AND VIBRATION OF SHAFTS. 
305 
should have zero righting moment, and in the second, an infinite righting moment. 
In other words, in the one case there would be no tendency to make the pulley 
deviate from its natural plane of rotation, and in the other, any such tendency would 
be met by an infinite moment tending immediately to right it. In either case, there¬ 
fore—assuming whirling to take place at the speeds given by the limiting values of 
ac®—it would whirl in such a manner that the pulley still rotates in a plane perpen¬ 
dicular to the original alignment of the shaft. 
In fact, the period of ivhirling, corresponding to the inferior limit of the speed, is 
identical ivith the naturcd p)eriod of vibration of the light shcft under the given 
conditions. 
This may be easily proved independently.'^ 
The superior limit is double the inferior limit. 
The inferior limit may be taken as a first approximation to the period of whirl. 
25. Referring to equation [B], § 23, by giving cjk difterent values likely to be met 
with in practice, we get, for each value of cjk, a relation between w, the angular 
velocity of whirl, and c, the overhanging portion. Knowing, therefore, the particular 
value of c, the value of co may be readily calculated. 
The following are the results obtained in this manner from equation [B] :— 
* This may be seen as follows :— 
If W be the weight of the pulley, and e the force necessary to deflect it one foot, then t (the time of 
lateral vibration) is ^Tr^tWjge). To get e, if P be the load acting at a distance c from the shoulder, as 
in fig. 12, M the bending moment at a distance x from the shoulder, then 
M = P^, 
fZ2y/d.-!;2 = M/EI =Pa;/EI, 
where E and I have the same meaning as in the text. 
Hence, 
y = 
6EI 
-f- Aaj B, 
where A and B are constants of integration. When a; = 0, y = 0, and dy/da; = 0; whence B = 0, 
A = 0, and 
y = Pa;3/6EI. 
The deflection, therefore, at the weight is Pc®/6EI, and P = e when this is unity. Hence f = 6El/c^ 
and, therefore, 
t = natural period of lateral vibration 
= 27r (Wc*/6yEI). 
Whence 
a- = iTTjt = g (6yEI/Wcb = P732v/(yBI/Wcb. 
2 R 
MDCCCXCIV.—A. 
