FINITE ELEMENT ANALYSIS OF FRICTIONALLY EXCITED THERMOELASTIC INSTABILITY

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1 Journal of Thermal Stresses ISSN: (Print) X (Online) Journal homepage: FINITE ELEMENT ANALYSIS OF FRICTIONALLY EXCITED THERMOELASTIC INSTABILITY Shuqin Du, P. Zagrodzki, J. R. Barber & G. M. Hulbert To cite this article: Shuqin Du, P. Zagrodzki, J. R. Barber & G. M. Hulbert (1997) FINITE ELEMENT ANALYSIS OF FRICTIONALLY EXCITED THERMOELASTIC INSTABILITY, Journal of Thermal Stresses, 20:2, , DOI: / To link to this article: Published online: 27 Apr Submit your article to this journal Article views: 143 View related articles Citing articles: 34 View citing articles Full Terms & Conditions of access and use can be found at Download by: [University of Michigan] Date: 03 January 2018, At: 08:00

2 FINITE ELEMENT ANALYSIS OF FRICTIONALLY EXCITED THERMOELASTIC INSTABILITY Shuqin Du, P. Zagrodzki, J. R. Barber, and G. M. Hulbert Department ofmechanical Engineering andapplied Mechanics University ofmichigan Ann Arbor, Michigan, USA The frictional heat generated during braking causes thermoelastic distortion that modifies the contact pressure distribution. If the sliding speed is sufficiently high, this can lead to frictionally excited thermoelastic instability, characterized by major nonuniformities in pressure and temperature. In automotive applications, a particular area of concern is the relation between thermoelastically induced hot spots in the brake disks and noise and vibration in the brake system. The critical sliding speed can be found by examining the conditions under which a perturbation in the temperature and stress fields can grow in time. The growth has exponential character, and subject to certain restrictions, the growth rate b is found to be real. The critical speed then corresponds to a condition at which b = 0 and hence at which there is a steady-state solution involving nonuniform contact pressure. We first treat the heat sources Q at the contact nodes as given and use standard finite element analysis (FEA) to determine the corresponding nodal contact forces P. The heat balance equation Q = jvp, wheref is the coefficient of friction, then defines a linear eigenvalue problem for the critical speed V. The method is found to give good estimates for the critical speed in test cases with a relatively coarse mesh. It is generally better conditioned and more computationally efficient than a direct finite element simulation of the system in time. Results are presented for several examples related to automotive practice and show that the flexural rigidityofthe friction pad assembly has a major effect on the critical speed. It is well known that thermoelastic distortion due to frictional heating can initiate a form of instability in sliding systems known as thermoelastic instability (TEl). Briefly, the distortion affects the contact pressure distribution, which in turn affects the distribution of frictional heating and hence the thermoelastic distortion [1-3]. This feedback process is unstable if the sliding speed is sufficiently high and can lead to localization of the load in a small region of the nominal contact area. The resulting high local temperatures and thermal stresses have various undesirable effects such as material transformations, thermal cracking, and brake fade. The earliest rigorous investigation of TEl for a simple sliding system is due to Dow and Burton [2], who introduced the idea of determining the conditions under Received 30 July 1996; accepted 18 August The authors are pleased to acknowledge support from the Ford Motor Company and from the National Science Foundation under contract number CMS P. Zagrodzki is on leave from the Institute of Transport, Warsaw University of Technology, Koszykowa 75, Warsaw, Poland. Address correspondence to Professor James Barber, Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI jbarber@engin.umich.edu Journal of Thermal Stresses, 20: , 1997 Copyright 1997 Taylor & Francis /97 $

