Figure 5.

Topological Mechanics of Diatomic Lattices with Nonreciprocal Elastic Springs. (a–c) Three configurations of a diatomic lattice model. Nonreciprocal springs made of the developed metamaterial are considered such that the stiffness is different when the spring is stretched/compressed from two opposite ends ($k_{A\to B}>k_{B\to A}$). The free body diagrams of the spring forces and the inertia forces are represented when (a) atoms vibrate to the right, (b) when atoms vibrate to the left, and (c) when each atom vibrates opposite to its two neighbor-atoms. (d) The band structure $\omega \left(q\right)$ of the diatomic lattice for different values of the nonreciprocal elasticity parameter, $ϵ=0.5$ (left), $ϵ=0$ (middle), and $ϵ=-0.5$ (right). (e) The acoustic and optical frequencies at different wavenumbers, $q=0,\pi /2$ and $\pi$, versus the nonreciprocal elasticity parameter $ϵ$. The insets represent the Floquet–Bloch eigenmodes. (f) Contour plots of the acoustic and optic eigenmodes $\mathrm{\Psi }\left(q\right)$ in the complex plan for different values of the nonreciprocal elasticity parameter $ϵ$.