34 Proceedings of the Koyal Society of Edinburgh. [Sess. 
where n and m are constants depending upon the type of section. The 
values of n and m for the more important commercial sections were found 
to be as follows 
Table II. 
Type of Section. 
n . 
m . 
I 
2-00 
60 
Channel ... 
2*30 
40 
Tee 
2-30 
25 
Angle .... 
2-30 
18 
II. The Torsional Vibration of Beams. 
When a bar or beam is subjected to torsional vibration the frequency 
of vibration can be readily determined, as will be shown later by equating 
the potential energy of the vibrating system at the point of extreme 
angular displacement to the kinetic energy of the system in its mean 
position. Two cases arise, 
(a) The unloaded beam. 
(b) The loaded beam. 
The first of these is of very little importance in the subject with which 
we are dealing, as the conditions necessary for the production of torsional 
vibrations in an unloaded beam are seldom if ever reproduced in practice, 
and, moreover, the natural frequency of vibration of such a beam is so high 
as to preclude any possibility of resonance with machinery in operation. 
The problem of the loaded beam is, however, of some importance, and 
it is intended to treat of this in some detail. 
III. “ Fixed-Free ” Beam with Single Load at Free End. 
Let the beam AB (fig. 1) be rigidly fixed at one end and free to rotate 
about the axis 00 at the other end. Furthermore, let the mass moment 
of inertia of the beam be negligible compared with the moment of inertia 
of the load w whose eccentricity is r. 
Let be the angular displacement due to the statically applied 
torque T ( = Wr), and let 0 be the angular displacement on each side of the 
mean position when the system is vibrating. 
From formula (1) we obtain 
CJ 
L 
(3) 
where — is the static torque necessary to produce unit angular displace- 
0i 
