Topological Edge States in Quasiperiodic Locally Resonant Metastructures

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Topological Edge States in Quasiperiodic Locally Resonant Metastructures

Yiwei Xia, Alper Erturk, and Massimo Ruzzene

Phys. Rev. Applied 13, 014023 – Published 14 January 2020

Abstract

We investigate the dynamic behavior and topology of quasiperiodic resonant metastructures. We show that the quasiperiodic arrangement of resonators introduces frequency band gaps in addition to the locally resonant band gap defined by the natural frequency of the resonators. The concept is illustrated on a beam with an array of mechanical resonators. Numerical evaluation of the spectrum as a function of the quasiperiodic arrangement of resonators reveals a structure reminiscent of a Hofstadter butterfly and allows the study of key topological properties. The results illustrate the occurrence of additional band gaps that are topologically nontrivial and that host edge-localized modes in finite structures. The occurrence of these gaps and of the associated edge states is demonstrated experimentally by measuring the frequency response of the beam and by evaluating the spatial distribution of selected operational deflection shapes. The results unveil the potential of deterministic quasiperiodic structural designs to induce wave localization and attenuation over multiple frequency bands, which may find applications in vibration isolation and energy harvesting, among others.

Received 3 September 2019
Revised 5 December 2019

DOI:https://doi.org/10.1103/PhysRevApplied.13.014023

Physics Subject Headings (PhySH)

Acoustic metamaterials Acoustic signal processing & noise Acoustic wave phenomena Band gap Edge states Mechanical & acoustical properties Topological phases of matter

Energy applications

Condensed Matter, Materials & Applied PhysicsGeneral Physics

Authors & Affiliations

Yiwei Xia ^1,*, Alper Erturk ¹, and Massimo Ruzzene²

¹George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta 30332, USA
²Department of Mechanical Engineering, University of Colorado Boulder, Boulder 80309, USA

^*yxia63@gatech.edu

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Vol. 13, Iss. 1 — January 2020

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Images

Figure 1
The projection operation for placement of the local resonators according to the procedure described in Ref. [25]).
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Figure 2
(a) Bulk (black) and finite-beam (red) spectra as a function of $θ$ . The finite-beam spectrum, obtained for a finite simply supported beams with 30 resonators, shows the presence of modes spanning the nontrivial gaps. The blue shaded area highlights the LR band gap, which is estimated according to the formula derived in Ref. [7]. (b) A detail of the spectrum showing four frequencies and corresponding bulk and edge-localized modes (the black curve represents the deflection of the beam, while the red circles denote the displacements of the resonators). (c) A detail of the spectrum showing three labeled nontrivial topological band gaps with an increasing number of topological modes (blue dashed lines separate regions between commensurate values of $θ$ , while the blue shaded area highlights the region between $θ = 2 / 30$ and $θ = 3 / 30$ ). (d) The IDS as a function of $θ$ exhibits sharp linear jumps at the band gaps. The slopes of three of these lines [highlighted by the white dashed lines and corresponding to the three gaps labeled in (c)] are equal to $m = 1, 2, 3$ , while that for the LR band gap (highlighted by the red dashed line) is $m = 0$ , which indicates that this band is topologically trivial.
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Figure 3
(a) The bulk spectrum (black) and the finite spectrum for a clamped-free beam with 30 resonators (red). (b) The numerical frequency-response function of the beam spatially averaged between 20% and 30% of the beam span: the color map evolving from blue to red corresponds to the log scale of the magnitude. The blue regions highlight the low-response ranges corresponding to the band gaps. (c) The numerical frequency-response function of the beam spatially averaged between 90% and 100% of the beam span: the response near the beam tip highlights the presence of resonances within the gaps, which correspond to edge states. The vertical white lines in (b) and (c) correspond to the values of $θ = 0.175$ and $θ = 0.25$ considered in the experiments.
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Figure 4
The experimental setup, a cantilever beam with 30 resonators: (a) the front view of the beam; (b) a close-up of the resonators; and (c) view of the tip of the beam, excited by a electrodynamics shaker.
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Figure 5
(a),(e) Details of the numerical bulk spectrum, with vertical magenta lines corresponding to $θ = 0.175$ and $θ = 0.25$ . The green and red dashed lines show the theoretical boundaries of the LR and nontrivial topological band gaps, respectively. (b)–(d) The experimental results for $θ = 0.175$ . (f)–(h) The experimental results for $θ = 0.25$ . The magnitude of the beam frequency response is spatially averaged between 20% and 30% (b),(f) and between 90% and 100% (c),(g) of the beam span. The green and red shaded areas highlight the theoretical LR and topological band gaps. (d),(h) The measured deflection shapes of the beam. Modes “I,” “III,” and “IV” are bulk modes at frequencies before the LR gap, between the LR and topological band gap, and after the topological band gap, respectively. The corresponding frequencies are marked by the green diamond, blue square, and magenta asterisk in (b), (c), (f), and (g). The mode labeled as “II” in (d) and (h) is edge localized, its frequency falls in the topologically nontrivial gap, and it is marked by a red circle in (c) and (g).
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