﻿id Critical Speeds of Rotors. 



137 



Inserting this on the right-hand side of the equation and 

 sol vino- we get 



y=— ' il-f - , M „gos 2 cot y 

 ,y a [_ cr — 4M&) 2 J 



= -^'i 2 { 



1 + 



€/'M 



cos 2w/ 



(36) 



Cl --4:C0 2 J' ' 



There is thus a double frequency vibration about the 

 statically flexed position, which has a maximum when 

 (0= 2 c u that is, at half the critical speed. 



It is evident that such a motion must have a tendency to 

 arise if a rotor is unsymmetrical as regards its rigidity, for 

 in such a case when the shaft rotates the deflexion will be a 

 maximum or minimum twice every revolution, and if the 

 frequency of the consequent up and down motion is equal to 

 the critical speed there will be resonance ; this will be the 

 case whether the rotor is perfectly balanced or not. 



It is thus possible for a perfectly balanced rotor which 

 would be quite steady at the critical speed to show marked 

 vibration at half that speed. If the normal running speed is 

 above the critical the forces called into play at half the 

 critical speed will be very small and may give no appreciable 

 effect, but if the running speed is in the neighbourhood of 

 half the critical speed vibration might arise. 



Fig. 4. 



u 



,--Ti£t 



15. It is worth while to examine the motion a little more 

 fully as there will evidently be some vibration in the hori- 

 zontal plane also. Let C (fig. 4) be the position of the 

 centre line of the shaft and OA, OB two axes at right 

 angles rotating about with the same angular velocity co as 

 rotor. Let the co-ordinates of C be a and b with respect to 

 OA and OB and u and v the corresponding velocities along 

 those axes. 



