中文摘要：

研究了随机参激作用下一个非线性碰撞振动系统的随机响应.基于Krylov-Bogoliubov平均法,借助第一类改进的Bessel函数,得到了决定平凡解的几乎确定稳定性的最大Lyapunov指数.模拟结果发现,碰撞振动系统的最大Lyapunov指数特性不同于一般的非碰撞系统,其最小值为负.同时,在确定性情形下,得到了骨架曲线方程和不稳定区域的临界方程.进一步,利用矩方法,讨论了系统的一阶和二阶非平凡稳态矩,发现了碰撞振动系统中有频率岛现象的存在.最后,借助FokkerPlanck-Kolmogorov方程,利用有限差分法,讨论了碰撞振动系统中存在的随机跳现象.在随机强度较小时,稳态概率密度集中于响应振幅的非平凡分支;但是随着随机强度的增加,平凡稳态解的概率会变大.

英文摘要：

The stochastic responses of nonlinear vibro-impact systems under random parametric excitation were investigated. Based on the Krylov-Bogoliubov averaging method,the largest Lyapunov exponent deciding the almost sure stability of the trivial solution was derived. R esults showthat the characteristics of the largest Lyapunov exponent of the vibro-impact system was different from that of the system without impact. Meanwhile,the backbone curve and the critical equation for the unstable region were also derived in the deterministic case. Then,the 1st- and 2nd-order non-trivial steady-state moments of the system were discussed,and the frequency island phenomenon was also found. Finally,the phenomenon of stochastic jump was analyzed via the finite difference method. The basic jump phenomena indicate that,under the conditions of system parameters within a smaller bandwidth,the most probable motion is around the non-trivial branch of the amplitude response curve,whereas within a larger bandwidth,the most probable motion is around the trivial one of the amplitude response curve.