Hairspring frequency formula/rule of thumb

Discussion in 'Watch Repair' started by gwynplaine19, Mar 8, 2016.

  1. This site uses cookies. By continuing to use this site, you are agreeing to our use of cookies. Learn More.
  1. gwynplaine19

    gwynplaine19 Registered User

    Feb 8, 2014
    81
    0
    0
    Dear group,

    I was wondering if anybody knows the formula for calculating the frequency of a hairspring by its physical dimension/properties. I remember having seen it once somewhere that it has to do with:
    1) the inherent strength of the material
    2) the length of the spring
    3) the number of coils that are packed along a given radius
    4) the radius of the collet

    A bit on the context of the question: I am right now in the midst of an experiment with a very old verge alarm clock, which first came with a clearly modified "modern" hairspring. It had many coils tightly packed, but the last coil was reshaped to a much larger radius to fit into the placement of hairspring stud. This original configuration was losing about half an hour over night.

    Since it was a modern replacement, I went ahead with a few "aggressive" experiments, and as it turned out, those factors above seem to work in less than expected fashions:

    1) I tried to shorned the hairspring. My experience with modern hairspring before had been that if a complete coil is cut, one may have a variation of maybe 2 minute per hour. In this case, I cut out about 3/4 of a coil, but the result was a whopping +5 minute per hour, and that on top of the original deficit, means that the 3/4 of a coil shorter caused a gain of almost 10 minutes

    2) I tried to even out the coils so that they are not all tightly packed close to the collet, into a shape that would resemble an original old hairspring with few coils, but this seems to have little impact on the overall rate--maybe slowing it just a bit, but no more than 1 per hour

    3) I used a piece of fine sand paper to carefully sand down the hairspring somewhat. While I roughly counted the number of sanding motions I used, and tried to go linear, the result seemed to be rather non-linear. Initially, the sanding slowly decreased the rate, but after a point, it started to decrease much faster

    It would be really cool if someone could give a formula that would fit with above.

    Thanks!

    Alex
     
  2. Smudgy

    Smudgy Registered User

    May 20, 2003
    2,872
    23
    38
    Country Flag:
    Region Flag:
    Regular springs are categorized based on the spring constant, represented by the letter K. Formulae dealing with springs using the spring constant can be found in any beginners physics text in the mass and energy section. I'm not sure how you would translate that to use with hairsprings. Hairsprings are categorized by the cgs number (cgs means centimeter, gram, second and just says that it uses the metric system). They can also be found using older systems that relate to the cgs system in the way that dennison, ligne, or inch relate to mm. When looking for a hairspring you would usually just look on a table and see what cgs number (or other scale that is represented) corresponds to a balance of whatever diameter you are working with and the type of escapement (anchor or cylinder). I don't know where you would locate a table that would identify what cgs number you would need for a verge. The one verge escapement I needed to make a hairspring for was done by trial and error. It took a remarkably small hairspring to make it. If you were looking for a place to start researching the information to build a table I think 'Watch Adjustment' by Jendritzki would be a good place to start. I think he discusses how the cgs system tables are developed and the formulae involved.
     
  3. gwynplaine19

    gwynplaine19 Registered User

    Feb 8, 2014
    81
    0
    0
    Hi Smudgy, you've exactly identified the issues that I face--I was previously a math/physics major, so familiar with the usual formulas for spring, harmonic oscillators and such, but hairspring was one thing that I didn't study in college :) I could swear that I saw it once in a text book, but could not find it any more.

    Indeed, for verge, I have also gone with trial and error, usually going for defunct movement/balance assembly and gauging roughly by the weight and diameter of the balance wheel to get something approximate. Usually, with few coils, it seems that the long regulator arc is able to get things approximately accurate, but in this instance, my challenge is that I have run out of easily useable verge hairspring, and was trying to salvage the modern replacement that came originally, so my estimates seemed to be off
     
  4. karlmansson

    karlmansson Registered User

    Apr 20, 2013
    2,604
    113
    63
    Doctor
    Linköping, Sweden
    Country Flag:
    One reason for the rate changing a disproportionate amount when you cut coils at anything other than full turns from the original studding Point is that you will have effectively altered the relation between pinning Point at the collet and the studding Point at the terminal curve. The relation between the two is a compromise in a flat hairspring to allow positional variation to be minimal over the positions in which a Watch is normally worn. Changing it will move the irregularities to the positions where you actually wear the Watch. This will show in the vertical positions so if you are testing your Watch in a horizontal one it shouldn't have any effect. It has to do with the center of mass for the spring and long and short strokes in some way, I still haven't figured it out completely myself... But the effects are quite noticable!

