, 1994]. Coincident, Quadrature

and Peak are single degree-of-freedom methods and make no estimate of modal

damping. For this reason they were not used except for an initial examination of mode

shapes. Polynomial and Global methods are suitable for single or multiple degree-of-

freedom fitting and can estimate the modal damping, modal frequency and mode

shapes. The Polynomial and Global methods use the Rational Fraction Least Squares

method to identify the modal parameters and were used throughout this investigation.

Figure 3.1: Schematic of the experimental set-up.

FFT

Analyser

Computer

Power amplifier

Charge

amplifiers

Impedance

head

Shaker

Accelerometer

Test

component

Signal generator Ch. A Ch. B

39

3.4 Component Testing

The rotor, pad, caliper housing and anchor bracket were tested and analysed

individually. The remaining components, namely the piston and pins, were not tested.

Testing with either shaker or impact excitation would not be practical due to their small

size.

Testing of the individual components was conducted with free-free boundary

conditions. This was implemented by placing the components on a sheet of foam

insulation during the testing. The rigid body natural frequencies of the component on

the foam mat were several orders of magnitude lower than the lowest structural

resonance of any component so coupling between the test component and base support

was negligible.

3.5 Rotor

3.5.1 Brake Rotor Mode Shape Descriptions

Brake rotors may appear to be relatively thin discs on first appearance, and it may seem

that the dynamic characteristics could be described by bending (out-of-plane) vibration

modes, similar to a circular plate. In reality a brake rotor possesses significantly more

complex geometry than a thin circular plate. A drum-in-hat (DIH) rotor, as shown in

Figure 3.2, is substantially thick and has a hat section that introduces asymmetry in

cross section. The hat section in particular makes it more difficult to assign a

numbering scheme to all brake rotor modes. Therefore the mode descriptions that

follow are concerned mainly with motions of the main annular disc region containing

the friction surfaces, referred to as the friction disc.

40

Figure 3.2 Cross section of a solid drum-in-hat (DIH) rear rotor.

Out-of-plane modes involve rotor bending motion and nodal regions that appear as

either diametrical or circumferential lines, as displayed in Figure 3.3. Here the regions

designated with a “+” and “-” represent regions that are 180˚ out of phase. Two

numbers, m and n, can be used to describe the modes shape in a similar manner to those

devised for circular plates as described in Rossing and Fletcher (1995). The first

number indicates the number of nodal diameters and the second number indicates the

number of nodal circumferences. The primary bending modes involve primarily nodal

diameters and commonly identified merely by the ND (nodal diameter) number.

top-hat section

Service brake

friction faces

friction disc

park brake

friction surface

41

(a) (5,0) Mode

5 nodal diameters

0 nodal circumferences

(b) (0,2) Mode

0 nodal diameters

2 nodal circumferences

(c) (5,1) Mode

5 nodal diameters

1 nodal circumference

Figure 3.3 Bending mode descriptions

In-plane vibration modes are somewhat more complicated in that a number of different

types exist. With reference to Figure 3.4, in-plane modes can be classified as either

tangential or radial modes. Tangential modes are also often also called longitudinal or

circumferential modes, and radial modes are also known as star modes.

Tangential in-plane modes occur in the two distinct forms, compression and shear (also

called racking), as shown in Figure 3.4 (b) and 3.4 (c) respectively. Section 3.5.5

investigates the detection of rotor in-plane modes more specifically.

+

+

+

+

+

+

_

_

_

_

+ + + +

+ _ _

_

+

_

+ +

_

_

+

+ +

+

_ _

_ _

42

(a) Radial in-plane mode with 3

nodal diameters

(b) Tangential in-plane

compression mode with 2 nodal

diameters

(c) Tangential in-plane shear

mode, 1st order.

Figure 3.4 In-plane mode descriptions

It is also possible for the equivalent modes to exist in the top hat region of the rotor.

This leads to the additional complication found in some rotor modes, with coupling

between the disc and top hat modes. Top hat modes are designated in the same way as

the disc modes, but with a prefix of TH.

3.5.2 Rotor Test Grid

The grid for the rotor, displayed in Figure 3.5, consisted of 384 measurement points

arranged along 48 lines radiating from the centre of the rotor at an angular spacing of

7.5˚. Each line featured 8 measurement points as shown in Figure 3.6. Table 3.2 lists

the coordinates for the rotor grid points.

Measurements were taken at the response points in either the local r (radial) and z

(axial) direction, while the excitation was applied with the shaker in the z direction at

point 6, shown in Figure 3.5. As a result, direct measurement of only out-of-plane and

radial in-plane modes was possible. The tangential in-plane modes can be inferred by

the accompanying deformation in the radial or, possibly, out-of-plane direction, and

with the aid of a validated finite element model. Indeed, the 1st and 2

nd tangential in-

plane modes were identified this way. However, due to the relative lack of response, a

43

direct measurement of tangential deformation is not possible while exciting and

measuring in the r and z directions. This issue is addressed further in sections 3.5.5.

Table 3.2: Coordinates for the rotor grid.

Point no. r (mm) θ (°) z (mm) Direction

1 + 8n 40 34 z

2 + 8n 80 34 z

3 + 8n 103 28 r

4 + 8n 103 8 r

5 + 8n 110 0 z

6 + 8n 125 0 z

7 + 8n 140 0 z

8 + 8n 144

7.5n,

n = 0,..,47

-8 r

Figure 3.5: Brake rotor experimental grid (384 points). Point numbering is

omitted for clarity.

Excitation

point

44

X X

X XX

X

X

X

1 2

3

4

5

6 7

8

r

z

Figure 3.6: Cross sectional view of brake rotor showing location of grid

points. Each line radiating from the centre of rotor contains 8 points.

Excitation was applied at point 6 in the z direction.

3.5.3 Rotor Test Results

The spatially averaged FRF traces for the free rotor, over 384 points, are displayed in

Figure 3.7. The modal peaks are numbered from 1 to 27. These traces were used to

identify modal peaks for the curve fitting process and the magnitudes of the modal

peaks were also taken from the averaged FRFs.

45

Figure 3.7: Spatially averaged rotor FRF traces.

Table 3.3 provides a summary of the modal parameters as found by STAR Modal.

Modal damping values range from under 0.1% for the top hat modes through to 0.4%

for the RI 5 mode. Most of the out-of-plane modes have a damping value between 0.1%

to .15%. It is also clear the ND bending modes (nos. 1,3,8,11,14,18,24) have the

highest modal peaks.

The 1st and 2

nd tangential in-plane modes (no. 16 and 26) were also identified, although

the modal peak for the 2nd

mode does not appear very clearly in the spectrum. These

were established through observing the accompanying radial and top hat deformations

46

and comparing these with a finite element model.

Table 3.3 Rotor modal parameters. Log magnitude is with a reference value of 1

m/Ns2. Mode shapes refer to nodal diameters and circumferences of bending

modes except where prefix is added. TH indicates deformation in predominantly

in top hat region, RI is radial in-plane and CI is circumferential in-plane.

Mode No. Mode shape Freq

(Hz)

Damp factor

(%)

Log Mag

(dB)

1 (2,0) 994 0.37 15.3

2 (0,1) 2010 0.19 21.9

3 (3,0) 2430 0.18 27.1

4 (0,2) 2550 0.22 18.7

5 (1,1) 2690 0.15 21.5

6 RI 2 2900 0.15 19.6

7 (1,2) 3200 0.16 17.6

8 (4,0) 3800 0.14 31.9

9 TH(2,1) 4230 0.15 10.2

10 RI 3 4630 0.22 17.8

11 (5,0) 5290 0.13 31.9

12 TH(1,1) + RI 1 5730 0.10 14.5

13 TH(2,2) 6770 0.12 19.1

14 (6,0) 6990 0.11 35.8

15 RI 4 7120 0.12 14.3

16 TI 1 7840 0.17 12.8

17 TH(0,2) 8130 0.11 23.2

18 (7,0) 8900 0.11 40.8

19 TH(3,2) 9090 0.07 23.2

20 TH(1,2) 9230 0.10 27.2

21 TH RI 0 9770 0.08 13.8

22 RI 5 10060 0.40 21.0

23 TH RI 3 TH(3,1) 10560 0.08 21.9

24 (8,0) 10990 0.11 44.9

25 TH(5,1) 11620 0.08 21.5

26 TI 2 12130 0.40 15.3

27 (4,3) 12360 0.07 32.2

3.5.4 In-Plane Mode Detection

Pure longitudinal waves can occur only in solids where the dimensions in all directions

are greater than the wavelength (Cremer et al, 1988). This is not the case with the rotor

since the thickness is small compared to the wavelengths of the modes of interest.

Consequently, the in-plane waves that occur are actually quasi-longitudinal and are

accompanied by deformation in the lateral direction (see Figure 3.8). This makes it

possible to measure some in-plane motion while measuring in the out-of-plane

direction.

47

Figure 3.8 (a) Out-of-plane vibration, (b)In-plane vibration. Note that the

longitudinal compression is accompanied by expansion in the lateral

direction and longitudinal expansion is accompanied by contraction.

A number of experiments were conducted to examine how well in-plane modes could

be detected by measuring in the out-of-plane direction. Matsuzaki and Izumihara

(1983) discussed a method for specifically detecting modes where in-plane vibration is

present. This can be accomplished by measuring the phase difference between the

opposite faces of the rotor. If the surfaces of the rotor have out of plane motions that

are 180˚ out of phase then the mode must have in-plane deformation. All bending

modes would have the faces of the rotor in phase.

In principle, a convenient method of identifying which are the in-plane modes would be

to mount two accelerometers directly opposite one another on the rotor surface. If

broadband excitation is applied the phase can be examined to see at which frequencies

the in-plane modes occur. Unfortunately, this method did not work with this brake

rotor and a possible explanation of why follows.

Significant induced bending can occur during in-plane vibration. Consider the rotor

shown in figure 3.9. Since the rotor is not symmetrical with respect to the plane of the

friction surfaces, this will cause significant secondary bending in addition to the in-

plane deformation. During testing it was found that the two faces of the disc surface

were in phase. This is in direct contrast to the results published by Matzusaki and

Izumihara (1983). However, their test was conducted on a vented rotor which are

usually considerably thicker than the solid type tested here. Since flexural rigidity is

proportional to the 3rd

power of section thickness, it would be reasonable to expect the

bending in their investigation to be significantly reduced compared to the rotor in this

study, making it possible to detect the phase difference between the faces.

48

Figure 3.9 Rotor in an in-plane vibration mode. The top hat provides

unsymmetrical support to the disc itself and may cause out of plane

deformation.

3.5.5 In-Plane Measurement

Experiments were also conducted to directly measure the circumferential in-plane

modes, which the shaker tests with the previous grid could not detect directly. The set

up for these experiments is shown in Figure 3.10. A steel cube of 12 mm side length

was fixed via a steel stud to the outside edge of the rotor to enable the accelerometer to

measure in the circumferential direction. An impact hammer was used to provide the

excitation directly into the in-plane direction. The rest of the experimental set up

remains unchanged from the earlier tests. Here the hammer is striking the edge of the

rotor at an angle of approximately 45°. By using the hammer in this way it is not

possible to accurately measure the magnitude of the FRFs. However, the phase

component remains intact allowing the mode shape to be determined.

The grid for this test consisted of 12 points located around the outer edge of the rotor.

The response point remains unchanged due to its mechanical fastening so a roving

exitiation point was used. The number of wavelengths for the circumferential modes in

the frequency range of interest is at most two, so it was sufficient to use only 12

measurement points around the rotor circumference to resolve the mode shapes.

49

Figure 3.10 In-plane test configuration.

Analysis was conducted for these tests without the STAR Modal software. The

objective here was to be able to visualise the circumferential modes shapes with no

specialised analysis software. The test and analysis procedure that was utilised was as

follows:

1. Measure FRFs at 12 equally spaced around the circumference using a roving

excitation and fixed response point.

2. Identify modal peaks within the FRFs. This can be done by looking for the

magnitude peaks in individual FRFs or by spatially averaging FRFs from all test

points. The averaged FRFs are more reliable than looking at individual FRFs,

but it requires data to be exported from the FFT analyser for analysis.

3. Record magnitude and phase of each point from the FFT analyser for the modal

frequencies identified in step 2. Sort the magnitudes and phase by frequency to

capture complex-valued displacement information for each mode.

4. Set the phase equal to either 1 or -1 signifying in phase or out of phase. This is

to enable the modes to be displayed as normal modes, ie, with all points either in

phase or 180˚ out of phase. Note that the level of damping for the rotor is small,

so normal modes are expected. It is best to add an offset such as 22.5˚ or 67.5˚

to the phase since the relative phase values tend to sit around multiples of 90˚.

This will reduce potential phase errors due to noise if many of the phase values

are scattered around multiples of 90˚. For example, all points with -157.5° <

50

phase ≤ 22.5° are set to 1, and all points with 22.5°< phase ≤ 202.5° are set to -1.

5. Multiply the magnitude by the rounded phase from step 4 for each point and

normalise it by the largest magnitude. The result is a single value for each point

ready for plotting.

6. Repeat steps 4 and 5 for each modal frequency.

This procedure was applied to identify the 1st and 2

nd tangential in-plane modes of the

rotor. Figure 3.11 shows the result for the 1st mode graphically. It is clear that the

displacement represents one wavelength quite well. However, great care needs to be

applied when using this procedure. The phase values can be subject to considerable

scatter and the value chosen for phase offset rounding becomes very important. It is

often best done on a case-by-case basis after inspecting the FRF data. This limitation

makes it time consuming to analyse each mode and automating the process would

require careful implementation. As a result, the procedure should be used with caution

when determining mode shapes and is best used in conjunction with a finite element

model of the structure. This way the finite element model can act as a guide to what

frequency and shape is expected.

51

Figure 3.11 Mode shape for the first circumferential in-plane mode (7840

Hz). The y-axis represents the normalised displacement from the mean

position around the circumference. The value in this case was actually

accelerance, but, once normalised, it is equivalent to displacement.

3.6 Pad

The brake pad, displayed in Figure 3.12 (a), is made of two different materials. The

backing plate is steel and the friction material is a specialised composite material. A test

grid was applied to the backing plate for these experiments. The grid consisted of 25

points with the only direction of measurement being the out-of-plane direction (see

Figure 3.12 (b)). The excitation was applied with a shaker to point 8. Appendix A

displays the coordinate table for the 25 points.

52

(a) (b)

Figure 3.12 (a) Brake pad, (b) Brake pad experimental grid geometry (25

points). The direction of excitation and measurement in the out-of-plane

direction

Modal parameters for the brake pad are shown in table 3.4. Six modes were identified

from the averaged FRF data displayed in Figure 3.15 (a). The mode shapes are defined

in a manner similar to the bending modes of a rectangular plate. The pair of numbers m,

n represent the number of nodal lines in the lateral and longitudinal direction

respectively. The modal damping is high in comparison to the other brake system

components. This is due to the high structural damping of the friction material.

53

Table 3.4 Modal parameters of the brake pad.

Mode No. Mode Shape Freq (Hz) Damp factor

(%)

1 (2,0) 2340 0.96

2 (1,1) 5160 1.02

3 (3,0) 6200 1.34

4 tab only 7460 0.97

5 (2,1) 7760 0.84

6 (4,0) 10090 0.62

3.7 Caliper Housing

The caliper houses the piston within the brake system and is made of cast aluminium

(Figure 3.13 (a)). The grid consisted of 65 points covering the outer surface of the

housing as shown in figure 3.13 (b). Measurements were taken in all three coordinates

to try and capture as much of the vibration behaviour as possible. A comprehensive

table of the coordinates and measurement directions can be found in Appendix A.

(a) (b)

Figure 3.13 (a) Caliper housing. Also seen are the slide pins which were

not analysed as part of these experiments. (b) Caliper housing experimental

grid geometry (65 points).

The caliper housing was found to have 13 vibration modes as tabulated in Table 3.5.

No convenient scheme for identifying the mode shapes was conceived, so the modes are

simply identified by consecutive numbers. It can be seen that the modal damping

values for the caliper are mostly low owing to the light structural damping of

aluminium. The averaged FRF data is displayed in Figure 3.15 (b).

54

Table 3.5: Caliper housing modal parameters.

Mode

No.

Freq

(Hz)

Damp factor

(%)

Mode

No.

Freq

(Hz)

Damp factor

(%)

1 2810 0.21 8 8580 0.10

2 3120 1.20 9 9170 0.10

3 3790 0.43 10 10240 0.08

4 4420 0.19 11 10900 0.05

5 6310 0.15 12 11580 0.10

6 7500 0.07 13 12710 0.04

7 8330 0.07

3.8 Anchor Bracket

The anchor bracket is a cast iron component used for housing the caliper and pads in the

brake assembly. The anchor is shown in Figure 3.14 (a) and the test grid is shown in

Figure 3.15 (b). The grid features 36 points measuring in the local y and z directions.

Excitation was applied at point 1 in the y direction.

(a) (b)

Figure 3.14 (a) Anchor bracket, (b) Anchor Bracket experimental grid

geometry (36 points)

A total of 24 modes were identified from the averaged FRF data in Figure 3.15 (c) and

presented in Table 3.6. Again, as with the caliper, no convenient way of classifying the

modes exists so they are identified by number only. Cast iron has relatively high

structural damping resulting in damping factors higher than those found for the caliper.

55

Table 3.6 Anchor bracket modal Parameters.

Mode

No.

Freq

(Hz)

Damp factor

(%)

Mode

No.

Freq (Hz) Damp factor

(%)

1 990 3.4 13 7280 0.09

2 1020 0.56 14 8020 0.33

3 1200 2.3 15 8180 0.16

4 1710 0.29 16 8670 0.73

5 2810 0.17 17 8920 0.30

6 3170 0.19 18 9170 0.10

7 3260 0.17 19 9970 0.11

8 4340 0.19 20 10510 0.38

9 5290 0.32 21 10870 0.54

10 5940 0.20 22 11230 0.09

11 6610 0.10 23 12370 0.50

12 6830 0.15 24 12460 0.11

56

Figure 3.15 Spatially averaged FRF data. (a) Pad, (b) caliper housing, and

(c) anchor bracket.

3.9 Assembled Brake System

Testing of the assembled brake system was conducted including as much of the original

brake and suspension components as possible. Small feet were attached to the hub

carrier where the lower rear suspension arms would normally be attached. This allowed

the half shaft and hub assembly to be fitted, along with the rotor, anchor bracket and

caliper. The relationship between the components is shown in Figure 3.16.

57

Figure 3.16 Assembled brake system test configuration.

Three assembled test conditions were investigated:

1. Mounted rotor: Rotor only mounted on the carrier hub.

2. Assembled no pressure: Assembled condition but no line pressure.

3. Assembled 20 bar: Assembled condition with 20 bar of line pressure applied.

Measurements were taken using the majority of the rotor grid points. The caliper

partially obscured the rotor surface meaning only 324 measurements were taken instead

of 384. The measurements were confined to the rotor surface. With reference to the

rotor measurement points defined in Table 3.2, points 3 to 8 on lines n = 16 .. 25 were

omitted.

The experimental set up otherwise was the same as for the rotor. The excitation was

applied with the shaker at point 6 in the z (axial) direction. Again, the frequency range

of interest was 0 to 12.8 kHz.

Figure 3.17 shows the resulting averaged FRF values for the 3 assembled conditions as

well as for the rotor. The modes are numbered using the same number as for the rotor

and, where possible, the equivalent modes are identified. The modal parameters

obtained for these modes are tabulated in Table 3.7.

The mounted rotor condition tends to increase modal frequencies due to the additional

stiffness at the top hat. The change was least for the bending modes, where there is

little top hat deformation. Some of the modes where deformation was restricted mainly

58

to the top hat were not found at all due to the direct coupling of the top hat to the carrier

hub. Damping for most of the modes increased also, with the greatest effect visible in

the low frequency modes. A plot of damping values is displayed in Figure 3.18. Here

the change of damping for the lower frequency modes is plain to see.

The assembled no pressure condition continued the trends seen in the mounted rotor

case. Even though there was no line pressure, the pads were still resting against the

rotor surface. This explains the further increases in stiffness and damping that was seen

for this condition.

The greatest effects were seen when a typical braking pressure was applied. Now there

was a large increase in contact stiffness between the components, driving the

frequencies higher. Also, there was a marked increase in damping. The modal peaks

were significantly attenuated and had taken on a much wider profile. This is seen

clearly in Figure 3.18, where the damping ratio had become much higher for all modes.

These effects appear to be greater with the bending modes than the in-plane modes

where out of plane deformation is smaller.

The first bending mode is an exception in that its frequency was reduced in the

assembled 20 bar condition. This is most likely due to the caliper being located close to

an antinode for this mode and the mass loading effect was more pronounced than the

increased stiffening or damping.

59

(a)

-60

-30

0

30

60

0 3200 6400 9600 12800

1

2

3

4

56 7

8

9

10

11

1213

14

15 16

17

18

19

20

21

22 23

24

25 26

27

(b)

-60

-30

0

30

60

0 3200 6400 9600 12800

13

4

56

8

10

11 14

1516

18

19

22

24

26

Figure 3.17 Comparison of spatially averaged FRF plots. (a) free rotor, (b)

mounted rotor.

Acc

eler

ance

(dB

re:

1 m

/Ns2

)

Frequency (Hz)

Acc

eler

ance

(dB

re:

1m

/Ns2

)

Frequency (Hz)

60

(c)

-60

-30

0

30

60

0 3200 6400 9600 12800

13

4

56

8

10

11 14

1516

18

19

22

24

26

(d)

-60

-30

0

30

60

0 3200 6400 9600 12800

13

56

8

1011

14

16

18

19 22

24

Figure 3.17 Comparison of spatially averaged FRF plots. (c) assembled no

pressure, (d) assembled 20 bar.

Acc

eler

ance

(dB

: re

1 m

/Ns2

) A

ccel

eran

ce

(dB

re:

1 m

/Ns2

)

Frequency (Hz)

Frequency (Hz)

61

Table 3.7 Modal parameters for the free rotor and assembled conditions.

Free Rotor Mounted Rotor Assemb. - No P Assemb. - 20 Bar

Mode

No.

Mode

Shape

Freq

(Hz)

Damp

(%)

Freq

(Hz)

Damp

(%)

Freq

(Hz)

Damp

(%)

Freq

(Hz)

Damp

(%)

1 (2,0) 994 0.37 1030 0.74 1060 1.0 860 4.3

2 (0,1) 2010 0.19

3 (3,0) 2430 0.18 2460 0.60 2480 0.44 2690 1.7

4 (0,2) 2550 0.22 2680 0.74 2680 1.6

5 (1,1) 2690 0.15 3070 0.28 3100 0.69 3120 1.1

6 RI 2 2900 0.15 3150 0.36 3160 0.63 3220 1.5

7 (1,2) 3200 0.16

8 (4,0) 3800 0.14 3810 0.15 3870 0.63 4050 1.6

9 TH(2,1) 4230 0.15

10 RI 3 4630 0.22 4680 0.21 4720 0.45 4810 1.0

11 (5,0) 5290 0.13 5290 0.13 5330 0.36 5350 1.5

12 TH(1,1)+ RI1 5730 0.10

13 TH(2,2) 6770 0.12

14 (6,0) 6990 0.11 6980 0.09 7020 0.16 7020 0.61

15 RI 4 7120 0.12 7170 0.14 7180 0.20

16 CI 1 7840 0.17 7840 0.26 7830 0.35 7850 0.60

17 TH(0,2) 8130 0.11

18 (7,0) 8900 0.11 8890 0.09 8910 0.14 9020 1.3

19 TH(3,2) 9090 0.07 8480 0.28 8490 0.35 8460 0.82

20 TH(1,2) 9230 0.10

21 TH RI 0 9770 0.08

22 RI 5 10060 0.40 10120 0.15 10130 0.19 10210 0.60

23 TH RI3 TH(3,1) 10560 0.08

24 (8,0) 10990 0.11 11010 0.23 11050 0.28 11140 0.61

25 TH(5,1) 11620 0.08

26 CI2 12130 0.44 15.3 26 CI2 12130 0.44 15.3

27 (4,3) 12360 0.07

62

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0 2000 4000 6000 8000 10000 12000 14000

Frequency (Hz)

Dam

pin

g F

acto

r (%

)

Free Rotor

Mounted Rotor

Assembled - No P

Assembled - 20 Bar

Figure 3.18 Comparison of the damping factors for the free rotor and 3

assembled conditions.

3.10 Modal Analysis Summary

An experimental modal analysis has been conducted on a typical passenger car rear

brake system. The modal parameters of the rotor, pad, caliper housing and anchor

bracket have been determined using the STAR Modal commercial modal analysis

software package. It was found that these components had 27, 6, 13 and 24 vibration

modes respectively between 0 and 12.8 kHz.

Specific tests were also conducted using an impact hammer to implement a straight

forward testing procedure to find the circumferential in-plane modes of the rotor. This

procedure was successful in determining the 1st and 2

nd circumferential in-plane modes.

However, the procedure must be used with great care and its applicability as a stand-

alone technique is limited. It was not possible to detect in-plane modes by simply

measuring the phase relationship between the two faces of the rotor because any in-

plane deformation also induces significant bending. However, it may be possible to use

63

this technique to detect in-plane modes with a thicker rotor since the induced bending is

reduced.

The complete brake assembly was also analysed in 3 assembled conditions. The

assembled conditions tend to increase the stiffness of the rotor as well as increasing the

modal damping, resulting in an increase in modal frequencies and an attenuation of

modal peaks. The effects were most significant when pressure was applied to the

system.

Each of the components studied contributes to the generation of noise. The pad/rotor

interface provides the excitation and energy source for squeal noise, but all of the

components participate in system vibration and radiation of sound. Hence it is

important to characterise the dynamic responses of the components and assembly,

which have been the primary focus of this chapter.

3.11 Brake System Noise Evaluation

The evaluation of the noise performance of the brake system was conducted using a

brake noise dynamometer. A brake noise dynamometer is a test rig that can apply

representative brake loads to the brake system in a laboratory environment. The inertial

load, which represents what the brake would experience if fitted to the vehicle, is

applied either as simulated inertia with an electric motor, directly with flywheels, or a

combination of the two.

A variety of stops are applied including deceleration snubs and constant speed drag.

Initial brake temperature (IBT), line pressure and angular velocity are monitored during

each stop and sound pressure level (SPL) measurements are made by means of a

microphone. The semi-anechoic test chamber is 3m x 3m x 2.4m with the microphone

mounted 50 cm from the brake in direct line-of-sight from the brake. Ideally it would

be desirable to measure sound power due to the directivity of the high frequency sound,

but this is not possible for a transient event like brake squeal with a single microphone.

It is typical to run the brake through a matrix of speed, temperature and pressure

conditions to investigate noise performance in a wide range of operating conditions.

On the PBR’s Rubore drag-type brake noise dynamometer, test spec TS 576 was used

for the evaluation, which is based on an extended European AK noise matrix test. The

test procedure called for 1400 individual braking applications with pressures from 0 to

64

30 bars, and initial brake temperature from 50 to 300°C, except for a small fade section

where IBT is in excess of 350°C. The speed range represented includes 0 to 100 km/h,

although primarily it focuses on 0 to 10 km/h given the importance of low speed noise

performance.

During each one of the braking application, an A-weighted, peak hold, sound pressure

level (SPL) spectrum is recorded from within the test chamber with a ½” free-field

microphone. This signal is input for an FFT analyser, and the SPL spectrum is

calculated using the parameters shown in Table 3.8

Table 3.8 Parameters for determining the SPL from the test chamber during a

noise test.

Parameter Value

Frequency range 20 kHz Baseband

Lines 800

Resolution 25 Hz

Window Hanning

Overlap 50%

Averaging Peak hold

No. averages 25/sec

Once the stop is completed the peak value of the FFT is examined. If it exceeds

70dB(A) it is counted as a noisy stop and the stop number, SPL and frequency are

logged. Further stop data can also be logged if required including, but not limited to,

line pressure, IBT, initial speed and effective µ.

The baseline noise performance of the brake system is shown in Figure 3.19 (a) and (b).

Each spot marked on the figures represents a noisy brake stop event, with sound

pressure level plotted against frequency. The curves running through the background

marked R9 to R4 (lowest to highest) represent approximations to a commonly used

vehicle driver subjective scale where R10 is no noise and R1 is unbearable noise. For

this test procedure, R9 is considered acceptable, R8 is borderline, and R7 or worse is

unacceptable.

The primary frequencies of interest are 6-6.5 kHz, 7.5-8 kHz and 11.5-12 kHz. These

frequencies correspond with the problem frequencies reported from vehicle testing. The

65

noise occurrences above 12 kHz were not observed in vehicle testing and were not

considered to be amongst the main problem frequencies.

This illustrates the lack of repeatability between tests of a brake system with different

frequencies apparent in each respective result. The noise occurrence at 11.5-12.0 kHz

represents the greatest risk since it occurs significantly in both tests. The analysis of

11.5-12.0 kHz will be the main focus of the analysis presented in later chapter of this

thesis.

