Kaplan 



and a similar term will occur in the pitch equation, proportional to 

 X, given by 



Mc = - £^y_ e'^^s'(e) de • y_ (e - eg)e''^s'(e) di . i. (69) 



All of the above expressions can be combined to produce the coupled 

 surge and pitch motion equations of the spar buoy, together with the 

 heave motion equation. The effects of the mooring are included in 

 terms of the appropriate degrees of freedom to allow computation of 

 the complete system response, which will be the end product of the 

 program. 



The disc-shaped buoy hull is analyzed as a case of a shallow 

 draft vessel. The section in the water is a circular cylinder with a 

 small draft compared to the cylinder diameter, and the form is 

 axisymmetric. The hydrodynaxnic and hydrostatic forces are found 

 using the shallow draft approximation, as in [13] , together with other 

 simplified representations for the wave-induced forces. Because of 

 symmetry relations, where the disc- shaped buoy is assumed to be 

 circular shape, some of the coefficients in the basic equations of 

 motion, Eqs, (54) - (56), are immediately evident: 



(70) 



a-29 = ^36 - ^ ' 



Specific values of certain other coefficients are readily evaluated 

 for a circular discus shape, and they are given below. Assuming 

 that the discus buoy is a cylinder of radius R, aind draft d' , the 

 following heave restoration coefficient value is found: 



2 



a-26= Pg^^ • ^^^^ 



For the case of the pitch restoring moment, the basic term (cor- 

 responding to the hydrostatic portion) of the coefficient a^g is 

 obtained from the expression 



Mj,= - W|GM|e (72) 



where 



W = weight of buoy 



IgmI = metacentric height 



1060 



i 



