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Thermal Deformation Analysis of Automotive Disc Brake

Squeal

BY

Muhammad Zahir Hassan

Submitted in accordance with the requirements for the degree of

Doctor of Philosophy

The University of Leeds

School of Mechanical Engineering

July 2009

The candidate confirms that the work submitted is his own and that appropriate credit has been given where reference has been made to the work of others

This copy has been supplied on the understanding that it is copyright material and that no quotation from the thesis may be published without proper acknowledgement

Abstract -

Automotive disc brake squeal has been a major concern in warranty issues and a challenging problem for many years. A variety of tools have been developed which include both experimental studies and numerical modelling technique to tackle the problem. The aim of this project is to develop a validated thermo-mechanical finite element model considering both the mechanical structural compliance and thermal effects in the dynamic instability of a disc brake system leading to squeal. A key issue in the process is to investigate the structural deformation of the brake components due to the combined effect of thermal expansion and contact loading between pad and disc when subjected to temperature change during a typical braking cycle. A new methodology is introduced whereby a fully coupled transient thermo-mechanical analysis is carried out to provide the temperature and contact distributions within the brake before executing an instability analysis using the complex eigenvalue method. A case study is carried out based on a typical passenger car brake as it undergoes a partial simulation of the SAE J2521 drag braking noise test. The actuation pressure, coefficient of friction and vehicle travelling speed are all considered to derive the temperature dependent contact pressure distributions making allowance for the "rotating heat source" effect. An experimental investigation using a brake dynamometer is also carried out to measuring the squealing noise and thermal deformation which leads to a validation of the results predicted by the numerical modelling. It is demonstrated that the fully coupled thermo-mechanical FE model enhances understanding of the time dependent non-linear contact behaviour at the friction interface. This, in turn, demonstrates the fugitive nature of brake squeal through the system eigenvalues that appear and disappear as a function of temperature throughout the braking period. Parametric studies on the geometrical effect and materials of brake components determine the contribution of each of these factors to brake squeal. The approach therefore can be use as a predictive tool to evaluate disc brake squeal using finite element method.

CHAPTER ONE

Introduction

1.1Overview

Disc brake squeal still continues to be a major concern for the automotive industry despite the efforts to reduce its occurrence during the past decades. It has been the subject of both experimental and numerical modelling since 1930s (Jarvis and Mills, 1963). The continuously evolving expectations related to vehicle performance have resulted in the car manufacturer having to strive to provide not only a competitive and efficient braking system, but also a 'quiet' braking system (Mohd Ripin, 1995).

The elimination of brake squeal noise is very important as it causes discomfort of the vehicle occupants as well as any pedestrians. Squeal problems may cause the car manufacturer substantial revenue loss from warranty claims associated with the quality of noise produced by the brake despite the fact that the brake remains fully functional and safe (Lang and Smales, 1983).

The trade-offs involved in this process continue to challenge engineers to understand and control brake noise and vibration phenomena. From a theoretical perspective, disc brake squeal can be classified as a form of friction-induced vibration (North, 1972). The characteristic and the understanding of this problem are complicated by the fact that it is a transient phenomenon. The disc rotor, while acting like a speaker, is a rotating component and the assembled brake combines many components with complex interfaces (Limpert, 1999).

There are three general categories of noise associated with automotive disc brakes. These categories are classified according to the frequency range in which the noise

occurs. Low frequency disc brake noise typically occurs in the frequency range between 100 and 1000 H z (Kinkaid et al.,2003). The noise types-in this category are known as grunt, groan, grind and moan (Millner, 1978; Papinniemi et al., 2002).

Squeal is defined as brake noise occurs above 1000 Hz. For the frequency bandwidth

1000 Hz to 5 kHz,the noise generated is classified as low frequency squeal (Dunlap et al., 1999).High frequency squeal is classified as squeal occurring above 5 kHz (Crolla and Lang, 1991). Low frequency squeal is distinct from low frequency noise since the squeal noise is strongly related to the elastic vibration of the brake rotor and its associated natural frequencies (Lang and Smales, 1983).

1.2Research Background

In recent years, finite element analyses (FEA) have become a standard and favourable methodology through which to study brake squeal (Kinkaid et al., 2003). This is because the finite element analysis is able to rapidly simulate brake squeal events for any changes made in the design of the brake components compared to experimental work (Papinniemi et al., 2002). Experimental programmes increase the cost as well as the duration of the design and development process (Yumoto and Okarnura, 2006).

