M^=P X + I F x (i) (26) 



1=1 



dt< 



d 2 z 



—f- " P« + P., + I F z (i) (27) 



dt' 



1=1 

 n 



I -=4 = T, + T + ^ M B (i) (28) 



dt: 



1=1 



where M is the mass, and I is the moment of inertia of the floating 

 body. In equations (26), (27), and (28), (P x , ? z , T d ), (P g , T g ), and 

 (F (i), F (i), Mg(i)) are the hydrodynamic effects, the hydrostatic effects, 

 and the effects of the ith mooring line on the sway, heave, and roll modes, 

 respectively. The mass, M, of the floating body and the moment, I, about 

 the center of gravity may be expressed in terms of the fluid density, p, and 

 a characteristic length, h , as 



M = pv x h2 (29) 



I = pv 2 h2 (30) 



By assuming that the center of gravity of the body is on the vertical line 

 through the center of the waterline, CC 1 , in equilibrium condition and by 

 denoting the length of the waterline on the structure by 21 , the second 

 moment of the floating plane about the center, I , and the immersed volume 

 of the body, V, are given by 



^-■J' 1 ? (31) 



V = v 3 h2 (32) 



In equations (29), (30), and, (32), v , v , and v are constants, depend- 

 ing on the shape and the density distribution of the floating body, and the 

 initial tension of the mooring lines. 



The hydrodynamic forces and moment about the center of gravity are given 

 by taking pressure and moment integrals on the immersed surface. 



P 

 P x = / cos(v, x) ds (33) 



S 

 P 

 P z = / cos(v, z) ds (34) 



S 

 P 

 T d = '/ [-(x - x Q ) cos(v, z) + (z - z Q ) cos(v, x)] ds (35) 



S 



The restoring vertical force and moment due to the hydrostatic pressure are 

 given as 



46 



