A nite-element method is developed for determining the critical sliding speed for thermoelastic instability of an axisymmetric clutch or brake. Linear perturbations on the constant-speed solution are sought that vary sinusoidally in the circumferential direction and grow exponentially in time. These factors cancel in the governing thermoelastic and heat-conduction equations, leading to a linear eigenvalue problem on the two-dimensional cross-sectional domain for the exponential growth rate for each Fourier wavenumber. The imaginary part of this growth rate corresponds to a migration of the perturbation in the circumferential direction.The algorithm is tested against an analytical solution for a layer sliding between two half-planes and gives excellent agreement, for both the critical speed and the migration speed. Criteria are developed to determine the mesh re nement required to give an adequate discrete description of the thermal boundary layer adjacent to the sliding interface. The method is then used to determine the unstable mode and critical speed in geometries approximating current multi-disc clutch practice.

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a b s t r a c tSliding systems with frictional heating exhibit thermoelastic instability (TEI) when the sliding speed exceeds the critical value. TEI can lead to hot spots on contact surfaces and is generally of great practical importance in friction brakes and clutches. The phenomenon is well defined in terms of the theory of stability with a classic perturbation approach being commonly used. While the perturbation analysis determines the stability limit, recent interest extends further towards exploration of the unstable behavior. This is motivated by practical reasons, namely by the fact that many common friction brakes and clutches operate instantaneously at speeds exceeding the critical speed for TEI, i.e. in the unstable regime. In order to determine a transient solution, possible mechanisms of excitation of unstable modes of different nature need to be accurately defined and quantified. These mechanisms are normally not considered in stability analysis of the steady-state where an initial perturbation of the thermoelastic field is assumed. In many realistic situations, however, there is no indication of the existence of meaningful initial temperature variation. Lack of full understanding of these mechanisms has perhaps limited broader industrial applications of recent theoretical advances in TEI. In this paper a method of solving the transient thermoelastic process in frictional systems using finite element spatial discretization and modal superposition is presented. Then mechanisms that excite the unstable thermoelastic modes other than the initial perturbation of temperature are studied. The role of the background process (corresponding to nominal applied loads) in the excitation is shown in a clear form and illustrated by practical examples for automotive friction clutches. It is demonstrated, in particular, that while for some geometries and configurations of the sliding system the imperfections determine the excitation of unstable modes, with other configurations strong excitation occurs even in the absence of imperfections.