Unstable Motion of Free Spar Buoys in Waves 



Equation (II-4. 2) is only approximate for roll- sway cou - 

 pling terms have not been taken into account (cf . previous paragraph 

 for linear terms and Morison - O'Brien formula for non linear ones). 

 Moreover Morison - O'Brien formula is only approximate . The 

 coefficient C is actually time -dependent and the value to be attribu- 

 ted to this coefficient varies in a large range [6] . Lastly , the 

 finiteness of the buoy slenderness (H/D) is not taken into account 

 (end effects) . 



One should note than equation (II-4. 2) does not include 

 the non linearity due to the static restoring moment T ( ^p ) .In 

 fact , the exact expression of P ( f> ) is : 



r {p) = - m g [l +a + g3L_; fa + tf f f 



with 



= 4* 3 



4(W- 

 tg vp 



sin if 



W = 



m 

 P 



There is very little difference between this expression 



and / \ . „ 



- mg (r + a) sin <p 



even for angles approaching <p = 60° (relative variation is about 



3 / lOO) . 



V - APPROXIMATE EQUATIONS FOR HEAVE-PITCH-ROLL, 

 MOTIONS - 



Figures n° 11 , 12, 13 relative to the frequency respon- 

 se of version n° 1 1 show that, in the linear case,the dynamic system 

 associated to the buoy (in the sense of paragraph 1-6, 2) may be 

 approximated by a second order differential system . This result 

 suggests that we may substitute to equations (II-2.. 3) , (II-3. l) and 

 (II-4.2) the equations 



(II-5.1) 



d 2 Z . 



(m+mzz) , 2 +Nj 

 dt 



dZ 



dt 



+ pgS(o) 



Z + f(s) 



PgS(o) 



1-C kHQo(k) 

 P 



I (t) 



1039 



