Ultraharmonics and Subharmonics in the Rolling Motion of a Ship: Steady-state Solution

In this paper the roll 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 sub-multiple of the natural roll frequency of the ship. The approximate analysis of the equation of motion, carried out with the Bogoliubov-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, ultra- and sub-harmonics respectively. In these nonlinear resonance regions the ship has a steady state rolling amplitude which could be much greater than that predictable by linear theory. The analytical predictions are compared with the numerical results obtained by solving the equation of motion with the Runge-Kutta method. The agreement of the frequency response curves appears to be optimum in the whole interval of frequency, up to considerable rolling amplitudes.