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
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
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.
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.
I have made detailed measurements of the metronomes behaviour, mostly for
average frequencies of 208 ticks per minute, see reference . 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 . There
the final state was either antiphase synchronization or "oscillator death".
It is also somewhat different than was found in an older study 
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.
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 . 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
---------- = A - B Sin[theta]
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.
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 . 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  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 .
Because the metronomes synchronize inphase, they provide a useful physical
representation of biological synchronization in heart cells or fireflies
. 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  which is often used to describe biological
If you study some aspects of the metronome system, please let me know what
- 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.
- 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.
- 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.
- S.H. Strogatz,
Nonlinear Dynamics and Chaos (Perseus Publishing, Cambridge, 1994).
Excellent textbook which discusses some elementary features of synchronization.
- S.H. Strogatz and I. Stewart, Coupled oscillators and biological
synchronization, Sci. Am. 269, 102-209 (1993).
- Y. Kuramoto, Chemical Oscillations, Waves and Turbulence
(Springer, Berlin) (1984).