Manipulating Waves with LEGO® Bricks:

A Versatile Experimental Platform for Metamaterial Architectures

Paolo Celli, Stefano Gonella

Department of Civil, Environmental, and Geo- EngineeringUniversity of Minnesota, Minneapolis, MN 55455, USA

In this work, we discuss a versatile, fully-reconfigurable experimental platform for the investi-gation of phononic phenomena in metamaterial architectures. The approach revolves aroundthe use of 3D laser vibrometry to reconstruct global and local wavefield features in specimensobtained through simple arrangements of LEGO® bricks on a thin baseplate. The agility bywhich it is possible to reconfigure the brick patterns into a nearly endless spectrum of topologiesmakes this an effective approach for rapid experimental verification and proof of concept of newideas, as well as a powerful didactic tool, in the arena of phononic crystals and metamaterialsengineering. We use our platform to provide a compelling visual illustration of important spa-tial wave manipulation effects (waveguiding and seismic isolation), and to elucidate importantdichotomies between Bragg-based and locally-resonant bandgap mechanisms.

Over the past two decades, acousto-elastic phononic crys-tals and metamaterials have received increasing attentiondue to their unique wave manipulation capabilities. Per-haps the most well-known property is the ability to openphononic bandgaps [1–5], i.e., frequency intervals of forbid-den wave propagation. Another distinctive feature is thefrequency-dependent anisotropy (or directivity) observed intheir spatial wave patterns [6–8] and the ability to featurea negative refractive index [9–11]. A number of metamate-rial architectures have also been proposed to design acousticlenses [12–14] and attain subwavelength imaging [15, 16].

In recent years, the field of phononics has seen a signifi-cant increase in the amount of experimental work [17]. Inthe realm of metamaterials design, the need for experimentsis pressing on two levels. At the conceptual developmentstage, it is beneficial to rely on a set of fast and economical,yet accurate experiments to obtain a quick proof of conceptof new ideas, or to compare and rank multiple candidate con-figurations. At the later stages of the process, more precise,ad hoc experiments are required to fully characterize the se-lected configuration and to verify its actual implementabil-ity as a device. For proof of concept, it is often too costlyand time consuming to work with specimens fabricated withadvanced manufacturing methods, especially when a largenumber of configurations must be tested; it may be con-venient to work with laboratory materials characterized bylow cost, fast manufacturing time and high reconfigurability.These attributes are also crucial in the development of pro-totypes for didactic demonstrations. 3D-printed materials,for example, display many of these characteristics; however,they are typically affected by unreliable mechanical proper-ties and high geometric variability; moreover, their response

is often tainted by high levels of damping that are detrimen-tal to the establishment of noise-free wavefields.

Inspired by these considerations, we here explore a versa-tile platform for rapid verification of phononic phenomena inmetamaterial architectures, based on the reversible assemblyof patterns of LEGO® bricks on a thin baseplate. LEGO®

components have been widely used, especially for didacticpurposes, in a variety of scientific environments, includingrobotics [18], biological sciences [19, 20] and physics [21, 22].In our approach, bricks of different size and shape can beplaced at designated locations on a baseplate as to produce avariety of periodic or disordered stub patterns with differentlength scales. Thanks to the non-permanent brick-plate con-tacts, it is possible to seamlessly transition between differenttopologies, effectively switching between radically differenttests, in a matter of minutes. Note that the acrylonitrile bu-tadiene styrene (ABS) baseplate, which serves as the wavepropagation support, displays relatively low viscosity, whichresults in low damping and moderate wave attenuation.

For our measurements, we rely on the sensing flexibility ofa 3D Scanning Laser Doppler Vibrometer (SLDV), which al-lows simultaneous in-plane and out-of-plane wavefield recon-struction. Thanks to their sensitivity and frequency band-width, and in light of the benefits of non-contact sensing,laser vibrometers have gained increasing popularity for theexperimental analysis of phononic structures [23–28]. Here,the scanning vibrometer contributes to the sensing side ofthe problem a dimension of reconfigurability that comple-ments the flexibility that is achieved, at the specimen fabri-cation level, by using bricks as building blocks.