The moving contact interfaces underneath the disc-pad challenge engineers to extend the brake squeal modelling with the incorporation of both thermal and structural effects (Hassan et al., 2008). Up to now, most researchers have omitted to include the effect of structural deformation and temperature dependent material properties when studying brake squeal. This is due to the complexity as well as the computing load involved in the inclusion of the thermal effects in brake instability studies (Ouyang et al., 2005). Most of the studies related to thermal effects focus on low frequency vibration, i.e. brake judder (Kubota et al., 1998).

To address the problem, a validated fully coupled FE model with respect to brake squeal analysis is required. The FE model should improved the existing FE modelling approach through integrated studies of time dependent thermo-mechanical effects within a non-linear contact analysis followed by brake instability analysis using the most common numerical prediction technique, namely the complex eigenvalue method. Computational and experimental methods must be combined to correlate and validate the results obtained from the models (Ouyang et al., 2005). Good correlation

2

between the finite element modelling and experimental results through dynamometer testing will deliver positive outcomes towards the research targets.(Ioannidis et al., 2005).

1.3Aim and Objective of Research

1.3.1 Aim

The overall aim of this research is to develop a fully coupled thermo-mechanical FE model to investigate brake system stability that includes the structural deformation of the brake components when subject to thermal effects. The SAE J2521 drag brake test considers different stages of brake instability particularly in relation to the temperature distribution including both heating and cooling effects. The development of a fully coupled thermo-mechanical model will also take account of the so-called "rotating heat source" effect in order to predict the non-axisymmetric rotor temperature distributions that occur in practice. This will lead to a representative FE model in which instability is predicted throughout the braking period.

The study will consider four major aspects in the modelling of the brake squeal phenomenon. These are the development of validated 3-dimensional FE models through free-free modal analysis, non-linear thermo-mechanical contact pressure analysis, instability prediction thorough complex eigenvalue analysis and parametric studies based on the brake design. Experimental work using a brake dynamometer which measure the thermal characteristics and squeal is required for the comparison and validation of the numerical modelling results.

1.3.2 Objectives

The objectives of the research are as follows:

1.To develop a validated 3-dimensional finite element model of a typical brake assembly by conducting free-free modal analysis in order to validate the finite element models by correlation of natural frequency data extracted from the finite element model with the results from experimental modal analysis (EMA).

3

..

11. To simulate the thermo-mechanical contact pressure distribution at the disc and pad interface with frictional contact conditions under'the dynamic drag braking define by SAE 52521 test schedule.

iii.To conduct instability analysis of the disc brake system using complex eigenvalue analysis (CEA).

iv.To simulate the effects of material properties and geometrical changes of the brake components on the occurrence of squeal noise.

v.To correlate the predictions of the finite element model with the results of experimental investigations carried out using the Leeds Brake Dynamometer.

1.4Organisation of Thesis

The remainder of this thesis is compromised of eight further chapters as summarised below.

Chapter 2: A review of literature relevant to the present study comprising brake system fundamentals, brake noise in general and in particular squeal noise.

Chapter 3: The new methodology, proposed through the integrated FE approach is described. This uses a fully coupled thermo-mechanical instability simulation technique to study the dynamic behaviour of a typical automotive brake system during different periods within a braking event.

Chapter 4: Experimental free-free modal analysis of the brake system components. This determines the individual brake component's free vibration behaviour and these results are used to develop appropriate FEmodels.

Chapter 5: This experimental section discusses the laboratory based brake dynamometer test facility and data acquisition system that is used to carried out the drag brake test. The experimental work evaluates the brake system performance by measuring the temperature, disc distortion and squeal noise.

Chapter 6: A non-linear thermo-mechanical contact analysis investigates the disc-pad non-linear transient thermal and contact pressure distribution under specific drag braking operating conditions. The effect of contact pressure distribution under heating and cooling conditions is reported for different contact interface properties. The results (for example, contact condition) are presented as a function of time. This chapter also discusses the effect of disc deformation during the heating-up process. A thorough study gives an insight into the extent of disc coning during the braking process. A correlation between the FE model prediction and experimental results is presented.

Chapter 7: Squeal prediction through complex eigenvalue analysis is presented as a function of time during the braking cycle. The unstable complex modes and frequencies are extracted throughout the braking period. These results lead to better understanding of the fugitive nature of the squeal problem.

