## Abstract

The paper studies interface pressure distributions and thermal contact resistance (TCR) of a large automotive bolted joint. The research was initiated by the need to determine accurately conductive heat dissipation from a commercial vehicle disc brake. The main area of interest was the conduction between the grey cast iron disc and the spheroidal graphite cast iron wheel carrier. The bolt clamp forces and interface pressure distributions were investigated theoretically and experimentally. Finite-element analyses and pressure-sensitive paper experiments provided very similar interface pressure distributions. TCR change with interface pressure was studied experimentally, by conducting numerous temperature measurements. The derived linear relationship is of generic nature, enabling the calculation of the TCR for a variety of engineering bolted joints, over a wide range of interface pressures.

## 1. Introduction

Thermal contact resistance (TCR) is of vital importance in determining heat flow through a bolted joint. The research was initiated by the need to determine accurately conductive heat dissipation from a commercial vehicle (CV) disc brake. The main area of interest was the conduction between the grey cast iron disc and the spheroidal graphite (SG) cast iron wheel carrier (see figure 1). This type of bolted joint is common to many engineering applications. Published brake thermal analyses deal with conductive heat dissipation inadequately. Since the contribution of this mode to the brake cooling is, in most service conditions, lower than convective and radiative heat dissipation, the contribution of conductive cooling is either neglected or oversimplified. This was mainly the result of lack of adequate data, since conductive cooling can be substantial. Conductive heat dissipation is speed insensitive (unlike convection), therefore, for braking applications at low speeds and high disc temperatures (such conditions are found during drag braking on long downhill descents, or in repeated brake applications), conduction becomes a very important mode of heat dissipation and can be responsible for up to 20% of the total heat dissipation (Voller 2003). For accurate temperature predictions, it is vital to have an accurate model that includes conductive heat dissipation. This is particularly important when considering heat soaking into undesirable areas/components such as bearings, brake fluid and tyres, where excessive temperatures may have catastrophic consequences.

In this research, numerous experimental and theoretical studies have been performed in order to determine bolt clamp force, interface pressure distribution and TCR. The ultimate aim was to establish a theoretical procedure for reliably predicting TCR, which will enable heat flow calculations through bolted joints of a similar type in a variety of engineering applications, not restricting the findings to the considered automotive assembly. The research started with the detailed study of available literature data, the information considered most important being reviewed in §2.

## 2. Thermal contact resistance studies

In the current literature, much of the work in the determination of TCR has assumed that the contact pressure is uniform at the interface. To predict the TCR across a given area, the number and average size of the micro-contacts that make up the apparent contact area have been predicted. Greenwood & Williamson (1966) proposed one of the first models to predict contact area, known as the GW model, which can be summarized as an elastic micro-contact model. The GW model has been modified by McCool (1986) using a random process model of rough surfaces. The modified version of the GW model has been applied by McWaid & Marshall (1992) to predict TCR, and results compare well with experimental data. However, for common joints found in engineering applications, such as bolted or riveted joints, resulting interface pressure distribution is highly non-uniform in most cases.

Gould & Mikic (1971) considered bolted joints with smooth surfaces. The contact areas and pressure distribution were investigated using finite-element (FE) methods and good agreement was achieved with measurements. Roca & Mikic (1972) took the effect of surface roughness into account and found that the total TCR can increase or decrease with roughness due to the changes in the macro- and microscopic contact. Mittelbach *et al*. (1994) presented thermal conductance and interface pressure distribution data for a bolted joint. It was shown that there is little influence of the plate thickness on the area of contact. However, the thermal conductance is a function of the plate thickness ratio.

The determination of TCR of bolted joints has concentrated mainly on the contact resistance between the plates. Other heat transfer paths, such as through the bolts, has been mostly neglected. This problem has been approached by Mantelli & Yovanovich (1998) in a sensitivity study considering the resistance paths and determining the controlling parameters of TCR.

