If a,p = but a22_ ¥ 0, Equations [U9a,b] give v = 0, F = ©/a . 

 Then [i+6a] requires hmto = (l/ag^^) - BS and also, to remove 6, CS = 0. 

 IfCyo, S=0; ifC=0, any S may occur together with the proper u. 

 Equation [i+6b] merely gives the value of G. 



If a = but a # 0, then Equations [l+9a,b] give 9 = 0, G = v/a . 

 From [i+6b], hmu^ = l/a , LBS = 0. Equation [U6a] gives F. 



If a = a^-, = 0, then v = 6 = and, hence, by Equations [U6a,b] 

 F = G = 0; and by Equations [l+7a,b] v = 6^ = 0. Thus, if s = s^ = a,- = 

 ap, = 0, there can be no harmonic vibration at the w for which these 

 parameters were calculated. 



Examination now shows that all of these conclusions as to u and S 

 follow also from Equations [53c, d] when D = and the parameters have the 

 specified values. 



The general validity of Equations [53c, d] is thus established. 



6.k DISCUSSION OF DAMPING EFFECTS 



Components of force or moment on the foil that are proportional to 

 velocities have a damping effect, positive or negative. Included here are 

 the force - c-,v and the components - BSv and CSB in F^ , assumed to act 

 through the 6 axis, whose velocity is v. Then there are also the moment 

 - Cq6 and the terms - LBSv and - E^Se in M , acting at the velocity 6. 

 These forces and moments do work on the beam- foil system at the total 

 rate (see Section 8.3 of Reference 3 and Appendix H of Reference h) : 



[- (c^ + BS) V + CSe] V + (-02© - LBSv - E Se) 6 



58 