Chapter 8: Parametric studies of brake component geometry and material properties establish the importance of these parameters with respect to squeal events. This section discusses the relationship between brake operating conditions, brake geometry, material properties and squeal noise generation or propensity.

Chapter 9: Conclusions are drawn from the overall findings of the research along with recommendations for future work.

CHAPTER TWO

Literature Review

2.1Introduction

The subject of brake squeal has generated a considerable volume of literature which includes a number of theories that have been formulated to explain the mechanisms of brake squeal. Studies on disc brake squeal involve two major areas of study: disc-pad interaction and friction-induced vibration.

This chapter begins with an introduction to the automotive disc brake system, to give an overview of disc brake components and their function. The distinct categories of automotive brake noise are presented according to the frequency range in which they occur. A review of brake squeal literature is then presented that explains the disc brake squeal phenomenon. The scientific findings are categorised into theoretical, finite element (FE) and experimental approaches to tackle and analyse brake squeal problems. In the FE section, a review on disc-pad contact pressure distribution, which has become essential feature of brake squeal study, is presented. A review of brake thermal analysis and brake instability analysis is also described. The subsequent section discusses experimental investigations that have been employed to tackle squeal problems followed by research reviews publication related to the area of brake noise. Finally, summary of the existing approaches are provided together with their limitations for tackling real brake noise issues. The structure of the chapter is shown in

Figure 2-1.

Introduction to Automotive

Braking System

I

Description of Automotive

 

I

Brake Noise

 

 

 

v

 

 

 

 

Low Frequency

Low Frequency

 

High Frequency

Brake Noise

Brake Squeal

 

Brake Squeal

 

i

 

L

b

Research Work

4

Approach

Theoretical

Experimental

Finite Element

Research

Work

Analysis

I

I

I

 

Brake Noise Research Reviews

 

 

x

 

 

Summary

 

Figure 2-1:Overview of Literature Review

2.2Automotive Disc Brake System

2.2.1 Historical Background

Elmer Ambrose Sperry pioneered the development of the clutch type disc brake in 1898 as cited in Hughes (1971). The design featured an electromagnetically actuated disc where braking torque is generated when the brake magnet disc is pressed into contact with another disc known as the brake disc.

The first patented model of disc brake was registered by Frederick William Lanchester in 1902 (Cited in Newcomb and Spurr, 1967). The disc brake, consisting of a sheet

metal rotor connected to one of the rear wheels of a vehicle and is pinched at the wheel edges in order to slow down the vehicle. This invention predates the early type of sports car disc brake system design in the early 1950s which traces its developments to Dunlop, Girling and the Lockheed Corporation (Newcomb and Spun, 1967). The sport-type design of brake disc is similar to the present type of disc brake which can be found on most road vehicles (Harper, 1998). The biggest problem that Lanchester encountered was noise as metal-to-metal contact between the copper linings and metal disc caused an intense screech that sent chills through anyone within earshot. Since then, the materials and actuation method used in this early brake have been greatly modified and improved.

The most important contribution towards the widespread used of brake discs is due to enhanced safety regulations throughout the world. For instance, most cars in the United States are equipped with front brake discs due to Federal Motor Vehicle Safety Standards (FMVSS) 105 regulations, which require all cars to meet standard stopping distance and brake fade requirements (Oppenheimer, 1977). This is because disc brakes satisfy the braking power specification with superior water resistance and fade performance compared to drum brakes.

2.2.2 Brake Components and Functions

A disc brake system usually consists of a brake disc rotor, two brake pads and a calliper (SAE International, 2007). The combination of these components allows the rotating wheel to experience severe braking in a short stopping distance. The braking surface is the area on which the braking action of the friction material takes place (Limpert, 1999).

Figure 2-2 illustrates the disc brake system components on a passenger vehicle. The centre part of the brake disc has a circular aperture, which is locates on the wheel hub (SAE International, 2007). It is surrounded by a number of holes for the wheel bolts. The brake disc rotates along with the wheel. The normal load, produced when the brake is actuated result in the generation of an in-plane friction force at the disc-pad interface. This in turn produces a brake torque about the centre of rotation of the wheel as shown in Figure 2-3. The reaction to the brake torque is seen in the brake force, between the tyre and ground, that slow the vehicle.

