Rh  ¼ bhMvh;

measure the swellness, the heights of the sea surface has to

be computed。 A moving grid is attached with the ship as

hi;j=ðL m WÞ;

shown  in  Fig。 2。  To  generate  the  grid,  the  ship  body is

vertically projected onto the sea surface (the X–Z plane), and the bounding box of the projection is calculated。 Then a uniform grid of LxW cells is superposed on the bounding box。 The grid-lines are aligned with the Xs and Zs axes of the local coordinate system of the ship。 The gap between two grid-lines is set to 1 meter。 When evaluating the height field at a grid point, we assume the ship is not presented and the height field is calculated by using the wave model presented in the previous section。 Once the height fields in the moving grid are computed, the swellness of the water is decided by multiplying the average height field with the area of the grid, and the force for ship motions is computed。

i¼—L=2  j¼—W=2

Fh ¼ KhHaA — Rh;

where Rh is the resistance which is computed by multiplying the moment of the ship, Mvh, with the resistance coefficient of water, bh。 The mass of ship is represented by M and vh is the current heave velocity。 The variable Ha is the average height field, and hi,j are  the height fields at the grid points。 The volume of the water swelled under the ship is HaA, where A is the area of the grid。 The force for heave is computed by multiplying the swellness of water with the coefficient Kh。 The net force Fh is obtained  by subtracting  the resistance  from  the   force。

The two coefficients, bh and Kh are determined by ship shapes and can be modified by users。 Once the net force has been calculated, the acceleration and velocity are computed and updated by:

ah  ¼ Fh=M;

vh  ¼ vh þ ahDt;

where ah and vh are the acceleration and new velocity respectively, and Dt is the time step size。 Then, the magnitude of heave is calculated  by:

heave ¼ vhDt: ð1Þ

The resistance coefficient bh is critical for heave motion。 It can be used to simulate a floating object vibrating on a still water surface。 In our system, we assume that the ship stops heave quickly if waves disappear。 Therefore, we set bh = 1。3,  and  the  heave  motion  is  over-damped。  The      ship

Fig。 3 Classification of height fields for pitch。 The height fields in the first half have positive weights while the weights of the height fields of the other half are  negative

becomes steady fast once no external force acting on    it。

where ap

is the angular acceleration and  hp

is the angle of

3。2Pitch

Assume pitch is driven by the difference of height field between the front and rear halves of the ship。 This height difference is multiplied with the grid area to compute the variation of water volume, and then modulated with a coef- ficient, Kp to estimate the force for pitch。 When the ship pitches, damping effect of water will restore the ship position from oscillation。 The damping force is proportional to the velocity of pitch but exerting on the ship in opposite direction。 Totally, the net force for pitch is determined as follows:

pitch。 If the damping coefficient bp is greater than but close to 1。0, for example bp = 1。3, then the ship will pitch one or two times if the sea surface suddenly becomes still。 This phenomenon matches real ship behaviors。 Thus we adopt this method for specifying the damping  coefficient。

The inertia moment, Ip, plays the same role as the mass M does in translation motions (Benson 1996)。 To compute Ip, we assume the mass center of the ship is at its geo- metrical center G。 The ship is regarded as a beam structure shown in part (a) of Fig。 4。 The beam is composed of l cross-sections of mass, mi。 Assume the mass of each cross- section is a constant m, and the total ship mass M = ml。