Machinery placed on Upper Floors of Buildings. 397 



where C is the denominator in eq. 2 ; when C = 0, the 

 amplitudes of displacement may become large and the 



Eev.permin. Fig. 2. 



4000 



2000 



1000 

 800 



600 



200 



100 





























*<? 



















* 

































































































•04 



•06 -08 -10 



Deflection of 



spring , 



•20 

 cm. or inch. 



•80 10 



critical speeds will be found ; the solution for co 2 when 

 the quadratic equation C = is solved, is 



, fa a b~] , Tt o 4a?> ~p 



« 



where L= (a/M + a/m + 5/M), and so 



" 2 = i[^ 2 + "« 2 + ^ 2 ] ±i(L 2 -4o, a W) § - • (5) 



This means that there will be two critical speeds, such 

 that ft) 1 2 a) 2 ' i = ft) a 2 ft)6 2 , or k x k 2 = l'O ; we shall now seek to find 

 out in what relation co x and o> 2 stand to (o a and a> b . The 

 natural period of a mass m oscillating upon a spring 

 which requires a force a to compress or deflect it unit dis- 

 tance, is 2ir{jnja)^ ; the oscillations per second =(a/m)* ; (27r), 

 &) a = 27T (frequency) = (a/m)K In our problem this gives 

 the critical speed for the motor if it is connected to an 



Phil. Mag. S. 6. Vol. 38. No. 225. Sept. 1919. 2 E 



