Time-Periodic Solutions of Driven-Damped Trimer Granular Crystals

TL;DR

This study investigates time-periodic solutions in driven-damped trimer granular crystals, revealing complex bifurcation structures and chaotic dynamics through theoretical analysis and experimental visualization.

Contribution

It introduces the analysis of nonlinear surface modes and bifurcation structures in trimer granular crystals under harmonic boundary actuation, highlighting differences from dimer chains.

Findings

Bifurcation of nonlinear surface modes in spectral gaps

Transition to chaos with increasing driving amplitude

Complex bifurcation diagrams with loops of solution branches

Abstract

In this work, we consider time-periodic structures of trimer granular crystals consisting of alternate chrome steel and tungsten carbide spherical particles yielding a spatial periodicity of three. The configuration at the left boundary is driven by a harmonic in-time actuation with given amplitude and frequency while the right one is a fixed wall. Similar to the case of a dimer chain, the combination of dissipation, driving of the boundary, and intrinsic nonlinearity leads to complex dynamics. For fixed driving frequencies in each of the spectral gaps, we find that the nonlinear surface modes and the states dictated by the linear drive collide in a saddle-node bifurcation as the driving amplitude is increased, beyond which the dynamics of the system become chaotic. While the bifurcation structure is similar for solutions within the first and second gap, those in the first gap appear to be less robust. We also conduct a continuation in driving frequency, where it is apparent that the nonlinearity of the system results in a complex bifurcation diagram, involving an intricate set of loops of branches, especially within the spectral gap. The theoretical findings are qualitatively corroborated by the experimental full-field visualization of the time-periodic structures.