Nonreciprocal phase shifts in a nonlinear periodic waveguide

TL;DR

The paper investigates nonreciprocal vibration transmission in a nonlinear periodic waveguide, with phase nonreciprocity arising from unit-cell asymmetry and cubic grounding nonlinearity. Using harmonic balance in the weakly nonlinear regime, it analyzes forward/back phase differences, $\Delta\phi$, under equal amplitudes and maps how $\mu$, $P$, and $\zeta_g$ shape nonreciprocal phase shifts. It further demonstrates reciprocity restoration by introducing a second symmetry-breaking parameter $r_g$, showing that $R_N=0$ and $\Delta\phi=0$ can be achieved near resonance. These results highlight phase as a key design knob for nonlinear nonreciprocity and reveal that symmetry breaking is necessary but not sufficient, with implications for phase-controlled devices in imperfect structures.

Abstract

We explore nonreciprocal vibration transmission in a nonlinear periodic waveguide. Nonlinearity and asymmetry, the two necessary requirements for nonreciprocity, are both introduced within the unit cell of the periodic waveguide. We focus primarily on the contribution of phase to the nonreciprocal steady-state response of the system. To highlight the phase effects, which are rarely discussed in the literature, we investigate response regimes in which nonreciprocity is solely due to nonreciprocal phase shifts: when the locations of the source and receiver are interchanged, the amplitude of transmitted vibrations remains unchanged but the transmitted phases are not equal. We present a computational analysis of this state of phase nonreciprocity in the weakly nonlinear frequency-preserving response regime, where we characterize the response using its nonreciprocal phase shift. This allows us to systematically find a set of system parameters (including two symmetry-breaking parameters) that lead to reciprocal nonlinear response in a system with broken mirror symmetry. In other words, we show that breaking the mirror symmetry of a passive nonlinear waveguide is a necessary but insufficient condition for nonreciprocal dynamics to exist. Our findings highlight the important role of phase in nonlinear nonreciprocity and showcase the potential of asymmetry to serve as an additional design parameter.

Figures (9)

• Figure 1: Schematic of the finite periodic structure. The red dashed box shows the unit cell, which consists of two linear oscillators coupled with a linear stiffness and grounded by a nonlinear spring and a linear damping.
• Figure 2: Output norms (Eq. \ref{['Norms']}) at $P=0.15$ for the system with mirror symmetry ($\mu=1$)
• Figure 3: Frequency response curves at $P=0.15$ for the system with broken mirror symmetry, $\mu=2$. The grey circles show points at which $N^F=N^B$. The black dots indicate unstable regions in the response.
• Figure 4: Nonreciprocal dynamics for $P=0.15$ and $\mu=2$. (a) The reciprocity norm, $R$, with the markers indicating $N^F=N^B$. Diamonds denote phase nonreciprocity $(N^F=N^B,~\phi^F\ne\phi^B)$ and circles denote frequencies at which the response of the system is almost reciprocal $(N^F=N^B,~\phi^F\approx\phi^B)$. (b) The time-domain response at the red circle, $\omega_f=1.025$. (c) The time-domain response at the blue diamond, $\omega_f=0.99$.
• Figure 5: Locus of phase nonreciprocity ($N^F=N^B$) as a function of the asymmetry parameter, $\mu$, at $P=0.15$. (a) Normalized reciprocity bias, $R_N$, and the phase shift between the forward and backward configurations, $\Delta\phi=\phi^F-\phi^B$. (b) The forcing frequency, $\omega_f$, at which phase nonreciprocity occurs.