Figure 2-2: Brake system components: 1) Piston 2) Calliper 3) Pad 4) Brake Disc

Retainer 5) Rotor Disc 6) Wheel Hub (Flickr, 2008)

Brake Torque

Disc

 

gular- locity

Brakingforce -,

Direction of

Motion

transmitted to the

 

vehicle

 

Axle load

BrakefForce ryre-Road Normal Reaction

Figure 2-3: Schematic diagram of forces and moment acting on wheel

9

The ventilated brake rotor is a one-piece casting with cooling fins between the two braking surfaces. This enables air to circulate between the braking surfaces, making them less sensitive to heat build-up and more resistant to fade. Dirt and water do not generally affect the braking action since contaminants are thrown off by the centrifugal action of the brake disc or scraped off by the pads. In addition, the equal clamping action of the two brake pads tends to ensure uniform, straight-line stops (SAE International, 2007).

The two main functions of the brake rotor are the transmission of mechanical force and the dissipation of heat, produced when functioning at both medium and high temperature (Limpert, 1999). This means that the materials used for brake discs must be able to support high temperatures (Grieve et al., 1998). The rotor material must be cost effective, allowing for potential reductions in weight as well as for the stability of the components (Ioannidis et al., 2005).

A solid disc brake consists of a rubbing surface and a top hat section (Bae and Wicket, 2000). The section that connects these two parts is known as the neck (Yumoto and Okamura, 2006). The rubbing surface section is the area where a tangential friction force is generated when the disc interacts with a stationary pad to stop the moving vehicle. The disc rubbing surface area is sometimes known as the cheek (Grieve et al., 1998; Koetniyom, 2000). A top hat section is connected to the disc rubbing surface and mounted to the vehicle wheel hub.

A disc brake which has separate inboard and outboard rubbing surfaces with cooling vanes or fins in between is known as a ventilated rotor. These vanes allow the air to flow through the structure and cool the rubbing surfaces during and after all braking events. There are two types of ventilated disc: front-vented and back-vented. Figure 2-4(a) shows the front-vented type of disc brake, where the top hat section is connected by the neck to the outboard rubbing surface. On the other hand, a neck that connects the inboard rubbing surface with the top hat section as shown in Figure 2-4(b) creates what is known as a back-vented disc (SAE International,2007).

I

Rubbing

Neck

(a) Front-Vented Disc

(b) Back-Vented Disc

Figure 2-4: Typical front and back-vented disc (Koetniyom, 2000)

The friction between the pad and disc plays a decisive role in defining the amount of pedal force required to obtain a given rate of deceleration (Papinniemi et al., 2002). This factor is also important in designing brakes for the balanced operation of a vehicle as brake imbalance can lead to a yaw torque about the vertical that could compromise the vehicle stability. The additional task for the brake disc is to induce air movement, as air moving over the rubbing surface of the disc reduces the heats build up (Valvano and Lee, 2000).

The brake pad is designed to rub against the disc surface leading to diminution of vehicle speed, thereby converting mechanical work into thermal energy. The structure of the pads can be very complex (Nicholson, 1995). They can consist of different materials or numerous parallel layers as shown in Figure 2-5. There are many different types of friction material on the market which can be classified into the following categories: semi-metallic (SM), non-asbestos organic (NAO) and sintered metal (Anderson, 1992). A friction material is mounted to a rigid metal backplate using adhesive. A substrate material is sometimes located in between the friction material and backplate. The main function of the substrate material is to act as a thermal insulator that will prevent an excessive flow of heat towards the piston and brake fluid as well as damping vibration. The backplate distributes the force exerted by the piston over the pad contact surface (Lee et al., 2003a). An anti noise layer or shim located

behind the backing plate minimises the transmission of vibrations produced during braking action.

Backplate

Anti-noise

Friction layer

Material

Figure 2-5: Brake pad construction (Nicholson, 1995)

The calliper, which contains one or more pistons, holds the two brake pads on either side of the rotor. The movement of the pistons is controlled by a hydraulic system (SAE International, 2007). When hydraulic pressure is applied by pressing the brake pedal, the piston is pushed forward to press the inner pad against the rotor while the housing is pushed in the opposite direction to press the outer pad against the rotor, hence generating a hydraulic clamp around the rotor. For the fixed (non-floating) calliper type of disc brake system, each piston presses the brake pad against its respective side of the brake rubbing surface, as shown in Figure 2-6(a). Meanwhile the floating calliper housing, which is designed to slide on its support, reacts by shifting and pushing the pads against both sides of the disc as shown in Figure 2-6(b) (SAE International, 2007).

