Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Breathers and other time-periodic solutions in an array of cantilevers decorated with magnets

1 Department of Mathematics, Bowdoin College, Brunswick, Maine 04011, USA
2 Department of Mechanical and Process Engineering (D-MAVT), Swiss Federal Institute of Technology (ETH), 8092 Zürich, Switzerland
3 Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA, USA
4 Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-4515, USA

$^\dagger$This contribution is part of the Special Issue: Hamiltonian Lattice Dynamics
  Guest Editors: Simone Paleari; Tiziano Penati
  Link: http://www.aimspress.com/newsinfo/1165.html

Special Issues: Hamiltonian Lattice Dynamics

In this article, the existence, stability and bifurcation structure of time-periodic solutions (including ones that also have the property of spatial localization, i.e., breathers) are studied in an array of cantilevers that have magnetic tips. The repelling magnetic tips are responsible for the intersite nonlinearity of the system, whereas the cantilevers are responsible for the onsite (potentially nonlinear) force. The relevant model is of the mixed Fermi-Pasta-Ulam-Tsingou and Klein-Gordon type with both damping and driving. In the case of base excitation, we provide experimental results to validate the model. In particular, we identify regions of bistability in the model and in the experiment, which agree with minimal tuning of the system parameters. We carry out additional numerical explorations in order to contrast the base excitation problem with the boundary excitation problem and the problem with a single mass defect. We find that the base excitation problem is more stable than the boundary excitation problem and that breathers are possible in the defect system. The effect of an onsite nonlinearity is also considered, where it is shown that bistability is possible for both softening and hardening cubic nonlinearities.
  Figure/Table
  Supplementary
  Article Metrics

Keywords breather; time-periodic; Fermi-Pasta-Ulam-Tsingou; Klein-Gordon; cantilever array; magnet

Citation: Christopher Chong, Andre Foehr, Efstathios G. Charalampidis, Panayotis G. Kevrekidis, Chiara Daraio. Breathers and other time-periodic solutions in an array of cantilevers decorated with magnets. Mathematics in Engineering, 2019, 1(3): 489-507. doi: 10.3934/mine.2019.3.489

References

  • 1. Gallavotti G (2008) The Fermi–Pasta–Ulam Problem: A Status Report. Heidelberg: Springer-Verlag.
  • 2. Kevrekidis PG (2011) Non-linear waves in lattices: Past, present, future. IMA J Appl Math 76: 389–423.    
  • 3. Lederer F, Stegeman GI, Christodoulides DN, et al. (2008) Discrete solitons in optics. Phys Rep 463: 1–126.    
  • 4. Binder P, Abraimov D, Ustinov AV, et al. (2000) Observation of breathers in Josephson ladders. Phys Rev Lett 84: 745–748.    
  • 5. Trías E, Mazo JJ, Orlando TP (2000) Discrete breathers in nonlinear lattices: Experimental detection in a Josephson array. Phys Rev Lett 84: 741–744.    
  • 6. English LQ, Sato M, Sievers AJ (2003) Modulational instability of nonlinear spin waves in easy-axis antiferromagnetic chains. II. Influence of sample shape on intrinsic localized modes and dynamic spin defects. Phys Rev B 67: 024403.
  • 7. Schwarz UT, English LQ, Sievers AJ (1999) Experimental generation and observation of intrinsic localized spin wave modes in an antiferromagnet. Phys Rev Lett 83: 223–226.    
  • 8. Swanson BI, Brozik JA, Love SP, et al. (1999) Observation of intrinsically localized modes in a discrete low-dimensional material. Phys Rev Lett 82: 3288–3291.    
  • 9. Peyrard M (2004) Nonlinear dynamics and statistical physics of DNA. Nonlinearity 17: R1–R40.    
  • 10. Morsch O, Oberthaler M (2006) Dynamics of Bose-Einstein condensates in optical lattices. Rev Mod Phys 78: 179–215.    
  • 11. Nesterenko VF (2001) Dynamics of Heterogeneous Materials. New York: Springer-Verlag, USA.
  • 12. Chong C, Kevrekidis P (2018) Coherent Structures in Granular Crystals: From Experiment and Modelling to Computation and Mathematical Analysis. Heidelberg: Springer.
  • 13. Starosvetsky Y, Jayaprakash KR, Hasan MA, et al. (2017) Dynamics and Acoustics of Ordered Granular Media. Singapore: World Scientific.
  • 14. Rosas A, Lindenberg K (2018) Pulse propagation in granular chains. Phys Rep 735: 1–37.    
  • 15. Chong C, Porter MA, Kevrekidis PG, et al. (2017) Nonlinear coherent structures in granular crystals. J Phys Condens Matter 29: 413002.    
  • 16. Sen S, Hong J, Bang J, et al. (2008) Solitary waves in the granular chain. Phys Rep 462: 21–66.    
  • 17. Bonanomi L, Theocharis G, Daraio C (2015) Wave propagation in granular chains with local resonances. Phys Rev E 91: 033208.    
  • 18. Lydon J, Serra-Garcia M, Daraio C (2014) Local to extended transitions of resonant defect modes. Phys Rev Lett 113: 185503.    
  • 19. James G (2011) Nonlinear waves in Newton's cradle and the discrete p-Schrödinger equation. Math Models Methods Appl Sci 21: 2335–2377.    
  • 20. Kimura M, Hikihara T (2008) Stability change of intrinsic localized mode in finite nonlinear coupled oscillators. Phys Lett A 372: 4592–4595.    
  • 21. Sato M, Hubbard BE, Sievers AJ (2006) Colloquium: Nonlinear energy localization and its manipulation in micromechanical oscillator arrays. Rev Mod Phys 78: 137–157.    
  • 22. Erturk A, Inman DJ (2011) Piezoelectric Energy Harvesting. Hoboken: Wiley & Sons.
  • 23. Giannoulis J, Mielke A (2004) The nonlinear Schrödinger equation as a macroscopic limit for an oscillator chain with cubic nonlinearities. Nonlinearity 17: 551–565.    
  • 24. Theocharis G, Kavousanakis M, Kevrekidis PG, et al. (2009) Localized breathing modes in granular crystals with defects. Phys Rev E 80: 066601.    
  • 25. Boechler N, Theocharis G, Daraio C (2011) Bifurcation based acoustic switching and rectification. Nat Mater 10: 665–668.    
  • 26. Huang G, Shi ZP, Xu Z (1993) Asymmetric intrinsic localized modes in a homogeneous lattice with cubic and quartic anharmonicity. Phys Rev B 47: 14561–14564.    
  • 27. Nadkarni N, Arrieta AF, Chong C, et al. (2016) Unidirectional transition waves in bistable lattices. Phys Rev Lett 116: 244501.    

 

Reader Comments

your name: *   your email: *  

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved