358 
MR. S. DURKERLEY ON THE WHIRLING 
61. The only alternative method is to consider the effects of the shaft (ivhatever he 
its mode of su'p'porh) and each of the pulleys [ivhatever he their number, position, and 
size) separately, and so ohtahi the whirling speed for each on the assumption that all 
the others are neglected. By means of an empirical formula the whirling speed, ivhen 
the effects of the shaft and of all the pulleys are taken into account, may he calculated 
from the separately calculated ivhirling speeds. 
62. The particular form of the empirical formula was found as follows ;— 
If a weight be supported by a spring which requires e pounds to stretch it one 
foot, then the number of vibrations which that weight makes per second is 
N, = ff{ge/Wf 
The number of vibrations which a second weight Wg (attached at the same point 
of the spring as the weight W;^) makes is 
N.,= y(i/e/W,); 
and the number which the combined weight (Wj + Wg) would make is 
N= =i 
V lw, + wj v/lNj-'+Nr') x/(N,= +N,2) 
In the same manner this formula would be strictly accurate in the case of a rod, 
however supported, provided that any concentrated loads which it might carry could 
be supposed concentrated at the same point. For example, if three loads be con¬ 
centrated at the same point of the rod (the effect of the rod being neglected), and 
if the number of vibrations which each makes per second, when assumed independent 
of the others, be N^, Ng, Ng, then the number of vibrations of the three together will be 
hbNgNg 
^(Ng^Ng^ -f NVNd -f Nd-Ng^) ’ 
and so on for any number of loads. 
If, however, the loads be concentrated at different points, the above formula will 
not be strictly true; for, in addition to the number of vibrations varying inversely as 
the square root of the weight, the value of e will vary with some function of the 
distance of the weight from the point of support. 
In the same manner, in the whirling of shafts, if N^, Ng, Ng be the whirling speeds 
due to three pulleys when each is considered independently of the remaining two, we 
have (§§ 25, 27, 32, 36, 37, 41, 42, 48, 52, 53 and 57), since w oc ^^(I/Wc^) and 
therefore as cTj ff(Wc^), where d = diameter of shaft, 
Ni = (^1 
ch 
v"(Wicd) ’ 
N.. 
cP 
d~ 
WWW) ’ WWsW ’ 
