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Vehicle Disc Brake Roughness Noise -Experimental Study of the Interior Noise and

Vibro-Acoustic Modelling of Suspension Systems

Eskil Lindberg

Licentiate Thesis

Stockholm 2011Material and Structural Acoustics Group

The Marcus Wallenberg Laboratoriet for Sound and Vibration ResearchDepartment of Aeronautical and Vehicle Engineering

Postal address Visiting address ContactRoyal Institute of Technology Teknikringen 8 Tel: +46 8 790 76 10MWL/AVE Stockholm Email: [email protected] 44 StockholmSweden

Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan i Stock-holm framläggs till offentlig granskning för avläggande av teknologie licentiatexamenfredag den 30 September 2011, 13:00 i sal MWL74, Teknikringen 8, KTH, Stockholm.

TRITA-AVE-2011:63ISSN-1651-7660ISBN-978-91-7501-096-0

c© Eskil Lindberg, 2011

Vehicle Disc Brake Roughness Noise - Experimental Study of the Interior Noiseand Vibro-Acoustic Modelling of Suspension SystemsEskil Lindberg

Material and Structural Acoustics GroupThe Marcus Wallenberg Laboratoriet for Sound and Vibration ResearchDepartment of Aeronautical and Vehicle EngineeringRoyal Institute of Technology

Abstract

Prediction of vehicle disc brake roughness noise is a non-trivial challenge. In fact,neither the source mechanisms, nor the transfer paths are so far well understood. Tra-ditionally, disc brake noise problems are studied as part of the friction-induced noisefield, where the source is considered to be a more or less local phenomenon related tothe brake disc and brake pad. However, for the roughness noise of interest here thisviewpoint is not adequate when attempting to solve the interior noise problem sincethe transfer of vibro-energy from the brake into the vehicle body is a crucial aspect andplays an important role in the understanding and solution to the problem. The vibro-acoustic energy transfer associated with the brake roughness noise is a problem wheregeometrical complexity and material combinations, including rubber bushings, pose anintricate modelling problem. Additionally, system altering effects from moving partsand loadings are important, e.g. due to the steering or brake systems. In addition,the source mechanisms themselves must also be understood to be able to solve theproblem. The current work constitutes a combined experimental and theoretical inves-tigation, aiming at an increased understanding of the source, the transfer paths and howthey are affected by change in the operational state. The experimental study of the ve-hicle disc brake roughness noise, is based on measurements conducted in a laboratoryusing a complete passenger car. It is found that the interior noise is a structural-bornebroadband noise event well correlated to vehicle speed and brake pressure. The resultssuggest that the friction source may be divided into vibrations created in the sliding di-rection and vibrations created normal to the contact plane, where the sliding directionlevels appear to be proportional to brake pressure according to Coulomb’s friction law;the vibration level in the normal direction of the contact plane on the other hand hasbehaviour proportional to Hertz contact theory. The measurements also indicate that

the brake force created carried by the suspension system when braking will also al-ter the vibro-acoustic response of the system. To facilitate the theoretical simulations,an approach for modelling of the suspension system is developed. The vibro-acoustictransfer path model developed is using a modal based on the Craig-Bampton methodwhere a restriction on the coupling modes is suggested. The approach suggested usesundeformed coupling interfaces, to couple structures of fundamentally different stiff-ness such as may be the case in a vehicle suspension system where for instance rubberbushings are combined with steel linking arms. The approach show great potential inreducing computational cost compared to the classical Craig-Bampton method.

Keywords: Acoustics; Disc brake; Surface topography; Wire brush noise; Rough-ness noise; Component mode synthesis; Vehicle suspension;

iv

Dissertation

A Licentiate of Technology is an intermediate Swedish academic degree that can beobtained half-way between the MSc and the PhD. While less formal than a DoctoralDissertation, examination for the degree includes writing a thesis and a public presen-tation.

The work presented in this Licentiate was carried out at the Department of Aeronauti-cal and Vehicle Engineering, the Royal Institute of Technology (KTH) in Stockholm,Sweden.

This thesis consists of two parts. The first part gives an overview of the research witha summary of the performed work. The second part collects the following scientificarticles:

Paper A. E. Lindberg, N-E Hörlin and P. Göransson "An Experimental Study ofInterior Vehicle Roughness Noise from Disc Brake Systems", To be submitted to Wear.

Paper B. E. Lindberg, N-E Hörlin and P. Göransson, "Component Mode Synthe-sis Using Undeformed Interface Coupling Modes to Connect Soft and Stiff Substruc-tures", To be submitted to Vehicle System Dynamics.

v

Acknowledgements

• First of all I would like to thank my supervisors Nils-Erik Hörlin and PeterGöransson! Thank you for your support and guidance. Nisse, thank you foralways challenging me and my ideas! After a meeting with you I often have thesame feeling as after a five hour exam.

• Big thanks to Kent Lindgren and Danilo Prelevic for invaluable help with mymeasurements! Without you, Kent, these measurements wouldn’t be possible.

• Thanks to everyone who has tried to help me with the strange letters and there tome totally illogical order! Just to mention a few: Mathias, Jacques and Eleonora.

• To all industrial partners at SAAB automobile and Opel thank you for a goodcooperation!

• Tack Mamma, Pappa och Ida!

Contents

I Overview and Summary 1

1 Introduction 31.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Social motivation of thesis . . . . . . . . . . . . . . . . . . . . . . . 41.3 Commercial motivation of this thesis . . . . . . . . . . . . . . . . . . 41.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Brake noise and friction-induced sound and vibrations 72.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Friction-induces noise . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Brake noise classification . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 Roughness noise theory . . . . . . . . . . . . . . . . . . . . 9

3 Measurement 113.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1.1 External roller and brake pump . . . . . . . . . . . . . . . . . 113.1.2 Shaker measurement . . . . . . . . . . . . . . . . . . . . . . 133.1.3 External load . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.1 Sources of error . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.2 Brake pressure . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.3 System loading . . . . . . . . . . . . . . . . . . . . . . . . . 183.2.4 Vehicle speed . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Conclusions and findings . . . . . . . . . . . . . . . . . . . . . . . . 253.3.1 Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

ix

4 Suspension modelling 274.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2.1 General problem . . . . . . . . . . . . . . . . . . . . . . . . 294.2.2 Change of basis . . . . . . . . . . . . . . . . . . . . . . . . . 304.2.3 Local modes . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 334.3.1 Test structure . . . . . . . . . . . . . . . . . . . . . . . . . . 334.3.2 Vibro-acoustic response . . . . . . . . . . . . . . . . . . . . 344.3.3 Evaluation of the approach . . . . . . . . . . . . . . . . . . . 35

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5 Outlook 39

II Appended Papers 45

x

Part I

Overview and Summary

1

CHAPTER 1

Introduction

This chapter gives the motivations for the research effort put into this work. It gives abrief introduction to scope of the thesis, and what the aim of the work has been. Thechapter ends with the outline of the thesis.

1.1 BackgroundResearch is the search for new and novel knowledge, where the scientific study is theactive, systematic and methodical process of accumulating this knowledge. The meth-ods used in the scientific study can be many e.g. cartography, case study, classification,experience, experiment, interview, mathematical model, simulation, statistical analy-sis and ethnography. The findings that are discussed in this thesis are mainly based ontwo of these methods i.e., experiments and mathematical models. This thesis consistsof two appended papers A and B; in paper A an experimental study was conductedand in paper B a mathematical approach was developed. Often a distinction betweenbasic- and appliedresearch is made. Basic research is the scientific study which issaid to be driven by the curiosity of the researcher with the primary goal of extendingthe human knowledge and theoretical understanding of the building blocks that buildup our existence. Applied research on the other hand is goal oriented, the goals canbe commercial or social, set by e.g. companies, states or unions. This is an appliedresearch thesis, where both commercial and social interests have motivated the study,the goals was set by both a company (SAAB automobile AB) and the Swedish state(Vinnova/Green Car program). The overall goal of the state is to help the Swedishvehicle industry to develop environmentally friendlier vehicles. The industrial goal is

3

4 1 Introduction

to solve a specific problem and find tools and methods to cope with these. Bearing inmind the applied scope of this research the personal motivation of the author has beenthe strong curiosity to gain new knowledge and understanding through a systematicand methodical process.

