Abstract
The qualitative behaviour of the response of a ship rolling in longitudinal waves whose amplitude or frequency (parameters) are slowly varied is presented. An analytical and numerical technique is used to predict the qualitative change taking place in the stable solutions of a ship model as one of the parameters is slowly changed. The analysis took into consideration linear, cubic and quantic terms in the polynomial expansion of the relative roll angle. The damping moment consists of the linear term associated with radiation and viscous damping and a cubic term due to frictional resistance and eddies behind bilge keels and hard bilge corners. Two methods (the averaging and the multiple time scales) are used to investigate a first-order approximate analytical solution. The modulation equations of the amplitudes and phases are obtained. These equations are used to determine steady state solutions. Numerical calculations are presented which illustrate the behaviour of the steady state response amplitude as a function of the detuning parameter. The stability of the proposed solution is determined applying Liapunov's first method. The effects of different parameters on the system behaviour are investigated numerically. The results obtained by the two methods are in excellent agreement.
Published Version
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