Abstract: The paper presents the formulation of the proposed new set of motion equations and the associated terms in the equations together with a solution procedure. The mathematical model is then applied to simulating the dynamic behaviour of a purse-seiner fishing vessel and of a container-ship, for which very extensive input and seakeeping test data for a following-sea environment is available. The two ships are numerically tested for various speeds, wave steepnesses, wave lengths and loading conditions. Comparisons have been made between the new axes system and the traditional axes systems in order to determine the difference in the simulated behaviour for the two cases. The effect of memory with respect to heading angles is investigated and comparisons of model predictions with and without memory effects are presented.

　

 The large motions performed by a ship during operation in a following-sea environment, whether such motions are caused by waves, by some control action or their combination, can have a critical effect on the safety of the vessel.

　

 As it is well known, when a vessel encounters waves from astern, usually three different modes, which lead to dangerous situations, are observed. The first is the so-called pure loss of stability on a wave crest where there is not enough buoyancy under the free surface to keep the vessel upright. This is an essentially static mode but capsize happens suddenly and abruptly. However, the majority of the research work has been focused on the other two modes which involve the dynamic stability of the vessel. One of these modes is the low cycle resonance which is caused by the parametric built-up of large roll motion, whilst the other is a combined mode that involves strongly the manoeuvring motion of the ship and is known as broaching. The latter mode is often realised in relatively high Froude numbers and increased wave-steepness leading where the ship experiences the so-called "surf-riding" condition. When the frequency of encounter is low, the vessel restoring force provided by rudder could be inadequate to impede the increasing oscillatory yawing motion. This dangerous situation termed as "broaching-to" could lead a vessel to capsize. In other terms, when a periodic motion becomes less stable a ship can be attracted by an unstable "surf riding" point as a saddle and then repelled with a violent yaw motion despite maximum opposite rudder. In astern seas. coupling between surge and lateral motions cannot be avoided therefore it is essential to tale both surf riding and broaching onto account.

　

 Vassalos and Maimun [1], giving a brief review of individual mechanism likely to contribute to broaching, confirmed using numerical and experimental investigations that coupling between longitudinal and lateral motion are of paramount importance for broaching as well as the ensuing extreme vessel behaviour. Spyrou [2], [3] has described in detail the interesting dynamics of broaching using 4 DOF mathematical model.

　

 In an attempt to enhance numerical prediction of the motion of ships in astern seas, The Specialist Committee on Prediction of Extreme Ship Motions and Capsizing [4] set up by ITTC carried out the benchmark tests. Umeda and Renilson [5] gave a detailed account of the benchmark study and it was concluded that degrees of freedom; and frequency effect or so called "memory effect" were seen to be the most important elements affecting the accuracy of numerical models

　

 Deriving from this background, the aim of this paper is to describe numerical code, focusing on a fully non-linear coupled 6 DOF numerical model with frequency dependent coefficients, incorporating memory effects in waves with a new axis system that allows straightforward combination between seakeeping and manoeuvring models whilst accounting for extreme motions. The model in its current form takes into account the effect of waves through the incident and diffraction components with control action through the rudder and the propeller. The model focuses on two important aspects: The first one, is to express the equations of motion with respect to a new "horizontal body axes" system which allows easier handling large angles in pitch (measured between the ship's longitudinal axis and the horizontal plane) which may be realised during operation in steep waves. The second important aspect is to investigate how the behaviour of the ship is influenced by the consideration of this frequency-dependence of the hydrodynamic coefficients. Since the frequency of encounter in following seas is quite low, it is currently quite common to use "zero-frequency" constant hydrodynamic coefficients, as it is normal practise in calm-water manoeuvring calculations. However, wave effects associated with the unsteady motion of the hull at the free surface and vortices which are shed from the oscillating hull, especially when a ship has very large heading angle, indicates that the convolution terms (representing the so-called "memory effects") cannot be negligible. Such convolution terms are currently being incorporated in order to improve the prediction of the behaviour of the vessel at encounter frequencies which are not very near to zero. The details of numerical model and its verification against ITTC benchmark tests along with further experimental studies carried out to improve its accuracy were presented by Ayaz et al. [6]. [7].

　

 Within the above framework the paper presents the formulation of the proposed new set of motion equations and the associated terms in the equations together with a solution procedure. The implementation of the proposed memory effects is also described. The mathematical model is then applied to simulating the dynamic behaviour of a purse-seiner fishing vessel and of a containership. The two ships are numerically tested for various speeds, wave steepnesses, wave lengths. Comparisons are made between the new axes system and the traditional axes systems in order to determine the difference in the simulated behaviour for the two cases. The memory effect of with respect to heading angles is investigated and comparisons of model predictions with and without this are presented. It is believed these improvements to the numerical model have good potential for providing a more rational basis for predicting those dangerous conditions which a ship could face in extreme following and quartering seas, whilst offering new insights in linking ship behaviour to ship design parameters.

　

2.1 Equations of Motion

　

 The Horizontal Body Axes is quite a common system and it has been used in many other studies of ship manoeuvring [8]. The difference of the current numerical model. from aforementioned conventional methods is that no assumptions are made for small angles in order to solve motions equations more accurately.

　

 In deriving the basic equations of motion, normally three different coordinate systems are used as shown in Figure 6. The first is an Earth fixed system, defined by 0-ξηζ. The second is a general body axes which is fixed in the ship with the origin G being located at the centre of gravity of the ship defined by G-xyz. The third is the Horizontal body axes fixed in the ship with the origin at G and defined by G-x'y'z'.

　

 Newton's law of dynamics describes the equations of motion for a ship having six degrees of freedom and under the action of certain external forces with forward speed. It can be formulated for translations and rotations in vector form as follows.

　

　

where m is the mass of a ship, HG the momentum about the centre of gravity,ω the angular velocity, VG the linear velocity, XF the external force vector and XM the moment vector.

　

 The terms on the right hand side of Equation (1) are the external forces acting on the hull and they can be divided into hydrostatic and hydrodynamic components. In order to describe the situation of the ship in the earth fixed axes, it is normal to use a transformation of equation (1) in terms of Eulerian angles ψ,θ,Ψ which are defined as the rotations about the body fixed axes (Figure 1).

　

 The transformation between body axes and horizontal body axes in terms of the Eulerian angles is as follows

　

　

 The transformation between the angular velocity of the horizontal system and the body axis system is shown as

　

　

 In order to derive the equations of motion in a practical form some approximations are necessary. Firstly,because of symmetry and because the origin is located at the centre of gravity, it is assumed that IYZ=O, IXY=0, IXZ=O and in the horizontal system IyyIzz.However, in order to simplify these equations the following expressions are used. Substituting P for