I have recently read "A Magnetic Gear Clock, Part One" by Weston Bye in the winter 2011 issue of Digital Machinist. It is an excellent article and I enjoyed it very much. It has given me an idea to make an even more mysterious clock where there is also no obvious means of powering the toothless gears that turn the hands. That combined with gears that do not even touch each other should make a very interesting clock. Just what I need, another project: thanks Weston.
In the article I was intrigued by a statement Weston made there. "In clock making it is customary, using odd and even numbered gear combinations, to design the gear ratios such that each tooth eventually meshes with every other tooth on the corresponding gear, distributing wear equally around both gears." I have seen this before in many sources and have often seen practical examples of it in real gear trains, such as the back gears on my lathe. I think a more precise statement of this design principal would be that a pair of meshing gears should have tooth counts that do not contain any common factors. An example would be 40:21 instead of 40:20: 40 factors into 2 x 2 x 2 x 5 while 21 would be 3 x 7 so there are no common factors. This will ensure that each tooth on the 21 tooth gear will mesh with a different space on the 40 tooth gear for 40 times without repeating any and on the 41st turn it will start on the first space in the sequence and repeat the pattern. Thus the wear will be completely even for all teeth on each gear with no combinations receiving any more wear than any others. I believe in mathematics this is known as non commensurate numbers.
Anyway, I started to wonder, is this really possible with the gear ratios needed for a mechanical clock. Seconds to minutes requires a 60:1 ratio and minutes to hours a 12:1 or 24:1 ratio, both nice even numbers. Indeed, is it possible for any possible gear ratio, or perhaps it can only be accomplished for a select few? So I worked on it for a bit.
Starting with the 60:1 ratio for the second to minute hands, I first factored the 60 to the prime factors or 2 x 2 x 3 x 5. In my youth I disassembled anything I could get my hands on, including clocks and watches so I knew that they had to use multiple gear pairs in the train to allow the hands to be concentric. The motion had to go outwards from this axis and then come exactly back to the same, original axis. This could be accomplished with two pairs of gears only if they had the exact same spacing from one shaft to the other and this meant that if the modulus of the gear was the same that the tooth counts would also necessairly be the same. In this case it would have to be the square root of 60 which is 7.745... Not a possible number of teeth on a real gear so that was simply not done. What all clock makers did do was use three sets of meshing gears in the string. This would allow the shafts to form a triangle with it's three sides determined by the distances between the three pairs of meshing gears. Thus, I needed three ratios that would combine to equal the 60:1 overall ratio that I needed. Since there are four prime factors in 60, I simply needed to combine two of them to give me three separate ratios. Others are possible but I choose to combine the two 2s which would give me three ratios that are as close to equal as possible. 2 x 2 is 4 so I had 1:3, 1:4, and 1:5. I find it more mathematical to express them as fractions so I wrote
1/3, 1/4, 1/5
as my starting point.
To get actual, real world gear ratios I knew that I needed higher numbers because a one tooth gear is not practical. Yea, I know it can be done, but it does present difficulties and is generally not done. So, again relying on my experience with demolishing clocks, I knew that clocks often had pinions with as few as six teeth or to be more precise, lantern pinions with six pins for teeth. So six was my minimum tooth count. But how to proceed? If I just multiplied each of the above ratios by six, it would work and give the correct overall ratio. But it would not have the desired incommensurate individual ratios.
6/18, 6/24, 6/30
So it is no good for what is needed here. Some brief experimentation showed that some kind of systematic attack was needed. But first some considerations of gear trains.
The ratio of a pair of gears is not changed if both the numerator and denominator (the tooth count of each gear) are multiplied by the same number. Thus a 3:1 ratio can be obtained by gears with any of these numbers of teeth:
1/3, 2/6, 7/21, 10/30, 111/333, or any of an infinite number of others as long as the denominator is three times the numerator.
Another thing is that the total gear ratio in a train of gears, such as this is equal to the product of all the numerators divided by the product of all the denominators. In other words, what is of interest here is the place where each factor appears (which exact gear it is on) is not important. The only important thing in determining the overall ratio is that the numerators or driving gears be kept the same and the denominators or driven gears also be kept the same. Thus, the following two gear trains will both produce a 12:1 ratio:
6/18, 8/32 (2 x 3)/(2 x 3 x 3), (2 x 2 x 2)/(2 x 2 x 2 x 2 x 2)
4/12, 12/48 (2 x 2)/(2 x 2 x 3), (2 x 2 x 3)/(2 x 2 x 2 x 2 x 3)
because both reduce to 1/12. I swapped factors between the individual fractions in both numerators and denominators, but the overall combination of factors in each remained the same so the final ratio remained the same.
Now since we are actually interested in this overall ratio, we can multiple the overall numerator and the overall denominator by the same factor and the ratio stays the same (first principal above). Using the above example, we can add a factor of 11 to both in the example above:
(5 x 6 x 11) / (18 x 20 x 11) = 1/12.
And we can rearrange things like this:
(5 x 11)/20 / 6/(18 x 11)
and still get the exact same ratio of 1/12 (second principal above).
So to obtain incommensurate ratios in our clock gear train I tried multiplying the denominator of one pair and the numerator of another pair (the next pair) by the same factor. This will not change the overall ratio, but it can give us a way of introducing incommensurate ratios by using factors that are not commensurate with the numerator of the first ratio and the denominator of the second. A bit confusing, but here is an example:
<continued in second post>