有界随机噪声参数激励下碰撞系统的矩稳定性

MOMENT STABILITY OF A SINGLE-DEGREE-OF-FREEDOM LINEAR VIBROIMPACT SYSTEM TO A BOUNDARY BANDOM PARAMETRIC 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. 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. 利用伊藤法则给出了系统一、二阶矩满足的常微分方程,根据微分方程的稳定性理论得到了系统一阶矩稳定充分必要条件的解析表达式和二阶矩稳定充分必要条件的数值算法,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,无论是相对于一阶矩还是二阶矩的稳定性,随着随机激励振幅变大,系统的稳定性区域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形时,系统的稳定性区域变得最小. 当随机噪声强度逐渐变小趋于零时,由二种矩稳定性给出的稳定性区域变得一致. 在一定的参数区域内,随机噪声使得系统稳定化.

This paper investigated the resonance response and moment stability of a single-degree-of-freedom linear vibroimpact oscillator with a one-sided barrier to boundary random parametric excitation. The analysis is based on a special Zhuravlev transformation, which reduces the system to one without impacts, or velocity jumps, thereby permitting the applications of asymptotic averaging over the period for slowly varying random process. By using the It＇s differential rule, the differential equations ruling the time evolution of the first and second order re- sponse moments were obtained. The necessary and sufficient conditions of stability for the first and second order moments are that the matrix of the coefficients of the differential equations ruling the moments have complex ei- genvalues with negative real parts. The analytical expression of the stability condition of the first order moment was obtained, while the results of the second order moment stability were given numerically. Some numerical simulations and graphs were presented for representative cases. It is founded that, when the amplitude of the parametric excitation increases, the stability regions will reduce either for the first order moment or the second order moment stability. The stability regions will reduce to the minimum value if the detuning parameter tends to zero. The stability regions based on different order moments will become identical when the intensity of the random disturbance increases to zero. In some cases the stochastic excitation stabilizes the system.