Area of the unstable solution of rolling equation : jumps of the oscillations amplitude

The most interesting motion of the ship is rolling. This is because the rolling amplitudes are much bigger than amplitudes of other degrees of freedom and under resonance conditions, which can exceed 40º. In such a case, when the maximum of the righting arm curve is placed at relatively small angles, the roll equation reveals a strongly nonlinear character and bistability areas as well as an area of unstable solutions of the roll equation occurs. Together with the appearance of the above-mentioned areas, amplitude jumps are possible. In the study, the case of strongly nonlinear rolling is analysed. For the purpose of numerical simulations, the 1DOF mathematical model of rolling with damping dependent on amplitude and frequency is used. The article presents the roll spectrum including the bistability areas and the area of unstable solutions for one loading condition of the offshore support vessel. It is demonstrated that for strongly nonlinear rolling, rolling with two different amplitudes for the same value of excitation is possible. It is also shown that transitions (jumps) between these amplitudes are possible too. A few scenarios of jumps of the rolling amplitude within the region of unstable solutions of the rolling equation are presented. The presented rolling scenarios show that under some circumstances rolling can be observed as chaotic