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Nonlinear Pulse Equipartition in Weakly Coupled Ordered Granular Chains With No Precompression

[+] Author and Article Information
Yuli Starosvetsky

Department of Mechanical Engineering,
Technion—Israel Institute of Technology,
Technion City,
Haifa 32000, Israel
e-mail: staryuli@technion.ac.il

M. Arif Hasan

e-mail: mhasan@illinois.edu

Alexander F. Vakakis

e-mail: avakakis@illinois.eduDepartment of Mechanical Science and Engineering,
University of Illinois at Urbana–Champaign,
1206 West Green Street,
Urbana, IL 61822

Contributed by the Design Engineering Division of ASME for publication in the Journal of Computational and Nonlinear Dynamics. Manuscript received September 26, 2011; final manuscript received January 10, 2013; published online March 21, 2013. Assoc. Editor: D. Dane Quinn.

J. Comput. Nonlinear Dynam 8(3), 034504 (Mar 21, 2013) (6 pages) Paper No: CND-11-1160; doi: 10.1115/1.4023863 History: Received September 26, 2011; Revised January 10, 2013

We report on the strongly nonlinear dynamics of an array of weakly coupled, noncompressed, parallel granular chains subject to a local initial impulse. The motion of the granules in each chain is constrained to be in one direction that coincides with the orientation of the chain. We show that in spite of the fact that the applied impulse is applied to one of the granular chains, the resulting pulse that initially propagates only in the excited chain gets gradually equipartitioned between its neighboring chains and eventually in all chains of the array. In particular, the initially strongly localized state of energy distribution evolves towards a final stationary state of formation of identical solitary waves that propagate in each one of the chains. These solitary waves are synchronized and have identical speeds. We show that the phenomenon of primary pulse equipartition between the weakly coupled granular chains can be fully reproduced in coupled binary models that constitute a significantly simpler model that captures the main qualitative features of the dynamics of the granular array. The results reported herein are of major practical significance since it indicates that the weakly coupled array of granular chains is a medium in which an initially localized excitation gets gradually defocused, resulting in drastic reduction of propagating pulses as they are equipartitioned among all chains.

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References

Figures

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Fig. 9

Binary collision model for studying pulse equipartition

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Fig. 8

Responses of the beads of the two chains of the array with N = 2 and for initial excitation of the first chain with an impulse of magnitude V = 1

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Fig. 7

Response of the 90th bead of the directly excited (a), first absorbing (b) and second absorbing chain (c), depicting the formation of Nesterenko solitary waves due to equipartition, for the system with N = 3, V = 1, α = 1, ε = 0.1

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Fig. 6

Responses of the first 100 beads of the three chains of the array with N = 3 and α = 1, ε = 0.1 for initial excitation of first chain with an impulse of magnitude V = 1

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Fig. 5

Responses of the beads of the two chains of the array with N = 2 and for initial excitation of the first chain with an impulse of magnitude V = 1

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Fig. 4

Response of the 55th bead of the directly excited (a) and absorbing chain (b) depicting the formation of Nesterenko solitary waves due to equipartition, for the system with N = 2, V = 1, α = 5, ε = 0.1

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Fig. 3

Responses of the first 100 beads of the two chains of the array with N = 2 and α = 5, ε = 0.1 for initial excitation of the first chain with an impulse of magnitude V = 1

Grahic Jump Location
Fig. 2

Response of the 55th bead of the directly excited (a) and absorbing chain (b) depicting the formation of Nesterenko solitary waves due to equipartition, for the system with N = 2, V = 1, α = 1, ε = 0.1

Grahic Jump Location
Fig. 1

Responses of the first 100 beads of the two chains of the array with N = 2 and α = 1, ε = 0.1 for initial excitation of the first chain with an impulse of magnitude V = 1

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Fig. 10

Transient response of the fifth bead of (a) the excited chain and (b) the absorbing chain of the binary model before the stationary state and pulse equipartition are reached

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Fig. 11

The stationary state and pulse equipartition in the binary collision model: Response of the 25th bead of (a) the excited chain and (b) the absorbing chain

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