Duality,Hidden Symmetry and Dynamic Isomerism in 2D Hinge Structures

TL;DR

This paper uncovers the origin of duality in 2D hinge structures, introduces a design principle for self-dual structures, and demonstrates their potential for reflectionless waveguides and broad applicability in Hamiltonian systems.

Contribution

It clarifies the origin of duality in 2D mechanical networks and proposes a general design principle for self-dual structures with potential applications.

Findings

Duality originates from PCI symmetry of hinges.

Dissimilar structures can have identical dynamic modes.

Design of reflectionless waveguides using duality.

Abstract

Recently, a new type of duality was reported in some deformable mechanical networks which exhibit Kramers-like degeneracy in phononic spectrum at the self-dual point. In this work, we clarify the origin of this duality and propose a design principle of 2D self-dual structures with arbitrary complexity. We find that this duality originates from the (PCI) symmetry of the hinge, which belongs to a more general end-fixed scaling transformation. This symmetry gives the structure an extra degree of freedom without modifying its dynamics. This results in , i.e., dissimilar 2D mechanical structures, either periodic or aperiodic, having identical dynamic modes, based on which we demonstrate a new type of wave-guide without reflection or loss. Moreover, the PCI symmetry allows us to design various 2D periodic isostatic networks with hinge duality. At last, by further studying a 2D non-mechanical magnonic system, we show that the duality and the associated hidden symmetry should exist in a broad range of Hamiltonian systems.