Longitudinal wave band gaps in metamaterial-based elastic rods containing multi-degree-of-freedom resonators

Wave propagation and vibration transmission in metamaterial-based elastic rods containing periodically attached multi-degree-of-freedom spring–mass resonators are investigated. A methodology based on a combination of the spectral element (SE) method and the Bloch theorem is developed, yielding an explicit formulation for the complex band structure calculation. The effects of resonator parameters on the band gap behavior are investigated by employing the attenuation constant surface plots, which display information on the location, the width and the attenuation performance of all band gaps. It is found that Bragg-type and resonance-type gaps co-exist in these systems. In some special situations, exact coupling between Bragg and resonance gaps occurs, giving rise to super-wide coupled gaps. The advantage of multi-degree-of-freedom resonators in achieving multiband and/or broadband gaps in metamaterial-based rods is demonstrated. Band gap formation mechanisms are further examined by analytical and physical models, providing explicit formulae to locate the band edge frequencies of all the band gaps.