Vibrational properties of quasi-periodic beam structures

Abstract

Quasi-periodic structures have been widely studied, notably in the atomic vibration domain. In this paper a beam structure based on Octagonal quasi-periodic tiling is studied. We provide a complete description of its vibrational response, including the density of its vibrational states, a detailed description of its vibration modes, and the computation of the dynamical structure factor (spectral density of energy) for transverse and for longitudinal waves. It is shown that quasi-periodic structures exhibit localized low frequency vibration modes that are due to resonant vibrations of isolated patterns in the quasi-periodic structure, but in opposite, high-frequency modes are (non-trivially) extended. Moreover, the paper shows the possible existence of band gaps in the vibrational response of periodic and quasi-periodic beam lattices as a function of the ratio between the bending and the tensile stiffness of the beams.