Nonlinear resonances and antiresonances of a forced sonic vacuum
Abstract
We consider a harmonically driven acoustic medium in the form of a (finite length) highly nonlinear granular crystal with an amplitude- and frequency-dependent boundary drive. Despite the absence of a linear spectrum in the system, we identify resonant periodic propagation whereby the crystal responds at integer multiples of the drive period and observe that this can lead to local maxima of transmitted force at its fixed boundary. In addition, we identify and discuss minima of the transmitted force (“antiresonances”) between these resonances. Representative one-parameter complex bifurcation diagrams involve period doublings and Neimark-Sacker bifurcations as well as multiple isolas (e.g., of period-3, -4, or -5 solutions entrained by the forcing). We combine them in a more detailed, two-parameter bifurcation diagram describing the stability of such responses to both frequency and amplitude variations of the drive. This picture supports a notion of a (purely) “nonlinear spectrum” in a system which allows no sound wave propagation (due to zero sound speed: the so-called sonic vacuum). As a result, we rationalize this behavior in terms of purely nonlinear building blocks: apparent traveling and standing nonlinear waves.
- Authors:
-
- Princeton Univ., Princeton, NJ (United States)
- Univ. of Illinois at Urbana-Champaign, Urbana, IL (United States)
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Univ. of Massachusetts, Amherst, MA (United States)
- Publication Date:
- Research Org.:
- Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
- Sponsoring Org.:
- USDOE
- OSTI Identifier:
- 1246363
- Alternate Identifier(s):
- OSTI ID: 1233896
- Report Number(s):
- LA-UR-15-24544
Journal ID: ISSN 1539-3755; PLEEE8
- Grant/Contract Number:
- US ARO W911NF-09-1-0436; FA9550-12-1-0332; AC52-06NA25396
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
- Additional Journal Information:
- Journal Volume: 92; Journal Issue: 6; Journal ID: ISSN 1539-3755
- Publisher:
- American Physical Society (APS)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 36 MATERIALS SCIENCE; 97 MATHEMATICS AND COMPUTING
Citation Formats
Pozharskiy, D., Zhang, Y., Williams, M. O., McFarland, D. M., Kevrekidis, P. G., Vakakis, A. F., and Kevrekidis, I. G. Nonlinear resonances and antiresonances of a forced sonic vacuum. United States: N. p., 2015.
Web. doi:10.1103/PhysRevE.92.063203.
Pozharskiy, D., Zhang, Y., Williams, M. O., McFarland, D. M., Kevrekidis, P. G., Vakakis, A. F., & Kevrekidis, I. G. Nonlinear resonances and antiresonances of a forced sonic vacuum. United States. https://doi.org/10.1103/PhysRevE.92.063203
Pozharskiy, D., Zhang, Y., Williams, M. O., McFarland, D. M., Kevrekidis, P. G., Vakakis, A. F., and Kevrekidis, I. G. Wed .
"Nonlinear resonances and antiresonances of a forced sonic vacuum". United States. https://doi.org/10.1103/PhysRevE.92.063203. https://www.osti.gov/servlets/purl/1246363.
@article{osti_1246363,
title = {Nonlinear resonances and antiresonances of a forced sonic vacuum},
author = {Pozharskiy, D. and Zhang, Y. and Williams, M. O. and McFarland, D. M. and Kevrekidis, P. G. and Vakakis, A. F. and Kevrekidis, I. G.},
abstractNote = {We consider a harmonically driven acoustic medium in the form of a (finite length) highly nonlinear granular crystal with an amplitude- and frequency-dependent boundary drive. Despite the absence of a linear spectrum in the system, we identify resonant periodic propagation whereby the crystal responds at integer multiples of the drive period and observe that this can lead to local maxima of transmitted force at its fixed boundary. In addition, we identify and discuss minima of the transmitted force (“antiresonances”) between these resonances. Representative one-parameter complex bifurcation diagrams involve period doublings and Neimark-Sacker bifurcations as well as multiple isolas (e.g., of period-3, -4, or -5 solutions entrained by the forcing). We combine them in a more detailed, two-parameter bifurcation diagram describing the stability of such responses to both frequency and amplitude variations of the drive. This picture supports a notion of a (purely) “nonlinear spectrum” in a system which allows no sound wave propagation (due to zero sound speed: the so-called sonic vacuum). As a result, we rationalize this behavior in terms of purely nonlinear building blocks: apparent traveling and standing nonlinear waves.},
doi = {10.1103/PhysRevE.92.063203},
journal = {Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics},
number = 6,
volume = 92,
place = {United States},
year = {Wed Dec 23 00:00:00 EST 2015},
month = {Wed Dec 23 00:00:00 EST 2015}
}
Free Publicly Available Full Text
Publisher's Version of Record
Other availability
Search WorldCat to find libraries that may hold this journal
Cited by: 9 works
Citation information provided by
Web of Science
Web of Science
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.
