Probability Density Analysis of Nonlinear Random Ship Rolling

Ship rolling in random waves is a complicated nonlinear motion that contributes substantially to ship instability and capsizing. The finite element method (FEM) is employed in this paper to solve the Fokker Planck (FP) equations numerically for homoclinic and heteroclinic ship rolling under random waves described as periodic and Gaussian white noise excitations. The transient joint probability density functions (PDFs) and marginal PDFs of the rolling responses are also obtained. The effects of stimulation strength on ship rolling are further investigated from a probabilistic standpoint. The homoclinic ship rolling has two rolling states, the connection between the two peaks of the PDF is observed when the periodic excitation amplitude or the noise intensity is large, and the PDF is remarkably distributed in phase space. These phenomena increase the possibility of a random jump in ship motion states and the uncertainty of ship rolling, and the ship may lose stability due to unforeseeable facts or conditions. Meanwhile, only one rolling state is observed when the ship is in heteroclinic rolling. As the periodic excitation amplitude grows, the PDF concentration increases and drifts away from the beginning location, suggesting that the ship rolling substantially changes in a cycle and its stability is low. The PDF becomes increasingly uniform and covers a large region as the noise intensity increases, reducing the certainty of ship rolling and navigation safety. The current numerical solutions and analyses may be applied to evaluate the stability of a rolling ship in irregular waves and capsize mechanisms.