    And on the sanding aspect. A hairspring relies on being of equal strength throughout its length. It needs to develop concentrically, with no weaker areas, in order to maintain center of mass and to allow the spring to adapt to forces applied to it in the same way, regardless of load (ie. symmetry doesn't change with higher load). So thinning the spring out in one section would probably yeild some unpredictable results over different amplitudes. Also, if you thin the spring out Close to the terminal curve you will also get issues with the regulator pins. The pins need to be a very exact distance from each other in order to act as a termination Point for the Active portion of the spring. If you widen the gap on either side of the spring by thinning it out you will have increased the distance the spring needs to travel sideways Before it is met by a pin. So the Active portion of the spring remains longer for a longer amount of time, further slowing the rate of the oscillator.
     
    Berry Greene likes this.
  5. gmorse

    gmorse Registered User
    NAWCC Member

    Jan 7, 2011
    11,246
    1,554
    113
    Male
    Retired from Xerox
    Breamore, Hampshire, UK
    Country Flag:
    Region Flag:
    Hi Alex,

    If it's any help, a verge that I worked on recently had a steel hairspring of 3.5 turns, with a height of 0.2 mm and a thickness of 0.05 mm. The balance frequency was 17,280 bph. This spring had to be replaced, so I used a later pocket watch spring with the same cross-section but more turns, and manipulated the coils to give the right overall diameter with those 3.5 turns. It was a delicate job, and not to be rushed!

    Regards,

    Graham
    Regards,

    Graham
     
  6. gwynplaine19

    gwynplaine19 Registered User

    Feb 8, 2014
    81
    0
    0
    Thanks both! I have now managed to get the spring to within a range that can be regulated--I did the sanding with the hairspring under my finger tip and then gently moving across the sand paper--so still not going to be entirely uniform but the best that I can do--and I think a similar method was described in de carle's book? In any event, in reflection, I start to suspect that the big impact I got from the coil cutting initially vs. previous experience was because of the following:
    --in a typical modern hairspring, there are many coils, and the cutting of 1 will only have a say about 10% reduction on the overall lenth, and whatever physics that's still left in me leads me to believe that the relationship is likely square root, so we are talking about only about 5% of an increase in the rate.
    --on the hairspring that I was working on however, I think a few coils had already been removed by the previous repair person, and because it was also reshaped, the outer coil was probably longer than a coil, and we could be looking at maybe 20% or 25% reduction of the overall lenth, which would translate into a 15% increase in the rate.

    In any event, the exact formula is still elusive for me, though I think I've now gotten more experience with hairspring manipulation :)

    @Graham, you might remember an older post I had--a very old oignon watch that I was working on was running 12 hours in over 9 hours. I think both you and I suspected that the hairspring was a later replacement--it had a large number of coils (though it looked very old, and very big, more than 2CM in radius), so i put in a self-made hairspring with just 1 coil, and produced a very slow beat to get the whole watch into the right rate. However, lately, I had a chance to reopen the whole watch and counted the wheels. Call it a hunch or something else, it dawned on me, the problem might have been with the motion work--which you also suspected as being replacement. There is one wheel which had a 8 leave pinion and 32 leave wheel, and I repaced the wheel with 40 leaves, and thus slowing down the turning of the single minute hand by a rate of 4 to 5, which got the watch now very "accurate".

    What still strikes as very interesting is that this watch had wheels with 44 leaves and 50 leaves--numbers that strike me as not very common in more modern watches--probably because once we need a wheel to turn 1 revolution in a minute, the gear ratios need to work it out to more whole multiples of 60. Just for information, on this one, the contrate wheel turned one revolution in something like between 47 and 48 seconds... Graham, I wonder if you have encountered this kind of gear ratio--was it common on such older watches. Any reason that they may be made such--to make the meshing work well for given sizes of gears and gear configuration? I don't have another single hand verge to compare so I don't know whether this was common for that type.

    Thanks again!

    Alex
     
  7. gmorse

    gmorse Registered User
    NAWCC Member

    Jan 7, 2011
    11,246
    1,554
    113
    Male
    Retired from Xerox
    Breamore, Hampshire, UK
    Country Flag:
    Region Flag:
    Hi Alex,

    Frequencies were very variable, and there didn't seem to be any standard. I've seen 16,900, 17,280, 17,188, 18,000, 17,576, 14,400, and even 16,333.3 recurring. Pinions of 6 leaves usually have wheels of 50 to 56 teeth, whereas pinions of 8 leaves have wheels of 60 to 64 teeth as a rule, and not always even numbers of teeth either.

