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The authors realize a programmable mechanical metamaterial using a generalized rotor-chain lattice with two rotational degrees of freedom per unit cell that concurrently supports both kinematic and band topology. This architecture bridges a fundamental physical divide, enabling simultaneous and independent control of zero-frequency floppy modes and finite-frequency wave transport within the same medium. Using scanning laser Doppler vibrometry, they experimentally confirm the coexistence of a boundary floppy mode and a domain-wall state that undergo topological transitions at a shared symmetry-controlled critical geometry.
Static mechanical deformation and high-frequency wave routing no longer require mutually exclusive hardware: a single programmable metamaterial can now execute both via independent, coexisting topological channels.
Topological mechanical metamaterials generally derive their robustness from one of two distinct mechanisms: kinematic topology, which localizes zero-frequency floppy modes through geometric compatibility, or band topology, which localizes finite-frequency waves through topological bandgaps. Because these mechanisms arise from fundamentally different physical principles, mechanical systems are typically engineered to exploit either static or dynamic topological functionality, but not both simultaneously. Here we demonstrate that a single mechanical metamaterial can host two independent topological channels governing static and dynamic response within the same architecture. Using a generalized rotor-chain lattice with two coupled rotational degrees of freedom per unit cell, we realize the coexistence of a topological polarization mode and a finite-frequency topological band mode. We show that the two channels are governed by distinct invariants and can be programmed independently through geometry, angular asymmetry, and stiffness dimerization. Remarkably, despite being governed by distinct topological invariants and controlling different physical responses, the two channels undergo topological transitions at a common symmetry-controlled critical geometry. Experimental measurements using scanning laser Doppler vibrometry confirm simultaneous localization of a boundary floppy mode and a finite-frequency domain-wall state in the same structure. These results establish dual topological channels as a general design principle for multifunctional mechanical metamaterials and demonstrate how static deformation and dynamic wave transport can be programmed independently within a single architecture.