Mooring and Positioning of Vehicles in a Seaway 



^24^ ^ ^25^ "^ ^26^ ^ ^27® + ^ZB^ + ^%^ = ^m + ^^ (55) 

 St'^ + V + ^55^ + Se^ + ^37^* + ^8^ + ^9^ = M^ + M^ (56) 



where the mooring forces are represented in general form and the 

 wave forces can be represented as sinusoidal functions of time for 

 different wave frequencies. The mooring forces will depend upon 

 the mooring arrangement (i.e. number of cables, attachment point, 

 etc.), whereas the functional form and degrees of freedom in the 

 force representation depend upon the geometric arrangement. 



For the surge degree of freedom, the coupling with the pitch 

 equation, and vice versa, occurs as a result of hydrodynamic 

 inertial coupling (potential flow theory) and hence the symmetry 

 relation ajy = a,, is attained. This result is due to the equivalence 

 of the off-diagonal terms of the added mass tensor representation 

 of inertial forces. With the longitudinal force mx assumed to act 

 through^the center of buoyancy (CB) of the hull, a pitch moment 

 m|BG|x occurs, i.e. a,. = m|BG| = a _ where |-BG| is the 

 vertical distance between the CB and the CO (center of gravity). 

 The remiaining terms in the surge equation are a.. = m and some 

 estimate for surge damping a.-. 



The surge damping can be represented in a number of ways, 

 either linearly with allowance for the current by means of perturba- 

 tion theory, or in a nonlinear form as a drag coefficient representa- 

 tion, etc. (see [ 10] for more details). Ordinarily, this surge 

 damping term is not very important in its influence on the resulting 

 ship or buoy motions since there is no natural resonant response 

 in surge. However, in the present case of a moored buoy there is a 

 restraining surge force from the mooring cable and there may be 

 some resonant surge motion. Thus , the proper inclusion of the 

 surge damping force on the buoy hull can be important for dynamic 

 behavior calculations* 



The mooring cable forces acting on the buoy hull are con- 

 sidered separately further ahead in this study. They are inaportant 

 since such forces affect the buoy motions, and the buoy motions in 

 turn determine the boundary conditions as well as the input excitation 

 for the cable dynamics. The techniques for inclusion of these effects 

 in the overall mathematical model are considered later in this inves- 

 tigation. 



A spar buoy hull form is axisymmetric about the vertical 

 axis and hence motion analysis can be carried out for the three 

 degrees of freedom with slender body theory techniques used in the 

 analysis of the hydrodynamic action on a long slender spar form. 



m*z = - pgSg • z + zj + z^ + z^ (57) 



1057 



