﻿128 Mr, 0. Kodgers on the Vibration 



and if B = M^ 2 2 , 



c ^h 



\/w ~N 



(12) 



and comparing with (4) we thus get : 



C\ Aug 



c 2 ~~ I 



We thus find that 



First critical speed (transverse vibration) 

 Stationary Second critical speed (oscillation) 



_ Radius of Gryration for the cross moment of inertia 

 Half the distance between the bearing centres 



This is a useful formula for calculating the stationary 

 second critical speed when the first is known, for cases 

 where the centre of gravity is midway between the 

 bearings. 



It shows that with cylindrical rotors of this type the 

 second critical speed must always be considerably above 

 the first, and the only instance in normal designs in which 

 the second critical speed could be lower than the first would 

 be that of a flywheel mounted on a short shaft. 



Section II. — Transverse Vibrations — First Critical 



Speed. 



A. Neglecting Junctional Resistance. 



1. It will simplify the treatment of this question if we 

 first consider the case of a rotor unimpeded by frictional 

 resistances set up by the air and then treat separately the 

 effects produced by friction. 



The conditions obtaining when a rotor is not perfectly 

 balanced and is rotating are illustrated in fig. 3, where 

 represents the position of the centre line c f the bearings, 

 and C the deflected position of the centre line of the shaft, 

 while Gr shows the position of the centre of gravity of the 

 rotor. thus gives the undeflected position of the shaft 

 centre line and 00 = r the shaft deflexion at any instant, 

 while 0Gr = <? is the error in the centering of the rotor; 

 M<7 is the weight of the rotor acting vertically downwards. 



The rotation of the rotor about its centre line, i. e. the 

 rotation imparted by the prime mover, is represented by 

 the motion of G around 0, i. e. by the rate of change of 0. 



