# Formation of Local Resonance Band Gaps in Finite Acoustic Metamaterials: A Closed-form Transfer Function Model

## TL;DR

This paper introduces a transfer function model to analyze the formation of band gaps in finite locally resonant acoustic metamaterials, revealing the fundamental mechanisms and how band gaps evolve with the number of cells.

## Contribution

It provides a closed-form transfer function approach for understanding and tuning band gaps in finite metamaterials, bridging finite and infinite structure analyses.

## Key findings

- Band gaps are characterized by conjugate zeros at resonator frequencies.
- Two poles flank the band gap, which is free of poles.
- The model's predictions align with Bloch-wave analysis for infinite structures.

## Abstract

The objective of this paper is to use transfer functions to comprehend the formation of band gaps in locally resonant acoustic metamaterials. Identifying a recursive approach for any number of serially arranged locally resonant mass in mass cells, a closed form expression for the transfer function is derived. Analysis of the end-to-end transfer function helps identify the fundamental mechanism for the band gap formation in a finite metamaterial. This mechanism includes (a) repeated complex conjugate zeros located at the natural frequency of the individual local resonators, (b) the presence of two poles which flank the band gap, and (c) the absence of poles in the band-gap. Analysis of the finite cell dynamics are compared to the Bloch-wave analysis of infinitely long metamaterials to confirm the theoretical limits of the band gap estimated by the transfer function modeling. The analysis also explains how the band gap evolves as the number of cells in the metamaterial chain increases and highlights how the response varies depending on the chosen sensing location along the length of the metamaterial. The proposed transfer function approach to compute and evaluate band gaps in locally resonant structures provides a framework for the exploitation of control techniques to modify and tune band gaps in finite metamaterial realizations.