66

(a) Test result 1

(b) Test result 2

Figure 3.19 Baseline noise performance of the Ford Falcon AUII rear brake

system

67

Chapter 4

Finite Element Modal Analysis

4.1 Introduction

In chapter 3, the experimental testing of the individual brake system components and

the complete assembly to establish their dynamical properties was described along with

the testing of brake squeal on a brake noise dynamometer. The focus for the remainder

of this thesis is the analysis of brake squeal propensity using finite element analysis

(FEA).

The objective of this chapter is to describe the development of a FEA model of the

brake assembly; firstly through creating “tuned” individual component models,

followed by assembly into the complete assembled model with appropriate boundary

conditions.

Chapter 5 will describe how the FEA model is used for predicting brake squeal, chapter

6 on some additional analytical methods for investigating behaviour within the brake

system during squeal and chapter 7 describes a parametric study conducted with the

FEA models. Chapters 8 and 9 present the application of these techniques in an

industrial setting.

4.2 Application of FEA to Dynamical Problems

Finite element analysis (FEA) has become a standard analytical tool across many areas

of engineering. Together with the advance in modern computing capacity, FEA has

made it possible to analyse structures with a great deal of accuracy and detail beyond

the hand calculations that were possible in the past.

Essentially the FEA procedure is to convert a continuous physical structure into a

discrete model featuring a finite number of degrees-of-freedom (DOF). In mathematical

terms the structure is represented as a linear system of n equations, where n is the

number of DOF. These equations are treated as a matrix equation of degree n, and

powerful tools from matrix algebra are used for the solution procedures. Detailed

68

descriptions of the formulation of FEA models and solution techniques are described by

Bathe [Bathe, 1996].

The equations of motion for a structure can be expressed in the general form

[ ]{ } [ ]{ } [ ]{ } { }PuKuCuM =++ &&& (4.1)

where [M], [C] and [K] are the mass, damping and stiffness matrices, and { }u { }u& and

{ }u&& are the displacement, velocity and acceleration vectors respectively. The vector

{ }P represents the external loads applied to the structure.

The mass, damping and stiffness matrices represent the distribution of each throughout

the structure. Although it is possible to directly define each of these, they are usually

defined with the aid of a commercial FEA code or pre-processor.

While the stiffness and mass matrices are relatively easy to determine for most physical

structures, the exact nature and distribution of the damping is less simple. Various

methods are used for modelling damping, often purely out of mathematical

convenience, but for the moment only an undamped system will be considered.

Equation (4.1) can be simplified considerably when considering the free vibration of a

structure. No external load vector is required and the effects of damping are negligible

on both the modal frequencies and mode shapes for lightly damped structures such as

the brake components. Subsequently equation (4.1) can be simplified to

[ ]{ } [ ]{ } 0=+ uKuM && (4.2)

To determine the natural frequencies of the structure, sinusoidal motions are assumed,

with the vectors satisfying equation (4.2) of the form

{ } { } tj

iiieuu

ω= (4.3)

where { }iu is the ith

mode shape and ωi is the corresponding modal frequency.

Equation (4.2) can then be rewritten as a linear eigenvalue problem

[ ]{ } 02

=− ii uMK ω (4.4)

If the matrix on the left in equation (4.4) is invertible, then the only possible solution is

the trivial solution, which implies no motion. It follows that the matrix should be non-

invertible which is the case when

69

[ ] 0det 2=− MK ω (4.5)

The resulting eigenvalues are the squares of the modal frequencies of the structure and

the eigenvectors are the mode shapes, and provide a mathematical description of the

dynamical characteristics of the system. For an in depth treatment of the analytical

modal analysis, Inman (1994) or Bathe (1996) can be consulted.

For further dynamical analyses, such as forced response or transient response analysis,

the general form of the equation of motion is required (equation (4.1)). For system with

a large number of DOF, solution procedures will usually be modally based, that is, the

eigenvectors provide a subspace with a smaller number of DOF, thus greatly reducing

the time and space requirements of solving such problems.

4.3 Modelling Approach

MSC.Nastran, a commercially available FEA code, was chosen to analyse the brake

system because of its well developed dynamic analysis capabilities. As described in

chapter 1, brake squeal noise is associated with unstable vibration modes of the system.

The complex eigenvalue analysis technique forms the basis of the stability analysis for

predicting brake squeal in this thesis. MSC.Nastran is one of the few codes that has the

capability to perform complex eigenvalue analysis and has been used extensively for

this reason in brake noise prediction. Chapter 5 will describe in detail the complex

eigenvalue procedure and how it is applied to a brake system.

The FEA model, which was developed using the pre-processor MSC.Patran, is large and

complicated. In total there are in excess of 40000 DOF, six individual components and

seven different material properties. It is simply not possible to create an assembled

model of such a complex system in a single step and expect it to accurately reflect the

dynamical characteristics of the physical system.

A more sensible approach is to create the model in stages that allow comparison with

experimental results more readily. The basic procedure used is shown schematically in

Figure 4.1.

70

Figure 4.1 Flow chart for generating a validated FEA assembly

Two types of experimental results from chapter 4 are used for the comparative

purposes; experimentally determined modal parameters and driving-point frequency

response functions (FRFs).

4.4 Individual Component models

All of the components in the brake system utilised hexahedral (6-sided) brick elements

of the Nastran type CHEXA. While this element type can accommodate 8 to 20 nodes

(also called grid points in Nastran nomenclature), only 8-node elements were used in

the models, as shown in figure 4.2. Each node has only the three translational degrees

of freedom defined; rotational degrees of freedom are not included for 3D continuous

Tuned

component

models

Assembled

model

Component

mesh

Tuned

assembly

Geometry

clean-up

Component

test data

Component interfaces Apply boundary

conditions

Mesh

generation

Tune material

properties

Assembly test data

CAD data

71

elements. The detail of the geometry at the pad / rotor interface requires a relatively

high mesh density, sufficient for mesh convergence without the using higher order

elements.

Figure 4.2 8-node CHEXA element showing grid points 1 through 8.

The geometry for the individual components was supplied by the manufacturer as 3-D

solid models in IGES format. Considerable simplification and clean up of the models

were required before any meshing could be performed to remove unnecessary details.

For example, small fillet radii and other unnecessarily small features were removed

because, while they may contribute considerably to the local stress distribution, they

have negligible impact on the stiffness, and hence modal properties, of the components.

Coincident node meshing is required on all of the component interfaces that use linear

spring connection. This greatly increases the time taken to mesh a model since

automatic meshing, utilising tetrahedral elements, is not possible. At this time, no

commercially available pre-processor or mesher features hexahedral free meshing

capability, so each solid section needs to be iso-parameteric, that is, a solid with 6 faces.

Large sections of mesh with a uniform character can be meshed simultaneously, but

often it was required to create elements on an individual basis to account for detail in

the geometry. Three dimensional hexahedral meshing thus tends to be tedious, and

even relatively minor modifications to components can often be time consuming.

72

4.4.1 Brake rotor

Despite being the largest and dynamically the most significant component in the brake

system, the brake rotor is relatively simple to model for the most part. Its axisymmetric

geometry allowed the bulk of the hexahedral elements to be created as a revolved

extrusion from a radial plane of shell elements as shown in Figure 4.3. The final mesh,

which can be seen in Figure 4.4 (a), contained 4916 8-node brick elements and 7042

nodes.

Figure 4.3 Rotor mesh generation by revolving a cross sectional plane of

shell elements about the rotor rotation axis.

Elements in the region at the pad rotor interface were modified after the brake pad was

meshed to form the coincident node interface to allow the necessary connection to the

brake pads. The top-hat region of the rotor was also modified to account for the

mounting bolt holes.

(a) (b)

Figure 4.4 (a) Rotor mesh of 4916 8-node brick elements. (b) Zoomed in

detail of the pad interface region of the rotor

73

Establishing the correct material properties or “tuning” each component is important to

ensure the dynamical properties agree with those of the physical component. This is

particularly important for the rotor, which is made of grey cast iron. The elastic

modulus of cast irons varies from below 100 GPa through to the values close to that of

steel at approximately 200 GPa. Grey cast iron is particularly variable in properties

depending upon its carbon and, to a lesser degree, silicon content [Malosh, 1998,

Chatterley, 1999]. The type of casting method utilised for production can also vary the

final characteristics of the material, so the most reliable way to set the material

properties is to tune them to the experimentally determined modal properties.

Determining the density is straight forward; the volume of the rotor is known from the

CAD geometry, so the density is adjusted to the measured mass. Young’s modulus is

based largely on the dynamic response of the rotor. The flexural rigidity of the rotor

will scale in proportion to the elastic modulus, so modal frequencies will scale in

proportion to the square root of modulus. Poisson’s ratio is the last variable to be

adjusted, but it has a much smaller effect than the mass or the modulus.

The final material properties for the rotor, based on the tuning process, are listed in

Table 4.1. No damping was included when a FEA normal modal analysis was run. The

level of damping determined for the components in Chapter 3 has negligible impact on

the modal frequencies or mode shapes.

Table 4.1 Final material properties for the brake rotor FEA model.

Density (T/mm3) Modulus (MPa) Poisson’s Ratio

7.10 x 10-9

118 x 103

0.32

Table 4.2 displays a comparison of the free rotor modal frequencies from the FEA

model compared to those found in the experimental modal analysis. The mode shape

descriptions are the same as those described in section Chapter 3.5. It can be seen that

the FEA modal frequencies agree to within 5% except for the (0,1) mode.

An additional comparison was made between an experimentally measured driving point

FRF and the forced response from the FEA model as shown in Figure 4.5. The % error

values for the modal frequencies shown the Table 4.2 can be seen graphically. A value

of 0.2 % structural damping was applied to the FEA model which corresponds to an

average modal damping value found for the rotor in Chapter 3. The magnitudes of the

74

peaks in Figure 4.5 are in close agreement using this damping value. The shift in

frequency values increases at high frequencies since the limitations in mesh resolution

will become more pronounced as wavelengths decrease. However, the mesh resolution

of 12mm is adequate for the frequency range of interest, and experimental and FEA

modal frequencies agree to within 5%.

Note that an attempt was made to minimise the difference between FEA and

experimental model frequencies for all modes in Table 4.2. Certain modes with

prominent high peaks in Figure 4.5 are shifted uniformly in frequency in one direction.

However, considering some modes with smaller peaks, it can be seen that the final

tuned properties and subsequent frequency shift are a compromise between modes.

Table 4.2 Comparison of the experimental and FEA modal analysis of the free

brake rotor.

Mode Modal Frequency (Hz ) Error (%)

Shape Experimental FEA (2,0) 994 964 -2.9

(0,1) 2010 2214 10.1

(3,0) 2430 2388 -1.7

(0,2) 2440 2626 3.0

(1,1) 2690 2764 2.8

RI 2 2900 2943 1.4

(1,2) 3200 3234 1.0

(4,0) 3800 3734 -1.8

TH(2,1) 4230 4166 -1.4

RI 3 4630 4674 1.0

(5,0) 5290 5157 -2.5

(2,2) n/a 5167 n/a

(0,3) n/a 5748 n/a

TH(1,1) + RI 1 5730 5769 0.7

TH(3,1) n/a 7383 n/a

TH(2,2) 6770 6774 0.1

(6,0) 6990 6774 -3.1

RI 4 7120 7042 -1.1

CI 1 7840 7733 -1.4

TH(0,2) 8130 7862 -3.3

(7,0) 8900 8603 -3.3

TH(3,2) 9090 8687 -4.4

TH(4,1) n/a 8881 n/a

TH(1,2) 9230 9067 -1.8

TH RI 0 9770 9953 1.9

RI 5 10060 9894 -1.7

TH(2,2) n/a 10405 n/a

TH RI 3 TH(3,1) 10560 10554 -0.1

(8,0) 10990 10621 -3.4

Axial shear n/a 10836 n/a

TH(5,1) 11620 11623 0.0

CI 2 12130 11840 -2.4

(4,2) 12360 11834 -4.3

9,0 n/a 12795 n/a

75

-30

-20

-10

0

10

20

30

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Frequency (Hz)

Experiment

FEM

Figure 4.5 Comparison between experimental and FEA predicted driving

point FRF for the free rotor with 0.2% structural damping applied to the

FEA model.

4.4.2 Anchor Bracket

The anchor bracket was, along with the caliper housing, the more difficult component

for which to generate the FEA mesh. In total, 1434 8-node brick elements were used in

the final model which is shown in Figure 4.6. Symmetry exists about a mirror plane

parallel to the x-y plane which halves the modelling effort, but nevertheless the

geometry is complicated by the intersection of beam like structures of different cross

sectional profiles. The majority of the mesh was created in small sections, often

element by element.

dB

(R

e: 1

m/N

s2)

Frequency (Hz)

76

Figure 4.6 Final mesh for the anchor bracket featuring 1434 8-node brick

elements.

Selecting material properties for the anchor bracket, which is made of nodular cast iron,

proceeded in much the same way as for the rotor. The density is chosen such that the

FEA and measured masses agree, and Young’s modulus and Poisson’s ratio are adjusted

to match modal frequencies between FEA and the experimental modal analysis results.

Final material properties for the anchor bracket are shown in Table 4.3 and a

comparison of modal frequencies in Table 4.4. As was the case with the experimental

mode shapes, no convenient way of naming the modes was devised owing to the

complexity of the resulting motions.

Table 4.3 Final material properties for the anchor bracket FEA model.


7.40 x 10-9

165 x 103

0.30

The error between the modal frequencies predicted by FEA and the experimentally

measured values are within 3% with the exception of the mode predicted at 936 Hz.

77


anchor bracket below 10 kHz.

Modal Frequency (Hz ) Error (%)

Experimental FEA 990 988 -0.2

1020 936 -8.3

1710 1718 0.4

2810 2766 -1.6

3170 3215 1.4

3260 3213 -1.4

4340 4312 -0.6

5290 5283 -0.1

5940 5944 0.1

6610 6772 2.5

6830 6947 1.7

7280 7395 1.6

8020 8153 1.7

8180 8374 2.4

8920 9097 2.0

Figure 4.7 is a plot of predicted driving point FRF compared to the experimentally

measured case, and the agreement between the peaks is good in terms of frequency.

The FEA results do not have structural damping added since the effect on natural

frequencies is negligible at structural damping levels below 1%.

Figure 4.7 Comparison of experimental and FEA predicted driving point

FRF for the free anchor bracket.

0.01

0.1

1

10

100

1000

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Frequency (Hz)

Experiment

FEM

Acc

eler

ance

(m

/Ns2

)

Frequency (Hz)

78

4.4.3 Caliper Housing

Meshing the caliper housing also proved to be quite time consuming owing to the

complexity of the geometry. Again, as was the case with the anchor bracket, the caliper

is symmetric at the x-y plane, as shown in Figure 4.8. A total of 1648 8-node brick

elements were used in the structure.

Figure 4.8 Caliper mesh consisting of 1648 8-node brick elements.

The caliper was manufactured from aluminium and the final material properties for the

FEA model was again tuned to the experimental modal frequencies. Material properties

are displayed in Table 4.5 and a modal frequency comparison in Table 4.6. Figure 4.9

is the driving point FRF comparison. It can be seen from the FRF that there is a

considerable discrepancy in the behaviour around modal peaks between the

experimental and FEA results. The FEA model was undamped, which significantly

changes the response near the modal peaks. The modal frequency agreement is still

quite acceptable.

Table 4.5 Final material properties for the caliper FEA model.


2.65 x 10-9

68 x 103

0.33

79


caliper housing.


Experimental FEA 2810 2763 -1.7

3120 -

3790 3674 -3.1

4420 4218 -4.6

6310 6277 -0.5

7500 7277 -3.0

- 7512 -

8330 8428 1.2

8580 8614 0.4

9170 9373 2.2

10240 10229 -0.1

10900 11063 1.5

11580 11586 0.0

12710 12859 1.2

0.01

0.1

1

10

100

1000

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Frequency (Hz)

Experiment

FEM

Figure 4.9 Driving point FRF comparison between experimental and FEA

results for the free caliper.

4.4.4. Brake Pad

The brake pad is comprised of two sections that are bonded together; the backing plate

and the lining. The inner and outer pads are shown in Figure 4.10. Meshing of each

pad was relatively straight forward since the majority could be extruded from a single

Acc

eler

ance

(m

/Ns2

)

Frequency (Hz)

80

plane of shell elements. However, consideration needed to be given to the coincident

nodes that are required where the pads interface with the piston for the inner pad, and

caliper for the outer pad. Some elements at these interface points are modified for this

reason, so the pads are almost, but not exactly, identical to each other.

(a) (b)

Figure 4.10 (a) Outer pad, (b) inner pad. Both pads feature 430 8-node

brick elements.

Each section of each pad has considerably different material properties. The backing

plates are steel, and the material properties correspond to mild steel. However, the

composition of the linings is not exactly known. Linings are typically made from a

dozen or more constituents, the exact specifications of which are not released by the

manufacturer. Typical lining properties are shown in Table 4.7 as quoted by Eriksson

and Jacobson (2000).

Table 4.7 Example composition of a friction material

Material %

Matrix 19

Fibres 30

Fillers 8

Lubricants 38

Others 5

Determining the properties for lining was done by first setting the backing plate to be

steel, and then modifying the lining properties until agreement with measured modal

frequencies was found. The assumption of standard steel properties for the backing

plate is valid since the properties of mild steels are well known and do not vary. To

complicate matters, the linings are not isotropic in general. Further, the modulus and

damping levels of the lining material are not constant, but can be a function of

81

frequency and amplitude. While some test data was available on the response

characteristics of the material, using isotropic material was found to give acceptable

results. Table 4.8 is a summary of the material properties used for the analysis of the

pad.

Table 4.8 Summary of material properties for the brake pad


Lining 2.6 x 10-9

5.0 x 103 0.3

Backing plate 7.86 x 10-9

207 x 103 0.3

Modal frequency comparisons are displayed in Table 4.9. It can be seen that the

isotropic material properties provide good agreement with the experimentally

determined modal frequencies.

Table 4.9 Comparison of the experimental and FEA modal frequencies of the

brake pad.


Experimental FEA

2340 2375 1.5

5160 4998 -3.1

6200 6385 3.0

7760 7536 -2.9

10090 10281 1.9

Figure 4.11 is a comparison of the driving point FRF between the experimental results

and the FEA model with isotropic material properties for the lining. Structural damping

of 1% was added to the FEA model and corresponds to the approximate levels of modal

damping found for the pads in Chapter 3.

82

0.1

1

10

100

1000

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Frequency (Hz)

Experiment

FEM

Figure 4.11 Driving point FRF comparison between the brake pad

experimental and FEA results. The FEA model had isotropic lining material

properties and 1% structural damping added.

4.5 Mounted Rotor

In chapter 3, modal testing of the brake system included testing the brake rotor mounted

on its rotor. Figure 3.16 shows the experimental test configuration.

In this study the rotor was fully constrained in all DOFs at the mounting bolt holes as

shown in Figure 4.12. Fully constraining the rotor at the hub also greatly reduced the

modelling time since the models of the hub and suspension components do not have to

be generated and validated.

Acc

eler

ance

(m

/Ns2

)

Frequency (Hz)

83

(a) (b)

Figure 4.12 (a) The bolt holes on the rotor were fully constrained in all

DOFs. (b) Close up detail of the constraints shown as a wireframe.

Figure 4.13 displays the driving point FRF for the mounted rotor predicted by FEA

compared to the FRF obtained form measuring the rotor mounted on its hub. The

responses agree to a similar level to those of the plane rotor in terms of modal

frequencies and magnitudes of the peaks.

0.001

0.01

0.1

1

10

100

1000

10000

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Experiment

FEM

Figure 4.13 Comparison of driving point FRF for the mounted rotor below

10 kHz.

Frequency (Hz)

Acc

eler

ance

(m

/Ns2

)

84

4.6 Assembled Models

4.6.1 Component Interfaces

Many different methods can be used in FEA modelling for the contact between

components. These methods are (in order of simplest to most complex):

• Merged nodes

• Multi-point constraints

• Linear springs

• Contact elements

Merged nodes are shared between neighbouring elements so that the components are

effectively fused together. While this is simple, it does not allow for any type of

interfacial property, such as contact stiffness or damping to be applied.

Contact surfaces are by far the most advanced method of coupling components together.

Sophisticated contact surface models exist that allow some level of motion between

components, which include specifying normal and tangential contact stiffness. Here a

more realistic representation of what is a highly non-linear feature can be applied.

Unfortunately, it requires considerable computational effort compared to the other three

methods. Chapter 8 will focus on using contact elements at interfaces with the

commercial FEA code HKS Abaqus.

Components within the finite element model of the brake system within this thesis were

connected using the two remaining methods, multi-point constraints and linear springs,

as described below.

4.6.1.1 Multi-point constraints

Multi-point constraints (MPCs) are general tool for directly coupling selected degrees-

of-freedom (DOFs) in FEA models. They are often used in places where discontinuities

in geometry or FEA mesh. They lend themselves particularly well to connecting

rotating or sliding components.

The brake assembly consists of two areas where components slide within one another.

Firstly, the piston slides within the caliper housing bore when brake line pressure is

applied and forces the inner brake pad against the brake rotor. Secondly, the caliper

85

guide pins, which are rigidly connected to the caliper housing, slide within the anchor

bracket to allow force distribution to the outer pad. In both of these cases two of the

three translational DOF are coupled, as are all the rotational DOF. An MPC is a

straight-forward manner of implementing these sliding connections. The procedure is

illustrated with the piston / caliper interface.

An annular ring of piston nodes is defined as a rigid body element (Nastran element

RBE2). The ring then behaves as a rigid body and has its motion linked to a node at its

centre as shown in Figure 4.14.

A rigid body ring is defined for the caliper which is also linked to a node at its centre.

The two central nodes are directly coupled in all DOF except the sliding direction. A

spring can be used to couple the sliding direction. This allows the piston to slide freely

within the caliper, but provides load transfer when displacement in any other direction

occurs.

Figure 4.14: Implementation of an MPC for creating a sliding connection.

The annular ring on the piston is connected to a central node, as is an

annular ring from the caliper. The central nodes are directly coupled in all

DOFs.

MPCS are also defined to link the caliper slide pins to the anchor bracket in the same

manner. All DOFs are directly coupled except for the sliding in the axial direction.

The slide pins are also linked to the caliper via MPCs. However, in this case a solidly

bolted joint is simulated rather than a sliding one. Hence the MPCs are coupled in all

DOFs with no provision for relative axial displacement.

Nodes on

housing tied to

central node

Nodes on

piston tied to

central node

Central nose tied

in all DOF except

sliding direction

86

The main disadvantage of using MPC connections is they can introduce some additional

stiffness into the components in the regions where MPC is applied. For example, the

annular ring MPCs that are applied to the caliper bore and anchor bracket guide pin

holes provide more rigidity that that of the component alone. However, these regions

are relatively small and would have little impact on overall deformation.

4.6.1.2 Linear springs

The bulk of the contact within the FEA model utilised linear springs. They are a simple

and effective way of allowing load to be transferred between two connected components

and to prevent penetration. While contact itself is a highly non-linear process, within

the small displacements that occur at an interface during vibration, a linear contact

stiffness has been assumed. Figure 4.15 shows the interface between two adjacent

components. A requirement for linear spring connection is coincident node meshing

between components.

Figure 4.15: Schematic diagram of nodes on adjacent components

connected with linear springs. Note: the gap between the components is

illustrative only, and the nodes are coincident within the FEA model.

Selection of an appropriate contact stiffness is a critical part of using linear springs to

connect components. No completely satisfactory method exists to experimentally

determine what this stiffness should be, although a variety of methods have been used.

Liles (1989) used transfer frequency response functions (FRFs) between components to

assess the energy transfer between components. Park et al (2001) used experimental

penetration measurements and material properties to estimate the contact stiffness.

87

Nack (2002) has used surface asperity measurements but that has proved difficult to

implement in practice.

The method used in this study is based on the contact stiffness formulation used by

Ansys 5.7 for initial contact stiffness (Ansys User’s Manual, 2000). The calculation is

based on the material properties and geometry of the elements that are being connected.

The relationship between the (softer) element involved and the contact stiffness is given

by

∑=

n

i i

isj

V

A

n

Kfk

2

(4.6)

where Ai and Vi are the contact area and volume of the ith

element surrounding the jth

node on the contact surface, K is the bulk modulus of the material and fs is a penalty

factor, usually in the range of .01 to 10.

Alternatively, for a friction interface with largely uniform element contact face area and

element volume, equation 4.6 can be expressed as

avg

avg

avg

s

V

A

n

Kfk

2

= (4.7)

where navg is the average number of nodes per element , Aavg is the average elemental

contact face area, and Vavg is the average elemental volume. This sets the contact

stiffness of all the springs across the whole interface rather than setting each one

individually.

The connection between the components, using linear springs and MPCs is shown

schematically in Figure 4.16.

88

Figure 4.16 Component interface connection schematic

4.6.1.3 Linear vs. non-linear static analysis

Nastran provides a variety of solution sequences covering a range of static and dynamic

analyses. Static solutions are available in the both linear and non-linear forms. The

fundamental difference between these types of solutions is that the stiffness matrix may

change as loads are incrementally applied during a non-linear analysis, whereas it

remains constant throughout a linear analysis.

Non-linear solutions are usually used wherever significant non-linearity is present in the

physical system being modelled. Three sources of non-linearity can be applied using

FEA models.

1. Large deformations

2. Material non-linearity

3. Contact

Each of these will cause the stiffness matrix in a finite element model to change from

one load condition to the next during an analysis. It is usual to apply loads in small

increments and iteratively check that the internal system forces and displacements

provide an equilibrium solution given the current external loads to within some small

error or convergence criteria.

Large deformation non-linearity follows from displacements that are large enough to

cause significant variation in geometrical relationships between nodes. Material non-

linearity is caused by non-constant elastic modulus of the material, such as when the

yield stress level is exceeded. Finally, contact causes changes at the structural

Disc

rotor

Outer

pad

Inner

pad

Piston Caliper

housing

Anchor

bracket

Ground

Ground

Springs MPC

MPC Springs Springs Springs

Springs

89

boundaries or component interfaces as loads are added and removed from the system.

Each of these may cause changes in the stiffness matrix.

A linear solution, on the other hand, assumes that no such changes are occurring in the

structure as the load is applied. The geometry, material properties and boundary

conditions do not change with load. Hence there is no requirement for the system

matrix to be updated as the load is applied. The full load is applied and the final state of

the structure can be determined immediately.

The computational cost of performing a non-linear solution can be much greater than a

linear solution. It is therefore preferable to use linear solutions where ever possible

after considering the non-linearity to be expected from the structure under the expected

load condition.

In the case of a brake system, large deformations and material non-linearity are not

significant concerns due to the small structural deformations and low stress levels

respectively. However, contact between components can make the system extremely

non-linear as the contact between components changes.

When using spring elements to connect components, little reason exists for using a non-

linear solution. Unlike contact elements, where nodes at the contact interface can

become (or cease to be) active intermittently through a load step, linear springs remain

active all of the time. Hence the inherent non-linearity of a contact problem cannot be

captured with a single step approach.

However, an iterative linear static procedure was used for the static analysis to capture

the final contact state more accurately. Successive static solutions followed by the

removal of springs under tension effectively achieve a non-linear solution using a model

with purely linear elements.

4.1.6.4 Spring interface tuning

Scope exists for tuning of springs connections in the case of the direct spring

connections. Table 4.10 shows a summary of the baseline model set up for the

assembly.

90

Table 4.10: Interface tuning parameters for baseline model

Interface K

(MPa)

Aavg

(mm2)

Vavg

(mm3)

navg fs k

(N/mm)

Rotor / lining 4200 20 97 1.1 0.1 2 x 103

Inner Pad BP /

piston 68000 35 308 2.0 10 1.4 x 10

6

Outer Pad BP /

housing 67000 35 77 1.7 10 6.0 x 10

6

Pad BP / anchor (y) 138000 13 19 4.0 10 3.0 x 106

Pad BP / anchor (z) 138000 20 40 2.0 10 7.0 x 106

A key step in using springs at a connection interface is to examine the spring state under

a static load condition. When an interface is under load, different areas may be forced

into compression and other areas may be under tension. Springs behave such that any

relative motion between the “ends” of the springs (the node points that define the

element) results in a reactive spring force. While this is indeed the aim of the spring

under compressive loading, it makes no physical sense for a contact spring to support a

tensile load.

To ensure a realistic contact interaction, the springs which are under tensile loading

must be removed. Hence the static solution is an iterative procedure. All springs are

activated initially and a static solution obtained. Springs that are found to in tension are

then removed and a static solution is again obtained. This is repeated until an

equilibrium solution is obtained with no springs under tension. This is then considered

to be the base static state around which the dynamic perturbation analyses are

performed.

