Group 4

Group Members: Luis Jover, Vlad Levenfeld, Brad Taylor, & Jeff Tithof

Synchronicity of coupled metronomes.

Background

The phenomenon of synchronization is found throughout nature and in many applications of different fields of science. Examples of synchronization include the flashing of firefly populations, the firing of pacemaker cells in the heart, applications of Josephson junctions, coupled fiber laser arrays, and perhaps even epileptic seizures. Many instances of synchronization in nature and applications in science become extremely complicated as soon as multiple variables or several coupled oscillators are introduced. However, the pervasive presence of synchronization in nature and science motivates work towards a deep understanding of such phenomena. So, as with any system that is to be scientifically understood, the simplest case is the best place to begin; for synchronization, the case of two identical, coupled oscillators is perhaps the simplest.

Research into the mutual synchonization of two identical coupled oscillators is believed to date back to the work of Christiaan Huygens in the 1600's, in which he discovered and investigated the synchronization of two pendulum clocks hanging on a common support beam. Although his goal was to solve the "Longitude Problem" at the time, he did reach some insightful conclusions about the nature of this system, eventually concluding that small movements of the support beam with every oscillation were responsible for the antiphase synchronization that consistently occurred.

Significant progress has been made in explaining simple instances of synchronization. In 2002, Bennett et al. \cite{Bennett} constructed an apparatus meant to recreate the setup decribed by Christiaan Huygens. The experimental and numerical results agree qualitatively with records from Huygens's research. Concurrently, Pantaleone \cite{Pantaleone} investigated a similar setup in which metronomes were placed on a common support which was free to roll across cylinders, allowing coupling through the low-friction horizontal translation of the platform. Pantaleone's theoretical analysis provided agrees with the experimental observations reported.

In 2009, Ulrichs et al. \cite{Ulrichs} reported agreement with Pantaleone's results in computer simulations. Additionally, these simulations were run for coupling of up to 100 metronomes and reportedly showed chaotic and hyperchaotic behavior. A paper by Borrero-Echeverry and Wiesenfeld \cite{Borrero}, currently in the process of publication, describes a theoretical model which encompasses the behavior of both types of oscillators often studied, clocks and metronomes, which often contrast in the phase of synchronization encountered. This paper attributes the difference in synchronization phase to differences in parameters of the oscillators.

An arguably strong theoretical understanding of the $N=2$ case has been developed in recent years; it seems appropriate to expand this theory to a larger number of coupled oscillators. This is the goal of the current proposal.

Theory and Simulation

The equations of motion for two coupled pendulums or metronomes are the same and are available in \cite{Bennett} and \cite{Borrero}. These equations may easily be extended to $N$ metronomes:

\begin{equation} \ddot\phi_j+b\dot\phi_j+\frac{g}{l}\sin\phi_j=-\frac{1}{l}\ddot X \cos\phi_j+F_j \end{equation}

\begin{equation} (M+m)\ddot X+B\dot X = -ml\frac{d^2}{dt^2}(\sin\phi_1+\sin\phi_2+ ... +\sin\phi_N) \end{equation} where $\phi$ is the angular displacement of the jth pendulum, $b$ is the pivot damping coefficient, $g$ is the acceleration due to terrestrial gravity, $l$ is the metronome length, $X$ is the linear displacement of the platform, $F$ is the impulsive drive, $M$ is the platform mass, $m$ is the metronome rod mass, $B$ is the platform friction coefficient, and the dots represent differentiation with respect to time.

References

\bibitem{Strogatz} Steven H. Strogatz, {\it Nonlinear Dynamics and Chaos}, Westview, Cambridge (1994).

\bibitem{Bennett}

M. Bennett, M. F. Schatz, H. Rockwood, and K. Wiesenfeld, ``Huygens's clocks,"

Proc. R. Soc. Lond. A 458, 563-579 (2002).

\bibitem{Pantaleone} J. Pantaleone, ``Synchronization of metronomes," American Journal of Physics 70, 992-1000 (2002).

\bibitem{Ulrichs} H. Ulrichs, A. Mann, and U. Parlitz, ``Synchronization and chaotic dynamics of coupled mechanical metronomes," Chaos 19, 043120 (2009).

\bibitem{Borrero} Daniel Borrero-Echeverry and Kurt Wiesenfeld, ``Huygens (and Others) Revisted," Publication in progress, Obtained through personal contact (2011).

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