
Transverse Vibrations of Rotating Shafts. d37 
radius, is less than the values recorded in the two last lines of 
Table X., the stress-difference will exceed 2 tons on the square 
inch betore the shaft whirls. 
MATHEMATICAL APPENDIX. 
§ 40. The kinetic energy of a body rotating about an axis 
through its C.G. is given “by 
Tr=$(1,@,? + I,@5° + 1305”) , 
where I,, I,, I; are the principal moments of imertia, and 
@), @, @; the component angular velocities about the three 
principal axes. Supposing the body one of revolution, and 
that it rotates with angular velocity » about a fixed direction 
with which its axis of figure makes a small angle @, then 
@, =o Cos 6, @,=o@ sin 0, w;=8, 
and 
T,.=4{o7l,— (1,—1,)o’ sin’é +I, 2672. Bae tance 1 bs 
If the body be a flat disk 1,=2ZI,; and if @ be very small 
sin @ may be replaced by @, thus leading to 
T,=4071, +41(P—o6). . . . . (2) 
This result is applicable to a plate-shaped pulley carried 
on a rotating shaft, at a place where the tangent to the axis 
of the shaft is inclined at an angle @ to its undisturbed position. 
It is of course additional to the energy of the mass supposed 
collected at its centre of gravity. Unless the thickness of the 
pulley is small, variations of @ throughout it may not be 
negligible ; ; and unless it closely resembles a cylinder of revo- 
lution, it may be necessary to allow for J, not being double I,. 
In the text, and subsequently in the Appendix, I’ is used 
fer 1... 
§ 41. For illustrative purposes I shall take the case of a 
massless shaft of length /, supported at its two ends A and B 
(AB=l), carrying a load of mass M, and inertia I’, at an 
intermediate point C (AC=a, BC=S). 
For AC we measure w from A, and for BC we measure «’ 
from BR, At any time ¢ suppose C to be at a distance z from 
AB, in the wy plane, and let the tangent at C to the axis of 
the shaft make an angle @ with AB. Then we must have 
At-¢=0, y= y/dz =O, 
ee, 125 dy/dv=6, | 
ir’ =); y ary (dal), 
e =h. y =z, dif /da’ = —8@. 
