﻿126 Mr. 0. Rodgers on the Vibration 



The solution is 



^MVl^y^ ■ ■ ■ m 



where Nx and y ± are constants the values o£ which depend 

 on the initial conditions. The vibration therefore takes 

 place about the statically deflected position as a centre, and 



with a frequency of vibration of — c l9 where 



c '=Vm w 



This vibration takes place in a vertical plane and may be 

 considered as the resultant of two vectors rotating in oppo- 

 site directions, each with an angular velocity of \/ ^ . If 



o- and M are expressed in eg s. or f.p.s. units, this angular 

 velocity will be in radians per second and since from (1), 



^ is numerically equal to -^ , the speed of either of these 

 vectors in revs, per minute will be _: — A / — . If, further 



2-7T V y Q 



g and y are in c.g.s. units we have the formula 



WTJ ,, 60 /Ml 300 . , . /K . 



where y is the static deflexion in cm. 



3. It will be seen afterwards that, as is well known, this 

 formula gives the first critical speed in R.P.M. ; this is to 

 be expected, as the out-of-balance forces will then resonate 

 with the natural free vibrations, with the result that the 

 latter will become of considerable magnitude. 



B. Oscillatory Vibrations. 



1. If the rotor is twisted about its centre of gravity so 

 that the deflexion is in a vertical plane, and is allowed to 

 oscillate freely, the motion is represented by 



B^ + /c^ = 0, (6) 



where B is the cross moment of inertia, that is, the moment 

 of inertia of the rotor about a line through the centre of 

 gravity at right angles to the shaft, yjr is the angle through 



