Stochastic averaging for SDOF strongly nonlinear system under combined harmonic and Poisson white noise excitations

Abstract A stochastic averaging method for single degree of freedom(SDOF)strongly nonlinear system under combined harmonic and Poisson white noise excitations is proposed by using generalized harmonic functions. The averaged stochastic differential equations (SDEs) and the Generalized Fokker-Plank-Kolmogorov (GFPK) equation for the stationary joint probability density of amplitude and phase difference are derived. Then the reduced GFPK equation is solved by using finite difference method and the successive over relation (SOR) method to obtain the approximate stationary probability density of the response. This method is applied to Duffing oscillator subject to combined harmonic and Poisson white noise excitations. In the case of primary resonance, the stochastic bifurcation of the Duffing oscillator under combined harmonic and Poisson white noise excitations as the system parameters change are examined using the stationary probability density of amplitude. At last, the analytical results obtained from the proposed method are verified by those from the Monte Carlo numerical simulation.