Clocks and Gear Trains: Design Technique

[Paul Alciatore]

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)

and

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>

[Paul Alciatore]

<continued from above>

Start with our original set of three ratios:

1/3, 1/4, 1/5

and look at the first and second fractions, 1/3 and 1/4. I want a number to multiply the first denominator (3) and the second numerator (1) by and that number should not be commensurate with the first numerator (1) and the second denominator (4). I also want a number that is 6 or greater so the second fraction or gear pair will have at least 6 teeth in the numerator. 6 does not work because 4 and 6 are both evenly divisible by 2. The next number is 7 and it does work because 7 is prime or 7 = 1 x 7. I don't worry about the 1 in the first numerator because it will be changed in the third step below. Anyway, multiplying by 7 we get the following new set of ratios:

1/(3 x 7), (1 x 7)/4, 1/5

or

1/21, 7/4, 1/5

All three of these ratios are non commensurate as required but two of them still have single tooth numerators so more is needed. As a second step I consider the second and third ratios, 7/4 and 1/5 and repeat the same procedure described above. I need a factor that is non commensurate with the 7 in the numerator of the second fraction and the 5 in the denominator of the third. Again we start with 6 as the smallest number of teeth desired and it will work because it has no common factors with either 7 or 5. 6 = 2 x 3 while 7 and 5 are both primes so there are no common factors. Multiplying as above I get:

1/21, 7/(4 x 6), (1 x 6)/5

or

1/21, 7/24, 6/5

What remains now is the numerator of the first fraction and the denominator of the last, both are less than our minimum number of 6 teeth so we do the same thing again between these two fractions. We need a number that is not commensurate with either 6 or 21. The numbers 6, 7, 8, 9, and 10 will not work as each of these shares a common factor with either 6 or 21 or both of them. So 11, a prime number, is the lowest number that will work and we use it:

(1 x 11)/21, 7/24, 6/(5 x 11)

or

11/21, 7/24, 6/55

Now each of these three gear ratios is non commensurate so each of them will provide completely even wear on the gears. And the overall ratio is still 1/60:

(11 x 7 x 6) / (21 x 24 x 55) = 462 / 27720 = 1/60

This is not the only possible solution to this design problem. I could have started with a different combination of the four factors of 60, such as 1/2, 1/5, 1/6 or 1/2, 1/2, 1/15. I could have listed the three ratios in a different order and chosen a different order for working with pairs of ratios. I could have chosen higher factors to multiply by. All of these choices could yield different final ratios, all of which would work. Thus there is not just a single way of accomplishing this, but many, actually an infinite number of solutions.

Thus it is possible to generate three non commensurate ratios that multiply to a desired, exact overall ratio. This procedure should work in ALL cases where two or more gear pairs can be used to generate a desired, exact, overall gear ratio. It is obviously not possible to do so if only a single pair of gears is used because you can only multiply both gears by the same number and if they are commensurate to start, they will be commensurate when multiplied by any factor.

So with two or more gear pairs in the chain it is always possible to design any overall gear ratio using non commensurate gear pairs for even wear.

As another example, the 12:1 ratio for the minute/hour hands can be accomplished as follows:

1/2, 1/2, 1/3

Using a factor of 7 on the first two fractions you get:

1/14, 7/2, 1/3

And a factor of 11 on the second and third fractions:

1/14, 7/22, 11/3

And finally a factor of 9 on the first and last fractions:

9/14, 7/22, 11/27

Proof of the pudding: (9 x 7 x 11) / (14 x 22 x 27) = 1/12

So this is the part that all the texts that talk about even gear tooth wear leave out. No wonder they do, considering the length of it.

[Paul Alciatore]

It didn't occur to me until after I had posted the above that a clock that used three sets of reduction gears between the second and minute hands would have them running in opposite directions. So a real clock would have either two or four sets to have both hands going the same way, clockwise of course. This does not change the principles involved and the same solution still works. But if only two pairs of gears are used in the train, them they must be of different modulus or DP to allow the shafts to be at the same spacing. With four pairs no such problem exists.

