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.
Skip Nav Destination
,
,
,
,
,
Article navigation
April 2022
Research Article|
April 13 2022
Coupled metronomes on a moving platform with Coulomb friction Available to Purchase
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
Search for other works by this author on:
Lars Q. English
;
Lars Q. English
b)
2
Department of Physics and Astronomy, Dickinson College
, Carlisle, Pennsylvania 17013, USA
Search for other works by this author on:
Emma Behta;
Emma Behta
c)
2
Department of Physics and Astronomy, Dickinson College
, Carlisle, Pennsylvania 17013, USA
Search for other works by this author on:
Hillel Finder;
Hillel Finder
d)
2
Department of Physics and Astronomy, Dickinson College
, Carlisle, Pennsylvania 17013, USA
Search for other works by this author on:
Alice N. Nadeau
;
Alice N. Nadeau
e)
3
Department of Mathematics, Cornell University
, Ithaca, New York 14853, USA
Search for other works by this author on:
Steven H. Strogatz
Steven H. Strogatz
b)
3
Department of Mathematics, Cornell University
, Ithaca, New York 14853, USA
Search for other works by this author on:
Guillermo H. Goldsztein
1,a)
Lars Q. English
2,b)
Emma Behta
2,c)
Hillel Finder
2,d)
Alice N. Nadeau
3,e)
Steven H. Strogatz
3,b)
1
School of Mathematics, Georgia Institute of Technology
, Atlanta, Georgia 30332, USA
2
Department of Physics and Astronomy, Dickinson College
, Carlisle, Pennsylvania 17013, USA
3
Department of Mathematics, Cornell University
, Ithaca, New York 14853, USA
a)
Electronic mail: [email protected]
c)
Electronic mail: [email protected]
d)
Electronic mail: [email protected]
e)
Electronic mail: [email protected]
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
Download citation file:
Pay-Per-View Access
$40.00
Sign In
You could not be signed in. Please check your credentials and make sure you have an active account and try again.
Citing articles via
Reservoir computing with the minimum description length principle
Antony Mizzi, Michael Small, et al.
Recent achievements in nonlinear dynamics, synchronization, and networks
Dibakar Ghosh, Norbert Marwan, et al.
Data-driven nonlinear model reduction to spectral submanifolds via oblique projection
Leonardo Bettini, Bálint Kaszás, et al.
Related Content
Synchronization and chaotic dynamics of coupled mechanical metronomes
Chaos (December 2009)
Anti-phase synchronization of two coupled mechanical metronomes
Chaos (June 2012)
Synchronization of clocks and metronomes: A perturbation analysis based on multiple timescales
Chaos (February 2021)
Synchronization of metronomes
Am. J. Phys. (October 2002)
Human-technology interfaces with the tactile metronome
J. Acoust. Soc. Am. (September 2018)