﻿150 Mr. C. Rodgers on the Vibration 



speed of the machine, but may also be of double frequency 

 arising through any of the causes we have discussed. 

 Further, when the normal frequency oscillation has estab- 

 lished itself, a double frequency oscillation may start up 

 owing to bipolar asymmetry or some of the other causes 

 mentioned in connexion with the transverse vibration. It 

 is therefore necessary to consider in the oscillatory motion 

 forces both of the actual frequency of rotation and of 

 double frequency. 



5. We have found that the transverse motion can, with 

 sufficient accuracy, be considered the same as for a perfectly 

 balanced machine with an out-of -balance force attached to 

 it. In the same way we shall treat the oscillatory motion as 

 being due to an out-of-balance couple of the frequency corre- 

 sponding to that of rotation or a multiple of that frequency 

 acting on an otherwise perfectly balanced machine. 



6. In the diagram fig. 5 let G be the centre of gravity of 

 the rotor and GL the direction of the centre line of the rotor 



twisted from its normal position by an angle f = L(}Z, where 

 GZ is the direction of the centre line when not vibrating. 



The direction cosines of the centre line GL with the axes 

 GX, GY, and GZ are respectively f , 77, and J ; if L is a 

 point at unit distance along the shaft from the centre G, 

 then J, 7), and f are also the co-ordinates of the point L, as 

 shown on the diagram. 



If the moments of inertia of the rotor about the shaft 

 centre line and about the line at right angles to it, through 



