Exploring Topology of 1D Quasiperiodic Metastructures through Modulated LEGO Resonators
EExploring Topology of 1D Quasiperiodic Metastructures through Modulated LEGOResonators
Matheus I. N. Rosa, Yuning Guo, and Massimo Ruzzene
P. Rady Department of Mechanical Engineering,University of Colorado Boulder, Boulder CO 80309 (Dated: January 25, 2021)We investigate the dynamics and topology of metastructures with quasiperiodically modulatedlocal resonances. The concept is implemented on a LEGO beam featuring an array of tunablepillar-cone resonators. The versatility of the platform allows the experimental mapping of thebeam’s Hofstadter-like resonant spectrum, which is the first reported observation for an elasticmedium. The non-trivial spectral gaps are classified by evaluating the integrated density of statesof the bulk bands, which is experimentally verified through the observation of topological edge stateslocalized at the boundaries. Results also show that the spatial location of the edge states can bevaried through the selection of the phase of the resonator’s modulation law. The presented resultsopen new pathways for the design of metastructures with novel functionalities going beyond thoseencountered in periodic media by exploiting aperiodic patterning of local resonances and suggest asimple, viable platform for the observation of a variety of topological phenomena.
The discovery of topological insulators [1] has attractedsignificant interest from the metamaterials communitydue to promising prospects for robust wave localiza-tion and transport. [2–9] Recent efforts focus on ex-ploring higher dimensional topological effects in lowerdimensional systems by exploiting virtual dimensionsin parameter space. [10–14] Indeed, edge states com-monly attributed to the Quantum Hall Effect in 2D sys-tems [4] have been illustrated in 1D periodic [15, 16] andquasiperiodic [17–23] systems, while 4D and 6D Quan-tum hall phases have been observed in 2D [24–26] and3D [14, 27] lattices. In addition to opening avenues forthe exploration of novel topological wave physics phe-nomena, these investigations are also promising for tech-nological applications and devices. For example, topo-logical pumps as originally envisioned by Thouless [28]were recently implemented, [16, 17, 29–39] suggestingnew mechanisms for robust energy transport in systemsof a single spatial dimension.Among the many types of elastic metamaterials, lo-cally resonant metastructures are particularly interestingdue to the possibility of affecting dispersion at subwave-lengths. [40–46] Recent studies have explored the effectsof aperiodicity and disorder [47–50] for bandgap widen-ing and producing rainbow effects. For example, elasticbeams with arrays of identical resonators located accord-ing to quasi-periodic patterns investigated in [22] wereshown to feature additional spectral gaps hosting topo-logical edge states. These were produced at no additionalcost or increase in mass when compared to the nominalperiodic configurations. Thus, quasi-periodic patterningof locally resonant metastructures may open new avenuesfor wave localization or attenuation in multiple bands,and for extending the behavior of periodic configurations.In this letter, we investigate locally resonant metas-tructures whose resonating attachments are tuned ac-cording to a quasi-periodic modulation law. We employ a LEGO elastic beam with pillar-cone resonators (Fig. 1),whose resonant frequencies are readily adjusted by slid-ing the cones along the pillars. LEGO bricks of this typewere already employed in prior works to explore the ef-fects of disorder in locally resonant metamaterials [48, 51]through an experimental platform that is also suitable forinvestigations in the the context of quasi-periodic media.Indeed, the versatility of the platform enables the ex-perimental mapping of the Hofstadter-like resonant spec-trum of the beam, which to our knowledge is presentedhere for the first time for an elastic medium. Numeri-cal simulations are conducted and allow for predictionsof the spectral gaps, along with their topological classi-fication based on the framework presented in [18], whilethe experimental observation of an Hofstadter’s spectrumin acoustic waveguides is reported in [20]. Our exper-iments also illustrate the presence of topological edgestates spanning the gaps, which are localized at one ofthe boundaries of the beam. In contrast to the investiga-tions presented in [22], this work considers equally spacedresonators whose resonant frequencies are modulated in-stead of their spacing. This alternative approach may beadvantageous especially for tunable devices, whereby themodulation of local properties such as the resonant fre-quency of piezoelectric shunt circuits [52–54] may be em-ployed for versatile platforms without the need of physi-cal reconfiguration.The considered elastic LEGO beam (gray solid inFig. 1) is equipped with an array of resonators of equalspacing a , whose resonance frequencies are modified bysliding the cones (blue) along the pillars (black). Theheight h n of the cone in the n -th resonator is assignedaccording to the law h n = h + ∆ h sin(2 πθn + φ ) , (1)where h and ∆ h denote the offset and amplitude of a r X i v : . [ phy s i c s . a pp - ph ] J a n FIG. 1. Schematic of LEGO beam with pillar-cone resonators.A modulation of the cones’ heights h n = h +∆ h sin(2 πθn + φ )is employed, represented by the dashed red line. The figureillustrates a periodic domain obtained with θ = 1 /