3 186 S. DU ET AL. which a perturbed thermoelastic field can grow exponentially in time. This method has since been used extensively to determine the critical sliding speed for a variety of geometries, mostly involving bodies of infinite extent. The results generally underestimate the experimentally observed critical speed for practical systems, and there is evidence that this is due to the influence of the finite dimensions of the system. Lee and Barber [4] considered the case of an infinite layer sliding between two half-spaces and demonstrated that the finite thickness of the layer can raise the critical speed significantly and have a major effect on the nature of the dominant perturbation. Thus, if practical brakes and clutches are to be designed against TEl-related failure, it is essential to find some way of taking into account the finite geometry of the sliding system. An obvious approach to this problem would be to simulate the transient behavior of the finite geometry system, using a finite element description. A parametric investigation of the behavior would then enable stability boundaries to be determined. However, such an effort would be extremely computer-intensive since the solution of the transient heat conduction equation with reasonably fine meshing involves the use of a very small time increment for numerical stability and convergence. This is manageable in problems with two spatial dimensions [5] but would pose considerable difficulties in the three-dimensional geometry needed to give a realistic approximation to a practical brake system. A preferable alternative is to apply the finite element method (FEM) directly to Burton's perturbation method. We first formalize the method by postulating the existence of a small perturbation on the steady state that grows exponentially in time. This leads to an eigenvalue problem for the growth rate b, and we anticipate a denumerably infinite set of eigenvalues with corresponding eigenfunctions describing the perturbation fields. An arbitrary initial perturbation could be expanded as an eigenfunction series, and hence instability is indicated if anyone of the eigenvalues is positive or has positive real part. The problem is further simplified if we can assume that instability is governed by a real eigenvalue since in that case the stability boundary is determined by the presence of an eigenvalue of zero. An exponential growth rate of zero corresponds to a nontrivial perturbation in equilibrium that satisfies the steady-state equations. In this case, the method is equivalent to the determination of critical loads for structural stability by seeking conditions under which nontrivial equilibrium solutions exist. This approach was first suggested by Yeo [6], who developed it in the context of the related static contact problem, where instability results from the pressure dependence of an interfacial contact resistance. It is usual to consider Burton's method as a means of determining the stability of a steady state of the system. However, it is equally applicable to transient processes as long as the contact area does not change with time, since in that case the homogeneous perturbation problem has the same boundary conditions as it would have for a steady process. This is not the case for the static thermoelastic contact problem of Yeo and Barber [6], where changes in contact pressure affect stability through the local gradient of contact resistance.

4 FRICflONALLY EXCITED THERMOELASTIC INSTABILITY 187 STATEMENT OF THE PROBLEM We consider the plane strain system shown in Figure 1, in which an elastic body 0,1 slides at constant speed V in the z-direction (normal to the plane of the figure) against a second elastic body 0,2' The boundary of o' y is denoted by T, ("Y = 1,2) and the (plane) contact surface by f e Tractions are applied in the region T, U f 2 f e such as to ensure contact over the entire region fe' i.e., where u y n is the outward normal displacement component of body "y. Coulomb friction is assumed to occur at the contact surface, so that for each body where f is the coefficient of friction; unl' unn are tangential and normal in-plane stresses, respectively; and U n z is the out of plane tangential stress. No additional constraints are imposed on the applied tractions, except the equilibrium condition implied by the requirement of constant velocity. Indeed, the tractions can be permitted to change in a fairly general way with time. In the contact region fe' the local shear tractions will cause frictional heating, which flows into the bodies; so the local heat generated at the contact surface is denoted (2) q =fvp on r; (3) where we define the contact pressure p = -Unn on f e Continuity of temperature is assumed across the contact interface. We supposed that the other boundaries are insulated, hence the local heat fluxes are (4) / 0, Figure I. Two-body sliding system. Relative motion occurs normal to the plane of the figure.

