﻿and Critical Speeds of Rotors. 145 



The equations of motion are therefore 



Mai -I- M/Xci; + ax = Ma) 2 e cos tot, 



M.y + Mfiy + ay = Mco 2 e sin atf — M#. 



Electrical engineers will notice the similarity between 

 these equations and 



Jj'q -f ~Rq + p- g = E sin o)£ + E 0j 



which holds for a circuit comprising an inductance L, a 

 resistance JEfc, a capacity K, an alternating E.M.F. of 

 maximum value E, and a steady E.M.F. E , q being the 

 charge in the condenser at any time. Thus, mass is equiva- 

 lent to inductance, capacity to deflexion per unit force, and 

 applied E.M.F. to applied mechanical force. 



The solutions of these equations are, as is well-known : — 



q = ~$e~ t > T sin (pt — <f>) 



E • / , ■ i E \ 



H r /-. it- t — ro — TV^r sl n ( cot — tan zmy 7 — 



V{(l/Ka)-La)) 2 +li 2 } V 1/Kft) — La>/ 



+ KE , (45) 



^ = Ne _i/T sin (pt — <f>) 



Mco 2 e . / ' , _, Maw \ 



+ ^{(,-M.f+MV^} sm r " tan " ^Mo?j 



-M<?/<r, (46) 



that is, 



r/ = Xe~ f/T sin pt — <j> 



+ /7-5 — -072 , » 2 sin ( wt -tan" 1 2 A6> 2 ) — ^/q 2 , 



, • • • ( 47 ) 



where 2L 2M 



^ = Vl/LK-1/T 2 or W/M-l/T 2 , 



There is therefore in both cases a free vibration having a 

 frequency slightly less than the natural frequency of the 

 system, but independent of the frequency of the applied 

 E.M.F. or of the speed of the rotor. This vibration is 

 damped out by friction. 



Phil. Mag. S. 6. Vol. 44. No. 259. July 1922, L 



