Numerical study of a container ship model for the uncoupled parametric rolling

Parametric roll on ships is a resonance phenomenon whose onset causes heavy roll oscillations leading to dangerous situations for the ship, the cargo and the crew. It affects both small vessels with marginal stability and large container ships if four conditions are met simultaneously: the wave length and ship length are approximately equal, the frequency of encounter wave is twice the roll natural frequency, the ship's roll damping is low enough and the wave height exceeds a limit value characteristic to each vessel. For a container ship, the first two conditions result in a time-varying geometry of the submerged hull and thus to a periodic variation of the transverse stability. In the last two decades, several mathematical models for parametric rolling have been proposed by scientists. In the paper, we cancelled the couplings terms between heave, pitch and roll in a 3 DOF model proposed by Neves and Rodriguez to obtain an uncoupled 1 DOF model for the parametric roll. Even if it seems too simplistic, such a model based on Mathieu type equation is a good start to capture the basic features of the analysed phenomenon. There are no less than seven parameters that can influence the ship stability and the appearance of large roll amplitudes. By modifying two of these parameters one by one and keeping the others constant, a series of colour plots have been drown that capture the role played by the ship's parameters in obtaining dangerous roll amplitudes or too short times to take countermeasures. Analysing the instability regions suggested by these colour plots, one can imagine control measures to reduce the roll amplitudes, such as changing the forward velocity or the heading angle. The data used in the paper have been obtained from experiments with a container ship model in a towing tank followed by expansion to a full scale ship.