Shafts in the Neighbourhood of a Whirling Sjieed. 311 



It is worthy o£ note, however, that it* a shaft be run 

 quickly through its whirling speed to a higher working 

 speed, there may not be time for serious vibration to take 

 place ; in fact, the value of t may be so small that the 

 amplitude of this forced vibration ^acot may never be great. 



8. Apart from a knowledge of the numerical value of b in 

 any particular calculation, we cannot obtain the exact value 

 of the amplitude of vibration ; but, as we have seen, it is 

 only in the close proximity of the whirling speed that the 

 damping seriously modifies the amplitude, and we may omit 

 it at all except such speeds. 



Also when not too close to the whirling speed we may 

 regard the displacements of the mass centre and elastic 

 centre as taking place along the same radius-vector from 

 the axis of rotation, which likewise may be obtained from 

 the general solution by putting b = 0. 



Let u be the ultimate amplitude of the vibration at the 

 elastic centre, then 



maw 2 . maco 2 



\/(c-mco 2 ) 2 + b 2 co 2 ' ±{c-mw 2 ) 

 Put k — \J — = speed of free vibration. 

 If a) < k, we have 



a 



u = 



1^ 1_ 



2 k 2 



CO 



The centrifugal force is 



F = mco 2 (n + a)= — =- =mk 2 u. 



If Q) > k, we have 











a 











u 





1? 









1 





1 











F 





CD 2 





F= 



: mco 2 (u 



— 



a) = 



1 



ma 



1 



CO 2 



and ¥ = ma> 2 (u — a)= -z —=mk 2 u. 



Plotting these results against the speed <w, we see that, 



