The mathematics behind a pocetwatch?

calchan

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Dec 8, 2009
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hello, so i am taking this math - 124 class and we have an option to do a project paper instead of doing online assignments. the project paper is suppose to focus on real life example of mathematics in the real world. i would like to make focus on machnical watches for this project, but i don't know where to start. any soggestion would be nice, but i do understand that this is not a kind of question that gets a really strait forward answer. i am just asking for a part of a mechanical watch that is a good example of a placation of mathematics.
 

Smudgy

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May 20, 2003
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The escapement would be a good area to use, there's a lot of mathematics involved.
 

DaveyG

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Mar 21, 2005
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Smudgy is right and Henry B Fried's book The Watch Escapement is a good place to review that topic. The whole design of a watch is based upon mathematics, there is the design of the plate layout so that wheels and pinions mesh correctly, the design of the teeth of wheels and the leaves of pinions so that the friction between the two is minimised, the diameters of the wheels and pinions so that they fit properly between the plates, the number of teeth on wheels and leaves on pinions to achieve the desired beat rate and the relationship between the gear train, balance weight and the springs to achieve the same thing. All that and more just on the time side - don't even think of repeating work.

It never ceases to amaze me, when reading some of the old books on watch and clock making, the depth and breadth of mathematical knowledge displayed by the clock and watch designer/maker of the 17th & 18th centuries. It includes algebra, geometry, trigonometry, metallurgy et al and all in a practical as well as a theoretical sense. Just imagine John Harrison, a carpenter by trade, whose lifetime work involved him in all of these skills which he, presumably, taught himself as the needs arose.

There are countless books, both old and new, to help you with your paper but you have picked a very big subject if you propose to cover the whole of a watch or clock. I wish you evry success and the very best of luck and enjoyment in you chosen task.
 

convertible

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Nov 8, 2002
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A guy named Rawlins or possibly Rawlings wrote a fine book called The Mathematics of Clocks and Watches--I'm away from my library and can't be sure I have the title right. You could no doubt borrow it from the NAWCC library.

Joe Jones
 

Dr. Jon

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This is a very wide range. A simple project is the count train.

The theory of the isochronal balance spring is a lot more complex. This problem was solved in the later part of the 19th century by Phillips and reprints of his work are easily available. Its pretty rough going so it woudl be quite a project.

Something not quite as dense is the subject of watch accuracy. One measure you might examine is the Allan variance. Lots of information on this is on line and you can do it for a watch or clock on a spreadsheet.

If you want to get into he depths you can look at Horlogerie théorique by Grossmann.

This is pretty deep and it is probably more than you want to tackle especially since the book is rare and only available in French. I suspect this is the source of the equations used in todays CAD CAM watch design software.

A nice simpel project is to work out the theory of the reqruied accuracy for navigation. In 1714 The English Governmentset up a prize to determine Longitude. You can work out what was wring with the way it was set up and what the detailed requirements should have been.

Another thing you might look at is the point scoring methods used in Swiss and English observatory trials and see if you can convert these to navigational or railroad requirements. Milham's book Time and Timekeepers has the information you would need.
 

Neuron

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Nov 4, 2010
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Great topic!

Timekeeping devices (burning candles, hourglasses, mechanical clocks and watches, sundials, etc.) all rely on the ability to measure the passage of time by correlating that with some measurable physical process (the movement of celestial bodies, combustion, gravity's effect on sand flowing through an oriface, or the disipation of other sources of energy (springs, weights, etc.) through a mechanism that allows the energy to be transformed into movement that is metered out at a predictable and regular rate (as in all mechanical and even electronic watches).

Understanding how these devices work involves not only math, but also physics.
 

calchan

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Dec 8, 2009
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Thank you for all of your suggestions. some of these stuff are way above my head but i will try to get a basic understanding to complete this project. i have 3-4 months to finish so i have plenty of time. :D
 

bbodnyk

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Aug 14, 2009
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I recently bought a Microset watch timer that allows me to record a watches rate over time and save the data to my computer. I'm thinking a discrete fourier analysis on the watch rate data might show patterns that relate to actual problems in a watch.

So I'm thinking a useful project would be analysing watch rate data specifically a fourier analysis.

Bruce
 

Dr. Jon

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I suggest you take a look at The Science of Timekeeping.

at
http://www.allanstime.com/Publications/DWA/Science_Timekeeping/TheScienceOfTimekeeping.pdf

This covers Fourier analysis and how to acquire data as well as the Allan Variance.

It is also a very good read.

One thing you should know about Fourier Analysis is that most implementations use the Fast Fourier Treansform (FFT). This is perfectly valid but requries that the number of samples be a power of two. Thus you need 2, 4 8 16 32 or other powers. You need ot plan you data taking to get the right number of samples.


You will get an output for half the number of samples. Thus if you have 32 samples you get 16 values. You will need to take all your sample at regular intervals. There are other Fourier transform methods that allow you to use almost any number of samples but these are harder to use.

If you use software like Mathcad or Matlab or most other canned math routines this is what yuo get.

Usually a Fourier transform will produce a complex result, at is most of the values will be a mix or real and imaginary numbers. You will ahve to learn how to deal with this.
Since you will be working with samples you will be taking a Discrete Fourier transform or DFT.

One prroblem with DFT's is that they alias. this means that a frequency above yor limit can appear at a changed value.

Suppose you sample your watch once a day. The highest frequency you can get relaibly is one half cycle per day. Suppose your watch has a frequency at .75 cycles per day. Aliasing will make it show up at 0.25 cycles per day. Since you are takign math and this is your project you should go study this yourself.

Buzzwords are Shannon Sampling theorem, Nyquist limtis aliasing and anti aliasing.

There is a sub set of this in which you can "pad" yuor sampel size to get a numerb of samples that let yuo use the FFT. This is legitimate if you state you are doign this and do it accoriding to convenrtions.

The Allan reference assumes you know about this stuff but it's not so hard to learn.

If you are going to do a Fourier Analysis you should check your watch at least twice a day so you can at least measure a one cycle per day frequency. Pick times that are convenient and you will have to be fairly careful about doing it at about the same time(s).

Be sure to wind the watch at about the same time and keep it constant temeprature or vary it the same everyday.

I hope this helps and that I have not bored you with stuff you already know.
 

calchan

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Dec 8, 2009
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i think i will be doing my research paper on the wheel train, is that what you meant by count train Dr. Jon? also i am having some trouble fining some online sources that actually talk about the math and statistics of it. also i am not the best at math so i am trying to find the simplest topic for this paper.
 

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