Fractional modeling of nonlinear ship rolling dynamics under evolutionary and irregular sea-wave loads

This paper presents an efficient semi-analytical methodology for determining the non-stationary response of ships undergoing nonlinear rolling under stochastic sea-wave excitations. The dynamic response is captured through a comprehensive and physically consistent nonlinear formulation that incorporates both softening and hardening restoring moment characteristics, non-conventional fractional-order hydrodynamic damping mechanisms that naturally embed memory effects, and non-stationary stochastic wave loads representative of complex maritime environments. By leveraging a refined blend of stochastic averaging and statistical linearization the proposed stochastic fractional-order framework delivers computationally efficient, time-dependent second-order response statistics together with fully non-stationary roll angle amplitude probability-density surfaces. Tailored weighting factors and fractional derivative orders allow flexible tuning of restoring and damping characteristics to represent a wide range of relevant ship roll dynamics. Numerical analyses across a range of case studies, validated against benchmark Monte Carlo simulations, demonstrate the robustness and efficiency of the proposed methodology, underscoring its potential for assessing ship rolling response and seakeeping performance under dynamic and uncertain maritime conditions.