On pure Willis coupling: Brillouin-zone and finite-lattice analysesa)

Momentum bias within one-dimensional elastic media introduces non-reciprocal phenomena in the propagation of elastic waves, manifesting as Willis coupling in the governing wave equation. To the best of our knowledge, the implications of Willis coupling being the sole coupling mechanism within elastic media and the understanding of non-reciprocity through finite structure analysis remain unexplored. In this paper, a non-reciprocal wave phenomenon with pure Willis coupling is synthesized using feedback control in mechanical lattices consisting of masses, grounded springs, and linear actuators. The proposed system presents a unique lack of physical connections between masses, allowing for pure Willis coupling through feedback forces exerted by the linear actuators. The dynamical behavior of a unit cell segment from an infinite lattice chain is considered, and the emergent non-reciprocal dispersion relation is detailed, including an analytical quantification of the Brillouin-zone translation resulting from pure Willis non-reciprocity. Analytical derivations of the eigenpairs of a finite lattice configuration are established, and the theoretical analyses reveal the mechanism of non-reciprocity through the lens of the lattice's natural frequencies and corresponding mode shapes.