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.

Uncertainty quantification of tunable elastic metamaterials using polynomial chaos

“…The presence of uncertainties in an EM's system parameters hinders its ability to fulfil the desired performance. In here, we focus on the effect of such uncertainties on the EM's dispersion patterns (band structure), dynamic response and its ability to sustain Bragg and local resonance band gaps [20,25]. For a conceptual understanding, we consider three types of one-dimensional (1D) EMs: (1) A chain of identical masses connected via springs of alternating stiffnesses, (2) A monatomic lattice with a periodicity in the form of alternating grounded springs, and (3) A locally resonant metamaterial (LRM) with grounded springs.…”

“…The poles of the system can then be found from the determinant of D(s) and are indifferent to the excitation and measurement (sensing) locations. Analytical expressions for the aforementioned transfer functions for all three EM types considered here and free-free boundary conditions have been recently reported and the derivations are therefore omitted here for brevity [20,21]. However, final expressions for Z(s) and P (s) for the MV, EF, and LRM cases are summarized in Table III. The frequency response functions (FRFs) of the three EMs are graphically presented in Figure 4 for ϑ = ±0.5ϑ max and n = 40 cells.…”

“…extended frequency ranges of forbidden wave propagation; a hallmark feature of phononic, acoustic and elastic metamaterials [15][16][17][18][19]. Such gaps are theoretically predicted in infinitely long periodic structures and are derived from unit-cell-based models which often fail to capture truncation and boundary effects in finite realizations of such metamaterials; the ramifications of which have been extensively reviewed in literature [20][21][22][23]. As such, the * Corresponding author: mnouh@buffalo.edu analysis conducted here spans both infinite and finite configurations of three distinct types of elastic metamaterials, namely (1) Periodic structures with periodic variations -reminiscent of traditional phononic crystals, (2) A lattice with a periodic elastic foundation, and (3) A locally resonant metamaterial.…”

Non-reciprocal wave phenomena in energy self-reliant gyric metamaterials

“…1a, which represents the building block of a periodic gyric structure. Unlike conventional elastic metamaterials where the outer body is connected to an internal resonator and transmits a force to it 37,38 , the interaction between the outer and inner components here is rather a gyroscopic moment emerging as a result of the change in the total angular momentum vector of the gyric body (i.e. B + R).…”

Estimating dispersion curves from Frequency Response Functions via Vector-Fitting

“…For commonly used homogeneous and heterogeneous materials, dispersion relations have been documented in various handbooks and textbooks [7,8]. Numerical methods, such as time and frequency domain spectral element method [9][10][11][12], the semi-analytical finite element method [13,14], the wave finite element method [15,16], and transfer function and transfer matrix methods [17,18] have also been developed to describe wave propagation along uniform and periodically varying waveguides. For realistic wave propagation characteristics to be obtained, accurate description of material and geometric characteristics of the waveguide is required [19].…”