286 
MR. S. DUNKERLEY ON THE WHIRLING 
Let 
M = bending moment at a distance x from the origin, and let the deflection 
at this point be y. 
C = centrifugal force per unit length of shaft. 
I = geometrical moment of inertia of a cross-section of the shaft about a 
diam. 
E = Young’s Modulus for the shaft, 
CO = angular velocity of shaft. 
vj = weight of shaft in lbs. per foot run. 
W = weight, in lbs., of any pulley which the shaft carries. 
I' = some moment of inertia of the pulley yet to be determined. 
Neglecting the dead weight of the shaft, the ordinary equations of the beam 
give us 
cim/dx'^ =C .(1), 
and 
cry/dx^ = M/EI .( 2 ), 
whence 
d^/ _ G _ 1 
~ El “ El 
where 
m = (ivco^/ffEI)K 
w 
(J 
oj y j = m y 
( 3 ), 
Equation (3) holds between every pair of singular points, that is to say, between 
bearings and pulleys. 
At a point of support, the diflerence of shearing force on the two sides must clearly 
e(]ual the pressure, that is, 
f/R/c/a’ — dEjdx = P.(4), 
where R and L are the bending moments to the right and left of the support, and 
P is the pressure on the support. 
At a load consisting of a revolving weight W, this equation becomes (neglecting 
the dead weight of the pulley) 
dEldx — dL/dx = W Ig.oj'y .(5). 
A further equation may be obtained by considering the “centrifugal couple” 
tending to straighten the shaft. The moment of the centrifugal forces about a 
diametral line in the plane of the pulley and passing through its centre of gravity is 
I'co^.dyjdx where I' = A — B, 
and 
A = mass-moment of inertia of pulley about an axis through its centre 
of gravity perpendicular to its plane, and 
