Watch rate. Where's my reasoning going wrong?

[VWatchie]

Considering that most mechanical/automatic watches are not perfectly isochronous, would it be reasonable to say that the rate of a mechanical/automatic watch gradually slows as the mainspring gradually winds down? If true, what makes it slow down? I have a somewhat solid understanding of how a mechanical watch works, and when I reason about it, I conclude that the opposite ought to be true. That is, that the rate should increase as the mainspring winds down.

The way I reason is that as the mainspring winds down, less and less power is transferred to the escape wheel teeth pushing the pallet stones. This in turn decreases the amplitude of the balance which (unless perfectly isochronous), at least near the end of the power reserve, should make the balance swing faster as the arc (distance) of swing (amplitude) decreases making the rate increase. However, the opposite appears to be happening.

Where’s my reasoning going wrong?
 

[watchweasol]

Hi Watchie,  nothing wrong with that reasoning at all , and I for one would agree with it. As the power diminishes as you state the arc of vibration will also diminish proportionatly, maintaining the rate of the watch due to the shorter arc speeding up the watch. I am sure others will have an opinion as that is what the site is for.

[clockboy]

As I understand it watches do not work that way. A watch needs a certain amount of energy to run & the energy release is regulated by the escape. Its only when the mainspring is nearly fully un-wound does the mainspring effect the running of the watch. With automatics this of cause does not apply. 

[FLwatchguy73]

You're forgetting two very basic principles of physics here. Friction and Mass. No matter how polished your jewels and pivots are, or how slippery your lubricants are, there will always be a minute amount of friction. Specifically where the teeth of the gear train mesh with each other along with the barrel, mainspring and arbor. Add to that the inherent friction of the pinion and the hour and minute wheels and you have quite a bit of friction to overcome. All watches have a minimum amount of power required to drive the works at the correct amplitude. When the spring winds down, friction takes over. Finally is the mass of everything in the gear train but most of all the mass of the balance wheel. That alone is enough to bog the works down at very low power. An engineer who knows the mathematics involved should be able to calculate all of this information and give you the optimal power needed to properly drive an average watch movement at the correct amplitude.

Nearly forgot, the balance spring itself also becomes a source of resistance at very low power. It can cause a tiny bit of feedback resistance that has to be overcome.

Edited April 25, 2020 by FLwatchguy73

[VWatchie]

49 minutes ago, watchweasol said:

Hi Watchie,  nothing wrong with that reasoning at all , and I for one would agree with it. As the power diminishes as you state the arc of vibration will also diminish proportionatly, maintaining the rate of the watch due to the shorter arc speeding up the watch. I am sure others will have an opinion as that is what the site is for.

Hmm... I have a Unitas 6325 watch that I wind daily, just before I go to bed. I wear it perhaps a couple of days per week and usually it's just a few seconds off per day. However, if I forget to wind it, it runs for nearly 50 hours, but when I discover that I have forgotten to wind it (usually within 48 hours) it has lost almost 10 minutes. I'd really like to understand the mechanics behind the rate dropping in this case 

[VWatchie]

Thanks for the input @FLwatchguy73. Very interesting and facts way too easy to overlook. However, I still don't understand why it would slow down and not increase the rate as is the case with my Unitas 6325 (see my previous post).

EDIT: Hmm... let me think about that a little more...

EDIT: I think I'm beginning to get it. The friction will make the escape wheel turn more slowly. That seems more than plausible. Again, thanks!

Edited April 25, 2020 by VWatchie

[watchweasol]

Hi as FLWatchguy remarked I think we are into a design engineers calculations in order to have a complete answer, I guess that during that stage the thickness of the wheels Number of teeth weight of the wheels and the friction coeficients are all computed to maximise the accuracy and minimise the friction .  

[FLwatchguy73]

I think I see what you're getting at, logically with less power the balance should swing less distance and therefore have a faster beat rate. However, the balance doesn't swing faster at a shorter swing with low power, it swings slower. Again, the power is reduced. If you adjust the amplitude by moving the regulator, you are shortening the effective length of the balance spring which in turn speeds up the balance. With reduced power, the effective length of the spring hasn't changed any, you just aren't pushing the balance hard enough to give it a full swing. It can't logically move faster with less power, it just won't move as far. The balance spring exerts a return force on the balance at the end of its swing. It's proportional to the amount of power applied to it by the escapement.  I guess this is where logic and physics clash, lol.

