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

Abstract

Above a certain critical speed, sliding systems with frictional heating such as brakes and clutches can exhibit thermoelastic instability (TEI) in which non-uniform perturbations develop in the pressure and temperature fields. A method is described in which the transient thermomechanical behavior of such systems is approximated by a reduced order model, describing few dominating perturbations or eigenfunctions. The goal is to construct a mathematical model of the system with 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 behavior be generated by a sequence of operations in which the sliding speed is piecewise constant. Both models are evaluated by comparison with direct numerical simulation and prove to give good accuracy with a dramatic reduction in computation time. A method is also described for solving the unperturbed thermoelastic contact problem with frictional heating, which involves non-homogenous equations. A solution method is explored based on the concept of superimposing a steady state solution to an eigenfunction expansion of an equivalent homogenous problem. The difference between the initial temperature field and the steady state distribution acts as an initial disturbance that can grow yielding a non-uniform distribution in the temperature and contact pressure. This method was evaluated by comparison with direct numerical simulation and provides an excellent computational efficiency. A reduced order model, in which few dominating eigenfunctions are retained in the expansion series, gives an excellent approximation especially during the growth phase of a clutch or brake engagement when the sliding speed is above the critical speed. The contact area may shift or change in size causing the thermoelastic contact problem to be non-linear. An approximate transient solution is described in which the contact area is treated as a piecewise constant in time. This is investigated in the context of a typical clutch problem for the maximum temperature reached by the system. A typical clutch system operates above the critical speed, causing reduction in the contact area and, therefore, high local contact. A parametric study is conducted for better clutch performance. Short stopping time is shown to result in higher temperature caused by the high rate of heat generation. The effect of a multiple clutch engagement cycles is explored, in which the temperature is found to decay to a uniform state between the engagements. Finally, the cooling fluid, found in wet clutch systems, plays an important role in preventing temperature accumulation between engagement cycles.