Efficient Monte Carlo simulation and Grim effective wave model for predicting the extreme response of a vessel rolling in random head seas

In this paper, a computationally efficient Monte Carlo simulation (MCS) approach is introduced in order to estimate the extreme response statistics of parametric rolling in random longitudinal seas. Basically, parametric roll is caused by sufficiently large oscillations of the roll restoring moment occurring within certain frequencies of wave encounter (approximately twice the natural roll frequency). The concept of the Grim effective wave is applied herein in order to approximate the variation of the restoring moment in random waves. A fourth order linear filter is introduced to model the random effective wave amplitude process, which is assumed to be the driving process for the variation of the restoring term as well as for the stochastic nonlinear system of the roll motion by application of the Grim effective wave approximation. For the stochastic dynamical system, the roll response is a random process and an extrapolation procedure is developed for estimating the extreme values of the response statistics by assuming regular behavior in the tail region of the mean upcrossing rate. The rationality of the linear filter model and the feasibility of an efficient MCS method based on extrapolation techniques for predicting the extreme roll response are illustrated. Furthermore, the phenomenon of parametric roll in random seas as well as the effect of vessel speed on the stochastic roll response are investigated.