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Nonlinear normal modes and band zones in granular chains with no pre-compression

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Abstract

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 (Adv. Appl. Mech. 9:155–242, 1966), 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.

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Authors and Affiliations

  1. Department of Mechanical Science and Engineering, University of Illinois at Urbana Champaign, 1206 W. Green Street, Urbana, IL, 61822, USA

    K. R. Jayaprakash, Yuli Starosvetsky & Alexander F. Vakakis

  2. Department of Aerospace and Mechanical Engineering, University of Liege, 1, Chemin des Chevreuils (B52/3), 4000, Liege, Belgium

    Maxime Peeters & Gaetan Kerschen

Authors
  1. K. R. Jayaprakash
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  2. Yuli Starosvetsky
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  3. Alexander F. Vakakis
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  4. Maxime Peeters
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  5. Gaetan Kerschen
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Correspondence to K. R. Jayaprakash.

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Jayaprakash, K.R., Starosvetsky, Y., Vakakis, A.F. et al. Nonlinear normal modes and band zones in granular chains with no pre-compression. Nonlinear Dyn 63, 359–385 (2011). https://doi.org/10.1007/s11071-010-9809-0

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