Analytical Determination of Coupled Bending-Torsion Vibrations of Cantilever Beams by Means of Station Functions,

A method based on the concept of Station Functions is presented for calculating the modes and the frequencies of nonuniform cantilever beams vibrating in torsion, bending, and coupled bending-torsion motion. The method combines some of the advantages of the Rayleigh-Ritz and Stodola methods, in that a continuous loading function for the beam is used, with the advantages of the influence-coefficient method, in that the continuous loading function is obtained in terms of the displacements at a finite number of stations along the beam. The Station Functions were derived for a number of stations ranging from one to eight. The deflections were obtained in terms of the physical properties of the beam and Station Numbers, which are general in nature and which have been tabulated for easy reference. Examples were worked out in detail comparisons were made with exact theoretical results. For a uniform cantilever beam with n stations, the first n modes and frequencies were in good agreement with the theoretically exact values. The effect of coupling between bending and torsion was shown to reduce the first natural frequency to a value below that which it would have if there were no coupling.