   Effective particles and classification of periodic orbits of homogeneous granular chains with no pre-compression; ; et al in Physical Review. E : Statistical, Nonlinear, and Soft Matter Physics (2012), 85 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 ... [more ▼] 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. [less ▲] Detailed reference viewed: 13 (3 ULg)Nonlinear normal modes and band zones in granular chains with no precompression; ; et al in Nonlinear Dynamics (2011), 63 We study standing waves (nonlinear normal modes—NNMs) and band zones in finite granular chains composed of spherical granular beads in Hertzian contact, with fixed boundary conditions. Although these are ... [more ▼] We study standing waves (nonlinear normal modes—NNMs) and band zones in finite granular chains composed of spherical granular beads in Hertzian contact, with fixed boundary conditions. Although these are homogeneous dynamical systems in the notation of Rosenberg, we show that the discontinuous nature of the dynamics leads to interesting effects such as separation between beads, NNMs that appear as traveling waves (these are characterized as pseudo-waves), and localization phenomena. In the limit of infinite extent, we study band zones, i.e., pass and stop bands in the frequency–energy plane of these dynamical systems, and classify the essentially nonlinear responses that occur in these bands. Moreover, we show how the topologies of these bands significantly affect the forced dynamics of these granular media subject to narrowband excitations. This work provides a classification of the coherent (regular) intrinsic dynamics of one-dimensional homogeneous granular chains with no pre-compression, and provides a rigorous theoretical foundation for further systematic study of the dynamics of granular systems, e.g., the effects of disorders or clearances, discrete breathers, nonlinear localized modes, and high-frequency scattering by local disorders. Moreover, it contributes toward the design of granular media as shock protectors, and in the passive mitigation of transmission of unwanted disturbances. [less ▲] Detailed reference viewed: 8 (2 ULg) |
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