﻿130 Mr. C. Rodgers on the Vibration 



in magnitude when the angular velocity n is equal to c x or 

 \f y. , and, further, that the variation in the magnitude 

 of r consists of a free vibration having a periodicity of 



l /in 



2tt V c?-n 2 ' 



This result for the periodicity of the free vibration would 

 lead to the conclusions that when the machine is not rotating 

 (n = 0) the periodicity is c l5 the same as for the stationary 

 transverse vibration, and that when running at the critical 

 speed (n = ci) the periodicity is zero. There would thus be 

 some intermediate speed where the periodicity of the free 

 vibration corresponds with the running speed, and resonance 

 might take place. This would occur when 



c 2 — n 2 = n 2 



We should thus be led to expect marked vibration when 



the running speed is — ^ x the critical speed. 



This conclusion and the argument on which it is based are, 

 however, erroneous. In the first place, the assumption is 

 made that co or 6 is constant and further the condition 

 is omitted that, as all the forces pass through O, the angular 

 velocity about must be constant, or r 2 = h, say. The 

 correct equations for the free vibrations are thus : 



jV( Cl »-#)r=b,i 



r 2 = h.) 



This does not admit of direct solution *, and it is simpler 

 to use rectangular co-ordinates, as we shall now proceed 

 to do. 



* The solution is, however, well known and is given in books on 

 Dynamics dealing- with Central Forces : — 



If p is the length of the perpendicular from the centre of force on the 

 tangent to the path, it is known that 



h 2 dp 



— —■ — err, 



p 6 ar 



giving — = a constant — err 2 , 



which is the pedal equation of a central ellipse. 



