Bragg band gaps associated with infinite phononic crystals are predicted using wave dispersion models. This paper departs from the Bloch-wave solution and presents a comprehensive dynamic systems analysis of finite phononic systems. Closed form transfer functions are derived for two systems where phononic effects are achieved by periodic variation of material property and boundary conditions. Using band structures, differences in dispersion characteristics are highlighted and followed by an analytical derivation of the eigenvalues. The latter is used to derive the end-to-end transfer function of a finite phononic crystal as a function of any given parameters. The analysis reveals intriguing features that explain the evolution of Bragg band gaps in the frequency response. It quantifies how the split of eigenvalues into sub- and super-band-gap natural frequencies contribute to band gap formation. The unique distribution of poles allows the closely packed sub-band gap natural frequencies to achieve maximum attenuation in the Bode response. At that point, the impact of the super-band-gap frequencies on the opposing side becomes significant causing the attenuation to fade and the band gap to come to an end. Finally, the effect of splitting the poles further apart is presented in both phononic systems, with material and boundary condition periodicities.

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.

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Most of the existing acoustic metamaterials rely on architected structures with fixed configurations, and thus, their properties cannot be modulated once the structures are fabricated. Emerging active acoustic metamaterials highlight a promising opportunity to on-demand switch property states; however, they typically require tethered loads, such as mechanical compression or pneumatic actuation. Using untethered physical stimuli to actively switch property states of acoustic metamaterials remains largely unexplored. Here, inspired by the sharkskin denticles, we present a class of active acoustic metamaterials whose configurations can be on-demand switched via untethered magnetic fields, thus enabling active switching of acoustic transmission, wave guiding, logic operation, and reciprocity. The key mechanism relies on magnetically deformable Mie resonator pillar (MRP) arrays that can be tuned between vertical and bent states corresponding to the acoustic forbidding and conducting, respectively. The MRPs are made of a magnetoactive elastomer and feature wavy air channels to enable an artificial Mie resonance within a designed frequency regime. The Mie resonance induces an acoustic bandgap, which is closed when pillars are selectively bent by a sufficiently large magnetic field. These magnetoactive MRPs are further harnessed to design stimuli-controlled reconfigurable acoustic switches, logic gates, and diodes. Capable of creating the first generation of untethered-stimuli-induced active acoustic metadevices, the present paradigm may find broad engineering applications, ranging from noise control and audio modulation to sonic camouflage.

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