[Marc]

I think I'm right in saying that if you could eliminate all energy loss from a balance wheel then for any given temperature its rate would remain constant regardless of its amplitude and it would oscillate with a perfect conversion between potential energy, which is at a maximum at the point where its direction reverses and for an instant the balance is stationary (kinetic energy zero) and the hair spring has maximum deformation from its rest state, and kinetic energy which is at a maximum (highest rotational velocity) at the instant the balance passes through the rest state point of the hair spring (zero potential energy).

However, even before we introduce the rest of the escapement into the system there energy losses, mainly through friction, but also through hysteresis in the h/s, both processes which ultimately cause energy to escape from the system as heat. So you have to top the system up with energy via the escapement in order to keep it going, a process that introduces even more friction and therefore even more energy losses.

The escapement transfers energy into the balance wheel via the interaction between the pallet fork and the impulse pin. This process only happens for the duration of the interaction between these two parts (i.e. the lift angle of the escapement) and during this part of the balance oscillation the balance wheel is accelerated by the input of energy as it rotates through the lift angle. The lift angle is a fixed parameter and does not change in relation to the changes in amplitude. However, during the engagement between the pallet fork and the impulse pin, frictional losses in the system are at their highest.

This effectively means that the overall rate of the balance system is made up of two components. The period of acceleration as the wheel rotates through the lift angle (rl), and the period of swing through which there is no outside interference (r).

r can be considered a constant determined by the physical parameters of the balance and h/s. However, rl will vary because as the m/s winds down and provides less torque, less energy will be delivered into the balance through the lift angle and therefore the balance will accelerate less. rl at full wind will be higher than rl at half wind.

Also the influence of rl over the overall rate will increase as the amplitude reduces.

When the amplitude is at say 310 degrees at full wind in an escapement with a lift angle of 52 degrees, it effectively operates as a free oscillator (without outside interference) for 83.23% of its swing, its rate will be r for 83.23% of its oscillation. During the remaining 16.77% of the oscillation, as it rotates through the lift angle its rate will be rl, so its overall rate will be (83.23%r+16.77rl). The escapement delivers enough energy into the system for the overall rate to remain constant.

If the balance amplitude drops to 180 degrees at half wind (the lift angle doesn't change) then it effectively operates as a free oscillator for only 71.11% of its swing, so r only operates for 71.11% of the time which means that rl now influences 28.89% of the overall rate, so the overall rate is (71.11%r+28.89&rl). But rl is now slower than it was at full wind so it's impact on the overall rate is further amplified, and the overall rate will drop even further.

If the balance amplitude drops as low as 52 degrees at a quarter wind though then the overall rate will be 100%rl, and rl will be further reduced by the fact that at a quarter wind the m/s delivers even less energy. So the watch slows even further.

The ideal is to keep the torque from the m/s as constant as possible over as much of it's wind as possible so that the rate remains constant. However, once you get to the threshold below which the m/s torque starts to drop appreciably the rate will start to drop.

If you've read this far then congratulations. If you've managed to follow it as well then I'm impressed 

This is just me thinking the system variables through though and coming up with a mechanism that seems to work. I know that it's not perfect as I know of scenarios in which very low amplitudes can result in an increased rate.

In the true spirit of "The Scientific Method" please anyone, shoot my mechanism down.

[clockboy]

2 hours ago, VWatchie said:

Hmm... I have a Unitas 6325 watch that I wind daily, just before I go to bed. I wear it perhaps a couple of days per week and usually it's just a few seconds off per day. However, if I forget to wind it, it runs for nearly 50 hours, but when I discover that I have forgotten to wind it (usually within 48 hours) it has lost almost 10 minutes. I'd really like to understand the mechanics behind the rate dropping in this case 

One reason might be what position the watch is resting in. I have read that some have found if a watch is running a touch fast they help the regulation by resting it face down over night and the reverse if the watch it is running slow. Hence why watches are regulated in different positions to get a mean average. 

[FLwatchguy73]

46 minutes ago, Marc said:

I think I'm right in saying that if you could eliminate all energy loss from a balance wheel then for any given temperature its rate would remain constant regardless of its amplitude and it would oscillate with a perfect conversion between potential energy, which is at a maximum at the point where its direction reverses and for an instant the balance is stationary (kinetic energy zero) and the hair spring has maximum deformation from its rest state, and kinetic energy which is at a maximum (highest rotational velocity) at the instant the balance passes through the rest state point of the hair spring (zero potential energy).

