396 Mr. A. B. Eason on Critical Speeds of 



Every floor is elastic and bends under its own and super- 

 incumbent weights, and will oscillate if displaced from the 

 position of equilibrium. Let this system of springs and 

 masses be displaced from the equilibrium position owing to 

 the force P acting on the motor. 

 Choosing the following symbols : 



x = displacement of m from the point of equilibrium at 



time t. 

 y = displacement of M from the point of equilibrium at 



time t. 

 a = force to compress the motor spring unit distance. 

 b = force to compress the floor spring unit distance. 

 co = angular velocity of the motor, assumed constnnt. 

 o)„ = resonant speed of the motor resting on spring 1 alone. 

 u>t — resonant speed of floor vibration resting on spring 2. 

 (These resonant speeds correspond to the angular 

 velocity of harmonic motion whose period — the 

 natural period of the masses on their springs. 

 co = 2irn, « = frequency per second.) 

 a = mco a 2 , b = Mcob 2 . 

 ©!, G> 2 = the critical speeds of the motor, such that the system 

 as a whole is in resonance, and dangerous vibrations 

 may occur. These are ordinarily neither w a nor coi. 

 h — u\\u> a , h — tozl&b- 



The forces acting on m are P downwards and «(#—?/) 

 upwards. 



The forces acting on M are a{as— y) downwards and by 

 upwards. 



The equations of motion including the accelerations of m 

 and M are 



d 2 y d' 2 v 



M^=a(x-y)-b !/ , m-jp=F-a(x-.y). . (1) 



These equations neglect air friction which would bring 

 in a function of dyjdt probably of the form (dy/dt) 2 . We 

 assume that the oscillations of m and M will only be the 

 forced oscillations due to the impressed force P cos cot, as 

 the natural oscillations will soon be damped out by friction ; 

 the solutions are 



a?=P cos*rf-2 — - — - — ^7iv^^ w • ( 2 ) 



a- — {may- — a) (M&r — a — b) 

 y=P cosarf(-a/C), (3) 