A comparison was made between the preloaded static state of the friction interface from

the Nastran model to an Abaqus non-linear solution utilising contact elements, shown in

Figure 4.17. Each contact interface was examined to compare the areas where contact

and separation were occurring. The areas differ slightly, but the overall agreement is

reasonable.

The penalty factors, as given by fs in equation (4.6), were all set to 0.1 for the initial

comparison. It was found that excessive penetration was found at most of the interfaces

with metal to metal contact, ie all excluding the rotor / lining interface, and a penalty

factor of 10 offered better agreement. For the crucial rotor / lining interface a penalty

91

factor of 0.1 was found to be sufficient. The final contact stiffness values used for all of

the contact interfaces are listed in Table 4.10.

(a) Inner pad

(b) Outer pad

Figure 4.17 Overlays of the Nastran static solution and Abaqus non-linear

solution contact areas. The areas in red represent the footprint of the pad

from the Abaqus solution. The blue dots represent active nodes from the

Nastran solution

4.6.1.5 Friction interface

The interface that connects the brake rotor to the brake pad linings provides the friction

that is required in the braking operation. It also supplies the energy that drives the

structural vibration of the system. This will be discussed in greater detail in later

chapters. However, the basic method of implementing the friction coupling terms is

now presented.

92

Figure 4.18: Basic friction force diagram

Consider the simple two-degree-of-freedom model shown in Figure 4.18. A mass m is

in sliding contact with a conveyor belt; the contact stiffness is k and friction coefficient

µ. The friction force fF and the normal force fN act in the x and y directions respectively.

kyfN −= (4.8)

kyff NF µµ =−= (4.9)

Note that the friction force is one directional in the sense that normal forces result in

frictional forces, but not vice versa. This provides a complication for implementation

into a finite element model since it will result in unsymmetric terms in the system’s

stiffness matrix. Indeed, an unsymmetric stiffness matrix is a necessary condition for

the system to become unstable, the topic of chapter 5.

The friction coupling terms are input directly into the stiffness matrix in the FEA model

via Nastran direct matrix input (DMIG) terms. DMIG entires allow for the direct entry

of stiffness (or mass) to the system matrices in addition to those that are generated by

the finite element code from the node and element input data.

Unfortunately matrices solved in a linear static analysis in Nastran have to be

symmetric. The friction coupling (unsymmetric) terms cannot be added to the stiffness

matrix for the linear solution. Hence the iterative linear solution establishes the normal

contact distributions at the component interfaces including the effects of rotation.

The most significant effect of rotation and the friction force is to force the pad backplate

into contact with the trailing abutment in the rotation direction. It also removes any

fN

x

y

k

fF

m

93

contact for the leading abutment in the rotational direction. Hence, the contact springs

between the leading abutment and the pad backplate that act in the rotation direction

were removed manually.

4.7 Summary

The development of a large scale FEA model has been presented. A step by step

process was followed beginning working with individual component models, flowing

through to the completed assembly.

Initially the CAD geometry and imported, cleaned-up and meshed. Experimental modal

analysis results were used to “tune” the material properties of the components. Finally

the individual components were assembled and boundary conditions and component

interfaces were created. The final assembly forms the basis of the complex eigenvalue

stability analysis that follows in Chapter 5.

94

95

Chapter 5

Prediction of Unstable Modes

5.1 Introduction

In Chapter 4, the development of a validated model of a brake system using finite

element analysis (FEA) is described. The focus of this chapter is to use that model for

predicting the brake system’s brake squeal propensity.

It is commonly accepted amongst researchers that a brake system is in an unstable

vibration mode when it undergoes squeal. Various types of instability have been

described in the literature as described in Chapter 2.

• Stick-slip

• Negative slope µ vs v

• Sprag-slip

• Mode coupling

In principle, the prediction of brake squeal involves evaluating the stability of the brake

system. Two common methods for assessing the stability of a large scale finite element

model can be determined by observing the evolution in the time domain or by

determining the location of poles in the complex eigenvalue analysis.

The approach used in this thesis is complex eigenvalue analysis. In an undamped

system without friction all of the system modes will be normal modes. Addition of

frictional forces into the stiffness matrix of the FEA model allows the possibility of

modes to become coupled and to form a stable/unstable mode pair. This is a larger

scale application of the mode coupling instability described in section 2.3.3.

96

If the response of an unstable system is plotted in the time domain, it is seen to diverge

and the amplitude of vibration continues to grow without bound. In reality, a brake

system in self excited vibration is non linear and exhibits limit cycle behaviour. An

initial perturbation about the base state initially grows exponentially before dissipative

effects balance the friction work being converted into vibrational energy.

Figure 5.1 Single degree-of-freedom system with viscous damping

As described in many dynamics texts such as Inman (1994), a single degree of freedom

system, such as the one shown in Figure 5.1, with mass m, viscous damping c, and

spring stiffness k, the free vibration is given by the 2nd

order differential equation

0=++ kxxcxm &&& (5.1)

The time response can be expressed as a product of sinusoid and an exponential decay

)sin()( φωζω

+=−

tAetx d

t (5.2)

where

km

c

2=ζ

(5.3)

is the damping ratio,

m

k=ω

(5.4)

is the undamped natural frequency, and

21 ζωω −=d (5.5)

is the damped natural frequency. The amplitude A and phase φ are determined from the

initial conditions of the system.

m

c

k

x(t)

97

Examining the two parts to this solution reveals that the sinusoid simply represents

simple harmonic motion at a frequency of ωd. The exponential term provides the decay

and is given by the product ωζ. For solutions of the form given by equation 5.2, ζ can

take any value between 0 (undamped) and 1 (critically damped). Once ζ exceeds the

critical damping ratio of 1 it becomes overdamped and oscillatory motion does not

occur; the mass simply moves toward rest at the equilibrium position. A thorough

description of system behaviour can be found in Inman (1994).

It is a mathematical possibility for the damping ratio to take on a negative value, that is,

the damping coefficient c has a negative value. This implies that the damping force

applied on the mass does not resist the motion of the mass, but rather it acts in the same

direction as travel and reinforces the displacement from equilibrium. If this is the case,

the vibration amplitude grows with increasing time and the system is said to be

negatively damped. As was the case with positive damping, the damping ratio can take

on an absolute value between 0 and 1. If the magnitude of the damping ratio exceeds 1,

the oscillatory motion does not occur and the mass diverges exponentially away from

the equilibrium position.

Figure 5.2 shows these results graphically. Analytically, an unstable brake system

exhibits the negatively underdamped (Figure 5.2(c)) behaviour at the point when

instability occurs.

Analysis in the time domain is one of the methods that can be used to determine system

stability. However, for a large scale FEA model with tens of thousands of degrees-of-

freedom (DOF), considerable computer resources are required. Regardless of the type

of the solution method employed for an analysis, each time increment requires an

equilibrium solution to be obtained. Hence it can become prohibitively expensive

computationally if a long time period with many time increments is analysed.

98

Time

Dis

pla

cem

ent

Time

Dis

pla

cem

ent

(a) Positive underdamped (0 ≤ ζ < 1) (b) Positive overdamped (ζ > 1)

Time

Dis

pla

cem

ent

Time

Dis

pla

cem

ent

(c) Negative underdamped (-1 < ζ ≤ 0) (d) Negative overdamped (ζ <-1)

Time

Dis

pla

cem

ent

s

(e) Undamped

Figure 5.2: Response of an unstable SDOF system for various levels of

damping.

An alternative method is analysing system parameters in the frequency domain.

Equations of motion are solved as a matrix equation for their eigenvalues and

eigenvectors. The eigenvalues provide not only modal frequencies, but also the level of

damping within a mode. Negatively damped (unstable) modes can be readily identified

as potential squeal modes of the brake assembly. This chapter will present the complex

eigenvalue analysis and how it is implemented on a FEA model of the brake system.

99

5.2 Complex Eigenvalue Analysis

The equations of motion for a vibrating system in free vibration can be represented as

the following matrix equation

[ ]{ } [ ]{ } [ ]{ } 0=++ uKuCuM &&& (5.6)

where [M] is the mass matrix, [C] is the damping matrix, [K] is the stiffness matrix and

{ }u { }u& and { }u&& is the displacement vector and its time derivatives respectively. The

system matrices are established using FEA as detailed in Chapter 4.

This eigenvalue problem is solved to yield eigenvalues and eigenvectors that are

complex valued, that is, they have a real and an imaginary part. The eigenvalues take

the form

idiii jωωζλ ±−= (5.7)

where the real partii

ωζ is the damping of the complex mode and the imaginary part id

ω

is the frequency of the complex mode.

Plotting the eigenvalues, as shown in Figure 5.3, on the complex plane provides

information about the system’s modal parameters:

• Modal frequency – determined from the location along the imaginary axis.

• Modal damping – determined from the location along the real axis. The real part

of the eigenvalue is a product of the damping ratio and the undamped natural

frequency.

The stability of the mode is determined by simply examining the real part of the

eigenvalues. An eigenvalue with a positive real part is unstable, and appears on the

right hand side of the complex plane as indicated on Figure 5.3.

100

Figure 5.3 Location of an eigenvalue on the complex plane. Its position

provides the level of damping as well as the frequency

Each of the complex eigenvalues also is accompanied by a complex eigenvector or

mode shape. Complex modes differ from normal modes in that the displacements do

not occur perfectly in-phase or out-of-phase, so a real number is not sufficient for

expressing the displacement of any given node. The motion must now be described by

complex number which contains the relative phase of the node as well as the magnitude

of its displacement from the equilibrium position.

5.3 Implementation for a Brake System

Complex eigenvalue analysis forms one step of the larger modelling problem. The

modelling process consists of the following steps

1. Generation of a validated FEA model

2. Static analysis

3. Apply friction coupling

4. Complex eigenvalue analysis

Each of these will be described in the following sections.

x ζω

21 ςω −

cos-1

ζ

Re

Im

Unstable region

101

5.3.1 Generation of a validated FEA model

Individual component models and an assembled model of the system components were

created from the 3-D CAD models. Experimental modal testing results were used to

“tune” the material properties of the component models and validate the assembled

model, as described in the chapter 4.

5.3.2 Static analysis

A static load analysis was performed to obtain the system’s state under typical braking

pressure. Particular attention was directed to the contact conditions at the pad/rotor

interface. Linear springs are used between coincident nodes on the pad and rotor

surface. Static analyses were run iteratively with the static state of the system updated

in a quasi non-linear analysis.

At the end of the static analysis, the brake system is in a base static state that occurs

during a typical braking occurrence. The dynamic analysis that follows is a linear

perturbation around this base state.

5.3.3 Friction Model

Prior to the complex eigenvalue analysis, the friction coupling terms were inserted into

the system stiffness matrix [K]. A simple friction law of the following form was used

for the friction force with reference to Figure 4.18

kyff NF µµ =−= (4.9 repeated)

where fF is the friction force, fN is the normal force at the friction interface, µ is the

coefficient of friction, k is the contact stiffness and y is the displacement in the friction

interface normal direction.

This same Coulomb friction model is applied to all nodes at the pad / rotor friction

interface. With reference to Figure 5.4, the resulting friction force is a function of the

relative displacement of the node pair in the surface normal direction given by

)( )1( yiiyjx uukf+

−= µ (5.8)

102

where ui is the displacements of the ith

node.

Figure 5.4 Coincident node mesh at the pad / rotor interface. Note that it is

shown with a gap for clarity.

On applying this friction model brings with a number of assumptions:

1. The coefficient of friction is constant, and as such it is independent of normal

force and sliding velocity.

2. The sliding velocity is greater than the tangential vibration velocity, resulting in

a friction force that always acts in the same direction with no possibility of

reversing.

The friction forces expressed in equation (5.8) provides a forcing function for Equation

(5.6), which can be rearranged as

[ ]{ } [ ]{ } [ ]{ } { } [ ]{ }uKFuKuCuM ff ==++ &&& (5.9)

where [Ff] is the friction force and [Kf] is the friction stiffness matrix. Upon further

rearrangement, equation (5.9) can be written in homogeneous form

[ ]{ } [ ]{ } [ ]{ } 0=−++ uKKuCuM f&&& (5.10)

[Kf] couples forces in the friction interface normal direction to the tangential direction,

but not vice versa, and as a result is unsymmetric. Hence the new system stiffness

matrix [K - Kf], which includes the friction stiffness, is also unsymmetric. Indeed, it is a

necessary, but not sufficient, condition for the system expressed in equation (5.10) to

kj

ui+1

ui

x

y

103

have an unsymmetric stiffness matrix for unstable modes (complex eigenvalues with

positive real parts) to exist (Liles, 1989).

5.3.4 Implementation with MSC.Nastran

MSC.Nastran is a commercially available finite element analysis code which is

particularly well suited to performing dynamic analysis. It was one of a very limited

number of finite element codes that can perform a complex eigenvalue analysis. A

preliminary study using ANSYS, another commercially code available during the study,

revealed less than realistic results. MSC.Nastran v2001 was used exclusively for the

analysis presented in Chapters 4 through 7 of this thesis.

Unfortunately, the implementation of the contact interface using the spring elements

must be performed manually. Each contact pair must not only have the spring element

inserted, but four matrix entries are required in the system stiffness matrix to represent

the friction forces. So each pair of nodes that are in contact requires the following 5

additional lines to be added to the input deck. For example, the following five lines

define the friction contact between nodes 40001 and 50001.

CELAS2, 210001, 0.20E+04, 40001, 1, 50001, 1

DMIG, FSTIF, 50001, 1, 50001, 3, -.20E+04

DMIG, FSTIF, 40001, 1, 50001, 3, 0.20E+04

DMIG, FSTIF, 50001, 1, 40001, 3, 0.20E+04

DMIG, FSTIF, 40001, 1, 40001, 3, -.20E+04

The first line specifies an elastic spring (CELAS2) connection in the global 1 direction

which is the friction interface normal direction. The following 4 lines couple

displacements in the 1 direction to forces in the 3 (tangential at friction surface)

direction. Each friction stiffness line adds a stiffness entry to the matrix FSTIF with

MSC.Nastran direct matrix input (DMIG) cards. An example input deck can be found

in Appendix C.

A further addition to the MSC.Nastran input deck is to perform matrix arithmetic using

the Direct Matrix Abstraction Program (DMAP) facility. The following DMAP “alter”,

ie, a user written subroutine, introduces the coefficient of friction via the parameter

FRIC. Effectively, the matrix defined earlier as FSTIF is multiplied by FRIC. This

allows flexibility in solving for different friction levels since only one parameter needs

to be changed to re-run the analysis for another friction level.

104

COMPILE GMA SOUIN=MSCSOU NOLIST NOREF

ALTER 'MTRXIN' $

ADD K2PP, /K2PPX/V,Y,FRIC=(1.0,0.0) $

EQUIVX K2PPX/K2PP/-1 $

ENDALTER

…

…

K2PP = FSTIF

PARAM, FRIC, .5

Defining stiffness entries for each pair of nodes at the friction interface via manual text

input is not practical. Not only is it tedious in the extreme, but it is also prone to error.

To aid in the definition of the friction surface, a FORTRAN 77 code bdfread was

written to connect the coincident node pairs with contact springs and to apply the

friction coupling terms. bdfread was also adapted to apply contact springs at the other

spring interfaces within the FEA model. While these other interfaces required only the

spring element to be inserted without friction terms, automating the process help avoid

errors in the modelling process.

The code reads an existing MSC.Nastran input deck generated by MSC.Patran to find

coincident nodes with a specified search area. Coincidence is determined by measuring

the distance between nodes and if they fall below a specified tolerance (eg, 0.01 mm)

then the appropriate CELAS2 and DMIG entries are written to a text file. Each contact

interface is determined from an individual search area to ensure that the correct spring

directions and contact stiffnesses are used. Once all of the coincident nodes are found

for each interface and the data has been written to a text output file, then the

MSC.Nastran input deck is re-generated with the bdfread output appended from within

MSC.Patran. Appendix D contains the bdfread source code.

5.4 Brake System Analysis

The Complex Lanczos method was used within MSC.Nastran to solve the complex

eigenvalue problem on the brake system FEA model shown in Figure 5.5 (Nastran

Advanced Dynamics Guide, 2002). This solution scheme is particularly efficient for

larger models, and the default choice for large models with MSC.Nastran.

105

Figure 5.5 The assembled FEA model

This model of the brake system is simplified in that it does not include rotational effects.

However, as discussed in Section 3.11, the primary focus of the noise evaluation is low

speed applies where rotational effects are not significant.

The 108 eigenvalues extracted between zero and 12 kHz for the base brake system with

µ = 0.5 are plotted on the complex plane in Figure 5.6. In the baseline case no other

sources of damping are specified. All of the modes have zero damping (lie on the

imaginary axis) except where pairs of modes have become coupled and formed a stable

/ unstable pair. These result in the eigenvalues that occur in conjugate pairs that are

symmetrically located about the imaginary axis. In this case seven unstable modes can

be seen.

An alternative and somewhat more insightful way to express these results is to plot

damping vs frequency as shown in Figure 5.7. Essentially the same information is

available as for the eigenvalue plot, but now the frequency and damping levels can be

directly read off. Note that the seven modes with positive real parts now appear with

negative damping values.

106

Complex Eigenvalues of Base Brake System

0

10000

20000

30000

40000

50000

60000

70000

80000

-250 -200 -150 -100 -50 0 50 100 150 200 250

Real Part of Eigenvalue

Ima

gin

ary

Pa

rt o

f E

ige

nv

alu

e

Figure 5.6 108 eigenvalues extracted from the baseline brake system

plotted on the complex plane. The 7 unstable mode pairs appear as

symmetric pair about the imaginary axis.

Damping vs Frequncy of Base Brake System

-2.0%

-1.5%

-1.0%

-0.5%

0.0%

0.5%

1.0%

1.5%

2.0%

0 2000 4000 6000 8000 10000 12000 14000

Frequency (Hz)

Da

mp

ing

Figure 5.7 Damping factor vs. frequency for the base brake system analysis.

The seven unstable modes, their respective eigenvalues and levels of negative damping

are summarised in Table 5.1.

107

Table 5.1 Summary of unstable modes for the baseline brake system

Mode No. Eigenvalue Frequency (Hz) Damping ratio (%)

27 83.07 + 20871j 3322 -.398

43 219.0 +29283j 4661 -.748

54 27.29 + 37123j 5908 -.074

73 51.84 + 51951j 8268 -.099

79 64.46 + 55776j 8877 -.116

81 49.78 + 56426j 8981 -.088

105 215.4 + 74521j 11860 -.289

While there is no direct proportionality between squeal propensity and the level of

negative damping, it has been suggested that higher values tend to be associated with

modes that are most likely to squeal (Liles, 1989). Furthermore, the sound level emitted

from a squealing brake is also not indicated by negative damping level.

The unstable modes can be compared to the baseline noise performance of the system

reported in section 3.11. Unstable modes have been found that correspond closely to

the frequency ranges of interest at 6-6.5 kHz, 7.5-8 kHz and 11.5-12 kHz. Also

predicted were modes with significant levels of negative damping at 3.3 kHz and 4.7

kHz. The former has not occurred at all in the dynamometer screening, but some

occurrences at 4.5-5 kHz were observed. Several other unstable modes were also

predicted that have not presented a concern in testing. This is a significant issue with

the over predictive nature of complex eigenvalue in that not all of the modes predicted

will cause an actual squeal concern.

Mode shapes for the 7 unstable modes from Table 5.1 are displayed in Figures 5.8 to

5.14. Note the guide pins have been omitted from the assemblies since the highly

magnified deformation level leads to an unnatural appearance of the pins penetrating

adjacent components.

Mode 27 at 3322 Hz (Figure 5.8) shows considerable deformation across the many of

the main components of the system. The rotor appears to have a of (4,0) bending mode

present and the housing features predominately a single mode with bending across the

caliper bridge.

Mode 43 at 4661 Hz (Figure 5.9) features a (6.0) rotor bending mode. While some

deformation can be seen in the housing and bracket, no particular modes can be readily

identified by visual inspection.

108

Rotor deformation is prominent in mode 54 at 5908 Hz (Figure 5.10), but several modes

appear to be participating.

Mode 73 at 8268 Hz (Figure 5.11) has the majority of the deformation occurring in the

caliper assembly itself. Visually it is difficult to identify component modes, but it is

clear the housing, bracket and outer pad are deforming significantly.

High order rotor deformation and bracket deformations feature in both mode 79 at 8877

Hz (Figure 5.12) and mode 81 at 8981 Hz (Figure 5.13). Again, the specific component

modes are somewhat difficult to identify by visual inspection.

The final unstable mode, mode 105 at 11860 Hz (Figure 5.14) shows limited rotor

deformation, but considerable deformation of the caliper housing. The rotor modes that

are present appear to include a 2nd

in-plane tangential mode.

The visual inspection of the mode shapes of unstable modes is a limited method for

assess the behaviour of the assembly. Some insight can be gained into which

components are active and component modes may be participating, but it is clear a

deeper level of analysis is requires to thoroughly characterise the behaviour of the brake

system.

109

Figure 5.8 Mode 27 at 3322 Hz. Guide pins are not displayed.


110



111



112

Figure 5.14 Mode 105 at 11860Hz. Guide pins are not displayed.

5.5 Summary

The FEA assembly model was used to predict system unstable modes using the complex

eigenvalue procedure, of which there were 7. The assembled model did not include

rotational effects since it was reasoned they would be negligible at low speeds.

Correlation to the observed noise occurrences was good in the sense that unstable

modes corresponding to the observed squeal concerns were found. However, a number

of other spurious modes were also predicted.

Visual inspection of the unstable mode shapes highlighted the difficulty in identifying

active components and the participating mode shapes. The further analyses presented in

Chapter 6 aims to overcome this difficulty and allow unstable system modes to be

characterised at a deeper level.

113

Chapter 6

Numerical Methods for

Assessing Brake Squeal Propensity

6.1 Introduction

In Chapter 5, the use of complex eigenvalue analysis for determining is described if a

brake system is unstable and could potentially lead to a squeal problem. Unfortunately,

while complex eigenvalue analysis allows the assessment of system stability, it offers

no insight into what is occurring internally within the system or what modifications

should be undertaken to increase stability. Visual inspection of unstable system mode

shapes did not offer a sufficient level of understanding of the brake system behaviour.

In this chapter, three different analysis techniques will be introduced for the specific

purpose of understanding how the components within a system are behaving and which

components might be suitable for modification. The three methods are

1. Strain energy – where the distribution of strain energy within the brake system is

calculated.

2. Feed-in energy - where the amount of vibrational energy being added to the

brake system from each friction interface is calculated.

3. Modal participation – where the correlation between individual component

modes under free boundary conditions and within the coupled system is

calculated using the modal assurance criterion (MAC).

While each of these methods provides insight into the behaviour of the brake system

components, each has individual strengths and weaknesses that require them to work in

a complimentary manner. Strain energy distribution indicates how active the

component is in the overall vibration of the system, feed-in energy shows how much

energy is being added from each brake pad during a vibration cycle, and MAC helps

identify which individual component modes are participating in the system’s vibration

motion. This way it is possible to not only see which components should be modified,

but also which vibration modes need to be considered in addressing the stability

114

problem. Figure 6.1 provides a schematic overview of the full system analysis

including the methods to identify treatment options.

Modal testing verify

Experimental

testing by test drive/noise

FE Structural

Model

Complex

Eigenvalue Analysis

Methods to

identify components for

treatment

Unstable mode?

Yes

Evaluate

effectiveness of treatment

verify

Figure 6.1 Schematic of approach to reduce brake squeal propensity.

6.2 Strain energy

The vibration of a structure involves cyclic displacement about some equilibrium

position. Consider the simple case of an undamped single degree-of-freedom (SODF)

system with mass m and spring stiffness k as shown in Figure 6.2.

Figure 6.2 Undamped single degree-of-freedom system of mass m and

spring stiffness k.

In free vibration the governing equation of motion is given by

0=+ kxxm && (6.1)

which has a solution of the form

m

k

x(t)

115

)sin()( φω += tAtx (6.2)

where

m

k=ω (6.3)

is the undamped natural frequency. Both the amplitude A and phase φ depend on the

initial conditions.

During one cycle, the energy within the system is transferred between strain and kinetic

energy. The strain energy is the energy stored in the spring as elastic potential energy

given by

2

2

1kxU = (6.4)

where x is the displacement of the mass from its equilibrium position. The kinetic

energy is given by

2

2

1xmT &= (6.5)

where x& is the velocity of mass. Since there are no dissipative effects present, the total

energy of the system remains constant, and is the sum of the strain and kinetic energy.

That is

constantTU =+ (6.6)

At the equilibrium position (x = 0) the velocity is at its maximum and all of the

system’s energy is kinetic energy. At maximum displacement (x = A) the velocity is

zero and all of the system’s energy is strain energy.

Figure 6.3. Two degree-of-freedom system.

m1

k1

x1(t)

m2

k2

x2(t)

116

Figure 6.3 displays an undamped two degree-of-freedom system described by the

following matrix equation

[ ]{ } [ ]{ } { }0=+ xKxM && (6.7)

where

[ ]

=

2

1

0

0

m

mM (6.8)

is the mass matrix,

[ ]

−

−+=

22

221

kk

kkkK (6.9)

is the stiffness matrix, and{ }x , { }x& and { }x&& are the displacement, velocity and

acceleration vectors respectively.

Similar to the SDOF case, the resulting motions are sinusoidal. However, in this case

there are two possible natural frequencies and two possible displacement vectors. From

the modal point of view the system has two modes; two modal frequencies and two

mode shapes.

The strain and kinetic energies can be calculated in much the same way as the SDOF

case, except they are now vector equations. The strain energy is given by

{ } [ ]{ }xKxUT

2

1= (6.10)

and the kinetic energy by

{ } [ ]{ }xMxTT

&&2

1= (6.11)

where the susperscript T represent the vector transpose.

6.2.1 Viscous Work

The undamped SDOF system described in the preceding section does not exactly

correspond to a physical system. Equation (6.1) predicts oscillatory behaviour that will

continue at a constant amplitude indefinitely. In reality, physical systems are damped

and the amplitude of oscillation decays over time. Eventually, regardless of how small

the damping is, the system will come to rest.

117

The most commonly used, and a mathematically convenient, model for damping is

viscous damping, as seen in Figure 6.4.

Figure 6.4 Viscous damped SDOF system.

The equation of motion for free vibration now becomes

0=++ kxxcxm &&& (6.12)

where c is the damping coefficient. A thorough discussion on viscous damping and the

characterisation of system response can be found in references such as Inman [Inman,

1994] or other elementary dynamics texts.

The underdamped case is of particular interest for studying vibration problems, and

results in motion of the form

)sin()( φωζω

+=−

tAetx d

t (6.13)

where

km

c

2=ζ (6.14)

is the damping ratio which can take any value between 0 and 1, and

21 ζωω −=d (6.15)

is the damped natural frequency. Again, as with the undamped case, A and φ are

determined by the initial conditions.

The roots to equation (6.12) are complex valued and can be written in the form

djωζωλ ±−=2,1 (6.16)

The real part of the root gives the exponent for the damping term in equation (6.13) and

the complex part gives the damped natural frequency.

m

c

k

x(t)

118

A damper is the dissipative element in the system shown in Figure 6.4. The energy that

is dissipated during one period T can be calculated by considering the work done by the

damper

∫∫++

==Tt

t

x

xviscous dtxcdxxcW

1

1

1

1

22

&&π

(6.17)

If a system is driven at ωd, the steady state response is

)sin()( θω += tAtx d (6.18)

and the viscous work becomes

22

0

2)]sin([ AcdxtAcW ddviscous

d

ωπθωωπ

=+= ∫ (6.19)

Substituting equation (6.14) into equation (6.19) and recognising the real and imaginary

parts of equation (6.16), the energy dissipated from the system as viscous work can be

expressed as

2)Im()Re(2 AmWviscous λλπ−= (6.20)

Equation (20) applies to a single degree of freedom, but it can be readily extended to

any number of degrees of freedom if the displacement vectors and mass matrix are

known. In matrix form the viscous work for the ith

mode of a system is

{ } [ ]{ }i

T

iiiviscousi uMuW )Im()Re(2 λλπ−= (6.21)

6.3 Feed-in Energy

The energy source in a squealing brake system is the friction interface between the pad

and the rotor. The exact nature of the friction coupling is not simple and a single,

general model does not exist. However, a number of different models have been used to

analyse the instability that arises in squeal modes as detailed chapter 2.

Mode-coupling type instability has widely been used because it lends itself readily to

use in conjunction with finite element analysis (FEA). Here the friction interface is

modelled as coincident node pairs distributed across the pad / rotor interface. The

friction coupling is a pseudo forcing function whose magnitude depends on the contact

119

stiffness and relative displacement of each coincident node pair. This linear friction

model predicts that no energy is added to the system when normal (stable) modes are

present.

