Resonance in wristwatches

[basiecally]

TL;DR Summary

Dampening effects on mechanical oscillators, mechanics.

Hello!

I'm a new member looking to get help with a question I've been mulling over for quite some time. I'm not a physics student but I do watch repair and restorations.

For those that are not familiar, a mechanical wristwatch keeps time by the use of a "balance wheel" which is effectively a rotary pendulum. Where the period of a classic pendulum is affected by the length of the pendulum and the force of gravity, the period of the balance wheel is determined by the rotational inertia of the wheel paired with the return force of a spiral spring that is attached coaxially to the wheel in question.

The balance wheel then swings back and forth. The force to drive the wheel is applied each time the balance moves in either direction by a mechanism that I don't think will be relevant to my question but that I can elaborate on if needed. The force applied is by design as low as it can be in order to conserve energy from the watches driving mainspring. It is also in the interest of time keeping to keep a high amplitude of the oscillation as a higher rotational speed of the balance makes the balance wheel less susceptible to disturbances when wearing the watch, such as position changes or shocks.

The balance assembly is effectively a mechanical oscillator. My question stems from this issue: sometimes when a watch is damaged a section of the spiral spring (often referred to as "hairspring" due to how thin it usually is) has to be removed, thus shortening the spring and effecting its elastic modulus so that the period is reduced (shorter spring means stiffer in this case). This can then be compensated for by removing weight from the balance wheel rim. The result is a system with (hopefully) the same frequency as when you started. For older watches this is 18000 beats per hour (5Hz) and for more modern watches often 21000BHP.

I have been told by authorities within the watchmaking community that this is not a viable practice as it will change the amplitude of the balance when running. I have tried to understand this from a perspective of resonance but was none the wiser afterwards. I tried figuring out if this could be related to a Q value of this mechanical oscillator but was unable to understand in what way this system was dampened by the change in weight and/or change in spring stiffness. A comparison that came up in regards to pendulums was a pendulum running in air as compared to one running in oil. I failed to see a parallel to my application.

The idea is that a balance assembly is designed from factory to have a very specific combination of balance wheel inertia to spring stiffness in order to achieve the highest possible amplitude for the least amount of force applied to the wheel. And changing either will reduce the effectiveness. This, to me, sounds a lot like an oscillator tuned to a particular internal resonance. But I don't have the physics knowledge to understand how and why these things connect, and would greatly appreciate your help!

Is it sound reasoning to think that changing the mass (and thereby rotational inertia) of the balance wheel and the spring force to match would dampen an oscillator of this type?

I have attached a very good video from the 1950s of how a watch movement works. The part about the balance assembly starts at about 6min

Regards

 

[Tom.G]

TL;DR Summary: Dampening effects on mechanical oscillators, mechanics.

Hello!

I'm a new member looking to get help with a question I've been mulling over for quite some time. I'm not a physics student but I do watch repair and restorations.

For those that are not familiar, a mechanical wristwatch keeps time by the use of a "balance wheel" which is effectively a rotary pendulum. Where the period of a classic pendulum is affected by the length of the pendulum and the force of gravity, the period of the balance wheel is determined by the rotational inertia of the wheel paired with the return force of a spiral spring that is attached coaxially to the wheel in question.

The balance wheel then swings back and forth. The force to drive the wheel is applied each time the balance moves in either direction by a mechanism that I don't think will be relevant to my question but that I can elaborate on if needed. The force applied is by design as low as it can be in order to conserve energy from the watches driving mainspring. It is also in the interest of time keeping to keep a high amplitude of the oscillation as a higher rotational speed of the balance makes the balance wheel less susceptible to disturbances when wearing the watch, such as position changes or shocks.

The balance assembly is effectively a mechanical oscillator. My question stems from this issue: sometimes when a watch is damaged a section of the spiral spring (often referred to as "hairspring" due to how thin it usually is) has to be removed, thus shortening the spring and effecting its elastic modulus so that the period is reduced (shorter spring means stiffer in this case). This can then be compensated for by removing weight from the balance wheel rim. The result is a system with (hopefully) the same frequency as when you started. For older watches this is 18000 beats per hour (5Hz) and for more modern watches often 21000BHP.

I have been told by authorities within the watchmaking community that this is not a viable practice as it will change the amplitude of the balance when running. I have tried to understand this from a perspective of resonance but was none the wiser afterwards. I tried figuring out if this could be related to a Q value of this mechanical oscillator but was unable to understand in what way this system was dampened by the change in weight and/or change in spring stiffness. A comparison that came up in regards to pendulums was a pendulum running in air as compared to one running in oil. I failed to see a parallel to my application.

The idea is that a balance assembly is designed from factory to have a very specific combination of balance wheel inertia to spring stiffness in order to achieve the highest possible amplitude for the least amount of force applied to the wheel. And changing either will reduce the effectiveness. This, to me, sounds a lot like an oscillator tuned to a particular internal resonance. But I don't have the physics knowledge to understand how and why these things connect, and would greatly appreciate your help!