5 188 S. DU ET AL. The Transient Problem The problem defined above is necessarily transient because there is a positive heat flow into the bodies in the contact region and no heat flow elsewhere, so the temperature must rise with time. If the applied tractions are constant and have a resultant F in the negative x-direction, the problem might be expected to approach a quasi steady state where the temperature increases linearly with time, i.e., A fvft T(x,y,t) -> T(x,y) + (C + C ) (5) 1 z where c. y is the thermal capacity of body n~. Fortunately, we do not need to solve this transient problem in order to determine the stability of the system. Instead, we consider the conditions under which a small perturbation in the temperature field can grow exponentially in time, i.e., T(x,y,t) =eb'o(x,y) (6) As long as contact is retained throughout fe' the perturbed problem remains linear and superposition applies. In particular, the perturbation must satisfy the heat conduction and thermoelastic equations. Also, since the prescribed boundary conditions are assumed to be satisfied by the unperturbed solution, it follows that the perturbation must satisfy a corresponding set of homogeneous boundary conditions. The problem for the perturbation therefore has a trivial solution in which all the perturbed fields are zero (as we should expect since, ex hypothesi, the unperturbed solution is a solution of the problem), but for certain eigenvalues of the exponential growth rate b we anticipate nontrivial solutions. If any of these has a positive real part, we conclude that the unperturbed solution will be unstable. If we make the further assumption that instability is always associated with real eigenvalues of b, we can conclude that the stability boundary-i.e., the critical speed Va at which instability begins-is determined by the existence of an eigenvalue at the origin b = O. Referring to Eq. (6), we see that this is equivalent to the perturbation being independent of time and hence to the condition that the homogeneous steady state problem has a nontrivial solution. It is this eigenvalue problem that we shall formulate and solve in this article. The Eigenvalue Problem We first develop a suitable continuous model of the problem and then discretize it. The perturbation problem is defined by the homogeneous boundary conditions (1), (2) in the contact region and the requirement that all other surfaces be tractionfree and insulated. The eigenvalue Va is introduced only in the boundary condition (3), which has the form q(y)=fvap(y)

6 FRICTIONALLY EXCITED THERMOELASTIC INSTABILITY 189 This condition also represents the only influence of the elastic solution on the heat conduction problem and, therefore, is convenient to impose at the end of the derivation. We first treat the q(y) as known, in which case it becomes a straightforward heat conduction problem to determine the temperature field, recalling that for b = 0 the temperature is independent of time and that for the perturbation problem the heat flux is zero at all noncontacting surfaces. A sequential thermoelastic problem then permits us to determine the stress and displacement fields and, in particular, the contact pressures p(y) needed to satisfy the contact boundary conditions (1)-(3), the remaining surfaces being traction-free. Since the problem is linear, the resulting pressures are linear functions of the assumed heat fluxes, i.e., p =~q or q = '?J'p where ~ is a linear operator and '?J' =~-l is its inverse. Substituting for q from Eq. (7), we obtain p = fv~p or fvp = '?J'p which is a linear eigenvalue problem for l/fv or the product fv in continuous form. We next use the FEM to obtain the corresponding discrete form of this problem. FINITE ELEMENT FORMULATION The Heat Conduction Problem For a solution of the form (6) with zero growth rate b, the perturbation in the temperature field 0 must satisfy the steady-state heat conduction equations (8) (9) (10) with the boundary conditions in F, in f 1 U f 2 - F, (11) where n y denotes the unit outward normal vector to the boundary and K; is the thermal conductivity.

7 190 s. DU ET AL. The heat fluxes qy in f e are defined implicitly by the conditions of temperature continuity and energy conservation, i.e., in fe' where q is given by Eq. (3). Notice that the solution to the problem defined by Eqs. (10), (11) exists only if (12) (13) y = 1,2 (14) and in that case is nonunique, having the form O(x, y) + C, where C is an arbitrary constant. The constant C corresponds to a uniform expansion and does not influence the stress and, in particular, the contact traction for linear elastic materials with the stated boundary conditions. Therefore, we can restrict attention to the spatially variable part O(x, y) of the solution of Eq. (10). To solve this problem, we use the standard Galerkin finite element procedure. We multiply Eq. (10) by an arbitrary weight function wand integrate over Oy Using Green's theorem we obtain Substituting boundary condition (11) into Eq. (16), we obtain Adding the two equations (17) and using Eq. (13), we obtain y=i,2 (15) (16) (17) /,KVO.VwdO= /, qwdf n r, (18) where 0 = 0 1 U O 2, 0 represents the temperature field in the whole domain 0; and K denotes the conductivity K; appropriate for the particular subdomain Oy, y= 1,2. The unknown 0 is approximated by O(x,y) =W(x,y)O (19)

