﻿132 Mr. 0. Rodgers on the Vibration 



angular velocity of rotation is constant. If x 1 y' are the co- 

 ordinates of C, 



,v = x' 4- e cos o>£, 



giving the equations 



■// = t/ -\-e sin cut 



it +Ci'V =W^COSG)^ (20) 



y' +cfy' = oo~e sin t»£ — <?, .... (21) 

 the solution of which is : 



9 



x' = TX l sm{C l t-y 1 ) + -p^GQ8<ot, . . . (22) 

 v' = N 2 cos (CV — 7i) h 2 — -^sin w^— ^/c*! 2 . . (23) 



('l — CO 



5. These equations are the same as for a perfectly balanced 

 rotor with a weight attached to it of such small value as not 

 to affect the position of the centre of gravity, or the down- 

 ward pull due to gravity, but producing a force of Mco 2 e. 

 In other words, we can treat the unbalanced rotor as if it 

 were a perfectly balanced rotor with a force Mw 2 <? attached 

 to it, and as this mode of presentation is easier to follow than 

 the former, we shall employ it in the remainder of the paper. 



6. It will be seen that the solutions for the motions of the 

 centre of gravity and the centre line of the shaft are the 



same, except that the former has an amplitude - 2 — 2 , and 



CO" € 



the latter an amplitude of — ^ - 2 , so that they differ by the 



amount e ; this shows that 00 and CG are in the same 

 straight line when not running at the critical speed. 



7. The solutions show firstly that the motion takes place 

 about the position 



Ma q 



which we have seen is the statically deflected position of the 

 centre of gravity of a perfectly balanced machine. It is 

 sometimes contended that as the speed increases the rotor 

 shaft tends to straighten out, but there is no indication in the 

 present treatment that this is the case. 



8. The free vibration 



x — N t sin {i\t — 71), 



y = N 2 cos fa*- 7O, 



is the same as for a perfectly balanced rotor and has a 