Throughout this work, all the considered topologies areobtained as assemblies of cylindrical bricks on a baseplate.

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The baseplate features periodically-distributed studs—smallcylindrical protuberances where the bricks can be anchored.The arrangement of studs allows for several periodic con-figurations with various lattice constants. Note that thebrick-stud contact only relies on friction: no use of glue(which would undermine agile specimen reconfiguration) isrequired. A detail of bricks and baseplate is shown in Fig. 1a.In Fig. 1b we show the scanned surface (the backside of the

FIGURE 1: Experimental setup. (a) Detail of the base-plate with cylindrical bricks. (b) Scanned surface (backsideof the baseplate) and shaker. (c) Front view of the speci-men with the three scanning heads of the 3D-SLDV in thebackground.

plate), coated with a thin layer of reflective paint to increasethe quality of the optical signal. The excitation is impartedusing a Bruel & Kjaer Type 4809 shaker (also shown inFig. 1b), with a 0–20 kHz bandwidth. Sensing is performedby scanning the baseplate with a Polytec PSV-400-3D Scan-ning Laser Doppler Vibrometer (Fig. 1c). To probe the re-sponse over broad frequency ranges, we excite the specimenwith a pseudorandom waveform featuring a flat power spec-trum over the frequency range of interest. To improve thesignal-to-noise ratio, measurements at each scanned locationare repeated 20 times and averaged. A low-pass filter is usedto cut-off all spurious features in the signal above 20 kHz.

Our first objective is to use our combined fabricationand sensing platform to visually elucidate the phononicbandgap phenomenon. Phononic bandgaps are frequencyintervals in which wave propagation is forbidden. The gen-eration of bandgaps can be traced back to two main mecha-nisms: Bragg scattering and locally-resonant effects. Braggbandgaps are due to the destructive interference betweencascades of waves scattered at the internal interfaces in thematerial. They require periodicity in the structure and man-ifest when the wavelength of excitation approaches the char-acteristic size of the unit cell. Locally-resonant bandgaps area byproduct of energy-localization mechanisms; they takeplace when the frequency of excitation approaches the res-onance frequencies of some “microstructural” elements [5].

They are subwavelength in nature, i.e. they arise when thewavelengths are about one order of magnitude larger thanthe unit cell size. In the following, we leverage the versatilityof our experimental setup to illustrate the duality betweenthese mechanisms.

Our first specimen comprises 420 bricks arranged as shownin Figs. 2a and b. For convenience, we label this topol-ogy “dense”. Fig. 2c shows the frequency response function(FRF) calculated from surface measurements using the vi-brometer in the 0–14 kHz range; the average out-of-plane ve-locity vPC

z measured at the scan points inside the brick-filledarea (the actual phononic crystal zone) is divided by the av-erage velocity vbasez measured along the bottom edge of thescanned surface. We observe two bandgaps (shaded areas):a narrow band approximately in the 1.8–2.6 kHz range, anda wider one approximately in the 5.5–11 kHz range. Snap-shots of the steady-state response at selected frequencies areshown in Figs. 2d–h. In all plots, the velocity ranges from−0.4 vbasez,max to 0.4 vbasez,max, where vbasez,max is the maximum am-plitude recorded along the bottom edge of the scanned re-gion at each frequency. The wavefield response provides vi-sual evidence of both bandgaps: Figs. 2d, f, h, correspondingto propagation regions, feature appreciable oscillations overthe entire scanned area; Figs. 2e and g, corresponding toattenuation zones, feature nearly no motion in the crystalregion (bordered by the dashed line).

