Dynamic Green's functions in discrete flexural systems

TL;DR

This paper analyzes the dynamic behavior of discrete flexural systems using Green's functions, revealing novel effects like anisotropy and wave-guiding, supported by analytical and numerical methods.

Contribution

It introduces a general algorithm for constructing Green's functions and defect modes in discrete flexural lattices, enabling tailored wave control without resonant elements.

Findings

Demonstrates extreme dynamic anisotropy and non-reciprocity in flexural lattices

Shows ability to create localized defect modes and wave-guides

Provides a versatile algorithm for tuning lattice properties across frequencies

Abstract

The paper presents an analysis of the dynamic behaviour of discrete flexural systems composed of Euler--Bernoulli beams. The canonical object of study is the discrete Green's function, from which information regarding the dynamic response of the lattice under point loading by forces and moments can be obtained. Special attention is devoted to the interaction between flexural and torsional waves in a square lattice of Euler--Bernoulli beams, which is shown to yield a range of novel effects, including extreme dynamic anisotropy, non-reciprocity, wave-guiding, filtering, and the ability to create localised defect modes, all without the need for additional resonant elements or interfaces. The analytical study is complimented by numerical computations and finite element simulations, both of which are used to illustrate the effects predicted. A general algorithm is provided for constructing Green's functions as well as defect modes. This algorithm allows the tuning of the lattice to produce pass bands, band gaps, resonant modes, wave-guides, and defect modes, over any desired frequency range.