AND VIBRATION OF SHAFTS. 
315 
If, in equation [A], we put Z = 0, we get 
1“2 “I" ^ (a ~ ~ 1 — 0, 
the equation already obtained for the ease of an overhanging shaft fixed in direction 
at one end. (Case IX., p. 304.) 
From equation [A] we get 
73 _ 1 _ C — ^ {c + 1) 
«c ’ -jV (^ + X 0 ~ (^ + i 0 * 
so that (as in § 24, p. 305) for whirling to be at all possible 
ac'^ must be > 3c/c fi- I and < 12 (3c + 0/(3c -f- 4^). 
If ac^ be equal to the first or second of these quantities the corresponding valiTe 
of (x) gives the inferior or superior limit of the speed respectively. Moreover, the 
2)eriod of whirl corresponding to the inferior limit of speed is identical with the WAtural 
period of vibration of the light shaft under the given conditions. 
The superior limit is the inferior limit multiplied by some function of the position 
of the pulley, that is, some function of I and c. The 
superior limit = 2 X inferior limit X 
oc I 
If, as ill Case X., p. 308, we put 
a = cjk = ratio of overhanging portion to the radius of gyration, 
and 
h = cjl = ratio of overhanging portion to the span, 
then the solution to the equation [A] is 
= 3TT^ [(S'-- + I) - + 1 ) + v/ {(S?- + 1 ) - + 4)] , [B], 
32. The following are the results obtained from this equation by assuming certain 
values for ci and h, as in Case X., ^ 27, p. 309. The vertical columns give the value of 
6 for difterent values of a, the value of h being fixed, whilst the rows denote the 
value of 6 for difterent values of h, the value of a being kept the same. 
2 s 2 