Problems with the determination of conductive heat dissipation in friction brakes have been recognized in early studies; Cetinkale & Fishenden (1951) suggested that the contact coefficient is likely to change dramatically with time or with subsequent dismantling and reassembly during servicing. Morgan & Dennis (1972) found that conduction coefficients are extremely variable for the theoretical prediction of brake temperatures. Quantified values of the heat transfer rate through brake assemblies were determined by Fukano & Matsui (1986) by changing the apparent rate of heat transfer from the disc to the hub, and by comparing calculated temperatures with experimental values. For the particular vehicle and test conditions, the best match of calculated and experimental temperature results was achieved for the thermal contact conductance of 712 W m^{−2} K^{−1}. Obviously, the approach may have been adequate for solving the specific problem, but it is not sufficiently detailed, or of a generic nature. Sheridan *et al*. (1988) modelled the conductive heat transfer from the disc, via its flange, to the hub and wheel, by doubling the convective heat transfer coefficients used on the disc friction surfaces. The applied convective heat transfer coefficient was 200 W m^{−2} K^{−1}, and the modelling included temperature difference between the bolted joint and ambient air, not temperature difference at the joint interfaces. Again, a practical approach in modelling a particular design and duty, but of limited scope for wider applications.

There are other TCR studies of bolted assemblies and conductive heat dissipation studies from disc brakes. However, for the main area of interest, the conduction between a grey cast iron disc and a SG cast iron wheel carrier, the available data are insufficient for accurate thermal modelling. Therefore, it was considered necessary to study TCR in detail (see Voller 2003).

## 3. Commercial vehicle brake assembly

A heavy CV wheel assembly is shown in figure 1, consisting of the brake disc, wheel carrier, wheel and tyre (22.5 in. nominal diameter). Heat is generated at the brake disc friction surfaces and conducted through the disc hat section. The assembly provides two paths for conductive heat dissipation from the disc, one through the bearing assembly, the other through the wheel carrier. Heat transfer from the brake disc to the bearing must be avoided to ensure bearing temperatures are kept low. However, the wheel carrier has a substantial mass, which is approximately two-thirds of disc mass, and therefore can provide a very desirable conduction path from the brake disc. The outer faces of the wheel carrier and wheel have direct contact with cool, fast-flowing and turbulent air and these boundary conditions can provide substantial potential for heat dissipation, as discussed by Tirovic & Voller (2002).

It is apparent that the disc/wheel carrier interface condition will vary throughout the life of the vehicle. This will be investigated in separate research. However, in this paper, the most common condition has been considered to represent a ‘typical’ bolted joint state for most of the life of the vehicle. The components used in tests had some corrosion at the interface, formed from moisture penetration. The bolted joint would not normally be dismantled, except for disc or bearing replacements. This should be a relatively rare occurrence, considering the life expectancy of the components is well over 6 years.

Both component interfaces were machined, the surface finish measured in numerous places was quite uniform, the extreme readings showing Ra values between 1 and 3.3 μm. Surface finish and surface flatness were both satisfying adequate specifications. The disc and wheel carrier are bolted together using 10 bolts (M16). In order to study TCR, bolt clamp force and interface contact pressure distributions need to be analysed.

## 4. Bolt clamping force determination

The bolt clamp force can be calculated using the bolt force equations found in the literature, such as Shigley (1977). However, despite bolt and thread geometry being accurately defined, friction coefficients at the thread and under the bolt head (collar) can substantially vary. Therefore, for accurate bolt clamp force determination, experimental verification is necessary. The relationship between the bolt clamp force and tightening torque was determined by measuring the clamp force using a special low profile force washer load cell (connected to a digital metre display) and a torque wrench showing applied torque. The clamp force was measured for a range of bolt torques, between 50 and 300 Nm (in 50 Nm increments). The maximum torque of 300 Nm is the nominal bolt torque specified by the vehicle manufacturer. Six experiments were conducted for each bolt torque and the average bolt clamp force value (*F*) calculated.

The average interface contact pressure (*P*_{avg}) can be calculated by dividing the total clamp force (product of the number of bolts *n*_{b} and the individual bolt clamp force (*F*) with the interface contact area (*A*_{int}):(4.1)The values of bolt clamp force and average contact pressure, for the applied bolt torque, are included in table 1. As expected, average contact pressure is a linear function of bolt torque.

