A. Cardo

On damping models in free and forced rolling motion

Abstract In this paper two different models for the damping moment to introduce in the rolling equation of the ship are proposed. They contain two terms, respectively linear-quadratic and linear-cubic in the angular velocity, and furthermore they foresee a non-linear term representing the dependence of the damping from the heeling angle. These models constitute a generalization of all the models up to now used in the naval literature. With the Bogoliubov-Krilov asymptotic method approximate relations, describing the decay curve of the free oscillations and the maximum roll amplitude in synchronism condition, are obtained. The analysis shows that the choice of the more realistic damping model cannot be based on the simple verification of a good fitting of the free oscillation decay curves. It is necessary to examine also the behaviour of the forced oscillations in synchronism. Finally, a plan of experiments which allows the determination of separate values for the different non-linear damping coefficients is proposed.

Subharmonic oscillations in nonlinear rolling

Abstract In this paper the rolling motion of a ship is examined with particular regard to the possibility of obtaining oscillations which are subharmonic to the excitation frequency. Three different mechanisms are found to be responsible for this phenomenon, the importance of which has already been recognized in the context of ship stability. The first is related to a strong symmetric or non-symmetric nonlinearity in the righting arm. The second is linked to the harmonic composition of sea waves and the third to the well known parametric excitation caused by coupling between different ship motions in a following sea. The onset of subharmonics is related to a threshold value for the excitation strongly depending on damping. The more appropriate analytical methods for a theoretical study of each mechanism are suggested.

A MULTISCALE ANALYSIS OF NONLINEAR ROLLING

Abstract A new analysis of nonlinear rolling carried out by the multiscale perturbation method is herewith presented. The behaviour of a ship in a regular beam sea is considered and approximate analytical solutions in three nonlinear resonance regions are obtained. These concern the transient and the steady state roll oscillations. The latter fits in well with a previous one obtained through the averaging method and with the results of the numerical simulation. The obtained results appear to be particularly convenient due to their major mathematical simplicity. Moreover, they allow a simple estimation of the maximum roll amplitudes predictable for a given excitation intensity.