    Regards,

    Graham
     
    Berry Greene likes this.
  8. gwynplaine19

    gwynplaine19 Registered User

    Feb 8, 2014
    81
    0
    0
    How interesting. I counted perhaps 8280 to 8400 beats per hour, so I guess this is still a slow beater. Next time I service the watch, I will take a picture of the wheels and send your way.
     
  9. karlmansson

    karlmansson Registered User

    Apr 20, 2013
    2,604
    113
    63
    Doctor
    Linköping, Sweden
    Country Flag:
    Wait, so you sanded on the height of the spring, not the thickness...? That puts it in a different perspective. That won't affect the relation with the regulator pins. On the other hand it will make a small difference, the strength of the spring is determined to a much greater extent on the thickness than on the height.
     
    Berry Greene likes this.
  10. DeweyC

    DeweyC Registered User
    NAWCC Member

    Feb 5, 2007
    2,075
    750
    113
    Watchmaker
    Baltimore
    Country Flag:
    The CGS system for balance springs is explained in Jendritiski.
     
  11. gmorse

    gmorse Registered User
    NAWCC Member

    Jan 7, 2011
    11,246
    1,554
    113
    Male
    Retired from Xerox
    Breamore, Hampshire, UK
    Country Flag:
    Region Flag:
    Hi Alex,

    Do you know how to count a train to arrive at the correct beat rate?

    Regards,

    Graham
     
  12. gwynplaine19

    gwynplaine19 Registered User

    Feb 8, 2014
    81
    0
    0
    Hi Graham,

    I think I do, and I did count the train when I was fixing the watch last time, but I only noted down the different gear ratios on the computer, but forgot to save :( so now, I am down to counting the actual beats of the balance wheel... which left me pretty breathless after just 30 seconds :) I won't bother the watch now, and will only count it when I need to take it apart again later.
     
  13. gwynplaine19

    gwynplaine19 Registered User

    Feb 8, 2014
    81
    0
    0
    Hi karlmansson, yes I sanded down the height, not the thickness. You are probably right, and this is what I noticed, strength of spring is probably not linearly proportionate to the height, which is a bit counter intuitive, but that seems to have been the results of my experiment.

    On the other hand, the positioning difference you speak to is showing quite significantly. THe problem for me was that the original spring being a replacement probably had problem too, and I had to significantly altern the radius and enlarge the coils, so whatever previous configuration there was is now lost. when I test with watch flat, I can get a decently accurate rate of less than a minute an hour, but in vertical pendant up position, things start to slow down to maybe 2 minutes per hour. I will experiment a bit more now that you alerted me to this issue.
     
  14. Smudgy

    Smudgy Registered User

    May 20, 2003
    2,872
    23
    38
    Country Flag:
    Region Flag:
    Richard Watkins has translated a number of old texts about watchmaking and has them posted on his web page. They cover time periods that go back to before watches were machine made.
     
  15. gmorse

    gmorse Registered User
    NAWCC Member

    Jan 7, 2011
    11,246
    1,554
    113
    Male
    Retired from Xerox
    Breamore, Hampshire, UK
    Country Flag:
    Region Flag:
    Hi Alex,

    Just count every other beat, it will leave you with more breath!

    Regards,

    Graham
     
  16. gwynplaine19

    gwynplaine19 Registered User

    Feb 8, 2014
    81
    0
    0
    Hi Graham, that's very clever. Will try that later. BTW, I had a question--if I consider the balance action acoustically as a series of click-clack's, would you count click as 1 beat, clack as another beat or you would count the entire click-clack as 1 beat?
     
  17. karlmansson

    karlmansson Registered User

    Apr 20, 2013
    2,604
    113
    63
    Doctor
    Linköping, Sweden
    Country Flag:
    Each beat is really Three Component sounds, but you will percieve them as one "tick". Most mid 1900s watches were 18000 vibrations/hour, amounting to 5 "ticks"/second. The Three Component sounds and the time between them are used to calculate amplitude in modern timing Machines. The difference in time between ticks is used to measure beat error (ti-TICK-ti-TICK instead of tick-tick-tick-tick-).

