Flexural wave propagation in an elastic beam with periodic structure

The theory of flexural waves in an elastic beam with periodic structure is developed in terms of Floquet waves. Special relationships have been determined among the fundamental solutions of the governing equation. Two lemmas about the properties of the fundamental solutions are proved. With the help of these relations and lemmas, the analysis and classification of the dynamic nature of the problem is greatly simplified. We show that the flexural wave propagation in a periodic beam can be interpreted as the superposition of two pairs of waves propagating in opposite directions, of which one pair behaves as an attenuated wave. The dispersion spectrum of the second pair of waves shows the band structure, consisting of stopping bands and passing bands. Exploiting the symmetry of the structure, the dispersion equation at the end points of Brillouin zones is uncoupled into two simpler equations. These uncoupled equations represent the dispersion spectrum of waves which are either symmetric, or antisymmetric.