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The discovery of topological insulators in materials science revolutionized the concept of wave propagation by giving rise to the existence of edge modes that are immune to backscattering. Similarly, the tunability in waveguiding – including in-situ frequency modifications and path designation – can be highly useful in manipulating energy flow, which still remains an open challenge. Here we investigate topologically tunable mechanical metamaterials based on the quantum valley hall effect (QVHE) by utilizing the bi-stable Stewart platform (SP). Generally, topologically protected wave propagation can leverage two physical mechanisms: the quantum hall effect (QHE) and the quantum spin hall effect (QSHE). Compared to the QHE and the QSHE, the QVHE exploited in this study maintains the time reversal symmetry and can be achieved by using a relatively simple, passive system with one degree-of-freedom. The tunable system we propose and investigate in this study is made of a two-dimensional hexagon crystal and is composed of SPs at nodes connected by linear springs. Each building block can exhibit one of the two stable states of the SP, so that the C6 inversion symmetry of the lattice is broken while C3 symmetry is reserved. By changing the sequence of the bi-stable state in the SP, we can formulate two kinds of unit cells – marked as A and B – with different topological properties. Berry curvatures as well as corresponding eigenmodes are obtained to demonstrate the topological conversion between the two lattices. Then we conduct super-cell analysis by forming a 1-by-20 array of A and B unit cells. Band structure of the super-cell indicates the existence of edge modes over the while band gap, which appear at the interface of A and B unit cells. Based on this tunable property of bi-stable SP, we can easily form S-type and L-type (60 and 120 degree bents) topological paths in the 40-by-40 lattices without breaking the original geometry parameters. We then conduct the numerical simulations with these topological wave guides to verify the topological protection of the valley hall edge states from backscattering. The tunable system we proposed in this paper may pave a possible way to achieving tunability of topological metamaterials.
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