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 in which non-uniform perturbations develop in the pressure and temperature ÿelds. 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. ?

The influence of frictional facing thickness on the contact pressure distribution of multi-disc dry clutches

“…Al-Shabibi and Barber [2] investigated the thermomechanical performance of the sliding systems using an approximation approach (Reduced Order Model) which describes the dominant perturbation or eigenfunctions in one value or more. Their numerical model represented the sliding system with a modest number of nodes to find the temperature field, and the distribution of contact pressure.…”

Thermoelastic instability in friction clutches and brakes – Transient modal analysis revealing mechanisms of excitation of unstable modes

“…Dow and Burton, 1972;Lee and Barber, 1993a;Du et al, 1997;Yi et al, 2000;Decuzzi et al, 2001;Krempaszky and Lippmann, 2005). While the classic perturbation analysis determines stability limits for the system, recent interest is directed towards exploration of the unstable behavior (Zagrodzki, 1990;Kao et al, 2000;Zagrodzki et al, 2001;Al-Shabibi and Barber, 2002;Afferrante et al, 2003;Choi and Lee, 2004). This is motivated by practical considerations, namely by the fact that many common friction brakes and clutches operate instantaneously at speeds exceeding the critical speed for TEI, i.e.…”

“…Also, in many known studies of transient behavior, e.g. Al-Shabibi and Barber (2002), Afferrante et al (2003) and Voldrich (2007), an initial perturbation of temperature is assumed and the background process is not considered. Initial temperature variation seems to be unquestionable in some practical situations, for instance as a remnant of the preceding clutch or brake application.…”

“…The general solution of this problem is presented in analytical form. The semi-discrete problem is then reduced to an eigenvalue problem with eigenvalues representing the stability parameter, an approach similar to that used in Yi et al (2000) and Al-Shabibi and Barber (2002). Next, by transforming the problem to modal variables, the transient solution is expressed by modal superposition; this solution was presented by Zagrodzki (2003), and used by Li and Barber (2004) and Li and Barber (2008).…”

“…For convenience, the analysis is performed for constant sliding speed. Substantial contributions to the solution of problems with variable speed using modal analysis have been made by AlShabibi and Barber (2002) and then further enhanced by Li and Barber (2004) and Li and Barber (2008); the latter approach was proven to be particularly robust and it can be applied to the models used in this study.…”