The motions of foil and body will then be two dimensional. Let v 

 denote displacement of the axis of rotation in a direction perpendicular to 

 the foil and to the stream and let 6 and G denote the small rotations of 

 foil and body, respectively, about the axis. 6 and Q are zero in the 

 undisplaced position of the system and positive from a direction opposite 

 to that of the stream toward positive v; see Figure 2. 



Assume that a spring and also an internal damping mechanism are 



present, exerting turning moments -k (6- 6 ) -c (9- 6) on the foil 



and k(9-e)+c(6-e)on the body. Assume also an external damping 



force - c V acting on the foil along the same line as the lift F, . 

 a ^ 



In general, there is a force F, positive toward positive v, acting on 

 the body at the axis and a corresponding reaction -F acting on the foil. 

 The displacement of the center of mass of the body is v + h 6 . 



The primary equations of motion for foil and body will then be: 



mv = - F - c V + F^ ; 16 = - k ( 6 - 6. ) - c ( 6 - 6. ) + L (- c v + F^ ) 

 a L ^ '-" a Li 



m (v + h e ) = F ; 16 = k ( 6 -6 ) + c ( 6 -6 ) - h F 

 o oo oo o oo 



2 



Eliminating F and inserting F = - BSv + BS 6 gives: 



(m + ra^) V + h^m^e" + (c„ + BS ) v - BS^G = [T3a] 



o o O o a 



.... .2 



16 + c (6 -6 ) + k (6 -e ) + L [(c + BS) v - BS 6] = [73b] 



o o a 



(I + h^m )e -c(e-e)-k(6-0)+hmv=O [T3c] 



oooo o ooo 



A more simple equation may be obtained by subtracting L times Equation 

 [T3a] from the s\m of Equations [T3b] and [T3c], namely: 



82 



