308 Prof. H. H. Jeffcott on Lateral Vibration of Loaded 



At speeds in the neighbourhood of this value of w the 

 vibration, caused by a given eccentricity of mass centre, 

 may be excessivelv great. This is the whirling speed of the 

 shaft. 



3. Looking next at the value of /3 which determines the 

 phase of the displacement of the mass centre relatively to 

 that of the elastic centre, we see that since 



tan 8= — s, 



c—mw 



the value of yS is when &> is zero. The value of /3 changes 



IT 



with increase of co, until when m(o 2 = c i /3= x , or the phase 



of the' mass displacement has now shifted by — from that 

 corresponding to very slow rotation. 



Above this value of co, the phase changes further, and at 

 very high values of a), /3=tt. 



Now the value of b is usually small, and on closer exami- 

 nation it will be seen that the whole phase-change through 

 an angle it takes place practically entirely between a speed 

 very slightly below and another very slightly above the 

 critical speed. In other words, this change of phase takes 

 place mainly within a comparatively small number of 

 revolutions per minute on either side of the critical speed. 



Thus at speeds appreciably below the whirling speed the 

 shaft rotates with the mass centre farther from the axis of 

 rotation than the elastic centre, while at speeds appreciably 

 above the critical speed the mass centre is closer to the 

 axis. 



4. This result can easily be obtained directly by considering 

 the steady motion in these extreme cases and omitting the 

 damping altogether, as is well known. 



p.i 



k n 



._19_ 



Fia.2. 







Thus in fig. 2, let OE = ^, EM = a, where 0, E, M repre- 

 sent the axis of rotation, the elastic centre^ and the centre 

 of mass. Equating the centrifugal force to the restoring 

 elastic force acting on m in the assumed steady motion, we 



obtain s/ , mtfa 

 mcor{x-\-a)-=-cx, or x= j? 



