Flexural wave band gaps in multi-resonator elastic metamaterial Timoshenko beams

Abstract

Plane wave expansion (PWE) is one of the most commonly employed methods of calculating dispersion diagrams of phononic crystals. Nevertheless, the method has been used less frequently to deal with elastic metamaterials (EMs), such as beams and plates containing local resonators. Until now, PWE and extended plane wave expansion (EPWE) methods have not been used to calculate the dispersion diagrams of EM beams with multiple periodic arrays of attached multiple degrees of freedom (M-DOF) resonators. We propose PWE and EPWE formulations to analyse flexural wave band gaps in an EM Timoshenko beam with multiple periodic arrays of attached M-DOF resonators. These formulations can predict more accurate results at higher frequencies than previous formulations developed based on Euler-Bernoulli beam theory. A coupling between locally resonant and Bragg-type band gaps is observed considering two arrays of single degree of freedom (S-DOF) resonators. However, this coupling does not occur for a single array of 2-DOF resonators even if the mass ratio is not fixed. Multiple arrays of 2-DOF resonators achieve better attenuation performance than multiple arrays of S-DOF resonators at the first resonance frequencies. However, near the first Bragg frequency, the attenuation is higher for multiple arrays of S-DOF resonators. Mass ratio effect is also investigated and it is significant for design purposes. Multiple arrays of M-DOF resonators enlarge EM potential applications for vibration management