Resonance described through control theory

Nov 13, 2022,21:43 PM
 






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.


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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 m... 

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 stationar... 

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. 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 ... 

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 ... 

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 ... 

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 ... 

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 ... 

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 ... 

"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. ... 

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 tha... 

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 ... 

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 fr... 

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. Fu... 

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 ... 

I don't have a way do that while wearing the watch and moving around.

 
 By: BigFatPauli : November 16th, 2022-11:39
But, yes, I would I love to see a video like that. More so, I would like to see someone put one a machine able to measure the two seperate beat rates and amplitudes, be able to move that around, and see how long it takes for the balances to naturally re-s... 

Paul raises a good point though.

 
 By: Cozmopak : November 16th, 2022-22:37
I believe that the resonance system works while at rest, but I would like to see proof that it persists with the wrist in motion. That's important. Or if it doesn't persist, how long does it take for it to come back in sync?

I imagine it will decrease variance

 
 By: mezentius : November 15th, 2022-09:39
From factors caused by differences between individual balance wheels/springs such as manufacturing and setting imperfections, but not those caused by factors affecting both wheels equally such as position on the wrist... Unless perhaps Journe designs a re... 

I get it in concept but shouldn't all those factors be accounted for when the watch is made and regulated?

 
 By: Fastwong : November 15th, 2022-14:24
In particular, the dual balance systems are really hard to regulate as seen by the propensity of the Resonance watches needing to go back to service for tuning... And also Max Busser talked about how much of a pita it is because you can just stick it on a... 

To be honest,

 
 By: mezentius : November 16th, 2022-05:49
I never understood why one had to be slow and one had to be fast. I've always thought the point of resonance to cancel out variances within the operation of each balance - to improve precision, not accuracy. Intuitively, that should occur no matter if the... 

Just to clarify. you dont have a slow and a fast balance. They are regulated to the same frequency. Ideally +/- 0 sec on all 6 positions...

 
 By: ChristianDK : November 16th, 2022-15:29
The mentioning of the slow and fast balance refers to the situation where the watch receives a knock. this disturbance will cause the balance wheel to either accelerate or slow down briefly. Since the two balance wheels run in opposite direction they will... 

I understand they need to be regulated to the same frequency in order to achieve resonance but I think it's safe to assume that at the smallest level they are not identical because then it would all be pointless.

 
 By: Fastwong : November 16th, 2022-15:47
My point being, could one of the balances be overall more accurate than the other in which case having two balances is detrimental to time keeping? Given that a single balance is easier to regulate, practically a single balance might be the more accurate ... 

On the Micro level you are right, of course

 
 By: ChristianDK : November 16th, 2022-18:36
They can not be regulated to exactl same. Just close enough and then they will begin to synchronize and resonate. This does help timekeeping in that er evens out error and creates a more stable running. To your other question about what is more accurate, ... 

Yes, he is quite clear on this

 
 By: ChristianDK : November 17th, 2022-07:08
He considers a Tourbillon, first and foremost a thing of beauty and fascinating. He has said that installing a Tourbillon is like deliberately breaking a leg to put a casket on it to fix a problem. It solves a problem but it also creates more problems tha... 

Thanks for explaining!

 
 By: mezentius : November 16th, 2022-19:16
But this phenomenon makes me more doubtful. I would imagine the vibrational forces coupling the two balances together at the moment of impact would be orders of magnitude less than external shocks acting on the balances as they move, and so only would aff... 

They are difficult to regulate because both sides need to be regulated to almost perfection in all 6 positions. That is hard in any watch.

 
 By: ChristianDK : November 16th, 2022-15:19
From what I have been explained by a watchmaker a watch can be well regulated in 5 positions and the watchmaker can "hide" the irregularities in the 6th position. The Resonance is unforgiving in that way. you need regulating in 6 positions x 2 to be withi... 

The proof can be seen in the wearing of the watch.

 
 By: TheMadDruid : November 15th, 2022-11:32
It’s that elemental; and that difficult. Assuming you aren’t an owner try going to a Boutique and playing with one.

what a nice post Andrea

 
 By: ChristianDK : November 14th, 2022-15:49
I really appreciate it. (even if formulas are not my strong side ;-)) I dont know if you ever watched this lecture at HSNY. I hope you will find it interesting. The Resonance is a very stimulating and fascinating watch in so many levels. Your last words i... 

I did

 
 By: andrea~ : November 14th, 2022-17:12
It was really insightful and helped me develop the mathematical model for this simulation

It's a pleasure!

 
 By: andrea~ : November 14th, 2022-20:17
I am an engineer, I studied control systems in the past but now my focus is on materials science

Grande!

 
 By: massi. : November 14th, 2022-20:53

Grazie! 😊

 
 By: andrea~ : November 14th, 2022-20:55

Excellent, Grazie Andreas.

 
 By: ChristianDK : November 14th, 2022-23:03
Its a pleasure with such well informed argumentation. It reminds of the day I sat for some time with the watchmaker who assembles the Resonance and bombarded him with all kinds of questions. I asked him; "how can you be sure its not just two well regulate... 

It must have been incredible to see!

 
 By: andrea~ : November 14th, 2022-23:18
The resonance and the optimum are both fantastic pieces of horology and fascinating in how they approach chronometry

Wow, that sounds like quite a moment!

 
 By: Spangles - Dr. Tabby : November 15th, 2022-18:08

Yes, my friend. I will never forget.

 
 By: ChristianDK : November 15th, 2022-18:37
...  

Thank you, it's a pleasure!

 
 By: andrea~ : November 14th, 2022-20:59
The great thing about this is that even within science, watches have many facets that can be explored, from mechanics to materials and everything inbetween, it's an endless pit of interesting discoveries

Definitely:

 
 By: massi. : November 14th, 2022-21:06
I remember as a child I was amazed at how such a small thing, to look at with the loupe, could produce such a loud tick-tock! Unfortunately, many have fallen for knowledge! 😢

A price worth paying ;)

 
 By: andrea~ : November 14th, 2022-21:17
(actually it probably depends on what you break 😅)

Ha! I had forgotten all about this article. Thank you, A!

 
 By: ChristianDK : November 17th, 2022-07:16
It encapsulates perfectly the delta between the vivid imaginations of members of watch forums versus real watches and real watchmakers. Kudos to Jack Forster for writing this. A former moderator here BTW.

👍🏽

 
 By: cmmnsens : November 17th, 2022-17:09

Through the mainplate, really ?

 
 By: Steyr : November 19th, 2022-08:06
Well, I have no doubts that both balance wheels are synched. I've seen it, and I am totally OK with that conclusion. But "How ?" still bothers me I don't believe this can happen through the mainplate. The energy would have to follow quite a path to make i... 

Hi Steyer, that is indeed a super interesting question....

 
 By: ChristianDK : November 19th, 2022-18:30
I will refer you to an article written by Jack Forster for Revolution Magazine in 2014 on the matter. In the article he is referring to George Daniels book "The Art Of Breguet" and how Breguet was studying experiments with vacuum. This article was already...