308 
MR. S. DUNKERLEY ON THE ’WHIRLING 
Hence, 
¥ = 
3/ 
a.CC 
1 - a {ch'^-jZl) 
SO that, for whirling to be at all possible (see Case IX., § 24, p. 304), acV^/3/ must 
be > 1 and < cjc + c'Jc — 1. 
If ac^c'^j'Sl be equal to the first or second of these quantities, the corresponding 
value of CO is the inferior or superior limit of the speed respectively. Moreover, the 
'period of ivhirl corresponding to the inferior limit of speed is identical ivith the 
natural period of vibration of the light she ft under the given conditions^' 
The superior limit is the inferior limit inultiplied by some function of the position 
of the pulley. With the same pulley on the same shaft the superior limit = inferior 
limit X \/(c/c' + c jc — l). 
* Tins may be seen as follows :— 
If W be file weight of the pulley, and e the force necessary to deflect it one foot, then t (the time of 
lateral vibration) is ^Tr^tWlge). To get e, if P be the load acting at distances c, c' from the bearings, 
as in fig. 13, M the bending moment at a distance x from the shoulder, then (fig. 12) M = x.Wc'll from 
A to B, and {I — x) . W cjl from B to C. Hence 
Pyldx- = aj.lV/ZEI, from A to B, 
(l~y'Idx^ = (Z — x) . Pc/ZEI, from B to C, 
E and I haviiig the same meanings as in the text. 
We get, therefore, 
y — 7 ^ . w + + ^’ A to B.(1), 
Ieji o 
Z/ = +E* + ICfromBtoC .(2). 
When 
.T = 0, y = 0 : therefore E = 0.(3). 
When 
X = Z, y' = 0 ; 
therefore 
P'= - - E7.(4). 
3EI ^ ^ 
When 
;r = c, y = y', and dyjdx = dy'jd.v ; 
therefore 
(F-F') + (E-E')c-^ .|- = 0.(5), 
(E - E') - 
El 
(ti). 
From equ-ations (3), (4), (5), and (6), Ave get 