To investigate the dual nature of the observed bandgaps,we test their robustness against changes in topology. Herewe invoke the notion that Bragg bandgaps are sensitive tochanges in lattice spacing and to relaxation of the periodic-ity, while locally-resonant bandgaps only depend upon theavailability of resonating elements. In Fig. 3, the referencedense topology (Figs. 3a and b) is compared to two othertopologies that are assembled via rapid reconfiguration ofthe brick pattern. The configuration in Fig. 3c is obtainedby downsampling (by half) the original topology while re-taining a periodic arrangement. The corresponding FRFin Fig. 3d still displays two main bandgaps: the gap in theneighborhood of 2.2 kHz is preserved, albeit narrower than inthe dense topology; the larger gap has shifted towards lowerfrequencies (3.5–6 kHz range, approximately). Shifting to-wards lower frequencies (longer wavelengths) in response toincreases in lattice constant can be seen as the signatureof Bragg scattering, which indeed requires compatibility be-tween unit cell size and wavelength. Fig. 3e, in contrast, isobtained by rearranging the 210 bricks into a (MATLAB-generated) disordered pattern. The FRF in Fig. 3f high-lights the survival of the 2.2 kHz bandgap while the highergap has vanished. The persistence of the low frequency gapacross all topologies highlights its periodicity-independenceand hints at underlying locally-resonant mechanisms—anaspect recently investigated by Rupin et al. [28]. The in-tensity of the resonant mechanisms depends on the numberof resonators that can contribute to energy trapping. Thisconfirms the reduction in the first bandgap width in goingfrom the case of Fig. 3a to those of Figs. 3c, e.

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FIGURE 2: Bandgap analysis for a dense periodic brick topology. (a,b) Specimen with detail of the brick arrangement. (c)Frequency response function (average out-of-plane velocity normalized by the average value measured along the bottom edge).The shaded areas highlight the bandgap regions. (d-h) Laser-acquired wavefields sampled at five reference frequencies; thedashed lines delimit the region occupied by the bricks. (d) 1.5 kHz. (e) 2.25 kHz. (f) 3 kHz. (g) 7.25 kHz. (h) 13 kHz.

FIGURE 3: Effects of scale coarsening and periodicity re-laxation. (a) Dense periodic topology (420 bricks). (b)FRF for the topology in (a). (c) Coarse periodic topology(210 bricks). (d) FRF for the topology in (b). (e) Randomassembly of 210 bricks. (f) FRF for the topology in (e).

To substantiate our hypothesis regarding the locally-resonant nature of the 2.2 kHz bandgap, we attempt to re-construct experimentally the vibrational behavior of a singlebrick (mounted at the center of the baseplate) at different

frequencies falling inside and outside the bandgap deemedto be locally-resonant; a detail of the brick, coated in a thinlayer of reflective paint, is shown in Fig. 4a. For this task,we define a dense cylindrical scan grid on the surface of thebrick, as well as a coarser rectangular grid on the regionof the plate immediately surrounding the brick. The colorgiven to the scanned points is proportional to the RMS of thethree measured velocity components; at each frequency, thevelocity is normalized by the maximum velocity recordedover the entire scanned region at that frequency, to high-light the relative motion between brick and baseplate. Theblack dots mark the original position of the scan points.Fig. 4b shows the brick-and-plate motion at 1000 Hz (beforethe bandgap). We can see that the brick moves in the out-of-plane direction in phase with the plate substrate, whichundergoes significant deformation. Fig. 4c pinpoints a fre-quency near the onset of the bandgap, 1950 Hz: the brickundergoes large tilting motion, while the points on the plateare approximately still. Fig. 4d represents a post-gap fre-quency (3000 Hz) where, once again, we observe significantmotion of both the plate and the brick. We conclude that,only at the onset of the bandgap, we observe large relativemotion between stub and substrate: this provides evidenceof energy trapping and highlights the locally-resonant na-ture of the bandgap.

