Probabilistic solution of non-linear random ship roll motion by path integration

This paper investigates the probability density function (PDF) of non-linear random ship roll motion using a previously developed path integration method. The mathematical model of ship rolling motion consists of a linear-plus-cubic damping and a non-linear restoring moment in the form of odd-order polynomials up to fifth-order terms. In the path integration method, the interpolation scheme is based on the Gauss–Legendre quadrature integration rule and the short-time transition probability density function is formulated by short-time Gaussian approximation. The present work extends the path integration method to the case of non-linear random ship roll motion. Different values of non-linearity coefficient and excitation intensity are used to examine the effectiveness of the path integration method. Numerical analysis shows that the results of the path integration method agree well with the simulation results, even in the tail region. The path integration method is effective and it is simply implemented in the examined cases. Due to the presence of non-linear damping terms and non-linear restoring moment terms, the PDFs of roll angle and angular velocity exhibit highly non-Gaussian behaviors.