﻿and Critical Speeds of Rotors. 135 



As a first approximation we substitute in (30) the values 

 already found tor .t* and ?/, as given in (24) and (25), for the 

 forced vibration and thus obtain (neglecting u in comparison 

 with 1 radian) : 



2 



M/iyzV + o-tf 2 — 2 2 a = —<reg]c 2 cos cot, . . (31) 



giving for the forced vibration (the free vibration may be 

 ignored as it will be damped out) : 



e(c x 2 -co 2 ) ,_ OA 



Z ' = * 2 2 27 2/ 2 2\-gCOSO)t. . . (.32) 



At the critical speed co = c^ and ^ = 0, z. 0., the angular 

 velocity of rotation is constant ; at other speeds than those 

 in the neighbourhood of the critical, the term co 2 e 2 c 2 mav De 

 neglected, as e 2 is much smaller than kx 2 , so that 



u = — 7 -„ . - 9 co§ cot (33) 



faf CO 2 



The variation in the angular velocity has thus the same 

 frequency as that of the rotation itself, but is very small in 

 magnitude, as will be seen from the following figures. For 

 a turbo-generator rotor balanced to about 1 oz. at radius 

 A'i per ton weight of rotor, e\k x will be about 3 x 10 ~ 5 , and 

 for a machine to run at 3000 R.P.M., £ x will be about 

 50 cm., and co is, say, 2-7TX 50, 



3xl0~ 5 981 



U= ^r — x r—s — ^ — ^ cos cot 



oO 477-- x 50 x 50 



= 6xl0" 9 coso)£, 



that is the vibration is very small, the amplitude being only 

 6 X 10" 9 of one radian. 



It should be noted that u is proportional to e and to g ; 

 this variation therefore arises through the action of gravity 

 on the rotor when not perfectly balanced, and the variation 

 will be absent if the balance is perfect. 



Substituting the 'value of a in (28) and (29) in order to 

 find the effect o£ the irregularity on the displacement : — 



e 2 

 & + c 2 x = co 2 e cos cot + 7 2 g sin cot cos cot, 



e 2 

 y -f c{ 2 1/ = co 2 e sin cot— — q cos 2 cot — g, 



