If.S = but Equation [Tib' ] furnishes a positive value for w , then v and 

 6 will vibrate in a certain ratio, decreasing to zero with time if either 

 c. > or c > 0. If, then S is made slightly positive, the term BSv will 

 predominate over - BS 6 in Equation [69a] and will itself damp out the 



motion even ifc =c =0. AsS increases further, however, the damping 



2 

 effect may decrease because of the increase in the value of the - BS 9 



term until, finally, at a certain speed S harmonic vibration may again 



become possible. Then S is the critical flutter speed. As S increases 



c 



above S , the vibration will probably become imstable. 

 c 



In special cases further conclusions can be drawn, and it is these 

 conclusions that may give a certain interest to the case of a foil movmted 

 on a rigid base : 



(a) Assume 02 = but k2 > , L ^ , and either S > or C-, > or 

 both. Then the factor c, + BS can be canceled out of Equation [Tib']; it 

 is found when substituting for w that k k disappears from [Tla] ; finally 

 Equations [Tla,b'] can be written: 



P kp P Lmk.p p 



i/ = ^-— ; BS'^ = ^ = Lmo)'' [T2a,b] 



I + L^m I + L m 



Inspection of Equations [69a, b] shows why this case is so simple. If 



P 

 BS has the value specified in Equation [T2b], the two 6 terms in Equations 



[69a] cancel each other. Hence^ if k, > 0, v must either be zero or 



execute a damped oscillation ending in zero. Or, if k =0, v must 



ultimatel.v reduce to a constant value since the only other possible 



solution of [69a] is then the nonharmonic solution v «: e , where 



76 



