Dynamical reliability of internally resonant or non-resonant strongly nonlinear system under random excitations

The dynamical reliability of multi-degrees-of-freedom (MDOF) strongly nonlinear system under Gaussian white noise excitations is studied, including resonance and non-resonance. Firstly, the equations of motion of the original system with or without internal resonance are reduced to a set of Itô stochastic differential equations after stochastic averaging. Then, the backward Kolmogorov equation and the Pontryagin equation associated with the resonantly or non-resonantly averaged Itô stochastic differential equations, which determine the conditional reliability function and the mean first-passage time of the original random system, are constructed under appropriate boundary and (or) initial conditions, respectively. In particular, if the non-resonantly averaged system is completely decoupled, the conditional reliability function and the mean first-passage time of the original non-resonant system can be obtained by solving a set of simplified backward Kolmogorov equations. A system comprising two weakly coupled and strongly nonlinear mechanical oscillators is given as a concrete example to show the application of the proposed method. The 1:1 internal resonance or non-resonance is discussed. The corresponding high-dimensional backward Kolmogorov equation and Pontryagin equation are established and solved numerically. All theoretical results are validated by a Monte Carlo digital simulation.