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We explore nonreciprocal vibration transmission in a nonlinear periodicwaveguide. Nonlinearity and asymmetry, the two necessary requirements fornonreciprocity, are both introduced within the unit cell of the periodicwaveguide. We focus primarily on the contribution of phase to the nonreciprocalsteady-state response of the system. To highlight the phase effects, which arerarely discussed in the literature, we investigate response regimes in whichnonreciprocity is solely due to nonreciprocal phase shifts: when the locationsof the source and receiver are interchanged, the amplitude of transmittedvibrations remains unchanged but the transmitted phases are not equal. Wepresent a computational analysis of this state of phase nonreciprocity in theweakly nonlinear frequency-preserving response regime, where we characterizethe response using its nonreciprocal phase shift. This allows us tosystematically find a set of system parameters (including two symmetry-breakingparameters) that lead to reciprocal nonlinear response in a system with brokenmirror symmetry. In other words, we show that breaking the mirror symmetry of apassive nonlinear waveguide is a necessary but insufficient condition fornonreciprocal dynamics to exist. Our findings highlight the important role ofphase in nonlinear nonreciprocity and showcase the potential of asymmetry toserve as an additional design parameter.

Nonreciprocal phase shifts in a nonlinear periodic waveguide