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I am not a mathematician so be gentle with me but what kind of math equations (or what field of mathematics) would be needed for the following:

I have designs for an old-style mechanical device I am creating with a complex set of cams -- a total of 62 of them! and I am trying to figure out how to predict how frequently the cams will all synch simultaneously to their starting position (i.e.: how often the overall complex pattern will repeat).

animated cam illustration

  • Each cam has its own unique rotation speed (generally the larger diameter ones are slower to complete one full rotation) but that speed is 100% reliable as all of the cam shafts are fed by a single MAIN drive shaft that is then geared to the various cam shafts. Essentially each cam rotates at a percentage of the MAIN shaft's rotation speed. That percentage can be over or under 100% of MAIN.

  • Each cam has a "starting position" or "zero point" and when the device is first activated each cam is reset to zero point.

  • Each cam is built as a simple circle with a different/unique radius (there "cam" effect is caused by a wavy groove on the side of each cam that transmits a pattern to an embedded armature).

So what is the required math to take these various cams's data and predict how long (i.e.: how many rotations of the MAIN drive shaft -- one full rotation = one heartbeat if you will) it will take for all of the cams to return simultaneously to zero point? For simplicity's sake lets talk about the math for only a few (3? 4?) cams (I assume this math can be expanded later to address 62 cams or more).

O.M.Y.
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1 Answers1

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You want to work out the lowest common multiple of the periods of rotation of each cam.

Mathematically you are trying to solve:

$$p_i\times n_i=t \text{ for } i=1\to62$$ where $p_i$ is the period of cam $i$, $n_i$ is how many times it rotates (an integer) and $t$ is the time taken. You are looking to minimize $t$.

Consider an easier problem as an example: One cam take 2 minutes to rotate and a second takes 3 minutes to rotate and a third which takes 4 minutes to rotate. The lowest common multiple of 2, 3 and 4 you get 12. After a total of 12 minutes the first will have done exactly 6 rotations and the second done exactly 4 rotations and the third done exactly 3 rotations. No smaller amount of time will satisfy the problem.

EDIT: If you aren't sure how to find the lowest common multiple of 2 or more numbers just post a comment.

Ian Miller
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  • Interesting, It seems like a pretty simple concept but I guess it can be pretty involved finding the LCM for 62 variables. And while I am not a mathematician I think I remember LCM's and GCF's from my early college algebra days (he says as the sound of rust creaks in his brain) ... it has something to do with prime factors if I recall correctly. :) – O.M.Y. Jun 02 '16 at 05:12
  • Yes. First you can write each number as the product of powers of prime numbers. E.g. $45=3^2\times5$ Then you can then just find the largest power of each prime across all 62 numbers. E.g. if the numbers are $24,45,150$ then $24=2^3\times3, 45=3^2\times5, 150=2\times3\times5^2$ and so the LCM is $2^3\times3^2\times5^2=1800$ – Ian Miller Jun 02 '16 at 05:28
  • A quick google search turned up numerous sites for working out LCM. None could manage with 62 random numbers between 1 and 200. Google also turned up many 'prime factorization calculators' which would allow you to do one number at a time then look at the powers. – Ian Miller Jun 02 '16 at 05:38
  • Thanks Ian. I tried much the same and I also tried Excel's builtin =LCM() function which can handle 255 numbers as input but even using straight increments of 1 for my inputs the LCM function couldn't handle more than 40 numbers (result > 2^52). It seems this is going to require a minor bit of programming in a language that allows arbitrary-precision large numbers. – O.M.Y. Jun 02 '16 at 10:35
  • If you have the 62 numbers you could post them as a new question. There would be a number of users here who have arbitrary-precision calculation software. (I'm curious how mine would go :) ) – Ian Miller Jun 02 '16 at 12:13
  • I am still working out the bugs on the rotation speeds (some of the gearing is not quite right) but once it runs smoothly I can do that. Much appreciated.. – O.M.Y. Jun 02 '16 at 12:40