Controlled synchronization of pendula

An experimental study on synchronization of nonlinear oscillators with Huygens' coupling

“…It should be noticed that system (9) with input (13) has at least two invariant sets, namely the anti-phase invariant set Ω 1 := φ 1 = −φ 2 ,φ 1 = −φ 2 , x 3 = 0,ẋ 3 = 0 and the in-phase invariant set Ω 2 := φ 1 = φ 2 ,φ 1 =φ 2 , x 3 = 0,ẋ 3 = 0 , see [7]. In addition, the local stability of the set Ω 1 has been proven in [7].…”

“…assuming that no friction is present -equations of motion of this system are: The limit behaviour of the uncontrolled and undamped system, i.e. u 1 = u 2 = d 1 = d 2 = 0, can be analyzed as follows [7]. Consider the total energy of the system as a candidate Lyapunov function: …”

“…We can distinguish two approaches: the theoretical approach and the one oriented to obtain insight by experimental analysis. Some works related to the theoretical approach are presented in [7][8][9][10][11][12][13][14]. For the second approach several experimental setups have been created by researchers in order to reproduce the observations made by Huygens and find mathematical arguments for the synchronized motion of the clocks.…”

Effects of time delay in the synchronized motion of oscillators with Huygens’ coupling

“…the pendula receive the same input. The vibrations in the coupling bar, which can be modeled as a suspended rigid bar of one degree of freedom (Pogromsky et al, 2003), are influenced by the forces exerted by the pendula. Therefore, Huygens' model of pendulum clocks can be casted into the form (1)-(5).…”

Experimental Huygens Synchronization of Oscillators

“…In [5][6][7][8] an extended analysis of this phenomenon is presented. In [9] the authors present an experimental study of Huygens synchronization and finally, in [10,11] a study of the uncontrolled as well as the controlled Huygens experiment is presented.…”