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

K. R. Jayaprakash, Yuli Starosvetsky, Alexander F. Vakakis, Maxime Peeters, Gaetan Kerschen

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)359-385
Number of pages27
JournalNonlinear Dynamics
Volume63
Issue number3
DOIs
StatePublished - Feb 1 2011

Fingerprint

Normal Modes
Compression
Granular Media
Dynamical systems
Disorder
Dynamical system
Discrete Breathers
Nonlinear Response
Standing Wave
Clearance
Traveling Wave
Notation
Topology
Boundary conditions
Scattering
Shock
Excitation
Disturbance
Classify
Contact

Keywords

  • Band gaps
  • Granular media
  • Nonlinear normal modes

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Aerospace Engineering
  • Ocean Engineering
  • Mechanical Engineering
  • Applied Mathematics
  • Electrical and Electronic Engineering

Cite this

Nonlinear normal modes and band zones in granular chains with no pre-compression. / Jayaprakash, K. R.; Starosvetsky, Yuli; Vakakis, Alexander F.; Peeters, Maxime; Kerschen, Gaetan.

In: Nonlinear Dynamics, Vol. 63, No. 3, 01.02.2011, p. 359-385.

Research output: Contribution to journalArticle

Jayaprakash, K. R. ; Starosvetsky, Yuli ; Vakakis, Alexander F. ; Peeters, Maxime ; Kerschen, Gaetan. / Nonlinear normal modes and band zones in granular chains with no pre-compression. In: Nonlinear Dynamics. 2011 ; Vol. 63, No. 3. pp. 359-385.
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