时滞碰摩故障转子-机闸系统的随机稳定性与随机分岔

Stochastic stability and stochastic bifurcation of a rotor-box system with time delay and rubbing impact

本文研究了时滞碰摩故障转子-机闸系统的随机稳定性与随机分岔.考虑到转子系统从接收到处理故障信号需要一定时间以及转子所处环境的随机波动,本文建立了一个含有时滞的无穷维随机动力学模型,其中的位置分量和速度分量均存在时滞.本文首先利用中心流形法将系统转化为有限维随机微分方程,用随机平均法推导了方程响应振幅满足的伊藤方程,然后利用Lyapunov指数和奇异边界理论分析了方程的稳定与分岔条件,并求得其联合概率密度函数.最后,通过数值模拟联合概率密度函数,本文分析了转子系统的随机稳定性和随机分岔对时滞及碰摩系数的依赖性.结果表明,时滞和碰摩系数均可诱发系统的随机P-分岔,但二者的影响是相反的:时滞越大分岔现象越明显、稳定性越低,而碰摩系数越大则系统的稳定性越高.本研究可望为旋转机械的设计、运行及剩余寿命评估提供理论依据.

In this paper,we study the stochastic stability and stochastic bifurcation of a rotor-box system with time-delay and rubbing impact.Considering that there exists time-delay from receiving signal to processing signal as well as the stochastic fluctuation of external environment,we establish a stochastic and infinitedimensional dynamical system with time delay.First,the dynamical system is transformed into a finitedimensional stochastic differential equation by using the central manifold method,the corresponding Itôequation for the response amplitude of equation is deduced by using the stochastic averaging method.Second,The Itôequation is analyzed by using the Lyapunov exponential method and the singular boundary theory,conditions for the stochastic stability and stochastic bifurcation of the equation are obtained,and the joint probability density function is deduced.Finally,numerical simulation for the joint probability density function are implemented to explore the stochastic bifurcation of system.It is shown that either the time delay or the rubbing impact coefficient can result in the stochastic P-bifurcation of system with exactly opposite effect.On the one hand,with the increasing of time delay,the P-bifurcation phenomenon of system gets more and more obvious,which means that the stability of system is weakened.On the other hand,with the increasing of rubbing impact coefficient,the system becomes more and more stable.The obtained result is expected to help the design,operation and remaining life assessment of rotating machinery.