1.2 Social motivation of thesisThis is an applied study where vehicle suspension structural borne noise is studied. Thesocial motivation of this thesis set by the Swedish state was to help in the developmentof vehicles with reduced negative impact on the environment. Two environmental con-cerns, i.e. reducing noise pollution and energy consumption, prevail. The functionalaspect that ties these problems together is the structural mass. Both structure-bornesound and energy consumption can be argued to be correlated to the structural mass ofthe vehicle. Generally speaking high structural mass leads to less structure-borne noisebut higher energy/fuel consumption of the vehicle and vice versa. It can therefore beargued that there must be a trade-off between energy consumption and structural bornenoise in vehicles. However, problems of structure-borne noise are not only governedby the structural mass. In addition, damping and isolation treatments can be designedtogether with optimisation of stiffness and mass properties to open up profound pos-sibilities. Nevertheless, to minimise the structural mass for a given noise problem re-quires a deep understanding of the vibro-acoustic system and the source mechanisms.This is where this is aiming to contribute to the knowledge building. Initially an exper-imental study was conducted to determine if the problem at hand is a structure-borneproblem and to understand more of the complexity of the vibro-acoustic system. Pa-rameters affecting the vibro-acoustic source were varied to gain understanding of thesource mechanisms. This experimental study laid the foundation of the first part ofthis thesis (paper A). The second part of this thesis has been the development of amathematical modelling approach (paper B). The approach was developed to be ableto better understand the complex vibro-acoustic transfer of structural borne sound in avehicle suspension system.

1.3 Commercial motivation of this thesisThe focus of this work is directed towards, structure-borne sound in vehicles, suspen-sion system which is investigated with respect to a broadband friction-induced vibro-acoustic noise source. This specific brake noise phenomenon has many names, in the

1.4 Overview 5

vehicle industry it is usually referred to as wire brush noise, but in this thesis the termroughness noise will be used. The commercial motivation for this research project asset by SAAB automobile AB was to gain knowledge and understanding of the brakeroughness noise, in order to be able to treat it in an adequate way. Costumers have highdemands; e.g. on the interior acoustic environment, and the fuel consumptions of thevehicle, both strongly linked to the structural mass of the vehicles. It can therefore beconcluded that commercial and social goals coincide well, underlining the relevanceof the topic.

1.4 Overview

Starting this project it was clear that to adequately treat the interior problem of rough-ness noise three different open questions had to be understood: i) what is governingthe friction source, which parameters are affecting the generation of vibrations in thecontact? It was clear that there was a gap here where the common knowledge of thescientific community has not been able to explain this source phenomenon; ii) how isthe brake system assembly (calliper, pads, disc etc.) effecting the finale interior noise;iii) how do vibrations generated in the disc-pad interaction transfer from the brakesystem in to the vehicle.The first step of this project was to map the problem. This was done using a fullvehicle measurement setup. From this measurement it was concluded that the interiornoise problem is a nearly purely structure-borne noise problem. Hence, to understandthe transfer path the suspension system needed a vibro-acoustic model. Moreover,these measurements strongly indicated that the brake pressure would alter the transferpath, e.g. when loading the suspension system with a brake force. This suggestedthat the vibro-acoustic model of the suspension also has to take into account the staticpreload. Together with the frequency range of interest (up to 1 kHz) it was clear thatno existing model was suitable, hence a model had to be developed within this project.Therefore, the second step of this project was to develop a mathematical approach tomodel the suspension system. In this thesis a component mode synthesis approachusing undeformed interface coupling conditions is suggested. It has been shown thatthis approach may be an efficient and qualitative good procedure to model a systemwith suspension system properties (see Chapter 4 or paper B).This thesis discusses what is needed to treat this brake noise problem. The following issome of the outcomes from this thesis: i) the levels of vibro-acoustic source have dif-ferent proportionality to the applied brake pressure in different directions (see Chapter

6 1 Introduction

3 or paper A); ii) the vehicle speed and the brake pressure dependence on the interiornoise, together with system altering effects of the brake force (see Chapter 3 or paperA); iii) the gain and limitations of the suggested mathematical approach (see Chapter4 or paper B).

1.5 Outline• Chapter 2 gives an overview of the related research in the field of friction-

induced noise and vibrations, and disc brake noise in general.

• Chapter 3 presents an overview of the experimental study on the disc brakeroughness noise.

• Chapter 4 presents an overview of the mathematical modelling approach sug-gested in this thesis.

• Chapter 5 gives the outlook of the project.

• Appended paper A presents the complete results and description of the experi-mental study.

• Appended paper B presents the complete evaluation and description of the sug-gested mathematical approach.

CHAPTER 2

Brake noise and friction-inducedsound and vibrations

The experimental part of this thesis consists of an investigation of the broadband ve-hicle brake noise phenomenon referred to as wire brush noise, roughness noise orrubbing noise. This chapter aims at placing the investigated brake noise in a sensiblecontext with friction noise and other brake noise phenomena.

2.1 BackgroundThe subject of friction-induced sound and vibrations is truly multidisciplinary and havemany applications e.g. music acoustics (bow instruments), seismology and railwayacoustic (curve squeal). The common factor of all of these is that an unstable frictionforce creates a dynamic excitation of the interacting bodies. A friction force is a non-conservative entity and is governed by the tangential stresses created by the relativemotion of two bodies Shpenkov (1995). Friction forces may, from a micro-structuralviewpoint, be explained as adhesive junctions formed by asperities in contact, and theshear force needed to cause breakaway Sheng (2008). When asperities break loose,they release stored elastic energy, resulting in a vibro-acoustic response. In addition,the ploughing effects of abrasion wear Wriggers (2006) can also be a mechanism inthe vibro-acoustic source. The exact type of excitation is highly dependent on theproperties of the contact, such as the surface roughness, the sliding speed, and thecontact pressure Persson (2000).Various types of noise phenomena with different spectral frequency contents are as-sociated with friction, such as tonal and broadband noise, either due to feedback of

7

8 2 Brake noise and friction-induced sound and vibrations

structural resonances or surfaces roughness. In 1979, Yokoi and Nakai (1979) made aclassification of friction-induced noise to be either rubbing or squealing. Rubbing is abroadband phenomenon where a large frequency range is excited, whereas squealingis a high pitch tonal noise where only a narrow frequency band is excited.

2.2 Friction-induces noise

In the literature, a vast amount of knowledge can be found on the acoustic behaviour ofsome of the important local contact parameters, primarily from different experimentalstudies. For instance, Yokoi and Nakai (1979, 1980, 1981a,b, 1982) have shown ex-perimentally in a series of papers that noise levels (sound recorded close to the contactand vibrations of one of the contact bodies) have strong correlation with both surfaceroughness and sliding speed. Furthermore, these results have been confirmed by othersand further studied on similar effects for various types of materials and setups, Othmanand Elkholy (1990); Othman et al. (1990); Stoimenov et al. (2007); Ben Abdelouniset al. (2010); Zahouani et al. (2009); Jibiki et al. (2001).However, most publications concerning friction-induced noise focus understandablyon the squealing noise, perhaps due to its perceived annoyance. Squealing noise eventsare most of the time associated with a stick-slip phenomenon. The concept of stick-slip is sometimes used loosely but in general it stems from a variation in the frictionforce. It can originate from the difference of dynamic and static friction coefficient orvarying normal force caused by for example resonances, and can occur on many dif-ferent length scales. Therefore the source mechanism may be seen as a local or globalphenomenon of the surface in contact. For further discussion of stick-slip phenomenasee for instance Akay (2002); Sheng (2008); Persson (2000); Chen et al. (2005).

2.3 Brake noise classification

Despite the multitude of brake noise phenomena, it may be argued that all of themcan be classified as belonging either to squealing and/or roughness friction-inducednoises. However, it is also common to classify brake noise according to where inthe frequency spectra the noise phenomenon can be expected. For instance, Akay(2002) uses this kind of classification which serves as a good tool to find a commonname of the phenomenon and thus, the appropriate literature on the subject. In fact,there are many different names of brake noises, sometimes without a clear and unique

2.3 Brake noise classification 9

correspondence. Most attention concerning brake noise has been focused on the high-pitch tonal-noise phenomenon, such as, the disc brake squeal, (DBS) phenomenon (outof thousands see Chen et al. (2005); Papinniemi et al. (2002); Kinkaid et al. (2003);Hoffmann and Gaul (2008)). Another, scarcely studied, tonal brake noise problem isthe moan phenomenon (to the knowledge of the author the core of the literature isNack and Joshi (1995); Gugino et al. (2000); Wang et al. (2003); Kim et al. (2005);Kim and Park (2006); Hoffmann and Gaul (2008)). Common for these tonal brakenoise problems are that they are considered to have a strong dynamic coupling to thesupporting structure. This coupling is known to be a non-linear feedback betweenstructural dynamics and unsteady surface contact, Sheng (2008); Chen et al. (2005),where structural resonances create an unstable friction force that feeds energy back intothe resonance. For instance, DBS occurs when a circumferential mode of the disc istriggered, and a dynamic force is created at the same frequency as the mode, resultingin an unstable system. In addition, there are many other brake noise phenomena whichmay not arise from an unstable non-linear feedback phenomenon, which has to someextent been studied in literature, for example (hot and cold) judder and roughnessnoise (wire brush). Judder occurs when the disc exhibits non-uniform behaviour inthe circumferential direction, e.g. disc thickness variations Sheng (2008); Hoffmannand Gaul (2008). However, the roughness brake noise problem seems not to havebeen studied much in the literature. The source may be described as a broadband“rubbing/roughness/scratch” noise. Its broadband nature, compared to the tonal natureof for instance DBS, suggests a fundamental difference of the generating mechanisms.Perhaps the most prominent is that the roughness phenomenon does not exhibit a stronglink between structural resonances and the source mechanism itself. A supportingillustration comes from flow acoustics, where steady mean flow whistle noise, such asthe tones from a flute, are by necessity considered as non-linear feedback mechanisms.In contrast, noise sources of broadband character, are often considered to have a weaklink between system resonances and sound generating mechanism, e.g. the broadbandpart of a turbulent sound generated by a fan Carley and Fitzpatrick (2000); Powell(1961).The scope of the current thesis is to study the broadband disc brake noise problem;typically referred to as wire brush, rubbing or roughness noise.