8 FRICfIONALLY EXCITED THERMOELASTIC INSTABILITY 191 where W, 6 are vectors whose components are the piecewise shape functions W;(x, y) and the nodal temperatures ()i' respectively. In the Galerkin method, the shape functions W; are also used as weight functions w. Substituting Eq. (19) into Eq. (18) and replacing w by Uj leads to the (M X M) system of algebraic equations L6=Q* (20) where M is the total number of nodes, L= fnk(vw)tvwdfl (21) is the thermal conductivity matrix, and is a vector of nodal heat sources. Notice that although the heat source vector Q* has M components, only N of these are nonzero, corresponding to the N nodes in the contact area. For all other nodes, the shape functions Uj make zero contribution to the surface integral. We can therefore define a new vector Q of order N containing just these nonzero elements of Q*. The continuous distribution q(y) is approximated using the shape functions Uj, i.e., where We denotes the subset of N shape functions corresponding to the N contact nodes and q is a nodal heat flux vector of order N. Substituting Eq. (23) into Eq. (22), we obtain where Q=Wq (22) (23) (24) (25) is a matrix of dimension N X N. The Elastic Problem The constitutive equation for the elastic problem with thermal distortion is (J' = D(E - Eo) (26)

9 192 S. DU ET AL. where is the initial strain vector resulting from thermal expansion; (J", E are stress and strain vectors, respectively; and D, a are the elasticity matrix and thermal expansion coefficient, respectively. We first treat the two subdomains 01' O 2 as separate problems. The standard finite element treatment of the elastic problem, based on the variational principle [7], leads to the following equation of equilibrium for domain Oy (27) (28) where u, is a vector of nodal displacements, is the stiffness matrix, is a vector of nodal forces, B is the strain matrix [7], and P y * is a vector of nodal forces resulting from the unknown contact pressure. Substituting Eq. (27) into Eq. (30), we obtain where 'l'y is a matrix. The two problems (28) are coupled through the boundary condition (1), which states that the displacement vectors u 1 ' u 2 have common normal components in the contact region. However, rather than eliminating the dependent degrees of freedom between the two subproblems, we simply impose a penalty function on the relative normal displacements of corresponding nodes at the contact interface, which is equivalent to the connection of the two subdomains through a set of contact elements. This leads to an equation of the form (29) (30) (31) Ku=F (32) where u is a vector containing all the components of u" u 2, and F = '1'0 (33) The forces P; become internal forces in the assembled problem and therefore cancel in Eq. (33). Notice that in our case the only contribution to F is that resulting from thermal expansion.

10 FRIcnONALLY EXCITED THERMOELASTIC INSTABILITY 193 Equation (32) is solved with boundary conditions (2) and an additional condition constraining rigid body motion, after which the vector P; can be computed from Eqs. (28), (31).. In the vector P y *, only N components are nonzero, corresponding to the normal forces at the N contact nodes. We therefore define a vector P of order N containing just these nonzero elements of P;. The relation between P and the piecewise-continuous contact pressure distribution p(y) is analogous to that between the nodal heat sources and the heat flux defined in Eqs. (23), (24), i.e., and P = «Ilp where p is the vector of nodal pressures and «Il is defined by Eq. (25). Stability Analysis In the discrete representation Eqs. (7), (8) take the forms Q=fVP P=AQ respectively, where vectors Q and P were defined in previous subsections and A is a matrix to be determined. The discrete form of the eigenvalue problem (9) is (34) (35) (36) (37) P=fVAP (38) The vectors Q and P have dimension N. However, the components of these vectors are not independent because the heat flux has to satisfy condition (14) and the contact pressure must satisfy the equilibrium relations 1p(y)df = 0 r; The discrete forms of these relations are 1p(y)ydf=O r; (39) (40) N LPj=O i=1 N L PiYi=O i=1 Thus, Q has N - 1 independent components, P has N - 2 independent components, and relation (37) can be reduced to (41) (42)

