﻿142 Mr. 0. Rodgers on the Vibration 



classes may be noted, firstly where there is an irregularity 

 in the torque applied to the shaft, and secondly where there 

 is resonance with masses outside the machine. 



20. Irregularities in the torque driving the machine may 

 arise, for example, through variation in the steam admission 

 or through a fluctuating electrical load. 



The result of fluctuation in the torque will be a corre- 

 sponding fluctuation in the angular velocity of rotation so 

 that the angular position, instead of being cot, will be 

 cot-\- € sin pt, where e is a small angle and p is an angular 

 velocity corresponding to the frequency of the disturbance. 

 Then cos {e sin pt) =1 and sin (e sin pt) = e sin pt. The equa- 

 tions then become, taking the small quantities on to the 

 right-hand side : 



x -j- c 2 x = co 2 e (cos cot — e sin pt sin cot) , 



jj -f c-iy = (o 2 e (sin cot 4- e sin pt cos cot) — g ; 

 that is, 



x + c 2 x = &) 2 <?{cos cot + e/2 (cos o> +■ p . t — cos co — p . t) }, . (42) 

 y -\- c^ 2 y — co 2 e{sin cot + e/'2{>in co-rp . t-\-s\\\co — p . i) }. . (43) 



The main vibration is the same as before, but there are 

 two small vibrations superimposed ; the one has a frequency 

 corresponding to co-\-p and a maximum when <w = 6 , 1 — p, the 

 other a frequency corresponding to co— p and a maximum 

 when a>=c 1 + p. 



This shows that the vibration may have a maximum at 

 speeds corresponding to the sum of and to the difference 

 between the critical speed and the speed corresponding to 

 the frequency of the disturbing fluctuation. So that if dis- 

 turbed running show itself at such a speed that it cannot 

 be otherwise explained, a cause may be sought for in this 

 direction. 



21. The other variety of resonance mentioned is that 

 where, for example, the foundations are not sufficiently 

 rigid and the machine as a whole is vibrating so that there 

 is resonance between the rotor on its shaft and the machine 

 on its foundations. A similar case would be that of a machine 

 rotating in or near a building which itself shows marked 

 vibration, possibly in certain parts only, corresponding to 

 the vibration of the machine. Both these cases are similar 

 in principle and may be illustrated by supposing the whole 

 machine to be mounted on foundations having some elas- 

 ticity. If then M x is the mass of the rotor and M 2 the 



