﻿and Critical Speeds of Rotors. 127 



which the axis of the rotor at its centre of gravity is 

 deflected from the stationary position, and k is the torque 

 required to produce unit angular deflexion. 

 The solution is 



^ = N 2 sin(^/^- 72 ^ .... (7) 



where N 2 and y 2 are constants the values of which depend 

 on the initial conditions. The frequency of the oscillation 

 is therefore 



iiv b or L c " where 



'-a/b («) 



As already indicated, we cannot at once deduce from this 

 what will be the actual second critical speed, owing to the 

 gyrostatic effects, but the result is of importance, as it 

 simplifies the calculation of the actual second critical speed, 

 as will be shown later. We shall in what follows call c 2 the 

 stationary second critical speed. 



2. It should be pointed out that there is a simple relation 

 between c x and c 2 which greatly facilitates the calculation of 

 the stationary second critical speed in those cases where the 

 centre of gravity of the rotor is midway between the bearings. 

 If 21 is the distance between the bearing centres and P 2 the 

 force exerted by the deflected shaft on either bearing, 



/n/r = 2P 2 Z. 



The angle y]r is very small so that the force P 2 is the same as 

 would be required to depress the shaft through a distance 

 -tyl if the rotor were held rigidly. Now we have seen that 

 the force M</ at the centre of gravity causes a transverse 



deflexion of ?/ = , and as yfr is small, 



so that Mg — ayfrl, 



also /n/r = 2P 2 Z and F 2 = ±Mg ; 



therefore fcyjr = MpZ, 



so that K = a-l 2 (9) 



and (8) becomes 7 / a .■ 



C2==/ V B' ^ ' 



