ASYMPTOTIC-EXPANSION OF SHIP CAPSIZING IN RANDOM SEA WAVES .1. FIRST-ORDER APPROXIMATION

The first passage problem of ship non-linear roll oscillations in rand om sea waves is examined. The ship roll dynamics are described by a no n-linear stochastic differential equation which includes non-linear wa ve drag force and non-linear restoring moment. The non-linear restorin g moment is divided into a sine function plus a correction function. T he unperturbed motion of the ship is studied as a classical pendulum p roblem in terms of elliptic functions. The mean exit time of the pertu rbed ship motion is described by Pontryagin's partial differential equ ation. The method of asymptotic expansion is employed to solve this eq uation. Within the framework of first-order approximation, the analysi s reduces the Pontryagin equation into a second-order linear different ial equation with variable coefficients. These coefficients are functi ons 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 resolv able singularities. The dependence of the mean exit time on the initia l energy level, non-linear drag coefficient, and excitation spectral d ensity is graphically plotted. Second-order approximation is treated i n Part II of this two-part paper [Int. J. Non-Linear Mech. 30, 741-757 (1995)].