When the system enters an unstable system mode net vibrational energy is added to the

system during the vibration cycle. This energy, called feed-in energy, is added to the

system due to the relative displacement of the friction interface over the course of a

vibration cycle. The formulation presented here is essentially the same as Guan and

Huang [Guan and Huang, 2003].

Consider the simple two-degree-of-freedom model shown in Figure 6.5. A mass m is in

sliding contact with a conveyor belt; the contact stiffness is k and friction coefficient µ.

The friction force fF and the normal force fN act in the x and y directions respectively.

kyf N −= (6.22)

kyff NF µµ =−= (6.23)

Figure 6.5 A simple 2DOF system with sliding friction.

The work done on the mass during one cycle of oscillation is found by integrating the

force in the x direction with respect to distance

∫=cycle

Fx dxfE (6.24)

which, can be expressed in terms of time

fN

x

y

k

fF

120

dtdt

dxfE Fx ∫=

ωπ2

0 (6.25)

where ω is the frequency of oscillation. The motions in each coordinate are

)sin()( xx tAtx θω += (6.26)

)sin()( yy tAty θω += (6.27)

where Ai and θi are the amplitudes and phases respectively. Substituting equations

(6.26) and (6.27) into equation (6.25), the work done on the mass by the friction force is

dtttAkAE yxyxx ∫ ++=ωπ

θωθωωµ2

0)sin()cos( (6.28)

)sin( xyyxx AAkE θθπµ −= (6.29)

Figure 6.6 Phase plot of y vs. x displacement for the 2DOF system in Figure

6.5 with 0 < (θy - θx) < 90°.

Further insight into the importance of the phase of each displacement can be found by

examining Figure 6.6. Here the displacements in the y-direction lead the x-

displacement, which results in a clockwise trajectory. The inclination of the major axis

of the ellipse depends on the relative magnitudes of the x and y amplitudes and the

amount the y leads x.

x

y

1

2 Ax

Ay

-Ax

-Ay

3

4

121

The work added to the mass by friction can be found by integrating around the ellipse

points 1 through 4.

dxfdxfdxfdxfE FFFFx ∫∫∫∫ +++=1

4

4

3

3

2

2

1 (6.30)

Recall from equation (6.23) that the friction force is proportional to the displacement in

the y-direction. Hence, by inspection of the integrals 1 and 3 in equation (6.30) the

friction force and the direction of travel are opposite and the friction force is dissipative.

Integrals 2 and 4 have the force and motion in the same direction and add feed-in energy

to the mass.

After summing the areas, the feed-in energy is proportional to the area within the

ellipse. It is also worth noting that this result is general and does not depend upon the

equilibrium position of the mass.

For cases where the y displacement lags the x displacement, the trajectory reverses

direction and the feed-in energy can become dissipative overall. A summary of the

different cases of trajectories is given in Figure 6.7. Cases (a), (b) and (c) feature

frictional energy being added to the mass, (d) the energy is dissipated, and (e) and (f)

are normal modes.

Equation (6.29) can be extended to incorporate the feed-in energy from the difference in

motion between two coincident points on the friction interface rather than a single point

as shown in Figure 6.5. The feed-in energy from the relative motion of two points a and

b is

)sin( abxabyabyabxabx AAkE θθπµ −= (6.31)

where iabi baA −= is the amplitude of the difference between nodes a and b in the i

th

coordinate and iabi ba )arg( −=θ the phase of the difference.

122

(a) (b)

(c) (d)

(e) (f)

Figure 6.7 Phase plots for the system from Figure 6.5. (a) 0 < (θy - θx) <

90°, (b) 90° < (θy - θx) < 180°, (c) (θy - θx) = 90°, (d) -90° < (θy - θx) < 0, (e)

(θy - θx) = 0, (f) (θy - θx) = 180°.

6.3.1 Feed-in Energy vs. Viscous Work

An unstable system mode has an eigenvalue with a positive real part. From equation

(6.16) it is clear that the damping ratio of such a mode will be negative. So rather than

decaying, the system will exhibit an exponential growth. This doesn’t reflect physical

reality since the system will settle into a limit cycle vibration, but analytically the

system is negatively damped and will continue to grow.

x

y

x

y

x

y

x

y

x

y

x

y

123

In section 6.2.1 the energy dissipated by a viscous damper was discussed.

Mathematically, the same analysis can be applied to a system with negative damping, as

that with positive damping. The viscous “work” in this case takes on a negative value,

and represents energy that is being added to the system rather than dissipated by the

system.

The energy being added to the system can also be considered by calculating the feed-in

energy. Numerically the same value is attained. The energy being added to the system

by the friction interface is exactly the same as the energy being gained by the negative

viscous work.

6.4 Modal Participation with Modal Assurance Criterion

Modal assurance criterion (MAC) calculation provides a quantitative method of

assessing which component modes are significant in the overall system vibration. The

MAC, as the name implies, is used for comparing the correlation between modes shape

and often used for comparing numerical and experimental mode shapes. In the present

study it will be used for comparing FEA results together.

Mathematically, the MAC is a dot product between two vectors, {u1} and {u2}

normalised by their magnitudes

{ } { }

{ } { }( ){ } { }( )*

22

*

11

2*

21

21 ),(uuuu

uuuuMAC

TT

T

= (6.32)

where T indicates the vector transpose and

* indicates the complex conjugate of a

complex valued vector. The comparison of a real vector and a complex vector presents

no difficulty since the conjugate of a real vector is simply the vector itself.

MAC data was used primarily for determining which component modes are active in an

unstable system mode. For example, a system mode is often identified by the mode

shape of the rotor. The MAC allows the identification of the rotor mode shape more

easily because it does not rely on visual observation and a qualitative judgement. A

value of 1 indicates that two modes are identical (although they could be scaled) and a

value of 0 indicates that there is no correlation between the modes at all.

124

6.5 Example 4DOF System

To further highlight the application of the feed-in energy and viscous work, and the

relationship between the two, the 4DOF system in Figure 6.8 will be analysed. This

system is an extension of the simple 2DOF system analysed by Hoffman and colleagues

[Hoffman, 2002].

Figure 6.8 4DOF system with sliding friction.

The global stiffness is assembled using the direct approach from five elemental stiffness

matrices and the friction stiffness matrix. The elemental stiffness matrix for each

element is determined as follows.

Consider a spring aligned with the x’ axis of a rotated coordinate system as shown in

Figure 6.9 (a). The rotation angle between the x’-y’ and global x-y coordinate systems

is θ. The linear spring with a stiffness of k obeys Hooke’s law

lkf ∆= (6.33)

where k is a constant that defines the force f required for a displacement ∆l of the spring.

y2, fy2

k5

k3

k2 k1

m2

m1

α1

α2 y1, fy1

x1, fx1

x2, fx2

125

∆y

x

y’

x’

y

θ

k

∆x’

∆y’

∆y θ

(a) (b)

Figure 6.9 (a) A spring element undergoing a displacement of ∆y at one

end, and (b) displacement decomposed into the x’-y’ coordinate frame.

Note that the infinitesimal displacement does not change the angle θ.

Since the spring is aligned to the x’ axis, Hooke’s law for the spring is

'' xkf x ∆= (6.34)

where fx’ and ∆x’ are the forces and displacements in the x’ direction.

The spring undergoes an infinitesimal displacement of ∆y in the global y direction.

Referring to figure 6.9 (b), the displacement can be resolved into the x’ and y’

components

θ∆∆ sin' yx = (6.35)

θ∆∆ cos' yy = (6.36)

The spring force can be expressed in terms of fx and fy, which are the force components

in the global x-y directions

θcos'xx ff = (6.37)

θsin'xy ff = (6.38)

Substituting equations (6.34) and (6.35) into (6.37) and evaluating while ∆x = 0 yields

ykfxx ∆θθ

∆cossin

0=

= (6.39)

126

Similarly for fy

ykfxy ∆θ

∆

2

0sin=

= (6.40)

Equations (6.39) and (6.40) provide the relationship between a displacement in the y

direction and spring forces in the global x-y coordinate systems.

The same process can be followed to obtain the relationship between displacements in

the x direction and the x-y forces while ∆y = 0

xkfyx ∆θ

∆

2

0cos=

= (6.41)

xkfyy ∆θθ

∆cossin

0=

= (6.42)

Summing the forces in each direction gives the total force is each coordinate

ykxkfffxxyxx ∆θθ∆θ

∆∆cossincos2

00+=+=

== (6.43)

ykxkfffyxyyy ∆θ∆θθ

∆∆

2

00sincossin +=+=

== (6.44)

Expressing equations (6.43) and (6.44) as a matrix equation yields

=

y

x

kk

kk

f

f

y

x

∆

∆

θθθ

θθθ

2

2

sincossin

cossincos (6.45)

which is Hooke’s law in matrix form. The matrix in the equation is the elemental

stiffness matrix for the spring element.

For the 4DOF system shown in Figure 6.8, the global stiffness matrix will be four times

four in dimension since four-degrees-of-freedom are included the matrix. Each

elemental stiffness matrix can also be expressed in four by four form

=

2

2

1

1

44434241

34333231

24232221

14131211

2

2

1

1

y

x

y

x

kkkk

kkkk

kkkk

kkkk

f

f

f

f

y

x

y

x

(6.46)

The elemental stiffness matrices for the five spring elements in Figure 6.8 are

127

[ ]

=

0000

0000

00sincossin

00cossincos

1

2

1111

1111

2

1

1

ααα

ααα

kk

kk

k (6.47)

[ ]

=

0000

0000

00sincossin

00cossincos

2

2

2222

2222

2

2

2

ααα

ααα

kk

kk

k (6.48)

[ ]

−

−=

33

33

3

00

0000

00

0000

kk

kkk (6.49)

[ ]

=

0000

000

0000

0000

4

4k

k (6.50)

[ ]

=

5

5

000

0000

0000

0000

k

k (6.51)

The coupling at the friction interface is added by considering the spring stiffness as

shown in Figure 6.10. Constant contact is assumed, and also vibration velocity in the x-

direction is much smaller than the sliding velocity. Hence the friction force direction

does not reverse direction and is a function of displacements in the y-direction only.

)( 12311 yykff yx −=−= µµ (6.52)

)( 21322 yykff yx −=−= µµ (6.53)

This linear model of the friction force considers only small perturbations about the

equilibrium position.

128

Figure 6.10 Contact stiffness and forces at the friction interface.

Equations (6.52) and (6.53) can also be expressed as a friction stiffness matrix in terms

of the four degrees-of-freedom

[ ]

−

−

=

0000

00

`0000

00

33

33

kk

kk

k fµµ

µµ

(6.54)

The global stiff matrix is assembled as a summation of the elemental stiffness matrices

and the friction matrix

[ ] [ ] [ ] [ ] [ ] [ ] [ ]fkkkkkkK +++++= 54321 (6.55)

Note that all of the elemental stiffness matrices are symmetric. However, the friction

stiffness matrix is unsymmetric. It follows that the global stiffness matrix is also

unsymmetric.

Table 6.1 is a summary of the system parameters for the example analysis. The values

for k1, k2, k3, α1 and α2 are taken from the example given by Hoffman et al, in their

analysis of a 2DOF system [Hoffman, 2002].

y2, fy2

k3

y1, fy1 x1, fx1

x2, fx2

129

Table 6.1 Parameters for example 4DOF system

Parameter Value Parameter Value

k1 1786.)32(3

2≈− m1 2

k2 488.2)32(3

2≈+ m2 3

k3 333.13

4≈ α1 30°

k4 1 α2 150°

k5 1 µ 0.75

Inserting the values from Table 6.1 into equations (6.47) through (6.55), the global

stiffness and mass matrices are

[ ]

−

−

−=

333.20333.10

1110

333.1021

1002

K (6.56)

[ ]

=

3000

0300

0020

0002

M (6.57)

Note that the global stiffness matrix in equation (6.56) is unsymmetric which follows

from friction coupling terms.

Recall equation (6.7), which is the equation of motion for a multi-degree-of-freedom

system in matrix form

[ ]{ } [ ]{ } { }0=+ xKxM && (6.7)

The matrices [K] and [M] are inserted into equation (6.7) and a complex eigenvalue

extraction is performed to yield the eigenvalues in Table 6.2 and modes shapes in Table

6.3.

130

Table 6.2 Complex eigenvalues from example analysis

Mode Eigenvalue Frequency (Hz) Damping ratio

1 0.4665j .07424 0

2 0.5774j .09189 0

3 0.1109 + 1.137j .18093 -.09707

4 0.1109 – 1.137j .18093 .09707

Table 6.3 Mode shapes from example analysis

Mode x1 y1 x2 y2

1 -0.2169 0.4278 -0.2545 0.3394

2 0 0 0.5774 0

3 0.3929 -0.1364+0.3487j -0.1651+0.1486j 0.2201-0.1981j

4 0.3929 -0.1364-0.3487j -0.1651-0.1486j 0.2201+0.1981j

It can be seen that modes one and two are normal modes. The eigenvalue is purely

imaginary indicating the modes are undamped and the displacements for each degree of

freedom are purely real. By inspection, equation (6.21) yields zero viscous work since

the real parts of the eigenvalues are zero. Also by inspection, equation (6.31) shows the

feed-in energy is also zero since the relative phase terms between degrees of freedom is

either 0° or 180°, and the sine of π, or multiples thereof, are equal to zero.

Modes three and four are clearly complex modes, formed by the coupling of two

adjacent modes. Here the real parts of the eigenvalues indicate that the modes are

negatively damped and therefore unstable. The modal displacement vectors involve

complex motion, that is, there is a phase difference between degrees-of-freedom.

It is also apparent that the mode shapes of mode three and mode four are complex

conjugates of each other. Mode 3 is examined in mode detail and summarised in Table

6.4 and Table 6.5.

131

Table 6.4 Mode three data from example 4DOF analysis

DOF Real Imag Amplitude Phase (Degrees)

x1 0.3929 0 0.3929 0

y1 -0.1364 0.3487 0.3744 111.4

x2 -0.1651 0.1486 0.2221 138.0

y2 0.2201 -0.1981 0.2961 -41.99

Table 6.5 Mode three data from example analysis

x1-x2 y1-y2 A12x A12y θ12x(deg) θ12y(deg)

.5579-.1486j -.3565+.5468j 0.5774 0.6527 -14.91 123.1

The negative viscous work is calculated by using the amplitudes of the mode from

Table 6.4, mass matrix from equation (6.57), and the real and imaginary parts of the

eigenvalue from Table 6.2 and inserting them in equation (6.21).

{ } [ ]{ }i

T

iiiviscous uMuW )Im()Re(2 λλπ−= (6.21)

−=

2961.

2221.

3744.

3929.

3000

0300

0020

0002

2961.

2221.

3744.

3929.

)137.1)(1109(.2

T

viscousW π

792.0−=viscousW (6.58)

The feed-in energy is calculated by taking the stiffness and friction coefficient values

from Table 6.1 together with the relative magnitude and phase values from Table 6.5

and inserting them into equation (6.31).

)sin( 1212121212 xyyxx AAkE θθπµ −= (6.31)

])91.14[]1.123sin([)6527)(.5774)(.333.1)(75(.12 −−= πxE

792.012 =xE (6.59)

132

The equivalent viscous work, which represents the dissipation of energy in the system,

is negative because energy is being added via the feed-in energy; hence, they have the

same magnitude but opposite sign.

6.6 Analysis of Numerical Model

6.6.1 Model Description and Unstable Modes

A numerical model of a brake system was developed using finite element analysis

(FEA) as detailed in Chapters 4 and 5.

Figure 5.5 shows the complete brake system model, which was made from 9858 8-node

brick elements and 14813 nodes. The friction coupling for the model is provided by the

friction stiffness matrix formulated in much the same way as described for the 4DOF

model from section 6.5.

Analysis of the FEA model included varying some system parameter and observing

whether system modes become coupled and unstable. In particular, increasing the

coefficient of friction (µ) of the brake pad material tends to drive many system modes

toward instability, as well documented in the literature. This present analysis used

coefficients of friction of 0.5 and less, which corresponds to the range typically found in

actual brake systems. Setting the coefficient of friction to 0.5 leads to the maximum

number of unstable modes. Table 6.6 is a summary of the eigenvalues, frequencies and

negative damping present for the unstable modes from Chapter 5.

Table 6.6 Summary of unstable modes up to 12 kHz for the assembled brake

system, µ = 0.5.

Mode No. Eigenvalue Frequency (Hz) Damping ratio (%) 27 83.07 + 20871j 3322 -.398

43 219.0 +29283j 4661 -.748

54 27.29 + 37123j 5908 -.074

73 51.84 + 51951j 8268 -.099

79 64.46 + 55776j 8877 -.116

81 49.78 + 56426j 8981 -.088

105 215.4 + 74521j 11860 -.289

It is evident that modes shown in table 6.6 are unstable due to the positive real parts of

the eigenvalues and consequent negative damping. It is possible to track the motion of

the eigenvalues and the effect on the stability by the variation of some system

133

parameter. Figure 6.11 shows the change in eigenvalues for modes 104 and 105 due to

an increase in the coefficient of friction. The modes are driven closer in frequency and

become coupled at some critical value of µ between 0.35 and 0.4. The coupled mode

pair consists of a stable and unstable mode pair, as seen from the real parts of the each

eigenvalue.

Figure 6.11 Change in eigenvalues of modes 104 and 105 due to an

increase in the coefficient of friction.

6.6.2 Feed-in Energy for a Numerical Model

Unfortunately, while the real part of the eigenvalue indicates the level of instability in

the system mode, it does not provide any information about which part of the system

could be modified to stabilise the mode.

A greater understanding of the behaviour of various components within the system can

be obtained by calculating the feed-in energy for each unstable mode. The actual

magnitude can be determined by either direct calculation of the feed-in energy by

equation (6.31), or by calculating the equivalent viscous work by equation (6.21).

However, direct calculation of the feed-in energy was used because it can indicate

which brake pad contributes more energy, and this can then be used as a guide to

determine which part of the system could be modified.

11830

11850

11870

11890

11910

11930

-300 -200 -100 0 100 200 300

Real part

Fre

q (

Hz)

Mode 104

Mode 105

µ = 0.4µ = 0.35

µ = 0.3

µ = 0.25

µ = 0.2

µ = 0

µ = 0.45 µ = 0.5

µ = 0

µ = 0.2

µ = 0.25

µ = 0.3

µ = 0.35

µ = 0.4µ = 0.45µ = 0.5

134

Table 6.7 shows the feed-in energy for the unstable modes of the brake assembly. It can

be seen that the feed-in energy contribution is relatively evenly distributed to both pads

for modes 27, 43 and 79. For the remaining modes, a large difference exists between the

contributions between the pads. For modes 54, 81 and 105 the feed-in contribution for

one pad is negative, indicating that the friction work for the pad is acting to damp the

mode rather than feeding the instability. The potential for a dissipative effect to occur

was examined in section 6.3.

Table 6.7 Summary of feed-in energy for the unstable modes of the assembled

brake system, µ = 0.5.

Feed-in Energy (J) Mode

No.

Freq

(Hz) Inner

Pad

Outer

Pad Total

27 3322 0.32 0.47 0.80

43 4661 2.41 3.03 5.44

54 5908 0.60 -0.07 0.53

73 8268 1.49 0.03 1.52

79 8877 2.37 1.54 3.91

81 8981 7.44 -2.28 5.16

105 11860 -1.87 13.97 12.10

A larger contribution to the feed-in energy indicates a larger relative motion between

the pad and the disc. This relative motion between the pad and rotor interface is

dependant not only on pad and rotor deformation, but also on any deformation in other

supporting components. A large differential between the feed-in contributions between

the pads indicates which pad, and its interface with the anchor or caliper, needs to be

modified.

Feed-in energy can also be used to examine which regions of the pad/rotor interface are

driving the mode. Figure 6.12 and 6.13 plot the feed-in energy across the pad surfaces

for mode 27 and 105 respectively. For mode 27, small contributions from each pad can

be seen. The inner pad is favoured at one end and the outer favours the lower radial

edge. For mode 105 the central region of the outer pad is predominately driving the

instability. It is also apparent how much more feed-in energy is present in mode 105

compared to 27. Plotting the feed-in energy in this way can help to understand the

importance of regions of the pads and help evaluate potential changes to pad shape.

135

(a)

(b)

Figure 6.12 Feed-in energy across the pad surfaces for mode 27. (a) inner

pad, (b) outer pad.

mJ

mJ

136

(a)

(b)

Figure 6.13 Feed-in energy across the pad surfaces for mode 105. (a) inner

pad, (b) outer pad.

mJ

mJ

137

6.6.3 Strain Energy for a Numerical Model

To further assist the determination of which components need to be modified in

controlling an unstable mode, strain energy was calculated for each component in a

system mode. This allows individual components that are particularly active in an

unstable mode to be identified.

In section 6.2, the strain energy of a SDOF and 2DOF system were discussed. Equation

(6.10) is a matrix equation describing the total strain energy of a 2DOF system, but it

applies equally for any multi-degree-of-freedom (MDOF) system. However, some

additional calculation is required if the strain energy is to be broken down component

by component.

Initially, the strain energy of each individual element is calculated from the following

equation

{ } [ ]{ }*

2

1xkxU i

T

i = (6.60)

where [ki] is the elemental stiffness matrix, {x} is the displacement vector, and *

indicates the complex conjugate.

The elements for each component are identified and the strain energies are summed

∑=

i

icomponent UU (6.61)

This process is repeated for each component in the assembly.

Table 6.8 displays the strain energy distributions as a percentage for the unstable modes

of the brake system with µ = 0.5. The larger components are identified individually,

with the remaining components grouped as “other.”

A significant issue to address before the strain energy of each component is considered

is the difference in size and material composition of each component. This leads to a

considerable imbalance in the strain energy distribution throughout the system, making

it difficult to determine if the strain energy for any of the components is unusually high.

To make the comparison easier, the average strain energy distribution from all the 108

modes of the base system (µ = 0) is included in the table.

138

Graphically the results are easier to visualise. Figure 6.14 plots the distribution of strain

energy throughout the system for all unstable modes. The last column is the average

strain energy distribution for all 108 modes of the base system.

Table 6.8 Distribution of strain energy for the unstable modes of the assembled

brake system, µ = 0.5 . The average strain energy distribution for all modes 1st

108 modes from the base (µ = 0) system is included.

Mode Freq

(Hz)

Rotor

(%)

Anchor

(%)

Caliper

(%)

Inner

(%)

Outer

(%)

Other

(%)

Total

(%) 27 3322 54.1 7.5 17.3 0.8 2.5 17.8 100

43 4661 64.9 9.4 2.3 6.3 8.7 8.4 100

54 5908 63.6 3.5 5.2 11.6 1.5 14.6 100

73 8268 14.6 7.8 34.5 6.0 23.6 13.5 100

79 8877 77.7 10.4 1.5 3.3 3.9 3.2 100

81 8981 61.3 11.3 4.1 7.0 6.9 9.4 100

105 11860 20.9 14.1 34.4 8.7 16.0 5.9 100

avg. n/a 48.0 18.3 12.7 4.4 5.2 11.4 100

Baseline Model - µµµµ = 0.5

0%

20%

40%

60%

80%

100%

27 43 54 73 79 81 105 Avg

Unstable Mode Number

Other

Outer

Inner

Housing

Anchor

Rotor

Figure 6.14 Strain energy distribution for unstable modes of the baseline

system with µ = 0.5. The average strain energy distribution is calculated for

108 modes below 12 kHz for the baseline assembly with µ = 0

Figure 6.15 plots the distribution of strain energy throughout the system for modes 104

and 105 as a function of friction level. Note that the modes have become coupled at µ =

0.4, as is also seen from the eigenvalue plot in Figure 6.11, so the distributions for the

coupled modes are identical for 0.4 and 0.5.

139

Strain Energy Distribution - Mode 104

0%

20%

40%

60%

80%

100%

0 0.2 0.4 0.5 avg

Coefficient of Friction

Other

Outer

Inner

Housing

Anchor

Rotor

(a)

Strain Energy Distribution - Mode 105

0%

20%

40%

60%

80%

100%

0 0.2 0.4 0.5 avg

Coefficient of Friction

Other

Outer

Inner

Housing

Anchor

Rotor

(b)

Figure 6.15 Strain energy distribution for (a) mode 104, (b) mode 105. The

average strain energy distribution for 108 modes the base system (µ = 0) is

also shown in each chart.

6.6.4 Modal Participation for a Numerical Model

The modal assurance criterion (MAC) was used to assess the modal participation of

different component modes (see section 6.4). The manner in which it was used is as

140

follows. An eigenvector u of the system contains n subvectors, each corresponding to a

single component.

{ }

=

nu

u

u

uM

2

1

(6.62)

Each subvector is correlated to the component’s free modes using equation (6.32).

Space considerations prevent a summary of all MAC values between each respective

component in each unstable mode from being presented here. Chapter 7, which covers

the parametric study, will feature MAC data as required for the analysis of unstable

modes.

However, to enable the example analysis of Section 6.6.5, MAC data for mode 27 and

105 is shown in Figure 6.16 and 6.17 respectively.

141

Mode 27 - Rotor MAC

0

10

20

30

40

50

965 2943 2944 3732 3735

Freq (Hz)

MA

C (

%)

Mode 27 - Anchor MAC

0

5

10

15

20

25

30

988 1718 2766 3215

Freq (Hz)

MA

C (

%)

(a) (b)

Mode 27 - Caliper MAC

0

20

40

60

80

100

2763 3674

Freq (Hz)

MA

C (

%)

Mode 27 - Inner Pad MAC

0

5

10

15

20

25

30

2366 4972 6313

Freq (Hz)M

AC

(%

)

(c) (d)

Mode 27 - Outer Pad MAC

0

5

10

15

20

25

30

2375 4998 10281

Freq (Hz)

MA

C (

%)

(e)

Figure 6.16 Modal assurance criterion for the unstable mode 27 at 3322 Hz.

(a) Rotor, (b) anchor, (c) caliper, (d) inner pad and (e) outer pad. In each

case, only more significant modes are shown.

142

Mode 105 - Rotor MAC

0

10

20

30

40

6774 9063 10407 10553 11835 11838 11842

Freq (Hz)

MA

C (

%)


0

5

10

15

20

25

30

2766 9097 10744 11275

Freq (Hz)

MA

C (

%)

(a) (b)


0

10

20

30

40

11063 12859 13312 14359

Freq (Hz)

MA

C (

%)


0

10

20

30

40

6313 7490 11090 12510

Freq (Hz)

MA

C (

%)

(c) (d)


0

5

10

15

20

25

30

6385 7536 11108 12533

Freq (Hz)

MA

C (

%)

(e)

Figure 6.17 Modal assurance criterion for the unstable mode 105 at 11860

Hz. (a) Rotor, (b) anchor, (c) caliper, (d) inner pad and (e) outer pad. Only

more significant modes are shown

6.6.5 Example Unstable Mode Investigation

The aim of this chapter is not to provide a deep analysis of each system mode of the

brake system, but to provide a review of the analysis methods and application. To

highlight how the feed-in energy, strain energy and MAC values are used in the analysis

procedure, modes 27 and 105 will now be examined. These modes provide contrasting

underlying behaviour and show how different countermeasures may be required for

specific system modes.

143

It can seen from Table 6.8 or Figure 6.14 that mode 105 involves less rotor deformation

than average, slightly higher than average level for the anchor and considerably more

strain energy for the caliper housing and pads, in particular the outer pad.

Examining Table 6.7 it is clear that the feed-in energy is being supplied by the outer

pad. Further it can be seen from Figure 6.13 that the central region of the outer pad is

driving the squeal. The inner pad is displaying negative feed-in energy and hence is

actually acting in a dissipative manner. The feed-in energy contribution from the outer

pad is greater than the total feed-in energy of the system overall.

The likely candidates for modification in the case appear to be the outer pad and caliper

housing. Further insight can be gained by examining the MAC values in Figure 6.17.

The outer pad MAC values (Figure 6.17(e)) indicate 3 main contributors; 6385 Hz,

7536 Hz and 12533 Hz. Modes at 7536Hz and 12533Hz feature predominately bending

motion of the top spring tab, as shown in Figures 6.18(b) and 6.18(c). However, the

mode at 6385 Hz in Figure 6.18 (a) is a bending mode of the pad featuring significant

deformation of the whole backplate. Controlling this bending motion, for example by

adding backplate shims, would aid in reducing the instability of this mode.

144

(a) 6385 Hz

(b) 7536

(c) 12533 Hz

Figure 6.18 Mode shapes of free pad for 6383 Hz, 7536 Hz and 12533 Hz

Turning to the caliper (Figure 6.17(c)) reveals that there is a relatively high MAC

between the caliper mode shape and the free caliper mode located 11063 Hz with 27%.

This mode shape, which is shown in Figure 6.19, features considerable bending of the

guide pin ears. Also, some bending across the caliper bridge can be seen. An

adjustment to the caliper mass or stiffness in these areas could also improve the stability

of mode 105.

145

Figure 6.19 Deformed mode shape of free caliper at 11063 Hz.

