Effects of mass and inertia changes on the equations of motion of a vessel

The paper discusses the deduction of the equations of motion for a vessel with non constant mass and inertia characteristics. In author's knowledge, the topic is not covered in recent marine engineering literature, probably due to the very few possible applications. In fact the problem of a vessel with a variable mass and/or inertia is not common in marine engineering. Classic applications refer to submarines during immersion and emersion manoeuvres. A field of application of wider interest is the simulation of flooding and its influence on the damaged ship stability. Another possible application is the ship motion control by passive water tanks. The authors faced this kind of problem dealing with the simulation of an high speed craft; to assure the static stability at zero speed a large quantity of ballast water is loaded during deceleration and to assure good speed performance it is unloaded during acceleration, producing a rapid change in mass and inertia of the vessel. The paper focusses on the analytical derivation of the “inertial part” of the equations of motion, taking explicitly into account the variability of mass and inertia. The aim is to provide the theoretical basis for works concerning the numerical simulation of propulsion and motion for vessels. Among others, two important findings are noticed: • the derived equations, in matrix form, maintain the same structure of the constant mass equations but contain an additional term, • the Kirchhoff's equations hold also for the variable mass case. The derived equations of motion are suitable to be implemented into a simulation code able to represent both 'conventional' constant mass vessels, as well as variable mass/inertia vessels.