Eigenvalue solution of thermoelastic instability problems using Fourier reduction

A finite-element method is developed for determining the critical sliding s peed for thermoelastic instability of an axisymmetric clutch or brake. Line ar perturbations on the constant-speed solution are sought that vary sinuso idally in the circumferential direction and grow exponentially in time. The se factors cancel in the governing thermoelastic and heat-conduction equati ons, leading to a linear eigenvalue problem on the two-dimensional cross-se ctional domain for the exponential growth rate for each Fourier wavenumber. The imaginary part of this growth rate corresponds to a migration of the p erturbation 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 critica l speed and the migration speed. Criteria are developed to determine the me sh refinement required to give an adequate discrete description of the ther mal boundary layer adjacent to the sliding interface. The method is then us ed to determine the unstable mode and critical speed in geometries approxim ating current multi-disc clutch practice.