## 5. Finite-element analysis of contact pressure

In order to study contact pressure in more detail, SDRC I-DEAS software was used to create a FE model of the CV brake disc and wheel carrier assembly. Both the disc and wheel carrier were truncated to reduce model complexity, while maintaining accuracy, as shown in figure 2. Making use of the circumferential symmetry, a three-dimensional segment of 36° was modelled, which included one bolthole. The 10 bolts connecting the disc and the wheel carrier are not equally spaced, but grouped in five equally spaced ‘pairs’. Appropriate boundary conditions were applied. The disc and wheel carrier were modelled with 3900 solid linear brick elements and 286 gap elements, having a total of 4577 nodes. The disc was modelled with grey cast iron material properties and the wheel carrier was modelled with SG iron properties. At the interface of the disc and wheel carrier, gap elements were used to join the adjacent nodes. The gap elements used were node-to-node type, transmitting only compressive force perpendicular to the interface and allowing separation of contact surfaces. The initial surfaces (before bolt clamping forces were introduced) were considered to be flat, and no springs (in gap elements) were used. Since static contact was considered, no friction was modelled. Experience from similar analyses shows that the assumptions used are valid for such an application. Modelling static friction is possible, but ultimately, the resultant interface pressure distribution is almost identical. In such modelling, the relative movement (sliding) of interfaces can only be artificially included, since the user must define fixed nodes (that do not move), which will determine the direction of sliding. In practice, some relative movement of components is inevitable during assembling, which combined with gradual increase of clamp force makes the ‘non-friction’ approach the most suitable.

For interface pressure and TCR measurements, the disc and wheel carrier were bolted to the very stiff spin rig adapter (see figure 6). This enabled simplified (‘rigid’) but realistic modelling of the underside surface of the disc hat (see ‘clamped’ nodes in figure 2). To model the bolt clamp force, a continuous load was applied to 120 nodes under the bolt head on the surface of the wheel carrier (see bolt force, figure 2). At the symmetry planes of the model (36° apart), appropriate boundary conditions were introduced, ensuring nodes remained within these planes during loading.

Figure 3 shows the contour plot of the *ZZ* stress component (perpendicular to the interface), representing interface pressure distribution. The applied bolt force of 120 kN is resulting from the maximum torque applied to the bolt (300 Nm; see table 1). High interface pressure can be clearly seen around the hole, the maximum value being 124 MPa (the negative sign in figure 3 indicates compression). The pressure reduces away from the hole edge, with lowest values predicted towards the outside diameter of the contact area. Two distinctive areas can be established, area (1), ‘in the bolt proximity’, being relatively small area with high pressure (of the order of 95.0 MPa), and area (2), a larger area ‘between the bolts’, with low pressure (of the order of 32.3 MPa). The ratio of ‘typical’ pressures in these two areas is approximately 3:1. Compared with the average pressure of 56.2 MPa (see table 1), area (1) has 70% higher pressure, while area (2) has 40% lower pressure than average.

Further contact pressure analyses have been performed for the bolt torque values of 200 and 150 Nm. The local pressure change with bolt torque is shown in figure 4 for the two positions, at the bolt (1) and between the bolts (2). The relationship between the local pressure and bolt torque was found to be linear (confirming no separation is taking place), as shown in figure 4.

## 6. Contact pressure measurement

To verify the FE analyses and further investigate pressure distribution at the brake disc/wheel carrier interface, measurements of the contact pressure were conducted, by using pressure-sensitive paper. The change in colour of initially white pressure-sensitive paper, to shades of red, is directly proportional to the pressure at the particular area. The pressure-sensitive paper, supplied by Sensor Products, is available in four ranges of interface pressure. Using the information about the average disc/carrier interface pressure (56.2 MPa for maximum torque; see table 1), and FE pressure distribution results (up to 124 MPa; see figure 3), papers for two pressure ranges were selected. The medium-range paper, designed to measure interface pressures in the range between 9.7 and 49.0 MPa, and high-range paper, for measuring pressures above 49.0, up to 127.6 MPa.

Disc and wheel carrier clamping faces were cleaned and the pressure-sensitive paper (requiring careful handling) cut to the profile of the interface. The humidity and ambient temperature were recorded as this affects the colour intensity change of the paper and is required for the analysis of results. The pressure-sensitive paper was placed between the clamping faces and the components bolted together. The bolts were tightened first to 100 Nm, by tightening opposing bolts alternately, to ensure application of a uniform clamp force over the entire contact area. The bolts were finally tightened to the nominal torque (300 Nm) in the same order. The pressure-sensitive paper requires that the pressure be sustained for more than 10 s. After this period, the bolts were loosened in the order described for tightening and removed. The wheel carrier was carefully separated from the disc, and finally, the pressure-sensitive paper removed and sent to the supplier for processing. Computer analysis of the pressure-sensitive paper provides pressure distribution and magnitude. The system renders high resolution, colour-calibrated images that reflect pressure distribution at the interface. The results obtained from the analysis include two-dimensional and three-dimensional contour plots, as shown in figure 5 for the high-range paper.

