﻿140 Mr. C. Rodgers on the Vibration 



The equations then are, putting the small quantities on 

 the right-hand side : 



x + c 2 x = (o' 2 e cos cot — c^ex?, 



y ~t" G iy — ( ° 2e *> m Mi —9 ~~~ c i 2 ^/- 



Neglecting the small quantities, the forced vibration is as 

 before given by 



we 

 X=-^-—: 2 COS(Dt,. ..... (40) 



9 



co e 

 y=J*t^p ainht-g/d*. . . . . (41) 



Inserting these values on the right-hand side o£ the 

 original equation, we get after some reduction : — 



x + c 2 x = {co 2 e - %<'i 2 ep 3 } cos cot — ^cfep 3 cos Scot, 



ij + Cl 2 y= -g + cfey^pP+yJ) + {co 2 e + c l 2 ep{'2i/ 2 t fp 2 ) } sin cot 



— Ci 2 €p 2 g cos 2cot - r jc^ep 3 sin Scot, 



where p= — J = 



c/ — OT 



and i /o =gjc l 2 . 



Solving these equations we get : 



* = /> ll-jc^e ^^ J cos tot-\c v 2 €-^_ — 2 cos 3u*, 



Ci 2 € 



2 • p 2 yo cos % w t 



C 2 -± 



* c i 2 e 



+ i • ~9 — rr~i P z sin 'dot. 

 C] 2 - yco 2 r 



Examining these terms in turn we find that the centre of 

 motion is now at the point 



#=0, 



instead of the point # = 0, y=- - y . This indicates that the 

 centre of motion rises, i. e., the shaft straightens out slightly, 

 as the vibration increases. 