2.3.1 Roughness noise theory

The surface roughness and the sliding speed are two very important parameters whencharacterising roughness noise. How the surface roughness of the interacting bodies

10 2 Brake noise and friction-induced sound and vibrations

affects noise generation has been studied for quite some time. For instance, Yokoi andNakai (1982) used a so called pin-on-rim setup where they pressed a steel rod onto anunlubricated rotating disc. The surface roughness of the disc was varied between mea-surements, and they concluded that there is a correlation between increasing surfaceroughness and an increase in sound pressure and vibration levels in the system. Theyfound that the noise could be predicted with the simple formula, Eq. (2.1):

∆Lp (dB) = 20 log10( HHref

)m, (2.1)

where H is a statistical value of the surface roughness, and m = 0.8 for the overallvalue of the Sound Pressure Level, (SPL) and m = 1.2 for the peak SPL for the rod res-onance frequencies. They showed that there is a strong correlation between increasedsliding speed and the sound and vibration levels observed in the system. The relativesound pressure level change due to increasing speed could then be approximated bythe relation:

∆Lp (dB) = 20 log10( VVref

)n, (2.2)

where V is the sliding speed and n is a value that can range between 0.6 and 1.1.The correlation between surface roughness and noise has also been confirmed Oth-man and Elkholy (1990) and Ben Abdelounis et al. (2010). Furthermore, Othman andElkholy (1990) also stated that the correlation is independent of the contact sample sizeand material. As a matter of fact, also the correlation between speed and noise statedby Yokoi and Nakai (1982) has been confirmed by Smyth and Rice (2009); Ben Abde-lounis et al. (2010). Moreover, Smyth and Rice (2009) showed that the sliding speedhad no effect on the frequency content of the roughness noise, and Ben Abdelouniset al. (2010) showed that the noise dependency on the surface roughness and the slid-ing speed could be separated. They found that the problem may be modelled using thesame variables as Yokoi and Nakai (1982), but using 0.8 ≤m≤ 1.16 and 0.7 ≤n≤ 0.96,as

∆Lp (dB) = 20 log10(( VVref

)n ( HHref

)m). (2.3)

In addition, Othman et al. (1990) have stated that “The magnitude SPL is sensitive tovariation in contact load; increasing the contact load tends to increase the SPL and viceversa”. In that paper a spring stylus was run over a rough surface and the roughnesswas estimated from the sound generated.

CHAPTER 3

Measurement

This chapter present an experimental study of interior brake roughness noise. The ex-periments where conducted in a laboratory environment, the test object was a smallpassenger car with disc brakes. The study showed that the problem of interior rough-ness noise can be viewed as a structural-borne noise problem well correlated to boththe vehicle speed and the brake pressure.

3.1 Experimental setupThe experiments discussed here, were performed under laboratory conditions where asmall passenger car was put on rollers (see Fig. 3.1). The test object was selected for itssize and weight, based on the hypothesis that roughness noise is more prominent for alightweight vehicle. The main part of this study consisted of several noise and vibrationrecordings, for different vehicle speeds and brake forces. In each measurement thebrake force and vehicle speed was kept constant through the recording, and all noiseand vibration signals were acquired simultaneously. The sound pressure was recordedboth inside the passenger cabin and close to the brake system. The accelerations wererecorded at numerous locations in the brake system. For the sake of conciseness theresults herein are from measurements conducted while driving only the left front wheelof vehicle.

3.1.1 External roller and brake pumpThe vehicle tyre was driven by an external roller and the front left wheel was posi-tioned such that the tyre was captured in between two rotating smooth steel cylinders

11

12 3 Measurement

Figure 3.1: Photograph of the full vehicle experimental rig.

(diameter 151.5 mm) (see Fig. 3.2). The vehicle was oriented horizontally and thesteering kept in a non turning position. The roller was driven by an electric engine, en-capsulated in the roller casing, in order to eliminate the noise from the vehicle drivelineitself. In addition, disturbances of magnetic fields in measurement equipment causedby the high voltage frequency converter of the engine were controlled by shielding ofthe high voltage cables.Roughness brake noise problems are usually only of concern for low vehicle speedand low brake pressure; most likely due to the noise masking provided by other noisesources at higher speeds. Furthermore, for low vehicle speed, high brake pressure isseldom used for a long time, and thus would not be a problem. Consequently, in thisstudy only relatively low speed and low brake pressure where investigated.To ensure a constant and stable brake pressure in the brake line, an external brake pumpsystem was built. In these measurements, the brake line pressure was varied from 0 to7.25 Bar. In this interval, the background noise was low enough for the measurementof sound pressure to be meaningful.Increased air-borne background noise from tyre/roller interaction was a problem whena high torque of the rollers was necessary, that is high speed and brake pressure. This

3.1 Experimental setup 13

Figure 3.2: Photograph of the roller and tyre.

noise may arise from two nearly related forces, that is, the friction force (tangent direc-tion of tyre) and the adhesion force (radial direction of tyre). With an increased frictionforce, the strain in the tyre grows and hence more vibro-acoustic energy is released inthe slip phase. It has also been shown that adhesion forces in the tyre/road (roller)separation is a noise generating mechanism Andersson and Kropp (2009). However,Persson (2000) has argued that adhesion forces is not significant for lubricated contact.The problem with tyre noise in these measurements was thus minimised by spraying asmall quantity of water on the tyre. Hence, by reducing the friction coefficient and theadhesion bounds, the background noise was greatly reduced.

3.1.2 Shaker measurement

As part of the focus of the present work is to investigate the structure-borne sound,hence the vibro-acoustic behaviour of the suspension system, using a shaker excitationof the brake system was studied. This is quite difficult measurements as excitation ofthe brake calliper in all three coordinate directions is not straightforward to conductunder operational conditions.

14 3 Measurement

Figure 3.3: Photograph of the external load mounting device, one of the three shaker mount-ings (disc rotational axis, Y direction), accelerometer mountings (three on the cal-liper and on the brake pad, mounted in the disc rotational axis, Y)

A shaker (LDS model no. v203) was used in three different arrangements, excitingthe calliper in the three spatial directions. Z being the vertical direction and Y beingthe inward disc rotational axis direction; consequently, X is in the rear direction ofthe vehicle. Hence, the contact plane of disk and pad is parallel with the XZ-plane.Shaker measurements were only conducted when rollers were shut off. In Fig. 3.3 thearrangement when exciting the Y-direction is shown.

3.1.3 External load

The disc brake system typically consists of a disc, two pads and a calliper. The discfunction is to generate a motion proportional to the speed of the vehicle and the padsare designed to generate friction in the interaction to the disc. The calliper functionis to enable the hydraulic system to press the pads on to the disc while holding thepads fixed in position. The brake force generated from a vehicle can be quite large,and hence the calliper must distribute a high force on to the suspension system. Tosimulate the force on the calliper by a high brake pressure, a 110 kg weight was hung

3.2 Results and discussion 15

from the calliper. The weight was hung via wire to the calliper by a special designedconnector see Fig. 3.3.

3.2 Results and discussion

3.2.1 Sources of error

The measurements herein were performed in a laboratory environment using a roller todrive the left front wheel of a small passenger car; while an external brake pump wasused to control the pressure in the brake line. The roller was driven by an electricalengine. The electrical engine, roller and the tyre roller interaction are all unwantednoise sources, mainly contributing to the external background air-borne noise. Therealso exist effects where tyre/roller vibrations transfer through the system, these proba-bly being important mainly at low frequencies. At higher frequencies these vibrationsare assumed to be isolated from the car body by the softness of the tyre.The effect of the tyre on the final interior noise is not a trivial question to answer, inthe appended paper A an investigation on the tyre effect is presented. There a clearinfluence of the tyre on the final interior noise for a given excitation of the calliper isfound.A microphone was placed close to the roller in order to estimate the air-borne back-ground noise level from these sources. This investigation showed that for the condi-tions studied in this thesis (the speed and the brake pressure) the measurements resultsare of good quality and are not contaminated by outside air-borne noise sources (seeappended paper A for more details, discussion and concluding results). It was also con-cluded in paper A that the interior noise may be considered a purely structural-bornenoise problem.

