Reliability of chain-like MDOF nonlinear structural systems under Gaussian white noises

An approximate analytical method is proposed to estimate the reliability of chain-like multi-degree-of-freedom nonlinear structural systems under Gaussian white noise, which combines the stochastic averaging method and the two-step generalized elliptical coordinate transformation to bypass the challenge of solving high-dimensional backward Kolmogorov equations and evaluating high-dimensional domain integrals. It involves reducing the original system to a one-dimensional averaged stochastic differential equation using the stochastic averaging method, leading to the backward Kolmogorov equation associated with the total energy and time. The averaged drift and diffusion coefficients are determined by applying the two-step generalized elliptical coordinate transformation. Then, the approximate reliability function and the conditional probability density function of first-passage time are obtained by addressing the low-dimensional backward Kolmogorov equation. A chain-like 6-DOF nonlinear structural system is carried out to verify the accuracy of the proposed method. Results from both the proposed method and the Monte Carlo simulation indicate that the reliability is sensitive to the weak Gaussian white noise but not to nonlinear damping coefficients and nonlinear stiffness coupling coefficients. Moreover, the running time for results from the proposed method is more than 106\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10<^>{6}$$\end{document} times faster than the Monte Carlo simulation when the relevant integrals are given.