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Resonance described through control theory

 
 By: andrea~ : November 13th, 2022-21:43





Have you ever heard of a feedback loop? If you haven't, I can best describe it as a way in which a system can actively detect its output through a sensor and make changes to its variables in order to achieve and maintain a certain desired state automatically.



Control theory is a branch of mathematics mainly used in Electrical, Automation and Robotics engineering, extremely useful for the autonomous operation of devices.


Example: control systems are extremely prevalent in the aerospace field, used to maintain the orientation of satellites, to autopilot planes and missles and to program robots used for space exploration.


But I'm not here to discuss aerospace (unless we're talking Breitling 😉). Here's how control theory can be applied to mechanical watchmaking, in the form of the resonance synchronization found in clocks and watches by Breguet, Janvier and Journe among others.




We can describe the coupling of the two balance wheels of a Resonance through this system of ordinary differential equations (for the sake of simplicity I'll just leave the definition of some of the variables for the answers to any eventual questions, just know that Theta describes the angular position of the balances, omega is the angular velocity of such balances and k signifies the rigidity of the base plate, through which the sound waves propagate in order for the coupling to happen).


As we all know, our beloved mechanical watches don't contain any electronics, so the way in which the oscillators control one another can be described as passive reciprocal control. This makes the system "open", which means that it lacks a feedback loop.




An example of an open loop system





If we assume that the watch is kept wound, the oscillations of the balance wheels can be described through this graph: as you can see, the amplitude is kept constant and the oscillators are in phase opposition, in such a way they are able to compensate one another, thus stabilizing the system. (The simulation was performed using MATLAB and Simulink).


What I love about watchmaking is that it can be appreciated through many lenses, and science is one of them. Hopefully I was clear enough in my explanation. Feel free to ask any questions, I'll try to answer them if I can.

Gorgeous movement,

 
 By: InHavenPro : November 13th, 2022-23:55
inspiring science. Love it, Filip

Thank you!

 
 By: andrea~ : November 14th, 2022-07:19

The two oscillators are mechanically uncoupled, right?

 
 By: mezentius : November 14th, 2022-01:13
How do they compensate and stabilize each other? Through vibrations on the mainplate? Or maybe movements in the air?

Exactly

 
 By: andrea~ : November 14th, 2022-07:19
The mechanical coupling happens through vibrations, and the mainplate is the means to achieve it, since both oscillators are mounted on top of it.

That's very cool

 
 By: mezentius : November 14th, 2022-07:47
I wonder if they're using any special materials on the mainplate to enhance the coupling... I'd expect something more rigid would be better? No doubt shape would matter as well...

I guess I'm wondering how much this effect has been optimized with modern materials and techniques, or if it's currently at the stage of "here's something cool Breguet tried, let's see if it still works decently today"

You are right, the harder the material is, the better the coupling will be

 
 By: andrea~ : November 14th, 2022-08:04
No new materials are being used, the mainplate is either brass (found in early versions) or rose gold, but the results are good regardless.

 
 By: mezentius : November 14th, 2022-08:11
Haha, looks like I would have to get a brass resonance to satisfy my optimization tendencies then! That's very bad news for my wallet though, I guess I'll have to stick with admiring my little CS for now...

Hahaha I know, I love the resonance but the price is a little steep 🤣

 
 By: andrea~ : November 14th, 2022-08:51
A CS is a beautiful watch regardless

I think it is a phenomena called resonance

 
 By: Cookies : November 14th, 2022-08:20
Very interesting to read about it. I watched a youtube video on it and it is mindblowing how this mysterious phenomena happens.

It's exactly that

 
 By: andrea~ : November 14th, 2022-08:52
The resonance effect makes the balances couple by the mechanical waves that propagate through the mainplate

How does all this hold up with a moving wrist?

 
 By: BigFatPauli : November 14th, 2022-13:32
We've all seen the metronomes sync up on a table but has anyone tried it, say, in a moving car on a road?  I think we can all easily envision their inability to stay synced with each bump or turn.

Clock makers used the system in clocks, which are stationary.  

A hard knock can make it fall out of sync, but the balances will reestablish the the Resonance

 
 By: ChristianDK : November 14th, 2022-15:20
If you watch this lecture from HSNY you will see a video of the balances being disturbed and the go back in to resonance. www.youtube.com

How does it hold up on the wrist? In practical terms I can wear mine on the wrist for days without the seconds hands falling out of perfect sync. I think the longest has been 5 days for mine)  This would not be possible without resonance  if you know and observe how differently a movement can behave even from one day to the next depending on tthe conditions it is worn.


Sure it will... Just like metronomes in a car.

 
 By: BigFatPauli : November 14th, 2022-17:52

There's a difference

 
 By: andrea~ : November 14th, 2022-20:16
This analogy doesn't necessarily work since metronomes in a car would have to be fixed to the car and even then there's a huge difference with resonance in a wrist watch.
The effect of your wrist moving doesn't impact the vibrations of the oscillators in the same way and at the same intensity.
One more thing to consider: metronomes are just coupled pendulums, while balance wheels are circular, their oscillations are affected by external forces in different ways.

