Modelling friction-induced Vibrations using Spectral Methods

A series of holographic studies of disc brake squeal have revealed a number of significantfeatures [1] that it is hoped to demonstrate in mathematical models: the presence ofcircular travelling waves with definite mode order and frequency; relatively largedisplacement of in-plane components of the vibration; odd and even variation of the inplaneamplitude of vibration through the disc thickness; and significant changes in thewaveforms produced with varying pressure distribution on the brake pads. The modelling approach adopted is to construct partial differential equations representing simplified models of discs, pads and frictional interfaces – in both 2 and 3 dimensions – and to investigate numerically both steady state (eigenvalue) and dynamic (timestepping) solutions. To considerably reduce the computation time involved and to enable the rapid production of animated graphical output the authors have used the mathematical technique of spectral methods [2,3] rather than the more conventional finite element approach. This method for numerically solving partial differential equations has proved very successful in dynamic models that normally involve considerable computation time such as those occurring in meteorology and geophysics. It is shown that at least some of the observed effects of disc brake squeal can be predicted, and the various quantities in the model – coefficient of friction, speed of rotation, etc – varied to compare with the “real world.” It is hoped that this approach will be of general interest in the field of friction-induced vibration, as well as mathematical modelling in structural mechanics