Synchronization of Metronomes


A mechanical exemplification of the synchronization of coupled oscillators
are synchronized metronomes. Following J. Pantaleone, two or more metronomes are
placed on a freely moving base, causing an effectice coupling [1]. This
experiment is well suited for classroom demonstrations and presentations.
Furthermore, the famous equation of the Kuramotor model [2] (for 2 oscillators)

∂ ψ / ∂ t = Δ - β sin ψ

can be motivated from an analytical treatment of the coupled metronomes [1]. Here,
β characterizes the oscillator coupling, the frequency difference is given by Δ=(ω12)/ω,
and ψ denotes the phase difference of the oscillators. Obvisiously, phase synchronization
(∂ ψ / ∂ t=0) can be understood as a competition between the coupling and the frequency
difference.

Two uncoupled metronomes

The intrinsic frequency difference of the (more or less identical) metronomes
can be measured by observing the two uncoupled metronomes. Determining the
time between seemingly synchronizations the frequency difference Δ can be
estimated, here Δ=0.004.

Uncoupled metronomes (mov movie, 33MBytes)

In-phase synchronization of metronomes

According to [1] the metronomes are located on a light board which is
placed on two empty cans. In this way, the (positive) coupling leads to
an in-phase (ψ≈0) synchronization of the metronomes. Increasing the number
of metronomes, the system can be considered as a mechanical toy model for
biological systems exhibiting synchronization (such as the famous fire flies).

In-phase synchronization of 2 metronomes (mov movie, 22MBytes)
In-phase synchronization of 5 metronomes (mov movie, 23MBytes)

Anti-phase synchronization of 2 metronomes

Introducing a strong damping of the base movement, the effective coupling
parameter becomes negative. This leads to an anti-phase synchronization (ψ≈π).
As well known, anti-phase synchronization of two pendulum clocks was first
observed by C. Huygens, already in the 17century. In our experiment, the
base movement is suppressed by mats of foam plastic.

Anti-phase synchronization of 2 metronomes (mov movie, 28MBytes)


[1] J. Pantaleone, American Journal of Physics 70, p.992 (2002)
[2] Y. Kuramoto, Chemical Oscillations, Waves and Turbolence (Springer, Berlin, 1984)
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