As triggered by the discovery of unique topologically protected states in topological insulators, abundant demonstrations of the corresponding topological states have recently been conducted in diverse domains. As a unique feature of the topological protection, the topological states are immune to backscattering such as defects or disorders. This attracts many researchers to this discipline, because the robustness of the topological states is beneficial for wave-guiding and energy harvesting purposes. More recently, higher-order topological insulators have been theoretically proposed. Subsequent experimental demonstrations, e.g., topological corner states and non-propagating zero-dimensional modes, have been realized in *periodic* bosonic systems.  In this work, we numerically and experimentally demonstrate another type of special non-propagating zero-dimensional localized state, so-called a mechanical analogue of the Majorana bound states (MBSs), in the *non-periodic* yet ordered mechanical system. Interestingly, such a non-propagating state cannot be destroyed by local perturbation.  

We choose a ubiquitous design that consists of a thin aluminum plate with steel bolts mounted as local resonators. The distribution of the bolts is implemented by applying the Kekulé distortion to the original honeycomb lattice to form a vortex in the center of the whole structure. The bolted plate with the Kekulé distorted pattern is a mechanical implementation of the celebrated Jackiw-Rossi model for Dirac fermions in two-dimensions. This topological binding mechanism can be mapped to a magnetic flux vortex in a topological superconductor since the Jackiw-Rossi states, in fact, are topologically identical to MBSs in magnetic flux vortices in p-wave superconductors. Therefore, we can realize a mechanical analogue of MBSs in the mechanical setting. In simulation, the resonators are modeled as spring-mass attached on a thin plate. Then the transverse motion of plate can be dealt as per the classical plate theory. We use the plane-wave expansion method to obtain the band structures of all surrounding patterns. We confirm that the bound state exists at the Dirac frequency of the undistorted patterns. We ensure that there is a common bandgap of all surrounding distorted patterns for the prevention of energy leakage.

As for the full bolted plate simulation, the multiple scattering method is introduced to obtain the wave-transmission maps by giving a point excitation at the center of the bolted plate. In experiments, we use a circular piezoceramic actuator to excite the chirp signal of flexural waves. We measure the out-of-plane motion of the plate via laser vibrometry with the point-by-point synchronizing measurements. Then, through the data post-processing, we construct the wave-field of the entire bolted plate in the frequency range of interest. We observe a topological bound state in both simulations and experiments, which are in an excellent agreement. More importantly, we verify that the bound state is completely insensitive to particle-hole (PH) symmetric perturbations.  This is consistent with the well-known fact that MBSs are protected by PH symmetry. We specifically show the effect of local-mass perturbations and PH-symmetric perturbations on the topological bound state. We confirm that the Majorana-like bound state is indeed robust against the PH-symmetric perturbations and maintains pinned to Dirac frequency. We foresee that this finding will open a new avenue for wave manipulation and enrich the control of topological states.


Demonstration of the Majorana-Like Bound State in an Elastic Bolted Plate

Division / Sponsors: 
Aerospace, Bioengineering, Applied Mechanics, Design Engineering Division, Advanced Energy Systems, Manufacturing Engineering Division, Noise Control and Acoustics, Nondestructive Evaluation, Diagnosis, and Prognosis (NDPD), Nuclear Engineering Division, Process Industries Division, Fluids Engineering, Heat Transfer, Materials, Microelectromechanical Systems, Safety Engineering and Risk Analysis, Computers & Information in Engineering


Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2020/volume/84478

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