We consider the synchronization of two clocks and repeat Huygens’ experiment using real pendulum clocks in the same way as it was done originally, i.e., we hang two clocks on the same beam and observe the behavior of the pendulums. The clocks in the experiment have been selected in such a way to be as identical as possible. It has been observed that when the beam is allowed to move horizontally, the clocks can synchronize both in-phase and anti-phase. We perform computer simulations of the clocks' behavior to answer the question how the nonidentity of the clocks influences the synchronization process. We show that even the clocks with significantly different periods of oscillations can synchronize, but their periods are modified by the beam motion so they are no more accurate. Later we consider the synchronization of two clocks which are accurate (show the same time) but have pendulums with different masses. We show that such clocks hanging on the same beam beside the complete (in-phase) and antiphase synchronizations perform the third type of synchronization in which the difference of the pendulums’ displacements is a periodic function of time. We identify this period to be a few times larger than the period of pendulums’ oscillations in the case when the beam is at rest. Our approximate analytical analysis allows to derive the synchronizations conditions, explains the observed types of synchronizations and gives the approximate formula for both the pendulums’ amplitudes and the phase shift between them. We consider the energy balance in the system and show how the energy is transferred between pendulums via oscillating beam allowing pendulums’ synchronization. Finally, we study synchronization of a number of different pendulum clocks. Pendulums have the same period of oscillations so the clocks are accurate but have different masses. It has been shown that after a transient, different types of synchronization between pendulums can be observed;(i) the complete synchronization in which all pendulums behave identically, (ii) pendulums create three or five clusters of synchronized pendulums. Contrary to the case of identical clocks antiphase synchronization in pairs is not robust for an even number of clocks. 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. We finish with the idea how to use “pendulum clocks” to extract energy from the sea waves.
Language of the lecture: English
Duration of a lecture: 2 academic hour
Tutor of the course:
Prof. Tomasz Kapitaniak, Technical University of Lodz, Lodz, Poland.