Twisting Kelvin Cells for Enhanced Vibration Control

TL;DR

This study shows that modest symmetry-breaking twists of the Kelvin cell, while preserving topology, can create and tune low-frequency band gaps in lattice metamaterials without adding mass. A Bloch–Floquet framework reveals two attenuation mechanisms: a Bragg-type gap from periodicity and a polarization-dependent gap from longitudinal–torsional coupling, the latter manifested as avoided crossings in the dispersion. An analytically informed lumped-parameter model, coupled with a frequency-dependent viscoelastic material description, explains and predicts the observed band gaps and attenuation, which are validated by SLA-fabricated three-cell specimens showing up to 20 dB reduction in transmission. The results offer a practical, manufacturable design rule for lightweight vibration isolation, with potential extensions to multi-directional lattices, defect engineering, and reduced-order homogenizations.

Abstract

This work investigates the propagation of elastic waves in periodic Kelvin-cell chains, focusing on symmetry-breaking geometric modifications induced by twisting the cell's faces. By imposing such twists, the original lattice topology is preserved, while mirror symmetries are strategically broken through modifying a single geometric parameter, allowing wave characteristics to be adjusted without additional resonators or mass augmentation. The complex-valued Bloch-Floquet analysis reveals that twisting activates two distinct wave attenuation mechanisms: Bragg-type band gaps associated with periodicity-induced scattering, and polarization-dependent band gaps arising from longitudinal-torsional mode coupling and avoided crossings. To obtain qualitative and quantitative insight into these mechanisms, a simplified analytical model with coupled translational and rotational degrees of freedom is considered. The finite-element wave transmission calculations are experimentally validated on SLA-printed three-cell specimens, for which wave attenuation reaches up to 20 dB within the predicted band-gap frequencies. Note that high prediction accuracy requires accounting for viscoelastic material behavior, underscoring the importance of material behavior on the wave propagation characteristics. Overall, the findings show that modest geometric modifications to a classical Kelvin-cell lattice can enhance wave-filtering behavior, offering a tractable design strategy for vibration control in lightweight architected lattices.

Figures (13)

• Figure 1: Illustration of the twisting operation. (a) The reference (achiral) configuration of the Kelvin cell with the local coordinate system and rotation angles $\theta_j^{\hat{i}}$ defining the twisting of the square faces. (b) The configuration with one axial twist obtained by rotating the top and bottom square faces about the local $\hat{z}$-axes in opposite directions by $\theta_{z{+}}^{\hat{z}} = -\,\theta_{z{-}}^{\hat{z}} = 45^\circ$. (c) The configuration with three axial (chiral) twists with all square faces rotated by $\theta_j^{\hat{i}} = \pm 90^\circ$ in opposite directions. (a)-(c) The connectivity pattern is preserved in all cases, while the ligament orientations are modified. Colors are for visualization only.
• Figure 2: Twisted unit cells obtained by the coordinate transformations shown in Fig. \ref{['fig:2_Sketch_TwistingApproach']}: (a) reference Kelvin cell; (b) the cell with one axial twist; (c) the cell with three axial twists. Note that the halves of the top and bottom square faces are removed to avoid overlapping while tessellating the cells into chains, which results in a periodicity constant $a = h_c$ as introduced in Fig. \ref{['fig:2_Sketch_TwistingApproach']}.
• Figure 3: Lumped mass--spring models for studying the low-frequency dispersion and wave-interaction characteristics of the reference and twisted Kelvin cell chains. (a) Longitudinal--torsional model with translational $u_n$ (along the propagation axis) and rotational $\varphi_n$ (about the propagation axis) degrees of freedom. Dashed lines indicate the linear elastic coupling between the degrees of freedom, with stiffness $K_{\mathrm{lt}} = K_{\mathrm{tl}}$; the diatomic variant uses alternating stiffnesses $K_{\mathrm{l},1}$ and $K_{\mathrm{l},2}$. (b) Shear--bending model with transverse $v_n$ and rotational $\phi_n$ (about the axis normal to the shear direction) degrees of freedom. Dashed lines denote the coupling stiffness $K_{\mathrm{sb}} = K_{\mathrm{bs}}$.
• Figure 4: Dispersion band structure of the reference Kelvin-cell chain and the corresponding analytical model. (a) Dispersion relations predicted by the analytical models, Eqs. \ref{['eq:disp_relation']} and \ref{['eq:flexural']}, reproducing the principal features of the low-frequency behavior of the Kelvin-cell chain, including the relative spacing of the longitudinal, torsional, and flexural branches. Model parameters were adjusted heuristically for overall correspondence and are listed in Appendix \ref{['Appendix:Analytical']}. (b) Numerically estimated dispersion relation of the reference Kelvin-cell chain. Color-coding highlights the axial rotational polarization computed from Eq. \ref{['eq:curl_polarization']}, to identify the mode polarization. Evanescent modes with $\mathrm{Re}(k)=0$ are shown as black dots, while modes with $\mathrm{Re}(k)\neq0$, $\mathrm{Im}(k)\neq0$ are shown as gray squares.
• Figure 5: The design procedure of the twisted unit cell. (a) The Kelvin cell with a top face twisted by $\theta_{z{+}}^{\hat{z}} = 45^\circ$ about the vertical axis. As this operation creates non-matching cell faces, the cell is duplicated by a mirror-translation with respect to the shown horizontal plane. (b) Resulting unit cell with a doubled characteristic length used in the subsequent analysis.