where,X',Y',Z',K',M',N' are surge,sway,heave,roll,pitch,yaw external forces and moments.

　

 The above equations contain 12 variables. However, because of the nonlinearity of moment equations, those equations need to be solved by using some numerical method. The aforementioned variables are defined as follows:

　

　

where δ is rudder angle and ξ represents position of ship on the wave.

　

2.2 External Forces

　

 In the current numerical model, wave forces, manoeuvring (hull) forces, rudder and propeller forces will be taken into account in the estimation of the external forces, since they were found to be most important components of the excitation.

　

2.2.1. Wave Forces: In the calculation of the wave forces the fluid is assumed to be inviscid and incompressible and the flow is irrotational. Surface tension is neglected and infinite water depth is assumed.

　

Froude-Kyrlov Including Hydrostatic Forces: The generalized Froude-Kyrlov force vector is given by integration of pressure in the undisturbed wave system up to the instantaneous wetted surface:

　

　

The former is static. the latter is dynamic wave pressure, and may be written as;

　

　

where p is density and Φl is the potential associated with the incoming wave potential. Froude-Kyrlov forces and moment are written as:

　

　

where n is normal vector and r×n is vector fixed with respect to the centre of gravity.

　

 Diffraction Forces: In this study the method proposed by Ohkusu [9] is followed. For the purpose of the study, only the disturbance equation is used which can be written as follows,

　

　

Here, j=1,2,3 denotes sway, heave, roll respectively, ΦD indicates the disturbance of waves, and Nj is the normal vector. For the pitch and yaw, values of the heave and sway at each cross section are multiplied by the distance between the cross section and centre of gravity of the ship.

　

2.2.2. Manoeuvring (Hull) Forces: For the calculation of the manoeuvring forces, equations based on MMG model [10] Tasai [11] formulations were used. Surge, sway, roll, yaw, the manoeuvring coefficients were obtained from the model experiments. The heave and pitch radiation coefficients were found using Tasai's [11] empirical formulae.

　

　

where, XH, YH, ZH, KH, NH, MH, are surge, sway, heave, roll, pitch, yaw hull forces, respectively. Res(u), resistance force which is obtained from model test results, C(ψ), damping moment and zy, vertical coordinate of the centre of action of lateral force.

　

2.2.3 Rudder Forces: In the calculation of rudder forces the model of the Japanese Manoeuvring Group, [10] was used. The rudder forces and moments including rudder-to-hull interaction are as follows:

　

　

where, XR, YR, NR, KR, are surge, sway, yaw, roll rudder forces respectively, FN, rudder normal force, aH rudder -to-hull interaction coefficient, xH, longitudinal coordinate of the point of action of the rudder to hull interaction force, xR, zR, longitudinal and vertical coordinates of the rudder's centre of pressure .

　

2.2.4 Propulsive Forces: Propulsive forces are calculated using the following standard formulation. All the parameters were obtained from model tests results.

　

　

where, Xp, Yp, Np, are surge, sway, yaw propulsive forces, respectively, tp, thrust deduction at the propeller in forward motion. KT, thrust coefficient, D, propeller diameter and n, propeller rate of rotation.

　

2.2.5 Automatic Control: The following standard proportional-differential (PD) autopilot is employed in this model in order to keep the vessel on course :

　

　

δR is the actual rudder angle, ΨR is the desired heading angle, kl is yaw angle gain constant, k2 is yaw rate gain constant and tr is the time constant in rudder activation.