Figure 2-6: Brake system components (SAE International, 2007)

(a)Cross sectional view of Brake System: Fixed Calliper

(b)Cross sectional view of Brake System: Floating Calliper

2.3Description of Automotive Disc Brake System Noise

There are three general categories of noise associated with the automotive brake disc. These categories are classified according to the frequency range in which they occur (Dunlap et al., 1999).

i.Low Frequency Noise ( Less than 1000Hz )

ii.Low Frequency Squeal ( 1000Hz to 5000 Hz )

iii.High Frequency Squeal ( Greater than 5000 Hz )

2.3.1Low Frequency Brake Noise

Low frequency disc brake noise typically occurs in the frequency range between 100 and 1000Hz (Lang and Smales, 1983). Low frequency vibration has been attributed to variation of coefficient of friction with sliding velocity. This type of noise is caused by friction material excitation at the brake rotor and lining interface where the energy is transmitted as a vibration within the wheel comer which couples with other chassis

components punlap et al., 1999). The noise types in this category are known as judder, groan and hum.

Groan is defined as a semi-resonant vibration with frequency typically less than 100

Hz. Groan as well as hum may i~-~volvethe rigid body rotational modes of the calliper and local suspension parts. The typical frequency range for hum noise is between 200 and 400 Hz. Judder generally occurs at frequencies below 150 Hz but high-speed judder exists in the frequency range of between 150 to 250 HZ (Kubota et al., 1998). The judder phenomenon, which is felt rather than heard, is characterised by vehicle body and steering system vibrations. Cyclic variations in brake torque resulting from disc circumferential thickness variation cause this very low frequency non-resonant vibration.

2.3.2 Low Frequency Brake Squeal

For the frequency bandwidth 1000 Hz to 5000 Hz, the noise generated is classified as low frequency squeal (Fosberr~and Holubecki, 1955; Lang and Smales, 1983; Kinkaid et al., 2003). Squeal is related to vibration involving transverse motion of the disc and a bending or twisting motion of the brake pad. The disc mode order is often such that the wavelength of the mode is less than the arc length of the pad. The noise generation can be associated with frictional excitation coupled with a phenomenon known as modal 'locking' (Dunlap et al., 1999). The friction force at the interface couples these modes and produces an instability known as 'binary flutter' due to its similarity to aircraft wing flutter (Crolla and Lang, 1991).

2.3.3 High Frequency Brake Squeal

High frequency squeal is classified as squeal occurring above 5 kHz (Dunlap et al., 1999). The noise is produced by friction induced excitation causing coupled resonance of the rotor and other brake components. Higher frequency squeal is sometimes referred to as squeak (Lang and Smales, 1983).

2.4Theoretical and Multi-body Methods Approach

The first description of brake squeal noise by Mills (1938) as cited in Kinkaid et al.

(2003) shows how a simple model of an elastic rubbing system can be used to demonstrate the instability of a brake system. This simplified model shows a fragment of pad material rubbing against the disc connected to the calliper by means of a visco- elastic system, comprising a parallel combination of spring and a linear damper. This model assumes that the friction coefficient is not fixed and varies in a linear manner with the rotational speed of the disc relative to the pad as shown in Figure 2-7.

Linear damper

Figure 2-7: A model of non-linear stick-slip vibration (Mills (1938) as cited in Kinkaid et al. (2003))

From the simple rubbing surface model shown in Figure 2-7, the coefficient of friction is assumed to be:

where p, : is static friction coefficient,

A : is magnitude of the slope of the friction speed curve,

V :is conveyer speed,

x : is instantaneous velocity of mass.

The equation of motion of the rubbing block with mass M and applied normal force N can be presented as:

Substituting equation 2-1 into equation 2-2 generates the following equation:

Re-arranging the previous two equations generates:

Since h is always positive, equation 2-4 gives a negative damping term, which makes the system become unstable as illustrated in Figure 2-8. As a result, the negative slope of the friction curve is the main mechanism for the system to generate instability.

