A novel stochastic approach to investigate the probabilistic characteristics of the ship roll system with sinusoidal restoring force

A novel constrained optimization-oriented exponential–polynomial–closure (CO-EPC) method is developed to investigate the ship roll-motion with non-smooth damping and sinusoidal restoring moments under stochastic wave excitations. The ’non-smooth damping’ and ’sinusoidal restoring moments’ refer to the damping force and restoring force of the system, respectively, which are mathematically expressed using non-smooth and sinusoidal functions. The propagation of uncertainty in the system is governed by the Fokker–Plank–Kolmogorov (FPK) equation, as the presence of stochastic wave excitation induces Markovian behavior in the system responses. The solution of FPK must meet the non-negativity, normalization, and limitation attributes. To fulfill these essential requirements, a novel method has been developed. By deriving the moment functions of the non-smooth terms (for the damping moment part) and the sine terms (for the restoring moment part), the complicated FPK equation is solved. Through two case studies, it is found that the CO-EPC approach demonstrates superior efficiency compared to Monte Carlo simulation and produces more accurate solutions in contrast to the Gaussian closure method. These findings support the effectiveness of CO-EPC in studying the ship roll motion problem under stochastic wave excitations.