1915-16.] Torsional Vibration of Beams of Commercial Section. 37 
V. Effect of Inertia of Beam. 
If the mass moment of inertia of the beam cannot be neglected, but if it 
is small compared with the moment of inertia of the applied load, a correc- 
tion can be obtained on the assumption that the angle of twist is propor- 
tional to the distance from the fixed end of the beam. Thus in a “ fixed- 
free ” beam, if the angle of twist at the free end at any instant is 0, the 
angle of twist at any section of the beam distant x from the fixed end is 
CO 
O.j, where L is the total length of the beam, while the angular velocity 
at this point equals ® 
Hence if J is the mass moment of inertia of 
the whole beam about the axis of twist, the kinetic energy of a section at a 
point distant x from the fixed end and of thickness dx is given by 
- —dx 
2 L 
a; d6\ 2 
L dt) 
-i Ut)’** 
and the kinetic energy of the whole beam is 
1 J m /d0\ 2 
2 U\di 
Ji f dx 
Mm 
i.e. the dynamic effect of the beam is the same as a mass concentrated at 
the free end, and having a mass moment of inertia equal to i that of the 
beam. Hence if the effect of the beam is not negligible, the moment of 
inertia of the applied load must be increased by an amount equal to ^ the 
mass moment of inertia of the beam. 
Where the mass moment of inertia of the beam itself is commensurate 
with the moment of inertia of the applied load, the motion is somewhat 
complex, and, except for a beam of circular section, a solution is difficult 
to obtain ; * while for beams of commercial section, I, channel, angle, etc., 
the analytical solution would seem to be impossible. In practice, however, 
the only case of importance is that of the loaded beam, in which the 
dynamic effect of the beam itself is negligible, and in most practical cases 
the frequency of vibration will be given with sufficient accuracy by 
equation (5). 
* Proceedings Inst. Giv. Engrs vol. clxii, p. 382. 
