336 
MR. S. DUNKERLEY ON THE WHIRLING 
whirling speed, due to both pulleys, will be Nj/y/2 (see § 62), so that the resulting 
whirling speed for the two pulleys will be proportional to 
2-05y(^EI/W^3), 
But, since the two spans are equal and similarly loaded, it is clear that there is no 
bending moment on the middle bearing. Consequently, as in the case of an unloaded 
shaft, § 15, the spans will whirl independently of each other, and the actual speed 
of whirl will therefore be proportional to 
2'45y^(^EI/W^^).(see § 27), 
where E, I, W and I have the same values as before. Hence the whirling speed, as given 
by the formula used in the investigation, is only 84 j)er cent, of the actual whirling 
speed of the pulleys. When the shaft is also taken into account the difference 
between the two calculated will be decreased by an amount depending upon the 
relation between the whirling speed of the shaft, taken separately, and the whiiding 
speeds for the pulleys as calculated above. 
Reasoning in a similar way, we may conclude that when the spans are not similarly 
loaded, the whirling speeds as obtained in the investigation wall be less than the 
actual wdiirling speeds. In other words, the formula used to determine the resulting- 
speed of whirl errs on the right side. 
Case XIV, 
47. Shaft, length I, fixed in direction at each end and loaded with a 
PULLIW, WEIGHT W, AND MOMENT OF INERTIA I', AT DISTANCES Co FROM THE 
SHOULDERS. 
Thus- 
Fig. 18. 
- I, - 
Take the origin at A. Then from A to C we have (§ 21, p. 304, equation 2) 
y — ^ 03^ + " cc® + Ca- + D 
2 /' = ~ ^ + Ox + D'. 
and from C to B 
