Linear Nonreciprocal Dynamics of Coupled Modulated Systems

TL;DR

The paper addresses nonreciprocal vibration transmission in short, spatiotemporally modulated systems by analyzing a minimal two-degree-of-freedom model with phase-shifted grounding stiffness. An averaging method yields a quasi-periodic steady-state expressed through harmonic amplitudes, enabling direct assessment of forward/backward transmission and phase-based nonreciprocity. The study demonstrates that the modulation phase $\phi$ is the primary symmetry-breaking mechanism, with stronger modulation amplifying phase effects, shifting resonances and enriching the spectrum. It highlights that nonreciprocity can arise from phase differences even when transmitted energies are similar, and it outlines a pathway for designing phase-controlled nonreciprocal devices in vibroacoustic applications.

Abstract

Waveguides subject to spatiotemporal modulations are known to exhibit nonreciprocal vibration transmission, whereby interchanging the locations of the source and receiver change the end-to-end transmission characteristics. The scenario of typical interest is unidirectional transmission in long, weakly modulated systems: when transmission is possible in one direction only. Here, with a view toward expanding their potential application as devices, we explore the vibration characteristics of spatiotemporally modulated systems that are short and strongly modulated. Focusing on two coupled systems, we develop a methodology to investigate the nonreciprocal vibration characteristics of both weakly and strongly modulated systems. In particular, we highlight the contribution of phase to nonreciprocity, a feature that is often overlooked. We show that the difference between the transmitted phases is the main contributor to breaking reciprocity in short systems. We clarify the roles of primary and side-band resonances, and their overlaps, in breaking reciprocity. We discuss the influence of modulation amplitude and wavenumber on the resonances of the modulated system.

Figures (13)

• Figure 1: Schematic of the 2-DoF model of coupled modulated systems.
• Figure 2: Plots of (a) output norms, (b) difference between output norms and (c) reciprocity bias as functions of $\Omega_f$. System parameters: $K_c=0.6$, $\zeta=0.005$, $K_m=0.1$, $\Omega_m=0.2$, $\phi=\pi/2$ and $P=1$.
• Figure 3: Plots of the amplitudes of three components ($q\in\{-1,0,1\}$) of (a) forward output and (b) backward output, and (c) the amplitude difference of each component-pair.
• Figure 4: Plots of (a) the amplitude difference, (b) the phase difference and (c) the contribution of each component-pair to the reciprocity bias.
• Figure 5: Plots of (a) $N_{df}$ and (b) $R$ as functions of $\Omega_f$ and $\phi$.