Acoustic Nonreciprocity via Boundary-driven Spatiotemporal Self-modulation Enabled by Nonlinear Mechanical Mode Interaction

Abstract

Inspired by examples from electromagnetics, acoustic nonreciprocity refers to the unusual phenomenon where waves (sound, elastic, or mechanical) propagate through a material differently in one direction versus the opposite. Analogously to the electronic diode, acoustic nonreciprocity is of particular interest in the context of enabling acoustic logic and signal processing devices, as well as low-loss (backscattering immune) signal propagation. One approach to disrupt reciprocity is to break time-reversal symmetry via spatiotemporal (ST) manipulation of a material's properties through which a wave is propagating. However, all hitherto experimentally demonstrated examples of acoustic nonreciprocity via ST modulation have achieved said modulation by active, distributed control of the material. Such active, distributed control is problematic for implementations involving high frequencies, small length scales, or many modulated elements. Here, we present two related variations of an alternative approach, namely self-modulation via nonlinear mechanical mode interaction, induced by driving from the material boundary. The first system we study consists of a discrete material composed of designer nonlinear elements, wherein a longitudinal mechanical wave excited from the material boundary modulates the shear stiffness of the material such that nonreciprocal shear wave propagation should be achieved. For this system, we formulate the equations of motion for the discrete, nonlinear ST modulated spring-mass system, and calculate the wave dispersion therein approximated to several harmonic orders. Additionally, we outline geometric design constraints of the nonlinear springs and masses comprising the material, that enable control over the relative speed and coupling strength of longitudinal and shear waves, along with the nonreciprocal band structure. To validate our theoretical framework, we present both experimental measurements and numerical simulations. The second approach is based on the Aubry-André-Harper (AAH) model, originally described in the context of quantum electronics, wherein a lattice is modulated by periodic on-site potentials. In this study, we introduce a mechanical analog of the AAH model. The structure comprises a one-dimensional lattice featuring a zero-energy mode parametrized by an angle. Depending on the value of this angle, the system exhibits finite-frequency edge modes, as a consequence of the non-zero Chern numbers of the bulk bands. The nonlinear interaction between these finite frequency edge modes and the zero mode results in a ``self-pumping effect''. This phenomenon is where an excitation from the end with the edge mode activates the zero mode, subsequently transporting (pumping) the excitation from that side to the opposite side of the system via a bulk mode, leading to nonreciprocity. We begin our investigation by deriving the equations of motion for the system to map out the spectrum of the system as a function of the angle of its last rotor. Then we used discrete element model to simulate the time response of the system, later we use fast Fourier transform to analyze the system in the frequency domain.