However, even before we introduce the rest of the escapement into the system there energy losses, mainly through friction, but also through hysteresis in the h/s, both processes which ultimately cause energy to escape from the system as heat. So you have to top the system up with energy via the escapement in order to keep it going, a process that introduces even more friction and therefore even more energy losses.

The escapement transfers energy into the balance wheel via the interaction between the pallet fork and the impulse pin. This process only happens for the duration of the interaction between these two parts (i.e. the lift angle of the escapement) and during this part of the balance oscillation the balance wheel is accelerated by the input of energy as it rotates through the lift angle. The lift angle is a fixed parameter and does not change in relation to the changes in amplitude. However, during the engagement between the pallet fork and the impulse pin, frictional losses in the system are at their highest.

This effectively means that the overall rate of the balance system is made up of two components. The period of acceleration as the wheel rotates through the lift angle (rl), and the period of swing through which there is no outside interference (r).

r can be considered a constant determined by the physical parameters of the balance and h/s. However, rl will vary because as the m/s winds down and provides less torque, less energy will be delivered into the balance through the lift angle and therefore the balance will accelerate less. rl at full wind will be higher than rl at half wind.

Also the influence of rl over the overall rate will increase as the amplitude reduces.

When the amplitude is at say 310 degrees at full wind in an escapement with a lift angle of 52 degrees, it effectively operates as a free oscillator (without outside interference) for 83.23% of its swing, its rate will be r for 83.23% of its oscillation. During the remaining 16.77% of the oscillation, as it rotates through the lift angle its rate will be rl, so its overall rate will be (83.23%r+16.77rl). The escapement delivers enough energy into the system for the overall rate to remain constant.

If the balance amplitude drops to 180 degrees at half wind (the lift angle doesn't change) then it effectively operates as a free oscillator for only 71.11% of its swing, so r only operates for 71.11% of the time which means that rl now influences 28.89% of the overall rate, so the overall rate is (71.11%r+28.89&rl). But rl is now slower than it was at full wind so it's impact on the overall rate is further amplified, and the overall rate will drop even further.

If the balance amplitude drops as low as 52 degrees at a quarter wind though then the overall rate will be 100%rl, and rl will be further reduced by the fact that at a quarter wind the m/s delivers even less energy. So the watch slows even further.

The ideal is to keep the torque from the m/s as constant as possible over as much of it's wind as possible so that the rate remains constant. However, once you get to the threshold below which the m/s torque starts to drop appreciably the rate will start to drop.

If you've read this far then congratulations. If you've managed to follow it as well then I'm impressed 

This is just me thinking the system variables through though and coming up with a mechanism that seems to work. I know that it's not perfect as I know of scenarios in which very low amplitudes can result in an increased rate.

In the true spirit of "The Scientific Method" please anyone, shoot my mechanism down.

Isn't this exactly what I said in my post? And yes, I read and understood it all and I R smarter now! 

[Watchtime]

Friction means loss of energy, isn't it that simple or am I putting it too simple?

[Marc]

29 minutes ago, FLwatchguy73 said:

Isn't this exactly what I said in my post? 

Kind of...

I was trying to un-clash the "logic and the physics"

[FLwatchguy73]

2 minutes ago, Marc said:

Kind of...

I was trying to un-clash the "logic and the physics"

HEHE! You said it way more comprehensively than I could have! I'm grateful for it too.

[CaptCalvin]

I got a much simpler answer: index gap. Depending on the width of this gap, the lower the amplitude, the more time the hairspring spends floating between the gap, which = slower rate. 

[Jon]

The only time I have come across a gain in time as a watch runs down is in a Waltham 8 day car clock I recently serviced, which in effect is a big  pocket watch with twin mainsprings.

It kept good time for the first 6 days, then past that it started to gain time, because of the lowering amplitude, therefore a higher rate that the pallet fork was moving and turning the escape wheel. Between 6 days to 8 days it gained 5 minutes.

[canthus]

Going back to my uni days. Basic law was 'energy cannot be created or destroyed, it can only change form'. 

So as the spring energy drops that energy must be in a another form, ie friction, rotational inertia etc etc.

[HSL]

On 4/25/2020 at 10:41 AM, VWatchie said:

Where’s my reasoning going wrong?