Figure 6.20 2nd

Tangential in-plane rotor mode shape of free rotor at 11838

Hz.

146

Another aspect to consider within mode 105 is the nature of the rotor modes. The

highest MAC contribution in Figure 6.17(a) is from the rotor mode at 11838 Hz, a

tangential in-plane mode, shown in Figure 6.20. In addition, a number of out-of-plane

modes involved at low levels. The tangential in-plane modes can be particularly

problematic since they may be readily excited by the rotor/pad friction interface, further

reason to control the outer pad motions that are present.

Turning the attention to mode 27, the strain energy distribution from Figure 6.14

indicates that no components are greatly higher than average. The housing and the rotor

are higher than average, but neither is dominating system behaviour. On the other hand

it can be seen that the strain energy contributions of the pads are particularly low,

further highlighted by the feed-in energy values from Figure 6.12, which indicates

relatively little pad activity. The MAC contributions for the pads, as shown the Figure

6.16, do not indicate any particular dominant mode for what little motion there is. In

contrast to mode 105, controlling pad motion may not be a solution to stabilize mode

27.

Figure 6.16 indicates that one particular mode for each of the rotor and caliper housing

do contribute most significantly. The rotor mode at 2944 Hz, shown in Figure 6.21,

contributes 48% to this unstable mode, while Figure 6.22 displays the dominant housing

mode, 2763 Hz, with 92%. The caliper based counter measure could possibly address

the unstable mode by modifying the bridge region which is bending significantly. The

rotor could possibly be also modified in design to modify the behaviour of the 2944 Hz

mode.

Section 3.11 noted that a squeal concern was not found in the 3.3 kHz region, even

though mode 27 was unstable in the analysis. This may stem from the fact that the rotor

radial mode, as shown in Figure 6.21, would be difficult to excite by the friction

interface. Also, the lack of activity in the pads reduces the likeliness of the mode to

squeal.

On the other hand, mode 105 coincides with a significant noise concern between 11.5

and 12.0 kHz. Here the rotor mode involved is a tangential in-plane mode which is far

more likely to be excited by the friction interface. Also significant outer pad bending

participation provides a strong source of feed-in energy to the drive the squeal.

147

Figure 6.21 2nd

order radial in-plane rotor mode at 2944 Hz

Figure 6.22 Caliper housing mode shape at 2763 Hz.

6.7 Summary

Three additional analysis techniques for examining unstable brake system modes have

been presented in this chapter. All three offer complementary information in

determining how to address a specific brake noise concern.

• Feed-in energy analysis provides insight into how vibrational energy is being

added to the unstable mode. The energy attributed to be added by each pad can

be assessed, and even what specific areas of the pad / rotor interface itself can be

identified. This allows counter measures to be implemented that address

specific action of the pads and pad/rotor friction interface.

148

• Strain energy analysis shows the distribution of activity across the system

components. While the components differ considerably in size and material

properties, it is possible to gauge how active each component is by comparing to

an average of all system modes. Modifications should be aimed to address the

behaviour of the components deemed to be the most active in an unstable mode.

• Modal participation using the MAC indicates which component modes are most

dominant in the unstable system mode. This can provide clues on how a

component may be modified to most effectively address the unstable mode.

149

Chapter 7

Parametric Study

7.1 Introduction

Recent decades have a seen a large increase in the use of numerically based analysis

tools such as the finite element method. Previously it was not possible to analyse

complex structures in detail, and analysis was restricted to simple models. Hence the

major portion of engineering development involved experimental work – prototyping

and testing of development components. Now it has become common place to perform

considerable analysis before a single prototype has been manufactured.

The reliance on trial and error type development is particularly true in the brake NVH

area. Engineering judgement and experience that has been built up over many decades

are used to overcome noise issues as they appear during the development cycle. This

can be time consuming and expensive, particularly if noise issues arise after a vehicle

has been put into production. So any insight that can be gained from numerical

simulation work can be a great benefit to the time and money involved in the

development of a new braking system.

This present study applies the analysis techniques described earlier in Chapters 4 - 6 to a

production brake assembly. The purpose for this study was twofold. Firstly, to

illustrate how the analysis techniques are applied to a practical brake system and

secondly, to gain insight into how sensitive the squeal propensity is to various

parameters within the brake system.

The previous three chapters have detailed the development of a large scale finite

element analysis (FEA) model of a brake system, how it is used to predict the stability

of system modes, and methods used to investigate the unstable modes to try and identify

suitable modifications to maintain stability. In this chapter these techniques are used to

systematically investigate parameters to gauge their influence on system stability.

150

Table 7.1 Summary of parameters under investigation

Material Properties Justification

Parameter Datum Alternative

Friction

coefficient

n/a 0, 0.1, .., 0.6 in steps of 0.1 Max range of

friction coefficient

Rotor

Modulus

118 GPa 100 GPa 140 GPa Max range of

rotor modulus

Anchor

Modulus

165 GPa 140 GPa 200 GPa Modification of

anchor stiffness

Caliper

modulus

68 GPa 100 GPa Modification of

caliper stiffness

Pad lining

Modulus

5 GPa 2, 3, .., 10 GPa in steps of 1 GPa Max range of pad

normal modulus

Rotor

damping

0 % .2% Measured rotor

damping

Anchor

damping

0 % .2% Measured rotor

damping

Caliper

damping

0 % .1% Measured caliper

damping

Backplate

damping

0 % 1% 5% 10% Modification of

pad damping

Contact Distribution


Pad chamfer none 15 mm leading/trailing chamfers Commonly used

countermeasure

Pad shim -

pressure

none Piston/pad contact on one side only Commonly used

countermeasure

Heavy usage n/a Contact around edges of pad Simulate heavy

braking

Full contact n/a Contact over full pad face Simulate light

braking / new pad

Damping Shims


Damping

shim

none Composite

0% loss

Composite

50% loss

Composite

100% loss

Typical damping

values of damping

shims

151

7.2 Parameters Under Investigation

A brake assembly is a complex mechanical system incorporating many different

components of differing material and geometrical properties. This leads to a large

selection of possibly parameters that could be included in the study. However, it is not

practically feasible to vary and investigate the influence of every conceivable factor.

Hence the scope of this investigation is limited to three broad classes of parameters

1. Material property changes

2. Contact pressure distribution

3. Geometric changes

Table 7.1 summarises the material property changes that were investigated and the

reason why these parameters were chosen.

7.3 Baseline System

7.3.1 Complex Eigenvalues

The development and analysis of the baseline brake system is described in chapters 4, 5

and 6. Complex eigenvalue analysis revealed that 7 system modes were unstable in the

frequency range from 0 to 12 kHz with a friction level of µ = 0.5, the friction level used

for the majority of the analysis in previous chapters. The unstable modes are plotted as

negative damping verses frequency in Figure 7.1. Also plotted are the unstable modes

for µ = 0.3, 0.4 and 0.6, to illustrate how the instability increases with friction level.

Table 7.2 displays the unstable modes for a friction level of µ = 0.5 in tabular form.

There is some variation in the propensity of the modes to become unstable as a function

of friction level. The modes at 3.3, 4.7 and 8.9 kHz become unstable quite readily with

µ = 0.3, while the 5.9, 8.3 and 11.9 kHz require higher friction levels.

152

Baseline - Negative Damping Vs Frequency

0.0%

0.2%

0.4%

0.6%

0.8%

1.0%

0 2000 4000 6000 8000 10000 12000

Frequency (Hz)

Neg

ati

ve D

am

pin

g

µ = 0.6

µ = 0.5

µ = 0.4

µ = 0.3

Figure 7.1 Baseline system, unstable modes with µ varied from 0.3 to 0.6.

Table 7.2 Summary of unstable modes for baseline system with µ = 0.5.

Mode No. Eigenvalue Frequency (Hz) Damping ratio (%)

27 83.07 + 20871j 3322 -.398

43 219.0 +29283j 4661 -.748

54 27.29 + 37123j 5908 -.074

73 51.84 + 51951j 8268 -.099

79 64.46 + 55776j 8877 -.116

81 49.78 + 56426j 8981 -.088

105 215.4 + 74521j 11860 -.289

7.3.2 Baseline Strain Energy Distributions

The concept of strain energy was built up in Chapter 6. It provides a convenient method

for assessing how active each component in an unstable vibration mode is and for

helping to identify which components might be the most advantageous to modify.

It is useful to examine the distribution of strain energy for the baseline system. For each

of the 7 unstable modes displayed in Table 7.2, the strain energy distributions were

calculated and are shown in Figure 7.2. As was the case with the results shown in

chapter 6, an average strain energy distribution was calculated for the base system on

the basis of 108 system modes below 12 kHz with µ = 0.

A basic summary of the most active components is displayed in Table 7.3. Note that

each of these “most active” cases is determined by comparing the strain energy of a

153

component with its average value over all 108 modes. In outright terms, the rotor

usually has the highest strain energy because it is the largest component and possess

material properties such that any deformation is accompanied by the high levels of

strain energy. The real interest, however, is whether it is more active than usual, hence

the comparison to the average case.

In modes 73 and 105, the caliper housing possesses strain energy in excess of its typical

value. Modes 43, 54, 73, 81 and 105 all have higher than average activity on one or

both of the brake pads, and the rotor is higher than average for all modes except 73 and

105. It follows most treatments for addressing these unstable modes will likely look to

address rotor, pad and housing activity.

Baseline Model - µµµµ = 0.5

0%

20%

40%

60%

80%

100%

27 43 54 73 79 81 105 Avg

Unstable Mode Number

Other

Outer

Inner

Housing

Anchor

Rotor

Figure 7.2 Strain energy distribution for the 7 unstable modes of the

baseline system with µ = 0.5. The average strain energy distribution is

calculated for 108 modes below 12 kHz for the baseline assembly with µ =

0.

Table 7.3 Summary of most active components for each unstable mode.

Mode Most active components compared to average 27 Rotor

43 Rotor, Outer pad

54 Rotor, inner pad

73 Caliper housing, outer pad

79 Rotor

81 Rotor, inner pad, outer pad

105 Caliper housing, outer pad

154

7.3.3 Baseline Feed-in Energy

Feed-in energy values also calculated as described in chapter 6 for the baseline system

are shown in Table 7.4. The highest level of feed-in energy is for mode 105, followed

by modes 43, 81, 79 and 73. This provides an alternative method of looking at the

instability in brake system modes, since it is clear the order of feed-in per mode differs

from the negative damping values shown in Table 7.2. Feed-in energy must be positive

for a negatively damped mode, so either can indicate system instability. However, feed-

in energy is a measure of absolute vibrational energy addition to a system mode, while

negative damping indicates the rate of divergence from one cycle to the next. Hence

they are subtly different metrics of instability.

Table 7.4 also indicates the feed-in energy contribution of each pad, and can be used to

assess which pad should be the focus of treatment.

Table 7.4 Baseline feed-in energy values for the seven unstable modes for µ = 0.5.

Feed-in Energy (J) Mode

No.

Freq

(Hz) Inner

Pad

Outer

Pad Total

27 3322 0.32 0.47 0.80

43 4661 2.41 3.03 5.44

54 5908 0.60 -0.07 0.53

73 8268 1.49 0.03 1.52

79 8877 2.37 1.54 3.91

81 8981 7.44 -2.28 5.16

105 11860 -1.87 13.97 12.10

7.3.4 Baseline Component MAC Modal Participation

The modal assurance criterion (MAC) is used to gain further insight into the behaviour

of components with the baseline brake system. The MAC value indicates the levels of

correlation between each component within an unstable system mode and the respective

free modes of each component. Figures 7.3 through 7.9 display the MAC for each of

the main components of the brake system for the 7 unstable modes of the baseline

system

155

Mode 27 - Rotor MAC

0

10

20

30

40

50

965 2943 2944 3732 3735

Freq (Hz)

MA

C (

%)


0

5

10

15

20

25

30

988 1718 2766 3215

Freq (Hz)

MA

C (

%)

(a) (b)


0

20

40

60

80

100

2763 3674

Freq (Hz)

MA

C (

%)


0

5

10

15

20

25

30

2366 4972 6313

Freq (Hz)

MA

C (

%)

(c) (d)


0

5

10

15

20

25

30

2375 4998 10281

Freq (Hz)

MA

C (

%)

(e)

Figure 7.3 Mode 27 3322 Hz MAC values (a) Rotor, (b) anchor, (c) caliper,

(d) inner pad and (e) outer pad for µ = 0.5.

156

Mode 43 - Rotor MAC

0

15

30

45

60

75

4674 4674 5155

Freq (Hz)

MA

C (

%)


0

5

10

15

20

25

30

1718 4312 9097 10744

Freq (Hz)

MA

C (

%)

(a) (b)


0

10

20

30

40

50

60

2763 3674 4218 6277

Freq (Hz)

MA

C (

%)


0

10

20

30

40

50

60

2366 6313

Freq (Hz)

MA

C (

%)

(c) (d)


0

5

10

15

20

25

30

10281 11108 12533

Freq (Hz)

MA

C (

%)

(e)

Figure 7.4 Mode 43 4661 Hz MAC values(a) Rotor, (b) anchor, (c) caliper,

(d) inner pad and (e) outer pad.

157

Mode 54 - Rotor MAC

0

10

20

30

40

50

60

5153 5748

Freq (Hz)

MA

C (

%)


0

15

30

45

60

75

988 2766 5944 6947

Freq (Hz)

MA

C (

%)

(a) (b)


0

10

20

30

40

50

60

2763 7512 8614

Freq (Hz)

MA

C (

%)


0

10

20

30

40

50

60

2366 4972

Freq (Hz)

MA

C (

%)

(c) (d)


0

5

10

15

20

25

30

4998 7536

Freq (Hz)

MA

C (

%)

(e)



158

Mode 73 - Rotor MAC

0

5

10

15

20

25

30

2765 4165 6383 6773 8599

Freq (Hz)

MA

C (

%)


0

5

10

15

20

25

30

988 5944 9299 10004

Freq (Hz)

MA

C (

%)

(a) (b)


0

10

20

30

40

50

60

4218 7277 8428 9373

Freq (Hz)

MA

C (

%)


0

5

10

15

20

25

30

2366 4972

Freq (Hz)

MA

C (

%)

(c) (d)


0

10

20

30

40

50

60

2375 6385 7536

Freq (Hz)

MA

C (

%)

(e)



159

Mode 79 - Rotor MAC

0

5

10

15

20

25

30

4165 4166 6773 6774 8608 8686 8687

Freq (Hz)

MA

C (

%)

Mode 79 - Anchor

0

10

20

30

40

5283 6947 9299 12728

Freq (Hz)

MA

C (

%)

(a) (b)


0

5

10

15

20

25

30

7277 8428 9373

Freq (Hz)

MA

C (

%)


0

15

30

45

60

75

4972 7490 13033

Freq (Hz)

MA

C (

%)

(c) (d)


0

5

10

15

20

25

30

2375 6385 11108

Freq (Hz)

MA

C (

%)

(e)



160

Mode 81 - Rotor

0

10

20

30

40

4165 6777 8599 8608 8686 8687

Freq (Hz)

MA

C (

%)

Mode 81 - Anchor

0

5

10

15

20

25

30

5283 8153 8374 9097 11275

Freq (Hz)

MA

C (

%)

(a) (b)


0

5

10

15

20

25

30

4218 7277 8428 9373

Freq (Hz)

MA

C (

%)


0

5

10

15

20

25

30

4972 7490 10228

Freq (Hz)

MA

C (

%)

(c) (d)


0

5

10

15

20

25

30

2375 10281 11108

Freq (Hz)

MA

C (

%)

(e)



161

Mode 105 - Rotor MAC

0

10

20

30

40

6774 9063 10407 10553 11835 11838 11842

Freq (Hz)

MA

C (

%)


0

5

10

15

20

25

30

2766 9097 10744 11275

Freq (Hz)

MA

C (

%)

(a) (b)


0

10

20

30

40

11063 12859 13312 14359

Freq (Hz)

MA

C (

%)


0

10

20

30

40

6313 7490 11090 12510

Freq (Hz)

MA

C (

%)

(c) (d)


0

5

10

15

20

25

30

6385 7536 11108 12533

Freq (Hz)

MA

C (

%)

(e)

Figure 7.9 Mode 105 11860 Hz MAC values(a) Rotor, (b) anchor, (c)

caliper, (d) inner pad and (e) outer pad.

7.4 Material Properties Sensitivities

The bulk of the components within the brake system are manufactured from various cast

iron and aluminium alloys. As far as dynamical properties are concerned, this restricts

the amount of change that can occur if the same basic alloy is used for the respective

components.

The natural frequencies of a structure made from a uniform isotropic material scale in

proportion to the speed of sound in the structure. Since structural metals have Poisson’s

ratios of approximately 0.3, it follows that the natural frequencies can be expressed as

162

ρω

Ei

∝ (7.1)

where E is the elastic modulus of the material and ρ is the mass density.

For most common structural metals, the ratio of E to ρ is approximately 25 X 106

(m/s)2, giving a speed of sound of approximately 5000 m/s. This implies that an

individual component would have approximately the same modal frequencies regardless

of the metal it was made from. For example if the material for the caliper housing is

changed from aluminium to nodular cast iron, the increase in elastic modulus, and hence

stiffness, is proportionately offset by the increase in mass density, and the modal

frequencies remain largely unaltered. However, the static stiffness will have increased

greatly as it scales with modulus.

When the components are in an assembly, there may be a significant redistribution of

stiffness and mass throughout the structure if a component’s material is changed. This

in turn will change the natural frequencies of the assembly as a whole as well as

potentially varying the strain energy distribution during vibration. The stability of

different assembly modes may be significantly altered by a change in material.

Grey cast iron is the material used for the brake rotor in the brake system being studied.

It is chosen because it possesses properties that are beneficial for use in brake rotors

such as cost, ease of manufacture, wear resistance, lower relative density (compared to

steel), sufficient strength, relatively high structural damping, and most importantly high

thermal conductivity. Grey cast iron also possesses relatively low modulus and tensile

strength compared to other common structural irons and steels, and has limited use in

industry. The high thermal conductivity, due to the large flake carbon that precipitates

out of solution during casting, is the main reason for its wide spread use in brake rotors

(Chatterley and Macnaughton, 1999).

When manufactured the final form of the graphite with cast iron depends very heavily

on the carbon level of the alloy since this controls the type of carbon that forms with the

structure. This can also be strongly affected by the amounts of silicon in the alloy

because silicon also changes the types of carbon particles. Carbon content is usually

measured in carbon equivalent (CE), which considers not only the carbon content, but

also the content of the other elements such as silicon (Malosh, 1998)

163

Cast irons, and grey cast iron in particular, differ somewhat to steels and most other

structural metals in that the elastic modulus can be varied significantly by changing CE.

This allows rotors to be manufactured with an elastic modulus that runs from below 100

GPa through to approximately 140 GPa. The baseline FEA model had the rotor

modulus tuned to 118 GPa, so two additional cases were run where the modulus set to

the other extremes of the available range.

Rotor Modulus - Negative Damping Vs Frequency

0.0%

0.4%

0.8%

1.2%

1.6%

2.0%

2.4%

0 2000 4000 6000 8000 10000 12000

Frequency (Hz)

Neg

ati

ve D

am

pin

g

Baseline E = 118 GPa

E = 100 GPa

E = 140 GPa

Figure 7.10 Negative damping levels of system modes for different rotor

modulus levels, µ = 0.5.

Figure 7.10 displays the negative damping levels for the assembly with a change in

modulus with µ = 0.5. The impact of most of the system modes is large, since much of

the brake system’s dynamical properties are governed by the behaviour of the rotor.

This was highlighted by the strain energy in Table 7.3 where rotor activity was

significantly above the average in 5 of the 7 modes.

Compared to the baseline case, setting E = 100 GPa has little effect on the modes at 3.3

kHz, 4.7 kHz, and 9 kHz. However a strong new unstable mode appears at

approximately 7.5 kHz, which is in the range of one of the main concern frequencies

from section 3.11. The unstable modes at 5.9 kHz and 11.9 kHz corresponding to the

test frequencies of concern are suppressed, so the overall noise performance of the

system in testing may be significantly improved. Several other modes with small levels

of instability appear at around 10 to 10.5 kHz.

164

The E = 140 GPa case also significantly impacts the system. Of the main concern

frequencies in testing, the 5.9 and 8.3 kHz are suppressed, but little influence is seen on

the 11.9 kHz mode. Significant new unstable modes have appeared in between 4 and 6

kHz, with a particularly strong mode unstable mode appearing at 4.2 kHz.

Cast iron is also used for the anchor bracket, albeit nodular (also called ductile) cast iron

rather than grey cast iron. The carbon content of nodular irons is lower than grey cast

iron, and the carbon formation is in spheroidal or nodular form. Nodular iron has

tensile strengths of around 400 MPa as opposed to high carbon grey irons which may be

as low as 150 MPa. This, along with an elastic modulus of approximately 170 GPa,

makes them the preferred choice for use in cast iron components in low cost

applications where the hi thermal conductivity of grey iron is less important. The

higher strength is an obvious benefit in most applications, but also the higher modulus is

important since the stiffness of components is proportional to modulus and can be vital

in ensuring proper operation and wear of components. Nodular iron is also relatively

insensitive to small changes in carbon content, so the modulus does not vary

significantly from the region of 170 GPa. A value of 165 MPa was used for the

baseline assembly since this was determined from the tuned FEA bracket model.

Alternative analyses were run with the modulus of the cast iron set to 140 and 200 MPa

respectively with these results displayed in Figure 7.11. While these cases do not

necessarily represent a realistic range for nodular iron, and would require different types

of material, it does provide a convenient method for examining the impact of stiffness

of the bracket which could be achieved with geometry changes.

The anchor also has a significant affect on the stability of the system. Reducing the

modulus to 140 GPa reduced the number of unstable modes to just 2. On the other

hand, increasing the Young’s modulus to 200 GPa had less of an impact with unstable

modes still present at 3.3 kHz, 4.7 kHz, 5.9 kHz and 9 kHz. Several other new unstable

modes appeared also. Reducing bracket stiffness may reduce the brake squeal

propensity of the system overall.

The caliper housing is made from aluminium, with a modulus of 68 GPa determined

from the tuned FEA model used for the baseline assembly. Aluminium does not show

much variation in modulus through alloying, but an alternative case was run with the

caliper modulus set to 100 GPa as shown in Figure 7.12. As was the case with the

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bracket, this was done as a convenient method for determining the influence of

changing the housing stiffness without requiring geometry changes.

Anchor Modulus - Negative Damping Vs Frequency

0.0%

0.4%

0.8%

1.2%

1.6%

0 2000 4000 6000 8000 10000 12000

Frequency (Hz)

Neg

ati

ve D

am

pin

g

Baseline E = 165 GPa

E = 140 GPa

E = 200 GPa

Figure 7.11 Negative damping levels of system modes for different anchor

modulus levels, µ = 0.5

Many of the same concern frequencies remain, but strong instabilities are seen at 4.2

and 5,8 kHz. It should be noted that 11.9 kHz is no longer unstable, and caliper

stiffness was a potential cure indicated in chapter 6 for that mode. Overall it appears

that caliper stiffness may not be beneficial for overall system stability.

The friction material was the only non-metallic material used in the model of the brake

assembly. Friction material is inherently complex and is a mixture of dozens of

material as described in section 4.4.4. Nine different modulus levels were run, all with

isotropic material model. 5 GPa was used as the baseline, and 2 and 10 GPa were the

alternate extreme levels selected since they represent the approximate extremes found in

friction material. Figure 7.13 displays the lining change results.

The pad friction material results show a mix of trends. The mode at 3.3 kHz is

insensitive to pad modulus change. Referring to the strain energy in Figure 7.2 it can be

seen that pad deformation is very low for this mode rendering it insensitive to pad

modulus. A new mode appears altogether at 4 kHz that is strongly dependant on pad

modulus. The instability level (ie, negative damping) for most modes tends to increase

with a decrease in pad modulus. Increasing pad modulus reduces the overall number of

unstable modes and indicates that overall system stability would be enhanced with

166

higher modulus (stiffness) pads. The most probably physical explanation for this would

be reduction in pad deformation that would follow, hence a reduction in overall feed-in

energy values.

Caliper Modulus - Negative Damping Vs Frequency

0.0%

0.2%

0.4%

0.6%

0.8%

1.0%

1.2%

0 2000 4000 6000 8000 10000 12000Frequency (Hz)

Neg

ati

ve D

am

pin

g

Baseline E = 68 GPa

E = 100 GPa

Figure 7.12: Negative damping levels of system modes for reduced caliper

modulus, µ = 0.5

Pad Modulus - Negative Damping Vs Frequency

0.0%

0.6%

1.2%

1.8%

2.4%

3.0%

0 2000 4000 6000 8000 10000 12000

Frequency (Hz)

Neg

ati

ve D

am

pin

g

Baseline E = 5 GPa

E = 2 GPa

E = 3 GPa

E = 4 GPa

E = 6 GPa

E = 7 GPa

E = 8 GPa

E = 9 GPa

E = 10 GPa

Figure 7.13: Negative damping levels of system modes for changes in

friction material modulus, µ = 0.5

167

The baseline brake assembly analysis was conducted devoid of material damping. This

is mainly because the system is relatively lightly damped and therefore the modal

frequencies and mode shapes are largely unaffected. However, the experimental modal

analysis conducted in chapter 3 provides modal damping levels for the main

components.

The analysis was re-run with structural damping added to the main components, first

one at a time and for all at once, to see if this had any impact on the stability results.

Figure 7.14 displays the impact of structural damping being added to all of the

components. It is clear that the damping acts simply to reduce the negative damping.

Modes with little negative damping may become stable, and modes that are more

unstable have some measure of damping added. An additional slightly unstable mode

appears at 4.1 kHz which is not seen in the baseline case, although at low levels of

instability.

Damping Level - Negative Damping Vs Frequency

0.0%

0.2%

0.4%

0.6%

0.8%

1.0%

0 2000 4000 6000 8000 10000 12000

Frequency (Hz)

Neg

ati

ve D

am

pin

g

Baseline No Damping

Structural Damping

Figure 7.14: Structural damping with µ = 0.5

The use of these low levels of structural damping was considered to not significantly

affect the analyses in this study. The main impact is simply to reduce the instability of

system modes with little influence on the character of the modes themselves.

Further impact due to damping was modelled by applying relatively large levels of

structural damping to the pad backplates. This was done in an effort to simulate the

application of damping shims. Figure 7.15 shows that the application of backplate

168

damping has little affect on most modes. The exceptions are the modes at 7.3 kHz and

11.9 kHz. It is clear from strain energy distributions in Figure 7.2 that both of these

modes have a very high level of pad motion, so the effects of the damping is

maximised. In Chapter 6 it was shown that out pad motion was a key driver of the 11.9

kHz mode.

Backplate Damping - Negative Damping Vs Frequency

0.0%

0.2%

0.4%

0.6%

0.8%

1.0%

0 2000 4000 6000 8000 10000 12000Frequency (Hz)

Neg

ati

ve D

am

pin

g

No BP Damping

BP Damping = 1%

BP Damping = 5%

BP Damping = 10%

Figure 7.15: Backplate damping with µ = 0.5

The remaining components of the assembly, the guide pins and the piston, maintained

their baseline material properties through the study.

7.5 Contact Distribution Sensitivities

A key feature of brake system dynamics is the interaction of components at the various

interfaces. The most critical is the rotor/pad interface, but can also be important at other

interfaces such as the piston/inner backplate.

During the noise package development for a production brake system, many of the

modifications trialled are designed to alter the contact pressure distribution at the

rotor/pad interface. This is achieved usually through one of two ways; modification of

the footprint of the “puck” of friction material that contacts the rotor by use of different

chamfers or cuts, or by the use of slotted or cut shims between the inner backplate and

the piston.

169

Backplate

Friction

material

Removed

material

Figure 7.16 shows a puck that has been modified by the addition of chamfers that

directly change the area of contact between the pad and the rotor. Moving the contact

pressure backward from the leading edge of the pad is considered beneficial for system

stability, so the chamfer is symmetrically added to both the leading and trailing edges of

the pad to remove the need for “handed” pads, i.e., pads that have different geometry

between left and right hand fitment, or differences between inner and outer pads.

Figure 7.16 Cross section of a modified puck with landing and trailing

chamfers

The alternate way to modify the contact pressure distribution of the brake pad is to

apply cut shims to the backplate, an example of which is shown in Figure 7.17. While

backplate shims usually contain damping material to help decouple components and/or

add damping to the system, they also play a role in determining the contact pressure

between the piston and the pad. Slots change the centre of pressure between the piston

and pads, and it follows that the contact pressure between the pad and rotor will also be

modified. Under certain circumstances, slots may also decouple modes.

Figure 7.17 Slotted shim which removes pressure from one end of the

piston, helping to alter the contact pressure at the friction interface.

