Time variant reliability estimation of randomly excited uncertain dynamical systems by combined Markov chain splitting a
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O R I G I NA L
Oindrila Kanjilal
· C. S. Manohar
Time variant reliability estimation of randomly excited uncertain dynamical systems by combined Markov chain splitting and Girsanov’s transformation
Received: 3 April 2020 / Accepted: 25 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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. Keywords Time variant reliability · Random excitation · Uncertain system parameters · Monte Carlo simulation · Markov chain splitting · Girsanov transformation 1 Introduction Estimation of time variant reliability of dynamical systems, in which randomness exists in both the system parameter specifications and the external excitations, remains one of the most challenging problems in stochastic mechanics. This problem usually consists of determining the probability that the state of the dynamical This work was carried out when the first author was a research student at the Indian Institute of Science. Electronic supplementary material The online version of this article (https://doi.org/10.1007/s00419-020-01726-y) contains supplementary material, which is available to authorized users. O. Kanjilal (B) Engineering Risk Analysis Group, Technical University of Munich, Munich, Germany E-mail: [email protected] C. S. Manohar Department of Civil Engineering, Indian Institute of Science, Bangalore, India E-mail: manohar@iisc
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