Approximate Analysis of Limit Cycles in the Presence of Stochastic Excitations

In many cases the dynamic behaviour of technical systems is characterized by a non-stable equilibrium position. For instance flow-induced vibrations of elastically supported bodies, friction-induced vibrations of wheel-rail-systems or the lateral movement of railway vehicles have to be looked upon as self-sustained oscillations which can show distinct stable limit cycles. Frequently an additional stochastic excitation has to be taken into account, e.g. the flow turbulences, the roughness of the rail surface or disturbances of the wheel-rail-geometry. The mathematical description of these phenomena leads inevitably to non-linear differential equations with random excitations. The exact analysis of the probabilistic properties of the system leads to the Fokker-Planck-equations. For practical purposes, however, the application of this equation is to cumbersome. An alternative method consists in the Monte-Carlo-simulation, but this procedure consumes a lot of computing time if the obtained results must be reliable. In this situation linearization techniques have shown to be an extreme effective tool for an approximate analysis /1/. The aim of the present paper is to give a survey of those linearization methods which allow the investigation of stochastically disturbed limit-cycle systems.