外共振耦合Duffing-van der Pol系统的首次穿越

摘要

First-passage failure of two-DOF Duffing-van der Pol system with strong nonlinearity under com- bined harmonic and wide-band stochastic excitations was studied. In the case of external resonance, the equations of motion of the system are reduced to a set of averaged Itostochastic differential equations after stochastic averaging. Then, the backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the mean first-passage time are established. Under specified boundary and initial conditions, the method of finite difference is used to solve the high-dimensional backward Kolmogorov equation and Pontryagin equation to yield the conditional reliability function, the mean first-passage time and the conditional probability density function of the mean first-passage time. Different parameters are chosen to show the influence on the reliability and mean first-passage time of the original system. All theoretical results are verified by Monte Carlo simulation.%研究了谐和力与宽带噪声激励下二自由度强非线性Duffing-van der Pol系统的首次穿越问题．在外共振情形，应用随机平均法将系统动力学方程化为关于振幅与角变量的Ito随机微分方程．然后建立了系统的可靠性函数满足的后向Kolmogorov方程以及平均首次穿越时间满足的Pontryagin方程．在一定的边界条件和初始条件下，用有限差分法求解了这两个高维偏微分方程，得到系统的条件可靠性函数、平均首次穿越时间以及平均首次穿越时间的条件概率密度．讨论了不同参数对系统可靠性以及平均首次穿越时间的影响．用MonteCarlo数值模拟验证了理论方法的有效性．