﻿144 Mr. C. Rodgers on the Vibration 



putting co 2 = c 1 2 /2, 



c 1 2 -2 W{l + m) + r 2 2 } -f4c 2 2 = 0, 



that is c 2 = \ ' — g — c„ 



so that if, for example, M 2 were large in comparison with 

 Mi and m is therefore small, 



c 2 = c 1 /V2 = c l xO-707, 



but if, as in the former example, M 2 = 5M l5 



c 2 = CjL x 0-836. 



B. Transverse Vibrations with Friction. 



1. The frictional resistance opposing the motion of the 

 rotor may be considered to consist of two parts. The first 

 part opposes the rotation of the rotor about the centre 

 line, C, of the shaft, and this is counteracted by the torque 

 supplied by the turbine, or, if the turbine is cut off from the 

 steam supply, it tends to bring the set to rest ; it has no 

 retarding effect on the whirling. The second part opposes 

 the whirling only, and it is with this that we have to deal. 



2. It is not known how the frictional resistances opposing 

 the whirling vary with the speed, but it seems likely that 

 they vary with the square of the speed at least. We shall, 

 however, first consider the case where the resistance is 

 assumed proportional to the first power of the speed, as the 

 motion is then simpler to work put, and there is an inter- 

 esting electrical analogy, which enables the motion to be 

 more readily followed. 



The resistance to whirling is in opposition to the path of 



ds 

 the rotor centre, so that if -r- is the speed of the centre in 



any direction, the frictional resistance is M//,f , ) , and the 



(ds\ n dx 

 7 - ) t- and 



f(xs\'" av 

 M-fjulj.) -/-, that is, Mfzs^x and M/^" 1 ?/, where fi is a 



constant. 



3. In the particular case we are about to consider, n — 1, 

 and the components are therefore M/xx and Mfiy. 



