Synchronous states of slowly rotating pendula

Marcin Kapitaniak, Krzysztof Czolczynski, Przemysław Perlikowski, Andrzej Stefanski, Tomasz Kapitaniak*

*Corresponding author for this work

Research output: Contribution to journalReview articlepeer-review

47 Citations (Scopus)


Coupled systems that contain rotating elements are typical in physical, biological and engineering applications and for years have been the subject of intensive studies. One problem of scientific interest, which among others occurs in such systems is the phenomenon of synchronization of different rotating parts. Despite different initial conditions, after a sufficiently long transient, the rotating parts move in the same way - complete synchronization, or a permanent constant shift is established between their displacements, i.e.,the angles of rotation - phase synchronization. Synchronization occurs due to dependence of the periods of rotating elements motion and the displacement of the base on which these elements are mounted.We review the studies on the synchronization of rotating pendula and compare them with the results obtained for oscillating pendula. As an example we consider the dynamics of the system consisting of n pendula mounted on the movable beam. The pendula are excited by the external torques which are inversely proportional to the angular velocities of the pendula. As the result of such excitation each pendulum rotates around its axis of rotation. It has been assumed that all pendula rotate in the same direction or in the opposite directions. We consider the case of slowly rotating pendula and estimate the influence of the gravity on their motion. We classify the synchronous states of the identical pendula and observe how the parameters mismatch can influence them. We give evidence that synchronous states are robust as they exist in the wide range of system parameters and can be observed in a simple experiment.

Original languageEnglish
Pages (from-to)1-44
Number of pages44
JournalPhysics Reports
Issue number1
Early online date2 Mar 2014
Publication statusPublished - 1 Aug 2014

Bibliographical note

Funding Information:
This work has been supported by the Foundation for Polish Science, Team Programme—Project No. TEAM/2010/5/5.


  • Coupled systems
  • Rotating pendula
  • Synchronization


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