Objective Parametric rolling is a typical mode of dynamic instability, characterized by strong nonlinear dynamics and random wave excitation forces.International maritime organization (IMO) has issued the "Guidelines for assessment of the second generation intact stability of ships", which outlines the assessment methods and technical requirements. Predicting dynamic stability of ships under complex sea conditions is a key scientific challenge in the field of maritime safety. Stability criteria based on deterministic waves cannot accurately identify the instability domain in random wave environments.

Method In this paper, a single-degree-of-freedom equation for roll motion is established, incorporating both external excitation and parametric excitation, with nonlinear damping and restoring forces. Assuming that the ship's pitch and heave motions in waves are quasi-static processes, the roll restoring moment is calculated numerically using the strip theory. The righting arm is approximated with a polynomial expression for various wave directions, accurately capturing the characteristics of roll motion. The dynamic stability of nonlinear roll motion in random waves is investigated using a stochastic analysis method. The improved stochastic averaging method of energy envelope (ISAM-E) is introduced to account for the frequency component differences in rolling motion caused by both parametric and external excitation. ISAM-E is suitable for analyzing nonlinear roll motion under narrow-band spectra. Based on first-passage theory, the first-passage probability of stochastic roll motion is calculated under specified boundary and initial conditions.

Results Based on the first-passage probability approach, the C11 container ship is taken as an example to calculate the probabilities of the roll response exceeding 25° for full wave directions. The entire sea area is divided into three regions: high stability, medium stability, and low stability. This method effectively identifies the random sea conditions where the rolling response amplitude exceeds 25°.

Conclusion The occurrence mechanism of parametric rolling and dynamic stability assessment are thoroughly explored through the application of the stochastic averaging method and first-passage theory. The methods proposed in this paper significantly enhance computational efficiency without considering non-ergodicity. This approach provides valuable insights into the stochastic stability of roll motion under various wave directions.