Kinetic Friction

Coefficient ,us

b v

Sliding Speed

Figure 2-8: Kinetic friction coefficient as a linear function of sliding speed with negative slope

Spurr (1961) introduced the sprag-slip model since the experimental investigations following Mills's model description showed that the negative slope of the friction versus speed curve could not alone explain all the phenomena associated with brake squeal. Spurr came up with mechanism that includes a geometrically induced instability arising from the friction coefficient. The main difference between the Spurr and Mills model is that the sprag-slip mechanism does not require the negative slope of the friction versus curve model and also explains why the out-of-plane modes are excited by the in-plane friction force. However the Spurr model still could not explain real brake system squeal noise generation.

In 1963, Jarvis and Mills adapted the Spurr model to produce a further mathematical model of brake squeal. Their work shows that the instability of coupling between the two members (disc and pad) produces brake squeal by idealising the brake system as a simplified model of a cantilever-disc system. The cantilever is subjected to both normal force and friction force. Increase in normal force will increases the friction force which pushes the cantilever away. By using the Lagrange method, Jarvis and Mills proved that the cause of instability is not due solely to variation in the coefficient of friction. They proposed a complete model of the brake system, which included the calliper. The mathematical model proposed by Jarvis and Mills was validated by pin on disc experiment, which has become widely adapted equipment for the study of squeal characteristics as shown in Figure 2-9.

Figure 2-9: Jarvis and Mills cantilever-disc system model (Jarvis and Mills, 1963)

A better model developed by North (1972) can be regarded as an 8 degree of freedom model for the brake assembly with four major components; two brake pads, a disc and calliper as shown in Figure 2-10. These components were allowed to move in the transverse direction and also rotate.

In contrast to the North model, Millner (1978) developed a squeal model based on a fixed calliper disc brake. The main feature of this model is that it can examine the effect of centre of pressure between the piston and brake pad. Millner concluded that, as well as the geometry, instability could be excited in almost any contact configuration when the coefficient of friction is sufficiently high. He further demonstrated that the propensity for brake squeal can be reduced by moving the centre of pressure towards the trailing edge of the brake pad.

Rudolph and Popp (2001) published a 14 degree of freedom multi body brake assembly model as shown in Figure 2-11 in which there are 18 coupling elements between the rigid bodies. The force and moment coupling depends on the relative displacement and velocity between the two connected rigid bodies. An important feature of this model is that the coefficient of friction is assumed to depend on the brake line pressure at the initial temperature of the disc. The model was verified by the experimental data.

I

Y1

11111

Pad

Disc

+ ;Lo;

Pad

Figure 2-10: %degrees of freedom brake assembly model (North, 1972)

 

 

 

Free Canier

Supporting Carrier

 

 

b

 

18

 

 

 

 

 

 

15

 

Calliper

 

 

 

10

8

 

 

 

 

 

 

-

7 + 2

Outer Pad

-

-

1 )

Disc

-

4

13

Inner Pad

11 12

16

1

Figure 2-1 1: 14-Degrees of freedom brake assembly model (Rudolph and Popp 2001)

2.5Modelling Approach: Finite Element Analysis

The finite element (FE) method is the most popular of approaches used to simulate the performance of a brake system design (Liles, 1989; Lee et al., 2003a). The vast number of degree of freedom and high fidelity of an FE model has enabled the FEA to deliver a detailed representation of the brake system performance (Ouyang et al.,

2005). There are several types of analysis that have been used by researchers to evaluate a given design with a view to producing an early predictive tool with which to evaluate problems (Kinkaid et al., 2003). The following section presents the FE techniques used to tackle brake design issues which include thermal, contact pressure and instability analyses. The latter have used the complex eigenvalue and transient dynamic approaches.

2.5.1 Thermal Analysis

During braking, the kinetic energy of the moving vehicle is converted into thermal energy through friction at the disc and pad interface. Frictional heat is generated at the rubbing surface due to the interactions between the pad and disc. The disc absorbs up

19

to 90% of the generated heat energy by means of conduction away from the friction interface between disc and pad (Limpert, 1975). The energy is quickly dissipated to the surrounding air. Radiation also helps to dissipate the heat energy stored within the rotor when the temperature is high.