11 194 S. DU ET AL. where Q and P are vectors with N - 1 and N - 2 components, respectively, i.e., (43) and Ahas dimension (N - 2) X(N - 1). To find the coefficients of A, we first solve the heat conduction problem (20).for a set of N - 1 linearly independent vectors Qi, j = 1,..., N - 1. We then solve the corresponding N - 1 thermoelastic problems (31), (32) with the frictionless contact boundary condition (1) and all other boundaries tractionfree. Once the displacement vectors u i are determined, we follow the procedure described in previous subsection to find the corresponding nodal contact forces pi. From the applied vectors Qi and the resulting vectors pi we form the matrices [Q](N-I)X(N-I) and [P](N-2)X(N-I)" The matrix Ais then computed from the equation - = --I A = [p][q] Equation (36) implies that Q = jvp; hence, using Eq. (42), we have P=jVAP Using Eqs. (41), we obtain an expression for the component P N - 1 in terms of Pi' i = 1,..., N - 2, which we then use to eliminate this component on the right-hand side of Eq. (45), obtaining where Ais a new matrix of dimension (N - 2) X (N - 2). Finally, we rewrite Eq. (46) in the standard eigenvalue form as cp- jvp=o where C = A-I. This constitutes a linear eigenvalue problem for the critical sliding speed V o at the stability boundary, the eigenfunctions P defining the form of the corresponding unstable perturbation in contact pressure. Note that in the above formulation we do not use elemental stresses or heat fluxes but their discrete nodal counterparts, i.e., forces and heat sources. It is well known that the accuracy of FEM solutions for the latter is higher by one order than for the former [7]. Thus, the applied method ensures the highest accuracy available for the specific element type. (44) (45) (46) (47) RESULTS The method was first validated using the simple example of a rectangular block sliding against an insulated rigid plane, as shown in Figure 2. The block is assumed

12 FRICIlONALLY EXCITED THERMOELASTIC INSTABILITY 195 a b Figure 2. Rectangular block sliding against a rigid nonconducting wall. to be homogeneous and isotropic and to have width a (defining the contact length) and length b. The matrix C was determined as described in the last section using the FEM, with quadrilateral bilinear elements. Convergence of the algorithm was tested for the case of a square block (a = b) with an increasing number of nodes from 10 X 10 up to 100 X 100. Figure 3 shows the leading (lowest) eigenvalue as a function of the number of nodes N along each side. For this purpose, it is convenient to define a dimensionless eigenvalue as v; 7.2r , , ,-----,------, FEAsoluUon Converged soluuon (48) ~--~--==---~--~ L L: L L <, ' J Figure J. Convergence of the critical speed with increased mesh refinement.

13 196 S. DU ET AL. where E is Young's modulus. The results converge monotonically on the value Vo* = , no further change being obtained beyond N = 60. Taking this as the exact result, Figure 3 shows that 0.6% accuracy in the critical speed can be obtained for this configuration with a 10 X 10 mesh and 0.12% accuracy with a 20 X 20 mesh. Thus, for practical engineering purposes, a quite coarse mesh is sufficient to obtain good estimates of the critical speed. Effect of Aspect Ratio We define the aspect ratio as b r= a (49) so that large values of r correspond to contact of a strip on an end face and small values to contact of a thin layer. Figure 4 shows the first three eigenvalues as functions of the aspect ratio in the range 0.2 < r < 1. All the eigenvalues tend to a constant as r --> co, and in practice there is very little further change beyond r = 1. The explanation for this can be seen by considering the corresponding eigenfunctions. Figure 5 shows the contact pressure distribution corresponding to the first three eigenvalues for the case r = 1. Equilibrium considerations [Eqs. (41)J demand that the eigenfunctions 45r-----,r ,.--...,---..., ,----,----, 3 2 Figure 4. Effect of the aspect ratio on critical speed for the first three modes.

