Analytical Modeling of Ship Motion Control Considering Variable Cargo Mass and Time-Dependent External Forces

Abstract

For the analysis of the vessel's stability, maneuverability, and control under different operational conditions, accurate mathematical modeling of the ship's motion is required. Usually, conventional ship-motion models are developed for a given loading condition and thus assume that the total vessel mass and associated inertial properties remain constant during the maneuver under consideration. This assumption may prove restrictive in practical applications, such as cargo loading, unloading, or transfer, especially when external forces and moments are also time-varying. In the present study, an analytical model for the motion and control of a vessel is developed that explicitly accounts for variable cargo mass and time-dependent external forces and moments. The formulation is based on Newton's second law for the vessel, including the mass, the added hydrodynamic mass, the mass moments of inertia, the added moments of inertia of the water, the gravitational and buoyancy forces, the hydrodynamic resistance, and the external disturbing forces and moments. The resulting governing equations allow for the time variation of the principal vessel-motion parameters and provide an analytical framework for evaluating the coupled translational and rotational response of the vessel. Successive integration of the governing equations under prescribed initial conditions yields analytical expressions for translational velocity, displacement, angular velocity, and angular displacement. Parametric analyses are then performed to explore the effects of cargo mass, volumetric displacement, water density, vessel moment of inertia, and added hydrodynamic inertia on the vessel response. The results show that increasing cargo mass increases the roll response for a given heeling moment, while increasing vessel displacement and water density decrease the roll angle due to the increased restoring effect associated with buoyancy. The intrinsic and added moments of inertia of the vessel also significantly affect its rotational response. Also, the temporal results indicate that varying external loading can lead to substantial coupled changes in roll, vertical displacement, and trim angle. The proposed formulation provides a relatively simple analytical framework for studying vessel dynamics under varying loading and environmental conditions. It can serve as a basis for developing more comprehensive variable-mass ship-motion and control models.