Designing unique mechanical modes through an extension of the quantum hopping method

We extend the hopping method, traditionally used in condensed matter physics to calculate electron distribution in crystalline structures, to mechanical systems. This extension offers an efficient and convenient approach for constructing the dynamical matrix, D , analogous to the quantum Hamiltonian. Not only does our method enhance computational efficiency, but it also provides valuable insights into the design of metamaterials. By exploring three key quantum hopping perspectives-connectivity, hopping interactions, and on-site potentials-we establish a unique design framework for mechanical systems. This framework enables the realization of similar connectivity topology in both ordered and disordered systems, anisotropic Dirac cones with confined modes, and near-zero topological modes. Our approach opens up broad perspectives for the design of metamaterials in classical mechanical systems.