Phenomenon of Time Synchronization

The video above displays 5 coupled metronomes that all start at different times with different frequencies, but somehow synchronize in less than a minute. Perplexing right? Well, let's take rewind a bit...

Background:

In 1657, Christian Huygens invented the first working pendulum clock and continuously tried to improve it in order to create a reliable time device for maritime travel to solve the longitude problem(clocks often became out of time and slowly became inaccurate due to the rolling seas). His plan was to attach two clocks to a heavy, freely-moving mass in order to avoid the effect of the sea, but when he tested out this arrangement, he discovered that after about 30 minutes, the two clocks slowly became in sync and ticked at the same time. 

After several tests, he realized the reason for the synchronization was due to the clocks being attached to the same platform, and once they weren’t, they wouldn’t sync together. Huygens realized that mechanical vibrations were being transferred by the clocks to the platform which transferred to the other clock, making them coupled. 

In this Instructable, I will be displaying, explaining, and analyzing this phenomenon through a system of coupled metronomes and connecting it to real world applications. 

Terms to Know: 

Coupled: In classical mechanics, coupling is a connection between two oscillating systems

Frequency: the term frequency refers to the number of waves that pass a fixed point in unit time. It also describes the number of cycles or vibrations undergone during one unit of time by a body in periodic motion.

Synchronization: two nonidentical oscillators start to oscillate with the same frequency

Kuramoto Model: a mathematical model used to describing synchronization

3 Metronomes

Smooth Platform 

Two cylindrical objects

I found two metal cans and placed my foam board platform on top of the cans. Afterward, I spaced the metronomes an equal distance from each other where one was in the middle and two were on the sides. Afterwards, I started one metronome and continued to start the others where the metronomes were out of sync and all started with different frequencies. After a while, in about 30 seconds to a minute, the metronomes slowly sync together and become synchronized. I repeated this experiment for a total of 5 times, each time a different way the metronomes would tick in relation to one another, and the same phenomenon occurred every single time.

Now, let's start analyzing why this occurs...

Referencing the test videos in the previous step, it can be seen that the platform moves with the metronomes. Let’s use the movement of just two metronomes for now, which is displayed in the video above. If the two metronomes are in sync, when the metronome ticks to the left, the platform will move to the right. This can easily be explained by Newton's Third Law, where the action and reaction pair is the metronome to the moving platform system (action force to the left, reaction force to the right). However, if we add a third metronome that is out of sync with the two metronomes that are in sync, the movement of the board becomes unstable, but the overall movement of the center of mass still oscillates with respect to the two metronomes that are in sync. This is due to the overall force of the two in sync metronomes to be greater than the out of sync metronome. As time passes, since the overall movement of the board oscillates with respect to the two in sync metronomes, the out of sync metronome is given a “kick”, or mechanical motion, that affects its oscillation of its tick, speeding it up until it oscillates and syncs with the two other metronomes. 

A simple explanation would be like this: The overall movement of the platform follows the movement of the two synced metronomes(coupled metronomes), so the force of the single out of sync metronome is canceled out, yet it is also affected by the the remaining force of the platform, resulting in external mechanical energy that creates a force that speeds up the metronome and increases its frequency in order to make all three metronomes synchronized, allowing for the platform to oscillate with respect to the metronomes without disturbances. 

Evidence of this “external” mechanical energy is this video. The second part of the video shows two metronomes moving together in sync with the moving platform moving with respect to the system, but it can also be seen that the metronome that should be stationary is actually moving. This implies that the movement of the platform results in mechanical energy being transferred into the stationary metronome, causing it to oscillate. 

Now that it can be seen that there is an external, or extra force, acting on the out of sync metronome from the platform. This force is what causes the metronomes to eventually sync up, regardless of how out of sync they start. 

If the intuitive explanations do not make sense, here are some mathematical equations. I am going to cite Xuepeng Wang’s paper on coupled metronomes as the equations get a bit complicated and confusing.

