Numerical investigation of a two - degrees - of - freedom ship model for pitch - roll motion

This paper numerically investigates a two degrees of freedom harmonically excited pendulum system, known in the literature to be a good model for the coupling between the roll and the pitch ship motions. We concentrate mainly on the dangerous situation for the ship where the pitch frequency is almost twice the roll frequency and the excitation period is near the pitch period. In this case, although only the pitch excitation is taken into consideration, part of the energy in the pitch mode is transferred to the roll mode through the non - linear coupling leading to excessive resonant rolling amplitudes and, as a consequence, to ship capsizing. The stability of the system is studied using both frequency response curves and bifurcation diagrams. It was proven that for small external forcing often co-exist two periodic attractors and that the jump phenomena accomplish the transition from one attractor to the other. Periodic oscillations take place with the external frequency for pitch mode and half of the external frequency for roll mode. Mostly, the transient towards the steady state is slowly, stage in which the model behaves chaotically. For moderate and high forcing the system evolves quasiperiodically or chaotically, especially in the neighbourhood of the jump points and for resonant frequencies. The effects of damping on dynamics are also illustrated.