Synchronizing Metronomes
Sam Weeks
Department of Physics and Astronomy
Ithaca College
This equation includes angular acceleration and gravitational torque terms, similar to a simple pendulum. It also contains the Van Der Pol term, which guides the metronome angle to increase or decrease, and the fictitious force of the base, which couples the two metronomes.
Constants like angular frequency and the coupling parameter can then be substituted in and the equation can be applied to both metronomes 1 and 2 to give:
Making substitutions for the sum and difference of the angles:
The final equations become:
Figure 1.1: plots with and without initial angular velocity.
Figure 1.3: relationship between difference in angle vs synchronization time and coupling parameter vs synchronization time.
Figure 1.2: zoom in at beginning and end of plot with velocity. To demonstrate synchronization.
Project Statement
When metronomes are placed on a moving board, they will synchronize, regardless of initial conditions. To be considered “synchronized”, they must oscillate with the same period, max angle, and be either in phase or 2π radians out of phase. This project aims to show this by simulating two metronomes and demonstrating that they are able to synchronize when started at different initial angles or velocities.
Equations of Motion
Starting with the equation of motion for a metronome derived from Newton’s Second Law:
Solutions and Results
To determine whether this system synchronizes, both θ’s were plotted with respect to time. The first plot was done by starting the system with no initial velocity and setting the initial angles at a difference of 0.18 radians. Next, the system was started with an initial angular velocity of 1.5 rad/s to obtain the second plot. Zooming in to the beginning and end of that plot, it can be seen that the metronomes begin out of phase at different angles, but eventually synchronize 2π radians out of phase, oscillating at the same max angle. The last two plots express the relationship between ω vs synchronization time and β vs synchronization time.
Conclusions
As seen in the figures above, metronomes can properly synchronize under varying conditions. The synchronization time of metronomes depends on a few factors like angular velocity, difference in initial angles, and the coupling parameter. Including an initial angular velocity in the system makes it take much longer to synchronize than a system without initial velocity. It was also found that a larger difference in starting angles causes an exponential increase in synchronization time, but a larger coupling parameter causes an exponential decrease in synchronization time.
Acknowledgments
Dr. Matthew Sullivan and Dr. Colleen Countryman, Ithaca College Peers from PHYS 311: Analytical Mechanics
References
Pantaleone, J. (2002). Synchronization of metronomes. American Journal of Physics, 70(10), 992–1000. doi: 10.1119/1.1501118