Synchronization and chaotic dynamics of coupled mechanical metronomes

Ulrichs, Henning; Mann, Andreas; Parlitz, Ulrich

doi:10.1063/1.3266924

Cited by 71 publications

(60 citation statements)

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“…To explain why other cluster configurations are not possible go back to the approximation given by eq. (102). The pendula act on the beam M with the force which consists only of the first and the third harmonics of the pendula's oscillations frequency α.…”

Section: Discussionmentioning

confidence: 99%

“…Instead of the models with an impulse supply of energy to the clock pendulum (i-iii), a van der Pol oscillator model with a continuous supply of energy can be considered [74,18,87,102]. In this case the equation of motion of a single pendulum clock is as follows,…”

Section: (Iv)mentioning

confidence: 99%

“…Recently, this idea has been validated by a few groups of researchers who tested Huygens' idea [11,90,55,96,39,62,43,87,102,[31][32][33][34][35]. These studies do not give the definite answer to the question: what Huygens was able to observe.…”

Section: Introductionmentioning

confidence: 99%

“…Both in-phase ( Figure 3(b)) and antiphase (Figure 3(a)) synchronization of the metronomes have been observed. The synchronization problem for larger number of metronomes has been studied inUlrichs et al [102]andCzolczynski et al [31,32]. antiphase synchronization.…”

Section: Introductionmentioning

confidence: 99%

“…The problem of clocks synchronization is also studied by Blekhman [16] where the clocks have been modeled as van-der Pol's type self-excited oscillators (as in [87,102]). Blekhman also discusses Huygens' observations, and recounts the results of a laboratory reproduction of the coupled clocks as well as presenting a theoretical analysis of oscillators coupled through a common supporting frame.…”

Section: Introductionmentioning

confidence: 99%

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Synchronization of clocks

et al. 2012

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Abstract:In this report we recall the famous Huygens' experiment which gave the first evidence of the synchronization phenomenon. We consider the synchronization of two clocks which are accurate (show the same time) but have pendulawith different masses. It has been shown that such clocks hanging on the same beam can show the almost complete (in-phase) and almost antiphase synchronizations. By almost complete and almost antiphase synchronization we defined the periodic motion of the pendula in which the phase shift between the displacements of the pendula is respectively close (but not equal) to 0 or π.We give evidence that almost antiphase synchronization was the phenomenon observed by Huygens in XVII century. We support our numerical studies by considering the energy balance in the system and showing how the energy is transferred between the pendula via oscillating beam allowing the pendula's synchronization. Additionally we discuss the synchronization of a number of different pendulum clocks hanging from a horizontal beam which can roll on the parallel surface. It has been shown that after a transient, different types of synchronization between pendula can be observed;(i) the complete synchronization in which all pendula behave identically, (ii) pendula create three or five clusters of synchronized pendula. We derive the equations for the estimation of the phase differences between phase synchronized clusters. The evidence, why other configurations with a different number of clusters are not observed, is given.

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Section: Discussionmentioning

confidence: 99%

Section: (Iv)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Synchronization of clocks

et al. 2012

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Individual and collective behavior of vibrating motors interacting through a resonant plate

Mertens

Weaver

2010

Complexity

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Dynamics of oscillators coupled by a medium with adaptive impact

Daneshvar

2013

Complexity

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In this article, we study the dynamics of coupled oscillators. We use mechanical metronomes that are placed over a rigid base. The base moves by a motor in a one-dimensional direction and the movements of the base follow some functions of the phases of the metronomes (in other words, it is controlled to move according to a provided function). Because of the motor and the feedback, the phases of the metronomes affect the movements of the base, whereas on the other hand, when the base moves, it affects the phases of the metronomes in return. For a simple function for the base movement (such as y = γ x [rθ 1 + (1 − r)θ 2 ] in which y is the velocity of the base, γ x is a multiplier, r is a proportion, and θ 1 and θ 2 are phases of the metronomes), we show the effects on the dynamics of the oscillators. Then, we study how this function changes in time when its parameters adapt by a feedback. By numerical simulations and experimental tests, we show that the dynamic of the set of oscillators and the base tends to evolve towards a certain region. This region is close to a transition in dynamics of the oscillators, where more frequencies start to appear in the frequency spectra of the phases of the metronomes. We interpret this as an adaptation towards the edge of chaos.

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Synchronization and chaotic dynamics of coupled mechanical metronomes

Cited by 71 publications

References 10 publications

Synchronization of clocks

Synchronization of clocks

Individual and collective behavior of vibrating motors interacting through a resonant plate

Dynamics of oscillators coupled by a medium with adaptive impact

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