AND VIBRATION OR SHAFTS. 
2911 
CHAPTER IV.—SPECIAL CASES—LOADED SHAFTS. 
18. In considering shafts loaded with pulleys two methods may be adopted. 
First. The period of whirl may be calculated taking both the sliaft and pulleys 
into account together. 
Second. The period of whirl may be first calculated for the shaft, neglecting the 
pulleys, and then for the pulleys, neglecting the shaft. By irieans of an approximate 
formula, the period of whirl, taking both shaft and j^uHeys into account, may be 
calculated from the separately calculated periods of whirl. 
First Method oe Solution. 
Investigation shows that the first method leads to equations which are not solvable, 
so as to give results in a form convenient for actual use. 
The following two simple cases will illustrate this. 
Case VII. 
19. Overhanging shaft, length c, fixed in direction at one end, and 
LOADED WITH A PULLEY, WEIGHT W AND MOMENT OF INERTIA I', AT ITS FREE 
END, THE COMBINED EFFECTS OF BOTH SHAFT AND PULLEY BEING TAKEN INTO 
ACCOUNT. 
Thus- 
Fig. 10. 
we have 
pulley, 
whence 
(§ 7, p. 286, e(j[uation 3) for every point between the bearing and the 
d^yjdF^ = rtdy, where m = {ivoIlgVlY ; 
y = A cosh mx + B sinh 77ix + 0 cos 7nx + D sin mx. 
Taking the origin at the shoulder, we have at a singular point consisting of a 
concentrated load. 
dF\,jdx — dhjdx = Wjy.cj-y ; (§ 7, p. 286, equation 5). 
2 Q 2 
