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Proceedings of the Royal Society of Edinburgh. [Sess. 
III. — The Torsional Vibration of Beams of Commercial Section. 
By Ernest G. Ritchie, B.Sc., Engineering Department, University 
College, Dundee. Communicated by Professor A. H. Gibson. 
(MS. received November 13, 1915. Read December 20, 1915.) 
When an elastic body is constrained in any manner whatsoever, it is 
susceptible to vibration, by virtue of its elasticity, when disturbed from 
its position of equilibrium by an externally applied force. The period 
and amplitude of such vibration are dependent upon the mass and inertia 
of the system, the rigidity of the constraints, and upon the nature of the 
disturbing force. 
When a beam of commercial section is loaded centrally, and subjected 
to vibrations, the frequency of transverse vibration can be readily 
determined from a knowledge of the dimensions of the beam, its modulus 
of elasticity, and the conditions of loading. On the other hand, where 
the loading is eccentric, the transverse vibration is accompanied by a 
torsional vibration the frequency of which is very much lower than is 
indicated by the ordinary elastic theory, due to the inefficiency in torsion 
of beam sections other than circular. In practice it is not always possible 
to eliminate the eccentric loading of beams, as for instance where power 
is transmitted through countershafts supported from structural steel- work, 
and it is with the problem of the torsional vibration of such eccentrically 
loaded beams that it is proposed here to deal. 
I. The Torsion of Beam Sections. 
When a bar of circular section and of length L is subjected to the 
application of a static twisting-moment T, it may be readily shown that 
GJ= rJ f (1) 
where 6 is the angle of twist in radians, C is the modulus of rigidity, and 
J is the polar moment of inertia of the section. 
When a bar of non-circular section is subjected to a twisting-moment 
it has been demonstrated both theoretically and experimentally that the 
above formula does not truly represent the condition of affairs, the angle 
of twist for a given torque being greater than would be anticipated from 
formula (I). The discrepancy is largely due to the inefficiency in torsion 
