窄带随机噪声激励下多自由度非线性系统的响应

在随机振动的研究中，研究较多的是系统在宽带噪声作用下的响应问题，对于非线性系统特别是多自由度非线性系统在窄带随机噪声作用下的响应问题则研究较少。本文研究了三自由度非线性系统在窄带随机噪声激励下的主共振响应和稳定性问题。用多尺度法分离了系统的快变项，给出了系统响应的振幅和相位角满足的方程。用摄动法讨论了系统随机项对系统响应的影响。当随机扰动较小时，在一定的参数范围内，对应于不同的初值，系统具有两个均方响应值，随机饱和现象也存在。当随机扰动增大时，系统可从一个大的响应突跳为一个小的响应，或从一个小的响应突跳为一个大的响应，即存在随机跳跃现象。数值模拟表明本文提出的方法是有效的。

The Response Statistic of Multiple-Degree-of-Freedom Nonlinear System under Narrow-Band Random Excitation

In the theory of nonlinear random vibration, most results obtained so far are attributed to the response problems of nonlinear oscillators to wide-band random excitations. In relative, studies on the effect of narrow-band excitations on non-linear oscillators are quite limited. In this paper, the principal resonance of a three-degree-of-freedom nonlinear system to narrow-band random excitation is investigated. The method of multiple scales is used to determine the equations of modulation of amplitude and phase. The behavior, stability and bifurcation of steady state reponse are studied by means of qualitative analyses. The effects of damping, detuning, bandwidth, and magnitudes of random excitations are analyzed. The steady state solution can by obtained by the perturbation method and the moment method. The theoretical analyses are verified by numerical results. Theoretical analyses and numerical simulations show that when the intensity of the random excitation increase, the nontrivial steady state solution may change from a limit cycle to a diffused limit cycle. Under some conditions the systems may have two steady state solutions, the jumps and saturation may exist.