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