The pressure paper contour plots (figure 5) also indicate two distinctive areas, one with high pressure, around the bolt holes (1), and one with low pressure, between the bolts (2). The highest pressure is at the group of two holes between the jacking hole (smaller diameter hole). High pressure is also seen close to the internal diameter concentrating at the regions close to the boltholes. The pressure reduces from the internal diameter towards the outside diameter of the contact area. The boltholes continue to influence the pressure distribution close to the outside diameter. The low pressure, in the region of the outside diameter between the boltholes, is not registered in the shown contour plots (figure 5). The pressure in this region is below the threshold of the high-pressure paper, that to say, it is below 49 MPa. However, the medium paper provides more details in this region and the results have been carefully analysed. In order to reduce the amount of data presented, these results have not been shown.

## 7. Comparison of FE and measured interface pressure results

The measured pressure-sensitive paper results (figure 5) are very close to the stress results from the FE analysis (figure 3). Note that results in figure 5 are obtained from the high-range paper, which cannot measure pressures lower than 49 MPa. FE analyses and pressure-sensitive paper measurements results are compared in table 2 (for maximum bolt torque), at positions (1) and (2). The results show that the FE modelling can reliably predict pressure distribution in bolted brake component interfaces. This gives confidence in designing new bolted joints with predictable interface pressure distributions.

## 8. Thermal contact resistance measurement

### (a) Experimental set-up

The experimental part of the study was conducted using a specially developed spin rig (see Voller 2003). The rig has a simple, in-line arrangement of the disc/wheel assembly, shaft with adapter, torque transducer, speed sensor and electric motor. The spin rig has been designed for measuring all modes of brake disc heat dissipation (convection, conduction and radiation) and disc airflow characteristics. Experiments have been conducted on the CV brake disc and wheel carrier assembly (see figure 1) installed on to the spin rig shaft, schematically shown in figure 6. For TCR measurements, the shaft did not rotate.

The brake disc was heated using two hot air guns fitted to a heater box that shrouded the brake disc (not shown in figure 6). The heater box allowed hot air to flow over the surface of the disc providing uniform heating. The heating power could be controlled from 0 to 4 kW. The shaft adapter, disc and wheel carrier were insulated, as shown in figure 6, to prevent heat losses and ensure adequate heat flow. A thermal blanket placed over the test components and relatively low test temperatures (below 200 °C) further ensured very low thermal losses. For the worst case (maximum interface temperature), thermal losses have been calculated to be under 5% of the heat flow through the interface. The temperatures were measured in close proximity to the interface under steady-state conditions. Therefore, such relatively small losses were considered acceptable and no ‘compensation’ was introduced in processing the results.

Pressure distribution investigation, conducted in §7, indicated two distinctive areas, high pressure areas around the bolts, and low pressure areas between the bolts (see figure 3). Therefore, temperature distributions have been measured in these two areas, positions (1) and (2), as shown in figure 7 (note the heating box over the disc surface in figure 7*a*). At each position, four holes have been drilled into the disc and four into the carrier (in the radial direction, see figure 6). The axis of the first holes (closest to the interface) were 2 mm from the interface, with the remaining holes drilled every 4 mm. The hole diameter was sized to provide secure fitting of thermocouples. The hole depth was chosen to enable accurate measurement of temperature at the two distinctive areas (see figure 3). At each position, all eight holes were of the same depth. This was to ensure accurate radial position, providing measurement in the direction of conductive heat flow, minimizing the influence of any possible (despite insulation) convective or radiative heat dissipation.

The temperatures were measured using K-type welded tip glass fibre insulated thermocouples. Heat sink compound was applied to the thermocouple head to evacuate air and improve the contact at the measurement points. The temperature readings were taken when steady-state conditions were reached, allowing the temperatures of thermocouple tip and the surrounding material to equalize.

### (b) Experimental procedure

The fixing bolts were tightened to six torque levels (all bolts to the same torque), gradually increasing from 50 to 300 N m (as shown in table 1). The assembly was heated (see figure 6) with pre-set, constant power, until steady-state conditions were reached. By controlling the heating power, three temperatures levels were achieved, of approximately 70, 120 and 170 °C. The maximum temperature level (170 °C) can be considered to be relatively low, but it realistically represents service thermal loads in this region. All 16 temperatures were simultaneously logged. The criteria for reaching steady-state condition was set as a temperature change of less than 0.05 °C during 400 s.

