Frequency band structure of locally resonant periodic flexural beams suspended with force-moment resonators

Since the introduction of locally resonant (LR) phononic meta-materials and structures a decade ago, there has been a quest for wide band-gaps of elastic wave attenuation in a low frequency range of practical importance. We investigate periodic Euler-Bernoulli beams suspended with two degrees of freedom force-moment resonators. Based on mathematical analysis and calculations, we present their dispersive characteristics in flexural wave attenuation and propagation. As a prime focus, we identify the appearance a below-resonance band-gap of Bragg-scattering (BS) type - in addition to the normal post-resonance BS band-gap - and the dependence of their edge frequencies on the resonator parameters. Furthermore, we present a full characterization of four different groups of pass and stop frequency bands and six distinct types of transitions between them, with conditions for all band edge frequencies. Our results indicate potentially richer dispersion properties in LR periodic structures with resonators of multiple degrees of freedom than those with the conventional force-only (or moment-only) resonators.