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 thedisturbing 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 instancewhere 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 itis proposed here to deal.