THE NONLINEAR DYNAMICS OF SHIPS IN BROACHING

Ship survivability against capsize in heavy seas has become one of the areas of primary concern among ship researchers, designers and regulators in recent years. When a ship is subjected to the effect of large waves it may capsize according to a number of different scenarios, depending on the magnitude and direction of the wave excitation and the ship’s own capability to resist such excitations. Resonant or breaking waves approaching a ship from the side (“beamseas”) have a potential to excite large rolling which could result in capsize, especially if the intensive oscillation of the ship causes shift of cargo or, if a considerable quantity of green water is shipped on the deck. More dangerous still can be a group of steep and relatively long waves approaching a ship from the stern (“following-sea”). Waves of this kind are known to incur significant reductions in roll restoring capability (i.e. the tendency to return to the upright position) for many types of vessels and they may also instigate dangerous coupled motions. According to a popular classification, in following-seas a ship may capsize in at least three different ways: Pure-loss of stability is a sudden, non-oscillatory type capsize taking place around a wave crest due to slow passage from a region of the wave where roll restoring has become negative. Parametric instability is the gradual build-up of excessively large rolling created by a mechanism of internal forcing, the result of a fluctuating restoring that depends on where the ship lies in relation to the wave. The third distinctively identified, yet until recently not fully understood, type of capsize is broaching (Fig. 1). Broaching is an unintentional change in the horizontal-plane kinematics of a ship. Broadly, it may be described as the “loss of heading” by an actively steered ship, that is accompanied by an uncontrollable build-up of large deviation from the desired course. Broaching is more commonly arising in waves which come from behind and propagate in a direction forming a small angle, say 10-30 deg, with the longitudinal axis of the ship. Although the inception of broaching represents a problem of instability on the horizontal plane, capsize may be incurred at the postcritical stage due to development of large heel as energy is transferred into the roll direction. Broaching could happen to small as well as to larger ships. It is notable however that the dynamics involved do not seem to fit always into a single pattern. Frequently, broaching is manifested as a sudden divergent yaw, which peaks within a single wave length. Control is lost when the middle of the ship lies somewhere on the down-slope and nearer to the trough. In other cases there is a gradual, oscillatory-type build-up of yaw as successive waves impinge on the ship from behind. In moderate sea states a ship is more likely to broach-to if it runs with a high speed and is slowly overtaken by the waves. Broaching may also occur at lower speeds if the waves are very steep. In this paper we shall present an overview of our research on broaching which spans a period of five years and has taken place in Japan, the UK and Greece, on the basis of fellowships provided by, or linked with, the EU [113]. Key contribution of this research has been the association of specific nonlinear phenomena with broaching behaviour. Nonlinearity due to the effect of the wave force is a key factor when the surge dynamics are involved in the creation of broaching. Consideration of stiffness nonlinearity in the roll direction is also essential when studying the possibility of a capsize occurring in the wake of controllability loss. Simple analogues of the surge and roll dynamics are shown in Fig. 2, while some basic concepts about nonlinear systems are presented in the Appendix.