First-exit problem of MDOF strongly nonlinear oscillators under wide-band random excitations without internal resonances

Abstract

The first-exit problem of strongly nonlinear oscillators with multi-degrees-of-freedom (MDOF) is studied theoretically in this paper. The excitations are modelled as wide-band random noises. When there is no internal resonance, the equations of motion of the original system are reduced to a set of averaged Ito stochastic differential equations (SDEs) after stochastic averaging. Two partial differential equations (PDEs), namely the backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the mean first-exit time, are established under appropriate boundary and (or) initial conditions. The theoretical method is applied to study a 3-DOF strongly nonlinear system. Long expressions of drift and diffusion coefficients of the averaged Ito SDEs are obtained by using software Mathematica. The qualitative boundary conditions of the associated backward Kolmogorov equation and the Pontryagin equation are quantified by examining the behaviour of drift coefficients at the lower boundary. The method of finite difference is used to solve the corresponding high-dimensional PDEs. Monte Carlo simulation is performed to verify the validity of the proposed theoretical method.