Non-stationary stochastic vessel roll dynamics with hereditary fractional-order hydrodynamic damping and added-inertia nonlinearity

In this paper, an efficient semi-analytical methodology for the non-stationary stochastic analysis of nonlinear ship rolling in irregular seas is developed. To account for the underlying maritime complexity, the roll dynamics are described by a physically consistent fractional-order system that incorporates added inertia nonlinearity, cubic-quintic restoring characteristics, and nonlinear quadratic viscous damping. A defining feature of the model is the representation of hereditary hydrodynamic damping via a fractional derivative operator, which naturally captures the memory effects inherent in fluid-hull interactions. The non-stationary sea-wave excitation is modeled through a shaping-filter approach, specifically by means of a calibrated fourth-order filter combined with deterministic amplitude modulation, yielding a compact finite-dimensional Markovian representation. The proposed method leverages a refined blend of statistical linearization (SL), harmonic balancing (HB), and a time-varying covariance Lyapunov equation. Its effectiveness rests on a parsimonious yet physically consistent Markovian augmentation of the ship-roll dynamics. Local-in-time nonlinearities are treated via SL, while the fractional-memory contribution is embedded into an evolving equivalent-linear closure through time-dependent HB components. A key methodological refinement involves updating the representative carrier frequency used in the HB step solely from the statistically linearized backbone to avoid implicit coupling. The framework efficiently delivers estimations of time-dependent response variances, evolutionary power spectra, and roll-amplitude probability-density function surfaces. Validation against benchmark Monte Carlo simulations demonstrates a high level of agreement across various sea states and fractional orders. Ultimately, the proposed methodology offers a reliable, physically interpretable, and computationally efficient tool for stochastic ship-roll response and stability assessment in uncertain and dynamically evolving open sea environments.