﻿148 Mr. C. Rodgers on -the Vibration 



If <j> is a small angle this becomes : 



M/q 2 <£ + am/)=0, 



the solution of which indicates a periodic motion having a 

 time o£ vibration of 



T = 2tt^ 



Mk, 2 



= 2 *V§e < 53 > 



In an actual machine for 3000 R.P.M. we shall have 

 figures of the order of: — 7^ = 50 cm., k A \e — 3 x 10 4 , 

 c 1 = 27rx30, say, while r may be of the order of 1 mm., 

 so that 



T __ 2it /50 x 3 x 10 



2vrx30 V 0-1 



= l'4Xl0 2 sec. or about 2 mins. 



As the time of vibration is very long in comparison with 

 that of the other vibrations occurring, it will be almost 

 unaffected by the latter, and the assumption that the other 

 vibrations can be ignored, which was made in deducing 

 (52), is therefore justified. 



For larger values of e corresponding to less perfect balance 

 and for deflexions of greater magnitude, T will be corre- 

 spondingly less, and will be greater the more perfect the 

 balance. If friction is ignored, r becomes infinitely great 

 at the critical speed and T becomes zero, and although this 

 can never be the case in practice, it is clear that T may have 

 a value of two mins. or more down to something considerably 

 smaller. 



In other words, if the rotor is disturbed from its position 

 of equilibrium by any chance external cause, it may take a 

 considerable time to settle down, and during that period the 

 position of the mark on the shaft will vary considerably 

 from its normal position. 



7. At the critical speed the lag is 90°, and the vibration 

 is also a maximum, but the sharpness of this maximum will, 

 as indicated above, depend on the frictional resistance to 

 whirling. In addition to this it will also be influenced by 

 the condition and design of the bearings, as the oil in tha 

 bearings exercises a considerable damping influence and aL-o 

 introduces a further complication as follows : 



When the speed is low and the vibration therefore small 



