First-passage failure of linear oscillator with non-classical inelastic impact

The first-passage failure of linear oscillator with inelastic impact subjected to the additive and multiplicative random noises is investigated. The impact is described by the non-classical inelastic impact model, which is essentially different from the traditional impact model and can provide the whole information of the impact process. First of all, the impact force in the motion equation is replaced by the quasi-linear damping and nonlinear stiffness terms. Then, the stochastic averaging is adopted and the averaged Itô stochastic deferential equation of the total system energy is derived. Last, by solving the established backward Kolmogorov equation and Pontryagin equation from the averaged Itô equation numerically, the conditional reliability, the conditional probability density function (PDF) and the mean time of first-passage failure can be obtained. The comparison between the analytical results and those from Monte-Carlo simulation reveals the proposed procedure is effective. The influences of some system parameters are discussed in detail.