Shafts in the Neighbourhood of a Whirling Speed. 307 



It is seen from the foregoing values of x and y that the 

 terms representing the forced motion satisfy the relation 



x 2 +y 2 = constant. 



Accordingly the path of the elastic centre, when the 

 oscillatory motion subsides, is a circle round the axis of the 

 bearings of radius 



maco 2 



\/(c-mco 2 ) 2 + b 2 co 2 ' 



The co-ordinates of M being x-\- a cos cot and y + a sin cot, 

 and x and y having the values corresponding to the forced 

 motion as already given, it follows in the same way that the 

 path of M is also a circle round the axis of the bearings 

 when the steady motion is established. Thus 



maco 2 . . 



cos (cot — p) + a cos cot, 



v/( c -W) 2 + 6 2 G> 2 

 maco 2 



sin (cot— ft) + a sin cot, 



* \/(c-mco 2 ) 2 + b 2 co* 



OM 2 = tf' 2 + \j 2 — x 2 + y 2 + a 2 + 2a(x cos cot +y sin at), 



Now 



maco 2 _ 



x cos cot -\- y sm cot= . to o eos/3 



V (c— mco 2 )- -\-b 2 co 2 



and . bco 



tan/3= 2* 



Hence 



rw / c 2 + b 2 co 2 



V (c-mco 2 y + b*co 2 



If all the quantities save co are given the amplitude is 

 maximum, or 



\/(c-mco 2 f + b 2 co 2 

 is maximum, when 



2c 

 co= , . 



\/4wc — 2& 2 



This value of co approximates to the speed 



\/\mc— b 2 

 2n~i 

 corresponding to the free period, when b is small. 



