First-passage failure of single-degree-of-freedom nonlinear oscillators with fractional derivative

Abstract

The first-passage failure of single-degree-of-freedom nonlinear oscillators with fractional derivative under Gaussian white noise excitations is studied. First, the term associated with fractional derivative is approximately equivalent to amplitude-dependent quasi-linear damping and stiffness forces by using the generalized harmonic balance technique and the given system is replaced by an equivalent nonlinear stochastic system without fractional derivative. Then, the equivalent nonlinear stochastic system state is approximately represented by a one-dimensional diffusive process through stochastic averaging. The backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the conditional mean of first-passage time are established from the averaged Itô equation of the total energy and solved numerically, respectively. Finally, two examples are worked out in detail and the analytical results are validated by those from the Monte Carlo simulation of original systems.

Keywords

Equivalent nonlinear stochastic system first passage failure fractional derivative single-degree-of-freedom stochastic averaging method

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