Limits of stability of ships subjected to strong parametric excitation in longitudinal waves

The present paper describes an investigation in which a complete third order mathematical model is contemplated, describing coupled nonlinear motions of a ship in heave, roll and pitch. Due to the introduction of the third order terms, 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 in the category of a Hill’s equation. The present investigation is limited to head seas configurations. Considering that limits of stability are a practical and direct way of assessing the safety of a design, the paper focuses on the comparison of the limits of stability derived from second and third order models. Considering that analyticity is an important tool when handling complex stability issues, the stability limits of the corresponding Hill’s equation are derived and discussed. These new limits are then compared to the Mathieu-type stability limits. It is important to realize that a more complete description of the stability diagram shall provide the designer with a more reliable and precise basis for assessing the limits of stability, particularly in the case of ship designs in which large amplifications associated with parametric rolling are expected to occur.