3.2.2 Brake pressure

It is clear from the observations made in the experimental study that the interior rough-ness noise may be considered as a broadband phenomenon. The measurements showthat increased brake pressure lead to a broadband increase of the interior SPL. Thismay also be observed concerning the vibration levels of the brake pad (see paper A).In Fig. 3.4 the normalised total (0.1-1 kHz) levels of the acceleration of the calliper(X-,Y- and Z-directions) and interior SPL is plotted as a function of brake pressure,the corresponding vehicle speed is 2.9 km/h and all curves are normalised to the accel-

16 3 Measurement

0.5 1 2 4 80

5

10

15

20

25

Brake pressure (log10

scale) [bar]

Nor

mal

ised

Lev

el0.

1−1k

Hz [d

B]

Visualisation line ∝P

Visualisation line ∝P 2/3

Interior SPLCalliper

x

Callipery

Calliperz

Figure 3.4: Total normalised Levels between 0.1-1kHz , Function of brake pressure. InteriorSPL, calliper acceleration levels, circles interior SPL, calliper acceleration in Z-,Y-,X-direction, crosses, stars and triangles respectively, with the correspondingvehicle speed of 2.9 km/h, normalised with the acceleration level of the calliper Y-direction for a brake pressure of 1/2 bar and speed of 2.9 km/h, Two visualisationline are added, the dashed line show a 3 dB per doubling of brake pressure slopei.e. linearly proportional to the brake pressure (P), the dashdotted line slope isproportional to the brake pressure as ∝ P2/3.

eration level of the calliper in Y-direction for a brake pressure of 1/2 bar and speed of2.9 km/h. Two visualisation lines are added, the dashed line shows the corresponding3 dB per doubling of brake pressure slopes i.e. linearly proportional brake pressure(P), and the dashdotted line shows the slope that is proportional to the brake pressureas ∝ P2/3. Note that in Fig. 3.4 the brake pressure is plotted in a logarithmic scale.As discussed previously, increasing the brake pressure while keeping the sliding speedconstant will result in an increase of the interior noise in the vehicle and the vibrationsof the brake system.Based on the assumption that the vibro-acoustic frictional source is proportional to thestored elastic energy released in the breakaway when the asperities break loose, it couldbe argued that the source level should be linearly proportional to the contact pressureand hence increased by 3 dB per doubling of the contact pressure. Indeed, as may beseen in Fig. 3.4, the total (0.1-1 kHz) acceleration level of the calliper in the vertical(Z-) direction follows this quite well. On the other hand, the slopes of the interior SPL


and the calliper acceleration in the X- and Y-directions appear, to have slopes that areproportional to the brake pressure as ∝ P2/3. Keep in mind that the contact plane ofdisk and pad is the XZ-plane and the sliding direction is in the Z-direction.In an attempt to describe the vibro-acoustic source, one could argue that the vibra-tions levels generated in the contact zone should be proportional to the stiffness (orresistance to motion). The broadband character of this source may then be explainedas local stiffness variations in the contact zone, where the overall level is consideredto be proportional to the ”DC” component of the stiffness. Assuming that the ”DC”component of the stiffness in the normal direction could be represented by Hertz con-tact theory and the tangential resistance to motion by an elasto-plastic analogy of theCoulomb’s friction law, then, for Hertz contact theory the normal elastic contact stiff-ness is proportional to the contact force Johnson (1985) as Kn ∝ F2/3

n , where Kn is thenormal contact stiffness and Fn is the normal force. From the elasto-plastic analogy ofthe Coulomb’s friction law one could then argue that the friction force can be viewedas a plastic ”stiffness” as Kplastic

t ∝ Fnµ where µ is the friction coefficient. From thisthe following vibration level relation could then be formed,

∆La (dB) = 10 log10( PαPαref

), (3.1)

where Pref is an arbitrary reference contact pressure, and α = 2/3 for the levels in thenormal direction to the contact zone and α = 1 in the tangential (or sliding direction).If the contact between pads and disc could be described by Hertz theory and Coulomb’sfriction law then it is interesting to see that there seems to be a link between the normalcontact stiffness (Hertz theory) and the interior SPL. It should also be noted that theinterior SPL has a slope very similar to both the calliper accelerations in X- and Y-direction. Possible explanations for these behaviours, see Fig. 3.4, might be: i) most ofthe vibro-acoustic energy is realised in the asperity break loose in the sliding directionin the frictional contact, and due to the orientation (sliding direction coincide fairlywell with the vertical direction see Fig. 3.3) of the brake system the vertical directionis mostly excited, ii) the vibration levels of X- and Y-directions are not uniquely de-pendent on the normal contact stiffness, instead they might be dependent on the plasticfrictional contact stiffness, iii) the interior SPL is more related to calliper vibrationslevels in the X- and Y-directions than in the Z-direction . Possibly the suspension withshock absorber, isolates and absorbs, these vibrations more efficiently. Based on thetype of brake-disc-pad assembly and the suspension system (transfer path), and thehypothesis from Eq. (3.1), a relation for the interior SPL may be written as,

18 3 Measurement

∆Lp (dB) = 10 log10( P2/3+ γPP2/3

ref + γPref), (3.2)

where γ = Ht/Hn is a constant that gives a measure of the relative influence of eachexcitation direction to the interior noise Ht = Psound/Fexci

t and Hn = Psound/Fexcin . This

constant γ could be experimentally determined. From the shaker measurements dis-cussed in section 3.1.2 the constant was determined as,

γ =

√√√√∫ 1000

100|Ht( f )|2 d f∫ 1000

100|Hn( f )|2 d f

(3.3)

where f is the frequency in Hz and Hn and Ht is the measured transfer functions ofsound pressure over excitation force in Y- and Z-direction respectively. In the currentinvestigation γ was found to be 0.10. In Fig. 3.5 the interior SPL is plotted togetherwith three visualisation lines from Eq. (3.1) with α = 2/3, α = 1 and Eq. (3.2) usingthe experimental γ = 0.10. These data are recorded for the brake pressure range of2.5-5 bar this since this was the range in which the best agreement for the SPL couldbe found in Fig. 3.4. A clear correlation for the interior SPL may be observed inFig. 3.5 with the combined model. It should be noted that the extreme values, a zeroγ or an infinite γ correspond to a model proportional to only P2/3 or P respectively.In the current investigation the experimental γ was found to be small and hence theinterior SPL had a slope close to the curve for P2/3. However, having said that, for adifferent design of the brake and suspensions system a different γ would most likelyhave been found. Thus, from the combined Hertz contact theory and a Coulomb’sstiffness analogy model of the interior SPL, it is suggested that the orientation of thebrake and suspension system is an important design parameter for the reduction ofinterior roughness noise.

3.2.3 System loadingYet another operational parameter, the brake pressure has a big influence on both theinterior noise and acceleration levels of the brake system. This is quite obvious fromthe clear trend can be seen with increased brake pressure and increased acoustical re-sponse (see Fig. 10 paper A). The noise generation dependence of the contact pressurehas to the knowledge of the author not been extensively investigated before. Thus,the strong link between the brake pressure and the interior noise is a new and originalresult arising out of this work.


2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

Brake pressure [bar]

Nor

mal

ised

Lev

el0.

1−1k

Hz [d

B]

Visualisation line ∝PInterior SPL

Visualisation line ∝P 2/3+P 0.1

Visualisation line ∝P 2/3

Figure 3.5: Total normalised Levels between 0.1-1kHz , Function of brake pressure. InteriorSPL, with the corresponding vehicle speed of 2.9 km/h, normalised SPL for abrake pressure of 2.5 bar and speed of 2.9 km/h, Three visualisation line are added,the dashed line show a 3 dB per doubling of brake pressure slope i.e. linearlyproportional to the brake pressure (P), the dashdotted line slope is proportionalto the brake pressure as ∝ P2/3 and the solid line show the combined model inEq. (3.2).

