Strongly Nonlinear Beat Phenomena and Energy Exchanges in Weakly Coupled Granular Chains on Elastic Foundations

We study the dynamics of weakly interacting, strongly nonlinear one-dimensional granular chains mounted on elastic foundations. These chains are composed of a number of identical linearly elastic beads interacting with each other through Hertzian contact. No dissipative effects, such as plasticity or dry friction effects, are taken into account in our analysis. Assuming zero precompression between beads, the dynamics of the system under consideration is strongly (essentially) nonlinear, having no linear component. The complete absence of linear structural acoustics in these chains led to their characterization as “sonic vacua.” The two sources of strong nonlinearity in the considered granular chains are (i) the nonlinearizable Hertzian law interaction between adjacent beads in compression, and (ii) the possible separations between beads leading to bead collisions in the absence of compressive forces. In the current study we demonstrate that the weakly coupled granular chains possess complex dynamics leading to strong energy exchanges between them. Three different types of nonlinear beat phenomena are analytically studied, based on spatially periodic traveling waves, stationary breathers, and propagating breathers, respectively. We employ a complexification—averaging methodology that leads to smooth slow flow reduced models of the dynamics despite the discontinuous nature of the bead interactions. Verification of the derived analytical approximations with direct numerical simulations is also performed.