Kaplan 



tudinal force component at one end of the ship is represented by 



(CAx cos Q')cos a + (CAx cos Q')cos a = 2CAx cos a, 



and since an extension of the cable at one end of the ship requires a 

 contraction at the other end, a similar force occurs. These forces 

 are restoring forces and the net result is a longitudinal force in the 

 barge due to the moorings, given by 



X^ = 4C cos^ a ' X = - k^x, (16) 



where x is the surge displacement variable. 



In the case of sway displacement, the effective displacement 

 along the cable is y sin a, and combining components for net Y- 

 force on the barge, accounting for all the cables, leads to a net 

 mooring lateral force given by 



Xh, = - 4C sin^ a ' y = - kyy. (17) 



For yaw displacements, Y«^ L/2 where L is the ship length. The 

 lateral force at one end of the ship is then 



2C sin^a • y^j = CL sin^ a • ijj, (18) 



and the contribution to the yaw moment is 



CL sin^ a • ij; (y) = ^ CL^ sin^ a • ij; (19) 



at each end. Since the forces at each end are equal and opposite 

 (approximately, since the origin is not exactly at the ship center), 

 the net yawing moment acting on the barge is given by 



N^ = - CL^ sin^ a • 4; = - k^ilj (20) 



The variations in the force in the mooring cables due to the 

 motions of the barge can easily be found, since they are related 

 kinematically to the motions. It is seen that the longitudinal dis- 

 placement, X, and the net latercd displacements, y + (L/2)4' at 

 the bow and y - (I_/2)4j at the stern, can be combined to determine 

 the net variation in elongation of each mooring cable. The cable 

 displacements due to surging motion on the barge are x cos a, while 

 the cable displacement due to the motions of sway and yaw are' 

 [y±(L/2)i|j] sin a, according as the cable is at the bow or the stern. 



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