Is it sound reasoning to think that changing the mass (and thereby rotational inertia) of the balance wheel and the spring force to match would dampen an oscillator of this type?

I have attached a very good video from the 1950s of how a watch movement works. The part about the balance assembly starts at about 6min

Regards

I agree with your conclusion.
In Physics there is a formula for describing the relation between Force, Mass, and Acceleration: F=MA

Rearranging that yields A=F/M,
or Acceleration is proportional to Force divided by Mass. If you keep the ratio between Force and Mass the same, then the Acceleration will be the same.

There are a couple confounding elements though:
1) any friction will more rapidly slow down the lower moving mass
2) you may need to statically balance the Balance Wheel as shown at 16:50 in the video

For minor changes, I would expect 1) above could be ignored.
For 2) above I have no idea how important that is in the real world.

That's my 2-cents worth anyhow!

Cheers,
Tom

 

[Baluncore]

Welcome to PF.

The question seems to be, if you stiffen the spring, and then lighten the balance wheel to compensate, does the angle of oscillation reduce? If so why?

My question stems from this issue: sometimes when a watch is damaged a section of the spiral spring (often referred to as "hairspring" due to how thin it usually is) has to be removed, thus shortening the spring and effecting its elastic modulus so that the period is reduced (shorter spring means stiffer in this case). This can then be compensated for by removing weight from the balance wheel rim.

I think you have that backwards, the stiffer spring needs a heavier wheel to maintain the original period. Weight needs to be added to the wheel rim.

The system is an energy balance. When the wheel stops at the end of travel, the energy is stored in the hair spring. When the spring is relaxed, the energy is stored in the kinetic movement of the wheel, and that is the instant when the escapement can add energy to the system, without altering the period of oscillation. The amplitude needs to be sufficient to allow the escapement to operate without influencing the period. I would expect the escapement design to also regulate the amplitude.

 

[tech99]

The balance wheel could be considered analogous to an electrical resonant circuit containing capacitance and inductance. When you reduce stiffness and increase mass, whilst at the same time maintaining the same frequency, it is equivalent to reducing C and increasing L. If the loss resistance is in series with L and C, then high L and low C will give the smallest losses because the current in the circuit is small. By analogy, velocity is equivalent to current, and when you remove part of the spring you are increasing stiffness, equivalant to higher C and lower L (L being equivalant to mass). This causes increased current in the LC circuit, resulting in increased losses, whilst for the balance wheel in results in increased velocity, which increases air resistance. losses.

 

[sophiecentaur]

The question seems to be, if you stiffen the spring, and then lighten the balance wheel to compensate, does the angle of oscillation reduce? If so why?

Wouldn't a stiffer spring require a higher MI to compensate for the frequency shift (the most important issue). How important is the amplitude for a hairspring? Isn't SHM a good enough approximation?

It's a matter of how good is good enough. Buying new watch parts is not hard and would the cost saving of DIY be worth it?

 

[Baluncore]

Wouldn't a stiffer spring require a higher MI to compensate for the frequency shift (the most important issue).

Yes, the OP got it backwards, and that is why I wrote:

I think you have that backwards, the stiffer spring needs a heavier wheel to maintain the original period. Weight needs to be added to the wheel rim.

 

[sophiecentaur]

Yes, the OP got it backwards, and that is why I wrote:

. . . .Yes, I repeated your question and added the amplitude factor question. There are many other factors too. The geometry (slopes of faces etc.) of the escapement governs the amount of energy injected from the mainspring at every tick and, if the speed at the end of the cycle is different, the energy after the escapement will be different.

All that must imply that just shortening the hairspring will have a lot of knock-ons. The oscillator Q is a lot lower than that of a quartz crystal oscillator so the supporting 'circuit' would have more effect. Watchmakers are total gods to achieve the accuracy of the best chronometers.

 

[basiecally]

I agree with your conclusion.
In Physics there is a formula for describing the relation between Force, Mass, and Acceleration: F=MA

Rearranging that yields A=F/M,
or Acceleration is proportional to Force divided by Mass. If you keep the ratio between Force and Mass the same, then the Acceleration will be the same.

There are a couple confounding elements though:
1) any friction will more rapidly slow down the lower moving mass
2) you may need to statically balance the Balance Wheel as shown at 16:50 in the video

For minor changes, I would expect 1) above could be ignored.
For 2) above I have no idea how important that is in the real world.

That's my 2-cents worth anyhow!

Cheers,
Tom

Thank you for replying!

The force applied to the balance from the escapement is not altered though, only the force of the spring that causes the balance to return. I'm beginning to think that the escapement does play a role in this after all.

