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quadratic deviation...does it make sense?

  1. dialstatic

    dialstatic Feb 14, 2018

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    Here's something that's been bugging me for a while now. We tend to check the accuracy of our watches on specific moments. What happens when we do a more longitudinal check of the accuracy, and plot the results over time?

    I've done this for two watches so far. One is a hand wound movement, the other an automatic (but I've left it just lying there until it runs out, so it's essentially a hand wound movement for purposes of my question). In both instances, I noticed that the deviation becomes larger as the hairspring uncoils. And it's not linear, either: it looks more or less like a second degree polynomial (see below for an example of a 30110 movement - an IWC modified 2892-A2).

    Now my questions are:

    1. Is it logical that the amount of time a watch gains (or loses) as it becomes more unwound, increases? In other words, is the accuracy a function of the degree to which a watch is wound?

    2. Does is make mechanical sense that this should happen according to a quadratic equation?

    I'd sure appreciate any responses, but just t be on the safe side, I'm tagging @Archer in this ;)

    Schermafbeelding 2018-02-14 om 12.09.59.png
     
    Edited Feb 14, 2018
  2. JimInOz

    JimInOz "Helpful Hints from Heloise" of bracelet cleaning. Feb 14, 2018

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    Between the turns.
     
  3. JimInOz

    JimInOz "Helpful Hints from Heloise" of bracelet cleaning. Feb 14, 2018

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    Pardon my cryptic response, but a mainspring powered watch will be most accurate in that state.

    Al has posted here before on this subject, try a search for isochronism (I think).
     
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  4. Canuck

    Canuck Feb 14, 2018

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    Isochronism (Greek for same, chronos for time) refers to the ability of a highly adjusted chronometer to maintain a constant rate, regardless of the amplitude of the balance wheel. As the mainspring runs down, amplitude of the balance wheel tends to drop off, and on a watch that is not adjusted for isochronism, the rate may be expected to vary more than would be the case with a chronometer. Even highly adjusted chronometers may be expected to vary a bit, particularly when close to being run down. Over all condition of a highly adjusted watch has an effect on isochronism. Rules on isochronism go out the window if condition is not tops.
     
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  5. Euxinus

    Euxinus Feb 14, 2018

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    I would argue that saying the error follows a second degree polynomial is an erroneous way of thinking. Unless derived from some sort of first principals as to why it should be a second order polynomial we have no reason to think so. If you just look at r squared then fit a n+1 polynomial where n is the number of data points and you get a perfect r squared. Give more data points the trend could look linear with a fair degree of variation from the mean, or not.
     
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  6. dialstatic

    dialstatic Feb 14, 2018

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    True. I just thought maybe someone with more horological knowledge than I (which is to say, any amount) might immediately know of some such underlying mechanism.

    The post by @Canuck about isochronism, while not containing any clues as to why the behavior would be quadratic (which, as you state, it might very well not be), does sound as a plausible explanation for what I observed.
     
  7. Archer

    Archer Omega Qualified Watchmaker Feb 14, 2018

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    [​IMG]

    This refers to the period (rate) of an oscillator staying the same, regardless of the distance (amplitude) that oscillator travels.

    I don't know that there is accepted "typical" pattern to how the timing will change, other than to generalize that as the distance the balance has to travel becomes shorter, the period is likely to become shorter, so the watch will tend to speed up as it runs down as you have observed.

    However this is a very broad generalization, because how a specific watch that is adjusted a specific way will react as the torque from the mainspring lowers and the balance amplitude starts to drop, will depend on a lot of different factors. It will depend on various "disturbances" that are affecting the balance, and I don't mean external to the watch, but things in the design/execution of the specific watch that will affect how it behaves.

    One is if the balance is free sprung (variable inertia) or uses a regulator. Another is the shape/type of balance spring (flat v overcoil). Location of pinning point can be a factor. And then a watch may react very differently if left in a horizontal position compared to a vertical position. So if the watch is in a vertical position, then the C of G of the balance now comes into play, so if the C of G at rest is above the axis of the balance (the staff) then as the amplitude drops it will tend to slow down up. But if the C of G is below the axis at rest, the watch will tend to run faster as the amplitude drops - this is related to the balance either having positive or negative acceleration as the impulse pin travels through the lift angle.

    These are just a small number of factors that are involved in how the oscillator behaves - it gets quite complex from a physics point of view very quickly, and quite frankly in a typical watchmaking school students won't get into these issues any further than is practically necessary from a bench level, where you are expected to minimize these errors. I even dug out my old textbook (The Theory of Horology) and they really don't discuss any specific pattern, because there are just too many variables at play.

    In reality you could have two identical movements made by the same manufacturer, and due to the way they are adjusted one might gain as it runs down and one might lose time.

    Anyway, hope this helps.

    Cheers, Al
     
    Edited Feb 14, 2018
  8. sevenhelmet

    sevenhelmet Feb 14, 2018

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    You need to label the X axis of your plot. I'd also recommend trying this test with other watches, and accounting for variables such as temperature and position to see whether its significant enough to matter under practical circumstances. If you find a family of curves that conform to similar 2nd order functions, you may be on to something.

    That said, I have noticed that some watches often run faster- well, faster- the closer they get to the end of wind. That phenomenon can occasionally be used to keep watches close to the actual time if the piece predictably runs slightly slow close to full wind, and speeds up as the mainspring drops below some threshold like ~50% wind. However, my testing in the WatchTracker app hasn't given me any results that appear to show a quadratic pattern, and each piece is unique in how much it is affected by decreasing mainspring torque. In my case, it's more of a zen method. ;)
     
  9. Waltesefalcon

    Waltesefalcon Feb 14, 2018

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    This isn't quite on topic but I am fascinated by the engineering that has gone into watch making over the years. Isochronism has been known about for a long time and is why the fusee was originally invented. In a fusee movement the mainspring barrel does not directly drive the movement, it is connected via a chain to a fusee (essentially a cone shaped pulley) and the fusee drives the movement. As the mainspring unwinds and the torque becomes less the chain unwinds from around the fusee going from the narrow end (which decreases the amount of initial torque applied to the movement) to the wider end (gradually increasing the amount of torque derived from the mainspring). This counteracts the the loss of torque in the mainspring as it unwinds.
     
  10. Archer

    Archer Omega Qualified Watchmaker Feb 14, 2018

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    The fuse was only one method used to give more constant torque - another solution used quite a lot was the Geneva stopworks. That limited the section of the mainspring that was used to run the watch to the center section where the torque curve was more consistent.

    These days more constant torque is achieved by the shape and materials used for mainsprings. Springs are now shaped like an "S" in the unwound state, and the materials they are made of are much more stable. Very few watches use a fuse or stopworks, and if they do it's more for show than for go...
     
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