A coupled non-linear mathematical model of parametric resonance of ships in head seas

The present paper describes a non-linear third order coupled mathematical model of parametric resonance of ships in head seas. Coupling is contemplated by considering the restoring modes of heave, roll and pitch motions. Numerical simulations employing this new model are compared to experimental results corresponding to excessive motions of a transom stern fishing vessel in head seas. It is shown that this enhanced model matches its results with the experiments more closely than a second order model. It is shown that the new model, due to the introduction of the third order terms, entails qualitative differences when compared to the more commonly used second order model. The variational equation of the roll motion will not be in the form of a Mathieu equation. In fact, it is shown in the paper that the associated time-dependent equation falls into the category of a Hill equation. Additionally, a hardening effect is analytically derived, related to the third order coupling of modes and wave passage effects.Limits of stability corresponding to the linear variational equation of the coupled roll motion are analytically derived. Numerical limits of stability corresponding to the non-linear equations are computed and compared to the analytically derived limits.