[ 395 ] 



XXXV. Critical Speeds of Machinery placed on Upper Floors 

 of Buildings, as related to Vihration. By A. B. Eason, 

 M.A* 



I^HIS article discusses some of the relations between the 

 frequencies of vibration of floors and machinery placed 

 upon elastic supports on the floors, and suggests the principles 

 upon which one should seek to find suitable supports for the 

 machinery in order to prevent vibrations being communicated 

 to the building. 



The system under discussion consists of a motor resting 

 upon elastic supports such as rubber, cork, or springs, upon 

 a floor which is capable of deflexion. The mathematical 

 problem is to find the critical speed of two masses coupled 

 together by one spring, one mass being joined by another 

 spring to an immovable mass. The springs are assumed to 

 be weightless, which of course does not correspond to 

 practical cases, but the general solution of the problem is of 

 use to those dealing with the problem of preventing 

 vibrations from electrical or mechanical machinery pene- 

 trating to various parts of a building. The problem (fig. 1) 



Fio-. 1. 



MOTOR, mass m 



FLOOR, mass M 





p 



P-P c 



os wt 



( 



i 

 ^ 



1 





m\. 



\J 







v.. 



) > 





\ 







*(*■*) 



(,'• 





, 





\ M 





, N 



f 



\> 





: 



< 



nn 



ITT 



by 



It'll I 1 



is solved by Stodola (Steam Turbines, p. 355) and by Berger 

 (Gesund. Ingr. vol. xxxvi. p, 433, 1913). We have a motor 

 of mass m, subject to an internal unbalanced force P = P coso>f, 

 the angular velocity of the motor armature being w. 

 Between the motor and the floor is an elastic support, 

 represented by a spring, which requires a force a dynes (or 

 poundals) to deflect the spring through unit distance, 1 cm. 

 or 1 ft. (If the force =A kg. or lb., a — Ag.) The floor 

 has a certain deflexion due to its weight ami can be repre- 

 sented by a beam of mass M resting upon another spring. 

 * Communicated by the Author. 