14 FRICflONALLY EXCITED THERMOELASTIC INSTABIUTY 197 -< <1.3 I I I -,',,,,,, y......,., \ J I ",... -, 15 -lsi Mode --- 2nd Mode --- 3rd Mode Figure 5. Eigenfunctions (contact pressure perturbations) for the first three modes. have zero sum across the width and hence have approximately sinusoidal form. The perturbed fields satisfy the steady-state heat conduction equation and hence tend to decay exponentially with x at a rate related to the wavenumber of the variation through the width. It follows that the addition of material distant from the contact area has little effect on the eigenfunctions and hence on the eigenvalues. An approximation to the lowest eigenvalue for large r can be obtained by assuming that the corresponding eigenfunction is exactly sinusoidal. In that case, Burton's half-plane solution [18] yields the result 47TK V.=- o faea (50) where A is the wavelength of the perturbation. Writing A= a and substituting into Eq. (50), we obtain V o* = 47T = (51) The free boundaries of the block place a constraint on the mode shape, and it is therefore to be expected that the actual critical speed will be somewhat larger than this approximation. This is confirmed by the results of Figure 4, which show the first mode tending to a limiting value of at large r,

15 198 S. DU ET AL. We note from Figure 5 that the lowest eigenvalue is characterized by the eigenfunction with the smallest number of zero crossings (two) and that each successive eigenfunction has one additional zero crossing, consistent with the set of eigenfunctions comprising a complete set of orthogonal self-equilibrated functions on the contact region. When r < 0.38 a change in this behavior is observed, the lowest eigenvalue (and hence the critical speed of the system) being associated with an eigenfunction with three zero crossings. The corresponding curves in Figure 4 cross near r = This behavior is consistent with the results of Lee [4], who found that the thermoelastic instability for a layer sliding between two half-planes is associated with a sinusoidal form whose wavelength is related to the layer thickness. Similar results were also obtained by Yeo [9] for the related problem of the stability of thermoelastic contact between a block and a plane across a pressure-sensitive contact resistance. Three-Body Contact Figure 6 shows several geometries chosen to illustrate the application of the method to geometries closer to the conditions in an automotive disk brake. In Figure 60 a layer of cast iron slides between two pads of composition friction material, in Figure 6b the backs of the pads are reinforced with steel layers, in (a) r-- / / V V (b) - (e) (d) Figure 6. Test geometries related to automotive practice: (a) a layer between two coextensive pads of friction material, (b) pads reinforced by steel back plates, (c) disk extends beyond the pad surfaces, and (d) back plate flexibility increased by grooves.

16 FRICfIONALLY EXCITED THERMOELASTIC INSTABILITY 199 Table 1 Material properties Properties Disk Pad Back Plate Elastic modulus E (N/m') Thermal expansion a (j Cl Thermal conductivity K (W/mOC) Poisson's ratio v 1.25 X 1011 l.05 X X 10" l.ox X X Figure 6c the cast iron layer extends somewhat beyond the contact region, and in Figure 6d the reinforcement layer is interrupted by grooves in order to reduce overall bending stiffness. The pertinent properties of the materials are summarized in Table 1. Figure 7 shows the pressure distributions and the corresponding critical speeds associated with the first three modes for the system of Figure 6a. In each case, the 0.3r-----.:----, , , Va = Vo = Vo = o 5 10 Y o 5 o 5 10 Y Figure 7. Eigenfunctions for the first three modes of the system of Figure 6a. The upper and lower figures present the pressure perturbations on the left and right interfaces, PI' P" respectively.

