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

Christopher Chong; Andre Foehr; Efstathios G. Charalampidis; Panayotis G. Kevrekidis; Chiara Daraio; Christopher Chong; Andre Foehr; Efstathios G. Charalampidis; Panayotis G. Kevrekidis; Chiara Daraio

doi:10.3934/mine.2019.3.489

Mathematics in Engineering

2019, Volume 1, Issue 3: 489-507. doi: 10.3934/mine.2019.3.489

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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

Received: 11 December 2018 Accepted: 19 April 2019 Published: 03 June 2019

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.
- 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[J]. Mathematics in Engineering, 2019, 1(3): 489-507. doi: 10.3934/mine.2019.3.489

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Abstract

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.

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