[Weston Bye]

Wow, Paul, what an excellent treatise. While I understand the underlying principles, without having the need to arrive at such ratios in a practical application, I've never taken the time to puzzle it out as you have. Excellent work. Not only have you worked it out, but you have made it understandable to the reader. Copied and stored as a reference.

Thanks for the kind words on the article.

[Ed P]

"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."

If that is true why does the motion works in most clocks have a 1:1 and a 1:12 gear ratio?

Ed P

[jackary]

Is this a possible solution to incorporate the hunting tooth principle and say a 1:1 ratio using two 20 tooth gears. If a 19 tooth, for instance, is placed between the two twenty tooth gears then each 20 tooth wheel is meshing with a 19 tooth gear but the twenty tooth gears are rotating at 1:1 ratio. Of course they are revolving in opposing directions but that can be fixed in other ways.
Alan

[Stepside]

On the street clocks I have repaired the 12:1 reduction uses 4 gears. The first set running off the "minute hand" shaft has a 25 tooth gear turning a 75 tooth gear. The shaft with the 75 tooth gear has a 20 tooth gear that turns the 80 tooth gear on the hour hand "cannon tube". Since the 25 plus 75 equals 100 and the 20 plus 80 does the same there is just the one intermediate shaft. Neither of the ratios involved are "hunting ratios".

This might not be an issue for several reasons; First the RPM is so slow there should be no harmonics to worry about. Secondly because the street clocks have mirror image faces the hands on one side help the hands on the other side this keeping the gear loads quite low.

The clocks are 90 years old or better and if cared for at all the gears in question are still good. That said, the pinions in the main works tend to eat themselves. The pinions are 12 teeth and 15 teeth both running a high count wheels. I have never counted the teeth on the wheels to see if they are a hunting ratio or not. When I tear down the current project I will count the teeth to find out.

I would suspect that with well cut gears running as slow as a clock this is a non-issue. With a poorly cut gear that would mesh with the same tooth each time it could cause premature wear.

Paul and Wes Nice work both of you.

Pete

[gvasale]

most normal clock makers wouldn't dream of using a hunting tooth mechanism. Cost prohibitive (extra gears & arbors and bearing blocks(2) of whatever type,) It also doesn't really add the the longevity IMO. Major causes of wear are accumulated dust, somethimes other "crap." Slow speed of most all the works is a plus. In most cases the fastest moving wheel is the escape wheel turning about 1 rpm is majority of cases, sometimes slower. Watches, are different, but I've never looked into how many rpms the escapewheel makes in a 28,000 bph escape wheel, which is probably the fastest you're likely to encounter.

[Paul Alciatore]

If that is true why does the motion works in most clocks have a 1:1 and a 1:12 gear ratio?

Ed P

I am sure that clock makers are very practical. As others have said, economy is a major factor and the light loading and slow speed of clock gears may make this a non issue. I was just going from the original statement that some such mechanisms do consider this and I wondered just how possible it was. I was delighted to find that it was completely possible and I thought I would share that conclusion, for whatever it is worth. I am also sure that some clock makers have considered this, perhaps for very large or very expensive clocks.

Faster gear trains with higher loads, two factors that would lead to far more wear, would obviously be far better candidates for this technique. The surprising conclusion, for me anyway, is that you can achieve ANY gear ratio with non commensurate gears. You do not have to have an odd ratio or you can have any arbitrary odd ratio needed and still preserve this principle. You can start with any EXACT ratio needed and preserve it, precisely, in this design process. I was genuinely surprised by this conclusion and the ease of achieving it.

[Paul Alciatore]

Wow, Paul, what an excellent treatise. While I understand the underlying principles, without having the need to arrive at such ratios in a practical application, I've never taken the time to puzzle it out as you have. Excellent work. Not only have you worked it out, but you have made it understandable to the reader. Copied and stored as a reference.

Thanks for the kind words on the article.

Weston, thanks for the kind words on my effort.