Transient Response of MDOF Nonlinear Dynamical System under Stationary Random Excitation

Engineering structures under external loadings are generally modeled as multi-degree-of-freedom (MDOF) nonlinear dynamical systems under random excitations. Based on the response Markov assumption, MDOF dynamical system with [Formula: see text]-dimensional state variables generally results in high-dimensional Fokker–Planck (FP) equation with [Formula: see text] spatial dimensions and one time dimension. It has been challenging to solve high-dimensional FP equation, especially the one with time dimension. In this paper, the previously proposed technique state-space-split-exponential-polynomial closure (SSS-EPC) method is further generalized to consider probability density evolution process and applied for time-dependent FP equation. In the proposed solution procedure, after spatial dimensionality reduction treatment of high-dimensional FP equation, the approximate solution is generalized as a time-dependent function with only interested state variables. The herein developed generalization and enhancement make the objective of determining time-dependent non-stationary response PDF available. Meanwhile, Monte Carlo simulation (MCS) is conducted to validate the proposed solution procedure. Three examples of ten-degree-of-freedom and eight-degree-of-freedom MDOF nonlinear systems under random external and/or parametric excitations are investigated. It is found that the obtained results agree well with the simulated ones at both the transient and the stationary stages. Moreover, the computation time taken by the proposed procedure is only a fraction of the one taken by the simulated method.