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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In this paper, we explore an electromechanical metastructure consisting of a periodic array of piezoelectric bimorphs with resistive-inductive loads for simultaneous harvesting and attenuation of traveling wave energy. We develop fully coupled analytical models, i.e. an electroelastic transfer matrix method, and exploit both locally-resonant and Bragg band gaps to achieve a multifunctional metastructure which is capable for maximum energy conversion and vibration mitigation in a broadband fashion. Our analytical and numerical results show that the proposed metastructure can achieve energy harvesting efficiency up to 95% at the local resonance frequency of 3.18 kHz, while reaching about 51% at 5.8 kHz near the upper limit of the Bragg band gap. The broadband vibration mitigation performance based on 50% power attenuation is predicted as 1.8 and 1.1 kHz in the vicinity of the band gaps. The theoretical frameworks and the applicability of the proposed metastructure are validated using a full-scale experimental setup.

This note analytically investigates non-reciprocal wave dispersion in locally resonant acoustic metamaterials. Dispersion relations associated with space-time varying modulations of inertial and stiffness parameters of the base material and the resonant components are derived. It is shown that the resultant dispersion bias onsets intriguing features culminating in a break-up of both acoustic and optic propagation modes and one-way local resonance band gaps. The derived band structures are validated using the full transient displacement response of a finite metamaterial. A mathematical framework is presented to characterize power flow in the modulated acoustic metamaterials to quantify energy transmission patterns associated with the non-reciprocal response. Since local resonance band gaps are size-independent and frequency tunable, the outcome enables the synthesis of a new class of sub-wavelength low-frequency one-way wave guides.

This work presents a comprehensive mathematical treatment of phononic crystals (PCs) which comprise a finite lattice of repeated polyatomic unit cells. Wave dispersion in polyatomic lattices is susceptible to changes in the local arrangement of the monatoms (subcells) constituting the individual unit cell. We derive and interpret conditions leading to identical and contrasting band structures as well as the possibility of distinct eigenmodes as a result of cyclic and non-cyclic cellular permutations. Different modes associated with cyclic permutations yield topological invariance, which is assessed via the winding number of the complex eigenmode. Wave topology variations in the polyatomic PCs are quantified and conditions required to support edge modes in such lattices are established. Next, a transfer function analysis of finite polyatomic PCs is used to explain the formation of multiple Bragg band gaps as well as the emergence of truncation resonances within them. Anomalies arising from the truncation of the infinite lattice are further exploited to design mirror symmetrical edge modes in an extended lattice. We conclude with a generalized explanation of the band gap evolution mechanism based on the Bode plot analysis.