Hilbert Transform-Based Stochastic Averaging Technique for Determining the Survival Probability of Nonlinear Oscillators

A Hilbert transform based definition of the response amplitude of randomly excited nonlinear oscillators has been proposed recently to address certain limitations associated with the standard stochastic averaging solution treatment. In comparison to standard stochastic averaging, the requirement of a priori determination of an equivalent natural frequency is bypassed, yielding flexibility in the ensuing analysis and potentially higher accuracy. In this paper, relying on the Hilbert transform based stochastic averaging, a semianalytical technique is developed for determining the time-dependent survival probability and first-passage time probability density function of stochastically excited nonlinear oscillators, even endowed with fractional derivative terms. To this aim, a Galerkin scheme based on the orthogonality of the confluent hypergeometric functions is utilized to solve approximately the backward Kolmogorov partial differential equation governing the survival probability of the oscillator response. Further, two distinct approximations for the equivalent instantaneous natural frequency are introduced, while their accuracy with respect to oscillator first-passage time statistics is also assessed. The hardening Duffing and the bilinear stiffness nonlinear oscillators, both with and without fractional derivative elements, are considered in the numerical examples for ascertaining the reliability of the technique. For comparison, the analytical results are examined in relation to pertinent Monte Carlo simulation data.