signðiÞhi;j=ðL m WÞ;

Then the inertia moment can be computed   by:

i¼—L=2  j¼—W=2

Fp  ¼ KpAHp  — bpIpxp:

The first equation is used to compute the height field difference of the two halves of the grid。 Each height field is associated with a weight of 1 or -1, depending on the sign, sign(i), of its Xs coordinate。 The height fields with positive and negative weights are depicted in Fig。 3。 In the second equation, the force for pitch is calculated by multiplying the height difference with the grid area A, and scaling down by using the pitch coefficient, Kp。 The net force is obtained by subtracting the damping force from the force of sea water。 The damping force is determined by the damping coefficient, bp, the inertia of moment, Ip, and the pitch velocity, xp。 Then the Newton’s law for rotation can be applied to calculate the acceleration and velocity of pitch:

ap  ¼ Fp=Ip; ð2Þ

xp  ¼ xp  þ apDt; ð3Þ

beam structure for pitch

beam structure for roll

Fig。 4  Ship body is regarded as a beam for computing inertia

where br and xr are the damping coefficient and the velocity of roll; the angle of roll is represented by hr,  and Ir is the inertia moment of roll。 Since the magnitude of roll is usually small and the ship will stop roll quickly if

the  sea  surface  becomes  flat。  We  suggest  that damping

coefficient be set to 1。2。 To deduce Ir, the ship is treated as  a  beam  structure  with  w   cross-sections  as  shown in  part  (b)  of  Fig。 4。  Then,  by  using  a  similar method

¼ ðmlÞðl þ 1Þðl þ 2Þ=12;

for computing the inertia moment of pitch, Ir is approximated by:

In the first equation, we set l to be the ratio of ship length L to ship width W。 Then Ip is solved and approximated by using Ml2/12。

3。3Roll

The mathematical model of roll is similar to that of pitch。 We pide the grid by using the Xs axis。 The height fields on the right side are given a positive weight 1 while the height field on the other side are associated with a negative weight -1。 The computational grid for roll is displayed in Fig。 5。 The height difference between the two sides is calculated by summation over the grid。 The height difference is then multiplied with a roll coefficient, Kr, and the grid area, A, to estimate the force for roll。 The net force is obtained by subtracting the damping force from the force of roll。 Then the net force is used to compute the acceleration of roll。 The roll velocity is updated as the acceleration is available。 The roll angle is decided by multiplying the roll velocity with the time step size。

In this model, w is equal to the ratio of ship width to ship draft。 The roll of a ship is depicted in Fig。 6。 In part (a), the ship is in a rest posture。 When encountering waves, the ship roll about Xs  as shown in part   (b)。

4Motions induced by internal and external    force

The wave-induced motions can not change the position of the ship on the sea surface。 Only the powers of propellers, the forces of rudders, winds, and currents can move the ship to a new position。 These forces produce surge, sway, and yaw motions for the ship。 In this section, the models for the three motions are  deduced。

signðjÞhi;j=ðL m WÞ; ð5Þ

j¼—W=2  i¼—L=2