170

To incorporate these modifications into the FEA model would potentially require a

change the geometry of the pad and/or the addition of an additional component between

the pad and the piston. Neither of these changes were attempted with geometry change

since it was not considered necessary to capture the key factor, that is, modifying the

areas in contact with each other.

Two types of common noise control features were modelled. The first was a shim with

a slot that covered half of the piston contact area, either leading or trailing, was

simulated. Secondly, a 15 mm parallel chamfer on the leading and trailing edges of the

puck, the results of which is shown in Figure 7.19. Both the chamfer on the puck and

the slotted shim were implemented via simply changing the contact areas by

deactivating contact springs in the appropriate locations within the FEA model.

Stiffness Shims - Negative Damping vs Frequency

0.0%

0.4%

0.8%

1.2%

1.6%

2.0%

0 2000 4000 6000 8000 10000 12000

Frequency (Hz)

Neg

ati

ve D

am

pin

g

Baseline

Piston shim - leading slot

Piston shim - trailing slot

Hat shim

Figure 7.18: Shims applied to the assembly with µ = 0.5.

Figure 7.18 displays the results of using shims. Two variations of the piston shim were

run, one with the slot on the leading side and another with the slot on the trailing side of

the piston. Another modification investigated in this study was to examine the effects

of using a top hat shim. This is a steel plate fitted between the rotor and the wheel

which provides significant stiffening of the hat. This was condition was simulated by

constraining all DOFs the regions of the hat within the mounting bolts holes. Some

success has been achieved by stiffening the hat for controlling brake noise in some

brake systems where deformation of the hat is significant. The results are also

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displayed in Figure 7.18, and it is clear that hat shims and slotted shims have little

beneficial affect on stabilising most of the system modes.

Two further contact pressure distributions on the rotor/pad interface were investigated.

These follow from the nature of the contact between the pad and the rotor in a brake

system in typical use.

When pressure is applied to the brake the pads are forced onto the rotor, but in a non-

uniform way. The inner pad is forced in the central region by the piston, while the outer

pad is forced by the caliper fingers. Since the friction material is relatively soft and the

pad can deform, the pressure distribution can become non-uniform. The outer edges of

the pads tend to pull away from the rotor resulting in the spring tuning described in

section 4.6.1.4.

Contact Distribution - Negative Damping Vs Frequency

0.0%

0.4%

0.8%

1.2%

1.6%

2.0%

0 2000 4000 6000 8000 10000 12000Frequency (Hz)

Neg

ati

ve D

am

pin

g

Baseline

Chamfer

Edge Contact

Full Contact

Figure 7.19: Contact distribution with µ = 0.5

An additional effect of this non-uniform contact pressure is non-uniform wear. An

extreme example occurs when a vehicle descends along a mountain road and undergoes

repeated high torque brake applications is a short period. Wear towards the centre of

the pad is significantly higher in the short term before more moderate braking tends to

normalise the wear rate across the pad. If a low pressure stop is made in such a

configuration the contact area tends to reverse. The central regions have little on no

contact pressure, and the outer edges of the pad are mainly in contact. A simulation of

172

the condition was made where the usual spring locations at the pad/rotor interface were

reversed. That is, only the outer edges of the pads were in contact.

The final contact pressure distribution that was investigated was the case with brand

new pads at extremely low pressure. Here the pad is largely in uniform contact all over

the face. So for this simulation all of the contact springs at the rotor/pad interface were

activated.

It can be seen in figure 7.19 that full pad contact causes considerable instability

compared to the other contact configurations, and reduced contact configuration offers

improved stability. The edge contact case is only a short term effect in a real brake

system, but it appears to offer the best solution. Addition of chamfers to the leading and

trailing edge of the pads also increases stability compared to the baseline case.

7.6 Damping Shims

Application of damping shims to the pad backplates was an alternative method to

simulate damping shims to that discussed in section 7.4 where high levels of structural

damping were added to the pad backplates. This was initially simulated by putting a

layer of 2 dimensional shell elements across the pad back plates. Unfortunately, this

was found to be a completely ineffectual technique. This is because the damping shim

is a three dimensional structure where shear between layers contributes most of the

system damping and a layer of two dimensional shell elements on the backplates do not

capture this behaviour.

The final trials using the pad damping shims were to incorporate full three dimensional

damping shims onto the backplates. The typical damping shim is usually a multi layer

laminate with alternating rubber, steel and adhesive layers, with a typical example

shown in Figure 2.6 and material specification in Table 2.1.

Only one simplified shim material was modelled for this study. The structure, which

represents a simplified shim layout, is shown in Table 7.5

Table 7.5 Simplified shim structure used in the FEA study

Layer Material Thickness

(mm)

Modulus

(GPa)

Poisson’s

ratio Density

Structural

loss factor Piston side Steel 0.40 207 0.3 7.86 none

Pad side Rubber 0.40 0.5 0.49 1.5 0, 50, 100%

Total 0.80

173

The stability results are presented in Figure 7.20. It can be seen that adding the shim

alone without any damping can have a negative impact on the stability of many modes.

A new strongly unstable mode has appeared at 4 kHz and several other modes indicate

increases in instability with no damping. It must be highlighted that addition of the

shims changes the structural behaviour of the base system even without the addition of

damping due to the increase in thickness on the back of the pad and the addition of a

steel layer. The rubber layer also changes the interface been the caliper and pad

backplates.

Application of damping increases stability for all of the modes. However the

comparison is only valid to the undamped shim as a baseline rather than the shim-less

baseline for the reasons mentioned in the previous paragraph. With 100% loss factor,

ie., 50% of critical damping, applied to the shim rubber, it is seen that almost all of the

system modes are stable. Even the 4 kHz mode, which is unstable with the undamped

shim, has been reduced to almost zero instability. These results certainly indicate that

for any given shim structural arrangement, an increase in damping will improve stability

of the system.

It is also worth noting that the results with the shim differ considerably from that shown

in Figure 7.15 for the damped backplate. This is most likely due to the additional

effects of the shim as discussed in section 2.6, including decoupling of components and

shear damping effects. The damped backplates only add a small measure of damping to

the system, and do not adequately capture the physical effects of a full multi-layer

damping shim.

174

Shim Modelling: Negative Damping vs Frequency

0.0%

1.0%

2.0%

3.0%

4.0%

0 2000 4000 6000 8000 10000 12000

Frequency (Hz)

Ne

ga

tiv

e D

am

pin

g

gr5c5 Baseline

Composite - No damp

Composite - 50% Loss

Composite - 100% Loss

Figure 7:20 Negative damping levels of system modes for changes damping

shim µ = 0.5

7.7 Summary

This chapter has presented a study on some of the parameters that affect the noise

propensity of a brake system. Many of the structural modification change the behaviour

in a significant way, and many new unstable modes may be created in addition to

addressing some the existing unstable modes. In this respect, problem modes would be

treated on a case by case basis, and rarely will a modification improve all potential

unstable modes.

Increasing the coefficient of friction of the pad/rotor interface was found to increase the

instability in all system modes. Conversely reducing coefficient of friction may be

considered a general method for improving system stability, but in practice the

performance requirements of a brake system render this approach requires system size

increase to maintain torque capacity.

Reducing pad contact area was found to increase stability of the brake system. Parallel

chamfers offer more stability than the baseline pad case as a counter measure that could

be applied to a production brake system. Increasing pad modulus was found to increase

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system stability also, although some modes were insensitive to pad modulus due to the

lack of pad deformation.

Modelling of damping was done in numerous ways. Adding structural damping to the

components of the base system was found to slightly increase the stability of the system

modes without greatly changing their character. Simulating backplate damping shims

by adding damping to the backplate increased system stability, but failed to capture the

behaviour of a laminated shim. Adding a damping shim to the back plate was found to

be very affective due to more adequately capturing the behaviour of a physical shim, but

even the addition of a shim without damping can influence the behaviour of the baseline

system since it adds a further structural component.

176

177

Chapter 8

Comparison of Contact Modelling Methods

8.1 Introduction

In the previous four chapters of this thesis, the development of a finite element model

for squeal prediction was presented. The commercial finite element analysis (FEA)

code MSC.Nastran was used for the analysis.

A central feature of using Nastran is the requirement for the user to implement the

contact interfaces within the model. This not only includes the pad / rotor interface with

its crucial friction interface, but other interactions between all of the other components.

Linear springs were used to represent the contact stiffness at the interface, with the

stiffness calculated on the basis of the properties of the adjacent elements. Consequently

the analysis was fully linear.

In this chapter an alternative method is presented which makes use of the powerful

contact modelling capabilities of HKS Abaqus. Contact elements are incorporated into

the code that establishes the required contact conditions at the interfaces automatically,

greatly simplifying the modelling process allowing non-linear contact. The results of

reanalysing the complete assembly using Abaqus are presented and comparisons made

with the results obtained from Nastran.

8.2 Contact Elements

HKS Abaqus is a powerful, non-linear FEA code that features contact modelling with

contact elements. Various types are available and are described in full in the Abaqus

User’s Manual [Abaqus User’s Manual, 2003]. The types for structural analysis are:

• Deformable-deformable - used to model contact between bodies where elastic

deformation of both is important

• Rigid-deformable - typically used where elastic deformation on only one body is

considered such as a manufacturing forming process

• Tied - used to directly fix bodies together, or for joining dissimilar meshes

Each type of contact features strict master-slave relationship between the two

components. A master surface is defined along the outer surface of elements on one

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component, and a set of nodes on the other body are defined as slave nodes. The master

surface and slave nodes form a contact pair. A 2 dimensional example of a master

surface and slave nodes is shown in Figure 8.1 (a).

During analysis the relative position of the slave nodes compared to master surfaces is

monitored and the overriding condition is that slave nodes cannot penetrate through a

master surface. It is recommended that the master surface is set as the surface with

lower mesh density, or the body with the higher elastic modulus if the mesh densities

are similar.

A striking difference between the contact element formulation compared node to node

definition used in the Nastran analysis is that the actual contact between slave nodes and

their respective master surfaces does not need to be predefined. Slave nodes become

automatically associated with a master surface during the analysis, so the requirement to

have mesh with coincident nodes at the contact pair is removed.

Two types of contact are required for implementing the brake assembly model;

deformable-deformable and tied. A description of each of these follows.

8.2.1 Tied Contact

Tied contact is the simplest form of contact where nodes on the slave surface are simply

tied, or rigidly fixed, to the master surface. An initial contact clearance distance can be

defined and a numerical algorithm checks the projected area from the master surface in

the normal direction, as shown for a 2 dimensional case in Figure 8.1 (b). Any slave

node that falls into this area has its position modified to bring it to lie exactly on the

master surface as shown in Figure 8.1 (c).

The node is geometrically fixed (tied) to the surface, and its forces are distributed

directly onto the adjacent master surface nodes in proportion to their respective

geometric proximity. This operation is performed before the first increment of the first

step of an analysis that follows the activation of the contact pair, and the contact

locations remain fixed throughout the analysis.

In 3 dimensional contact analysis the principle is essentially the same. At the start of

the analysis the contact surface and clearance distance define a volume that checked for

slave nodes. Any node that lies in the volume is modified to bring it to lie on the master

surface and the resulting load on the slave node is distributed to the adjacent master

179

surface nodes.

(a) Initial contact condition showing slave nodes on body 1 and master surface on body 2

(b) Initial clearance with ∆ t used to find nodes requiring positional adjustment

(c) Contact condition after slave nodes within initial clearance have been adjusted

Figure 8.1 Master/slave contact in 2 dimensions.

A key point to note in this modification of slave nodes, and hence the elements formed

by these nodes, is that it is a zero strain operation. That is, the elements after

Master

Slave nodes

Body 2

Body 1

∆ t

180

modification possess zero strain and the modified slave element defines the initial

unloaded condition.

Abaqus also checks for overclosure before an analysis. This is the condition where one

or more of the slave nodes are geometrically located within the master surface. The

same procedure is applied here as for the initial contact clearance check. The location

of the nodes is modified to bring them to lie exactly on the master surface.

The use of the initial clearance setting of the contact pair is to allow for geometric

modelling error and the implied modelling intent is that any tied contact pairs are

supposed to have exact geometrically matching boundaries and fully constrained to each

other for the duration of the analysis.

8.2.2 Deformable-Deformable Contact

Deformable-deformable contact is applied between bodies that are coming into contact

during an analysis, but are not physically fixed to each other. Some similarities to the

tied contact exist, but also many differences.

The analysis commences by examining an area projected from the master surface for a

distance given by the initial contact clearance. Again, as for tied contact, slave nodes in

this area will be adjusted to lie exactly on the master surface. However, the node is not

tied to the master surface for the duration of the analysis. Also, slave nodes that are not

in this area, and therefore remain unadjusted, are not discounted from the contact pair.

These nodes may come into contact at a later stage of the analysis. However, the

geometric adjustment for nodes within the contact clearance is only performed once,

which is prior to the first analysis step after the contact pair is activated.

In deformable-deformable contact nodal forces from a slave node are again transferred

onto the master surface, but the node is not physically fixed in this location. A normal

contact stiffness is added between the node and the master surface nodes, and also a

tangential stiffness is determined on the basis of the interfacial forces and the friction

coefficient. This procedure is completed for every increment of the analysis and nodes

can come into and go out of contact as required on every step of the analysis.

A further distinction in elastic-elastic contact is whether contact slip distance is

negligible, or if it needs to be considered. In “small sliding” the relative slip

displacement between the slave and master surface is much smaller that the length of a

181

contact element. The load distribution onto the master element is unchanged and the

contact distribution is not recalculated after being established between a slave node and

its master element. The slave node is still free to move relative to the master element,

but the loads act as if it is still in the original location when contact was established.

Hence it is of great importance that the relative motion between a slave node and master

element is negligible otherwise the contact load distribution becomes invalid. Nodes

are still free to move out of contact in small sliding.

In “finite sliding” no limitation exists on the amount of relative motion between slave

nodes and their master surface. The full contact state is recalculated at every step and

the contact load redistribution is free to occur. This provides the most accurate tracking

of the contact conditions, but it comes at a high computational cost. Therefore it is not

recommended to use finite sliding at any contact pair unless the relative displacement

needs to be considered. The relevance of this point to brake system modelling will

become apparent shortly.

A further unique analysis feature in Abaqus analysis is the ability to apply velocity

boundary conditions to parts of the FE mesh specifically for contact analysis between

moving structures. Here the mesh itself is not physically moved, but Abaqus treats the

contact pair as though there is relative motion between the bodies. Tangential stiffness

at the boundary due to frictional forces will be added by the implied motions at the

contact pair. This lends itself readily for brake system analysis and in other applications

where sliding contact is being simulated but the underlying surface are not being

deformed

A velocity boundary condition is used in the brake system analysis. The mesh that

represents the disc rotor is given an angular velocity boundary condition around its

central axis. The mesh is not physically moved, but Abaqus treats it as though it is.

The rotor and pads form a contact pair of which the rotor is moving. On the basis of the

normal forces at the interface, and the velocity directions given, Abaqus will calculate a

relative sliding velocity as if the master surface and slave nodes were in relative motion.

This then determines in which direction the frictional forces will act since they point in

a direction opposite to the implied slip direction.

Velocity boundary conditions also allow velocity dependent frictional forces to be

applied which are not possible to implement in the linear contact spring method used in

182

the Nastran model. This is particularly useful for brake system analysis since it is

common for the coefficient of the friction between the rotor and the pad to increase at

lower velocities. A great many options are available for the analysis which can be

found in the Abaqus Users manual (2003).

8.2.3 Non-linear Static Analysis

The Nastran analysis that was presented in preceding chapters utilised linear static

solutions to determine the base state of the brake system during a braking application.

Linearity means that only a single determination of the system’s stiffness matrix and

therefore only a single matrix inversion during solution is required. Linear solutions

also imply that the structural response scales directly with applied loads.

Non-linearity arises when the behaviour of the system changes as load is applied. The

static solution for the brake system is non-linear due to the changing contact conditions

during pressure application. This non-linearity was captured using a manual iterative

procedure where by an initial static analysis is run and contact springs with physically

unrealistic tensile forces are removed. This procedure was repeated until a final

position and contact state which was devoid of tensile contact forces was determined.

This solution strategy is essentially non-linear but each solution was linear.

The forte of Abaqus is the solution of non-linear problems, be they contact, geometric

or material property driven non-linearity. A solution step is approached in small

increments which in themselves are solved iteratively. Thus the loads are applied

gradually with equilibrium achieved for intermediate loads ensuring the convergence to

the correct final state of the system. Abaqus uses an iterative solution technique, the

principle of which, and the relationship between steps, increments and iterations, are

described in the following paragraphs.

During a solution static equilibrium is evaluated by checking the internal nodal forces

are in balance with the externally applied loads, known as a force residual. If they are

not smaller than a small tolerance, then the system is non-linear and updating of the

stiffness matrix will be required.

An overall analysis is decomposed into individual steps that represent different stages of

the analysis. For example, the first step of a brake analysis may be to apply pressure to

a static brake and then step 2 may be to apply the rotor rotation and so on.

183

A solution step is by default assigned a time of 1 second, even though a measure of time

has no physical meaning for a static solution. An initial increment is chosen, for

example 0.05s, and a solution is attempted. This means a load of 0.05 times the overall

required load for the step is applied. If large loads are applied too quickly it can be very

difficult to achieve equilibrium for an increment since the system stiffness matrix

requires substantial updating from one load increment to the next.

The first iteration on the increment is attempted and a new displacement determined

using the current system stiffness matrix. An updated stiffness matrix is formulated and

internal forces calculated. If equilibrium is not achieved under the new displacement

condition, then another iteration is made. Several iterations, each with updates to the

stiffness matrix may be required to converge at each increment.

Once an increment has converged, then a new increment size is determined based on the

size of the current increment, the default value being 1.5 times the current increment

size if convergence occurred with 5 iterations. If more than 5 iterations are required

then the next increment will have the same size as the current one. If, however, the

current increment does not converge within a reasonable number of iterations, the

default being 12 attempts, then the size increment is “cut back” to 0.25 times the current

increment.

Increments are repeated until a time of 1s has been completed. The step is now

complete and the next step of the analysis will commence.

8.2.4 Contact Set-up and Solution Steps

The model employed for the brake system analysis in this present study has identical

geometry and meshing to the Nastran model considered in chapters 4 through 7. Two

types of 3-dimensional elements were used from the Abaqus element library:

• C3D6 – 1st order 3D continuous wedge element with 6 nodes.

• C3D8 – 1st order 3D continuous hexahedral element with 8 nodes.

Also used in the regions where contact occurs are the contact surface elements.

It is not necessary to implement finite sliding contact pairs at any of the locations of

contact within a brake system model. Most of these locations exhibit negligible relative

slip when the brake is loaded. The only exception is the rotor / pad interface where

184

obviously the rotor rotation leads to a high slip rate in a physical brake system.

However, in the contact analysis using Abaqus it is simply a case of defining a velocity

boundary condition to the rotor as described in the section 8.2.2. Table 8.1 displays a

summary of the parameters used for each of the contact interfaces within the brake

assembly.

Table 8.1 Contact interfaces in the Abaqus brake assembly model.

Interface Type Initial Clearance (mm) µµµµ

Rotor / lining Small sliding .005 0.5

Inner Pad BP / piston Small sliding .005 0.12

Outer Pad BP / housing Small sliding .005 0.12

Pad BP / anchor Small sliding .005 0.12

Piston / housing Small sliding .001 0.05

Pins / anchor Small sliding .001 0.05

Pins / housing Tied .01 -

The initial contact clearance values were chosen to be .005 mm for all of the pad

surfaces since these are all intended to have the surfaces resting against each other at the

commencement of the analysis and .005 mm represents the geometric resolution of the

geometry. The pad and piston surfaces use 0.001 mm since these surfaces are not

designed to be in initial contact and are modelled with a finite clearance. The value of

.001 mm ensures that no adjustment of nodes would occur at these contact surfaces

prior to analysis. The housing and pin clearance of .01 mm ensured all of the contact

surfaces will be fully adjusted and tied even if some nodes were up to .005 mm apart

due to modelling tolerance.

The overall solution strategy that was employed is not dissimilar to that used to solve

the Nastran models described in previous chapters. A static analysis establishes the

base state of the system under a typical brake load, and then a complex eigenvalue

solution was performed to determine system stability about this base state. However,

the steps used differ and 4 individual steps were used.

1. Static preload, non-linear static. Pressure is applied to the back of the piston

and inside the bore in the caliper housing. No rotation is applied to the rotor for

this step and the system reflects a stationary brake with pressure applied. This

185

allows non-linear static solver to more easily determine the contact conditions at

the rotor / pad, guide pin and piston interfaces without the complication

introduced by rotation. Solution stabilization, which involves applying artificial

damping to control rigid body motions, is applied to bodies that are not

constrained prior to contact being established. The damping is small enough not

to affect the final static solution when all the contact conditions have been

properly established.

2. Add rotation, non-linear static. A velocity boundary condition was added to

the rotor from the statically loaded state from step 1. The pads react to the

frictional forces at the rotor / pad interface and begin to translate until fully

captured by the pad abutment regions on the anchor bracket. The system

converged into its base state during a brake application. This provided the base

state for the analysis steps that followed.

3. Normal modes. The normal modes solution provides a subspace of modes to be

used for the complex eigenvalue solution in step 4. The number of modes

extracted was 160 covering a frequency range 0 to 18 kHz. The number of

modes in this step needed to be greater than the number of complex modes

requested for step 4 to adequately allow the complex modes to be represented.

4. Complex modes. Complex eigenvalue solution to provide the stability response

of the base statically loaded state. Abaqus implements only a sub-space

projection procedure, hence the requirement for the prior normal modes analysis.

In this analysis 120 complex modes were extracted, covering a range from 0 to

14.5 kHz.

8.3 Material Properties and Load Cases

Material property data for the Abaqus analyses is the same as that used for the Nastran

analysis. There is negligible difference between the normal modes analysis from

Nastran and Abaqus since the material property “tuning” from the Nastran model was

carried over. The properties are summarised in Table 8.2.

186

Table 8.2 Material properties for the assembled FEA model

Component Material Material

Model

Elastic

Modulus

(GPa)

Poisson’s

Ratio

Mass

Density

(T/m3)

Rotor Grey

Cast Iron Linear elastic 118 0.32 7.10

Anchor

Bracket

Nodular

Cast Iron Linear elastic 165 0.30 7.40

Caliper

Housing Aluminium Linear elastic 68 0.33 2.65

Friction

Material - Linear elastic 5 0.30 2.60

Pad

Backplates Steel Linear elastic 207 0.30 7.86

Guide Pins Steel Linear elastic 207 0.30 7.86

Piston Aluminium Linear elastic 70 0.33 2.70

Little non-linearity is present in the brake system other than the contact between

components. Varying the applied line pressure may change the areas of the contact

between the components, so a number of pressure levels were investigated.

Four levels of pressure were studied:

1. 10 Bars – represents light brake pressure in typical usage

2. 20 Bars – represents a moderate brake pressure

3. 50 Bars – represents pressure during heavy braking

4. 140 Bars – represents the limits of brake system safe usage pressure level

The simplest friction model, which represents Coulomb friction, was used for this study.

The friction coefficient is independent of the normal force and of the slip velocity. A

friction coefficient of 0.5 was used as it corresponds to the maximum level used in the

187

Nastran model and is approximately the maximum level to be expected for this brake

system in practice. Abaqus does provide more advanced friction modelling options, but

these were not employed for this study.

Given that the friction level is independent of the slip velocity, and no structural

properties were dependant upon rotor angular velocity, the actual magnitude of the

velocity boundary condition is of no consequence. Similarly to the Nastran analysis,

rotational effects are not modelled. The velocity merely defines a direction for the

frictional forces so 3 rad/s was used in all cases. A 10 rad/s case was run to confirm

that this was the case.

8.4 Analysis Results

8.4.1 Abaqus Results for varying Pressure

An initial comparison of the eigenvalues was performed for the 4 levels of pressure.

This was done to gauge the influence of the full spectrum of expected pressure on the

system stability. Figure 8.2 plots the eigenvalues in the complex plane in terms of

frequency and negative damping factor.

0.0%

0.5%

1.0%

1.5%

2.0%

0 2000 4000 6000 8000 10000 12000

Frequency (Hz)

Neg

ati

ve D

am

pin

g F

acto

r

10 Bars

20 Bars

50 Bars

140 Bars

Figure 8.2 Abaqus complex eigenvalue results at 4 different pressure levels.

188

A total of 12 separate unstable modes were identified between the 4 pressure cases.

Considerable overlap occurs between the load cases with most of the unstable modes

exhibiting similar levels of instability independent of the pressure level.

8.4.2 Abaqus vs Nastran Stability

One single Abaqus pressure case, at 20 Bars, was chosen for a deeper analysis of the

system behaviour. The direct comparison to the stability results from the Nastran

analysis is shown in Figure 8.3.

0.0%

0.5%

1.0%

1.5%

2.0%

0 2000 4000 6000 8000 10000 12000

Frequency (Hz)

Neg

ati

ve D

am

pin

g F

acto

r

Nastran

Abaqus

Figure 8.3 Comparison of complex eigenvalue results of Abaqus and

Nastran

Noted differences can be seen in the comparison of unstable system modes. Overall the

negative damping ratios for the Abaqus models indicate higher levels of instability than

the Nastran model. Also, some differences in the frequencies can be seen, but overall

each model predicts some unstable modes in or close to the frequency ranges of noise

concern. Table 8.3 provides a summary of the test results from Chapter 3.11, as well as

the Nastran and Abaqus prediction results.

189

Table 8.3 Summary of test, Nastran and Abaqus results

Test Concern Frequency Nastran Result Abaqus Result

4.5-5 kHz 4.7 kHz 5 kHz

6-6.5 kHz 5.9 kHz 6.5 kHz

7.5-8 kHz 8.3 kHz 8.4 kHz

11.5-12 kHz 11.9 kHz 11.7 kHz

The Abaqus model predicts higher levels of instabilities for all of the 4 concern

frequency ranges of interest except for the 11.7 kHz. Also, as was the case with the

Nastran model a number of other frequencies were predicted that do not correspond to

any observed test squeal concern. As with the Nastran analysis, this is due to inherent

linearization required to perform a modally based analysis which cannot include system

non-linearities and dissipative effects.

Further investigation of the system at a deeper level was conducted to see what other

differences lie in the predicted unstable modes.

8.4.3 Abaqus vs Nastran MAC

One of the analysis methods from chapter 6 were applied to the system to gain further

insight into its behaviour. MAC values were calculated between the modes in the

Abaqus assembly and the base component free modes. This allows a more accurate

assessment to be made if the modes predicted by either analysis methods offer

comparable modal results but with some frequency shifting, or if the modes predicted

are altogether different.

Figures 8.4 to 8.12 show the MAC values found for the unstable modes from the

Abaqus model. The equivalent results for the Nastran model are shown in Figures 7.3

to 7.9.

The first mode to compare is between the Abaqus mode at 5 kHz (Figure 8.7) to the

Nastran mode at 4.7 kHz (Figure 7.4). The rotor motion in each of these modes is

significantly different. The Abaqus model features a 5153 Hz (5,0) rotor bending mode

at 31%, whereas the Nastran model has the rotor in a 4674 Hz 3rd

order radial in-plane

mode at 70%. The Abaqus model rotor also has some 3rd

order radial in-plane motion,

but only at 17%.

The caliper housing motions are also significantly different, with the 2763 Hz mode in

the Abaqus model at 46%. The Nastran model has the caliper in a 4218Hz mode 52%.

190

It can be concluded that there are significant differences in the modes predicted in this

frequency range by the two models.

In the region of 6-6.5 kHz, the Abaqus and Nastran models predict modes at 6.5 and 5.9

kHz respectively, shown in Figures 8.8 and 7.5. Here the rotor modes are significantly

different also, with the Abaqus model predicting a 6771 Hz (6,0) bending mode at 36 %

against a Nastran prediction of a 5748 HZ (0,2) circumferential bending mode at 60%.

A striking difference in the other component MAC values is that the Abaqus model

does not show any component modes that are strongly dominant. The Nastran model,

on the other hand, shows that individual modes with dominant MAC values for the

anchor bracket (5944 Hz), housing (8614 Hz), inner pad (2366 Hz) and the outer pad

(7536 Hz). Again the conclusion must be that the modes predicted between the two

models do not correspond to each other.

For the region of 7.5 – 8.0 kHz the respective Abaqus and Nastran modes are 8.4 kHz

(Figure 8.9) and 8.3 kHz (Figure 7.5). Neither of these modes lies within the frequency

range of interest, but are the closest in each case.

Here the correlation for the rotor mode is good, with the models predicting the most

significant rotor contribution from the 8599 Hz (7,0) bending modes with MAC values

at 25% and 28% for the Abaqus and Nastran model respectively. The outer pad

motions correlate to 2375 Hz pad bending at 38% and 59%.

For the other components the correlation is quite poor again, with significantly different

contributions for the component modes.

