I-Min Yang 



the following form : 



T = c ( -^) n if AL > 

 L 



= if AL < 



where L is the moored length of a mooring line, the change in length 

 due to a tensile force T is AL, and c and n are two constants 

 depending on the type of mooring lines. The total restoring force and 

 moment due to all mooring lines have components in all three directions 

 of translation and rotation. 



Another restoring forces and moments come from fenders 

 which will be in action only when they are in contact with the ship. The 

 total restoring force has only one component in the sway direction and 

 the total restoring moment has components in the roll and yaw direc- 

 tions. 



Finally there are the exciting forces and moments due to water 

 waves. The waves are assumed to be sinusoidal standing or progressive 

 waves and have a unique frequency, then forces and moments on ship 

 sections can be obtained from certain wave potential. 



The six equations of motion for a moored ship are obtained by 

 balancing the various forces and moments discussed above and may be 

 represented in the following matrix form : 



where 



Mx + Cx + K x + f(x) = g(t) (1) 



M = virtual mass and moment of inertia matrix 



C = damping matrix 



K = stiffness matrix due to linear restoring forces and 

 o & 



moments 



f(x) = force vector due to nonlinear restoring forces and 

 moments 



g(t) = force vector related to water waves 

 = q.cos ((Jt +y .), i = 1, . . . , 6 



x = (x L> x 2> x 3 , Qj, 2 , Q 3 ) 



674 