Observing the shifting in the frequency of the peaks in Fig. 9(b), paper A, togetherwith the observation that different frequency bands are amplified more than others inFig. 9(a), paper A, indicate that there might also be a process where the vibro-acousticsystem is affected by the brake pressure. Hence, not only the source mechanisms are af-fected by the contact pressure but also the actual transmission paths between the sourceand the interior SPL. One possible reason for this behaviour shown in Fig. 9, paper A,could be the coupling conditions in the contact zone itself, that is the increased brakepressure leads to stronger coupling and hence the vibro-acoustic system is changed.Another hypothesis might be that an increased brake pressure will also lead to an in-creased brake force, and this force must be carried by the calliper. This may leadto pre-loading of bushings and geometrical non-linearity when different connectorschange relative position and thus also in this case changing the vibro-acoustic system.In Fig. 3.6 the acceleration levels of the outer brake pad are shown, with Fig. 3.6(a)showing a zoom of the three highest peaks that could be seen in Fig. 9(b), paper A, forthe three different non-zero brake pressures. In Fig. 3.6(b) the acceleration levels are

20 3 Measurement

plotted for three different situations, the thick black and the dashed lines are for the twomeasurements with a shaker excitation. The thick black line is for the case of a 110 kgexternal load and was used to simulate the brake force at high brake pressure. Thedashed line is for the case when no external load was used, for the same amplitude ofelectric signal to the shaker. In both cases the brake pressure was kept at 5 bar. Since,an adaptor was necessary to enable shaker excitation, the thin solid line in Fig. 3.6(b)is included to show that the system still is fairly intact, despite the external load. Thethin solid line showing the results from similar measurements as in Fig. 3.6(a) exceptthat the adaptor was used, when 5 bar brake pressure was used for a vehicle speed of1.9 km/h. For further information on the adaptor setup see paper A.Interestingly enough, when comparing the plots in Fig. 3.6, there is a shift upwards ofthe two highest (in frequency) peaks with the load (Fig. 3.6(b)) and this shift seemsto be in the same order of magnitude as may be observed in Fig. 3.6(a). Hence, thehypothesis that static brake force loading may change the vibro-acoustic system prop-erties appear to be valid.The static loading influence on the interior noise is highly dependent on the vibro-acoustic transfer path properties, thus governing how much noise are transferred intothe vehicle compartment for a given excitation. The effect of the static loading fromthe brake force is modelled using measured transfer functions (see Eq. (3.4)). Trans-fer functions were measured (calliper acceleration to interior sound pressure) with andwithout the 110 kg load and an estimate of the interior noise was made from accelera-tion of the calliper. Three different transfer functions were measured separately with ashaker, exciting the calliper in the three coordinate directions and simultaneously mea-suring the acceleration, in the direction of the excitation force, in the excitation pointand the interior sound pressure. The interior noise was estimated by using acceleration(all three coordinate directions) measured on the calliper for the case of 5 bar brakepressure and 1.3 km/h vehicle speed. The total level was estimated by treating eachcoordinate directions as being uncorrelated. In Fig. 3.7 the total level of the transferfunction estimation with and without the external load is shown together with the di-rectly measured interior SPL. The plot is zoomed in at the frequency region between500-700 Hz since it is the region where the strongest amplification of interior SPL canbe observed for increased brake pressure (see Fig. 9(a) paper A). The interior noisewas then estimated using

|pest|2 =

∣∣∣arolx

∣∣∣2 ∣∣∣Hshap, ax

∣∣∣2 +∣∣∣arol

y

∣∣∣2 ∣∣∣∣Hshap, ay

∣∣∣∣2 +∣∣∣arol

z

∣∣∣2 ∣∣∣Hshap, az

∣∣∣2 (3.4)

where pest is the estimated interior sound pressure and arol is the acceleration measuredin X, Y or Z direction when braking with the roller. Hsha

p, a is a transfer function between


550 600 650 700 75060

62

64

66

68

70

72

74

76

78

80

Frequency [Hz]

Leve

l [dB

re.

1 μ

m/s

2 ]

5.0 [bar]3.3 [bar]1.8 [bar]

(a) Acceleration levels of the outer pad in the disc rotationalaxis direction, when applying external pressure to the brakeliquid. dashed line 1.8 bar, thin line 3.3 bar and thick line5 bar. Vehicle speed 1.3 km/h. No adaptor, (Zoomin fromFig. 9(b) paper A, 5 Hz resolution

550 600 650 700 75070

72

74

76

78

80

82

84

86

88

90

Frequency [Hz]

Leve

l [dB

re.

1 μ

m/s

2 ]

Shaker, 0 [kg]Shaker, 110 [kg]Roller, 0 [kg]

(b) Acceleration levels of the outer pad in the disc rotationalaxis direction. Dashed line shaker excitation of brake calliperin disc rotational axis direction no external load, no rotation,5 bar pressure. Thick line shaker excitation of brake calliperin the disc rotational axis direction a 110 kg external load, norotation, 5 bar pressure. Thin line roller excitation, 5 bar. Allusing an adaptor, 5 Hz resolution

Figure 3.6: Graph comparing the frequency shifts, effect of an external force and the brakeforce, for the pad acceleration in normal direction to disc.

22 3 Measurement

interior sound pressure and driving point acceleration (X, Y or Z direction) either withstatic loading or with out loading, measured with a shaker. The estimated interiornoise for the two cases is compared with the measured interior noise from the samemeasurement as when arol was recorded in (see Fig. 3.7).There are of course limitations with this procedure of estimating the interior noise andthe assumptions made. The first assumption implies that the brake system (calliper,disc and pad assemble) could be treated as a rigid body suggesting that the movementof the system could be described using six degrees of freedom (DOF), the three spatialdirections and the three rotations around the corresponding axis. These DOFs couldideally be represented using six independent shaker measurements of the brake sys-tem. Here, only three independent shaker measurements were used. Secondly, assum-ing that the different shaker measurements could be conducted separately, also impliesthat all measured directions could be treated as being uncorrelated. These assumptionsmay appear very crude, since there will of course be correlation between the differentDOFs, and a three DOF representation of the motion of the brake system may be harshgeneralisation. But, interestingly enough the results presented in Fig. 3.7 gives a goodrepresentation of the interior noise considering all these simplifications. Furthermore,it appears that the model using the loaded transfer functions have a better representa-tion of the directly measured results. Hence, it could be argued that the system loadingeffect discussed above, have an influence on the transfer path problem. Moreover, ithas also been demonstrated that there is a clear link between vibrations in the calliperand the noise inside the vehicle.

3.2.4 Vehicle speed

To investigate the influence of the vehicle speed, Fig. 3.8 shows the sound pressure,inside the vehicle (a), and acceleration levels, of the brake pad (b), respectively when5 bar pressure is applied to the brake liquid for the three speeds 1.3, 1.9 and 2.9 km/h.The pad vibration plotted was measured in the Y-direction (inward disc rotational axisdirection). What can be noted in Fig. 3.8(a) is that changing the vehicle speed es-sentially corresponds to a broadband increase of the overall SPL inside the plottedfrequency range. Consequently, no frequency band seems to be affected more thananother, as was the case in Fig. 3.6 for different brake pressures. It is futhermore clearfrom Fig. 3.8(b) that the peaks in the frequency response do not shift with increasingspeed.Roughness noise is probably present in all vehicles with solid material friction brakes.However, it is mostly masked by other noise sources. To the knowledge of the author


500 550 600 650 70015

20

25

30

35

40

45

50

55

60

65

70

Frequency [Hz]

SP

L [d

B r

e. 2

0 μP

a]

MeasurementEstimation without loadEstimation with load

Figure 3.7: Measured and estimated SPL for 5 bar brake pressure and 1.3 km/h using the adap-tor. Thick line, direct measurement. Thin line, estimation using “unload” transferfunction. Dashed line, estimation using “load” transfer function.

most of the cases where wire brush (roughness) noise is reported as a problem arefor low vehicle speed when background levels are lower, so the effects of tyre-roadand other sources are of course important in how the noise event is perceived. Inthis investigation the masking effects were minimised to allow for a better picture thegeneration of the excitation itself.One of the goals of the present work, was to see how the interior roughness brake noisein the vehicle correlates to the sliding speed. Furthermore, it was interestedly to studyhow results from experimental studies of simplified setups such as the pin-on-rim setupby Yokoi and Nakai (1982), may be related to the problem of an entire brake systemincluding the vibro-acoustic transfer path problem. In the literature it may be foundthat statistical values describing the surface roughness are important parameters thatgovern the frictional noise source mechanism, but, measurement results in literaturehave also suggested that surface roughness parameters and sliding speed parametersmay be independently studied, Ben Abdelounis et al. (2010). In other words, slidingspeed as a noise generating parameter may be studied without knowledge of the param-eters describing the surface roughness. In addition, the size of the contact pairs shouldnot affect the behaviour. It may be seen in Fig. 3.8 that there is indeed a broadbandincrease of both interior noise levels and brake system vibrations. As stated by Smythand Rice (2009), the noise dependence on sliding speed should not affect the frequency

24 3 Measurement

100 200 300 400 500 600 700 80025

30

35

40

45

50

55

60

65

70

Frequency [Hz]

SP

L [d

B r

e. 2

0 μP

a]

2.9 [km/h]1.9 [km/h]1.3 [km/h]

(a) Sound pressure levels

100 200 300 400 500 600 700 80068

70

72

74

76

78

80

82

84

86

88

90

Frequency [Hz]

Leve

l [dB

re.