The final frequency range of comparison is the 11.5-12 kHz range where the Abaqus

model mode is at 11.7 kHz (Figure 8.12) and the Nastran mode is at 11.9 kHz (7.9).

Here the main rotor modes predicted by most models correspond well, with the 2nd

tangential in-plane mode at 11838 Hz seen in both models (Abaqus 24%, Nastran 40%)

and also a mode at 10407 Hz (Abaqus 33%, Nastran 19%).

The remaining components show significant differences again, but that isn’t altogether

unexpected in that modes featuring in-plane motion can be relatively insensitive to

caliper assembly.

Overall the correlation between the models seems relatively poor. It is notable that the

Abaqus predicted modes more often than not feature the rotor in a diametrical bending

191

mode and often little defined contribution from other component modes.

The root of these differences stems from the differences in modelling the component

interfaces. The non-linear contact element formulation of Abaqus allows more of the

non-linearity of the static analysis to be captured, even if the eigenvalue analyses

require linearization. In Section 7.5 it was seen that the complex eigenvalue analysis

results were sensitive to the areas of contact at the pad / rotor interface, and Figure 4.17

shows some differences were found between these areas from Nastran and Abaqus. It

seems likely that the contact modelling method of Abaqus provides more realistic

contact distribution to be captured and it is expected that the Abaqus model more

closely resembles the actual case in the brake system.

192

Mode 13 - Rotor MAC

0

10

20

30

40

50

7732 11838Freq (Hz)

MA

C (

%)


0

10

20

30

40

50

936 988 2766 2811

Freq (Hz)

MA

C (

%)

(a) (b)


0

10

20

30

40

50

2763 3674 6277 7512

Freq (Hz)

MA

C (

%)


0

10

20

30

40

50

2366 4972

Freq (Hz)

MA

C (

%)

(c) (d)


0

10

20

30

40

50

4998 11108

Freq (Hz)

MA

C (

%)

(e)

Figure 8.4 Mode 13 2308 Hz MAC values (a) rotor, (b) anchor, (c) caliper,


193

Mode 17 - Rotor MAC

0

20

40

60

80

2626 3231

Freq (Hz)

MA

C (

%)


0

20

40

60

80

1718 1728 2766 2811 3215

Freq (Hz)

MA

C (

%)

(a) (b)


0

20

40

60

80

2763 3674 8614

Freq (Hz)

MA

C (

%)

Mode 17 - Inner MAC

0

20

40

60

80

2366 4972 11090 12510

Freq (Hz)

MA

C (

%)

(c) (d)


0

20

40

60

80

2375 13095

Freq (Hz)

MA

C (

%)

(e)



194

Mode 25 - Rotor MAC

0

20

40

60

80

2765 3231 5769

Freq (Hz)

MA

C (

%)


0

20

40

60

80

988 1718 2766 2811

Freq (Hz)

MA

C (

%)

(a) (b)


0

20

40

60

80

2763 3674 7512

Freq (Hz)

MA

C (

%)


0

20

40

60

80

4972 6313 7490 11090

Freq (Hz)

MA

C (

%)

(c) (d)

Mode 25- Outer Pad MAC

0

20

40

60

80

2375 4998 7536 10281

Freq (Hz)

MA

C (

%)

(e)



195

Mode 35 - Rotor MAC

0

10

20

30

40

50

4674 5153 5155 5168

Freq (Hz)

MA

C (

%)


0

10

20

30

40

50

936 988 3215 5283 5944 6947 8374

Freq (Hz)

MA

C (

%)

(a) (b)


0

10

20

30

40

50

2763 3674 8428

Freq (Hz)

MA

C (

%)

Mode 35- Inner Pad MAC

0

10

20

30

40

50

2366 4972 11090 12510 13033

Freq (Hz)

MA

C (

%)

(c) (d)


0

10

20

30

40

50

4998 7536 11108 13095

Freq (Hz)

MA

C (

%)

(e)



196

Mode 45 - Rotor MAC

0

10

20

30

40

50

965 2388 3735 6771

Freq (Hz)

MA

C (

%)


0

10

20

30

40

50

936 988 4312 5944 6772 7395 10004

Freq (Hz)

MA

C (

%)

(a) (b)


0

10

20

30

40

50

3674 4218 7277 8428 8614

Freq (Hz)

MA

C (

%)


0

10

20

30

40

50

2366 4972 6313 11090

Freq (Hz)

MA

C (

%)

(c) (d)


0

10

20

30

40

50

2375 4998 6385 7536 12533

Freq (Hz)

MA

C (

%)

(e)



197

Mode 60 - Rotor MAC

0

10

20

30

40

50

4165 5168 6383 6773 7732 7734 8599 10623

Freq (Hz)

MA

C (

%)


0

10

20

30

40

50

1718 4312 5283 6947 7395

Freq (Hz)

MA

C (

%)

(a) (b)


0

10

20

30

40

50

3674 7277 7512 8428

Freq (Hz)

MA

C (

%)


0

10

20

30

40

50

2366 4972 6313

Freq (Hz)

MA

C (

%)

(c) (d)


0

10

20

30

40

50

2375 7536 10281 12533

Freq (Hz)

MA

C (

%)

(e)



198

Mode 74 - Rotor MAC

0

20

40

60

80

965 2388 3735 5155 6771 8599 9897

Freq (Hz)

MA

C (

%)


0

20

40

60

80

988 8153 9097 9299 10744 11275 12728

Freq (Hz)

MA

C (

%)

(a) (b)


0

20

40

60

80

2763 7512 8428 9373 11586 13312

Freq (Hz)

MA

C (

%)


0

20

40

60

80

2366 4972 7490 10228

Freq (Hz)

MA

C (

%)

(c) (d)


0

20

40

60

80

7536 10281 12533

Freq (Hz)

MA

C (

%)

(e)

Figure 8.10 Mode 74 9816 Hz MAC values (a) rotor, (b) anchor, (c)

caliper, (d) inner pad and (e) outer pad.

199

Mode 86 - Rotor MAC

0

20

40

60

80

6777 7862 8607 10618 10623 12786

Freq (Hz)

MA

C (

%)


0

20

40

60

80

6947 7395 8153 9097 10744 12559 14385 14879

Freq (Hz)

MA

C (

%)

(a) (b)


0

20

40

60

80

6277 7512 10229 12859 14359

Freq (Hz)

MA

C (

%)


0

20

40

60

80

4972 12510 13033

Freq (Hz)

MA

C (

%)

(c) (d)

Mode 86 -Outer Pad MAC

0

20

40

60

80

4998 6385 10281 12533 13095

Freq (Hz)

MA

C (

%)

(e)

Figure 8.11 Mode 86 11238 Hz MAC values (a) rotor, (b) anchor, (c) caliper, (d) inner

pad and (e) outer pad.

200

Mode 90 - Rotor MAC

0

10

20

30

40

50

6774 10407 10553 11838 12851

Freq (Hz)

MA

C (

%)


0

10

20

30

40

50

7395 9097 9299 10744 14385 14879

Freq (Hz)

MA

C (

%)

(a) (b)


0

10

20

30

40

50

2763 3674 4218 6277 8614 11586

Freq (Hz)

MA

C (

%)


0

10

20

30

40

50

4972 6313 11090

Freq (Hz)

MA

C (

%)

(c) (d)


0

10

20

30

40

50

10281 11108 12533

Freq (Hz)

MA

C (

%)

(e)

Figure 8.12 Mode 90 11694 Hz MAC values (a) rotor, (b) anchor, (c) caliper, (d) inner

pad and (e) outer pad.

8.5 Summary

A complex eigenvalue analysis of the brake system has been presented using an

alternate FEA code, Abaqus. The primary contribution of Abaqus is that it is practically

easy to implement during the modelling process. Further analysis was applied in the

form of MAC calculations to determine the component modes active in the system.

This allows a direct comparison to be made to the Nastran results from Chapters 6 and

7.

Both analysis codes predict a number of unstable modes that correspond to the

201

frequency ranges of interest. However, comparing MAC values shows considerable

differences in the underlying modal contribution to the respective unstable modes.

The differences in the predictions stem from the differences in modelling the component

interfaces. The non-linear contact element formulation of Abaqus allows a more

realistic contact distribution to be captured between the pad and rotor interface, and it is

expected that the Abaqus model more closely resembles the actual case in the brake

system. A comparison to static pressure distribution at pad / rotor interface would be a

good starting point for future work on an assembly. Future work could also include

investigating alternative contact and material property correlation methods to better

understand what analysis method is best, and to improve material modelling input data.

Good correlation was found for the modes at 11.7 and 11.9 kHz between Abaqus and

Nastran respectively, a mode that strongly features 2nd

tangential in-plane motion. This

highlights the problem frequency for the model that was confirmed by experimental

testing. This type of mode is notoriously insensitive to the caliper assembly.

202

203

Chapter 9

Applications to Rotor Design in an

Industrial Environment

9.1 Introduction

In the previous chapter Abaqus was applied to the brake system assembly that formed

the basis of the investigations in this thesis. It was found that considerable differences

were noted in the unstable modes that were predicted by the two analysis codes.

However, in the case of the high frequency squeal mode featuring 2nd tangential in-

plane motion, both analyses predicited comparable system behaviour. In this chapter,

Abaqus was applied in an industrial environment to improve the design of a brake rotor

of a brake system with a known 2nd tangential in-plane related noise concern.

High frequency brake squeal, occurring above approximately 5 kHz, often involves the

tangential in-plane vibration modes of the brake rotor. Noise associated with the

tangential in-plane modes are notoriously difficult to control with typical brake squeal

countermeasures such as backplate shims or modifications to the caliper and bracket

assembly.

This chapter details an investigation into the design of a rotor for front disc brake

system as found on a typical larger passenger vehicle, and how the design can be

improved to reduce the occurrence of rotor related high frequency squeal. Note that this

brake system is not the same as that studied earlier in this thesis, since the industrial

focus required a system that was in current development and development of the

previous system was already completed.

9.2 Brake system Under Investigation

The brake system that was the focus of this investigation is shown in Figure 9.1. It

features a twin piston aluminium caliper of sliding design that is typical for a front

brake of a large passenger sedan, and is considerably larger than that studied in previous

chapters. The rotor, as displayed in Figure 9.2, is a vented design with large cooling

vanes and also features cross drilled holes.

204

Figure 9.1 Brake system under investigation in this chapter.

Figure 9.2 Layout of the brake rotor to identify main features

During braking considerable weight transfer occurs toward the front of a vehicle. As

such it is common for the front axle to support up to 80% for vehicle total brake

capacity. This is reflected in the size of the brake under consideration in this study, the

basic specifications for which are displayed in Table 9.1. This can be compared to the

specifications for the brake shown in Table 1.1, which was for a similarly sized vehicle,

but on a rear axle as opposed to the front application considered here. Also quite

different is the friction material, which is a European style semi-metallic design. The

Cross-drilled

holes

Cheeks

Swan-neck

Hat section

Friction

faces

Vanes

205

friction level is a little higher than that considered earlier, and can be more prone to

noise.

Table 9.1 Brake system specification

Vehicle Type Large passenger sedan

Engine V8 – 210 kW

Drive Wheels Front

Rolling Radius 320 mm

GVM 2150 kg

Static Weight Distribution Front 55%, Rear 45 %

Equivalent Brake Inertia 75 kg m2

Rotor Diameter 323 mm

Rotor Mass 9.9 kg

Effective Radius 141 mm

Piston Diameter 2 x 45.0 mm (equiv. 63.6mm)

9.3 Noise Evaluation

The evaluation of the noise performance of the brake system was conducted using a

brake noise dynamometer. Section 3.11 discussed TS 576, the AK noise matrix based

test, which was used in the evaluation of the noise performance of the AU II rear brake

rear brake system. In this case the TS 640 test procedure was used which is an extended

version of the AK matrix noise procedure to include an additional cold section to the

test.

The addition of the cold section to the test has been found to add considerably to the test

procedure’s ability to uncover more of the potential noise concerns. This is particularly

true of “early morning sickness” type of noise that often occurs on first application at

low speed and pressure, such as an early morning apply departing a driveway.

The test procedure called for 2557 individual braking applications in total between the

warm and cold sections with the main sections summarised in Table 9.2.

206

Table 9.2 PBR TS640 test procedure summary.

Section Stops no. Temperature Pressure Speed

Break-in 32 100 3 MPa 80 km/h

Bedding 30 100 0.2 – 3 MPa 80 km/h

Warm Section 1784 50 – 300 0.2 – 3 MPa 3 – 80 km/h

Fade 15 100 – 450 0.4 G* 100 km/h

Recovery 18 100 3 MPa 80 km/h

Cold Section 678 0 – 50 0.2 – 3 MPa 3 – 10 km/h

The brake end of the dynamometer runs in an environmentally controlled test chamber.

The chamber is 3 m x 3 m in size and is fitted with sound absorbing material on the

walls to ensure background noise spectrum remains under 60 dB(A) between 1.0 and 20

kHz. The sound recording equipment is the same as was shown in Table 3.7

Determining the overall acceptability result for TS640 is done in a straight forward

quantitative manner. A cumulative noise distribution is calculated that notes the

percentage of stops above a given SPL level in both the warm and cold sections of the

test respectively. The acceptance criteria are given in Table 9.3, but are usually

displayed graphically.

Table 9.3 Cumulative Noise occurrence acceptability for TS640. Each of the

warm and cold sections are calculated and assessed separately.

70 dB(A) 80 dB(A) 90 dB(A) 100 dB(A)

Pass ≤ 10 % ≤ 3 % ≤ 0.8 % ≤ 0.2 %

Marginal 10%< SPL ≤20% 3%< SPL ≤6% 0.8%< SPL ≤1.6% 0.2%< SPL ≤0.4%

Fail > 20 % > 6 % > 1.6 % > 0.4 %

The noise performance of this brake system is displayed graphically in Figure 9.3.

Figures 9.3 (a) and (b) show the SPL levels vs. frequency for the noisy stops in the

warm and cold sections of the test respectively. Figure 9.3 (c) displays the cumulative

distribution of noisy stops. The two dashed lines represent the boundary of acceptable

and unacceptable noise in the cumulative result according to the criteria shown in Table

9.3. The area between the two dashed lines represents the marginal zone.

207

70

80

90

100

110

120

2000 4000 6000 8000 10000 12000 14000 16000 18000

Frequency [Hz]

SP

L (

dB

[A])

(a)

70

80

90

100

110

120

2000 4000 6000 8000 10000 12000 14000 16000 18000

Frequency [Hz]

SP

L (

dB

[A])

0.1

1

10

100

70 80 90 100 110 120

Noise Level (dB[A])

No

ise

Cu

mu

lati

ve (

%)

(b) (c)

Figure 9.3 Noise dynamometer results for problem brake system with the

baseline rotor design. (a) warm section, (b) cold section and (c) cumulative

result. A pass result is that the cumulative result remains below the lower

dashed line.

It is clear from Figure 9.3 that the baseline brake system failed the brake noise screening

on the noise dynamometer, with 33% of stops registering 70 dB(A) or above in the cold

section instead of the allowable 10%. The result in the warm section was marginal.

It can also be seen that the noise is exclusively registered at 9.4 kHz in the cold section,

and the same frequency contributes significantly in the warm section. Thus control of

the 9.4 kHz noise is of critical importance for controlling the noise in this brake system.

Warm

Cold

208

9.4 Mode Description

Section 3.5 provided an overview of the types of modes that are present within a brake

rotor. Of those, the tangential in-plane compression modes are of significant

importance in this study. Tangential in-plane modes do contribute to a large number of

brake noise, as has been well documented in the literature, as found by researchers such

a Matsuzaki and Izumihara (1993), Dunlap et al (1999), and a variety of works by Chen

and co-workers (Chen et al, 2000, Chen et al, 2003, Chen et al, 2004).

Figure 9.4 shows the frequency response function (FRF) plots for the brake rotor under

free-free boundary conditions. Two FRFs were recorded from the rotor using an impact

hammer and a small accelerometer. One was taken from the rotor friction surface in a

normal direction to measure the out-of-plane response, and the second one was taken by

affixing a small aluminium block to the outer circumference of the rotor and measuring

in a tangential direction. Note that the main peaks in the bending FRF are for modes

where the rotor is in a purely diametrical bending mode and the main peaks for the

tangential direction come from the tangential in-plane compression modes. The nature

of the modes located at these peaks is known by correlation to a finite element method

free modes analysis.

There has been some suggestion that noise related to in-plane noise occurs when there is

an alignment of bending modes and tangential in-plane modes. Researchers such as

Dunlap et al (1999) and Chen et al (2002) have proposed that a tangential in-plane

compression wave and an adjacent diametrical bending mode become coupled and is the

root cause of this type of noise. The squeal noise that is generated usually correlates

closely to the frequency of the in-plane mode. This appears to be well supported from

experimental evidence such as operating deflection shape (ODS) measurements from

the respective directions during a squeal event (Chen et al, 2004). Modal coupling is

not likely to occur if the in-plane mode lies toward the centre of the frequency band

from one diametrical bending mode to the next (Chen, 2002).

209

-40

-20

0

20

40

60

0 2000 4000 6000 8000 10000 12000

Frequency (Hz)

Ac

ce

lera

nce

(d

B r

e:

1 m

/Ns

^2

)

Figure 9.4 FRF in the out-of-plane direction

-40

-20

0

20

40

60

0 2000 4000 6000 8000 10000 12000

Frequency (Hz)

Acc

ele

ran

ce

(d

B r

e:

1 m

/Ns

^2)

Figure 9.5 FRF in the tangential in-plane direction

It can be seen from Figure 9.5 that the 2nd tangential in-plane mode is well spaced

between the adjacent bending modes. Yet a significant noise problem exists that is

Tang. in-plane

compression modes

Diametrical out-of-

plane modes

210

coincident in frequency with the 2nd tangential in-plane mode of the brake rotor. It

appears that the accepted wisdom of spacing of modes does not hold in this case.

9.5 Stability Prediction

Complex eigenvalue analysis was the focus of Chapter 5 and describes the methods

used for this analysis. Potential squeal modes can be identified analytically as having

positive real parts to their eigenvalues signifying negative damping.

Typically large scale finite element models, such as that used in earlier chapters, include

the brake rotor, a complete caliper assembly, and possibly the steering knuckle and

suspension components. However, since this investigation is concerned mainly with in-

plane modes that might not be strongly influenced by the caliper geometry, and also at

higher frequencies where the influence of steering and suspension components is

negligible, a highly simplified model with only the brake rotor and pads was used. The

mesh used for the analysis can be seen in Figure 9.6, and contains 52000 10-node

tetrahedral elements with 92000 nodes.

Figure 9.6 Simplified FE model used to perform the analysis.

The boundary conditions used for the model are as follows. The brake pad abutments

were constrained in all degrees-of-freedom (DOF) except the friction surface normal

direction, and the rotor was constrained in all DOF at the mounting bolt holes. A

pressure of 1 MPa was applied across the backplate and a non-linear static analysis was

performed that included both a preload step and a rotation step following the same

211

methodology as in Section 8.2.4. Complex eigenvalues were then extracted using the

subspace projection method.

(a)

Baseline Damping

0.00%

0.05%

0.10%

0.15%

0.20%

0 0.1 0.2 0.3 0.4 0.5 0.6

Mu

(b)

Baseline

9420

9430

9440

9450

9460

9470

9480

9490

9500

9510

9520

0 0.1 0.2 0.3 0.4 0.5 0.6

Mu

Fre

qu

en

cy (

Hz)

Figure 9.7 Unstable modes in the region of the 2nd tangential in-plane

mode. (a) modal frequencies, (b) negative damping level.

The baseline model contains a pair of modes that become coupled at 9.4 kHz. Both

involve 2nd tangential in-plane compression modes within the rotor. The rotor itself

possesses axial symmetry, and in the absence of other components or imperfections a

repeated pair of modes, or doublet modes, would exist, both at the same frequency.

µµµµ

µµµµ

212

When included in an assembly this symmetry is disturbed and the frequencies of the

repeated modes deviate slightly from each other.

Figure 9.7 shows the frequency of the original modes with the 2nd tangential in-plane

motion. As the friction level is increased this pair of modes was driven closer in

frequency until the modes become coupled at the onset of instability. This critical

friction coefficient µ was 0.2. Figure 9.6 (b) displays the level of negative damping of

the unstable mode. Initially it is zero as the system is stable, but it grows once the

system has become unstable as mu is increased.

9.6 Rotor Modification

Two modified designs, shown in Figure 9.9, were proposed in an effort to reduce the

coupling of the repeated in-plane modes. Both of the modifications concerned the hat

region of the rotor only so as not to change the friction disc itself. This also means that

the actual modal spacing of the rotor is largely unchanged, since it is primarily

determined by the dimensions of the friction disc. The two proposals were:

1. An addition of 3 ribs spaced evenly around the swan-neck region.

2. Replacement of the swan-neck all together with a conical design.

Both of these modifications aim to alter the interaction of the in-plane modes through

the hat section of the rotor.

213

(a)

(b)

Figure 9.8 (a) Mode shape of unstable 9458Hz mode, (b) Rotor mode shape

for 9473 Hz.

214

(a)

(b)

Figure 9.9 Proposed rotor modifications. (a) sides of hat replaced with

conical section, (b) three stiffeners added to the swan neck.

Inserting the modified rotors into the FEM model and re-running the complex

eigenvalue analysis showed both of these modifications to be an improvement. The

tendency for the in-plane modes to become coupled is shown in Figure 9.10. Both

designs exhibit the same tendency for the repeated 2nd in-plane modes to diverge as the

coefficient of friction µ is increased.

On the basis of these positive analytical results, prototypes of both proposed designs

were manufactured. Figure 9.11 and 912 shows the summary noise test results for the

new hat and 3 stiffeners designs respectively, and it can be seen that neither system

produced any noise at 9.4 kHz. Comparing to the baseline system (Figure 9.3) it can be

seen there is an increase in occurrences at other frequencies. This could be a little

misleading because the noise dynamometer only recorded the highest peak during a

particular stop. It is, therefore, possible that the earlier high peaks at 9.4 kHz was

masking some of the noise at other frequencies at lower levels. Regardless of whether

this is the case or not, the rotors have removed the 9.4 kHz noise altogether and reduced

the overall noise levels into the acceptable range. This is a very positive result given

that it is practically impossible to build a completely noiseless brake system.

215

(a)

New Hat

9420

9440

9460

9480

9500

9520

0 0.1 0.2 0.3 0.4 0.5 0.6

Mu

Fre

qu

en

cy

(H

z)

(b)

3 Stiffeners

9420

9440

9460

9480

9500

9520

0 0.1 0.2 0.3 0.4 0.5 0.6

Mu

Fre

qu

en

cy

(H

z)

Figure 9.10 2nd in-plane modes of the modified rotors as a function of µ. (a)

new hat, (b) stiffened rotor. Note that damping is not shown because it is

equal to zero while the system is stable.

µµµµ

µµµµ

216

(a)

(b)

Figure 9.11 New hat design rotor noise dynamometer results (a) SPL vs

Frequency, (b) cumulative occurrence.

Warm

Cold

Warm

Cold

217

(a)

(b)

Figure 9.12 3 Stiffeners design rotor noise dynamometer results (a) SPL vs

Frequency, (b) cumulative occurrence

Warm

Cold

Warm

Cold

218

The rotor designs were also subject to a range of structural and thermal analyses to

ensure their suitability to put into service. Neither rotor design showed significant

degradation with regard to the performance requirements and both were deemed

suitable for production. The stiffened rotor has since entered production and returned

good noise performance in the field.

9.7 Summary

The application of complex eigenvalue analysis in an industrial environment was

presented in this Chapter. The analysis showed that the root of the instability was

coupling of repeated tangential in-plane modes, which was confirmed by MAC showing

contributing rotor modes.

Modifications to the swan-neck region of the rotor hat were analysed and two designs

were found that did not exhibit the tendency to become coupled with the friction level

studied. One of the proposed designs featuring 3 lug “stiffeners” has since been put into

production with good noise performance in the field.

219

Chapter 10

Conclusions and Future Work

10.1 Conclusions

This thesis has presented analysis of automotive disc brake squeal based around large

scale FEA models. Initial model set-up was validated against experimental modal

analysis results before the model was used for prediction of potential squeal modes,

which appear as unstable system vibration modes in a complex eigenvalue analysis.

Deeper analysis of system modes was conducted using feed-in energy analysis,

assessing strain energy distributions across the system components, and Modal

Assurance Criterions (MAC) based modal participation analysis. A parametric study

followed where the key factors that influenced unstable system modes was assessed.

An alternative commercial FEA code, Abaqus, using an alternative contact modelling

formulation was put to use in place of MSC.Nastran which had been used in the

preceding analyses in the thesis. A comparison of system stability predicted by

MSC.Nastran and Abaqus was presented, before Abaqus was used in a study to improve

the design of a disc brake rotor in an industrial environment.

An examination of the literature on brake squeal analysis was presented in Chapter 2.

Three main methods of investigation have been reviewed: analytical, experimental and

numerical. The analytical models presented include those commonly used to describe

brake squeal. These models do provide some insight into the nature of brake squeal,

and it can be envisaged how certain types of phenomena act in brake squeal. However

none of these models relate to any specific brake system, so have limited relevance to

addressing an observed squeal concern on a brake. The experimental methods

presented bear much greater relevance to addressing a specific brake squeal concern.

Unfortunately the experimental approaches also do not thoroughly describe the

behaviour of the brake system during squeal, mostly due to the difficulty in adequate

measurement of the relevant effects. The numerical approach provides a larger scale

adaptation of the simple models that allow insight into mechanisms and relevance to a

specific brake system through the use of a large DOF model of a brake system.

Determining the characteristics of a brake system, as described in Chapter 3, revealed

the complexity within a brake system. These results were successfully used in

220

developing the base FEA model in Chapter 4. Tuning of the individual components was

quite straight forward, but assembly required a considerable number of connections

types and interfaces. Reasonable agreement was found in Chapter 5 between the

baseline stability predictions and the observed squeal on the car and brake noise

dynamometer.

The tools presented in Chapter 6 proved to be very useful for deeper probing on the

brake system. While feed-in energy analysis could be used in-place of complex

eigenvalue analysis to assess the level of instability at the onset of squeal, this is not its

leading feature. Feed-in energy can be used to assess which pad is providing more of

the energy to drive the squeal, and can even be used to give very specific information

about which parts of a disc / rotor interface is most important. This should prove useful

for tuning pad designs for reducing brake squeal. Strain energy distributions do provide

good insight into which components are most active in an unstable brake system mode.

However, it needs to be applied with caution due to the differences in size and material

properties of the components so comparison to an average baseline needs to be

performed to make the comparisons valid. MAC based modal participation analysis is

used to identify component modes are most active in an unstable brake system mode.

Chapter 6 highlights the importance of the three analysis methods. Two system modes

where analysed that were both potential squeal modes. However, only one of these

modes corresponds to a concern frequency on the vehicle. Key differences in their

behaviour were identified by examining the feed-in energy, strain energy distributions

and MAC values. The difference in the squeal performance of the two comparison

unstable modes was attributed to the difference in the nature of the brake rotor modes

and their potential to become excited by the pad/rotor friction interface.

Chapter 7 presented a study on some of the parameters that affect the noise propensity

of a brake system. Many of the structural modifications change the behaviour in a

significant way, and many new unstable modes may be created in addition to addressing

some of the existing unstable modes. Reducing pad contact area was found to increase

stability of the brake system, as was increasing pad modulus. Adding structural damping

to the components of the base system was found to slightly increase the stability of the

system modes without greatly changing their character. Simulating backplate damping

shims by adding damping to the backplate increased system stability, but failed to

capture the behaviour of a laminated shim. Adding a damping shim to the back plate

221

was found to be very effective due to more adequately capturing the behaviour of a

physical shim, but even the addition of a shim without damping can influence the

behaviour of the baseline system since it adds further structural modification to the

system.

Chapter 8 presented a comparison on brake squeal analysis using Abaqus and

MSC.Nastran. Both analysis codes predict a number of unstable modes that correspond

to the frequency ranges of interest. However, comparing MAC values shows

considerable differences in the underlying component modal contribution to the

respective unstable modes. The non-linear contact element formulation of Abaqus

allows a more realistic contact distribution to be captured between the pad and rotor

interface, and it is expected that the Abaqus model more closely resembles the actual

case in the brake system. Good correlation was found for the modes at 11.7 and 11.9

kHz between Abaqus and Nastran respectively, a mode that strongly features 2nd

tangential in-plane motion.

Chapter 9 further used Abaqus to assess the design of a brake rotor. The analysis

showed that the root of the instability was coupling of repeated tangential in-plane

modes, which was confirmed by MAC showing contributing rotor modes.

Modifications to the swan-neck region of the rotor hat were analysed and two designs

were found that did not exhibit the tendency to become coupled with the friction level

studied. One of the proposed designs, featuring 3 lug “stiffeners”, has since been put

into production with good noise performance in the field.

