Rolling Response in Synchronism: Transient and Steady-state Behaviour

The rolling motion of a ship has been described by a nonlinear differential equation explicitly taking into account the nonlinearities of restoring and damping. The equation has been solved by means of a perturbation method in the zone of main resonance and the first ultraharmonic relative to a regular beam sea. The approximate analytical solutions allow expressions for the frequency response and moreover an estimation of the maximum amplitudes foreseeable as a function of the excitation intensity. These are obtained through algebraic equations from which approximated solutions are obtained in an explicit form. In this manner, simple formulas are obtained the excitation to the maximum rolling by means of the nonlinear damping coefficients. They constitute an improvement with respect to those till now used. A comparison with the results of numerical simulation shows a quite good agreement in a wide range of values for the parameters