[Paper Review] Tailored Topological Edge Waves via Chiral Hierarchical Metamaterials

Jacopo Maria De Ponti, Luca Iorio+4 more|arXiv (Cornell University)|Oct 3, 2022

Metamaterials and Metasurfaces Applications6 citations

TL;DR

This paper proposes a chiral hierarchical metamaterial lattice that enables precise, tunable control of topological edge waves in elastic frames by embedding micro-resonators with graded mass distributions. By preserving the overall mass and symmetry-breaking via spatially varying resonator lengths, the authors achieve robust, frequency-localized edge modes and demonstrate a topological rainbow effect, experimentally validated through SLDV measurements and FEM simulations, enabling targeted energy delivery for vibration control and energy harvesting.

ABSTRACT

Precise manipulation of the direction and re-direction of vibrational wave energy is a key demand in wave physics and engineering. We consider the paradigm of a finite frame-like structure and the requirement to channel energy away from critical regions, leaving them vibration-free, and redirect energy along edges towards energy concentrators for damping or energy harvesting. We design an exemplar frame metamaterial, combining two distinct areas of wave physics. Firstly, topological edge states taking an unconventional tetrachiral lattice. We control these highly localised protected edge states leveraging a hierarchy of scales through the addition of micro-resonators that impose tuneable symmetry breaking and reconfigurable mass. This allows us to achieve precise positional control in the macro-scale frame lattice, thereby opening opportunities for robust signal transport and vibration control. Experiments, theory, simulation are all utilised to provide a comprehensive analysis and interpretation of the physics.

Motivation & Objective

• To enable precise spatial and frequency control of vibrational energy in elastic structures for applications in vibration isolation and energy harvesting.
• To overcome limitations in conventional waveguiding by leveraging topological edge states that are robust against disorder and defects.
• To design a hierarchical metamaterial with tunable edge mode dispersion through graded micro-resonators without altering total mass or band gap.
• To demonstrate the topological rainbow effect by spatially grading resonator lengths to localize different frequency components along the edge.
• To validate the theoretical and numerical predictions through experimental measurements using a Scanning Laser Doppler Vibrometer (SLDV).

Proposed method

• The authors design a tetrachiral lattice unit cell with embedded lateral micro-resonators whose lengths are spatially graded to tune the edge mode dispersion while maintaining constant total added mass.
• The system leverages passive topological protection via broken space inversion symmetry (SIS) while preserving time-reversal symmetry (TRS), enabling robust edge states in the bulk band gap.
• The dispersion properties of the edge mode are engineered by varying the effective stiffness and mass distribution in the resonators, enabling control over group and phase velocities.
• A finite element method (FEM) simulation is used to model the full structure, including the edge wave propagation and frequency-dependent localization.
• Experimental validation is performed using a real prototype suspended on a frame, excited via an electrodynamic shaker, with displacement fields measured using a Scanning Laser Doppler Vibrometer (SLDV).
• Theoretical analysis is supported by Fourier transforms of time-domain responses to confirm the existence and localization of edge modes.

Experimental results

Research questions

• RQ1 Can topological edge states in a tetrachiral lattice be precisely controlled in space and frequency using hierarchical micro-resonators without altering the total mass or band gap?
• RQ2 How does spatial grading of resonator lengths affect the dispersion of topological edge modes and enable frequency-dependent localization?
• RQ3 To what extent can the topological rainbow effect be engineered in a passive, symmetry-preserving elastic metamaterial system?
• RQ4 How does the system maintain robustness against defects and disorder while enabling tunable wave localization?
• RQ5 What is the agreement between numerical simulations, experimental measurements, and theoretical predictions for edge wave behavior in this hierarchical design?

Key findings

• The experimental displacement field measured via SLDV shows strong agreement with FEM simulations, confirming the existence and localization of topological edge modes.
• The edge mode remains confined within the bulk band gap across all configurations, with no significant frequency shift, preserving topological protection.
• A topological rainbow effect is successfully demonstrated by grading resonator lengths from 5 mm to 13 mm across five cells, leading to frequency-dependent spatial localization along the edge.
• The group velocity of the edge wave is tuned such that higher frequencies propagate slower and localize closer to the excitation point, consistent with the rainbow effect.
• The system maintains mirror symmetry and supports bidirectional edge wave propagation, enabling symmetric energy delivery to concentrators.
• The use of graded resonator lengths without changing the total mass ensures adiabatic variation of homogenized properties, preserving the topological nature of the edge state.