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



immovable body by the spring support 1. The floor may 

 be considered to be an elastic beam which deflects under its 

 own weight. If it is set in oscillation by a large enough 

 impressed force with a period equal lo its natural period, 

 dangerous vibrations may occur ; this critical frequency is 

 (o b =(b/M)K The natural period of a weight M which 

 deflects a weightless spring by / units is 2ir{f/g)i; the 

 frequency of the oscillations per minute = 187//* (if / is in 

 inches), or 300//* (if /is in cm.) and g = 32'2 or 981 respec- 

 tively. The same law holds good approximately for any 

 body on an elastic support, so that we get the critical speed 

 related to the amount of displacement, shown in fig. 2 : if 

 one wants a critical speed of 3000 rev. per min., then the 

 motor should be placed on a support which deflects 0*1 mm. 

 in consequence of the load. 



Now let us return to the question of critical speeds for 

 the whole system which we call (o 1 = k 1 co a and (o 2 = k 2 co b ; to 

 assist the calculations put m/M = i(r, and ci) a /co b = (p. r J hen 

 from eq. 5, after dividing by oo b 2 , 



h x and * 2 = i[(l + f)<£ 2 + l]±i(L 2 -4</> 2 )*. . (6) 



The way in which h x and k 2 vai T w ^ n ^ ie various values 

 of yjr and <j> is shown in fig. 3. The ordinates of the concave 

 curve show values of k u k 2 when m/M is the variable, but 

 (Oa/cob or w a /-H 6 =l'0. The series of convex curves to the 

 left show k 1 and k 2 with the ratio n a jn b as a variable for 

 particular values of m/M. In the curves njn b is put instead 

 of (o a /o) b , as the expression rev. per min. is more usual than 

 the expression angular velocity. There is no need to discuss 

 the curves in fig. 3 as the effect of altering the ratio m/M 

 or n a /n b is quite obvious. 



In practical cases we may have a motor of mass m placed 

 on an elastic support* such as cork pads, rubber, &c, which 

 can be altered in size or thickness ; these pads rest on a 

 floor of mass M, the deflexion of which is /, with a critical 

 frequency of oscillation n b . Suppose that the motor causes 

 obnoxious vibrations in the floor and that the motor speed 

 is n, and that this speed is near the resonant speed for the 

 whole system. If the motor causes big vibrations at all 

 speeds the only cure is to improve the motor ; but usually 

 such vibrations will only occur at particular speeds. In the 

 above case the one factor which can easily be altered is the 

 elastic support under the motor, the characteristic quality of 



* See Elec. Eev. vol. 84. p. 689, 1919 (June 21st). for references to 

 various supports. 



