Asymptotic expansion of ship capsizing in random sea waves—I. First-order approximation

Abstract

The first passage problem of ship non-linear roll oscillations in random sea waves is examined. The ship roll dynamics are described by a non-linear stochastic differential equation which includes non-linear wave drag force and non-linear restoring moment. The non-linear restoring moment is divided into a sine function plus a correction function. The unperturbed motion of the ship is studied as a classical pendulum problem in terms of elliptic functions. The mean exit time of the perturbed ship motion is described by Pontryagin's partial differential equation. The method of asymptotic expansion is employed to solve this equation. Within the framework of first-order approximation, the analysis reduces the Pontryagin equation into a second-order linear differential equation with variable coefficients. These coefficients are functions of the energy level of the ship. The solution of this equation is obtained in a closed form and is found to be well behaved, with resolvable singularities. The dependence of the mean exit time on the initial energy level, non-linear drag coefficient, and excitation spectral density is graphically plotted. Second-order approximation is treated in Part II of this two-part paper.