Synchronization of Metronomes

Picture of coupled metronomes

Introduction
Synchronization is the process where two or more systems interact with each other and come to move together.  Biology abounds with examples of synchronization: cells in the heart beat together, audiences often applaud together, fireflies in South-East Asia flash in synchrony, cicada emerge together, etc.. The earliest known scientific discussion of synchronization dates back to 1657 when Christian Huygens built the first working pendulum clock. Huygens studied systems of two pendulum clocks mounted on a common base.  He observed that the clocks would swing at the same frequency and 180 degrees out of phase.  This motion was robust, after a disturbance the synchronized motion came back in about half an hour.  Huygens spent some time exploring this curious phenomena.  Today, synchronization is a fundamental theme in nonlinear phenomena and is currently a popular topic of research.

Apparatus

A version of Huygens original system provides an elegant classroom demonstration of how systems synchronize. The system consists of two pendulum metronomes resting on a light wooden board which sits on two empty soda cans, see picture above. The metronomes used are Wittner's Super-Mini-Taktell (Series 880), and were chosen because they were inexpensive and had a light base.  The base supports were empty beverage containers (Mountain Dew, 12 fl. oz).

Make sure that the table is clean and level.  The cans should have no dents in them so that they roll smoothly and evenly.  Soda cans tend to have their center of mass a little off axis (causing them to oscillate on their own when placed on a table).  To reduce the can oscillation effects, andso make the system as simple as possible, I removed the cans "pop-top's" and a little of the surrounding metal on their ends. I recently tried using plastic soda bottles and these works just as well, with no trimming required.

Setup
 Here are a few suggestions on how to get started.  Set both metronomes to the same frequency setting.  I usually use presstisimo (208 ticks per minute) which is the fastest setting.  Deflect the two pendulum bobs in opposite direction (180 degrees out of phase) and let go.  Typically, the metronomes will oscillate near this "antiphase" state for 10-20 seconds, then over the course of a minute, the system will evolve away from this to a steady state of inphase synchronization---i.e. only a very small phase difference between the oscillators. Once you observe what this synchronized behaviour looks like, then try increasing the difference in the natural (uncoupled) frequencies and observing what happens.  The frequency difference between the pendulums must be set with a little care.  If the natural frequencies of the two pendulums differ by more than a few percent,  synchronization will not occur.  However the metronomes that I used allowed very precise, accurate, reproducible frequency settings.

For classroom demonstrations I always first place the metronomes on the table and show the class that the uncoupled oscillators do not synchronize but just oscillate quasiperiodically.  Then I place the oscillators on the movable platform and demonstrate synchronization.

Observations
I have made detailed measurements of the metronomes behaviour, mostly for average frequencies of 208 ticks per minute, see reference [1].  In general, for small frequency differences the metronomes quickly synchronize and remain in a phase locked state until the metronomes' springs wind down.  The final synchronized state always had only a small phase difference, i.e. inphase synchronization.  The motion of the base is quite noticeable, with an oscillation amplitude of order a couple millimeters, so that the reason for the coupling of the metronomes is obvious.  Phase differences between the metronomes can be heard as time lag between the metronomes clicks.  When the metronomes are synchronized, the larger the natural frequency difference between the oscillators the larger the time lag between the ticks in the synchronized state.  Thus the ticks  provide an easy way to quantify the relative motion of the pendulum bobs.

In the synchronized state, it is quite noticeable that the two pendulums oscillate with different amplitudes.  The larger the difference in the natural frequencies, the larger the amplitude difference.  The oscillator with the larger natural frequency has the larger swing amplitude.  

It is interesting that the inphase synchronization found here is different than was found in a recent reproduction of Huygens original system [3].  There the final state was either antiphase synchronization or "oscillator death".  It is also somewhat different than was found in an older study [2] where they reported finding both inphase and antiphase synchronization to be simultaneously stable.   I found that the antiphase synchronization state can be made stable in the metronome system if the cans are placed on a wet table---i.e. enhancing the damping associated with the base motion.  Also, antiphase synchronization and other types of synchronization are possible at larger average frequencies.

Explanation
As one metronome's pendulum bob moves to the right, this pushes the base to the left (because of momentum conservation).  The base moving to the left then pushes the other metronome's pendulum bob to the right---i.e. in the same direction as the first pendulum.  Thus the slightly faster pendulum gives a kick (through the base) to the slower metronome causing the slower pendulum to speed up.

The coupled metronome system may be described mathematically in a straightforward manner [1].  This model simplifies considerably in the limit that the coupling between the metronomes is small (when the base is heavy).  Then the dynamics of two metronomes are described by the approximate evolution equation:

d theta                            
  ----------  = A - B Sin[theta]
d t                              

where theta is the phase difference between the oscillators, A denotes the relative frequency difference and B the coupling.  Choosing some arbitrary values for these quantities (A=0.2, B=1), the nature of the evolution is illustrated on a phase portrait.  

Plot of phase evolution versus phase
The arrows illustrate the direction of the evolution, and the solid and open circles denote the stable and unstable fixed points.  This simple equation accurately describes the measured phase difference between the metronomes at small coupling.  Also, the model qualitatively explains why as the phase difference (A) is increased the synchronized state disappears.  However to accurately describe many of the fine details of the system the other degrees of freedom (the pendulum amplitudes) become relevant and one needs the more complete model [1].  For example, the above equation breaks down when describing the approach to synchronization (which is oscillatory), the precise location of the synchronization threshold, and the behaviour above the threshold (no bottleneck).

The metronome system described here differs from previous attempts [2, 3] to study coupled pendulums in a fundamental way.  The metronome system has negligible damping of the base motion, while in a recent study [3] the base damping appears to be the largest damping in the problem. More importantly, the oscillation amplitude of the metronomes pendulum is quite large, about 45 degrees.  Nonlinear effects from the large swing amplitude  destabilize the antiphase synchronization [1].

Further explorations
Because the metronomes synchronize inphase, they provide a useful physical representation of biological synchronization in heart cells or fireflies [5]. The number of metronomes coupled together can easily be increased beyond the two discussed here, just put more metronomes on the platform.  It would be interesting to study a larger system and compare the results to the well known Kuramoto model [6] which is often used to describe biological oscillators.  

If you study some aspects of the metronome system, please let me know what you find.

References
  1. J. Pantaleone, Synchronization of Metronomes, American Journal of Physics 70 (10) 992-1000 October (2002). My article studying the couple metronome system experimentally and theoretically.
  2. I.I. Blekhman,  Synchronization in Science and Technology, (ASME Press, New York 1988).  Experimental and theoretical studies of Huygens apparatus are discussed in this textbook.
  3. B. Bennett, M.F. Schatz, H. Rockwood and K. Wisenfeld, Huygens' clocks, Proc. Roy.  Soc., (2001).  Recent study of a version of Huygens apparatus using pendulum clocks mounted on a heavy, wheeled cart.
  4. S.H. Strogatz Nonlinear Dynamics and Chaos (Perseus Publishing, Cambridge, 1994). Excellent textbook which discusses some elementary features of synchronization.
  5. S.H. Strogatz and I. Stewart, Coupled oscillators and biological synchronization, Sci. Am. 269, 102-209 (1993).
  6. Y. Kuramoto,  Chemical Oscillations, Waves and Turbulence (Springer, Berlin) (1984).