Prediction of the surface temperature of a disc brake as well as that of the pad is complicated (Thuresson, 2000). This difficulty is due to the complex interactions between the thermal and mechanical behaviour of the brake components (Ouyang et al., 2005). The finite element method (FEM) is the most popular and widely used tool to simulate the influence of temperature in brake system design and performance. Bolt (1989) defined several numerical procedures for thermal analysis using the FE method to investigate disc brake performance. He revealed that FEA is the fastest and most accurate way to study the influence of temperature at the early design stage of a brake system.

There are three main categories of FE thermal modelling discussed in the following subsections: heat transfer analysis, thermal stress analysis and coupled thermo- mechanical analysis.

2.5.1.1 Heat Transfer Analysis

The non-uniform heat flux input to the disc brake is calculated from the non-uniform pressure distribution, friction coefficient and sliding velocity along the disc-pad interface. The amount of heat flux that flow into each component depends on the disc and pad material (Thuresson, 2004). The value of calculated heat flux is applied to the

FE model to examine the resulting temperature distributions (Koetniyom, 2000).

Limpert (1975) studied the thermal performance of a solid disc brake during braking. The heat flux for the disc was derived from the coefficient of friction and heat proportioning between the rotor and pad under uniform pressure loading. Experimental work was carried out to supported the results obtained from the theoretical calculations. During a series of brake applications, the results obtained from the experimental work and the theoretical calculations were well correlated.

Sheridan et al. (1988) studied different techniques for the thermal modelling of a disc brake ranging from a simple axisymmetric FE model to a complex 3-dimensional FE model. The paper also reviewed methods to calculate the thermal boundary conditions of the model. The effect of energy input and output as well as the material properties, such as thermal conductivity and specific heat, had a significant influence on the temperature response. Their findings suggested that 90% of the heat generated during braking is transferred by convection to the ambient air.

Huang and Chen (2006) constructed a 3-dimensional FE model of a disc brake to investigate its cooling performance in terms of the design parameters and boundary conditions. Each surface of the rotor was subjected to different values of convection heat transfer coefficient obtained from theoretical calculations. The neck fillet radius, which connects the rubbing plate to the top hat, was modified to study its influence on the cooling process during braking. The simulation shows that the highest temperature appeared at the mean radius location of the rubbing surface for both outboard and inboard side and this magnitude significantly depended on the fillet radius of the neck.

Sun (2006) developed a thermal model of disc brake system using ABAQUS by combining this with computational fluid dynamics (CFD) and a FORTRAN user subroutine to study the brake equilibrium temperature rise. The thermal model was correlated to physical test data during braking under mountain test schedules to study the effect of rotor, dust shield, wheel, wheel cover and air deflector on the brake performance. The simulation indicated that the modifications made to the rotor, dust shield and wheel geometry greatly improved the brake cooling performance when the brake temperature rise was between 370°C to 480°C. The results also showed that the number of vanes and the air deflector design also influenced the brake equilibrium temperature rise.

2.5.1.2 Thermal Stress Analysis

A thermal stress analysis of a brake disc uses the temperatures predicted in the preceding heat transfer analysis (Koetniyom, 2000). The nodal temperatures derived from the heat transfer analysis provide temperature variation input to the structural model to carry out the thermal stress analysis.

Timtner (1979) investigated the effect of changes in the geometry of a solid brake disc; looked at the top hat section depth and thickness as well as the undercut depth, which influence the disc brake coning, using a linear elastic FE model. ~ i r ~ t lhey,carried out the heat transfer analysis to obtain the nodal temperatures of the disc brake and then transferred these temperatures to the structural model to conduct the thermal stress analysis. The numerical results show that the degree of brake disc coning is reduced when the top hat section depth, thickness and undercut depth fire increased. He concluded that the greatest influence on brake disc coning was the value of hat thickness.

Fukano and Matsui (1986) used the same technique as Timtner (1979) to conduct an elastic thermal stress analysis of an automotive disc brake. The material properties of the disc, made out of cast iron, were temperature dependent. The outcome of their investigation predicted cracking in the top hat section of the disc, which was close to high stress concentrations around the bolt holes. This is due to fgct that maximum tensile stress in the disc exceeded the tensile strength of the cast iron. Experimental work showed good agreement with predicted cracking obtained from the ~umerical modelling results.

