调制白噪声激励下时滞非线性系统的瞬态响应

本文研究了调制白噪声激励下多自由度时滞非线性系统的近似瞬态响应概率密度．首先，由系统当前状态与时滞状态的关系，将原时滞系统近似等效为无时滞系统．然后，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．该方程的解可通过级数式表示，基函数为幅值相关正交函数，系数为时间函数．应用Galerkin方法，系数可由一阶线性微分方程组解得，从而得出幅值响应的瞬态概率密度、状态空间概率密度及幅值统计矩的半解析表达式．最后，以调制白噪声激励下阻尼耦合的二自由度Duffing-vanderPol振子系统为例，验证其求解过程，并讨论不同时滞的影响．

Transient probability densities of nonlinear system with time delay subject to modulated white noise excitation

In the present paper, the transient probability densities of nonlinear multi-degree-of-freedom system with time delay subject to modulated white noise excitation are investigated. Firstly, the time-delay stochastic system is approximately replaced by the corresponding non-time-delay system through appropriate relations between the current states and the delay states. Then, the stochastic averaging method based on generalized harmonic functions is adopted to reduce the dimension of the equivalent system, and the corresponding Fokker-Planck-Kolmogorov equation with regard to transient probability density is directly obtained. This equation is solved by expressing the transient probability density as multiple series in terms of a set of properly state-dependent orthogonal basis functions with time-dependent coefficients. According to Galerkin method, the time-dependent coefficients can be solved from a set of first-order linear differential equations. Finally, the semi-analytical formulae of the transient probability density of the amplitude response as well as the transient probability of the state response and the statistic moments of the amplitude response can be obtained. To illustrate the proposed procedure, coupled two-degree-of- freedom Duffing-van der Pol oscillators with time delay subject to modulated white noise excitation was given as an example. The effects of time delay on the responses are discussed as well.