Clustering of Non-Identical Clocks

We study synchronization of a number of different pendulum clocks hanging from a horizontal beam which can roll on the parallel surface. The results previously obtained for n identical clocks [Czolczynski et al., Prog. Theor. Phys. 122 (2009), 1027] are generalized for the case of non-identical clocks. Pendula 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 pendula can be observed; (i) the complete synchronization in which all pendula behave identically, (ii) pendula create three or five clusters of synchronized pendula. Contrary to the case of identical clocks antiphase synchronization in pairs is riot 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.