Works referenced in this record:
Discrete breathers — Advances in theory and applications
journal, October 2008
- Flach, Sergej; Gorbach, Andrey V.
- Physics Reports, Vol. 467, Issue 1-3
Discrete analogue of the Burgers equation
journal, October 2012
- Ben-Naim, E.; Krapivsky, P. L.
- Journal of Physics A: Mathematical and Theoretical, Vol. 45, Issue 45
Compactons: Solitons with finite wavelength
journal, February 1993
- Rosenau, Philip; Hyman, James M.
- Physical Review Letters, Vol. 70, Issue 5
Nonlinear Dispersion and Compact Structures
journal, September 1994
- Rosenau, Philip
- Physical Review Letters, Vol. 73, Issue 13
Multidimensional Compactons
journal, January 2007
- Rosenau, Philip; Hyman, James M.; Staley, Martin
- Physical Review Letters, Vol. 98, Issue 2
Shape profile of compactlike discrete breathers in nonlinear dispersive lattice systems
journal, December 2001
- Dey, B.; Eleftheriou, M.; Flach, S.
- Physical Review E, Vol. 65, Issue 1
Dynamics of Heterogeneous Materials
book, January 2001
- Nesterenko, Vitali F.
- Shock Wave and High Pressure Phenomena
Solitary waves in the granular chain
journal, June 2008
- Sen, S.; Hong, J.; Bang, J.
- Physics Reports, Vol. 462, Issue 2
Non-linear waves in lattices: past, present, future
journal, April 2011
- Kevrekidis, P. G.
- IMA Journal of Applied Mathematics, Vol. 76, Issue 3
Propagation of nonlinear compression pulses in granular media
journal, January 1984
- Nesterenko, V. F.
- Journal of Applied Mechanics and Technical Physics, Vol. 24, Issue 5
The fragmentation of a line of balls by an impact
journal, September 1999
- Hinch, E. J.; Saint–Jean, S.
- Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, Vol. 455, Issue 1989
Asymptotic solution for solitary waves in a chain of elastic spheres
journal, May 1999
- Chatterjee, Anindya
- Physical Review E, Vol. 59, Issue 5
On the solitary wave pulse in a chain of beads
journal, January 2005
- English, J. M.; Pego, R. L.
- Proceedings of the American Mathematical Society, Vol. 133, Issue 6, p. 1763-1768
Compactons and chaos in strongly nonlinear lattices
journal, February 2009
- Ahnert, Karsten; Pikovsky, Arkady
- Physical Review E, Vol. 79, Issue 2
Nonlinear resonances and energy transfer in finite granular chains
journal, February 2015
- Lydon, Joseph; Theocharis, Georgios; Daraio, Chiara
- Physical Review E, Vol. 91, Issue 2
Frequency bands of strongly nonlinear homogeneous granular systems
journal, July 2013
- Lydon, Joseph; Jayaprakash, K. R.; Ngo, Duc
- Physical Review E, Vol. 88, Issue 1
New family of solitary waves in granular dimer chains with no precompression
journal, March 2011
- Jayaprakash, K. R.; Starosvetsky, Yuli; Vakakis, Alexander F.
- Physical Review E, Vol. 83, Issue 3
Observation of Two-Wave Structure in Strongly Nonlinear Dissipative Granular Chains
journal, April 2007
- Rosas, Alexandre; Romero, Aldo H.; Nesterenko, Vitali F.
