2 ^1 



(a) c^ > 0; k, > 0, c, = 0, S = 0: u) = — •,6 = 



2 ' 1 ■ ' 1 m 



2 ^2 



(b) c^ = 0, k^ > 0: 0)'' = — 



Case (b) is not a typical flutter case since S may have any value, but 



it may serve to illustrate in a simple case a typical feature of flutter 



2 

 vibrations in that the term BS 6 forces a vibration of v in shifted phase, 



the energy abstracted by damping being furnished by the stream. This 



feature may be worth exploring in detail. 



By properly choosing the zero of time, 6 can be expressed thus: 



6 = C sin ojt ; CO = k /I 

 where C denotes an arbitrary constant. The accompanying forced vibration 

 of V can be written, in terms of amplitude factors a and 6 that remain to 

 be fovind, as follows: 



V = aC sin ut + 6C cos u)t 

 hence 



V = - ojBC sin (jot + ooaC cos wt 



■•2 2 



V = - 10 aC sin cot - oo 6C cos wt 



When these expressions are substituted into the mv equation, the sin tot 

 and cos lot terms must balance separately. Hence, after canceling out 

 C sin (ot or C cos tot: 



2 2 



(k - mio ) a - (c + BS) (o6 = BS (for sin tot) 



2 



(c + BS) too + (k - mtj ) 6 = (for cos tot) 



78 



