Pure low-frequency and pure high-frequency ship motion equations in regular waves

Abstract

The problem of ship motions in waves can be seen as a coupled problem of manoeuvring and seakeeping. The former is a low-frequency motion problem in terms of time scale of ship motion, whereas the latter is relatively a high-frequency motion problem. They can be separately predicted by solving a group of low-frequency ship motion equations and a group of high-frequency ship motion equations. However, the theoretical basis behind both usual groups of motion equations may be questionable. In our previous study, a time-averaged method was introduced to derive the pure low-frequency ship motion equations in regular waves. The new equations have a few additional terms which are the mean inertial forces and moments due to wave-induced high-frequency motions, compared with the usual equations. The low-frequency ship motion equations were then solved to predict ship manoeuvring in waves by means of a modelling approach. It has been demonstrated that these additional terms are non-negligible for manoeuvring prediction, if amplitudes of high-frequency motions are large enough, e.g. in the condition of long wave length and large wave amplitude. In this study, we will give out the more complete pure low-frequency ship motion equations in regular waves and some new explanations. The pure high-frequency ship motion equations are also derived by subtracting the pure low-frequency ship motion equations from transient ship motion equations. On the aspect of manoeuvring prediction, there are two improvements, as compared with before. One is that the mean inertial forces and moments due to second-order high-frequency motions are retained in the pure low-frequency motion equations. The other is that the effect of longitudinal ship speed is taken into account in the modelling approach. As will be shown, the prediction accuracy is improved.