velocities and position of the body as functions of time. Nonlinear 

 terms of two types are retained in this system of equations: 



(1) The {to} X {loj} and [B]~ terms 



(2) Nonlinear terms in the relationships between the external 

 forces and either the platform motions or the wave ampli- 

 tudes . 



The Mooring System . A cable or chain mooring device is usually 

 thought of as exerting a force on the moored ship or platform which 

 depends only on position. The dynamic characteristics of the line 

 itself and fluid forces exerted on it by virtue of its motion 

 relative to the water are usually ignored. For moderate water 

 depths, line tensions, and wave motion, this assumption is reasonably 

 good. In many realistic cases, however, the fluid and dynamic 

 effects become important in just the cases of greatest interest, 

 i.e., the case of survival of ship and mooring in extreme sea 

 conditions. The following development is based upon the static 

 model of the mooring, but the procedures may be easily extended 

 to more complete models. 



Assume that the force exerted by the mooring line on the 

 platform at the point of attachment may be expressed by three 

 components in the global oxyz coordinate system. In a linearized 

 motion analysis, these will be identical to the components in 

 the platform coordinate system. The coordinates of the point of 

 attachment are given by (X , Y , Z ) in the platform system. 

 Now, assume that the force versus displacement characteristics of 

 this end are also known. For a linear motions analysis, the forces 

 may be related to the displacements of the mooring line by a 

 3x3 stiffness matrix [k ] . 



{F^} = [kj . {x^} . (12) 



{F } = xyz - forces by mooring line on platform 



[k ] = stiffness matrix, 

 a 



{x } = xyz - displacements of end of mooring line, 



137 



