288 
MR. S. DUNKERLEY OR THE WHIRLIRG 
Ill addition to these equations we shall get equations according to the manner in 
which tlie shaft is supported at the ends. If it merely rest on the bearing, so that 
the bearing exercises no restraint on its direction, the bending moment at that point 
is zero, that is, d/yjdx^ = 0, and, therefore, 
A cosh mx + B sinh mx — C cos mx — D sin mx = 0 . , . (15), 
On the other hand, if the bearing be so long that it practically guides the direction 
of the shaft, in other words, if the shaft be fixed in direction, then we have dyjdx = 0, 
or 
A sinh mx + B cosh mx — C sin mx + D cos mx =0 . . . (16). 
It will be found that, in every case, equations (8) to (16), inclusive, are sufficient in 
number to allow of the elimination of the ratios A : B : C : D : A' : B' : &c. 
The resulting equation will give a relation between the whirling speed, size and 
weight of the pulleys, diameter of the shaft, &c., that relation depending on the 
manner in which the shaft is supported and loaded. 
The proper value of x has, of course, to be substituted, in the above equations, for 
any particular singular point. 
The values of the constants A, B, C, D at the ends of a shaft are zero. 
CHAPTER III.—SPECIAL CASES—URLOAGED SHAFTS. 
Case I. 
9, OVERHANGIITG ShAFT, LENGTH C, FIXED IN DIRECTION AT ONE END. 
Thus 
Fig. 8. 
—c -->, 
We have (§ 7, p. 286, equation 3) d^yjdx^ — m*y, where m = whence 
y = A. cosh mx + B sinh mx -f- C cos mx + D sin mx. 
Taking the origin at the shoulder, we have, when x = 0, 
y = 0, dyjdx — 0, 
and, when x = c, 
d^yjdx^ = 0 , d^’yjdx^ — 0 
(shearing force zero). Hence we get 
