Coupling between torsional and bending modes of vibration in cantilever beams

Abstract

The work presented is a theoretical and experimental investigation into the effect of coupling between torsional and flexural modes of vibration in cantilever beams. Two theoretical approaches are made: firstly, a finite element idealisation in which the beam is divided into a number of elements which possess cubic variations of deflection and rotation along their lengths; and, secondly, a more analytical method in which the simultaneous differential equations of motion are solved directly by application of Laplace Transforms. The investigation is restricted to the simplest cross section which possesses a single axis of symmetry, the isosceles triangle. The exact solutions for the torsion and flexure of such sections of general shape do not as yet exist, but use is made of several approximate techniques which have been developed for this section for the calculation of the torsional stiffness and position of the centre of flexure. It is shown experimentally and theoretically that torsional oscillations do not talce place about the centre of flexure ( or centre of torsion ), but about a point which may be considered to be coincident with the centrold. It is shown theoretically and confirmed experimentally that the effect of coupling on either mode is extremely small unless the original frequencies of torsional and flexural motion are almost coincident, in which case the two frequencies are separated into coupled modes which possess both torsion al and flexural characteristics. The effect is not, however, as significant as the coupling of flexural modes due to pretwist in sections which approximate to those of turbine blades.