36 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Solving this differential equation, we obtain 
ws 
where t is the periodic time of a complete vibration, from which, if n is the 
frequency of vibration, i.e. the number of vibrations per second, 
1-/C 
2t/V IJ 
CJ 
IL' 
For circular sections the above formula applies rigidly, while for sections 
other than circular the effective value of J, as obtained from Tables I and 
II, is to be substituted for J, giving 
n = 
1 l CJ ' 
2W1L 
(5) 
It may be noted that equation (5) gives the free or natural frequency of 
vibration of a loaded beam whether the loading is central or eccentric, I 
being in either case the mass moment of inertia of the load. 
If a loaded beam is in communication with an external periodic disturb- 
ing force the frequency of which is given by equation (5), torsional vibra- 
tions will be instituted the amplitude of which will gradually increase until 
rupture takes place. If the frequency of the disturbing force is not 
identical with n, but nearly equal to it, vibration will still occur, and a 
sufficient number of repetitions of the disturbing force may ensue to cause 
fracture. 
IV. “ Fixed-Fixed ” Beam with Single Load. 
If the beam of length L is fixed at both ends and loaded at the middle, 
the frequency of vibration is to be taken as that of a “ fixed-free ” beam of 
half the length and subjected at its free end to a load having half the 
moment of inertia of the given load, i.e. 
CJ' 
IL 
( 6 ) 
When the point of application of the load is not symmetrical with respect 
to the fixed ends of the beam, the frequency may be got in a similar 
manner. Thus if a beam of length L has a load of moment of inertia I 
situated at a point distant a from one end, the frequency of vibration is 
the same as a “ fixed-free ” beam of length a having a load whose moment of 
inertia is if — at its free end, or a beam of length (L — a) and a load 
whose moment of inertia equals I 
a 
L' 
