Approximate WPI based Survival Probability Determination of Nonlinear Oscillator under Combined Excitation

Abstract

An analytical method for determining stochastic response and survival probability of nonlinear oscillators subjected to combined evolutionary nonstationary excitation and periodic excitation is developed. This is done by the variational formulation of the analytical Wiener path integral (WPI) technique [1] in conjunction with a stochastic averaging/linearization treatment of the problem [2]. Specifically, the response of the system is approximately decomposed into a combination of a periodic and a zero-mean stochastic components. Thus, the equation of motion is cast into two coupled nonlinear differential equations with deterministic and stochastic components [3]. Relying on the stochastic averaging/linearization treatment, the differential equation with stochastic component is cast into an equivalent linear one with time-varying stiffness and damping elements. Next, based on the concept of Wiener path integral, the approximate analytical expression for the joint transition probability density function (PDF) is obtained for small time intervals. The analytical response of nonlinear system can be derived by short-time joint transition PDF in conjunction with the discrete form of ChapmanﾖKolmogorov (CﾖK) equation. Due to the decomposition of the response, the survival probability problem with fixed barriers subject to combined excitation is transformed to a problem of equivalent oscillator with time-varying barriers under the stochastic excitation, which can be easily solved by the proposed method. The hardening Duffing oscillator is considered as the numerical example. Comparisons with pertinent Monte Carlo simulation (MCS) data demonstrate the reliability of the developed technique.

Reference

[1] I.A. Kougioumtzoglou, P.D. Spanos, An analytical Wiener path integral technique for non-stationary response determination of nonlinear oscillators, Probab. Eng. Mech. 28 (2012) 125ﾖ131.

[2] I.A. Kougioumtzoglou, P.D. Spanos, An approximate approach for nonlinear system response determination under evolutionary stochastic excitation, Current Sci. (2009) 1203ﾖ1211.

[3] F. Kong, P.D. Spanos, Stochastic response of hysteresis system under combined periodic and stochastic excitation via the statistical linearization method. ASME J. Appl. Mech. 88(5) (2021) 051008.