A practical criterion which can be used for assessing the course-keeping capability of a ship in following waves, is proposed and evaluated. Presently, it accounts for regular waves and it accrued from an analytical estimation of the course instability region’s boundary by applying the method of harmonic balance. The calculation is performed with regard to the third-order yaw equation, derived from a classic sway-yaw-rudder model of ship maneuvering motions, with time-dependent coefficients at two places, which is like a Mathieu-type equation extended to third order. The proposed analytical criterion was evaluated thoroughly against simulations with regard to this sway-yaw model and it was found to be adequately accurate. A supplementary quasi-static yaw stability criterion (fitting to ship operation with frequency of encounter, with respect to the waves, close to zero) was also considered in order to determine which one yields more stringent requirements, for various operating conditions. The proposed criterion could be an extra vulnerability check for broaching-to, in the context of the Second Generation Intact Stability Criteria.


The difficulties of steering of ships in following seas have received the attention of the research community since more than 70 years ago (Davidson 1948). A relevant direction of research refers to the avoidance of the broaching-to instability and, in particular, to the development of practical criteria that could ensure sufficient course-keeping capability for a ship encountering steep following waves. Some classic works on this topic, such as those of DuCane and Goodrich (1962), Wahab and Swaan (1964), and Motora et al. (1981), were focused on the quasi-static condition of a ship on the wave which could be practically realized if the ship was advancing with speed equal to the wave celerity (zero frequency of encounter). As implied, these works were essentially focused on the avoidance of a type of broaching-to instability that is preceded by the realization of surf-riding, a phenomenon where the ship is forced to move with the wave, usually riding a downslope.

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