AND VIBRATION OF SHAFTS. 
309 
If c = c, tliat is to say, if the pulley be placed in the middle of the span, the 
superior and inferior limits coincide, and the pulley, at all speeds, revolves in a plane 
perpendicular to the original alignment of the shaft. Whatever be the size of the 
pulley, the period of whirling is the same as the “ natural period of vibration.” 
The solution to equation [A] may be put in the form 
1-iJ “^(I-&)3 
[B], 
in which 
a = c/k = ratio of the distance of the pulley from the nearer bearing to the radius of 
gyration, and 
h = cjl — ratio of the distance of the pulley from the nearer bearing to the whole 
span. 
Assuming certain values for a and h, results might be obtained giving relations 
between w, W, c. In the equation [B], the ratio h fixes the position of the pulley on 
.V Pc^ 
F' = 
6EI 
E' = - — (z + —) 
•6FA\ 21) 
wEence, substituting in (2) we get 
, _ Pc /Za;2 _ arn _ Pea; /, c^ \ Pc^ 
' /El V 2 6 / 3EIV ^ 2l) ^ GEI 
Hence, when 
deflection 
This is unity when P = e, wherefore 
Hence 
X = c, 
. Pc2c'2 
■ 3/EI 
3ZEI 
O I o 
c^c *’ 
/ = 2. 
V ge V -dqmi 
The corresponding value of uo in text is 
therefore 
or 
as in text. 
'dgim 
_ 27r _ / 3g'ZEI 
’“T ~ V Wc 2 c '2 ’ 
/W<^2\ /c2y2\ _ 
l^EI/V3zj 
ac2c'2/3Z = 1, 
(7) . 
( 8 ) , 
