Time-Periodic Solutions of Driven-Damped Trimer Granular Crystals

Charalampidis, E. G.; Li, F.; Chong, C.; Yang, J.; Kevrekidis, P. G.

doi:10.1155/2015/830978

Title: Time-Periodic Solutions of Driven-Damped Trimer Granular Crystals

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Abstract

In this work, we consider time-periodic structures of granular crystals consisting of alternate chrome steel (S) and tungsten carbide (W) spherical particles where each unit cell follows the pattern of a 2:1 trimer: S-W-S. The configuration at the left boundary is driven by a harmonic in-time actuation with given amplitude and frequency while the right one is a fixed wall. Similar to the case of a dimer chain, the combination of dissipation, driving of the boundary, and intrinsic nonlinearity leads to complex dynamics. For fixed driving frequencies in each of the spectral gaps, we find that the nonlinear surface modes and the states dictated by the linear drive collide in a saddle-node bifurcation as the driving amplitude is increased, beyond which the dynamics of the system becomes chaotic. While the bifurcation structure is similar for solutions within the first and second gap, those in the first gap appear to be less robust. We also conduct a continuation in driving frequency, where it is apparent that the nonlinearity of the system results in a complex bifurcation diagram, involving an intricate set of loops of branches, especially within the spectral gap. The theoretical findings are qualitatively corroborated by the experimental full-field visualizationmore » of the time-periodic structures.« less

Authors:

Charalampidis, E. G. ^[1]; Li, F. ^[2]; Chong, C. ^[3]; Yang, J. ^[2]; Kevrekidis, P. G. ^[4]

School of Civil Engineering, Faculty of Engineering, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece, Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-4515, USA
Aeronautics & Astronautics, University of Washington, Seattle, WA 98195-2400, USA
Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-4515, USA, Department of Mechanical and Process Engineering (D-MAVT), Swiss Federal Institute of Technology (ETH), 8092 Zürich, Switzerland
Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-4515, USA, Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87544, USA

Publication Date:: Thu Jan 01 00:00:00 EST 2015

Research Org.:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Org.:: USDOE

OSTI Identifier:: 1198644

Alternate Identifier(s):: OSTI ID: 1233184

Report Number(s):: LA-UR-14-28454
Journal ID: ISSN 1024-123X; PII: 830978; 830978

Grant/Contract Number:: IRSES-606096; CMMI1000337; DMS1312856; FA9550-12-1-0332; CMMI1414748; N000141410388; ESC-A 06-14; AC52-06NA25396

Resource Type:: Published Article

Journal Name:: Mathematical Problems in Engineering

Additional Journal Information:: Journal Name: Mathematical Problems in Engineering Journal Volume: 2015; Journal ID: ISSN 1024-123X

Publisher:: Hindawi Publishing Corporation

Country of Publication:: Egypt

Language:: English

Subject:: 36 MATERIALS SCIENCE; 97 MATHEMATICS AND COMPUTING; granular crystals; time-periodic orbits; discrete breathers; Floquet multipliers

Citation Formats


                    Charalampidis, E. G., Li, F., Chong, C., Yang, J., and Kevrekidis, P. G. Time-Periodic Solutions of Driven-Damped Trimer Granular Crystals.  Egypt: N. p., 2015. 
Web.  doi:10.1155/2015/830978.

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                    Charalampidis, E. G., Li, F., Chong, C., Yang, J., & Kevrekidis, P. G. Time-Periodic Solutions of Driven-Damped Trimer Granular Crystals.  Egypt.  https://doi.org/10.1155/2015/830978

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                    Charalampidis, E. G., Li, F., Chong, C., Yang, J., and Kevrekidis, P. G. Thu .  
"Time-Periodic Solutions of Driven-Damped Trimer Granular Crystals".  Egypt.  https://doi.org/10.1155/2015/830978.

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@article{osti_1198644,

  title        = {Time-Periodic Solutions of Driven-Damped Trimer Granular Crystals},

  author       = {Charalampidis, E. G. and Li, F. and Chong, C. and Yang, J. and Kevrekidis, P. G.},

  abstractNote = {In this work, we consider time-periodic structures of granular crystals consisting of alternate chrome steel (S) and tungsten carbide (W) spherical particles where each unit cell follows the pattern of a 2:1 trimer: S-W-S. The configuration at the left boundary is driven by a harmonic in-time actuation with given amplitude and frequency while the right one is a fixed wall. Similar to the case of a dimer chain, the combination of dissipation, driving of the boundary, and intrinsic nonlinearity leads to complex dynamics. For fixed driving frequencies in each of the spectral gaps, we find that the nonlinear surface modes and the states dictated by the linear drive collide in a saddle-node bifurcation as the driving amplitude is increased, beyond which the dynamics of the system becomes chaotic. While the bifurcation structure is similar for solutions within the first and second gap, those in the first gap appear to be less robust. We also conduct a continuation in driving frequency, where it is apparent that the nonlinearity of the system results in a complex bifurcation diagram, involving an intricate set of loops of branches, especially within the spectral gap. The theoretical findings are qualitatively corroborated by the experimental full-field visualization of the time-periodic structures.},

  doi          = {10.1155/2015/830978},

  journal      = {Mathematical Problems in Engineering},

  number       = ,

  volume       = 2015,

  place        = {Egypt},

  year         = {Thu Jan 01 00:00:00 EST 2015},

  month        = {Thu Jan 01 00:00:00 EST 2015}

}

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