Since the displacement of the center of mass of the foil is v - LQ , 

 then the equations of motion for the foil will be: 



m (v - Le) = - k V - c V + F 

 i 1 L) 



le = - k e - c e + L ( - k v - c v + f^) 



The second equation may be simplified by subtracting from it L times the 

 first equation. Then, inserting also the value of F. , the equations of 

 motion take the form: 



mv - Lme + k,v + (c-|_ + BS) v - BS^e = [69a] 



(I + L^m) e - Lmv + k^Q + CpO = [69b] 



The V and 6 motions are thus coupled together inertially through the Lm 

 terms . 



To search for harmonic motion, make the usual mathematical assumption 



i tot i ^"^ 



that V o: 6 =^ e . Canceling e res\ilts in the equations: 



[-to^m + k + ito (c, + BS) ] V + (o) Lm - BS ) 6=0 [TOa] 



i/hmv + [-uF (I + L^m) + k + icoc ] 6 = [70b] 



Equating to zero the determinant of these two equations gives: 



[-w^m + k-j^ + iu) (c^ + BS)] [-t/ (l + L^m) + k^ + iwc^] 



-oj^Lm (w^Lm - BS^) = 



74 



