A General Theory for Bandgap Estimation in Locally Resonant Metastructures

TL;DR

This paper develops a general operator-based modal analysis framework to estimate bandgaps in finite locally resonant metastructures, providing closed-form expressions and insights into the effects of resonator number and distribution.

Contribution

It extends modal analysis to finite metastructures with boundary conditions, deriving a universal bandgap expression independent of vibration type, and explores optimal resonator configurations.

Findings

Bandgap expression depends only on added mass ratio and target frequency.

Number of resonators needed increases with target frequency and mass ratio.

Optimal finite resonator number can produce wider bandgaps than infinite resonator models.

Abstract

Locally resonant metamaterials are characterized by bandgaps at wavelengths that are much larger than the lattice size, enabling low-frequency vibration attenuation. Typically, bandgap analyses and predictions rely on the assumption of traveling waves in an infinite medium, and do not take advantage of modal representations typically used for the analysis of the dynamic behavior of finite structures. Recently, we developed a method for understanding the locally resonant bandgap in uniform finite metamaterial beams using modal analysis. Here we extend that framework to general locally resonant metastructures with specified boundary conditions using a general operator formulation. Using this approach, along with the assumption of an infinite number of resonators tuned to the same frequency, the frequency range of the locally resonant bandgap is easily derived in closed form. Furthermore, the bandgap expression is shown to be the same regardless of the type of vibration problem under consideration, depending only on the added mass ratio and target frequency. It is shown that the number of resonators required for the bandgap to appear increases with the target frequency range, i.e. respective modal neighborhood. Furthermore, it is observed that there is an optimal, finite number of resonators which gives a bandgap that is wider than the infinite-absorber bandgap, and that the optimal number of resonators increases with target frequency and added mass ratio. As the number of resonators becomes sufficiently large, the bandgap converges to the derived infinite-absorber bandgap. The derived bandgap edge frequencies are shown to agree with results from dispersion analysis using the plane wave expansion method. Numerical and experimental investigations are performed regarding the effects of mass ratio, non-uniform spacing of resonators, and parameter variations among the resonators.