Synchronization of Coupled Mechanical Oscillators: Moving Toward a Mechanical Realization of the Kuramoto Model Linda Lee Kennedy-Columbus Public Schools Dr. Barbara S. Andereck- Ohio Wesleyan University Summer Research REU/RET-2007 Coupling and Synchronization Identical metronomes with similar frequencies were started 180˚ out of phase. Metronomes coupled and relaxed into synchronization because the center of mass of the system remained fixed, allowing the board to oscillate. The phase difference oscillated and diminished with time. Metronomes were set at photogates and started with an initial 25˚-50˚ angle in anti- synchronous position. They were allowed to run until a synchronous mode settled in and the data collection was complete (about 100 s). Modifications were made after analysis of data. These may have included board materials, additional mass, initial angle, metronome choices, metronome settings, damping materials, or photogate positions. *theoretical values in green and red for metronomes one and two* We found, after careful observation, that the metronomes were changing their amplitudes which caused their periods to vary because they are non-linear oscillators. This affect had not been noted in similar experiments published previously. Research Cycle Findings We observed that additional coupling was needed for theory to match experiment. After ruling out flexing of the board and vibrations, we determined that vertical motion due to the requirement that the center of mass stay fixed was non- negligible. This motion was verified by analyzing a video frame by frame. When we added the vertical center-of-mass motion to the equation, theory agreed with experiment. Key Parameters Goals For Future Explorations Identify the optimal board set-up. Increase the number of metronomes to better simulate the Kuramoto Model. Pantaleone, J. American Journal of Physics, Vol. 70, No. 10 Harvard Physics Lecture Series http://people.deas.harvard.edu/~jones/cscie129/nu_lectures/lecture3%20/ho_coupled/ho_coupled.html OWU Student Research Paper http://physics.owu.edu/StudentResearch/2005/BryanDaniels/results.html OWU Student Research Paper Ellis, C.D., Background Research of Coupled Oscillators (2007, unpublished) Strogatz, S.H., Nonlinear Dynamics and Chaos (Addison-Wesley, Reading, MA, 1994) Pikovsky, A; M. Roenblum, J. Kurths, Synchronization (Cambridge University Press, United Kingdom, 2001) References We had four key parameters, three of which were known. These were β (coupling), θ 0 (driving angle) and ∆ (phase difference). A fourth parameter, μ (van der Pol strength), was an unknown constant related to the driving mechanism of the metronome. Acknowledgements We would like to recognize the contributions of Professor Thomas Dillman, who developed and modified the LabView program we used. C. D. Ellis-OWU undergrad Supported by REU/RET NSF Grant 0648751 Analyze Using Theoretical Model Make Adjustments To Research Design Collect Data From Metronome Pairs Analyze Using Theoretical Model Make Adjustments To Research Design Collect Data From Metronome Pairs The Kuramoto Model (shown as a formula above) has been used to theoretically describe physical, biological, and chemical systems. Newton’s Second Law for Metronomes (scaled by J. Pantaleone) d 2 θ 1 /dt 2 + (1 + ∆ ) sin θ 1 + μ((θ 1 / θ 0 ) 2 -1) dθ 1 /dt -β cos θ 1 d 2 (sinθ 1 +sinθ 2 )/dt 2 = 0 This comparison of the theoretical and experimental values shows the phase difference between the two metronomes during one run. d 2 θ 1 /dt 2 + (1 + ∆ ) sin θ 1 + μ((θ 1 / θ 0 ) 2 -1) dθ 1 /dt -β cos θ 1 d 2 (sin θ 1 +sin θ 2 )/dt 2 + β sin θ 1 d 2 (cos θ 1 +cos θ 2 )/dt 2 = 0 How the metronomes synchronized: Additional Coupling Found:
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Synchronization of Coupled Mechanical Oscillators:Moving Toward a Mechanical Realization of the Kuramoto Model
Linda Lee Kennedy-Columbus Public Schools Dr. Barbara S. Andereck- Ohio Wesleyan UniversitySummer Research REU/RET-2007
Coupling and Synchronization
Identical metronomes with similar frequencies were started 180˚ out of phase.Metronomes coupled and relaxed into synchronization because the center of mass of the system remained fixed, allowing the board to oscillate.The phase difference oscillated and diminished with time.
Metronomes were set at photogates and started with an initial 25˚-50˚ angle in anti-synchronous position. They were allowed to run until a synchronous mode settled in and the data collection was complete (about 100 s).Modifications were made after analysis of data. These may have included board materials, additional mass, initial angle, metronome choices, metronome settings, damping materials, or photogate positions.
*theoretical values in green and red for metronomes one and two*
We found, after careful observation, that the metronomes were changing their amplitudes which caused their periods to vary because they are non-linear oscillators.
This affect had not been noted in similar experiments published previously.
Research Cycle
Findings
We observed that additional coupling was needed for theory to match experiment. After ruling out flexing of the board and vibrations, we determined that vertical motion due to the requirement that the center of mass stay fixed was non-negligible. This motion was verified by analyzing a video frame by frame.
When we added the vertical center-of-mass motion to the equation, theory agreed with experiment.
Key Parameters
Goals For Future ExplorationsIdentify the optimal board set-up.Increase the number of metronomes to better simulate the Kuramoto Model.
Pantaleone, J. American Journal of Physics, Vol. 70, No. 10Harvard Physics Lecture Series http://people.deas.harvard.edu/~jones/cscie129/nu_lectures/lecture3%20/ho_coupled/ho_coupled.htmlOWU Student Research Paper http://physics.owu.edu/StudentResearch/2005/BryanDaniels/results.htmlOWU Student Research Paper Ellis, C.D., Background Research of Coupled Oscillators (2007, unpublished) Strogatz, S.H., Nonlinear Dynamics and Chaos (Addison-Wesley, Reading, MA, 1994)Pikovsky, A; M. Roenblum, J. Kurths, Synchronization (Cambridge University Press, United Kingdom, 2001)
References
We had four key parameters, three of which were known. These were β (coupling), θ0 (driving angle) and ∆ (phase difference). A fourth parameter, μ (van der Pol strength), was an unknown constant related to the driving mechanism of the metronome.
AcknowledgementsWe would like to recognize the contributions of Professor Thomas Dillman, who developed and modified the LabView program we used.
C. D. Ellis-OWU undergrad
Supported by REU/RET NSF Grant 0648751
Analyze Using Theoretical
Model
Make Adjustments
ToResearch Design
Collect DataFrom
Metronome Pairs
Analyze Using Theoretical
Model
Make Adjustments
ToResearch Design
Collect DataFrom
Metronome Pairs
The Kuramoto Model (shown as a formula above) has been used to theoretically describe physical, biological, and chemical systems.
Newton’s Second Law for Metronomes (scaled by J. Pantaleone)
d2θ1/dt2 + (1 + ∆ ) sin θ1 + μ((θ1 / θ0)2-1) dθ1/dt -β cos θ1 d2(sinθ1+sinθ2)/dt2 = 0
This comparison of the theoretical and experimental values shows the phase difference between the two metronomes during one run.
d2θ1/dt2 + (1 + ∆ ) sin θ1 + μ((θ1 / θ0)2-1) dθ1/dt -β cos θ1 d2(sin θ1+sin θ2)/dt2 + β sin θ1 d2(cos θ1+cos θ2)/dt2 = 0
How the metronomes synchronized:
Additional Coupling Found:
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