Wave and vibration attenuation in graded elastic metamaterial beams with local resonators

This study investigated the bending band gaps in an Euler–Bernoulli metamaterial beam with attached mass–spring resonators. The position and mass of the resonators were considered following three different configurations, given by the arithmetic, geometric, and quadratic progressions. With the extended plane wave expansion (EPWE), wave finite element (WFE), and wave spectral element (WSE) methods, complex dispersion diagrams were obtained, where the band gaps due to Bragg scattering and local resonance were analyzed. From the study of vibration via forced response, the results are confirmed also for finite structures. A coupling between locally resonant and first Bragg-type band gaps ($\sim 461\mathrm{Hz}$) was observed considering a set of $N=10$ resonators, increasing the wave attenuation region. The wave propagation and forced response simulations showed that the grading of the resonators’ positions can modulate the coupling between local resonance and Bragg band gaps, demonstrating the potential to enhance attenuation by leveraging the natural vibration frequency of graded resonators. The influence of the resonator mass was studied through parametric diagrams, where the change of the smallest part of the imaginary component of Bloch wave vector with the increase of the ratio between the mass of the resonators and the unit cell of the bare beam was observed. The dispersion diagrams and forced responses indicated that the best dynamic performance in terms of wave and vibration attenuation was obtained for simultaneous geometric progression in the resonator’s positions and arithmetic progression in the resonator’s mass, respectively.