Wavenumber-space band clipping in nonlinear periodic structures

TL;DR

This paper introduces a multiple scales analytical framework to understand wavenumber shifts in nonlinear periodic structures under boundary excitations, revealing a novel band clipping effect and implications for tunable bandgaps.

Contribution

It develops a new analytical approach to model wavenumber shifts in nonlinear structures under boundary excitations, addressing limitations of previous frequency shift models.

Findings

Wavenumber shifts lead to a band clipping phenomenon in nonlinear chains.

The framework extends to locally-resonant structures, enabling bandgap tuning.

Cubic nonlinearity in internal springs allows for effective bandgap control.

Abstract

In weakly nonlinear systems, the main effect of cubic nonlinearity on wave propagation is an amplitude-dependent correction of the dispersion relation. This phenomenon can manifest either as a frequency shift or as a wavenumber shift depending on whether the excitation is prescribed as a initial condition or as a boundary condition, respectively. Several models have been proposed to capture the frequency shifts observed when the system is subjected to harmonic initial excitations. However, these models are not compatible with harmonic boundary excitations, which represent the conditions encountered in most practical applications. To overcome this limitation, we present a multiple scales framework to analytically capture the wavenumber shift experienced by dispersion relation of nonlinear monatomic chains under harmonic boundary excitations. We demonstrate that the wavenumber shifts result in an unusual dispersion correction effect, which we term wavenumber-space band clipping. We then extend the framework to locally-resonant periodic structures to explore the implications of this phenomenon on bandgap tunability. We show that the tuning capability is available if the cubic nonlinearity is deployed in the internal springs supporting the resonators.