Stochastic dynamic analysis of nonlinear MDOF systems with chaotic motion under combined additive and multiplicative excitation

Abstract

Stochastic response and global dynamic analyses of structures with chaotic motion are challenging issues, especially for nonlinear multi-degree-of-freedom (MDOF) system excited by combined additive and multiplicative excitation. In this paper, a novel direct probability integral method (DPIM) is extended to address these challenges. Firstly, the probability density integral equation (PDIE) of MDOF system under combined excitation is established. By using the DPIM to solve the decoupled deterministic dynamic equation and PDIE successively, the stochastic responses are then achieved, and the stochastic bifurcation of system under combined excitation is explored. Moreover, the important equivalent relationship between PDIE and Dostupov–Pugachev differential equation is derived, exhibiting the superiority of PDIE for MDOF system under additive and multiplicative excitation. Due to the lack of effective numerical tool for global dynamic analysis of nonlinear MDOF stochastic system, another aim of this study is to propose a DPIM-based strategy to attack this problem from the view of probability. As the generalized stochastic basin, the ϵ-committor is introduced, and global integrity measure (GIM) is utilized for evaluating the stability of stochastic basin quantitatively. Finally, two examples of nonlinear systems under combined excitation, including nonlinear MDOF coupled ship model with chaotic motion under random oblique wave, demonstrate the effectiveness of DPIM. It is shown that the safety basin of system under combined excitation can be effectively described in a probabilistic way. The dramatic effect of initial disturbance on stochastic ship system is revealed, i.e., the stochastic safety basin is broken up to a series of discretized regions with increasing of intensity of initial disturbance, resulting in the decreasing of system stability.