His paper: https://arxiv.org/pdf/1805.00416.pdf

Above, you can see an animation that displays the motion of one metronome. We can represent this motion as circular motion. We can set the starting point as 0 degrees, and each time the metronome ticks, or reaches one side, it will have “rotated” or moved 180 degrees, which is half a circle, and when it hits the other side and ticks, then it will have “rotated” or moved 360 degrees, which is a full circular rotation, and is back to 0 degrees. If we increase the frequency of the metronome, then the dot will move faster in the circle. Now, let's consider a system of two metronomes with this representation. In the next animation, it can be seen that the two metronome dots rotating are not in sync as the metronomes are not in sync and have different frequencies. The second animation depicts two metronomes with the same frequency, but are not in phase. This specific animation is two metronomes that are in antiphase as if they were in phase, the dots would be moving together as one.

Now that we have this knowledge, we can analyze the Kuramoto model. From this previous example, we can use this information to create an intuitive explanation for the Kuramoto model with the use of the dots rotating in a circle. It says that the rate that each dot goes around the circle equals its natural frequency plus a certain amount of how far it is from the other dot, which is determined by the coupling strength, K. (Depicted in third animation)

A better explanation for this model goes like this: Imagine you're running with some of your friends on a circular track, but maybe one of your friends is faster than you and another slower. The speed of how fast each individual is your natural frequency. Now, let’s say your faster friend tells you and your slower friend to speed up, and if they are willing to go a bit slower while you and your other friend decide to speed up, all of you will eventually run at the same pace with each other. This push by your other friend is the certain amount of how far you are to the other “dots”, or friends in this case and is the second part of the Kuramoto model. However, the coupling strength can be described as how strong your friendship is with one another. If you're not very good friends with one another, synchronization will never occur as your “coupling strength” is too weak, which is the letter K in the equation. (Depicted in fourth animation)

Here is a link to a nice simulation and simple explanation of the Kuramoto model: https://www.complexity-explorables.org/explorables/ride-my-kuramotocycle/

The fireflies of Southeast Asia are able to flash their lights together even though individually, they all flash at their own frequencies and there are thousands of them, yet they are somehow able to synchronize their flashes of light together. This is due to how strong their coupling strength is to each other. Based on the Kuramoto model, this means that each firefly would have an effect on every single other firefly. I want to reference a simulation that I found online that perfectly demonstrates the coupling of fireflies flashing: https://ncase.me/fireflies/

In this simulation, if you switch on the option to nudge neighbors, the fireflies will begin to start flashing together, and eventually, it can be seen that the fireflies will all sync their flashes together at the end. If we increase the slide to make the nudges, we are increasing the coupling strength, causing a much more synchronized pattern for the fireflies. Time synchronization is easily explained by the Kuramoto model in this case since in the simulation and in real life, the coupling strength of the fireflies is strong enough for a synchronized flashing of each flies light.

If we use the Kuramoto model and apply it to our daily lives, it seems as if everything in our lives somehow are tied together. Coupling strength could be anything, from a slight nudge from a friend or some sound waves of moving shoes or clapping. Take audience members clapping in Budapest for example. In the video above, notice the phase transition where the members suddenly become in phase and synchronized. Imagine yourself being an audience member. The clapping of your neighbors around along with the “coupling strength” between everyone eventually leads to synchronized clapping from the audience, which is represented by the Kuramoto model.

Additionally, we can freeze water. When water freezes in a bottle, it does not freeze a piece at a time. The freezing of all the molecules of water tends to synchronize and the water usually freezes together in a quick amount of time due to the coupling strength between the molecules. The effect of each molecule on each other is determined by its coupling strength, but after a certain point, the coupling strength will reach its critical point. In this case with water, the critical coupling strength is determined by the temperature of the water. At this point, the molecules are able to somehow lock their phases in time to freeze together, which is what we call synchronization. 

Time synchronization is a phenomenon that occurs every single day, whether it’s in nature, your home, or at work. What I find most intriguing and appealing is how universal time synchronization is and how it can be applied to our lives without even realizing it. Synchronization occurs at every single scale and point in nature, from subatomic particles to the cosmos and uses natural communication, from electrical to chemical to gravitational to mechanical interactions. Wherever nature is, time synchronization will always somehow occur, whether in our solar system or just birds chirping. I learned a lot from the science fair project and I hope you learned something cool and new as well. This was a really interesting topic and was extremely fun to research and understand such a unique and intriguing topic. I hope you enjoyed this Instructable and feel free to ask any questions if you have any! Thank you so much for reading and have a great day! :)

Big thanks to the Youtuber, Veritasium, for the inspiration for this project and some of the animations used