Analytical Evaluation of First Passage Failure for Nonlinear Ship Rolling Excited by Random Wave

The first passage failure of nonlinear ship rolling excited by random wave is evaluated through two novel procedures. The stochastic averaging technique based on the generalized harmonic functions and the diffusion process theory are adopted in the first procedure. The amplitude response is approximated by a Markov diffusion process, while the conditional reliability function and moments of the first passage time are derived by numerically solving the backward Kolmogorov equation and Pontryagin equation, respectively. The second procedure resorts to the stochastic averaging technique and the Laplace integral method. The transient probability density of the amplitude response is expressed through the probability current, and the reliability function is established by integrating the transient probability density on the safety domain. The asymptotic solutions for the conditional reliability function and moments of the first passage time are derived through the Laplace integral method. The first procedure is hybrid method while the second is entirely analytical. The applicability and accuracy of the proposed two procedures are illustrated by comparing to the results from Monte Carlo simulations. The effective parameter ranges of the analytical expression are exhibited with special emphasis.