    You will see different terms being used such as "vibration", "oscillation", "beat" and "frequency". They all mean one slope of a sine wave, so the distance between two valleys of a sine is in fact two beats, oscillations or beats. I this is because this is the only way to measure beat acoustically in a movement: measuring time between ticks.
    This then translates to when you are vibrating a new hairspring. You either match the oscillation to a balance of known rate or you Count every other vibration (ie. the return of the balance arms to their original position) and multiply by two. That is because it's simply too hard to Count five times a second.
     
    Berry Greene likes this.
  18. gmorse

    gmorse Registered User
    NAWCC Member

    Jan 7, 2011
    11,246
    1,554
    113
    Male
    Retired from Xerox
    Breamore, Hampshire, UK
    Country Flag:
    Region Flag:
    Hi Alex,

    The three components mentioned by Karl are drop, impulse and unlock, with the loudest usually being the drop. The impulse is not clearly enough delineated to be of any use. This website shows it graphically, and incidentally is the home of one of the best and most comprehensive electronic timing systems that I've come across.

    Regards,

    Graham
     
    Berry Greene likes this.
  19. gwynplaine19

    gwynplaine19 Registered User

    Feb 8, 2014
    81
    0
    0
  20. Berry Greene

    Berry Greene Registered User

    Oct 2, 2017
    436
    24
    18
    Male
    Retired electronics engineer
    Chichester
    Country Flag:
    Region Flag:
    Huge respect to both Karl Mansson & Graham Morse for this little discourse on tic-ing and toc-ing and full acknowledgement of how hard it can be to make the words work right. Just out of interest I should explain my electronics background has me look at a cycle or Hertz as one complete back & forth or up & down iteration. For me it comes from the generator or alternator and describes the voltage you get from each 360 deg turn plotted against time.
    Ah but a balance wheel doesn't do complete turns does it! It does a positive (CW) swing, stalls, reverses, passes through a would be relaxed "zero" position, and then sets off in its reverse or negative (CCW) excursion. There might be all sorts of low level acoustic noises but the one we are all familiar with is the "Tic" & the "Toc."
    On careful re-consideration - and this is your fault BTW - I am wondering where my Tics and Tocs are really coming from. I had always thought it was the escape wheel. When it lands on the (what's name) faces of the pallets. You wouldn't want to hear much from the fork & impulse roller would you? Where else then?
    To recap. For me a complete beat is a Tic & a Toc. IS THAT WRONG THEN? In a typical (18000bph) watch 5Hz (or 5c/s). I am calling a complete Tic & Toc an "Iteration" For every complete one the train moves twice... Yes or No?
    I've got a frequency counter here. If I place a sensitive microphone (with preamp) on its input - maybe I can measure this for myself without the need of expensive outlay. Or should I use a piezo pick up? You have me wondering. Can you shine any more even brighter light on old dim-wit here? With full respect to you, Sincerely, BjG
     
  21. karlmansson

    karlmansson Registered User

    Apr 20, 2013
    2,604
    113
    63
    Doctor
    Linköping, Sweden
    Country Flag:
    Hello Berry!

    As the ”tick” occurs at the baseline of the graph, if you will, the time between ticks is really offset by half an oscillation given that the escapement is in beat. But given that factor, the time between ticks should be equal to a full oscillation.

    The three components of the tick are: 1. The roller jewel hitting the fork horn. 2. The escape wheel tooth dropping onto the other fork jewel. 3. The lever hitting the other banking.

    The last paragraph applies to the Swiss lever escapement.

    Regards
    Karl
     
    Berry Greene likes this.
  22. gmorse

    gmorse Registered User
    NAWCC Member

    Jan 7, 2011
    11,246
    1,554
    113
    Male
    Retired from Xerox
    Breamore, Hampshire, UK
    Country Flag:
    Region Flag:
    #22 gmorse, Jan 15, 2020
    Last edited: Jan 15, 2020
    Hi Berry,

    If you go and look at the link in my post #18 above, I think you'll see what goes on in the waveform of the escapement. Although the balance doesn't rotate continuously, it does move through more than 360 degrees in the course of each complete cycle, and in the process receives two impulses, one from each pallet.

    The loudest noise is usually the drop and the next loudest is the unlock; the impulse is there, but isn't significant for timing purposes.

    This is why you multiply the wheel tooth count by two when calculating beat rates.

    Regards,

    Graham
     
    Berry Greene likes this.
  23. Berry Greene

    Berry Greene Registered User

    Oct 2, 2017
    436
    24
    18
    Male
    Retired electronics engineer
    Chichester
    Country Flag:
    Region Flag:
    Wow - so quick! Thanks gents very much appreciated. I have that link up. Very good too. No more questions..... Yet!
    Best regards, BjG
     

Share This Page