1 μ

m/s

2 ]

2.9 [km/h]1.9 [km/h]1.3 [km/h]

(b) Acceleration levels of brake pad,Y-direction

Figure 3.8: Levels for different speeds. thick black line 1.3 km/h, dashed line 1.9 km/h andthick gray line 2.9 km/h. Brake pressure all cases 5 bar.

3.3 Conclusions and findings 25

1 2 3 4 5 6 70

2

4

6

8

10

12

14

16

18

Roller speed [km/h]

Nor

mal

ised

SP

L 0.1−

1kH

z [dB

]

Ben Abdelounis et al.Yokoi and Nakai1.3 [bar]5.0 [bar]

Figure 3.9: Total normalised SPL between 0.1-1 kHz as a function of roller speed. Crosses,interior noise for a brake pressure of 1.3 bar. Circles, interior noise for a brakepressure of 5 bar. Dashed line, limits from Eq. (2.3). Dotted line, limits fromEq. (2.2). Normalisation total SPL of individual signal and 1.3 km/h.

content, but only increasing the overall level of the noise, which is verified for the cur-rent problem see Fig. 3.8. It may also be argued that sliding speed dependence of thenoise problem is simpler to model than the brake pressure effect, as there seems to belittle system altering effect associated with change of the speed. In Fig. 3.9 the interiortotal (0.1-1 kHz) normalised SPL is shown as a function of the vehicle speed for thebrake pressure 1.3 and 5 bar respectively, the normalisation chosen as the total SPLfrom the lowest speed using the same brake individual brake pressure. The limits fromthe two equations Eq. (2.2) (Yokoi and Nakai (1982)) and Eq. (2.3) (Ben Abdelouniset al. (2010)) are also included in the graph. In fact, it can be seen that even thoughthese formulas were derived from simplified measurements and the results from thisstudy is for a much more complex setup, the results correlate surprisingly well.

3.3 Conclusions and findings

Herein, the main results from measurements performed on a full vehicle laboratory testrig are presented. The test rig was designed by the author for the purpose of full bodyvehicle measurements of brake noise. Evidently it is possible to reproduce the effects

26 3 Measurement

of friction-induced noise in a vehicle using this test rig.

3.3.1 FindingsThe vibro-acoustic excitation of the roughness noise vibrations may be divided intotwo components. That is, i) vibro-acoustic excitation in the sliding direction, in thesemeasurements a linear proportionality to the brake-pressure was found, ii) the vibro-acoustic excitation in normal direction of the contact plane, in these measurements anon-linear proportionality to the brake-pressure was found (the brake pressure to thepower of 2/3 is proportional to the acceleration levels). The proportionalities in bothcases can be argued to be linked to the stiffness, where the resisting force (stiffness) inthe sliding direction is directly proportional to the brake pressure which is consistentwith Coulomb’s friction law. The contact stiffness in the normal direction is propor-tional to the contact pressure to the power of 2/3, according to the Hertz contact theory.Combined models of the Hertz contact theory and the Coulomb’s friction law, maythen be used to predict the relative total interior SPL change due to a change in brakepressure.

3.3.2 ConclusionsIt is concluded that the vehicle phenomenon of interior disc brake roughness noise(wire brush) is a purely structural-borne noise problem, hence not air-borne (see pa-per A). Moreover, the noise phenomenon is dependent on the brake pressure and thevehicle speed. Both interior SPL and brake system vibrations increase with increasingbrake pressure and sliding speed.Furthermore, increased brake pressure may lead to system-altering effects, and thisloading of the system has to be included to accurately model the problem.

CHAPTER 4

Suspension modelling

This chapter introduces a component mode synthesis approach where the undeformableattribute of an interface between a soft and a stiff connector is directly enabled in theformulation of the coupling condition. This approach of undeformed coupling inter-faces builds on the classical Craig-Bampton method. It is shown that the computationalcost can be greatly reduced using the undeformed coupling interfaces approach com-pared both to a direct finite element solution as well as the classical Craig-Bamptonmethod. This for a system build of components of the same properties as the rubberbushing/ linking arm assembly, the accuracy is from an engineering perspective good(less than 1% error).

4.1 BackgroundIt was concluded from the experimental study that disc brake roughness noise may beviewed as structural-borne noise phenomenon. Furthermore, system altering effects ofthe brake force was observed for the transfer path system. These two findings leadto the conclusion that a deeper knowledge of the vibro-acoustic system characteristicsof the vehicle suspension was needed. This part of the thesis introduce an approachto model multifaceted systems such as the suspension system of a vehicle, the rubberbushings and complex geometries are thus handled, using a finite element approachwhere the number of degrees of freedom is reduced to a small number of local normalmodes and a set of six undeformed interface coupling modes per coupling interfaces.This is found to lead to computationally fast and accurate modelling of systems suchas the vehicle suspension.

27

28 4 Suspension modelling

4.1.1 Introduction

Herein, a reduction technique for modelling of suspension systems is suggested, hav-ing the potential of overcoming the problem of an unnecessary large number of DOFsin the mathematical description of the vibro-acoustic field.Reduction methods in structural dynamics (vibro-acoustics) often use the approach ofmodes where a set of normal or eigenmodes are generated from an eigenvalue problem.From the theory of modal superposition, it is known that a reduced set of eigenmodescan be used to span an approximate solution of the original problem. The choice ofthe set of included modes in the approximated solution may come from a physicalmotivation. The normal choice is only to include modes with eigenfrequencies be-low a particular upper frequency limit. The theory of how reduced subsystems can becoupled together is commonly referred to as component mode synthesis (CMS). CMScan be described as a method where the local behaviour of individual substructuresare described by a set of reduced local eigenmodes. Force and displacement continu-ity between the substructures are enforced by a set of coupling functions (constraintmodes).There exists several different versions of the CMS method, the main distinction ofthe different formulations are the use of local eigenmodes; fixed interface, Craig andBampton (1968); free interfaces, (Rixen (2004); hybrid, MacNeal (1971)). Commonto all is that continuity between substructures is ensured by a static condensation. Thestatic solution of the inner DOFs is computed by successively prescribing either a unitdisplacement or unit force (fixed and free interfaces respectively) at the coupling inter-face one DOF at a time. This procedure generates a solution vector for each interfaceDOF and these vectors may be seen as coupling modes. Together with the local eigen-modes the solution of the fully coupled problem may now be spanned. For a morein-depth description of the different formulations, the reader is referred to the reviewpaper of De Klerk et al. (2008). A limitation of classical CMS methods such as theCraig-Bampton (C&B) method is that the reduction order is limited by the size ofthe coupling interface, hence this method is only appropriate for systems with smallcoupling interfaces, Junge et al. (2009); Tran (2009); Balmes (1996). Furthermore,non-conforming meshes between different components may appear e.g. because dif-ferent components are developed by different groups of engineers, Farhat and Geradin(1992). The problem with non-conforming mesh interfaces is usually dealt with byintroducing additional constraints in the form of so called Lagrange multipliers (seefor instance Rixen et al. (1998); Farhat and Geradin (1992)).Research on the interface reduction is a research area which has a big potential and

4.2 Theory 29

there have been some work published on the subject. For instance, Tran (2001, 2009);Balmes (1996); Herrmann et al. (2010). Most of the method uses static condensationsof the inner modes on the global problem. Thus, the new system is only comprised ofinterface DOFs, that may be used to generate yet another set of basis function by solv-ing the new eigenvalue problem. However, a physically motivated truncation criterionbased on the eigenfrequency cannot be used to choose a reduced set of interface DOFseigenmodes basis function. Instead, the method is limited to criteria’s as e.g. evaluationthe strain energy of each mode, Gérald et al. (2000) or singular value decompositionBalmes (2005); Herrmann et al. (2010).In this thesis a physically motivated technique of interfaces reduction is presented. Thetechnique is proposed for coupling between soft and stiff parts, such as the connectionto rubber bushings in the vehicle suspension system. The physical reasoning is basedon the assumption that an interface between a soft and a stiff part will have an approx-imately undeformed interface shape. Hence, the displacement of the interface can bedescribed by the six rigid ”body” motions (three orthogonal spatial directions and thethree rotations around these axes), van der Valk (2010).This formulation also allows for a substantial reduction of the original problem andcompletely eliminates the problem of non-conforming meshes, since only a conform-ing coordinate system is needed when generating the undeformed interface displace-ment functions.

