Analytical Investigation of Hopf Bifurcations Occurring in a 1DOF Sliding-Friction Oscillator With Application to Disc-Brake Vibrations

Abstract

This article deals with analytical investigations on stability and bifurcations due to declining dry friction characteristics in the sliding domain of a simple disc-brake model, which is commonly referred to as “mass-on-a-belt”-oscillator. Sliding friction is described in the sense of Coulomb as proportional to the normal force, but with a friction coefficient μS which depends on the relative velocity. For many common friction models this latter dependence on the relative velocity can be described by exponential functions. For such a characteristic the stability and bifurcation behavior is discussed. It is shown, that the system can undergo a subcritical Hopf-bifurcation from an unstable steady-state fixed point to an unstable limit cycle, which separates the basins of the stable steady-state fixed point and the self sustained stick-slip limit cycle. Therefore, only a local examination of the eigenvalues at the steady-state, as is the classical ansatz when investigating conditions for the onset of friction-induced vibrations, may not give the whole picture, since the stable region around the steady state fixed point may be rather small. The analytical results are verified by numerical simulations. Parameter values are chosen for a model which corresponds to a conventional disc-brake.