- (cj_ + BS) v^ + (C - LB) Sve - (c^ + Ej^S) 9^ 



This expression may be positive or negative. If, however, the Theodorsen 

 relations are used, namely: 



B = iTpb£„ 



C = B(|b-He) ; L=|b-e; \=B(|b+ej2 



then C - LB = B (b + 2e) , and the rate of work is: 



-c-^v^ - c^e^ - BS [v^ - (b + 2e) ve t/' 1 b + ej 6^ 



= - c^v^ - 026^ - BS V -( I b + e) ej 

 With these values, therefore, if c-l 2, 0, C2 i , and BS > , there is 



no gain but, in general, a loss of energy from the beam- foil system during 



harmonic motion. Somehow, such a loss must be compensated for through 



2 

 work done by the BS terms in F and Mg. 



If s = 0, the c, and Cp terms alone will caiise a loss of energy unless 



c = Cp = ; and when S = 0, the equations of motion are correct for any 



type of motion (whether harmonic or not). Consequently any existing 



motion must die out unless maintained by additional forces. The same 



conclusion will probably hold at first as S increases from 0. Eventually, 



however, the rate of loss may decreeise and become zero at a certain speed 



* The Theodorsen terms are derived only for harmonic motion. If, however, 

 S = 0, the Theodorsen terms disappear and, in general, the equations of 

 motion hold. General equations are needed to deal accurately with 

 damped motion. 



59 



