Synchronization of Huygens' clocks and the Poincaré method

Abstract

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 Poincaré 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.

Synchronization and Stability of Two Unbalanced Rotors with Fast Antirotation considering Energy Balance

“…The aforementioned investigation is mainly for the rotor-synchronization or exciter-synchronization in vibrating system, and the phenomenon of synchronization of pendulums hanging on a common moveable beam is another research subject by a number of authors. By the Poincare method and the small parameter method, Jovanovic and Koshkin have studied synchronization and stability of Huygens' clocks [12]. Recently, Koluda and Perlikowski derived the synchronization conditions and explained the energy transmission between double pendula via an oscillating beam with energy balance method [13][14][15].…”

An improved model for the classical Huygens׳ experiment on synchronization of pendulum clocks

“…1(b)) should be more flexible than the horizontal beam of the coupling structure to justify the horizontal mass-spring modelling of the coupling structure as has been applied in many references, see e.g. [11,15,16,[18][19][20]. With this condition, it is guaranteed that the first eigenmode of the structure corresponds to a horizontal displacement of the coupling bar.…”

“…During the last years, the study of Huygens' synchronization has regained attention either from a theoretical perspective [8][9][10][11][12][13][14][15][16][17] or by experimentation [18][19][20][21]. Furthermore, the study of Huygens' synchronization has been extended to the case of arbitrary second-order oscillators [22][23][24].…”

Synchronization of weakly nonlinear oscillators with Huygens' coupling

“…The amplitudes of these solutions are assumed to be complex, 7 i.e., a i ¼ r i e i/ i , i ¼ 1, 2, 3, 4, where a i 2 C; r i 2 R þ and / i 2 S 1 . In this way, it is easy to analyze phase synchronization by looking at the phase differences.…”

“…Since then, Huygens' synchronization has drawn the attention of many researchers. 3,7,20,21,28,33 However, a complete understanding of this phenomenon is still missing. Consequently, the present contribution aims to provide new insights into the intriguing synchronization phenomenon discovered by Huygens.…”