Roll dynamics of a ship sailing in large amplitude head waves
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Some ship types may show significant rolling when sailing in large-amplitude (near) head waves. The dynamics of the ship are such that the roll motion is affected by the elevation of the encountering waves. If the natural roll period (without forcing) is about half the period of the forcing by the waves, then a stationary solution will have an amplitude that is much larger than for other forcing frequencies. This phenomenon is called parametric resonance. For certain hull shape types the transverse stability may vary considerably due to the waves passing the ship. Moreover, near head waves will also have a direct effect on the roll dynamics. For these processes a differential equation model—a Mathieu type of equation—is formulated. Furthermore, the waves considered are of a type that is encountered in open seas. As a parameterization of these waves the Pierson–Moskowitz spectrum is used. The risk that the ship will reach a critical state is characterized by the time of arrival at this state, starting from an arbitrary pattern of the waves and the dynamic state of the vessel in the stationary situation. Large-scale Monte Carlo simulations of this process are carried out. The percentiles of the arrival time distribution indicate the risk of significant rolling to which the vessel is exposed. Furthermore, a method is proposed to estimate the maximum roll angle in a stationary state by taking into consideration only the part of the wave spectrum that relates to the state of parametric resonance. The result is compared with the outcome of the large-scale Monte Carlo simulations.
KeywordsCritical roll amplitude Parametric resonance Stochastic waves
The authors would like to thank the organizers and participants of the Study Group Mathematics with Industry for initiating a first exploration of the subject; see .
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