Space-time symmetry and nonreciprocal parametric resonance in mechanical systems

TL;DR

This paper develops a symmetry-based theoretical framework to analyze parametric resonance in space-time symmetric mechanical systems, enabling the design of nonreciprocal and one-way wave amplification in metamaterials.

Contribution

It introduces a novel symmetry-based approach to determine resonance conditions and nonreciprocal responses in space-time symmetric mechanical systems, extending understanding beyond individual oscillators.

Findings

Derived conditions for parametric resonance using symmetries.

Identified space-time symmetry constraints on system dynamics.

Proposed design principles for nonreciprocal mechanical metamaterials.

Abstract

Linear mechanical systems with time-modulated parameters can harbor oscillations with amplitudes that grow or decay exponentially with time due to the phenomenon of parametric resonance. While the resonance properties of individual oscillators are well understood, those of systems of coupled oscillators remain challenging to characterize. Here, we determine the parametric resonance conditions for time-modulated mechanical systems by exploiting the internal symmetries arising from the real-valued and symplectic nature of classical mechanics. We also determine how these conditions are further constrained when the system exhibits external symmetries. In particular, we analyze systems with space-time symmetry where the system remains invariant after a combination of discrete translation in both space and time. For such systems, we identify a combined space-time translation operator that provides more information about the dynamics of the system than the Floquet operator does, and use it to derive conditions for one-way amplification of traveling waves. Our exact theoretical framework based on symmetries enables the design of exotic responses such as nonreciprocal transport and one-way amplification in dynamic mechanical metamaterials, and is generalizable to all physical systems that obey space-time symmetry.