The investigation of brake squeal propensity via modally based analysis will require

some level of simplification in all cases. This does not, however, stop useful analysis

from being undertaken. The analysis of the brake rotor designs in Chapter 9 is one

example where a significantly simplified analysis provided a great deal of insight into

system behaviour and allowed alternate rotor designs to be developed.

It must also be noted that the results presented in this thesis draw heavily on

experimental and observed squeal results. At this time is it not practical to apply

complex eigenvalue analysis as an upfront design tool. However, once baseline

correlation has been established, the analyses presented allow detailed examination of

system behaviour. Utilising numerical analysis methods in conjunction with

222

experimental results provides a practical development path for addressing brake squeal

concerns.

10.2 Recommendations for Future Work

The majority of this thesis was focused on the analysis of a large scale MSC.Nastran

FEA model. The method for assessing system stability was modally based, which

requires linearisation of the system before the complex modes analysis. This limits the

modelling options for the friction interface which used linear springs and a Coulomb

friction model. The inherent non-linearity of the contact stiffness and sensitivity of the

friction coefficient to normal pressure and slip velocity are not captured. Abaqus allows

a non-linear solution of the static analysis which captures elements of the non-linear

contact stiffness; however, linearisation before the modal analysis again limits the

nature of the friction model.

Non-linear transient analysis provides an alternate path forward where greater

sophistication can be built into friction and contact stiffness modelling. Re-analysis of

the Abaqus assembly in the time domain may provide a truer representation of the

system behaviour. However, brake squeal is a highly transient phenomenon with highly

complex geometries, interfaces and materials, so whether it will be possible to

accurately capture the full system behaviour even with unlimited computer resources

remains to be seen.

Within the context complex eigenvalue analysis, the discrepancies between the

MSC.Nastran and Abaqus models could be investigated further. A detailed study using

a visualisation technique such as scanning laser vibrometry would provide insight into

which analysis code provides the more realistic system behaviour. This could also be

extended to include the non-linear transient analysis to assess the most useful analysis

method.

A particular area of interest is the tangential in-plane modes which were the basis of the

analysis in Chapter 9. The modelling results indicated almost pure in-plane modes

without coupling to an adjacent bending mode. Some out of plane deformation would

no doubt be required for the modes to radiate high-levels of sound. An investigation

into the sound radiation efficiency of the in-plane modes could indicate if modes of this

type could be responsible for the sound radiation without bending mode involvement.

223

Little work has been found in the literature dealing with the sound radiation of unstable

brake system modes. While stability analysis allows a propensity to the squeal to be

assessed, no direct correlation to the level of squeal can be drawn. Considerable scope

exists for investigation of the sound radiation behaviour of unstable brake system

modes.

224

225

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Noise’, SAE Paper, No. 2001-01-3126.

231

Yang, S., and Gibson, R.F. (1997), ‘Brake Vibration and Noise: Reviews, Comments,

and Proposals’, Int. J. of Materials and Product Technology, Vol. 12, Nos 4-6, pp 496-

513.

Zhang, L., Wang, A., Mayer, M. and Blaschke, P. (2003), ‘Component Contribution and

Eigenvalue Sensitivity Analysis for Brake Squeal’, SAE Paper No. 2003-01-3346.

233

Publications Arising From This Thesis

Journal Papers

1. Papinniemi, A.T., Lai J.C.S, Zhao, J. and Loader, L., “Brake squeal: a literature

review” Applied Acoustics, no. 67, 391-400, 2002.

Conference Papers

1. Papinniemi, A.T., Lai, J.C.S, and Zhao, J., ‘Vibration Analysis of Disc Brake

System’, Acoustics 2001, Canberra, ACT, Australia, 21-23 Nov., 2001.

2. Papinniemi, A.T., Lai, J.C.S, and Zhao, J., ‘Modelling of a Disc Brake System for

Investigating Brake Squeal’, Dearborn, Mi, USA, 19-21 Aug, 2002.

3. Papinniemi, A.T., Lai, J.C.S, and Zhao, J., ‘Numerical Modelling of Brake Squeal,

Wespac 8, Melbourne, VIC, Australia, 7-9 April, 2003.

4. Papinniemi, A.T. and Lai, J.C.S, ‘Comparison of Energy Based Methods for Brake

Squeal Propensity’, The 18th International Congress on Acoustics 2004, April 4-9,

Kyoto, Japan, 2004.

5. Papinniemi, A.T., Lai, J.C.S, and Zhao, J., “Vibro-acoustic Studies of Brake Squeal

Noise”, Acoustics 2004, Gold Coast, QLD, Australia, 3-5 Nov, 2004.

6. Papinniemi, A.T., Lai, J.C.S, and Zhao, J., ‘Towards Reducing Brake Squeal

Propensity’, Internoise 2005, Rio de Janeiro, Brazil, 7-10 Aug., 2005.

7. Papinniemi, A.T., Zhao J. and Lai, J.C.S, ‘A Study on In-plane Vibration Modes in

Disc Brake Squeal Noise’, Internoise 2005, Rio de Janeiro, Brazil, 7-10 Aug., 2005.

8. Papinniemi, A.T., Lai, J.C.S and Zhao J., ‘Numerical Prediction of Brake Squeal

Propensity – A Critical Review’, Internoise 2006, Honolulu, Hawaii, USA, 3-6

Dec., 2006.

9. Papinniemi, A.T., Lai, J.C.S, and Zhao, J., ‘Disc Brake Squeal: Progress and

Challenges, ICSV 14, Cairns, Australia, 9-12 July 2007.

234

Reports

1. Papinniemi, A.T. and Lai, J.C.S. “Experimental Modal Analysis of a Disc Brake

System”, Report AVU 0202, School of Aerospace and Mechanical Engineering,

UNSW@ADFA, 2002.

235

Appendix A: Measurement Grids

Table A.1: Coordinates for the rotor grid.

Point no. r (mm) θ (°) z (mm) Direction

1 + 8n 40 34 z

2 + 8n 80 34 z

3 + 8n 103 28 r

4 + 8n 103 8 r

5 + 8n 110 0 z

6 + 8n 125 0 z

7 + 8n 140 0 z

8 + 8n 144

7.5n,

n = 0,..,47

-8 r

Table A.2: Pad grid coordinates. Measurement direction is in the local z direction

for all points.

Point No. x (mm) y (mm) Point No. x (mm) y (mm)

1 0 9 14 54 35

2 0 22 15 54 52

3 15 9 16 70 0

4 15 22 17 70 9

5 28 0 18 70 22

6 28 28 19 70 32

7 38 0 20 80 0

8 38 9 21 80 28

9 38 22 22 93 9

10 38 32 23 93 22

11 54 0 24 108 9

12 54 9 25 108 22

13 54 22

236

Table A.3: Caliper housing grid coordinates and measurement direction.

Point

No.

x

(mm)

y

(mm)

z

(mm)

dir. Point

No.

x

(mm)

y

(mm)

z

(mm)

dir.

1 10 -12 0 x 34 -72 0 10 y

2 0 0 0 y 35 -72 10 -20 y

3 -10 0 0 y 36 -72 10 -80 y

4 -20 10 -20 y 37 -72 0 -90 -z

5 -20 10 -50 y 38 -90 -30 25 z

6 -20 10 -80 y 39 -90 -20 25 z

7 -20 0 -90 -z 40 -90 -10 25 z

8 -20 -15 -90 -z 41 -90 0 20 y

9 -20 -30 -90 -z 42 -90 0 10 y

10 -45 -20 25 z 43 -90 10 -20 y

11 -45 -10 25 z 44 -90 10 -50 y

12 -45 0 20 y 45 -90 10 -80 y

13 -45 0 10 y 46 -90 0 -90 -z

14 -45 10 -20 y 47 -100 -20 25 z

15 -45 10 -50 y 48 -100 -10 25 z

16 -45 10 -80 y 49 -100 0 20 y

17 -45 0 -90 -z 50 -100 0 10 y

18 -45 -15 -90 -z 51 -100 10 -20 y

19 -45 -30 -90 -z 52 -100 10 -50 y

20 -60 -30 25 z 53 -100 10 -80 y

21 -60 -20 25 z 54 -100 0 -90 -z

22 -60 -10 25 z 55 -100 -15 -90 -z

23 -60 0 20 y 56 -100 -30 -90 -z

24 -60 0 10 y 57 -125 10 -20 y

25 -60 10 -20 y 58 -125 10 -50 y

26 -60 10 -50 y 59 -125 10 -80 y

27 -60 10 -80 y 60 -125 0 -90 -z

28 -60 0 -90 -z 61 -125 -15 -90 -z

29 -72 -40 25 z 62 -125 -30 -90 -z

30 -72 -30 25 z 63 -135 0 0 y

31 -72 -20 25 z 64 -145 0 0 y

32 -72 -10 25 z 65 -155 -12 0 -x

33 -72 0 20 y

237

Table A.4: Anchor bracket grid coordinates and measurement direction.

Point

No.

x

(mm)

y

(mm)

z

(mm)

dir. Point

No.

x

(mm)

y

(mm)

z

(mm)

dir.

1 0 35 0 y 19 145 10 10 z

2 0 35 -17.5 y 20 120 0 20 y

3 0 35 -35 y 21 105 0 20 y

4 0 35 -50 -z 22 72 0 20 y

5 0 25 -50 -z 23 40 0 20 y

6 0 0 -50 -z 24 25 0 20 y

7 25 0 -50 y 25 0 10 10 z

8 40 0 -50 y 26 0 30 10 z

9 72 0 -50 y 27 25 -10 -70 -z

10 105 0 -50 y 28 40 -10 -70 -z

11 120 0 -50 y 29 72 -10 -60 -z

12 145 0 -50 -z 30 105 -10 -70 -z

13 145 25 -50 -z 31 120 -10 -70 -z

14 145 35 -50 -z 32 120 -10 30 z

15 145 35 -35 y 33 105 -10 30 z

16 145 35 -17.5 y 34 72 -10 30 z

17 145 35 0 y 35 40 -10 30 z

18 145 30 10 z 36 25 -10 30 z

239

Appendix B: Free Rotor Mode Shapes

1 - (2,0)

994 Hz

2 - (0,1)

2010 Hz

3 - (3,0)

2430 Hz

4 - (0,2)

2550 Hz

5 - (1,1)

2690 Hz

6 - RI 2

2900 Hz

7 - (1,2)

3200 Hz

8 - (4,0)

3800 Hz

9 - TH(2,0)

4230 Hz

240

10 – RI 3

4630 Hz

11 - (5,0)

5290 Hz

12 – TH(1,1) + RI 1

5730 Hz

13 – TH(2,1)

6770 Hz

14 - (6,0)

6990 Hz

15 - RI 4

7120 Hz

16 – CI 1

7840 Hz

17 - TH(0,2)

8130 Hz

18 - (7,0)

8900 Hz

241

Appendix C: Example Nastran Input Deck

Note: Element, grid and boundary conditions have been omitted except 1st and last

entries to save space.

$ NASTRAN input file created by the MSC MSC.Nastran input file

$ translator ( MSC.Patran 2001 r3 ) on December 17, 2002 at

17:27:22.

ASSIGN DBC='gr5c5.xdb', RECL=1024

$ Complex Eigenvalue Analysis, Direct Formulation, Database

SOL 107

$ Direct Text Input for Executive Control

COMPILE GMA SOUIN=MSCSOU NOLIST NOREF

ALTER 'MTRXIN' $

ADD K2PP, /K2PPX/V,Y,FRIC=(1.0,0.0) $

EQUIVX K2PPX/K2PP/-1 $

ENDALTER

compile sedrcvr souin=mscsou noref nolist

alter 'if.*gpfdr.*statics','' (1,0) $

IF ( GPFDR AND (STATICS OR APP='REIG' OR

APP='CEIGEN' OR APP1='FREQRESP') ) THEN $

alter 'if.*gpfdr.*statics'(2,0),''(2,0) $

IF ( GPFDR AND (STATICS OR APP='REIG' OR

APP='CEIGEN' OR APP1='FREQRESP') ) THEN $

CEND

SEALL = ALL

SUPER = ALL

TITLE = MSC.Nastran job created on 04-Dec-02 at 18:47:55

ECHO = NONE

MPC = 16

$ Direct Text Input for Global Case Control Data

K2PP = FSTIF

PARAM, FRIC, .5

SUBCASE 1

$ Subcase name : linst1

SUBTITLE=linst1

SPC = 2

LOAD = 2

DISPLACEMENT(PLOT,SORT1,REAL)=ALL

SPCFORCES(PLOT,SORT1,REAL)=ALL

STRESS(PLOT,SORT1,REAL,VONMISES,BILIN)=ALL

FORCE(SORT1,REAL,BILIN)=ALL

SUBCASE 2

$ Subcase name : cplx1

SUBTITLE=cplx1

CMETHOD = 1

SPC = 2

VECTOR(PLOT,SORT1,REAL)=ALL

SPCFORCES(PLOT,SORT1,REAL)=ALL

ESE(PLOT,THRESH=0.000000000001)=All

STATSUB = 1

$ Direct Text Input for this Subcase

BEGIN BULK

242

$ /me2/pi20/antti/nastran/griz_rear/analysis/gr5cplx3.res

$

$ DMIG Header

$

DMIG FSTIF 0 1 1 0

$

$ Springs for normal stiffness - area no. 1

$ k = 0.20E+04

$

CELAS2 210001 0.20E+04 40001 1 50001 1

CELAS2 210002 0.20E+04 40004 1 50216 1

……

……

CELAS2 210165 0.20E+04 46976 1 50692 1

CELAS2 210166 0.20E+04 46977 1 50694 1

$

$ Friction stiffness DMIGs for surface no. 1

$

DMIG FSTIF 50001 1 50001 3 -.20E+04

DMIG FSTIF 40001 1 50001 3 0.20E+04

DMIG FSTIF 50001 1 40001 3 0.20E+04

DMIG FSTIF 40001 1 40001 3 -.20E+04

$

DMIG FSTIF 50216 1 50216 3 -.20E+04

……

……

DMIG FSTIF 46977 1 46977 3 -.20E+04

$

PARAM POST 0

PARAM AUTOSPC YES

PARAM,NOCOMPS,-1

PARAM PRTMAXIM YES

EIGC 1 CLAN MAX

0. 120

$ Direct Text Input for Bulk Data

PARAM, PRGPST, NO

$ Elements and Element Properties for region : piston

PSOLID 1 1 0

$ Pset: "piston" will be imported as: "psolid.1"

CHEXA 20001 1 20018 20016 20015 20017 20036

20033

20034 20035

CHEXA 20002 1 20032 20031 20016 20018 20038

20037

20033 20036

……

……

CHEXA 60394 9 60534 60535 60471 60470 60652

60536

60476 60475

$ Referenced Material Records

$ Material Record : steel_pad

243

$ Description of Material : Date: 27-Jan-00 Time:

14:21:07

MAT1 7 207000. .3 7.86-9

$ Material Record : linning


14:21:07

MAT1 4 5000. .3 2.6-9

$ Material Record : cast_iron_rotor


14:21:07

MAT1 6 118000. .32 7.1-9

$ Material Record : cast_iron_anchor


14:21:07

MAT1 2 165000. .3 7.4-9

$ Material Record : alum_piston


14:21:07

MAT1 1 70000. .33 2.7-9

$ Material Record : steel_pin


14:21:07

MAT1 5 207000. .3 7.86-9

$ Material Record : alum_housing


14:21:07

MAT1 3 68000. .33 2.65-9

$ Multipoint Constraints of the Entire Model

MPCADD 16 3 6 9 12 15

RBE2 1 100101 123456 10699 10702 10703 10714

10725

10726 10727 10728 10729 10730 10731 10732

10733

10734 10735 10736 10737 10790 10796 10802

11983

11986 11992 12003 12004 12005 12006 12007

12008

12009 12010 12011 12012 12013 12060 12066

RBE2 2 100102 123456 20035 20036 20038 20041

20042

20044 20046 20048 20050 20052 20054 20055

20057

20059 20061 20064 20067 20068 20069 20071

20074

20075 20077 20079 20081 20083 20085 20087

20089

20092 20094 20096

MPC 3 100102 2 -1. 100101 2 1.

MPC 3 100102 3 -1. 100101 3 1.

MPC 3 100102 4 -1. 100101 4 1.

MPC 3 100102 5 -1. 100101 5 1.

MPC 3 100102 6 -1. 100101 6 1.

RBE2 4 100201 123456 10980 10981 10982 10983

10984

244

10985 10986 10987 10988 10989 10990 10991

10992

10993 10994 10995 10996 10997 10998 10999

11000

11001 11002 11003

RBE2 5 100202 123456 70022 70023 70027 70028

70029

70033 70034 70036 70039 70040 70041 70042

70066

70067 70068 70070 70071 70075 70076 70078

70081

70082 70083 70084

MPC 6 100202 1 -1. 100201 1 1.

MPC 6 100202 2 -1. 100201 2 1.

MPC 6 100202 3 -1. 100201 3 1.

MPC 6 100202 4 -1. 100201 4 1.

MPC 6 100202 5 -1. 100201 5 1.

MPC 6 100202 6 -1. 100201 6 1.

RBE2 7 100301 123456 12234 12235 12236 12237

12238

12239 12240 12241 12242 12243 12244 12245

12246

12247 12248 12249 12250 12251 12252 12253

12254

12255 12256 12257

RBE2 8 100302 123456 80022 80023 80027 80028

80029

80033 80034 80036 80039 80040 80041 80042

80066

80067 80068 80070 80071 80075 80076 80078

80081

80082 80083 80084

MPC 9 100302 1 -1. 100301 1 1.

MPC 9 100302 2 -1. 100301 2 1.

MPC 9 100302 3 -1. 100301 3 1.

MPC 9 100302 4 -1. 100301 4 1.

MPC 9 100302 5 -1. 100301 5 1.

MPC 9 100302 6 -1. 100301 6 1.

RBE2 10 100401 123456 30095 30098 30100 30102

30104

30106 30108 30110 30112 30114 30116 30118

30119

30120 30123 30125 30127 30129 30131 30133

30135

30137 30139 30141

RBE2 11 100402 123456 70184 70187 70188 70191

70192

70198 70199 70204 70210 70214 70215 70216

70217

70218 70221 70223 70225 70227 70229 70231

70233

70235 70237 70239

MPC 12 100402 2 -1. 100401 2 1.

MPC 12 100402 3 -1. 100401 3 1.

MPC 12 100402 4 -1. 100401 4 1.

245

MPC 12 100402 5 -1. 100401 5 1.

MPC 12 100402 6 -1. 100401 6 1.

RBE2 13 100501 123456 31337 31340 31342 31346

31348

31350 31352 31354 31356 31358 31360 31361

31362

31365 31367 31369 31371 31373 31375 31377

31379

31381 31383

RBE2 14 100502 123456 80184 80187 80188 80191

80192

80198 80199 80204 80210 80214 80215 80216

80217

80218 80221 80223 80225 80227 80229 80231

80233

80235 80237 80239

MPC 15 100502 2 -1. 100501 2 1.

MPC 15 100502 3 -1. 100501 3 1.

MPC 15 100502 4 -1. 100501 4 1.

MPC 15 100502 5 -1. 100501 5 1.

MPC 15 100502 6 -1. 100501 6 1.

$ Nodes of the Entire Model

GRID 10001 12.1 113. 41.25

GRID 10002 8.97052 113. 44.3795

……

……

GRID 100502 63.85 136. -72.035

$ Loads for Load Case : linst1

SPCADD 2 4 6

LOAD 2 1. 1. 1 1. 3

$ Displacement Constraints of Load Set : anch_constraint

SPC1 4 123 30447 30450 30453 30456 30459

30462

30465 30466 30471 30474 30477 30480 30483

30484

……

……

46385 46391 46397 46931 46940 46949

$ Loads for Load Case : new

$ Pressure Loads of Load Set : piston_press

PLOAD4 1 20081 14. 20234

20236

PLOAD4 1 20082 14. 20238

20235

……

……

PLOAD4 3 11562 14. 12300

12401

PLOAD4 3 11585 14. 12392

$ Referenced Coordinate Frames

ENDDATA a698e5a1

247

Appendix D: bdfread Source Code

c bdfread v2.1

c

c Reads the Nastran .bdf file and processes coincident nodes.

c Input file bdfread.in contains input data.

c Writes results file with formatting for appendage to final

c .bdf file.

c Write reoprt file stating the number if coincident grids

c and whether DMIGs were written

c

c

c Written by Antti Papinniemi 13/08/02

c

program bdfread

c Global variables

c vec1, mat1 - whole grid point ids, cordinates respectively

c vec2, mat2 - search regoin grid ids and coordinates

c vec3, mat3 - coicident grid ids and cordinates

c bounds - search cordinates x1,x2,y1,y2,z1,z2

c numgd - no of grids in model

c maxgd - no grids in search region

c numct - no contact pairs

c dir1, dir2 - contact surface normal, tangential directions

c nosets - number of contact sets

c offset - spring element id offset

c spstif - stiffnes of normal springs

c tol - search tolerance

c inpfl, outfl, resfl - input, output results files

real mat1(50000, 3), mat2(50000,3), mat3(10000,3),

bounds(24,6)

integer vec1(50000), vec2(50000), vec3(10000,3)

integer maxgd, numct, numgd

integer dir1, dir2

integer i, nosets, offset

real spstif, tol

character inpfl*16, outfl*16, resfl*16

c read executive control data

open (UNIT = 20, FILE = 'bdfread.in')

read(20,*)

read(20,*) inpfl

read(20,*) resfl

read(20,*) outfl

read(20,*) nosets

c open files

open (UNIT = 10, FILE = inpfl)

open (UNIT = 30, FILE = outfl)

open (UNIT = 40, FILE = resfl)

248

c write DMIG header, report header

write(40,500) '$', '$ DMIG Header','$'

write(40,501) 'DMIG', 'FSTIF', 0, 1, 1, 0

write(30,502) 'Report: ', inpfl

write(30,503) 'No of contact regoins: ', nosets

c read all grid points

call readgd(vec1, mat1, numgd)

c loop for each contact set

do 200 i = 1, nosets

c read contact set data

read(20,*)

read(20,*) spstif

read(20,*) tol

read(20,*) dir1,dir2

read(20,*) offset

read(20,*) bounds(i,1), bounds(i,2), bounds(i,3),

+ bounds(i,4), bounds(i,5), bounds(i,6)

c find grids in search region

call search(numgd,vec1,mat1, bounds(i,1), bounds(i,2),

bounds(i,3)

+ ,bounds(i,4),bounds(i,5), bounds(i,6),

+ vec2, mat2, maxgd)

c find coincident grids

call coinct(maxgd, tol, vec2, mat2, vec3, mat3, numct)

c write set results to output file

write(30,504) 'Contact set: ', i

write(30,505) numct , ' coicident pair(s) were found.'

c write springs for contact stiffness

call rtsprg(numct, dir1, vec3, spstif, offset, i)

c write DMIGs (for two sets only)

if (i .LE. 2) then

call rtdmig(numct, i, dir1, dir2, vec3, spstif)

if (numct .GT.1 0) write(30,506) 'DMIGs written'

endif

200 continue

c close files

close (10)

close (20)

close (30)

close (40)

c Format statements

500 format(A1,/,A13,/,A1)

501 format(A4,4X,A8,4(I8))

502 format(/,A8, A12)

503 format(/,A23, I4)

249

504 format(/,A13, I4)

505 format(4X,I6,A)

506 format(8X,A)

stop

end

c---------------------------------------------------------------

---------------

subroutine readgd(vec, mat, num)

c Reads all the grid points ids and cordinates

c

c inputs:

c none

c

c outputs:

c vec - gird point ids

c mat - grid point coordinates

c num - number of grids

c Argument declarations

integer vec(50000)

real mat(50000,3)

c Local variables

integer i, j, num

character line*80

c Read grid points

j = 0

do 100 i = 1, 50000

read(10, 511, END = 999) line

if (line(1:5) .EQ. 'GRID ') then

j = j + 1

backspace(10)

read(10,512) vec(j), mat(j,1),

+ mat(j,2), mat(j,3)

endif

if (line(1:5) .EQ. 'GRID*') then

j = j + 1

backspace(10)

read(10,513) vec(j), mat(j,1),

+ mat(j,2), mat(j,3)

endif

100 continue

999 num = j

c Format statments

511 format(A)

512 format(8X, I8, 8X, 3(E8))

513 format(8X, I16, 16X, 2(E16), /, 8X, E16)

250

return

end

c---------------------------------------------------------------

---------------

subroutine search(num,veca,mata,x1,x2,y1,y2,z1,z2,

+ vecb,matb,max)

c Finds grid points in search area and returns their ids and

coords

c inputs:

c num - number of total grid points in model

c veca - ids of all grids

c mata - coordinates of all grids

c x1,x2,y1,y2,z1,z2 - boundaries of search region

c

c outputs:

c vecb - ids of grids found in search region

c matb - coordinates of grids in search region

c max - number of contact pairs found


integer num, max

integer veca(50000), vecb(50000)

real mata(50000,2), matb(50000,3)

real x1,x2,y1,y2,z1,z2

c Local variables

integer i, j

j = 0

do 160 i = 1, num

if ((mata(i,1) .GE. x1) .AND. (mata(i,1) .LE. x2))

then

if ( (mata(i,2) .GE. y1) .AND. (mata(i,2) .LE. y2))

then

if ( (mata(i,3) .GE. z1) .AND. (mata(i,3) .LE.

z2)) then

j = j + 1

vecb(j) = veca(i)

matb(j,1) = mata(i,1)



endif

endif

endif

160 continue

max = j

return

end

c---------------------------------------------------------------

---------------

251

subroutine coinct(max, tol, veca, mata, vecb, matb, num)

c Finds the coincident grids in the remaining grid points

c inputs:

c max - number of remaining grid points

c tol - tolerance for coincidence

c veca - all remaining grid ids

c mata - cordinates of remaining grid points

c

c outputs:

c vecb - grid ids of contact pairs

c matb - cordinates of contact pairs

c num - the number of contact pairs


integer max, num

integer veca(50000), vecb(10000,3)

real mata(50000,2), matb(10000,3)

real tol

c Local variables

real temp

integer i, j, k

num = 0

do 130 i = 1, max-1

do 120 j = i+1, max

temp = sqrt((mata(j,1)-mata(i,1))**2 + (mata(j,2)

+ -mata(i,2))**2 + (mata(j,3)-mata(i,3))**2)

if (temp .LE. tol) then

num = num+1

vecb(num,1) = num

vecb(num,2) = veca(i)

vecb(num,3) = veca(j)

do 150 k = 1, 3

matb(num,k) = mata(i,k)

150 continue

endif

120 continue

130 continue

return

end

c---------------------------------------------------------------

---------------

subroutine rtsprg(num, dir, vec, stfnes, offset, count)

c Writes the spring entries for the nastran input file

c inputs:

c num - number of springs entries to write (1 per contact

pair)

c dir - spring direction

252

c vec - contact pair id numbers array

c stfnes - contact stifness value

c offset - offset for spring element ids (must be unique)

c count - contact set no

c

c outputs:

c none


integer num, dir, offset, count

integer vec(10000,3)

real stfnes

c Local variables

integer elemno

write(40,541) '$','$ Springs for normal stiffness - area

no. ', count

write(40,542) '$ k = ', stfnes, '$'

do 170 i = 1, num

elemno = offset+i

write(40,543)

'CELAS2',elemno,stfnes,vec(i,2),dir,vec(i,3),dir

170 continue

c Format statements

541 format(A1,/,A42, I4)

542 format(A6, E8.2,/,A1)

543 format(A6, 2X, I7, 1X, E8.2, 4(I8))

return

end

c---------------------------------------------------------------

---------------

subroutine rtdmig(num, type, dir1, dir2, vec, stfnes)

c Writes the DMIG entries for the nastran input file

c inputs:

c num - number of DMIGs to write (4 per contact pair)

c type - (1 or 2) opposite contact surfaces need to be

different type

c also specifies no of contact surfaces

c dir1, dir2 - normal and tangential force directions

respectively

c vec - contact pair id numbers array

c stfnes - spring stifness value for simulating normal force

c

c outputs:

c none


integer num, type, dir1, dir2

integer vec(10000,3)

253

real stfnes

c local variables

real stiff1, stiff2

if (type .EQ. 1) then

stiff1 = stfnes

stiff2 = -1*stfnes

else

stiff1 = -1*stfnes

stiff2 = stfnes

endif

write(40,553)'$','$ Friction stiffness DMIGs for surface

no.',type,'$'

do 180 i = 1, num

write(40,554) 'DMIG',

'FSTIF',vec(i,3),dir1,vec(i,3),dir2,stiff2







write(40,555) '$'

180 continue

c Format statements

553 format(A1,/,A42,I4,/,A1)

554 format(A4,4X,A8,2(I8),8X,I8,I7,1X,E8.2)

555 format(A1)

return

end

c---------------------------------------------------------------

---------------

Vibro-acoustic Studies of Brake Squeal Noise - UNSWorks

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