Figure 4

Topological mechanics of monoatomic lattices with nonreciprocal elastic springs. (a–c) Three configurations of a mass-spring system representing a monoatomic lattice (\(m\) indicates the atomic mass). 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 (i.e., \({k}_{A\to B}>{k}_{B\to A}\)). 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 two neighbor atoms vibrate in two opposite directions. (d) Representations of the different stiffnesses of the considered nonreciprocal springs under tension and compression. (e) Classical band structures \(\omega \left(q\right)\) (nondimensional frequency (\(\omega /{\omega }_{0}\)) versus nondimensional wavenumber (\(q\))) are obtained when atoms vibrate to the right with positive nonreciprocal elasticity (\(\epsilon >0\)). (f) Complex band structures are obtained when atoms vibrate to the left with negative nonreciprocal elasticity (\(\epsilon <0\)) (real band structures \(\omega \left({q}_{r}\right)\) (above) and imaginary band structures \(\omega \left({q}_{m}\right)\) (below)). (g) The band-gap (\(\Delta \omega\)) versus the nonreciprocal elasticity parameter (\(\epsilon\)).