Flexural wave propagation characteristics of metabeam with simultaneous acoustic black hole and local resonator

Abstract

This paper studies the flexural wave propagation in a metamaterial beam with the coupled acoustic black hole and local resonators (ABHLR). The main objectives are to derive the dynamic stiffness matrix of the coupled ABHLR unit cell and analyse its wave propagation characteristics. Initially, a closed-form solution of Euler–Bernoulli’s beam with an indented acoustic black hole is derived. From this, an exact dynamic stiffness matrix, incorporating local resonator effects, is derived using the spectral element method. Dispersion diagrams are generated by combining Bloch’s theorem with ABHLR transfer matrix. The coupled effects of acoustic black hole (ABH) and local resonator (LR) on the bandgap characteristics are investigated. The coupled ABHLR metamaterial shows either a positive or negative effect on bandgap formations, depending on the parametric values of the unit cell. As a positive effect, for some parametric values, the coupled ABHLR cell simultaneously produces bandgaps at lower frequencies (LR-based) and higher frequency ranges (ABH-based). Similarly, for other parametric value, as a negative effect, the bandgaps of ABHLR cell get merged but the pass bands appear on the overlapping region. This pass-band appears due to the constructive interference of ABH and LR, on each other. However, the attenuation bandgap with a higher attenuation rate is obtained on the overlapping region due to the destructive interference of LR and ABH effect. Finally, the global stiffness matrix of a finite beam with 30 unit cells is developed to analyse the frequency domain response and the results are validated with band gap analysis of the unit cell. The frequency domain response is also validated with finite element simulations using the commercial software. Results show that ABH and LR parameters have to be optimally designed to utilize the attenuation effects and avoid constructive interference due to coupling effects. The coupling of metamaterials can aid in vibration suppression and multiple-frequency filtering in mechanical and aerospace applications.