4.2 Theory

4.2.1 General problem

In Fig. 4.1 the system used in this section is defined to give an basic understanding ofthe theory behind the modelling approach. The vibro-acoustic displacement field u(x)is defined in the domain Ω which is entirely enclosed by the boundary ∂Ω. The bound-ary is subdivided into a coupling interface boundary Σ and the remaining boundary Γ.Λ is the union of boundary Γ and the domain Ω.The continuum mechanical displacement field u(x) may be written in a FE discretizedrepresentation. The displacement DOF vector may also be subdivided into UΣ andUΛ where UΣ corresponds to the nodal displacement vector belonging to the couplingboundary Σ and UΛ is the nodal displacement vector belonging to Λ. The correspond-ing matrix representation of the problem may be written as


Figure 4.1: The vibro-acoustic displacement u(x) and the body force g(x) in the domain Ω

where Σ = ∂Ω \ Γ, and Λ = Ω ∪ Γ. x = [x1; x2; x3] is the position in space

[ (KΣΣ KΣΛ

KTΣΛ

KΛΛ

)− ω2

(MΣΣ MΣΛ

MTΣΛ

MΛΛ

)] (UΣ

UΛ

)=

(FΣ

FΛ

), (4.1)

where K and M are the stiffness and mass matrices respectively, U and F are thedisplacement and force vectors respectively.

4.2.2 Change of basisThis section explains how to generate the (nodal) displacement vector representationof the basis functions used in the change basis CMS approach. The Classical C&Bsubstructuring method uses one set of basis function to span the local DOF of eachsubstructure, and another set of functions for the coupling of the structures. The twosets of functions will be referred here to as local modes and coupling modes.

4.2.3 Local modesThe local modes are generated by a constrained eigenvalue problem. On the couplinginterface Σ a homogeneous (zero) Dirichlet condition is imposed as UΣ = 0. Togetherwith a zero force conditions on Γ. Hence, the eigenvalue problem may be written as(

KΛΛ − ω2MΛΛ

)UΛ = 0 (4.2)

4.2 Theory 31

The solution of Eq. (4.2) gives the eigenvalues ω2n and eigenvectors φn. The total dis-

placement vectors of the constrained eigenvalue problem, including the constrainedcoupling interface DOFs as zeroes, is written as in Eq. (4.3) where each column corre-sponds to a displacement eigenvector.

Φ =

[0

φΛ,(1) · · ·φΛ,(n)

]=

[0ΦΛ

](4.3)

This is the first set of basis functions used in the projection from local DOFs to gener-alised DOFs.

Coupling modes

To allow for a kinematic coupling to an adjacent substructure, basis functions which arenon-zero on Σ are needed. These are constructed by static solutions of Mcoup different,linearly independent boundary value problems having different non-zero prescribeddisplacement conditions on Σ. These boundary constraints are imposed via the FE dis-placement vector UΣ = ψΣ,(m) where FΣ = 0 and FΛ = 0. Denoting the correspondingsolution of UΛ of the remaining displacement DOFs by UΛ = ψΛ,(m), the mth solutionis obtained by solving the static problem(

KΣΣ KΣΛ

KTΣΛ

KΛΛ

) (ψΣ,(m)ψΛ,(m)

)=

(00

)(4.4)

hence,

ψΛ,(m) = −K−1ΛΛKT

ΣΛψΣ,(m). (4.5)

The Mcoup solutions of Eq. (4.5) may be organised as shown in Eq. (4.6) where eachcolumn corresponds to the displacement vector (or ”mode”) for a given imposed dis-placement vector ψΣ,(m).

Ψ =

[ΨΣ

ΨΛ

]=

[ψΣ,(1) · · · ψΣ,(m)ψΛ,(1) · · · ψΛ,(m)

](4.6)

This is the second set of basic function used in the projection from local DOFs togeneralised DOFs.


Modal projection

If the two projection basis functions of the local and the coupling modes are assembledthe total projection basis Θ reads

Θ = [Ψ,Φ] =

[ΨΣ 0ΨΛ ΦΛ

]. (4.7)

The projection of each component may then be written as

S = ΘTKΘ, W = ΘTMΘ, (4.8)

andG = ΘTF, Umodal = ΘQ, (4.9)

with the following transformed system of generalised coordinates[ (SΣΣ 00 SΛΛ

)− ω2

(WΣΣ WΣΛ

WTΣΛ

WΛΛ

)] (QΣ

QΛ

)=

(GΣ

GΛ

). (4.10)

If the eigenvectors φΛ in the projection basis are mass normalised then WΛΛ is an diag-onal matrix with only ones in the diagonal, SΛΛ is a diagonal matrix with the eigenval-ues ω2

n in the diagonal. The mass and the stiffness submatrices of the interface DOFs ,WΣΣ and SΣΣ are full matrices, the coupling (interface or inner DOFs) mass submatrix(WΣΛ) is also full. Hence, also the global mass matrix W is full. However, the fastsolution properties of the diagonal matrix WΛΛ can still be utilised by condensing thesolution to interface DOFs.

Classical Craig-Bampton

In Eq. (4.10) the general form of the classical C&B method is formulated. For thespecial case when ΨΣ is chosen as the identity matrix I then Eq. (4.10) correspondexactly to the classical C&B method. That formulation is based on the calculation ofthe inner response (on Λ) for the case of successive displacing each DOF (on Σ) by aunity while keeping all other DOFs on the interface fixed. This procedure of repeatingthe calculation of Eq. (4.5) as many times at there are DOFs on Σ, is equivalent toreplacing ΨΣ with the unit matrix. Hence, no reduction of interface (Σ) DOFs is made,since multiplying by a square unit matrix will not change the dimension of the resultingmatrix .


Rigid interfaces six DOFs

So far only a general form of the classical C&B CMS has been presented. In thepresented theory the possibility of using any function that can be described on Σ hasbeen shown, this for the calculation of the static coupling modes ψ. As stated beforein the classical formulation of the C&B CMS, an unreduced interface formulation isused. Herein, a different set of displacement conditions are constructed to generatea different set of static coupling modes using Eq. (4.5). This approach allows for atruncation of the coupling interface.Herein, a UCI approach is introduced. The deformations of an interface between twoconnected bodies could be argued to be governed (among other parameters) by therelative stiffness of the bodies. If there is a large relative difference of the stiffness, thenthe coupling interface may translate and rotate, but keeping its original shape almostundeformed. The stiffer body interface behaves almost as a free boundary and thesofter body interfaces behaves similar to a spatially prescribed translation and rotation,that is, identical to the stiffer body translation and rotation.These assumptions of undeformed coupling interfaces allows for a restriction of cou-pling modes to represent six different rigid motions of the interface, e.g. three trans-lations and three rotations around a common rotation point. In all other aspects thesecoupling modes corresponds to the classical C&B coupling modes. The main ad-vantage of this approach is that the number of coupling modes are reduced from thenumber of FE DOFs associated to the coupling boundary Σ, to six. Another importantfeature is that mesh compatibility is not required, only the translation directions, rota-tion axes and rotation point have to be compatible. Formally, not even the geometryhas to be compatible.For a more in dept description of the CMS UCI approach see paper B.

4.3 Results and discussionIn order to evaluate the usefulness of the proposed approach a test structure is imple-mented. The question is whether the approach with the UCI may be used for vibro-acoustic modelling of built up substructures of fundamental different stiffness.

4.3.1 Test structureThe test structure herein is shown in Fig. 4.2. The properties of this test case arechosen to resemble the vibro-acoustic problem of triangular linking arm in a vehicle


Figure 4.2: The test structure used in this thesis consisting of seven substructures. Boundaryconditions are indicated by R (Rigid or Fixed) or F (unit surface force condition inall the spatial directions).

suspension system, the linking arm often being connected to the stiff vehicle body inthree positions via rubber bushings. The test structure consists of a cross (substructure3 in Fig. 4.2), in order to resemble the linking arm. The rubber bushings are includedas blocks (substructures 2, 4 and 6 in Fig. 4.2) and three connecting parts are included(substructures 1, 5 and 7 in Fig. 4.2) to mimic the stiffness of the vehicle body. A moredetailed description of the test object can be found in the appended paper B.

4.3.2 Vibro-acoustic response

From Figs. 5 and 6 in paper B, three different frequency regions may be recognised.These are defined as low (0-100 Hz), mid (100-600 Hz) and high (600-1000 Hz),mainly for evaluation purposes. In the low frequency range there are rigid body reso-nances behaviour of substructure 3. At mid frequencies the behaviour is mostly gov-erned by the mass law, and at high frequencies there are three elastic modes of the crossthat governs the behaviour. In Fig. 4.3 the solution (spatial root mean squared value(RMS) of the displacement magnitude of substructure 3) is shown for different Young’smodulus of the ”soft” connectors. When the relative Young’s modulus moves closer tounity, then the first rigid body resonances moves up in frequency (see Fig. 4.3).


