Time variant reliability estimation of randomly excited uncertain dynamical systems by combined Markov chain splitting and Girsanov’s transformation

Abstract

The problem of estimating the time variant reliability of randomly parametered dynamical systems subjected to random process excitations is considered. Two methods, based on Monte Carlo simulations, are proposed to tackle this problem. In both the methods, the target probability of failure is determined based on a two-step approach. In the first step, the failure probability conditional on the random variable vector modelling the system parameter uncertainties is considered. The unconditional probability of failure is determined in the second step, by computing the expectation of the conditional probability with respect to the random system parameters. In the first of the proposed methods, the conditional probability of failure is determined analytically, based on an approximation to the average rate of level crossing of the dynamic response across a specified safe threshold. An augmented space of random variables is subsequently introduced, and the unconditional probability of failure is estimated by using variance-reduced Monte Carlo simulations based on the Markov chain splitting methods. A further improvement is developed in the second method, in which, the conditional failure probability is estimated by using Girsanov’s transformation-based importance sampling, instead of the analytical approximation. Numerical studies on white noise-driven single degree of freedom linear/nonlinear oscillators and a benchmark multi-degree of freedom linear system under non-stationary filtered white noise excitation are presented. The probability of failure estimates obtained using the proposed methods shows reasonable agreement with the estimates from existing Monte Carlo simulation strategies.