Poising is one of those topics that is way more involved than it might appear at first look. New balance assemblies are poised statically as shown in the video but that is just the start. The next steps to assure positional accuracy of the watch (so that is will keep time regardless of how you hold it) is called dynamic poising. This is done by assessing the rate (frequency) of the watch in different positions. Material is then either added or removed from the balance rim in key positions to influence the rate of the watch in that position. If you are interested in digging deeper I can link to a good write up on the subject.

Regards

 

[basiecally]

Welcome to PF.

The question seems to be, if you stiffen the spring, and then lighten the balance wheel to compensate, does the angle of oscillation reduce? If so why?


I think you have that backwards, the stiffer spring needs a heavier wheel to maintain the original period. Weight needs to be added to the wheel rim.

The system is an energy balance. When the wheel stops at the end of travel, the energy is stored in the hair spring. When the spring is relaxed, the energy is stored in the kinetic movement of the wheel, and that is the instant when the escapement can add energy to the system, without altering the period of oscillation. The amplitude needs to be sufficient to allow the escapement to operate without influencing the period. I would expect the escapement design to also regulate the amplitude.

You are right! I did get that backwards. But I see you worked that out. In practice washers are added under screws on the balance rim in order to increase the rotational inertia of it, to compensate for the shorter spring.
To some extent the watch can also be "regulated" by moving a set of end stops for the spring to move against. It follows the spring curve and can be moved closer to or further away from the center of the spiral.

Thinking about this more I'm starting to suspect that the system before the balance, the escapement, is actually the root of this behaviour. The escapement in the video I linked is a swiss lever escapement, by far the most common escapement in both wrist and pocket watches the last 100 years. This escapement has what is called a "lift angle". This is the angular distance for a full vibration (from top to bottom of the sine curve) where the balance is in contact with the escapement. Meaning that for a rotation of (from a well serviced movement) upwards of 300 degrees, only a part of those 52 degrees is when the escapement has a chance to impart force on the balance. It will also be the point at which the balance has the highest angular velocity. All this to say: the time during which the power transfer from escapement to balance can occur is VERY small (a modern watch does this movement six times per second or more). I imagine a small change in the mass of the balance could affect the acceleration of the wheel during the impulse (power delivery) in a detrimental way.

Regards

 

[basiecally]

Wouldn't a stiffer spring require a higher MI to compensate for the frequency shift (the most important issue). How important is the amplitude for a hairspring? Isn't SHM a good enough approximation?

It's a matter of how good is good enough. Buying new watch parts is not hard and would the cost saving of DIY be worth it?

Sometimes when restoring older watches, parts are not readily available. Sometimes when you are able to find the parts they can be several hundred dollars for a single part, sometimes thousands. Many antique watchmakers develop amazing capabilities for recreating lost parts but the balance assemblies are one of those things that are a little bit of black magic... Lots of factors come into play when making springs the old way, hardening and tempering steel etc. So some way of keeping a watch a good time keeper, preferrably with original parts, is very appealing. Despite having to alter some of the existing parts.

I'm curious: do you buy watch parts yourself regularly?

Edit: I missed what you wrote about the amplitude the first time I read it. A high amplitude makes for a more stable time keeper. The higher rotational velocity overpowers the inherent inaccuracies in the escapement and balance design. I think in part is has to do with a gyroscopic effect. In part, this is why the invention and popularization of self winding watches provided an improvement in overall time keeping accuracy: so long as the watch is worn it will be working at maximum spring tension and the highest amplitude that the watch movement can achieve.

 

[basiecally]

The balance wheel could be considered analogous to an electrical resonant circuit containing capacitance and inductance. When you reduce stiffness and increase mass, whilst at the same time maintaining the same frequency, it is equivalent to reducing C and increasing L. If the loss resistance is in series with L and C, then high L and low C will give the smallest losses because the current in the circuit is small. By analogy, velocity is equivalent to current, and when you remove part of the spring you are increasing stiffness, equivalant to higher C and lower L (L being equivalant to mass). This causes increased current in the LC circuit, resulting in increased losses, whilst for the balance wheel in results in increased velocity, which increases air resistance. losses.

Thank you for this! Best analogy I've seen on this so far!

 

[sophiecentaur]

I'm curious: do you buy watch parts yourself regularly?

No but I did read a lot about watch mechanisms when I bought my Tissot self winder. Also, I had a friend who fixed watches and clocks. He was full of information and experience. I do some bits of metalworking but nothing as fine as clocks and watches; they are less tolerant to 'bodging', which tends to be my style.

A high amplitude makes for a more stable time keeper.

That could make sense. I am assuming that the oscillation is not pure SHM so the period would depend on the amplitude - hence what you say about self winding watches keeping good time because the drive source (hence amplitude) is more constant. High Q of the driven resonant element depends on the input energy pulses being minimal length with minimal energy input. Hence, I learned that the best watches have quiet ticks.