﻿136 Mr. C. Hodgers on the Vibration 



or 



1 e 2 

 x + c?x — cw 2 e cos oot-i --=— g sin 2g)£, 



V + Ci 2 y = a> 2 e sin cot- - -^ g cos 2o>*-^ 1 + -^j . 



The solution is (for the forced vibration) 



co 2 e cos cot 1 e 2 1 . _ , rt JN 



I= l?^ + 2V^i' SmH (34) 



&> 2 £ sin &>£ 1 e 2 1 _ g /^ 1 e 2 \ , „ N 



The irregularity has thus two effects on the main whirl : 

 firstly, the static deflexion is increased by a small amount, 

 and secondly, there is superimposed on the main whirl a 

 ripple o£ double frequency, which rises to a maximum at 

 half the critical speed. But the effect is very small, and may 

 not be noticeable ; in any case, as the double frequency effect 

 depends on e it cannot appear when the machine is well 

 balanced. 



There is, however, the possibility that a rotor which is not 

 perfectly balanced may show vibration at half the critical 

 speed due to the action of gravity, although gravity would 

 produce no such effect at the full critical speed. A vertical 

 spindle rotor is not of course subject to the action of gravity 

 in this sense, and if it vibrated specially at half the critical 

 speed, the cause must be sought elsewhere. 



13. We shall now consider some further possible causes of 

 subsidiary critical speeds or speeds where marked vibration 

 may appear other than the normal calculated critical speed. 



14. An important case is that of a rotor slotted for a 2-pole 

 winding or with a shaft in which a key-way is cut, where 

 the rigidity of the rotor is greater in one direction than in a 

 direction 90° away, so that if the shaft is rotating, the stiff- 

 ness in the direction of any one of the axes is not a constant er 

 but a 4- e cos 2cot, where e is small in comparison with cr. We 

 shall assume that the rotor is perfectly balanced (e = 0) and 

 to simplify the examination shall first consider the vertical 

 motion only. The equation is : 



M.y + cry=— M^— yecos2a>t. 



The first approximation is : 



y=— — = -#i- 



