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–Planck–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.

利用广义谐波函数，提出了单自由度（SDOF）强非线性系统在谐波和泊松组合白噪声激励下的随机平均方法。推导了振幅和相位差的平稳联合概率密度的平均随机微分方程（SDE）和广义Fokker-Planck-Kolmogorov（GFPK）方程。然后利用有限差分法和逐次相关法（SOR）求解简化的GFPK方程，得到响应的近似平稳概率密度。该方法适用于受到谐波和泊松白噪声组合激励的Duffing振荡器。在初级共振的情况下，利用振幅的平稳概率密度，研究了随着系统参数的变化，在谐波和泊松白噪声组合激励下，Duffing振荡器的随机分叉。最后，通过蒙特卡洛数值模拟验证了所提方法的分析结果。