Metamaterials are artificial assemblies of existing materials in certain configurations that provide unique features not seen in naturally occurring materials.
Image Credit: Gts/shutterstock.com
They show considerable potential in many applications, including vibration insulation, wave-guiding, noise control, and energy harvesting technologies.
While the foundation of metamaterials research is typically found in their periodic designs, research is now spreading into the realm of quasi-periodic metamaterials, where the absence of translational periodicity presents new difficulties and opportunities.
Our research employs reconfigurable LEGO parts and laser Doppler vibrometry to demonstrate the unusual properties of quasi-periodic materials, such as localized vibration modes and directional wave engineering.
In metamaterials research, resonators are commonly used to create so-called “locally resonant” metamaterials. Typically, arrays of identical resonators are mounted atop an elastic framework.
This results in the formation of band gaps, which are frequency regions where elastic wave propagation is prohibited, leading to an overall decrease in vibration levels around the resonance frequency of the attachments.
Several research groups are investigating this notion as an appealing approach to vibration and sound insulation materials.
The classical theory in this field is based on periodic arrangements made up of evenly spaced resonators tuned to the same frequency. Recently, there has been an increase in interest in exploring the characteristics of disordered and quasi-periodic resonator arrays, going beyond the periodic designs.
Figure 1. Experimental setup for vibration measurements of LEGO assemblies. Image Credit: Courtesy of Authors
The experiments use LEGO pieces as basic implementations of mechanical resonators (Fig. 1). Each resonator consists of a pillar (black) and a cone (blue). The resonance frequency is adjusted between a minimum and a maximum value of about 200 – 350 Hz by moving the cone along the pillar (as shown in Fig. 1).
These resonators are mounted to a base LEGO beam or plate in quasi-periodic combinations that vary in wave propagation and vibration characteristics.
An electrodynamic shaker is used to vibrate the samples, and a Polytec Scanning laser Doppler vibrometer (SLDV) is used to record the frequency response along the flat surface of the samples (the side without resonators).
Figure 2. Experimental results of 1D LEGO beams. Image Credit: Courtesy of Authors
In the first study, Polytec looked at one-dimensional elastic beams using quasi-periodic arrays of LEGO resonators.1
A total of 42 resonators were mounted to the LEGO beam, with their heights allocated according to a quasi-periodic pattern. The height hn of the nth resonator is calculated using the following equation: hn=h0+Δh sin(2πθn).
It comprises a mean height of h0, plus a sinusoidal variation of amplitude Δh and wavelength 1/θ. Changing the θ parameter creates a range of patterns, with rational θ values defining periodic patterns and irrational θ values defining quasi-periodic patterns.
University of Colorado researchers captured the frequency response variation as a function of θ by rearranging our sample and running several trials.
Fig. 2 shows a summary of the results in a color-map plot, with the color corresponding to the response averaged along the beam’s span as a function of θ (x-axis) and frequency (y-axis).
Yellow lines in the figure represent the change of the response peaks linked with the beam’s resonances, while blue regions correspond to band gaps that result in low vibration levels across the beam. The image creates a spectral map that resembles a Hofstadter Butterfly, thus the butterfly image.
The original butterfly refers to Douglas Hofstadter’s 1980s discovery of the fractal spectrum of electrons subjected to a strong magnetic field, which is related to the theory of the Quantum Hall Effect.
Interestingly, this work reproduces comparable features by employing reconfigurable elastic resonators rather than electronic systems with magnetic fields. This striking resemblance motivated the company’s featured publication in the Applied Physics Letters journal.1
Because of the limited size of the sample, it was able to capture a few features of the “fractal” spectrum. One of the distinguishing features of the Quantum Hall Effect is the formation of “edge states”, which are currents that propagate solely around the edge of a sample rather than into its core.
Similarly, on the elastic platform, this property shows as vibration modes concentrated at the beam’s edges.
Fig. 2 depicts an example of a localized mode at a frequency of about 450 Hz, comparing the numerical results produced by Finite Element Analysis (FEA) with the experimental mode shape observed by the vibrometer.
Figure 3. Experimental results of 2D LEGO plates for fixed frequency of 430 Hz. Image Credit: Courtesy of Authors
Excited by our discoveries, we investigated the properties of a two-dimensional square lattice of LEGOs mounted on a plate.2
The heights of the pillars are also modulated sinusoidally to produce quasi-periodic patterns. In this scenario, the relevant θ parameter represents the twist angle between the sinusoidal modulation and the fixed pillars in space.
Twisted bilayer electronic systems, which display remarkable features such as electronic superconductivity, inspired Polytec’s decision to achieve quasi-periodicity by twisting.
The researchers wanted to see how the twist angle of the modulation function, which causes different configurations of the pillars’ heights to generate moiré patterns, altered the elastic wave propagation within the LEGO plate.
It discovered that the addition of the LEGO resonators makes wave propagation in the plate highly directional at particular frequencies and that adjusting the twist angle may allow remarkable control over this behavior.
The twist angle, in particular, causes transitions from directional to non-directional wave propagation (or vice-versa), depending on the frequency. Fig. 3 shows one example with a frequency of 430 Hz, illustrating numerical findings and comparing them to experimental results for twist angles of 0, 15, 30, and 45 degrees.
In the untwisted case (θ=0), wave propagation in the plate is not directional, resulting in oval wavefronts. Hence, representing the wave field in reciprocal space as a function of wavenumbers kx, ky (obtained through Fourier Transformation) defines a closed elliptic contour.
By increasing the twist angle and rearranging the pillars, the wave contours at that fixed frequency widen and become “hyperbolic” contours, which describe extremely directional waves. In addition, the angle determines the direction of wave propagation.
This is a topological transition because the wave contours change from closed to open when non-directional wave propagation becomes directional. Polytec’s research article published in Advanced Science includes further examples.2
Our results demonstrate the extraordinary features of quasi-periodic metamaterials on a basic platform of LEGO pieces. The demonstrated characteristics are inspired by quantum matter concepts, which, when applied to elastic materials, result in unconventional wave localization and guiding features.
The LEGO implementation’s simplicity enables us to explore these complicated phenomena on table-top experimental platforms.
Moving ahead, the depicted principles can be applied to more complicated metamaterial implementations that integrate quasi-periodic patterns, potentially leading to novel materials with excellent vibration and wave propagation features.
References
- Rosa, M. I., Guo, Y., & Ruzzene, M. (2021). Exploring topology of 1D quasiperiodic metastructures through modulated LEGO resonators. Applied Physics Letters, 118(13), 131901. https://aip.scitation.org/doi/full/10.1063/5.0042294
- Yves, S., Rosa, M. I. N., Guo, Y., Gupta, M., Ruzzene, M., & Alù, A. (2022). Moiré‐Driven Topological Transitions and Extreme Anisotropy in Elastic Metasurfaces. Advanced Science, 9(13), 2200181. https://onlinelibrary.wiley.com/doi/full/10.1002/advs.202200181
This information has been sourced, reviewed and adapted from materials provided by Polytec.
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