The linearized mooring line force is then given, in terms of plat- 

 form motions by 



{F^} = [k^] [D]{x} . (16) 



The three forces {F } are transformed back to three centroidal 

 forces and three mom.ents about xyz by the transpose of [D] 



= [D]"^ • [k^] [D]{x} . (17) 



This enables us to define the centroidal 6x6 mooring line 

 stiffness matrix 



[k ] = [D]*^ • [k^] • [D], (18) 



from which, 



[F^] ^ [k^]{x} . (19) 



The mooring line spring constant matrix [k ] is defined according 



a 



to the type of mooring. A simple example will be described here, 

 a vertical taut cable or "tension leg" mooring line. The nomen- 

 clature for such a mooring is shown in Figure 12. 



The initial length of the cable in the vertical position is 

 L and the initial tension is T. In the displaced position, the 

 cable is inclined to a small angle 6 and stretched by a small 

 amount 6L. The new tension is given by 



T + 6T = T + k5L, (20) 



O O ''^^ / 



where k = elastic constant of the line. 

 Typically, k = -j^ , 



where A - cross sectional area of the line, 

 E = Young's modulus. 



The forces, in the horizontal and vertical directions, exerted 

 on the platform by the mooring line are 



139 