4, compris-ing 4 resonators per unit cell. the modulation. Such modulation can be visualized asthe sampling of a sinusoidal waveform h ( x ) = h +∆ h sin(2 πθx + φ ) (dashed red line in Fig. 1) at locations x n = n [16]. Alternatively, the law can also be visualizedas the projection from an array of circles. [19, 21, 22] Theparameter θ controls the periodicity of the modulation:rational θ values of the form p/q with co-prime p, q iden-tify periodic structures with q resonators per unit cell,while irrational θ values are associated with quasiperi-odic domains. The illustration in Fig. 1 exemplifies aperiodic domain with θ = 1 /
4, comprising 4 resonatorsper unit cell. The phase (or phason) φ does not affectthe periodicity of the domain, but is a parameter whichreveals the existence of edge states and defines their lo-calization at one of the two boundaries. [16, 38, 39]The spectral properties of the beam are first charac-terized for the case of uniform distribution of resonators( θ = 0) by conducting 3D finite element (FE) simulationsand subequent experimental verification. Following theprocess detailed in the Supplemental Materials (SM) [55],the local resonant gap produced by the pillar-cone res-onator is mapped as a function of the height of the cone h . This shows that the center frequency of the gapvaries approximately from 200Hz to 350Hz as h is variedfrom 0 to 30mm (other relevant physical parameters areprovided in [55]). The spectrum for the quasi-periodicmodulation (Eqn. 1) is then conveniently mapped byusing periodic approximants [18, 21]. To this end, theeigenfrequencies of a finite beam comprising N = 100resonators are computed by applying periodic boundaryconditions for θ varying in steps of 1 / h , ∆ h = 15mm areconsidered to explore the entire range of height variation(from 0 to 30mm). Results reported in Fig. 2(a) showthe Hofstadter-like spectrum of the beam as a functionof θ . For θ = 0, a single local resonant gap exists inthe f = { , } Hz range, which corresponds to thegap predicted by the analysis of uniform resonators with h = 15mm [55]. Interestingly, such local resonant gapis quickly transformed into a series of additional gapsat lower and higher frequencies as θ is varied through the reconfiguration of the cones’ heights defined by themodulation law of Eqn. 1. We note that in the analysiswe only include modes with a significant component ofmotion that is perpendicular to the beam axis (bendingmodes). These are separated from the other polariza-tions by considering a polarization factor that filters outpredominantly longitudinal or torsional modes. Defini-tion of the polarization factor, and of the modal filteringprocess are found in [55]The topological properties of the spectrum are re-vealed by computing the integrated density of states(IDS) [18, 21], which is displayed in Fig. 2(b). The ren-dering of the IDS highlights straight lines which are as-sociated with the spectral gaps. Non-horizontal lines in-dicate non-trivial gaps, identified by a nonzero Chernnumbers that are evaluated from the slope of the cor-responding gap line. [18, 20] The most prominant gapin Fig. 2(a) (rising up from 400-800 Hz approximately)is labeled by the fitting highlighted in Fig. 2(b) (whitedashed line), illustrating that IDS = 2 + θ for that gap,and thus its Chern number is C = 1.The experimental investigations have as a first goal themapping of the Hofstadter-like spectrum of Fig. 2(a). Abeam comprising N = 42 resonators is clamped at theright end and excited at the left end by an electrody-namic shaker [55]. A broadband pseudo-random signal inthe range f = { , } Hz is applied to excite the bend-ing motion of the beam. The motion is recorded by ascanning laser doppler vibrometer (SLDV) at a total of80 points aligned along the span of the beam. Finally,the transmission is calculated by computing the ratiobetween measured velocity at the measurement pointsto the beam velocity at the location of shaker, i.e. theinput point. Results presented in Fig. 3 compare thesimulation results computed via 3D FE (a) with experi-mental measurements (b). A total of 20 experiments areconducted for θ varying from 0 to 0 .
5, and results arepresented in the range θ ∈ [0 ,
1] for better visualization,which utilizes symmetry in the quasi-periodic pattern al-lowing mirroring the results obtained for θ ∈ [0 , . C = 1.Next, we experimentally demonstrate the existence oftopological edge states spanning the non-trivial gaps. Weconsider a representative case of θ = 0 .