A mechanical analogue of electromagnetic induction for waves in a chiral elastic structure

TL;DR

The paper addresses unidirectional wave propagation in a geometrically chiral elastic structure by introducing a central gyroscopic core (gyrocore helix) that combines geometric handedness with physical chirality. Using a Bloch-Floquet framework, it derives a dispersion relation from $\det[ extbf{C}(K)-oldsymbol{ olinebreak oldsymbol{ ext{ω}}}^2( extbf{I}- extbf{A})]=0$ and demonstrates that gyricity shifts dispersion branches in a spin- and frequency-dependent manner, with $oldsymbol{ ext{G}}=oldsymbol{ ext{γ}}/10$ linking the split eigenfrequencies to group velocity via $ ext{ω}^{\pm}- ext{ω}_0= ext{±}oldsymbol{ ext{G}}v_g$. The key finding is that the direction of energy transport at a fixed frequency can be controlled by the spinner’s vorticity, producing spatially invariant but energy-carrying modes at $Kd=0$, and thus enabling tunable, unidirectional waveforms. This mechanism offers a route to engineered elastic metamaterials for directional wave steering and energy harvesting, with broad implications for mechanical analogues of electromagnetic induction.

Abstract

Classical Faraday's law on electromagnetic induction states that a change of magnetic field through a coil wire induces a current in the wire. A mechanical analogue of the Lorentz force, induced by a magnetic field on an electric charge, is the gyroscopic force. Here, we demonstrate a mechanical analogy with a chiral elastic waveguide subjected to gyroscopic forcing. We study waves in an infinite mass-spring `gyrocore helix', which consists of a helix and a central line (gyroscopic elastic core). The helicoidal geometric chirality is considered in conjunction with a physical chirality, induced by gyroscopic forces. It is shown that the interplay between these two chiral inputs leads to the breaking of symmetry of the associated dispersion diagram, resulting in a unidirectional waveform with the direction of propagation being tunable through the gyricity.

Figures (11)

• Figure 1: A portion of the infinite 'gyrocore helix', where masses with gyroscopic spinners attached are depicted in blue. Use the Eigenglass application to view an enhanced three-dimensional representation.
• Figure 2: A close-up of the dispersion diagram for a right-handed gyrocore helix for (a) $\gamma = -10$ and (b) $\gamma = 10$. Note a right- and left-ward shift of the dispersion branches, respectively. The arrows indicate the higher (black) and lower (red) split eigenfrequencies at $Kd=\mp1$ which are shifted to $Kd=0$. Use the Eigenglass application (Supplementary Material) to see how the amount of deviation from the original dispersion curves is dependent on the spinner constant $\gamma$.
• Figure 3: An unwound two-dimensional representation of the infinite (three-dimensional) helix-like structure. The dashed lines indicate the connections between the generating unit cell $\Omega^{(0)}$ and its repeated copies below and above, namely the neighbouring unit cells $\Omega^{(-1)}$ and $\Omega^{(1)}$.
• Figure 4: A deconstruction of Figure \ref{['2D_helix-like_config']}, illustrating the four masses which have connections that extend outside of the generating unit cell $\Omega^{(0)}$. Springs and masses attributed to neighbouring unit cells are depicted by dashed lines and hollow circles
• Figure 5: A sparsity plot, indicating the entries of the stiffness matrix $\mathbf{C}(K)$ (black) superimposed with the sparsity of the spinners matrix $\mathbf{A}$ (blue). There are clearly $6M=72$ algebraic equations for the choice of $M=12$.