Nonuniform Isospectrals of Uniform Timoshenko Beams

Abstract

Spectrally equivalent systems are those that have the same free vibration natural frequencies for a given boundary condition. In this paper, we establish isospectrality between certain classes of nonuniform Timoshenko beams with a given uniform Timoshenko beam. We apply a transformation to convert the nondimensional coupled nonuniform Timoshenko beam equations from the frame of reference to a hypothetical frame of reference. The transformed equations are then combined by eliminating one of the variables. Specific material and geometric properties are chosen, and a few auxiliary variables are introduced to convert the transformed equation into the required form. If the coefficients of the transformed equation match with the required uniform equation, then the nonuniform beam is said to be isospectral to the uniform beam. The boundary configurations also change during this transformation. We present the constraints under which they are preserved. Frequency equivalence of the beams is confirmed by the finite element method. For the considered cases, examples of beams having a rectangular cross-section are also presented.