多自由度粘弹性非线性随机系统的瞬态响应

TRANSIENT RESPONSE OF NONLINEAR MULTI-DEGREE-OF-FREEDOM STOCHASTIC SYSTEM WITH VISCOELASTICITY

研究了高斯白噪声激励下多自由度粘弹性非线性系统的瞬态响应.首先,通过将粘弹性项对系统的作用近似地简化为对原系统阻尼部分以及刚度部分的修正,得到近似的不具粘弹性项的等效非线性随机系统.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.该方程的解可通过多重级数式表示,基函数为幅值相关正交函数,系数为时间函数.应用Galerkin方法,关于时间的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,以耦合的二自由度Duffing-van der Pol振子系统为例,通过与原系统数值模拟结果的比较分析验证了所提出的半解析方法的有效性,并讨论了粘弹性对系统响应的影响.

The transient response of a nonlinear multi-degree-of-freedom system with viscoelasticity subjected to Gaussian white noise excitation is investigated. Firstly, the effect of the viscoelasticity on the system is approximated by the modified damping and stiffness. The original system is replaced by a system without viscoelasticity. Then, the stochastic averaging method based on generalized harmonic functions is adopted to derive the averaged Fokker-Planck-Kolmogorov equation of amplitude transient joint probability density for each oscillator. This equation is solved by expressing the 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 as well as the transient probability of the state response and the statistical moments for the amplitude response is obtained. To illustrate the proposed procedure, coupled two-degree-of- freedom Duffing-van der Pol oscillators with viscoelasticity subjected to Gaussian white noise excitation is investigated as an example. The effect of viscoelasticity on the system response is initially discussed. Moreover, comparison with the simulation results of the original system indicates that the proposed procedure is accuracy and efficacy.