Effective particles and classification of the dynamics of homogeneous granular chains with no precompression

We develop a systematic methodology for classifying the periodic orbits of homogeneous ordered granular chains with no dissipation, under the assumption that all granules oscillate with the same frequency. The analysis is based on the idea of balancing linear momentum for sets of auxiliary models consisting of "effective particles." The auxiliary models may be defined for any given finite, ordered granular chain composed of n identical granules (beads) that interact with each other through strongly nonlinear Hertzian interaction law. In turn, the auxiliary models may be effectively used for theoretically predicting the total number of periodic orbits and the corresponding amplitude ratios of the granules. Good correspondence between the theoretical models and results of direct numerical simulations is reported. The results presented herein can be used to understand the complex intrinsic dynamics of ordered granular media, and to systematically study the generation of mode localization in these strongly nonlinear systems. The derived analytical models can be utilized to predict the response of the effective particles, and based on that, to predict primary pulse transmission in periodic layered media with granular interfaces. Moreover, our analysis can be extended to the general class of nonlinear chains of particles with smooth interacting potentials and possible separation between particles during the motion.