Application of Linear Filter and Moment Equation for Parametric Rolling in Irregular Longitudinal Waves

Parametric rolling is one of the dangerous dynamic phenomena. In order to discuss the safety of a vessel when a dangerous phenomenon occurs, it is important to estimate the probability of certain dynamical behavior of the ship with respect to a certain threshold level. In this paper, the moment values are obtained by solving the moment equations. Since the stochastic differential equation(SDE) is needed to obtain the moment equations, the Autoregressive Moving Average(ARMA) filter is used. The effective wave is modeled by using the 6th-order ARMA filter. In addition, the parametric excitation process is modeled by using a non-memory transformation obtained from the relationship between GM and wave elevation. The resulting system of equations is represented by the 8th-order It\^o stochastic differential equation, which consists of a second-order SDE for the ship motion and a 6th-order SDE for the effective wave. This system has nonlinear components. Therefore, the cumulant neglect closure method is used as higher-order moments need to be truncated. Furthermore, the probability density function of roll angle is determined by using moment values obtained from the SDE and the moment equation. Here, two types of the probability density function are suggested and have a good agreement.