Paulling 



roll displacement, x., is 



Fy = pgAv^z^x^ , (21) 



and for a small pitch displacement, x- 



Fy = - pgA^x^Xg . (22) 



Xy^, Zy^ are the coordinates of the center of gravity of A^. These 

 forces are included in the equations of motion as static restoring 

 or coupling force terms. 



The Restoring Forces 



Three types of restraints have been described in the Intro- 

 duction, dynamic positioning, spread array mooring, and vertical 

 tension leg mooring. 



A dynamic positioning system incorporates two principal 

 components: sensors for detecting deviations from the desired 

 position, and thrustors which may be activated automatically or 

 manually to exert a force tending to restore the structure to the 

 desired position* In the simplest system, the thrustors are actuated 

 without time lag to exert a force proportional to the displacement. 

 This would be termed a pure proportional controller. Real systems 

 seldom operate this simply but Incorporate tlmie lags, back lash, 

 and other non-Ideal characteristics. Increased sensitivity and 

 response may be built Into the system by having It sense velocity (rate 

 control) and acceleration. If the system can be approximated by 

 linear features , I.e. , If the applied thrust can be linearly related to 

 displacements, velocity, and acceleration of the structure, then the 

 control system constants may merely be Introduced In the force terms 

 of the equations of motion, (6) , as additions to the already defined 

 hydrodynamlc and hydrostatic terms. 



In a spread mooring system, several pretensloned anchor 

 lines are arrayed around the structure to hold It In the desired 

 location. If the structure moves from Its mean position, the tensions 

 In the anchor lines change and these changes may be related to the 

 geometry (catenary), elasticity, and hydrodynamlc properties of the 

 anchor lines. It Is usually permissible to neglect the hydrodynamlc 

 forces on the anchor lines and to approximate the force by a linear 

 relationship between force and displacements In the plane of the 

 anchor line. The displacements at the point of attachment of the 

 anchor line may be determined In terms of the coordinates of this 

 point for given displacements of the structure. These are resolved 

 Into horizontal and vertical displacements, x^ , y^ , In the plane of 

 the anchor line by a transformation similar to (12). The horizontal 

 and vertical forces exerted by the anchor line may then be expressed 



1098 



