﻿and Critical Speeds of Rotors. 133 



frequency the same as the frequency for stationary vibra- 

 tions, and, as we shall presently see, ii' expressed in R.P.M., 

 is the same as the first critical speed. It is, therefore, inde- 

 pendent of the speed of rotation of the rotor, and there is no 



possibility of resonance occurring at -y^ X the critical speed 



as suggested by the erroneous method mentioned earlier in 

 this section. The free vibration itself is thus represented by 

 two components having the same frequency, but different 

 amplitudes; it is therefore a central ellipse, the centre 



being at the point y= . 



It will be shown later that the free vibration is damped 

 out by friction, so that it has no importance in practice. 



9. The forced vibration for the centre C of the shaft is 

 given by 



x= —. — . 2 cos cot, (24) 



Ci — G) 



we 

 £=_-—_ sin cot-a/cj 2 . . . . (25) 



6*i — ft) 



This naturally has the frequency corresponding to the 

 angular velocity of rotation o>, and has a maximum value of 



we c , 



7. tor each axis. 



Cj — CO' 



The motion of the centre is thus a whirl, the radius or 

 amplitude of which is proportional to the out-of-balance and 

 is zero when the rotor is perfectly balanced. A perfectly 

 balanced rotor, therefore, cannot whirl in the manner ex- 

 pressed by equations (24) and (25). 1 



The amplitude of the whirl is also proportional to -j 2 



and it thus becomes a maximum when * 



(i)= +Ci. 



(The sign + merely indicates that the rotation may be in 

 either direction.) 



This value of co gives the first critical speed, which is thus 

 the same as the stationary frequency for transverse vibra- 

 tions. Reasons will be given later why the radius of whirl 

 does not become infinite at the critical speed, i. e., why the 

 shaft does not break when the rotor reaches this speed. 



