y = (c-, + BS)/m. In either case, ultimately v = and the term - Lmv 

 in Equation [69^] disappears; and the v and 6 equations are independent 

 of each other. According to Equation [69b], 6 vibrates steadily with to, 

 as given by Equation [72a]. 



(b) If k = and either c = or k = or both, and if also at 



Z 2 1 



2 r 



least one of c and S is positive, then u = according to Equation [Tib' , 

 and no harmonic vibration with to 9^ is possible. 



When k = and c = 0, solutions of Equations [69a, b] are easily 

 found in the form of finite series in powers of t. The simplest of these 

 may be mentioned; namely, 6 = a and, if k, 7^ , v = BS a /k ; or, if 



■Do2 + 



k, = but c, + BS > 0, V = + B, where a and B are arbitrary 



1 1 ci + BS 



/ '22 



constants and t denotes the time. Ifk i 0, Y = - BSv + BS 6 = BS a = 



1 ij 



• 2 • ? • • 



k V. If k = 0, (c + BS)v = BS a, so that BSv = BS a - c v and F = c v. 

 Ill 1 L 1 



Thios , in either case, a steady lift on the foil serves to balance another 



force . 



2 



(c) If L = 0, Equation [Tla] does not contain BS and it is more 



illuminating to return either to Equations [70a,b] or to the more specific 

 equations of motion. Equations [69a,b], which now read: 



• 2 



mv + k v + (c + BS) V = BS G 



1*6 + k e + c^e = 

 2 2 



Now there are Just two possibilities for steady harmonic vibration: 



* Here, this definition of p, is restricted to the case at hand and 

 should not be confused with the definition of m generally used 

 througihout this report. 



TT 



