Dern 



d 2 e 



(iyy+Jyy- m+mx J dt 2 n^ dt +mg 



[o) — 



zG 



r+a+ ~ g(s) 



e 



(II-5.3) 



= 2 mcr 



2 ,^ x ,2 , x d<P 



my cp \ d (p (o) 



Ixx+Jxx - 



m+myy, 



\_djp (o)_J 



/dt 2 +N ^ dt +N pV 7 



f =0 



where the coefficients in the left hand sides of the equations are 

 constants and where f (s) and g (s) are given by (H-2. 2) and (II-3. 2). 



In the sequel this particular form of the equations will 



be used 



VI - EXPLANATION OF THE DOUBLE REGIME IN THE HEAVING 

 MOTION - 



The presence of double regimes in a rolling motion has 

 already been investigated by various authors . The experimental 

 finding of this phenomenon was made by E.G.Barillon who explained 

 it by considering a non linear damping proportional to pJ" (r ^. 2) , 

 a forcing moment proportional toflK (m < 1 ) and a restoring moment 

 in the form T(yp ) - -Alp - B (p 3 (A and B are constant) [7] . The 

 rigorous mathematical explanation of the phenomenon was given by 

 R. Brard , starting from equation of the form : 



d 2 ^ /dP \ , , 



(II-6.1) -JT2 +P g(T-J-) + h (p) = F cos <r t 



where p is a constant , and g ( . ) and h ( . ) are analytical functions 

 [8 J , but numerous approximative methods exist [9 1 , fiol 



The double regime in the heaving motion of the buoy is an 

 analogous phenomenon , which can be explained at least qualitatively 

 by equation (II -5- 3) . Taking s (t) as the unknown in this equation , 



it becomes : 2 



d s , , ds 



(m+mzz) ,2 + NzV 



dt 

 P 



d t 



+ Pg S (o) F (s) 



= I (m+mzz) o- - pg S(o) C k H Qo (k) If coscrt-KrNzz / sin or t 



1040 