### (c) Experimental results

For illustration purposes, at position (2), figure 8 shows the average steady-state temperatures at the eight points, for the interface temperature level of approximately 70 °C. The temperatures are averaged from 100 consecutive temperature values logged at 0.25 Hz. The temperature results enable linear relationships to be established between the temperature and distance, as shown in figure 8. The temperature gradient (d*T*/d*x*) at the disc is −78.9 °C m^{−1}, and −191.3 °C m^{−1} at the carrier. The temperature gradients are proportional to the conductivity of the material. Higher temperature gradient in the carrier is a result of lower thermal conductivity of SG iron compared with grey iron. The temperature drop (Δ*T*_{int}) at the interface is 1.3 °C, as shown in figure 8 (note: Δ*T*_{int}=*T*_{D}−*T*_{C}).

## 9. Thermal contact resistance results

### (a) Data processing

So far, the term ‘TCR’ (*R*_{cond} (m^{2} K W^{−1})) has been used to describe the phenomenon of resistance to conductive heat transfer through the interface of two solid surfaces in contact. However, despite this term being most appropriate for describing the phenomenon, for calculations of conductive heat transfer, thermal contact conductance *h*_{cond} (W m^{−2} K^{−1}), is often more suitable. Thermal contact conductance is the reciprocal value of TCR coefficient (*h*_{cond}=1/*R*_{cond}). In brake thermal analyses, there is an additional benefit of expressing the conductive heat transfer using thermal conductance. This approach enables the conductive heat transfer coefficient (thermal conductance) to be directly compared with the convective heat transfer coefficient and (suitably derived) radiative heat transfer coefficient (all in W m^{−2} K^{−1}). In this manner, the individual contributions of the three heat transfer modes (conduction, convection and radiation) can be directly compared, and brake cooling effects investigated for variety of service duties and temperatures.

The average thermal contact conductance (*h*_{cond}) at the interface can be determined from the heat transfer equation:(9.1)where *Q*_{cond} is conductive heat transfer through the interface, *A*_{int} contact area, *T*_{D} disc interface temperature and *T*_{C} wheel carrier interface temperature (see figure 8). It is assumed that there are no heat losses in the proximity of the interface and all the heat (*Q*_{cond}) is conducted from the disc to the wheel carrier. For heat transfer calculations, the two flanges were considered to be continuous plates with no bolt holes and bolts. It is assumed that the combined thermal resistances of the bolt/washer/nut assembly is equal to the thermal resistance across the interface in the proximity of the bolt. Detailed analyses confirmed the validity of such an assumption.

The heat flow is determined by Fourier's law of heat conduction(9.2)where *k*_{D} is disc material thermal conductivity, and thermal gradient d*T*/d*x* is already known from figure 8 (d*T*/d*x*=−78.9 °C m^{−1}).

### (b) Thermal conductance as a function of interface pressure

By performing temperature measurements, and using equations (9.1) and (9.2), thermal contact conductance has been calculated for different bolt torque (i.e. different average interface contact pressure) values. Figure 9 shows the thermal conductance change, at two positions ((1) in the bolt proximity and (2) between the bolts), as a function of average interface pressure. Despite some scatter, the trend is a linear increase of *h*_{cond} with pressure. The two lines indicate that for the same average interface pressure, *h*_{cond} is higher in the proximity of the bolts (position (1)), than between the bolts (position (2)). Maximum values of thermal conductance have been determined for maximum bolt torque (300 N m), corresponding to an average pressure of 56.2 MPa. The values are 10 200 Wm^{−2} K^{−1} in the bolt proximity (position (1)) and 6350 W m^{−2} K^{−1} between the bolts (position (2)). At the lower end of the average interface pressure (at *ca*. 10 MPa), the thermal conductance values are much lower, 3700 W m^{−2} K^{−1} at position (1), and 2900 at position (2).

When the lines shown in figure 9 are extrapolated to zero average pressure, the *h*_{cond} values for the two positions converge to an almost identical value of around 2200 W m^{−2} K^{−1}. This can be expected because, for very low clamp force, the pressure distribution for nominally flat and smooth surfaces is quite uniform.

