Perfecta Chronograph / Hairspring Physics -- A Miscellany

[DouglasSkinner]

Here's picture of a WWII vintage chronograph I obtained c.1975.  I think I paid $35 for it then--certainly less than $50.  I actually wore it as a daily watch for many years.  Then it started to run erratically so I put it in a drawer and let it "sleep" for about 10 years.  In the late 90's I took it to a watchmaker in Chicago who said it could be restored, but it would cost!  Even then something inside me--akin to a doctor's impulse to heal--wanted this watch to live.  Besides, it has a nice rose gold case with the inscription "To Sid from Hank 1945 - 1946" on the back.  So I let him do the work.  It keeps great time.  I'm not familiar with the Perfecta brand and I don't know who made the movement (anyone know about them?).  The dial and hands are original.

 

Please forgive the background!  But part of the appeal of watchmaking is the theory behind horology.  I've been working on the hairspring starting from a paper by S. Goudsmit and Ming-Chen Wan, Introduction to the Problem of the Isochronous Hairspring in the Journal of Applied Physics (December 1940).  I tried working out the physics myself but without much success.  The physics of the Archimedes Spiral is--one might say--non-trivial.  There is at least one recent book out there which purports to work out the theory but it's really expensive.  I also have Charles Edgar Fritts book, The Balance Spring.  It's very complete and has a lot of practical information but its kind of densely written--requiring a lot of close study--but no mathematics.  Moreover it requires an understanding of some of its figures and I've never been good at (actually patient with) drawings, especially 19th century drawings.  (Can anyone explain to me Fig. 4?)

 

I'm fascinated with the idea of building a computer program that would locate graphically the point where a hairspring could be vibrated.  My impression is  that experienced watchmakers can often get very close by visual inspection, but for beginners like myself it would be very helpful if we could see such points for a lot of different hairsprings.  So, I've been using Mathematica along with its graphical processing functions to work from pictures of hairsprings.  Once I know the spring constant it should be possible, using the physics of the spiral, to work out such points mathematically--numerically, at least.  It goes like this: first, take a picture of your hairspring against a background of some standard hue.  Then use graphical processing to pull out the pixels of the hairspring, assign them coordinates, fitting them to a mathematical spiral and use the theory to locate the vibration point, which can then be plotted on the same picture.  That's the plan anyway but it may not be possible for reasons I haven't yet understood.

[frenchie]

This is intense.

My master's thesis was about shape recognition in an image and included a study of the different techniques you can use to clean up an image (this is the hard part) :)

Your idea sounds easily feasible and as long as the background is "clean", but not having done math like this in decades, I'm pretty useless with math now. I used mathlab at the time, but mathematica should work. There might be some modules on the Internets you can use out of the box to get you what you need. 

 

Some thoughts (keep in mind I have no idea what the math looks like) :

- you'll need some input from the user : width and thickness of the spring, at least.

- the software might be able to calculate the length for you, but it would need a point of reference somewhere.

 

Good luck, I look forward to seeing your progress !

[bobm12]

 

Good luck, I look forward to seeing your progress !

Me too!

[clockwatcher]

Douglas, from a quick scan of your workings I'd say that the value for Theta needs multiplying by a factor of 0.066* to account for the thrum angle inherent in the final circular portion of the spring. Also you need to take into account the infinitesimal surface hardening which can vary over the entire length and give unintended consequences.

I ran a theoretical vector diagram for the final portion of an annular spring in a railway buffer and forgot to account for the interaction between the grease and the rust, which gave me a wildly innacurate figure, so beware!

Aside from that, the doctor says I'm coming along nicely!

[clockwatcher]

That's a lovely old watch, by the way!

[Geo]

I too like the watch and could service it, but as for the rest, it's way beyond me.

I wish you well with your calculations Doug, and my the force (of the spring) be with you! :)

[Ado213]

Sounds really interesting and way above my level of calculus. The springs elasticity and composition would surely come into play, but not all springs are the same thickness and width or indeed the same material, so surely a test procedure on the to be vibrated Spring would need to be adopted prior to adding this result into the formula. As far as I am aware balance diameter and weight are accounted for within the formula.

As advances in 3D printing progress it may be possible to actually print your HS out of auto cad or the like, collet and pinning point along with the terminal curve all in one go. Now that would be fun!!

Good luck with the work, I really hope you crack it and kudos to your efforts