The most of your reasoning is correct, regardless on how the power is lost, there will be some parameters which are the same, like the mass of the balance and the lenght of the haispring, to make a philosofical approach without a lot of mumbo jumbo the moment of inertia which is supposed to generate the mechanical energy in the hairspring isn't high enough after a while. 
When the balance is changing direction the loss of torque due to a low amplitude will result in a slower acceleration of the balance wheel, remember Newtons second law of movement F= m*a where the F part can be translated into the torque described by hooks law.
So (a) which is the acceleration equals a= F/m , with lower torque you get a slower acceleration, a slower acceleration results in loss of time. Guess that was what you initially were woundering about?
Just a thought ...  

[JohnR725]

A lot of text here that I just skimmed through. Usually the simple one of why watch slows down as amplitude decreases is the regulator pins are too far apart. Which is why better grade watches don't have regulator pins and have free sprung balance wheels.

Then it might have been set up above but I like to think about lift angle as the angle at which the escapement screws up timekeeping. It's why the watch companies are always working on different escapement's that have a much smaller lift angle less screwup of timekeeping.. The two separate of things occur with the escapement. The balance wheel has to push on things and those things do not necessarily move at the same rate as the balance wheel so the balance wheel is slowed down rate wise and then it receives an impulse at a different rate. All of which operates at a rate loss.

Then there is the other thing that I find interesting? Some witschi timing machines have a time plot feature. That is they will plot the amplitude and timekeeping over a period of time like around 15 minutes. This is where you can see the effect of unequal power in the gear train for instance and what that effect has on balance wheel rate. Then the interesting thing of this is having time to variety of watches some of them like the Rolex I timed and I can't attach an image if you want. It very clearly has a wheel defect you can see that in the amplitude but almost no effect on timekeeping. Other usually cheaper watches the effect is dramatic. The normal timing machine doesn't show this because it kind of averages this all out.

[VWatchie]

22 hours ago, JohnR725 said:

Then the interesting thing of this is having time to variety of watches some of them like the Rolex I timed and I can't attach an image if you want.

If it's not too much trouble I'd like to see it.

22 hours ago, JohnR725 said:

Usually the simple one of why watch slows down as amplitude decreases is the regulator pins are too far apart.

So that's the second time this was stated in the thread. Only pointing it out as you "skimmed through".

Edited April 27, 2020 by VWatchie

[JohnR725]

6 minutes ago, VWatchie said:

So that's the second time this was stated in the thread. Only pointing it out as you "skimmed through".

Only two times how sad? Not as bad is one of the questions this morning before I even got the answer half a dozen people answered so it got answered multiple of times except I did add something they didn't have. It's actually quite common here that questions will get answered  multiple times basically the same thing for variety of reasons.

Then I have more images but they're not labeled these all have text on them. You also have to be careful because the machine will do an auto ranging so things that look really bad if you look at the numbers might not really be bad.

[VWatchie]

Not quite on topic but still closely related is the below video which I found quite fascinating. Like me, however, they to draw the same conclusion which in theory is correct but in practice erroneous: “With less force the balance spring can not be spun as far around its centre decreasing the time between beats and speeding the movement up”.

 

 

[VWatchie]

On 4/26/2020 at 10:27 PM, HSL said:

When the balance is changing direction the loss of torque due to a low amplitude will result in a slower acceleration of the balance wheel, remember Newtons second law of movement F= m*a

Put that way it makes it crystal clear to me and something I had completely overlooked! 

On 4/26/2020 at 11:44 PM, JohnR725 said:

Usually the simple one of why watch slows down as amplitude decreases is the regulator pins are too far apart.

That I'm still digesting, but I'm getting there I hope and guess... 

[HSL]

9 hours ago, VWatchie said:

Put that way it makes it crystal clear to me and something I had completely overlooked! 

Well if you think about it high end watches use a variable inertia balance, they don't have any regulator pins.. just micro regulator screws the balance is the main regulator in a watch.

[Nucejoe]

Inertia is the tendency of a mass body to resist change in its motion, it is an inherent property of mass but function of mass distribution throughout the spatial geomerty of a body and remains conatant for a given wheel. As balance wheel is lifted to a high amplitude hairspring exerts torque against inertia through longer distance, therefor the wheel reaches higher angular velocity, lower angular velocity is developed by lower amplitude. So the wheel dose low and high amplitudes is same time period. High amplitude is to enhance percision, accuracy can be achieved at low amplitudes too. High end watches deliver percision.