1 10 100 10000

20

40

60

80

100

120

Frequency [Hz]

Dis

plac

emen

t [dB

re.

1 p

m]

Erel

= 10−6

Erel

= 10−4

Erel

= 10−2

Erel

= 1

Figure 4.3: The vibro-acoustic response, integral of the total displacement of substructure 3for four different Young’s module of the soft connectors, presented as the relativeYoung’s modulus.

4.3.3 Evaluation of the approach

Relative Young’s modulus

In Fig. 4.4 the overall relative difference, in the respective previously defined frequencyranges (low, mid and high), is shown, for a changing stiffness of the ”soft” connectors.In other words, all material properties are kept constant except the Young’s modulus ofthe substructures 2, 4 and 6. What may be seen in Fig. 4.4 is that all results C&B andUCI have results that may be considered as accurate results from an engineering pointof view, with a deviation less than 1%, for relative Young’s modulus Erel less than 10−3

(relative Young’s modulus is defined as, Erel = Esoft/Estiff). As expected, the displace-ments at higher frequencies are the hardest to predict with the UCI approach. Thisis probably due to the fact that at high frequencies, the size of the interfaces becomesignificant in comparison to the wave length. In this test case, a fairly large interfacewas used. In real vehicle suspension systems, the interfaces between structures andbushings are usually smaller. The test case evaluated herein should only be viewed asa test of the possibility of modelling problems remotely similar to the built up structureof softer and stiffer parts, such as the vehicle suspensions. So far, results has only beenshown including all modes for the C&B and the UCI approach, which highlights theconsequences of the UCI approach compared to the C&B method. It should be noted


that the C&B method without any inner reduction should give exactly the same resultas the direct FE since only a projection is made.

10−6

10−5

10−4

10−3

10−2

10−1

100

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

Rel

ativ

e di

ffere

nce

[−]

Relative Young’s modulus [−]

C&BLow

UCILow

C&BHigh

UCIHigh

C&BMid

UCIMid

Figure 4.4: The total relative difference between the modal solutions and the direct solutionfor the three frequency bands, using all inner modes. Function of the stiffness ofthe ”soft” contactors (function of relative stiffness). C&B method (dashed lines)UCI approach (solid lines).

Inner reduction

In Fig. 4.5 the solution is calculated with the original material properties (Erel = 10−6),the total relative difference for the different frequency bands are shown as a function ofthe number of included normal modes of each substructure. In other words, the resultsare generated by calculating the response when having one normal mode included ineach substructure and then using two modes in each and so on until all modes are filledup in all substructures (the modes are organised by ascending eigenfrequency). FromFig. 4.5 it should be noted that; when using a low number of modes (under 100) thelow frequency representation is better than mid and mid is better than high, as expectedfrom a wave length consideration. Furthermore, it is clear that for a low number ofmodes the results for C&B and UCI are very similar up to around 100+ modes, exceptfor the low frequency range where the C&B result is somewhat better for very fewmodes. It seem like the UCI results has a cut on limit where any refined description ofthe inner problem no longer enhances the results, the limiting factor is most likely due


to the error induced by the interfaces reduction. However, in this specific case the limitof the relative difference due to truncation of the interface DOFs seem to be below anengineering error (∼1%).

1 10 100 100010

−12

10−10

10−8

10−6

10−4

10−2

100

Rel

ativ

e di

ffere

nce

[−]

Number of modes [−]

C&BLow

UCILow

C&BHigh

UCIHigh

C&BMid

UCIMid

Figure 4.5: The total relative difference between the modal solutions and the direct for thethree frequency bands, plotted as a function of the number of included modes innerin each substructure. C&B method (dashed lines) UCI approach (solid lines).

Time gain

As mentioned earlier there are possible benefits of the UCI, since strictly neither meshconformity nor geometrical conformity are required. However, the maybe biggest gainis the reduced computation effort needed with UCI approach. In Fig. 4.6 the nor-malised time consumption is shown as a function of the number of frequencies calcu-lated. The time is recorded for four different calculations different number of frequen-cies calculated, that is, 11, 101, 1001 and 10001. Seven different cases are evaluatedC&B and UCI for all, 10 and 100 modes and finally the direct FE solution. The giventime is the time to assemble the global problem for local stiffness and mass matrices,solve it and back substitute it to the original DOFs. Furthermore, the time is normalisedwith the total time of the direct solution for 11 frequencies calculated. What can beseen in Fig. 4.6 is that for all cases the UCI approach have a faster solution time thanthe C&B method, it is also clear that the more frequencies that are calculated the moreefficient the UCI approach is compared to both C&B and Direct solution. As expected,


10 100 1000 1000010

0

101

102

103

Nor

mal

ised

tim

e [−

]

Number of frequencies calculated [−]

C&Ball

UCIall

C&B10

UCI10

C&B100

UCI100

Direct FE

Figure 4.6: Normalised time consumption as a function number of frequencies calculated.C&B method (dashed lines). UCI approach (solid lines) for different number ofincluded inner modes (all, 100 and 10) and the direct solution (dashed-dotted line).

the smaller number of local inner modes used, the more efficient both C&B and UCIis compared to the direct solution.

4.4 ConclusionsA component mode synthesis approach that uses undeformed coupling interfaces, isproposed. The approach enables a significant reduction of the original problems, whereclassical CMS are limited to reduction of DOFs not associated with the coupling inter-faces. The approach also overcome any problem of non-conforming mesh of differentcomponents.It is concluded that specific systems may be modelled using the suggested approach,giving results that from an engineering point of view is valid. For the approach to bevalid the stiffness of the connecting bodies most be fundamentally different, such as,may be the case for the rubber bushing connected to a steel linking arm in a vehiclesuspension system.

CHAPTER 5

Outlook

There are several different steps and subprojects that may be done to progress towardsthe overall goal of understanding the full process of the interior brake disc roughnessnoise phenomenon.Thanks to this study it is now known that:• The interior noise is a structural-borne phenomenon where a broadband excita-

tion of vibrations is generated in the disc-pad interaction.

• Vehicle speed and brake pressure is well correlated with the noise and vibrationlevels.

• The excitation of vibration in the sliding direction and in the normal direction ofthe contact surfaces have different relations to the brake pressure.

• The brake force generated when applying a brake pressure can affect the systemproperties and hence the vibro-acoustic transfer path from brake to interior.

• There is a big computational efficiency gain to be won from modelling the ve-hicle suspension systems transfer path, using a Craig and Bampton componentmode synthesis technique using undeformed coupling modes (that is, the UCIapproach).

To be able to control brake generated interior roughness noise successfully in vehicledesign, understanding concerning the following aspects is needed:

• Understanding of the directivity (sliding- and normal to contact plane direction)of the broadband vibro-acoustic friction source better. When may Hertz contactmodels be used and when Coulomb’s friction law is appropriate, for what contactpressures/sliding speed/surface roughness are they valid?

39

40 5 Outlook

• Understanding of the brake disc assembly influence on the interior noise. Howdo brake system move? Which DOFs (rigid body, elastic) are needed to suffi-ciently describe the global motion of the brake system assembly?

• Understanding of the vibro-acoustic transfer path of the suspension system andthe possible loading effects better. Why do the transfer path change, is it mainlygeometrical non-linearity or is preloading of bushings important.

The following may be topics that could probably contribute to gain the necessaryknowledge and the tools to solve the above aspects:

• Pin-on-disc measurement combined with a contact model using both Hertz con-tact theory and Coulombs friction law. A pin-on-disc measurement apparatus isprobably the best and most practical to conduct pure measurements of a frictionphenomenon. With this device much of the complexity of the full vehicle brakesystem may be removed and the vibro-acoustic response and directivity maybe carefully studied for various materials, contact pressures and sliding speeds.These measurements, may be used as to develop and benchmark contact models.

• Component measurements, a more controlled measurement where the disc brakesystem more precisely may be studied. The idea of this measurement is to mapthe vibro-acoustic response of a brake system (Disc, pads calliper, etc.). Themeasurement would be conducted using a quiet and stable engine driving thedisc diretly where the brake system would be mounted in a rigid way.

• Update and further evaluate the CMS UCI code, include excising visco-elasticmodels of rubber bushings. Include other types of coupling modes to the alreadysix undeformed ones, for instants the coupling surface geometry could using ashell theory be generate different bending shapes for different boundary condi-tions. These shapes could then be used to calculate new coupling modes thatmay or may not improve the convergence of the approach.

References

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M. Yokoi, M. Nakai, A fundamental study on frictional noise (1st report, the generatingmechanism of rubbing noise and squeal noise)., Bulletin of JSME 22 (1979) 1665–1671.

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Part II

Appended Papers

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