Short-term dynamic instability of a nonlinear stochastic SDOF system

Abstract

Theory of marginally unstable stochastic systems, recently developed for linear systems with lumped parameters, is applied here to a nonlinear single-degree-of-freedom system. The name “marginal instability” is used for nominally stable systems with potential short-term instability due to temporal random variations of parameter(s) with brief excursions out of the stability domain. Prediction of the corresponding brief outbreaks in response is based on parabolic approximation of the parameter variation during excursion (asymptotic analysis of the Slepian model). The theory provides probabilistic description of the response (such as response probability density function (PDF) or solution to the first-passage problem) in terms of that of the parameter(s) variations. Systems with nonlinear restoring forces are considered in this Note with the use of quasiconservative version of averaging over response period. Presented results illustrate influence of nonlinearities on the response characteristics which are important for reliability evaluation.