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Synchronizing Pendulum Clocks

This simulation shows the anti-phase and the in-phase synchronization of two pendulum clocks. The interaction between the pendulums is mediated by a massless spring with stiffness constant . The pendulums are attached to two masses with mass constant , and is the damping of the masses. The parameters and are associated with the limit cycle of the individual pendulum clocks. In some cases, the clocks always synchronize in anti-phase. In other cases, depending on the initial conditions, the clocks can synchronize with the same phase or in anti-phase.

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DETAILS

The left and right pendulums have masses and and lengths and , respectively. Both points of attachment have mass . The coordinates and are the initial horizontal positions of the attachment points, and and are the initial angles of the pendulums, measured relative to the vertical. Besides the mass , the interaction or the synchronization parameters are the stiffness constant of the spring and the damping of the attachment points of the pendulums. The parameters and are related with the characteristics of the limit cycle in phase-space of each individual pendulum; is related to the speed with which each pendulum approaches the limit cycle, and is related to the amplitude of the limit cycle. For further details see "Synchronizing Huygens's Clocks" and "On the Problem of Synchronization of Identical Dynamical Systems: The Huygens's Clocks".

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