Demonstrating an In Situ Topological Band Transition in Cylindrical Granular Chains

We numerically investigate and experimentally demonstrate an in situ topological band transition in a highly tunable mechanical system made of cylindrical granular particles. This system allows us to tune its interparticle stiffness in a controllable way, simply by changing the contact angles between the cylinders. The spatial variation of particles’ stiffness results in an in situ transition of the system’s topology. This manifests as the emergence of a boundary mode in the finite system, which we observe experimentally via laser Doppler vibrometry. When two topologically different systems are placed adjacently, we analytically predict and computationally and experimentally demonstrate the existence of a finite-frequency topologically protected mode at their interface.

Figure 1

Schematic of the experimental setup. The top inset illustrates a cut section of the 3D-printed enclosure. The bottom inset shows the contact angles in the dimer chain and the representative spring-mass system.

Figure 2

Representative configurations of the infinite dimer chain to show its topological band transition as a function of $\alpha$. (a), (b), and (c) include the Bloch dispersion curves for $\alpha <{\alpha }_0$, $\alpha ={\alpha }_0$, and $\alpha >{\alpha }_0$, respectively. Bands are marked with the corresponding topological indices (0 and $\pi$) and the Bloch eigenmodes on their edges ([1, 1] and $[1,-1]$ for symmetric and antisymmetric oscillations in the dimer unit cell, respectively).

Figure 3

Topological transition and emergence of a boundary mode in the finite system. (a) Frequency spectrum evolution as a function of $\alpha$. Extracted experimental data (gray markers) match the edges of theoretical bulk bands (shaded area). Also, experiments (red markers) show the emergence of a boundary mode inside the band gap, which follows the trend of numerical simulations (black curve). (b) Zak phase of the band gap and its transition at $\alpha ={\alpha }_0$. (c)–(e) Numerical (black markers) and experimental (red markers) boundary mode profiles in terms of normalized velocity amplitude at $\alpha =5°$, 10°, and 15°.

Figure 4

Topologically protected modes arising from dimer configurations with (a) hard-hard and (b) soft-soft interfaces. Below are PSD plots obtained from the particles’ velocities measured experimentally. At the bottom are the TP mode shapes extracted from experiments (red markers) and eigenanalysis (black markers).