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Coming Up With Intermediate Gear Ratios

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ProfessorGuy

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I'm thinking about a simple 1-second pendulum clock with a 30-tooth, 1 rpm (second hand) escape wheel. I want to put in 3 more wheels, with the 3rd wheel running at 1/720 rpm (hour hand). There are plenty of intermediate ratios to choose from, with the most obvious being a train of 8:1, 9:1, and 10:1 (total 720:1). The problem with integer ratios is the same tooth on the wheel always hits the same tooth on the pinion so wear from imperfections is not spread out over all the teeth, but concentrates on just a few. So I want there to be NO INTEGER RATIOS.

No problem. I wrote a Java program in about 20 minutes that allowed me to search through various tooth counts to get what I was looking for. I had the program try lots of values that were still reasonable. So here's one possibility:

Hour wheel: 84 teeth. Mates with 8 leaved pinion on 2nd wheel. Ratio: 10.5.
2nd wheel: 64 teeth. Mates with 7 leaved pinion on 3rd wheel. Ratio: 9.143.
3rd wheel: 45 teeth. Mates with 6 leaved pinion on escapement. Ratio: 7.5.

But that's 3 different pinon cutters to make. So I thought maybe this one is easier:
98:7 , 60:7 , 42:7. This has the advantage that all the pinions are identical, but does introduce an integer ratio there at the end (42:7 = 6:1).

I'm considering buying a cutter set (probably 0.5 or 0.4 modulus). Then I thought if I constrained the pinions to at least 10 leaves, I wouldn't have to make any pinion cutters at all. Usually a cutter set of a particular modulus includes cutters down to 10 teeth. So I thought maybe:

120:13, 104:10, 75:10.

Now we're getting up there with 120 teeth on the hour wheel (only 1.5 degrees per tooth)!

What's the thinking here? What should I be looking for in my huge table of possible gear ratios? Any help would be greatly appreciated.
 

ProfessorGuy

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I've been looking up cutter sets and the ones I've found only go down to 12 leaves. So I looked carefully at the tooth ranges for a typical set: #1) 12-13 teeth; #2) 14-16 teeth, etc. If I want to minimize the number of cutters to buy, there are only 5 reasonable choices:

W1:135; W2:153; W3:192 --- P1:17; P2:18; P3:18 --- Final ratio: 720.0
W1:135; W2:170; W3:192 --- P1:17; P2:18; P3:20 --- Final ratio: 720.0
W1:135; W2:171; W3:192 --- P1:18; P2:18; P3:19 --- Final ratio: 720.0
W1:144; W2:170; W3:190 --- P1:17; P2:19; P3:20 --- Final ratio: 720.0
W1:150; W2:170; W3:192 --- P1:17; P2:20; P3:20 --- Final ratio: 720.0

These use only 2 cutters: cutter #8 (rack cutter) for teeth >134, and cutter #3 for tooth counts of 17-20. All the smaller wheels use at least 3 cutters. It makes me a little nervous to use a cutter right at the edge of its large range (135 teeth from a 135+ cutter). Should it?

How do you come up with your ratios?
 

Forty Niner

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#3
Your post is interesting. I applaud you for deciding to design and build a clock.
I haven't built a clock, but I work on them. Have you identified the ratios that typical 3600 BPH (beats per hour) clocks use? Lots of grandfather clocks have movements like this and they run for years before they need servicing. The servicing rarely includes faulty gears.

You know the ratios of the seconds hand to the minute hand and the minute hand to the hour hand. They are fixed and are integer ratios. Can't get away from that.

The type of wear that I see in clocks that are over 100 years old does not indicate gears with integer ratios cause problems. I understand the theory of hunting tooth gears in high speed turbo machinery, but is it necessary for the slow speed gears that are not even lubricated as used in a mechanical clock?
 

ProfessorGuy

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Your observation that integer ratios are not harmful, and are often used in long-running clocks, is confirmed by some of the great watchmakers in history. George Daniels says that a barrel of 96 teeth mating with a center pinion of 12 teeth (8:1 ratio) gives a near ideal transfer of power onto a beefy pinion with enough strength. He further mentions that no less than Bruguet used integer ratios in watches that have been running continuously for centuries.

It seems unlikely that my ratios are better than these tried (and tried and tried) and true solutions that are historically proven. Then again, isn't life all about curiosity and exploration?
 

brino

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Then again, isn't life all about curiosity and exploration?
Absolutely! My curiosity is why I have been quietly watching this thread.....although I felt I had no expertise to share.

One thought I had is that the use of integer ratios would probably be less of a concern if you knew the gears were all high-quality with near perfectly made teeth. However if a gear had an irregularity that continuously mated with the same tooth on an adjacent gear I could see an issue with wear, and perhaps a rhythmic spike in angular velocity. Whether that would affect a clock movement I don't know.

Non-integer ratios may spread any wear caused by tooth irregularities more evenly across all teeth.

All of the above is speculation, my own personal "thought experiment" I have no evidence to back any of it.
Interesting topic, I hope we get more informed input than mine.

-brino
 
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steve323

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#7
Professor Guy,

That is an interesting idea. I am just getting started. So far I have only made one wooden gear clock, but the concepts should be the same.

One thing that might help the analysis would be to allow the escapement to change. A 30 tooth escapement is great with a 1 second pendulum because the escapement will provide a second hand. If you don't need a second hand, then you have more options.

Here is one option with a 20 tooth escapement wheel. This will need to be divided by 90 to drive the minute hand.

20 tooth escapement with a 8 tooth pinion
81 tooth gear with a 9 tooth pinion
80 tooth minute hand gear

The first gear set has an 8:81 ratio. The second gear set has a 9:80 ratio. The escapement rotates every 40 seconds. The first gear rotates every 40/8*81=405 seconds. The minute hand rotates every 405/9*80=3600 seconds.

The gears driving the hour hand would be rotating at such a slow speed that you should be able to ignore any uneven wear.

Steve
 
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steve323

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Another way to generate the desired tooth counts is to always use different prime numbers for the pinions. Let's assume pinions with 11 and 13 teeth and go back to a 30 tooth escapement so we can have a second hand.

The gear train needs to divide by 60. We added 11 and 13 tooth pinions, so the total number of teeth in the wheels is 60*11*13=8580. We could do this with 78 and 110 tooth wheels. The 13 tooth pinion can't drive the 78 tooth wheel, so it needs to drive the 110 tooth wheel.

30 tooth escapement with a 13 tooth pinion
110 tooth wheel with an 11 tooth pinion
78 tooth minute hand gear

The gear ratios are 1:8.4615 and 7.0909.

You could do the same analysis with 13 and 17 tooth pinion, but the number of teeth in the wheels would increase. Maybe something like 13:102 and 17:130, for a total of 262 teeth to cut. The 11 and 13 ratios only have 212 teeth. A traditional 8:60 and 8:64 clock only has 140 teeth.

Steve
 
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