Non-rotating beams isospectral to rotating Rayleigh beams

Abstract

In this paper, we establish spectral equivalence between certain non-uniform rotating Rayleigh beams and a given uniform Rayleigh beam, and also between non-uniform non-rotating Rayleigh beams and a given uniform rotating Rayleigh beam. Spectrally equivalent systems are those that have the same spectra, i.e., the same free vibration natural frequencies for a given boundary condition. We use a transformation to convert the non-dimensional Rayleigh beam equations from the (x, W) frame of reference to a hypothetical (z, U) frame of reference. If the material and geometric properties of the beam are specific chosen functions of the two introduced auxiliary variables, then the transformation will be achieved. If the coefficients of the transformed equation match with the required equation, then the equation, on which the transformation is applied, is said to be isospectral to the required one. The mode shapes for these particular cases also match due to the nature of assumption on one of the introduced auxiliary variables. Frequency and mode shape equivalence of the beams is confirmed by the finite element method (FEM). For the considered cases, examples of beams having a rectangular cross-section are presented to show the application of our analysis.