Flexural wave propagation in a spatially periodic structure of articulated beams

This paper considers flexural wave propagation in a spatially periodic structure consisting of identical beams of finite length, as a unit, connected to adjoining ones by couplers giving restoring moment at junctions, and extending to both infinity. Motions of the beam are restricted in a plane, and two cases are considered in linear theory. One is concerned with rigid beams and the other elastic beams. For the former case, the difference and differential equations for the displacements are derived from the equations of motions for translation and rotation of each beam with boundary conditions at the junctions, and a linear dispersion relation is obtained. For the latter, the Bernoulli-Euler equation for flexural motions of each beam is solved with boundary conditions at the junctions. Then a transmission matrix is derived, which relates deflection in each unit to the one in adjoining units. By examining eigenvalues and eigenvectors of the matrix, propagation characteristics are discussed and Bloch wave functions as elementary functions for the deflection are obtained. It is shown that the dispersion relations exhibit stopping bands for the frequency, which is not found in the case of the rigid beams. Continuum approximation of the articulated beams is discussed to derive legitimate beam equations for long waves. Also discussed is the physical interpretation of the cutoff frequencies for the stopping bands.