﻿and Critical Speeds oj Rotoi-s. 139 



method of approximations ; it shows that there is a similar 

 motion of equal magnitude and 90° out of phase along the 

 horizontal axis, so that the motion is a circular whirl of 

 double frequency, which rises to a maximum at half the 

 critical speed. 



16. When the rotor is out of balance the equation for the 

 vertical motion is 



Mi/-f cn/ = M&> 2 £sin cot — Mg— ye cos 2cot. 



The first approximation is given by (25) and inserting this 

 on the right-hand side of the above equation we get : 



1YT 

 M y + crfj = Mmh sin cot — Mg + — e cos 2cot 



. e Mco 2 e , . . ' , / qo n 



+ 5" Tv/r «{sin m* --sin .•5ft>£|. \^) 

 ! (7 — Ivico 



The first three terms on the right correspond to the main 

 whirl and the double frequency whirl already dealt with. 

 The last term on the right will give in the solution a triple 

 frequency vibration, viz. : 



1 Mor<? e . 



_ n M 2 nil/I 2 sin 6wt i 



I ff-Mr a — 9 Mar 



which has a maximum value at ^ the critical speed. This 

 vibration is, however, proportional to e, the out-of-balance 

 force, and cannot arise in a perfectly balanced machine. 

 The remarks made as to the limited conditions under which 

 the double frequency vibration might arise apply with even 

 greater force to the triple frequencv vibration as the damping- 

 effect of friction will be correspondingly greater. 



17. Another case of interest is that in which covers or 

 sleeves are mounted on the rotor, or the rotor has slots in 

 the periphery for an exciting winding, closed by pressed- 

 in keys ; the closeness of these force tits will vary with the 

 deflexion, and the deflexion of the shaft may therefore be 

 not quite proportional to the force applied, i. e., the force to 

 produce a deflexion % will not be o\r, but say a(x + ex 3 ), 

 where e is small in comparison with unity. (The expression 

 for the force must contain odd powers of x only as the 

 rigidity is symmetrical, that is, the same numerically for the 

 same numerical value of x whether x is positive or nega- 

 tive ; if even powers were included this could not be the 

 case as an even power of x is always positive even if ,c itself 

 is negative.) 



