中文摘要：

基于等效非线性系统方法和突变理论,分析了随机参激下Duffing-Rayleigh碰撞振动系统的P-分岔.首先,借助非光滑变换和狄拉克函数将原碰撞振动系统转化为一个不含速度跳的新系统;接着,利用等效非线性系统方法得到了系统的稳态概率密度函数;然后,应用突变理论,得到了随机P-分岔发生的临界参数条件的解析表达式.最后,通过典型概率密度函数曲线和图像验证了结果的正确性.

英文摘要：

Vibroimpact dynamics has been widely studied by experts and scholars in the fields of physics, engineering and mathematics. Most of the researches focus on vibroimpact systems under deterministic excitations by using numerical methods. However, random excitation often exists in vibroimpact system, whose roles cannot be neglected, sometimes may be quite important. Stochastic bifurcation is one of the most critical parts of stochastic dynamics, but the relevant researches about vibroimpact system are rarely seen so far due to the fact that the analytical method has its inherent difficulty. This paper aims to investigate the P-bifurcations of a Duffing-Rayleigh vibroimpact system under stochastic parametric excitation based on an equivalent nonlinear system method and the catastrophe theory. Firstly, the original Duffing-Rayleigh vibroimpact system is transformed into a new system without velocity jump by using the nonsmooth transformation method and Dirac function. Then, the equivalent nonlinear system method is introduced to obtain the stationary probability density of the response. Finally, the explicit parameter conditions for stochastic P-bifurcations are derived based on the catastrophe theory. Besides, the effect of stochastic parametric excitation on the system response is also discussed.