Unstable Motion of Free Spar Buoys in Waves 



The pitch equation obtained by elimination of X in the 



above two equations is of order 3 . In order to simplify subsequent 



calculation we shall make use of the fact that the coefficients N 



xx 

 and N ~ are small . Neglecting them we obtain a second order 



equation in 6 . 



/ mx6 \ d 2 e de r_ 



Jlyy + Jyy- m+rn J^ ^ ee dt + p/<G*J 



/ _ mx6 



) dT/=2pftGP A c^xr m+mxx a ° X ]d^ 



The calculation of both integrals is performed in the 

 same way as for the heave equation . Neglecting second order terms 

 in s one gets : 



(II - 3.1) 



(. 



2 \ 2 

 mxG \d 6 





de 



Iyy+Jyy- , ]~77l +N6G j4 _ + mg t+a+ g (s) 



i' } 7 } m+mxx/ dt dt & l_ H _ 



zG 



= 2 mcr 



mx6 1 



Q, (k) + Qo (k) + — 



1 



m+mxx 



2C 



H 



df 

 Jdt 



where the symbols introduced are defined in the nomenclature . The 

 function g (s) is for all types : 



(II-3.2) 

 with h 



g(s) = 



if 



h 2 " 



^ h. 



2 = oo for the type 11 and 14 



The coefficients of equation (II- 3 . l) are the same as 

 n [1] 



= o(R) 



4 W 



N ee1' Zc p H t (0) F r [ Q / k ) 



1037 



