Coupled Bending Vibrations of Pretwisted Cantilever Beams

Abstract

The author describes a method of determining the vibrations of cantilever beams of non-uniform pretwist and cross-section. The known condition for orthogonality of normal modes for a beam in simple bending in one transverse plane is extended to that of coupled bending-bending modes of vibration in two transverse planes. This condition of orthogonality is then used in the Stodola process to determine the frequencies and modes of vibration for cantilever beams of uniform pretwist and cross-section. In this method the modes of vibration may be coupled or uncoupled. The resonant frequencies of the beam are determined in ascending order of magnitude and the modal patterns deduced without requiring any preliminary knowledge of the vibrational characteristics of the beam. It is considered that this is a considerable advantage on the alternative methods of solution. An analysis of experimental and theoretical results is made for three cantilever beams of uniform rectangular cross-section and angles of pretwist 0, 90·9 and 877° respectively. The frequencies and modal patterns of the first five modes of vibration were obtained experimentally and show good agreement with those calculated from the theory. The computed results for the first five frequencies are then given for a series of cantilever type beams in coupled bending-bending vibrations. The cantilevers considered had angles of pretwist from zero to 180° and were of uniform rectangular cross-section having breadth to depth ratios ranging from unity to 16. The calculated frequencies have been compared with known experimental results and show good agreement. It has also been shown that some of the larger errors may be anticipated for certain critical frequencies.