Unstable Motion of Free Spar Buoys in Waves 



and if it is assumed that h r^n , then 



(II-6.4) 



.^A 



V I (m+mzz)cr 2 - pg S(o) C k HQo (kf + <r 2 ISliz' 



fA 



V 



pg S(o) -o" (m+mzz)+pgS( 



•t 



«h| 



3T1h. 



4 



2N: 



+ <r 



TYPE 1 BUOY - 



From equation (II- 6 . 3) it is easy to see that for a given 

 frequency and a given wave amplitude , there exist one or three 

 positive roots s . By a method identical to that used in the following 

 paragraph , it can be verified that when there are 3 roots , one of 

 them is unstable . This explains the jump and hysteresis phenomena 

 observed when solving equation (II- 6. 2) on an analog computer . 

 Figure 17 presents the gain curve obtained (->>• versus f) , and shows 

 a good agreement with experiment . " 



TYPE 12 BUOY - 



Equation (II-6.4) is of degree 6 in s - It has therefore 

 six roots but it is difficult to see whether they are real positive . 

 Rather than solving directly this equation on an analog computer , we 

 solved it graphically , which permits to understand better the jump 

 in the gain curve of the relative motion (Figure 15) . 



The notations and terminology of [ 11 J are used here, 

 and it is assumed that the wave amplitude is small enough for s (t) to 

 be always less than h . In order to simplify the computations it is 

 also assumed that h = .In fact h = 0. 08 m and h = 0.42 m . 

 Setting h to zero will thus slightly modify the results . However the 



general shape of the phenomena will be kept 

 tion of equation (II -6. 2) is 



The describing func - 



H (ior.a) = Pg S(o) 



2 ., 



1- 



4 a 



+ 2 



3 " h 2 8 h 2 



(0) 

 or (m+mzz) - i cr Nzz 



where it has been set a _^_ s . for a while 



— A 



1043 



