Synchronization of Huygens' clocks and the Poincare method

We study two models of connected pendulum clocks synchronizing their oscillations, a phenomenon originally observed by Huygens. The oscillation angles are assumed to be small so that the pendulums are modeled by harmonic oscillators, clock escapements are modeled by the van der Pol terms. The mass ratio of the pendulum bobs to their casings is taken as a small parameter. Analytic conditions for existence and stability of synchronization regimes, and analytic expressions for their stable amplitudes and period corrections are derived using the Poincare theorem on existence of periodic solutions in autonomous quasi-linear systems. The anti-phase regime always exists and is stable under variation of the system parameters. The in-phase regime may exist and be stable, exist and be unstable, or not exist at all depending on parameter values. As the damping in the frame connecting the clocks is increased the in-phase stable amplitude and period are decreasing until the regime first destabilizes and then disappears. The results are most complete for the traditional three degrees of freedom model, where the clock casings and the frame are consolidated into a single mass.Comment: 23 pages, 8 figure

Synchronization of clocks

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Synchronization of two self-excited double pendula

We consider the synchronization of two self-excited double pendula. We show that such pendula hanging on the same beam can have four different synchronous configurations. Our approximate analytical analysis allows us to derive the synchronization conditions and explain the observed types of synchronization. We consider an energy balance in the system and describe how the energy is transferred between the pendula via the oscillating beam, allowing thus the pendula synchronization. Changes and stability ranges of the obtained solutions with increasing and decreasing masses of the pendula are shown using path-following

Impact force generator: self-synchronization and regularity of motion

Abstract Impacts in multibody mechanical systems are an object of interest for many scientists in the world. In this paper, we present a principle of operation of the impact force generator being an element of the rotor of the heat exchanger. In this machine, step disturbances of the rotational velocity of the generator cause rapid changes of the rotational velocity of the exchanger rotor, which leads to the intensi®cation of the heat exchange process. We show the phenomenon of self-synchronization, regular motion of the system, and in a special case: chaotic motion of the rotor.

Synchronous motion of two vertically excited planar elastic pendula

The dynamics of two planar elastic pendula mounted on the horizontally excited platform have been studied. We give evidence that the pendula can exhibit synchronous oscillatory and rotation motion and show that stable in-phase and anti-phase synchronous states always co-exist. The complete bifurcational scenario leading from synchronous to asynchronous motion is shown. We argue that our results are robust as they exist in the wide range of the system parameters.Comment: Submitte

Synchronization of slowly rotating pendulums

We study synchronization of a number of rotating pendulums mounted on a horizontal beam which can roll on the parallel surface. It has been shown that after the initial transient different states of pendulums' synchronization occur. We derive the analytical equations for the estimation of the phase differences between phase synchronized pendulums. After study of the basins of attraction of different synchronization states we argue that the observed phenomena are robust as they occur in the wide range of both initial conditions and system parameters