﻿and Critical Speeds of Motors. 143 



effective mass o£ the machine and that part of the founda- 

 tions which moves with it, and the forces required to give 

 unity deflexion are in the two cases cr, and a 2 respectively, 

 the equations of motion are as follows : 



Mii/! + o-, (y, - y 2 ) == M 1& rV sin pt, 



M 2 i/ 2 + 0-22/2 - o"i (yi ~ Vi) = 0. 

 giving for the amplitude of the forced vibration 



• • (44) 



where C\ 2 = 



a 



M,' 





u> 2 e(cx 



2 m + 



2 

 '2 ~ 



CO 2 ) 



CO 4 - 



■ (o 2 {Ci 2 m-\-c 2 



+ «I 



2 J + CiV 



9 



°2 



~M 2 ' 



in = 



M 2 



• 



Points of marked vibration may thus occur at either of 

 two frequencies given by putting the denominator = ; 



M 



these frequencies will therefore depend on the ratio ~ as 



well as on C\ and c 2 , and may thus have almost any values. 



For example, if c 2 = i\ and ?n = 0'2, i.e., the mass of the 

 machine and foundations is five times the mass of the 

 rotor — 



fi> s g(l-2c 1 2 ~o> 2 ) 

 • y ~a> 4 -co 2 (• 2 2 x2•2 + r I 4, 



and a maximum occurs when 



ta=c 1 xl'25 or c 1 x 0*80, 



that is, at speeds 25 per cent, above and 20 per cent, below 

 the calculated critical speed. 



If M 2 , the mass of the machine and foundations, is very 

 large in comparison with M 1? the mass of the rotor, the 

 denominator is very nearly equal to (co —c{) X (a> — c 2 ), which 

 shows that in such a case the two speeds where marked 

 vibration may occur nearly correspond to the natural fre- 

 quency of the rotor and of the machine and foundations 

 respectively. But as the numerator is also small the vibra- 

 tion might not appear if considerable friction is present. 



It: vibration should occur when ft) = c ] /v / 2, which is, as 

 mentioned above, sometimes thought to be a critical speed, 

 this might indicate that there was resonance with the founda- 

 tions or some structure outside the machine, in which case, 



