有界随机噪声激励下碰撞系统的稳定性

Stability of a single-degree-of-freedom linear vibroimpact system under boundary random noise excitation

研究了单自由度线性单边碰撞系统在有界随机噪声参数激励下系统的矩稳定性问题。采用Zhuravlev变换将碰撞系统转化为连续的非碰撞系统,然后用随机平均法得到了关于慢变量的随机微分方程。利用随机线性变化得到了控制第p阶Lyapunov指数的矩阵的特征值。在没有随机扰动的情况下,给出了系统最大Lyapunov指数的解析表达式;在有随机扰动的情况下,给出了系统任意阶矩Lyapunov指数的数值算法,并对理论结果用数值方法进行了仿真计算。理论分析和数值仿真表明:随着随机激励振幅变大,任何正数p阶矩稳定性区域都变小从而使得系统变得不稳定;而当调谐参数趋于零、系统达到参数主共振情况时,系统的稳定性区域变得最小;在一定的参数区域内,随机扰动使得系统稳定化。

The moment stabilities of single-degree-of-freedom linear vibroimpact oscillator with a one-sided barrier to boundary random parametric excitation are investigated. The analysis is based on a Zhuravlev transformation, which converts the system into one of non-vibroimpact, thereby permitting the applications of asymptotic averaging over the period for slowly varying random process of ordinary differentiation equation. The linear stochastic transformation is taken to obtain the eigen-value problem governing the p-th moment Lyapunov exponent. The analytical expression of the largest Lyapunov exponent is obtained in the case when there is no random disturbance, while the p-th moment Lyapunov exponent is obtained numerically in the case when the random disturbance exists. Some numerical simulations and graphs are presented for typical cases. It is found that when the amplitude of the parametric excitation increases, the stability regions will reduce whenever for any positive p-order moment stability. The stability regions will reduce to the minimum value if the detuning parameter tends to zero. In some range of parameter the random disturbance will stabilize the system.