Platonic Localisation: One Ring to Bind Them

TL;DR

This paper introduces an asymptotic model for localized flexural vibrations in elastic plates with structured rings containing resonators, deriving analytical conditions for resonance and demonstrating wave localization through simulations.

Contribution

It provides a novel asymptotic analytical model for wave localization in elastic plates with structured rings of resonators, including explicit formulas for resonator parameters.

Findings

Resonator parameters derived analytically.

Wave localization demonstrated in simulations.

Resonators can be tuned for negative inertia effects.

Abstract

In this paper, we present an asymptotic model describing localised flexural vibrations along a structured ring containing point masses or spring-mass resonators in an elastic plate. The values for the required masses and stiffnesses of resonators are derived in a closed analytical form. It is shown that spring-mass resonators can be tuned to produce a "negative inertia" input, which is used to enhance localisation of waveforms around the structured ring. Theoretical findings are accompanied by numerical simulations, which show exponentially localised and leaky modes for different frequency regimes.