﻿146 Mr. C. Rodgers on the Vibration 



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The forced vibration is permanent and has a value which 



is proportional to the applied E.M.F. or out-of -balance force. 



The vibration is a maximum when co = c u that is the critical 



speed is the same as the stationary critical, as in the case 



where friction is ignored ; the amplitude of the vibration at 



.Ee.. 

 the critical speed is p or — , that is, is equal to the applied 



E.M.F. or out-of-balance force and inversely proportional 

 to the resistance. 



The frequency of the forced vibration is the same as the 

 frequency of the E.M.F. or of the rotation, and the charge 

 or displacement lags behind the E.M.F. by an amount 

 depending on the frequency or speed and on the capacity 

 and inductance or elasticity of the shaft and mass of the 

 rotor. 



The lag is zero when the frequency is low, but increases 

 to 90° at the critical speed, which, as will be seen, is that 

 corresponding to the natural frequency of the system, while 

 at very high speeds the lag increases to 180°, in other words 

 the force is in opposition to the displacement. The change 

 is similar to that occurring when there is no friction except 

 that in the present case the change is gradual instead of 

 taking place suddenly at the critical speed. 



It will be noted that in both cases the vibration takes 

 place about the statically deflected position as a centre. 

 It is evident in both cases that a large static deflexion 

 would increase the tendency to break down, in the one case 

 by puncture or flashing over of the condenser, and in the 

 other by fracture of the shaft. 



4. If the resistance to whirling is proportional to the 

 square of the speed, that is n = 2, the equations are : 



x + fi'sx 4- CiX — co 2 e cos cot, 

 y + fisy + c 2 y = co' 2 e sin cot — g. 



The free vibration (i. e. the vibration when g = or the 

 rotor is perfectly balanced) cannot be expressed in simple 

 terms, but as it will be damped out, as before, it is not of 

 interest. 



The forced vibration (i. e. the vibration due to the rotor 

 being out-of-balance) is given by 



.£ = R cos (&>£ — <£), (48) 



y = R sin (cot - (/>) — g/ Cl 2 , .... (49) 



