Unstable Mot-Con of Free Spar Buoys in Waves 



difference between the real value of h (h = 0. 08 m) and the value 

 actually used in the computations (h =0) . 



From figures 18 to 22 , it is easy to determine the in - 

 fluence of wave on the value of a = s. . The equifrequency curves 

 are indeed independent from the waves amplitude , and the circle 

 radius is proportional to it . One can see , for example , that for 

 f = 0. 182 Hz the curve gain VS half-wave- amplitude must exhibit a 

 jump at J = . 1 28 m . 



In conclusion to this investigation of the double regimes , 

 figures 23 and 24 show the effect of h andj on the shape of gain 

 curves computed for the type 1 buoy . 



If we turn back to figure 15 we see that the above theory 

 does not explain the double regime in relative motion in the vicinity 

 of f = 2 f fi . One reason for this discrepancy is that we have assumed 

 that /d (t) ^ h (see beginning of this paragraph) . No attempt has 

 been made at explaining the double regime in the vicinity of f = 2 f fi 

 by discarding this hypothesis . 



VII - EXPLANATION OF ROLLING IN REGULAR WAVES - 



It has been known for many years that a ship moving in 

 longitudinal regular waves can perform rolling motions of large 

 amplitude [ 12 ] . In 1955 , Kerwin \_ 1 3 ] explained the motion by 

 the periodic variation of the restoring moment in rolling due to the 

 on-coming waves . The roll appears as an unstable solution of a 

 Mathieu equation . In 1959 , Paulling and Rosenberg [l4j have 

 shown that instabilities in the ship motion could be explained by the 

 effect of second order coupling terms in the equations of motion (see 

 also [_ 1 5 J and |_ 1 6 I). This latter work was pursued by M. R. Had - 

 dara for the case of a ship in oblique regular waves [lV] 



In the case of a spar buoy , Kervin's approach does not 

 apply , since the wave length is considered as large as compared to 

 the buoy diameter . However , the rolling motion can be explained by 

 the presence of a non linear coupling term between heaving and rol - 

 ling ( equation II- 3.3) . 



1045 



