﻿and Critical Speeds of Rotors. 147 



where R is the radius of whirl of the value : 



co e 



and tan<f> = \ 9 (51) 



r Ci— co z v y 



The lag of the displacement behind the force is in this case 

 proportional to the actual deflexion, and in this respect differs 

 from the result obtained in (47). This is of interest as it 

 shows that since the radius of whirling is for a given speed 

 dependent on the out-of-balance, the phase lag will be smaller 

 the more perfectly the machine is balanced ; in the former 

 case where the friction varied as the first power of the speed, 

 the lag was independent of the amount of out-of-balance. 



The maximum deflexion occurs, as before, when &> = c l5 that 

 is, when the speed is equal to the stationary critical speed. 



5. It is impossible to draw any conclusion from these 

 formulae as to the real angular advance corresponding to a 

 given speed, as it is not known how the frictional resistance 

 varies with the speed. We can, however, say that if the 

 machine is rotated first in one direction and then in 

 the other, the position corresponding to the out-of-balance 

 will be mid-way between the points of maximum deflexion. 

 When balancing a machine in the running condition it is 

 usual to hold a pencil or chalk against the shaft so that a 

 mark is made on the shaft at a point corresponding to the 

 maximum deflexion. If there were no friction and the speed 

 were not the critical speed, this mark would be in phase with 

 the heavy side of the rotor below the critical speed, and 180° 

 oat of phase with it if above the critical speed. But it will 

 be seen from (51) above that the actual position of the mark 

 depends both on the amount of friction and on the amount 

 of out-of-balance. At the critical speed the heavy side of 

 the rotor should be 90° out of phase with the mark on the 

 shaft, but the actual position will be uncertain, as the angle 

 varies rapidly with departure from the critical speed, and it 

 is not usually possible to judge exactly when the machine is 

 running at the critical speed. 



6. There is another reason why the position of the mark 

 on the shaft is somewhat uncertain. Referring to fig. 3, if 

 we ignore all other vibrations than that corresponding to the 

 variation in <j), we get by taking moments about G : 



MkM+crre sin 4> = (52) 



L2 



