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This question is about a "standard mechanical watch". I don't actually know much about mechanical watches, but I'm imagining a timepiece whose movement involves a lever escapement and balance wheel. Let's consider timepieces that are the size of a watch that one can wear on their arm as opposed to something scaled up, if that affects stability*.

A standard mechanical watch will fluctuate +/- a few seconds every day. This corresponds to a fractional frequency stability on the order of $10^{-5}$ or 10 ppm.

Why is the stability limited at $10^{-5}$? Obvious candidates are mechanical and thermal fluctuations experienced by the watch. Ok, what if the watch is maintained in a thermally controlled and vibration isolated environment (obviously no one is wearing the watch now). Then what would the stability be and what would it be limited by - $10^{-6}$, $10^{-7}$? Why?

I'm aware that the watch tick speed may change as a function of how wound the mainspring is. To resolve this attach a servo that continuously winds the watch at the same rate it becomes unwound so that (1) the mainspring maintains the same compression level and (2) there are not intermittent "violent" winding events that may affect stability.

The question is: with these mechanical and thermal controls what would the stability of such a mechanical watch be and what would be limiting that stability? Perhaps the residual instability would still be due to lingering mechanical and thermal fluctuations and stability would just continue to improve as you improve mechanical and thermal stability, but eventually there has to be another noise floor you hit.

One guess I have is that thermal Brownian motion will become limiting. That is, even if the temperature is held perfectly constant, there will still be thermal Brownian motion within the watch components that would result in temperature-dependent forces on the watch elements that would lead to frequency instability at some timescales.

Any technical references addressing these questions or showing, for example, Allan deviations for a standard mechanical watch would be appreciated.

*Though if stability can be improved by increasing size I would be interested to know that and to know why and to what limit.

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  • $\begingroup$ What accuracy do you actually need? do you really have any use of matching the atomic clocks? How about matching this for accuracy: news.mit.edu/2020/atomic-clock-time-precise-1216 $\endgroup$
    – Solar Mike
    Aug 27, 2022 at 20:10
  • $\begingroup$ Interested in stability, not accuracy. Yes, I realize there are many many more stable time keeping devices than mechanical watches. The most convenient example is the clock on my phone which is disciplined regularly to NIST references. Nonetheless, I am curious what physical effect specifically limits the stability of mechanical watches. $\endgroup$
    – Jagerber48
    Aug 27, 2022 at 20:34
  • $\begingroup$ It sounds like you are not interested in what limits the stability of mechanical watches as much as you are interested in what theoretically limits the stability of a mechanical watch. Honestly, it might never have a reached a point where you had to care worry about what theoretically limits the stability if you keep running into practical or manufacturing limitations (like manufacturing tolerances and accuracy) $\endgroup$
    – DKNguyen
    Aug 28, 2022 at 5:29
  • $\begingroup$ @DKNguyen no I’m interested in the theoretical limits of a regular mechanical watch as it is manufactured today. If engineering tolerances limit that I want to know which tolerance limits it and via what mechanism $\endgroup$
    – Jagerber48
    Aug 28, 2022 at 13:24

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Another contributor to mechanical watch instability is that the mechanical mechanisms inherently involve a small amount of friction.

This is both in the rotating wheels, and also in the escapement mechanism itself, where teeth are sliding over a catch on each cycle.

Mechanical watches are advertised as contained jeweled movements, where bearing surfaces are made from materials such as ruby and sapphire, to produce very hard and minimal friction surfaces. Nonetheless, there will be surfaces rubbing on each other.

Such rubbing is both a little bit unpredictable, and causes wear. As wear occurs the surfaces change, and so the friction will be different over time. And hence the watch speed is going to be variable.

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Accuracy will be determined by the quality and specs of the components.

Simply, better fitting and manufactured components will produce less variation. This is basically why a Lexus engine runs so smoothly. The same principles apply.

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  • $\begingroup$ Also, to mention that the kind of tolerances that go into high-end watches cannot be mass produced. You're going to be grinding and test-fitting those parts like grinding tenths or maybe even millionths of an inch. At these tolerances the effect of room ambient, your hands, breezes, and even lights is measurable (and might jam over temperature). These fits are smaller than what you can feel by hand and it might be impossible (or at least very difficult) to grind or measure these parts to make sure they are ground properly sometimes since they have unusual geometries and small dimensions. $\endgroup$
    – DKNguyen
    Aug 29, 2022 at 6:04
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    $\begingroup$ Accuracy is no problem. If a watch is running slow or fast there is an adjustment that can be made to balance wheel to bring the watch into sync with a more accurate and stable clock (like a NIST regulated quartz oscillator). But even if the watch is calibrated "perfectly" at 12:00, Jan 1 it will exhibit noise and drift in it's oscillation frequency that limits the long-term stability. I want to know what causes this noise and drift. I don't see how "mechanical tolerances" contribute to noise and drift. $\endgroup$
    – Jagerber48
    Aug 30, 2022 at 4:11

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