Transient solution of a thermoelastic instability problem using a reduced order model

Above a certain critical speed, sliding systems with frictional heating such as brakes and clutches can exhibit thermoelastic instability in which non-uniform perturbations develop in the pressure and temperature fields. A method is described in which the transient thermomechanical behaviour of such systems is approximated by a reduced order model, describing one or more dominant perturbations or eigenfunctions. The goal is to construct a mathematical model of the system with a modest number of degrees of freedom. If a single dominant perturbation is used, an integral expression can be written for the evolution of the perturbation with time. A more accurate description involving several terms requires that the transient behaviour be generated by a sequence of operations in which the sliding speed is piecewise constant. Both models are evaluated by comparison with a direct numerical simulation and prove to give good accuracy with a dramatic reduction in computing time.