Nonlinear Resonances and Antiresonances of a Forced Sonic Vacuum

TL;DR

This paper investigates nonlinear resonances and antiresonances in a driven sonic vacuum, revealing a nonlinear spectrum characterized by complex bifurcations and wave interactions in a system without linear sound propagation.

Contribution

It uncovers the existence of a nonlinear spectrum in a sonic vacuum through detailed bifurcation analysis and identifies resonant and antiresonant responses driven by nonlinear effects.

Findings

Resonant responses occur at integer multiples of the drive period.

Antiresonances correspond to minima in transmitted force.

Complex bifurcation structures include period doubling and Neimark-Sacker bifurcations.

Abstract

We consider a harmonically driven acoustic medium in the form of a (finite length) highly nonlinear granular crystal with an amplitude and frequency dependent boundary drive. Remarkably, despite the absence of a linear spectrum in the system, we identify resonant periodic propagation whereby the crystal responds at integer multiples of the drive period, and observe that they can lead to local maxima of transmitted force at its fixed boundary. In addition, we identify and discuss minima of the transmitted force ("antiresonances") in between these resonances. Representative one-parameter complex bifurcation diagrams involve period doublings, Neimark-Sacker bifurcations as well as multiple isolas (e.g. of period-3, -4 or -5 solutions entrained by the forcing). We combine them in a more detailed, two-parameter bifurcation diagram describing the stability of such responses to both frequency and amplitude variations of the drive. This picture supports an unprecedented example of a notion of a (purely) "nonlinear spectrum" in a system which allows no sound wave propagation (due to zero sound speed: the so-called sonic vacuum). We rationalize this behavior in terms of purely nonlinear building blocks: apparent traveling and standing nonlinear waves.