Reliability analysis of a class of stochastically excited nonlinear Markovian jump systems

It is inevitable for the nonlinear systems to suffer from external stochastic disturbance. Meanwhile, the components failure will bring abrupt changes in its substructures, which can be considered as the internal stochastic disturbance. It is demonstrated that the components failure performs random jumpy factors switching between a finite number of modes. This salient feature allows us to identify this type of dynamic behaviors as response of nonlinear hybrid systems undergoing Markovian jumps. In this paper a novel method is presented to analyze the dynamical reliability of multi-degree-of-freedom (DOF) nonlinear stochastic systems undergoing Markovian jumps. Firstly, the Markovian jump process is introduced to formulate the aforementioned systems as continuous-discrete hybrid systems. Secondly, a two-step approximate method is applied to convert the hybrid systems to one governed by a set of averaged Itô stochastic differential equations (SDEs) of mechanical energies. Then the associated averaged backward Kolmogorov equation and the Pontryagin equation are constructed and solved to yield the conditional reliability function and the mean first-passage time of the original system respectively. Finally, an example of two coupled Duffing oscillators is illustrated to compare the analytical results and those from Monte Carlo simulation to verify the effectiveness of the proposed method.