Unstable Motion of Free Spar Buoys in Waves 

 no rolling if and only if q < fi or 



S A 



We as'sume now that the natural frequency for heaving 



f is not too large as compared to the natural frequency for pitching 



f so that 

 



2f » f 



e z 



Then s A c=lj a for f = 2 f „ and the condition for no rolling reads : 

 (11-7.4) S A < *L 



(II-7.5) / A <2H£ f 1 - j 



(0) 



Unfortunately /? is not theoretically known since N pp 



is determined experimentally . Nevertheless for a given value of 



S» , (II-7. 5) shows that there is no rolling if the center of gravity 

 is located well below the center of buoyancy . 



VIII - EXPLANATION OF THE UNSTABLE PITCHING PHENOMENON. 



The unstable pitching observed with the type 1 buoy can 

 be explained qualitatively in the same manner as for rolling , by 

 reducing the problem to an equation where the left member is a 

 Mathieu equation . However there now exists a right member which 

 is a sinusoidal function of same frequency as the perturbation term 

 in the left member . It is known [ 19 1 that the regions of stability 

 and unstability are the same as for the equation without a right mem- 

 ber . The qualitative results established in paragraph VII are thus 

 valid . Figure 28 shows that the theoretical domain of instability does 

 coincide with the domain of instability determined experimentally . 



1057 



