Stability against capsizing in heavy seas is one of the fundamental requirements in ship design.
The second group employs the integration of hydrodynamic pressure acting on the ship's wetted surface to derive the external forces and moments (see, e.g., ). The first approach utilizes a mathematical development based on a Taylor expansion of the force function (see, e.g., ). In order to derive the hydrostatic and hydrodynamic forces and moments acting on the ship, two approaches have been used in the literature. The general equations of motion have been developed either by using Lagrange's equation (see, e.g., ) or by using Newton's second law (see, e.g., ). Generally, ships can experience three types of displacement motions (heave, sway or drift, and surge) and three angular motions (yaw, pitch, and roll) as shown in Figure 1. The paper includes an assessment of roll stochastic stability and probabilistic approaches used to estimate the probability of capsizing and parameter identification. The special cases of coupled roll-pitch and purely roll equations of motion are obtained from the general formulation. In the formulation of the restoring forces and moments, the influence of large-amplitude ship motions will be considered together with ocean wave loads. These are the inertia forces and moments, restoring forces and moments, and damping forces and moments with an emphasis to the roll damping moment. The ingredients of the formulation are comprised of three main components. The purpose of this paper is to present the coupled nonlinear equations of motion in heave, roll, and pitch based on physical grounds. In order to study the dynamic behavior of ships navigating in severe environmental conditions it is imperative to develop their governing equations of motion taking into account the inherent nonlinearity of large-amplitude ship motion.
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