Non-stationary nonzero mean probabilistic solutions of nonlinear stochastic oscillators subjected to both additive and multiplicative excitations

An extended exponential-polynomial-closure (EPC) method is presented in this paper for solving the non-stationary nonzero mean probabilistic solutions of nonlinear stochastic oscillators under both additive and multiplicative Gaussian white noise excitations. The main idea is to propose an exponential function of polynomial with time-variant coefficients which represents the non-stationary probability density function (PDF) of the responses of the nonlinear stochastic oscillator and to introduce a proper weighted function to solve the well-known Fokker–Planck–Kolmogorov (FPK) equation which governs the non-stationary probabilistic solutions of nonlinear stochastic oscillator. The numerical analysis illustrates that the extended EPC method is valid and efficient for achieving the non-stationary nonzero mean PDFs of the responses for the system with strong nonlinearity by comparing the results obtained by extended EPC method, equivalent linearization method and Monte Carlo simulation method. The PDFs of the responses are asymmetric about the nonzero mean values due to the existence of even nonlinearities.