- Physical Review Letters, Vol. 98, Issue 16
Damped-driven granular chains: An ideal playground for dark breathers and multibreathers
journal, March 2014
- Chong, C.; Li, F.; Yang, J.
- Physical Review E, Vol. 89, Issue 3
Nonlinear normal modes and band zones in granular chains with no pre-compression
journal, September 2010
- Jayaprakash, K. R.; Starosvetsky, Yuli; Vakakis, Alexander F.
- Nonlinear Dynamics, Vol. 63, Issue 3
Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study
journal, September 1982
- Aronson, D. G.; Chory, M. A.; Hall, G. R.
- Communications in Mathematical Physics, Vol. 83, Issue 3
Some common features of periodically forced reacting systems
journal, January 1986
- Kevrekidis, I. G.; Schmidt, L. D.; Aris, R.
- Chemical Engineering Science, Vol. 41, Issue 5
Entrainment regions for periodically forced oscillators
journal, March 1986
- Aronson, D. G.; McGehee, R. P.; Kevrekidis, I. G.
- Physical Review A, Vol. 33, Issue 3
Experimental study of nonlinear acoustic bands and propagating breathers in ordered granular media embedded in matrix
journal, December 2014
- Hasan, M. Arif; Cho, Shinhu; Remick, Kevin
- Granular Matter, Vol. 17, Issue 1
Chimera States for Coupled Oscillators
journal, October 2004
- Abrams, Daniel M.; Strogatz, Steven H.
- Physical Review Letters, Vol. 93, Issue 17
Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
book, January 1983
- Guckenheimer, John; Holmes, Philip
- Applied Mathematical Sciences
Nonlinear surface modes and Tamm states in periodic photonic structures
journal, November 2007
- Kivshar, Yuri S.; Molina, Mario I.
- Wave Motion, Vol. 45, Issue 1-2
NONLINEAR WAVES IN NEWTON'S CRADLE AND THE DISCRETE p-SCHRÖDINGER EQUATION
journal, November 2011
- James, Guillaume
- Mathematical Models and Methods in Applied Sciences, Vol. 21, Issue 11
Breathers in oscillator chains with Hertzian interactions
journal, May 2013
- James, Guillaume; Kevrekidis, Panayotis G.; Cuevas, Jesús
- Physica D: Nonlinear Phenomena, Vol. 251
Wave propagation in granular chains with local resonances
journal, March 2015
- Bonanomi, Luca; Theocharis, Georgios; Daraio, Chiara
- Physical Review E, Vol. 91, Issue 3
Granular metamaterials for vibration mitigation
journal, September 2013
- Gantzounis, G.; Serra-Garcia, M.; Homma, K.
- Journal of Applied Physics, Vol. 114, Issue 9
Highly Nonlinear Wave Propagation in Elastic Woodpile Periodic Structures
journal, March 2015
- Kim, E.; Li, F.; Chong, C.
- Physical Review Letters, Vol. 114, Issue 11
Energy equipartition in two-dimensional granular systems with spherical intruders
journal, March 2013
- Szelengowicz, I.; Hasan, M. A.; Starosvetsky, Y.
- Physical Review E, Vol. 87, Issue 3
Stress Wave Anisotropy in Centered Square Highly Nonlinear Granular Systems
journal, May 2012
- Leonard, A.; Daraio, C.
- Physical Review Letters, Vol. 108, Issue 21
Traveling waves in 2D hexagonal granular crystal lattices
journal, April 2014
- Leonard, A.; Chong, C.; Kevrekidis, P. G.
- Granular Matter, Vol. 16, Issue 4
Works referencing / citing this record:
Experimental Study of Nonlinear Resonances and Anti-Resonances in a Forced, Ordered Granular Chain
journal, October 2016
- Zhang, Y.; Pozharskiy, D.; McFarland, D. M.
- Experimental Mechanics, Vol. 57, Issue 4
Nonlinear Coherent Structures in Granular Crystals
text, January 2016
- Chong, C.; Porter, Mason A.; Kevrekidis, P. G.
- arXiv