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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-van der Pol振子系统为例,验证其求解过程,并讨论不同时滞的影响.
In this paper, we investigate the approximate transient response probability density of a nonlinear system with multi-degree-of-freedom excitation under white noise excitation.First, according to the relationship between the current state and the state of delay, the original time- Then, the average Fokker-Planck-Kolmogorov equation about the amplitude transient probability density is derived by using the stochastic averaging method based on the generalized harmonic function.The solution of the equation can be expressed by series formula and the basis function is the amplitude-dependent orthogonal function , The coefficient is a function of time.Using the Galerkin method, the coefficients can be solved by the first-order linear differential equations to obtain the semi-analytical expressions of the transient probability density, the state space probability density and the amplitude statistical moment of the amplitude response.Finally, Taking the Duffing-van der Pol system with two degrees of freedom damped by white noise as an example, the process of the solution is verified and the effects of different time delays are discussed.