Valvano and Lee (2000) utilized an ABAQUS program to investigate the thermal distortion of a brake rotor. The program is able to calculate thermal parameters based on measured data, using a brake dynamometer, for use in FE modelling. These parameter inputs were transported directly into ABAQUS and an alysed using both transient and steady-state conditions to calculate the temperature distribution and thermal stresses. The finite element analysis is divided into two categories: initial heat transfer analysis followed by the uncoupled thermal analysis th-at determines the thermal stresses and distortion of the brake rotor. This approach allaws the heat input and output of the heating and cooling process for a given brake system to be compared with the measured temperature and distortion data based on ex~edmentalwork. The method can also be a viable tool in the early stages of brake design t o reduce unwanted thermal distortion in the disc brake system.

Koetniyom (2000) investigated the thermal response of a cast i r o ~disc brake using FEA. He generated a complex thermo-mechanical model based on ulniaxial cyclic tests from samples cut from a Rover 200 ventilated disc. By using tlhe user developed

material subroutine in ABAQUS, the thermal responses of back and front-vented disc designs are compared and validated with the uniaxial cyclic test dat8. The results show that the back-vented disc suffers lower thermal distortion but with higher plastic strain accumulation at the vanes and neck section. This indicates that the back-vented disc is subjected to high temperature and plastic stresses that are most likely to cause thermal cracking under repeated thermal cycling.

Phan and Kondyles (2003) combined the computational fluid dynamics (CFD) technique with heat transfer analysis for the brake rotor development process. They carried out a thermal modelling technique to optimise the rotor design considering thermal film coefficient, rotor coning, temperature and thermal stress distribution. They found that the combined technique provided accurate thermal data to predict rotor thermal distortion so that the coning problems can be reduced at the early design stage.

Yumoto and Okamura (2006) constructed a disc brake system development program using computer aided engineering (CAE). The study of brake performance can be carried out by manipulating pull down menus in a graphical user interference (GUI) to conduct parametric design studies and FEA modelling. The methodology of the program satisfies the requirement of the brake performance analysis: thermal modelling,judder and squeal analysis. The program is divided into two main steps, the finite element stage followed by the design optimisation step. The pre-processor allows the user to define the geometry of the disc brake model, carry out the thermal analysis followed by the complex eigenvalue analysis. In the post processor stage, the program will automatically simulate the results. The program helps to reduce the time taken to design the brake system as well as improving the quality of the brake design.

2.5.1.3 Coupled Thermo-MechanicalAnalysis

Coupled thermo-mechanical problems have recently become one of the important research areas in brake performance. The coupled model predicts the temperatures as a function of time and derives a new contact pressure distribution when the disc and pad come into contact at the start of each time step (Hassan et al., 2008).

Brooks et al. (1994) investigated thermal judder using a fully coupled thermal structural model. The FE thermal model predicted the temperature as a function of time during the presence of heat flux. The temperature field was then transferred to the structural model to derive the contact pressure distribution. The change in contact pressure underneath the contact surface was then used to derive a new heat flux distribution to feed back to the thermal model and therefore the model was considered to be fully coupled. A parametric study was camed out to investigate the effect of friction material properties. The results show that friction material stiffness should be low as this generates a more uniform contact distribution for both temperature and contact pressure. A high thermal conductivity and a low coefficient of expansion also promote a uniform pressure distribution.

Yi et al. (2001) used a fully-coupled model to determine the critical speed using an eigenvalue formulation to study the thermo-elastic instability (TEI). The eigenvalue method predicts the critical speeds for the disc brake TEI by identifying the minor design changes related to the critical vehicle speed. An accompanying experiment is carried using an infrared detector where the results are used to validate the critical speed derived by the FE model. They concluded that fully coupled FE model is one of the best numerical approaches for TEI studies.

Thuresson (2002) camed out an investigation into the contact problem between two bodies in high speed sliding contact using quasi-static analysis. A thermo-mechanical wear model was developed to obtain the interface pressure and temperature distribution. The thermal analysis was camed out using the boundary element method and solved using a Newton-type method to investigate contact geometry subject to change in interface friction. He found that temperature and wear effects have a large impact on the disc-pad contact interface pressure distributions. At a low coefficient of wear, thermal expansion dominated the pressure distribution. In contrast, when the coefficient of wear is high, thermal expansion tends to have less effect on the distribution of contact pressure. Thuresson concluded that the model gave promising results and both temperature and pressure distributions are accurately determined by considering the effect of wear.

Recent studies by Zhu et al. (2008) simulate the SAE 52521 drag braking squeal test sequence using a sequentially coupled thermal structural FEmodel based on the Leeds