﻿(55) become 



and Critical Speeds of Rotors. 155 



— B?; — kv + Aw£= — 7(o 2 ±h\pcot, . . (59) 

 Bf+tff + Ao^O, (60) 



giving the forced vibration 



9 <> <> 2 



V ~TT (c 2 2 -jpV) 2 -mV> 4 



input. . . (61) 



sin 



The oscillatory motion is thus a whirl the nature of which 

 depends on the speed. The whirl is a maximum when the 

 denominator is zero, that is, at two speeds, one on each side 

 o£ the stationary critical, and given by 



a ^ = ^— (63) 



There is thus a possible further second critical speed, 

 corresponding to the + sign, lower than that already found 

 corresponding to the — sign. 



For the ordinary rotating couple the direction of whirling 

 is, of course, always in the direction of rotation, whether the 

 speed is above or below the critical, and this is indicated by 

 the fact that, as will be seen from equations (58), the sign of 

 the amplitude in both planes changes, showing also that the 

 phase of the motion has changed by 180°. But for an 

 alternating couple about the x axis only, as will be seen from 

 (61) and (62), the motion in the horizontal plane changes 

 sign at each of the critical speeds indicated by (63), while 

 the motion in the vertical plane has a further change of sign 

 when p 2 co 2 = c 2 2 . It will be seen, if these changes are fol- 

 lowed out, that the whirling is in one direction below the 

 stationary second critical and in the opposite direction above 

 that speed, while at the stationary second critical the motion 

 is in the horizontal plane only, that is, at right angles to the 

 applied torque. 



11. The following example is given to illustrate the appli- 

 cation of the above curves and formulse. 



A rotor consists of a solid cylinder 30 ins. diameter and 

 60 ins. long, running in bearings, 107 ins. between centres. 

 From the deflexion diagram, the deflexion at the centre of 

 gravity is, say, *0087 in. or '0221 cm. 



