有界随机噪声激励下碰撞系统的最大Lyapunov指数

Maximal Lyapunov exponent of a single-degree-of-freedom linear vibroimpact system to a boundary random parametric excitation

为了研究单自由度线性单边碰撞系统在有界随机噪声参数激励下的最大Lyapunov指数和稳定性问题,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。在没有随机扰动的情形下,给出了系统最大Lyapunov指数的值;在有随机扰动的情形下,通过求解FPK方程得到了系统的不变测度和最大Lyapunov指数的解析表达式。研究结果表明:随着系统阻尼项、有界随机噪声带宽、碰撞恢复系数的减少和有界随机噪声振幅的增大,最大Lyapunov指数增加;当随机激励的中心频率等于系统固有频率的两倍时,系统的Lyapunov指数达到最大,从而使系统变得更不稳定。根据系统的Lyapunov指数得到了系统稳定的充分必要条件,即当Lyapunov指数大于零时系统几乎必然不稳定,而当Lyapunov指数小于零时系统几乎必然稳定,Lyapunov指数等于零为系统的稳定性分叉点,并讨论了相应的稳定性分叉问题。

The resonance response and maximal Lyapunov exponent 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 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.The averaged equations are solved exactly and value of the maximal Lyapunov exponent is obtained in the case without random disorder.The FPK equations are solved exactly and the explicit asymptotic formulas for the maximal Lyapunov exponent and invariant measures are obtained for the case with random disorder.Theoretical analyses show that the maximal Lyapunov exponent will increase when the damping of the system,bandwidth of random excitation and restitution factor decrease.The maximal Lyapunov exponent will increase when the magnitudes of random excitation increase.The maximal Lyapunov exponent will reach the maximum values when the excitation frequency equals two times of the system frequency,therefore make the system become more unstable.The system will be almost sure stable(or unstable) if the maximal Lyapunov exponent is negative(or positive),therefore the stable bifurcation will be occur if the maximal Lyapunov exponent equal to zero and the stochastic stable bifurcation point can be obtained.