We can further exploit the reconfigurability of our spec-imens to illustrate (using the same brick set) a variety ofspatial manipulation phenomena, such as waveguiding andseismic isolation, at different wavelengths. The first topol-ogy, shown in Fig. 5a, is obtained by assembling a densesquare phononic crystal and by introducing a two-unit-cellwide snake-like defect path. Defect paths are known to act

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FIGURE 4: Experimental reconstruction of the motion of acylindrical brick at three frequencies in the neighborhood ofthe locally-resonant bandgap. (a) Detail of the brick. (b)Motion at 1000 Hz, before the presumed bandgap region.(b) Motion at 1950 Hz, near the onset of the bandgap. (d)Motion at 3000 Hz, above the bandgap.

as waveguides when the frequency of excitation falls insidea bandgap for the crystal. Consistently with the bandgapduality discussed above, waveguiding is attainable for wave-lengths that are comparable [29–31] or larger [25, 32] thanthe unit cell. In Fig. 5b we obtain subwavelength waveguid-ing (2512.5 Hz), while in Fig. 5c we report waveguiding atshorter wavelengths (7000 Hz, inside the Bragg gap).

We then proceed to rearrange the bricks into the topol-ogy shown in Fig. 5d, which implements a phononic crys-tal realization of the University of Minnesota M logo. Thewavefields obtained at 2362.5 Hz and 7000 Hz are shown inFig. 5e and Fig. 5f, respectively. We can see how the en-ergy is predominantly confined outside the crystal region;this effect is especially pronounced in the Bragg bandgap,where the response decays inside the M contour within 1–2cell layers. This configuration, in conjunction with the abil-ity to scan both interior and exterior of the crystal, providesan eloquent lab demonstration of the potentials of PCs forvibration and seismic isolation [33, 34].

As a summary, in this work we have illustrated how,through a simple assembly of LEGO® bricks, we can gen-erate metamaterial architectures with locally-resonant andBragg-based bandgaps. The reconfigurability of the brickpatterns, together with the availability of bricks of any sizeand shape, can be leveraged to assemble and test a plethoraof metamaterial architectures, thus serving as a versatileexperimental platform for proof of concept lab tests. Webelieve that the intuitive and tangible nature of toy-basedspecimens, along with the visualization capabilities of lasersensing, can provide a new dimension of intuitiveness to the

FIGURE 5: Experimental evidence of waveguiding and iso-lation effects. (a) Snake-like waveguide. (b) Waveguide re-sponse at 2512.5 Hz. (c) Waveguide response at 7000 Hz.(d) Metamaterial realization of the Minnesota M logo. (e)Image of the M at 2362.5 Hz. (f) Image of the M at 7000 Hz.

field of experimental phononics and act as perfect platformfor new and far-reaching teaching and outreach activities inthe realm of wave mechanics and metamaterials engineering.

The inspiration for this work sprouted from ideas origi-nated during the organization of outreach activities withina project sponsored by the National Science Foundation(grant CMMI-1266089). We are particularly indebted toNathan Bausman and Davide Cardella for their insight andassistance with the experiment setup. We also wish to thankJeff Druce and R. Ganesh for their valuable input.

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Supplementary material

The experimental results shown in this supplementary material section are designed to support and/or complement thosereported in the main manuscript.

Details on the experimental setup

To facilitate the reproduction of our experimental results, we here report details regarding the data acquisition settingswe chose in the Polytec PSV 9.0 Acquisition software. The acquisition is performed directly in the frequency domain (theFast Fourier Transform is performed automatically within the acquisition system by the software). As already mentioned,to eliminate non-repeatable noisy features, measurements are repeated 20 times at each scanned location. We also resortto a low-pass filter with cutoff at 23 kHz. As far as the vibrometer channel is concerned, we select a 5 V range and ACcoupling. We acquire in the 0–20 kHz frequency range, and we concentrate on the 0–14 kHz band in postprocessing. Thesampling frequency is fs = 51.2 kHz and the number of FFT lines is 1600, resulting in a frequency resolution of 12.5 Hz.The selected velocity decoder is the digital VD-08-10 mm/s/V, that allows acquisition up to 20 kHz. The excitation is apseudorandom waveform with maximum amplitude of 650 mV. The excitation signal is amplified using a Bruel & KjaerType 2718 Power Amplifier, with gain set to 30 dB.

