Ultraharmonics and subharmonics in the rolling motion of a ship: Steady-state solution 1

In the present paper the rolling motion of a ship with particular regard to the nonlinearity phenomena is examined. The behaviour in a regular beam sea is studied in cases where the encounter frequency is an integer-multiple or a sub-multiple of the natural frequency of the system.

The approximate analysis of the equation of motion, carried out with the Bogohubov-Krylov-Mitropolsky asymptotic method, shows that, apart from the synchronism, other resonance regions typical of nonlinear systems also exist. It deals with resonances of higher order, called ultraharmonic and subharmonic respectively. The origin of such phenomena is to be sought in the nonlinearity of the righting moment.

In these nonlinear resonance regions the ship has a steady-state response which could be much greater than that predictable by the linear theory. In particular, regarding the ultraharmonic resonance, we can see that it is always present and that both the maximum rolling amplitude and the frequency width of the corresponding zone strongly depend on damping. On the contrary, the subharmonic resonance appears only when the excitation intensity goes beyond a threshold level which is also dependent on damping. But, in this latter case, the amplitude of the response is practically independent from damping.

The analytical predictions are eventually compared to the numerical results obtained by solving the equation of motion with the Runge-Kutta-Gill method. The agreement of the frequency response curves appears to be optimum in the whole frequency range and goes right up to considerable rolling amplitudes.