Synchronization of weakly nonlinear oscillators with Huygens' coupling

Abstract

In this paper, the occurrence of synchronization in pairs of weakly nonlinear self-sustained oscillators that interact via Huygens' coupling, i.e., a suspended rigid bar, is treated. In the analysis, a generalized version of the classical Huygens' experiment of synchronization of two coupled pendulum clocks is considered, in which the clocks are replaced by arbitrary self-sustained oscillators. Sufficient conditions for the existence and stability of synchronous solutions in the coupled system are derived by using the Poincar e method. The obtained results are supported by computer simulations and experiments conducted on a dedicated experimental platform. It is demonstrated that the mass of the coupling bar is an important parameter with respect to the limit synchronous behaviour in the oscillators. Probably, the earliest writing on inanimate synchronization is due to the Dutch scientist Christiaan Huygens (1629-1695), who discovered that two pendulum clocks hanging from a common support show synchronized behaviour. 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. In particular, the following questions are addressed: given the Huygens system of coupled pendulum clocks, is it possible to replace the pendulum clocks by other types of second order nonlinear oscillators and still to observe the synchronized motion? Additionally, which is/are the key parameter(s) in the coupled system for the occurrence of in-phase, respectively, anti-phase, synchronization? These questions are answered by means of theoretical analysis, computer simulations, and experiments.

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

“…Kapitaniak et al explored synchronous states of slowly rotating pendula considering the oscillating beam moving in a single DOF [16,17]. Pena-Ramírez concerned the synchronized motion of Huygens' model building on the original work of Blekhman [18,19]. Yurchenko et al described synchronization of rotating parametric pendulums combining stochastic calculus [20].…”

The sympathy of two pendulum clocks: beyond Huygens’ observations

“…In the last years, Huygens’ synchronization has become a relevant topic among scientists and researchers. By designing novel experimental platforms 9 10 11 12 13 and/or by conducting theoretical analyzes 14 15 16 17 18 19 20 21 22 23 24 , further understanding about the exciting phenomenon described by Huygens has been obtained. In particular, the aforementioned studies somehow convey the same message: the key element in Huygens’ setup of pendulum clocks is the coupling structure and its mechanical properties.…”

“…Current models are an oversimplification of the real Huygens’ system, in part because the coupling structure has been assumed to be rigid, see e.g. 23 26 . However, the coupling structure used by Huygens, a wooden beam on the top of two chairs, is in fact a flexible body, which necessarily needs to be modeled by using a partial differential equation with suitable boundary conditions.…”

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

“…The local stability of the limit cycle of Fig. 5 can be investigated by using Proposition 3.3 in [33] from which it follows that the limit cycle is locally stable if the Jacobian of the map from xð0Þ to xð4Þ, see (24), has all its eigenvalues inside the unit disk and unstable if at least one of the eigenvalues is outside the unit disk. However, the stability analysis of the inphase or anti-phase synchronized motion is a difficult task and is considered beyond the scope of this paper.…”

“…Furthermore, the study of Huygens' synchronization has been extended to the case of arbitrary second-order oscillators [22][23][24]. In such works, the pendulum clocks have been replaced by arbitrary nonlinear oscillators.…”