Stochastic response analysis of the softening Duffing oscillator and ship capsizing probability determination via a numerical path integral approach

A numerical path integral approach is developed for determining the response and first-passage probability density functions (PDFs) of the softening Duffing oscillator under random excitation. Specifically, introducing a special form for the conditional response PDF and relying on a discrete version of the Chapman–Kolmogorov (C–K) equation, a rigorous study of the response amplitude process behavior is achieved. This is an approach which is novel compared to previous heuristic ones which assume response stationarity, and thus, neglect important aspects of the analysis such as the possible unbounded response behavior when the restoring force acquires negative values. Note that the softening Duffing oscillator with nonlinear damping has been widely used to model the nonlinear ship roll motion in beam seas. In this regard, the developed approach is applied for determining the capsizing probability of a ship model subject to non-white wave excitations. Comparisons with pertinent Monte Carlo simulation data demonstrate the reliability of the approach.