o k(c +BS)(I+I + h^m - h Lm ) + c BS^ [h m - L (m + m ) ] 



,,,^ _ a o o o o o o o o r . 1 



oj - _ [T5b'] 



(c + BS) I (I + h m )+c[(l + l)(m + m)+ h'^min ] 

 a ooo o ooo 



If cc =0 and S = 0, Equation [75a] gives, as the natural frequency 



of vibration of the undamped system without the stream: 



2 2k/ I (m + m^) \ 



0) = U = — 1 + ;; r 5 1 



o I \ I^ (m + m ) + h'^mm / 



\ o O O O ' 



It should be noted that the mathematical complexity of Equations 

 [75a,b] is no greater than that of Equations [Tla,b] in the case of the 

 foil attached elastically to an immobile base, although here three variables 

 (v, 6, 6 ) and three corresponding equations are involved as against two 



in the previous case. The reason for the simplicity here lies in the 



2 r , -. 



occurrence of the factor w in all three terms of Equation [T'^J. It is 



not clear in what general class of systems this sort of reduction in 



complexity will occur. 



The smallest positive value of S for which both [T5a] and [T5b] are 

 true will imdoubtedly be a critical speed for the inception of flutter. 

 At a lower speed any motion will be damped out due to the BSv term and, 

 perhaps, other causes ifS>0;or, ifS = 0, provided at least one of 

 the constants c and c is positive, 



The case: c = 0. If c = but c^^ + BS > 0, Equation [T5b'] gives: 



/I + I + h^m - Lh m 



[76: 



/I + i + n m - bn m \ 



k_[ o o o o o \ 



A I + h2m / 



o o o 



* cc„ = means c = or c„ = or both, 

 a a 



85 



