Research On Synchronization Phenomenon In Four Metronome Coupling System

In the past decades,the study of synchronization of coupled nonlinear oscillators was an active topic.Even now,the experiment designed by Huygens,or its variants,is still employed as an economic-yet-effective setup for exploring the collective dynamics of coupled oscillators,as well as for classroom demonstration of synchronization.Compared to the early pendulum clocks,the mechanical metronome was not only similar to the pendulum in terms of its working principle,but also precise and small and easy to operate.Research workers gradually began to replace the pendulum clock with a metronome in the study of complex coupled systems.Based on the coupled metronome system,we have studied the synchronization behavior of four coupled metronome in this paper.We have studied,experimentally and theoretically,the synchronization of four coupled identical mechanical metronomes,and observed four synchronization states,which include in-phase synchronization(IPS),anti-phase synchronization(APS),generalized splay state(GSS),and splay state(SS).GSS is a new kind of synchronous state that we observed.Under the influence of random noise,the general phase difference state is the bridge for transformation when each state transitions.Unlike previous scientists who used computers as the main means of tracking,we used audio signal detection to complete the recording of the state of the metronome system in the experiment.The method is simple and feasible.Measurement tools are very common in life and can easily be brought into classroom teaching.At the same time,the numerical simulation has also completed the study of the distribution of the states in the phase space of the attraction domain,which has guiding significance for us to choose the type of synchronization.For the study of four coupled non-identical metronome systems,we have observed in-phase synchronization,anti-phase synchronization and cluster's anti-phase synchronization in numerical simulation.For the in-phase and anti-phase synchronization states,the original system can be equivalent to two coupled metronome systems whose single metronome quality has doubled.We have studied the synchronization of two coupled mechanical metronomes in the experiment and theory,and found that by increasing the frequency mismatch between the metronomes,the probability for generating APS from the random initial conditions can be clearly increased.In the study of oscillator synchronization,a well-known feature is that the introduction of parameter mismatch will deteriorate synchronization,and in realistic situations,parameter mismatch is always unavoidable.Our present work reveals,from a different viewpoint,the constructive role of parameter mismatch on synchronization.That is,parameter mismatch facilitates APS.This finding provides new perspectives on the collective behavior of coupled oscillators,and might have implications to the functionality and operation of many realistic systems.What's more,we use paper clips to adjust the frequency mismatch between metronome,and the advantages are as follows.First,when the position of the clip is fixed,the natural frequency of the metronome will be determined(time independent),so is the frequency mismatch between the metronomes.Second,the positions of the clips can be adjusted continuously along the pendulum bob,making it possible to freely control the frequency mismatch.Finally,the frequency mismatch can be precisely measured,making it possible to evaluate the impact of frequency mismatch on synchronization in a quantitative manner.The study sheds lights on the collective behavior of coupled oscillators,and will be helpful to the classroom demonstration of synchronization.