Bloch waves in an array of elastically connected periodic slender structures

Abstract

This paper proposes the methodology to carry out the analysis of Bloch wave propagation in an array of vertically aligned and elastically connected structural elements such as beams, strings, plates, or other slender structures. The suggested approach is based on the Galerkin approximation and Floquet-Bloch theorem used in defining the eigenvalue problem and obtaining the band structure of the periodic systems. Special attention is devoted to the case of elastically connected Rayleigh beams with attached concentrated masses and wave propagation in the direction normal to the beam's length. A validation study is performed by using the finite element model and the frequency response function to confirm the accuracy of the solution obtained via the Galerkin approximation. Two configurations of unit cells, having two and three elastically connected beams with different geometrical and material properties, are considered in the numerical study. The effects of various parameters are investigated to reveal their influence on the frequency band structure and emergence of the zero-frequency bandgap. The results of this study demonstrates the tunability properties of the proposed periodic systems due to changes in values of concentrated masses, stiffness of the coupling medium or boundary conditions on structural elements within the unit cell.