中文摘要：

研究调制白噪声激励下，包含弱非线性阻尼及强非线性刚度的单自由度系统的近似瞬态响应概率密度．应用基于广义谐和函数的随机平均法，导出关于幅值瞬态概率密度的平均Fokker-Planck—Kolmogorov方程．该方程的解可近似表示为适当的正交基函数的级数和，其中系数是随时间变化的．应用Galerkin方法，这些系数可由一阶线性微分方程组解得，从而可得幅值响应的瞬态概率密度的半解析表达式及系统状态响应的瞬态概率密度和幅值的统计矩．以受调制白噪声激励的van der Pol-Duffing振子为例验证其求解过程，并讨论了线性阻尼系数及非线性刚度系数等系统参数对系统响应的影响．

英文摘要：

The nonstationary probability densities of system response of a single-degree-of-freedom system with lightly nonlinear damping and strongly nonlinear stiffness subject to dulated white noise excitation were studied. Using the stochastic averaging method based on the generalized harmonic functions, the averaged Fokker-Planck-Kolmogorov equation governing the nonstationary probability density of the amplitude was derived. The solution of the equation was approximated by a series expansion in terms of a set of properly selected basis functions with time-dependent coefficients. According to the Galerkin method, the time-dependent coefficients can be solved from a set of first-order linear differential equations. Then the semi-analytical formulae of the nonstationary probability density of the amplitude response as well as the nonstationary probability density of the state response and the statistic moments of the amplitude response can be obtained. A van der Pol-Duffing oscillator subject to modulated white noise was given as an example to illustrate the proposed procedures. The effects of the system parameters, such as linear damping coefficient and nonlinear stiffness coefficient, on the system response were discussed.