Using a combination of theory, experiment, and simulation, we revisit the dynamics of two coupled metronomes on a moving platform. Our experiments show that the platform’s motion is damped by a dry friction force of Coulomb type, not the viscous linear friction force that has often been assumed in the past. Prompted by this result, we develop a new mathematical model that builds on previously introduced models but departs from them in its treatment of friction on the platform. We analyze the model by a two-timescale analysis and derive the slow-flow equations that determine its long-term dynamics. The derivation of the slow flow is challenging due to the stick-slip motion of the platform in some parameter regimes. Simulations of the slow flow reveal various kinds of long-term behavior including in-phase and antiphase synchronization of identical metronomes, phase locking and phase drift of non-identical metronomes, and metronome suppression and death. In these latter two states, one or both of the metronomes come to swing at such low amplitude that they no longer engage their escapement mechanisms. We find good agreement between our theory, simulations, and experiments, but stress that our exploration is far from exhaustive. Indeed, much still remains to be learned about the dynamics of coupled metronomes, despite their simplicity and familiarity.
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April 2022
Research Article|
April 13 2022
Coupled metronomes on a moving platform with Coulomb friction
Special Collection:
Dynamics of Oscillator Populations
Guillermo H. Goldsztein;
Guillermo H. Goldsztein
a)
1
School of Mathematics, Georgia Institute of Technology
, Atlanta, Georgia 30332, USA
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Lars Q. English
;
Lars Q. English
b)
2
Department of Physics and Astronomy, Dickinson College
, Carlisle, Pennsylvania 17013, USA
b)Authors to whom correspondence should be addressed: englishl@dickinson.edu and strogatz@cornell.edu
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Emma Behta;
Emma Behta
c)
2
Department of Physics and Astronomy, Dickinson College
, Carlisle, Pennsylvania 17013, USA
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Hillel Finder;
Hillel Finder
d)
2
Department of Physics and Astronomy, Dickinson College
, Carlisle, Pennsylvania 17013, USA
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Alice N. Nadeau
;
Alice N. Nadeau
e)
3
Department of Mathematics, Cornell University
, Ithaca, New York 14853, USA
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Steven H. Strogatz
Steven H. Strogatz
b)
3
Department of Mathematics, Cornell University
, Ithaca, New York 14853, USA
b)Authors to whom correspondence should be addressed: englishl@dickinson.edu and strogatz@cornell.edu
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a)
Electronic mail: ggold@math.gatech.edu
b)Authors to whom correspondence should be addressed: englishl@dickinson.edu and strogatz@cornell.edu
c)
Electronic mail: behtae@dickinson.edu
d)
Electronic mail: finderh@dickinson.edu
e)
Electronic mail: a.nadeau@cornell.edu
Note: This article is part of the Focus Issue, Dynamics of Oscillator Populations.
Chaos 32, 043119 (2022)
Article history
Received:
January 13 2022
Accepted:
March 23 2022
Citation
Guillermo H. Goldsztein, Lars Q. English, Emma Behta, Hillel Finder, Alice N. Nadeau, Steven H. Strogatz; Coupled metronomes on a moving platform with Coulomb friction. Chaos 1 April 2022; 32 (4): 043119. https://doi.org/10.1063/5.0085216
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