Non-Reciprocity in Structures With Nonlinear Internal Hierarchy and Asymmetry

Acoustic reciprocity is a property of linear, time invariant systems in which the locations of the source of a forcing and the received signal can be interchanged with no change in the measured response. This work investigates the breaking of acoustic reciprocity using a hierarchical structure consisting of internally-scaled masses coupled with cubically nonlinear springs. Using both direct results and variable transformations of numerical simulations, energy transmission is shown to occur in the direction of decreasing scale but not vice versa, constituting the breaking of acoustic reciprocity locally. When a linear spring connects the smallest scale of such a structure to the largest scale of another identical structure, an asymmetrical lattice is formed. Because of the scale mixing and transient resonance capture that occurs within each unit cell, it is demonstrated through further numerical experiments that energy transmission occurs primarily in the direction associated with the nonlinear coupling from the large to the small scale, thus signifying the breaking of reciprocity globally. This nonlinear hierarchical structure exhibits strong amplitude-dependency in which reciprocity-breaking is associated with specific ranges of excitation amplitudes for both impulse and harmonic forcing.