Wave characterization of the baseplate

To further confirm the brick-dependent nature of the bandgaps discussed throughout this work, we analyze the responseof the baseplate with no bricks attached. The plate and a detail of the studs arrangement is shown in Figs. S1a and b. In

FIGURE S1: Steady-state analysis of the brick-free baseplate. (a,b) Baseplate and detail of the studs. (c) Frequency responsefunction (average out-of-plane velocity normalized by the average value measured along the bottom edge). (d-h) Laser-acquiredwavefields sampled at the same five reference frequencies analyzed in Figs. 2d–h. (d) 1.5 kHz. (e) 2.25 kHz. (f) 3 kHz. (g)7.25 kHz. (h) 13 kHz.

Fig. S1c we show the frequency response function of the baseplate (for the FRF we use the same output metric used forthe other topologies across the manuscript). Once we remove the bricks, we clearly see that no bandgaps are present in the0–14 kHz range. This aspect is corroborated by the snapshots of the plate’s steady-state response shown in Figs. S1d–h:the structure is able to propagate waves at all frequencies, including those that, in the dense topology, corresponded tobandgap regions (Figs. S1e and g, corresponding to 2.25 kHz and 7.25 kHz, respectively). Note that, due to the presenceof the small cylindrical studs, the baseplate itself is a periodic structure; however, we did not detect any stud-related gapin the considered frequency range, which suggests that the amplitude of the associated scattering events is negligible.

We now use the response of the pristine plate to visually quantify the wavelength of plate waves having frequenciesthat fall inside bandgaps observed in the dense topology. Note that, for the dense topology, the unit cell characteristicdimension is a = 1.2 cm. In Fig. S2a we zoom on the baseplate response at 2.25 kHz, corresponding to the locally-resonantgap. Note that in Fig. S2, for visualization purposes, the velocity amplitude bar has been modified to range from −vbasez,max

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FIGURE S2: Wavelength identification from the baseplate response. (a) Baseplate response at 2.25 kHz, inside the locally-resonant bandgap of the dense topology. (a) Baseplate response at 7.25 kHz, inside the Bragg bandgap of the dense topology.

to vbasez,max. From the drawn arrow, we can see that the characteristic wavelength of the response can be estimated tobe λ ' 5.4 cm, a value corresponding to about 6 unit cells. In Fig. S2a we report the baseplate response at 7.25 kHz,corresponding to the Bragg gap. In this case, the estimated wavelength is λ ' 2.4 cm, corresponding to 2 unit cells. Thisresult appears to support the notion that locally-resonant gaps are subwavelength in nature, while Bragg ones take placewhen the wavelength is comparable to the unit cell size.

Crystal topologies with point defects

We further expand the portfolio of architectures that we can study with this approach by considering two additionaltopologies, both obtained by introducing point defects of various sizes within a dense topology. The specimen in Fig. S3ais obtained by removing a single brick. Its response in the locally-resonant bandgap region (Fig. S3b) shows that the defect

FIGURE S3: Experimental evidence of defect modes in phononic crystals with point defects. (a) Topology with small defect,(b) its response at 2512.5 Hz and (c) its response at 7000 Hz. (d) Topology with large defect, (e) its response at 2512.5 Hz and(f) its response at 7000 Hz.

is not large enough to produce subwavelength energy localization. In the case of Fig. S3c, corresponding to the Bragg gap,we can see how the energy localizes at the defect site. Note that, in this set of results, we can see a spurious noisy featurein the wavefield data. The persistence of this feature across the frequency spectrum indicates that this is an artifact ofthe acquisition process, probably due to a failed laser measurement at one of the scan points, and not the signature ofany abnormal behavior of the structure. The topology in Fig. S3d features a 3× 2 defect, obtained by removing 6 bricks.In Figs. S3e and f, we can see that the defect is large enough to induce energy localization in both locally-resonant andBragg bandgap regimes.

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