Dynamics and Topology of Nonlinear and Aperiodic Metastructures

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Xia, Yiwei
Erturk, Alper
Ruzzene, Massimo
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Metamaterials and metastructures (i.e., metamaterial-based finite structures with specified boundary conditions) enable various properties that are not found in ordinary materials and structures. For example, locally resonant (LR) elastic/acoustic metastructures exhibit bandgaps at wavelengths much longer than the lattice size, enabling low-frequency vibration/noise attenuation. Most investigations in this domain have considered nominally identical linear resonators that are periodically attached to the structure. In these cases, the LR bandgap depends on the natural frequency of the resonators, and the bandwidth is limited by the added inertia and the electromechanical coupling in mechanical and electromechanical metastructures, respectively. In this work, the LR metastructure concept is expanded in two directions: one is including nonlinearity, specifically considering bistable attachments; the other one is including aperiodicity, namely by considering the resonators in quasiperiodic arrangements. First, numerical and experimental results show the amplitude-dependent enhancement of the attenuation bandwidth in LR metastructures via bistable attachments, and demonstrate that the vibration attenuation band offered by nonlinear interwell oscillations is substantially wider than the linear LR bandgap. Next, the dynamic behavior and topological properties of LR mechanical metastructures hosting quasiperiodic distributions of resonators are investigated. It is shown both numerically and experimentally that quasiperiodic placement of resonators introduces additional bandgaps that are topologically non-trivial and host protected edge-localized modes. The findings suggest the application of quasiperiodic resonators as an effective way to achieve attenuation over multiple frequency bands and to control vibration localization at desired frequencies. In analogy with mechanical LR metastructures with quasiperiodic arrangements of resonators, electromechanical LR metastructures with quasiperiodic resonant shunt circuits are explored. Topologically non-trivial bandgaps and associated edge-localized modes are observed experimentally in an electromechanical metastructure by leveraging digitally programmable synthetic impedance circuits. Additional studies investigate the transitions of localized mode through slow phase modulation, with experimental observation of temporal topological pumping in an electromechanical structure under stiffness modulation via variable negative capacitance circuits.
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