Optimization-Oriented EPC Approach for Analyzing the Stochastic Nonlinear Oscillators with Displacement-Multiplicative and Additive Excitations

Abstract

This paper presents a new approach named optimization-oriented exponential–polynomial-closure (OEPC) to study the behavior of stochastic nonlinear oscillators under both displacement-multiplicative and additive excitations that are Gaussian white noise. In addition to the original projection exponential–polynomial-closure (PEPC) solution procedure, the OEPC method provides an alternative procedure for solving the Fokker–Planck–Kolmogorov (FPK) equation. The OEPC method is formulated by constructing an objective function with the residue of FPK equation. By minimizing the objective function with a gradient-based method, the parameters in the exponential polynomial can be determined and the approximate PDF solution of the nonlinear random oscillator can be obtained. The solutions obtained through the OEPC approach are highly consistent with the available true solutions in special case or the Monte Carlo simulations (MCS). They are much more accurate than those obtained using the Gaussian closure method when the nonlinearity is strong. In addition, the OEPC method is computationally more efficient than MCS. The difference of the results from the OEPC and PEPC is compared and the advantage of the OEPC over PEPC is also shown when the system nonlinearity is strong and complicated.