There's a difference

 
 By: andrea~ : November 14th, 2022-20:16
This analogy doesn't necessarily work since metronomes in a car would have to be fixed to the car and even then there's a huge difference with resonance in a wrist watch.
The effect of your wrist moving doesn't impact the vibrations of the oscillators in the same way and at the same intensity.
One more thing to consider: metronomes are just coupled pendulums, while balance wheels are circular, their oscillations are affected by external forces in different ways.

You believe whatever you like. I know many are very passionate about the brand.

 
 By: BigFatPauli : November 14th, 2022-21:09
And after all, this is a purely passion based hobby: no one needs a mechanical watch.

I'm not a huge Journe fan

 
 By: andrea~ : November 14th, 2022-21:15
I'm personally not a mechanical engineer so my knowledge of the field is restricted, from what I can understand it seems like it does work, but again, I'm not qualified to talk about that aspect.

Being skeptic can be good of course, proof is important, I think that this matter transcends fandom and has more to do with the science behind the watch. 


Well, one thing to consider is how it is implemented by other brands...

 
 By: BigFatPauli : November 14th, 2022-22:02
Consider Dufour, for example, where he used a differential system.  That system as also adopted by MB&F.  Armin Strom uses a "Resonance Clutch Spring".  In all cases, the balances are physically connected which is probably more reliable than counting on vibration carried through a main plate on a moving wrist...

But, again, everyone should buy the watch they like and I know that journe owners will strongly disagree with me.  And that's okay with me; they should enjoy their watches without having to justify it.

There's a few differences

 
 By: andrea~ : November 14th, 2022-22:10
In the case of the Dufour, the watch isn't about the use of resonance, that's why it's not the same as the Journe.

The Resonance Clutch Spring is a nice feature, but not essential for resonance to happen. Resonance was actually discovered when Christiaan Huygens observed how the two pendulum clocks he had on the wall of his room synchronized after oscillating for a while. Just like the wall allowed for that coupling, the mainplate allows for the coupling of the balances in the Chronometre A Resonance. The actualy mechanical pairing is there, just like how the clutch spring is used in the Armin Strom.

In the mathematical system this is shown by the variable k, which represents the rigidity of the mainplate. The higher it is, the better the coupling will be.

"Resonance was actually discovered when Christiaan Huygens ...

 
 By: BigFatPauli : November 14th, 2022-23:26
...observed how the two pendulum clocks he had on the wall of his room synchronized after oscillating for a while. Just like the wall allowed for that coupling, the mainplate allows for the coupling of the balances in the Chronometre A Resonance. "

Sure.  I suspect the wall wasn't moving.  

As soon as I see an test showing that the resonance effect and happen when the coupling surface is moving in an irregular manner (like a wrist), I'll believe in it for a wristwatch like this (without a physical coupling like a Duality).  Until then, it's marketing mumbo-jumbo.

Mine works.

 
 By: TheMadDruid : November 15th, 2022-00:05
Have you ever played with one? Worn one? I wouldn’t be so condescending about something I hadn’t experienced myself.

I'm sorry if you feel I was condescending: that was never my intention.

 
 By: BigFatPauli : November 15th, 2022-00:26
All I am asking for is a simple proof of concept.  Can you provide one?

I think without some serious calculus

 
 By: mezentius : November 15th, 2022-03:35
You would not be able to provide a proof of concept. That is, I don't believe that a proof of concept would be "simple", and I don't believe that a simple explanation would be "proof".
That being said, I'm not a mechanical engineer but I would believe that the whole point of the setup is that the equilibrium in one in which the oscillators are synced, and thereby they would be self correcting.
There is also a concept that disturbances will only affect others if they are at similar frequencies. To a 3 hz oscillation, something like a wrist movement might be considered stationary across the timescale of that oscillation, and will not disturb them relative to each other.
Both these concepts have made me comfortable with the resonance effect, despite me not fully understanding it.
If I had a year, a series of Arduinos, raspberry Pi's, microphones, oscilloscopes and a vibration-isolating optics table I might be able to give you an answer as to how it performs regarding external shocks. But alas I do not.
That gives me an idea though... Anyone want to give a poor scientist a grant and a chronometre a resonance? wink

That's so true!

 
 By: andrea~ : November 15th, 2022-07:27
It's not an easy thing to prove, this is not my field, but everything makes sense to me.
I'd gladly lend you a resonance if I had it 😅

It actually should be an easy thing to prove....