17 1 200 S. DU ET AL. upper and lower figures present the pressure perturbations on the left and right interfaces, Pt' P2' respectively. The lowest critical speed (V o* = ) corresponds to an antisyrnmetric mode (Pt = -P2)' consistent with the results of Lee [4]. The second mode (Vo* = ) is also antisymmetric, but the third mode (Vo* = ) is symmetric (PI = P2)' The pressure distribution for the first mode of each of the systems of Figure 6 is shown in Figure 8. In each case, the perturbation is antisymmetric (PI = -P2); hence only the curves for PI are presented. The addition of steel back plates as in Figure 6b has a major effect on the critical speed, reducing it by almost a factor of two to V o* = The wavelength is increased by about 8%, but in the spirit of Burton's equation (50), the critical speed is probably more influenced by the increase in effective modulus due to the steel back plate. Essentially similar results were obtained for the geometry of Figure 6c, indicating that the extension of one of the two bodies in the contact plane has very little effect on the stability behavior. Clearly, the addition of a back plate increases the stiffness of the pad, enabling it to support an unstable perturbation at lower speeds. We might therefore expect that structural changes designed to restore flexibility would increase the critical speed. This is confirmed by results for the geometry of Figure 6d, where the addition of a groove increases bending flexibility in the direction parallel to the contact area and increases the critical speed to Vo* = 0.90n-an increase of 47% over the value for Figure 6b. 0.5r.-: ::';";"'"c-::-c:-:o (a) YO = (b) Vo = Y Y (d) (c) Vo= YO = =---:::---:,=---:: o Y Figure 8. Eigenfunctions (contact pressure perturbations) for the first (dominant) mode of the systems of Figures 6a-d. In each case, the perturbation is antisymmetric (PI = -P2)'

18 FRICTIONALLY EXCITED THERMOELASTIC INSTABILITY 201 CONCLUSIONS These results show that the numerical implementation of Burton's perturbation analysis for thermoelastic stability provides an extremely efficient method for determining the critical speed in sliding systems, for cases where instability is associated with real eigenvalues of exponential growth rate. By contrast, a direct numerical simulation of such a system would be very computationally intensive and require several runs to establish the critical sliding speed. A relatively coarse mesh is sufficient to obtain a good estimate of the first eigenvalue and, hence, of the critical sliding speed for the system. The method therefore bodes well for use in three-dimensional systems of significant complexity. The results show that the flexural rigidity of the friction pad assembly has a major effect on the critical speed. REFERENCES 1. J. R. Barber, Thermoelastic Instabilities in the Sliding of Conforming Solids, Proc. Roy. Soc. A, vol. 312, pp , T. A. Dow and R. A. Burton, Thermoelastic Instability of Sliding Contact in the Absence of Wear, Wear, vol. 19, pp , F. E. Kennedy and F. F. Ling, A Thermal, Thermoelastic and Wear Simulation of a High Energy Sliding Contact Problem, J. Lub. Tech., vol. 96, pp , Kwangjin Lee and J. R. Barber, Frictionally-Excited Thermoelastic Instability in Automotive Disk Brakes, ASME J. Tribology, vol. 115, pp , P. Zagrodzki, Analysis of Thermomechanical Phenomena in Multidisc Clutches and Brakes, Wear, vol. 140, pp , Taein Yeo and J. R. Barber, Finite Element Analysis of Thermoelastic Contact Stability, ASME J. Appl. Mech., vol. 61, pp , O. C. Zienkiewicz, The Finite Element Method, 3rd ed., McGraw-Hill, New York, R. A. Burton, V. Nerlikar, and S. R. Kilaparti, Thermoelastic Instability in a Seal-Like Configuration, Wear, vol. 24, pp , Taein Yeo and J. R. Barber, Finite Element Analysis of the Stability of Static Thermoelastic Contact, J. Thermal Stresses, vol. 19, pp , 1996.