From figure 9, the linear relationships between thermal conductance *h*_{cond} (W m^{−2} K^{−1}) and average interface pressure *P*_{avg} (MPa) can be established for the two positions:(9.3)(9.4)The differences in thermal conductance between the two areas (positions (1) and (2)) can only be the result of difference in local interface pressure, since all other parameters (surface finish and condition, temperature level) are identical. This is confirmed by studying local interface pressures, presented earlier (see tables 1 and 2, figures 3–5). Therefore, it is possible to calculate thermal conductance as a linear function of local interface pressure, as shown in equation (9.5) and figure 10.(9.5)where, as before, *h*_{cond} is the thermal conductance (W m^{−2} K^{−1}), but *P* is now local interface pressure (MPa). Although a linear relationship between thermal conductance and local contact pressure could have been expected, it was necessary to conduct measurements in at least two different areas, to ensure beyond any doubt that the relation is indeed linear on the entire interface.

## 10. Discussion of results

Equation (9.5) is a very useful tool for calculating thermal conductance in many technical applications, in particular for detailed prediction of local temperatures. However, for general engineering calculations of conductive heat transfer through the entire bolted joint, the average value of thermal conductance can also be of interest. Since the relationship (9.5) is linear, average thermal conductance for a bolted joint can be calculated from the average interface pressure(10.1)The average pressure (*P*_{avg}) can be easily calculated based on applied bolt torque, number of bolts and contact area, as per equation (4.1). That means the average thermal conductance can be efficiently predicted for a variety of components with similar materials and surface finish to the disc and wheel carrier. Particular care is required if the above equation is to be applied to the bolted joint where there is a separation at the interface, since such conditions have not been investigated.

In order to validate results further, in table 3, thermal conductance (*h*_{cond}) is compared with available published data. Unfortunately, very limited results for components of similar material, surface finish and interface pressure, were available. By using equation (10.1) an average thermal conductance of 2460 W m^{−2} K^{−1} was calculated for the interface pressure of 2 MPa (a similar value to the pressures quoted in literature).

Fukano & Matsui (1986), as discussed earlier, quoted an average thermal conductance of 712 W m^{−2} K^{−1}. Because many relevant details were not given, a number of estimates had to be made. After correction for the difference in interface pressure, the thermal conductance increases to 890 W m^{−2} K^{−1}, which is approximately one-third of the value determined by equation (10.1).

The highest thermal conductance is given by Mittelbach *et al*. (1994), for the aluminium interface and contact pressure of 2 MPa. The higher thermal conductance value (two times higher than the value calculated using equation (10.1)) is not a surprise, considering the lower Young's modulus and surface hardness of aluminium when compared with cast irons.

The stainless steel result, by Cengel (1998), shows an approximately 50% higher thermal conductance value than the analysed CV disc/carrier joint. This can also be expected when considering the lack of corrosion and ground surface finish.

## 11. Conclusions

Performed theoretical and experimental analyses of bolted joints enable reliable predictions of the clamp force as a function of torque applied, for a typical, large automotive joint. Average contact pressure can easily be calculated from the bolt force, however, interface pressure distribution can be very non-uniform, depending on the bolted joint design. Based on conducted FE studies and pressure-sensitive paper measurements, the interface pressure distributions can be predicted accurately and confidently. The bolted joint showed considerable interface pressure variations, despite the very strong, robust design, having thick flanges and large bolts arranged in relatively close proximity.

Numerous TCR measurements provided repeatable results and a detailed understanding of the TCR change with interface pressure. Linear relationships have been established, enabling the calculation of thermal conductance (or TCR) as a function of either average or local interface pressure. Therefore, even at early design stages, accurate theoretical prediction of conductive heat transfer across a variety of bolted joints can be made. This enables thermal studies and temperature predictions on either a detailed local level or a global estimate for the entire joint. The calculations can provide good results for a typical machined surface roughness (*Ra*=1–3.3 μm), over a wide range of interface pressures. Further work is in progress to determine the influence of other factors (temperature, surface condition) on TCR.

## Acknowledgments

The work presented in this paper was made possible by the assistance obtained from the EPSRC (Engineering and Physical Sciences Research Council), ArvinMeritor (UK) and Brunel University, Department of Mechanical Engineering. The authors are grateful for this help.

## Footnotes

- Received July 28, 2003.
- Accepted December 1, 2004.

- © 2005 The Royal Society