 
 By: BigFatPauli : November 15th, 2022-12:07
The whole concept is based on metronomes on a table syncing, or clocks on a wall. That's the proof of concept. Simple.
The fly in the ointment is that the table or wall are stationary and not having external, irregular forces acted on them (like a moving wrist). The same proof of concept needs to be done, but with a moving table or wall. To those that claim a balance isn't the same as a pendulum... It actually is:
en.wikipedia.org
It is analogous to the pendulum, according to Wiki so if someone claims otherwise, I need a better source than that.
To say that forces from your wrist don't impact the balance is also unfounded. Frist, if the watch is stopped, just slightly nudging the watch will cause the balance to move. Second, if the watch is ticking and it impacted, it will certainly impact the balances beat rate and/or amplitude (ask a watchmaker). Third, we know that how a watch is worn impacts performance (even the direction it is rested while on a table impacts performance) and we know it impacts performance by its effect on the balance. So watches are certainly impacted by outside forces.
The two balances aren't in the same location so the forces will different that impacts each balance...
But none of that matters. All I keep asking for a simple proof of concept: that the balances will "re-sync" after being out of sync from a nudge or quick flick of the wrist. No one can show this, it seems. And that's fine. I'm not trying to convince anyone that it's a good or bad watch, I'm just stating my personal view of it, so I'm not sure why others are trying to convince me otherwise.
The reality is that, in all industries, there can be a lot of "marketing" and ideas being oversold. Does anyone think that the world of watchmaking is totally exempt from this?

The balances are fixed in relative position to each other

 
 By: mezentius : November 15th, 2022-17:41
So translational movements such as "slightly nudging the watch" as you say will affect both equally. The movements which will act significantly differently on the wheels involve rotations about a center which is close to one balance wheel. The distance from the center of rotation to the wheels also must be on the same order as the distance between the wheels - the closer the balance wheels are, the less the rotation will affect them as the more they will behave like a point source.

So most movements, such as swinging an arm around your elbow, I would imagine will apply an fairly equal force to both balance wheels, unless you're wearing your watch at your elbow.

And as to a demonstration of "re-syncing", doesn't the watch do that every time you restart the watch and it falls into resonance after a few minutes?

You have already been given plenty of proof in this thread. But you chose to ignore the arguments, video clips ect and stick to your home made theories.

 
 By: ChristianDK : November 15th, 2022-17:49
You have been here before Pauli. I am sure no matter what arguments you are presented with, your disregard for this watch and FP Journe generally, will remain. 

When You own the watch, you know when it works or when it doesn’t. Just I’ve said all along. Funniest of all, I have never met an owner, past or present who has said: it is/was just a marketing scam. Experience with ownership says  otherwise. Only armchair analysts with No past or current experience to ownership. So I can understand MadDruid when he calls you condescending, as you write off his experience. You don’t hold ownership experience to much value and I have a problem with that as a moderator. I think it is exactly one of the most important aspects of WatchProSite that we hear the real experiences. I don’t have much more patience with that FYI. The discussions we have here need to be respectful. 

An other aspect, interested parties might consider in this discussion are the real watch makers who support this watch. Try to ask Rexhep Rexhepi what he thinks. He built the Resonance watch for years. Real first hand experience, not opinions. He has personally told me on two separate occasions, It is his grail watch! An other watch maker that immediately springs to mind is Anthony Randall who also has had a deep and long interest in FPJ Chronometre a Resonance. He is no fool either. These people may not be able to write a scientific document that can please Pauli but I would take their word as highly talented watchmakers over anyones half baked opinions.

That’s it’s from me.

Again, I am not asking for a scientific paper. I am just asking for a simple proof of concept.

 
 By: BigFatPauli : November 15th, 2022-23:40
I understand that people get defensive of this, but I wish that wasn't the case: I am just trying to understand how it is supposed to work and I am not attacking anyone, or their choice or watch.  You say I have been shown proof here...  I have seen people get mad, I have seen people say, "well it does work" or, "someone told me it works" but I have yet to see actual proof.

I just have a simple ask: a proof of concept.  I can't think of another watch making item that can't be shown via proof of concept, yet I have yet to see someone present if for this complication.  Doesn't that give you pause?

Don't you as an owner want to actually be shown it is working?  The reality is that the balances tick too quickly for our eyes to tell if they are truly synced and a model showing this working while moving would put all the debate to rest.  

If you feel that no proof is needed for you, or that you have sufficient proof, despite not being able to see it work, that's fine but I personally want more, and there is no need to get upset about things as I have said many times.  

We are all watch lovers here, right?  WPS is a place for people to share knowledge and I am humbling asking for this.  I am not a mathematician so formulas and other charts have little meaning to me here.  More over, NASA wasn't 100% certain we could land on the moon, even though they did all the math prior, until we ACTUALLY landed on the moon.  There are larger models of escapements, tourbillions, chronographs, automatic winding systems, keyless works - everything we see in watches - that all are proof of concept.  The resonance effect, within a moving object, seems to totally exclusive to this particular watch where no such model seems to be available. 

You can make a SLOMO video of the movement with an Iphone and then watch the replay.

 
 By: TheMadDruid : November 16th, 2022-11:15
Why don't you try that.
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