Prime number division without gearing?

[talvare]

Gentlemen,

Thanks for all the replies. The flaw in my thinking became obvious pretty quickly.

Ted

 

[silence dogood]

Look up Clickspring Antikythera fragment #2. He will show you how to make a single plate with your 127 holes. He did for 223 gear teeth. Just substitute holes for teeth.

 

[Asm109]

In the village press book "The Shop Wisdom of Rudy K", He describes a method of making dividing head wheels from scratch.
He gave 3 options depending on the machinery in your shop. 1. Rotary table, 2. Drill Press with circular table, 3. Plywood cut round to sit on drill press.
You also need some old fashioned drafting equipment like T square and triangles. and a long meter ruler.
You get a roll of adding machine paper and wrap it around your rotary table and cut it at the over lap so it fits perfect.
Lay that out on your drafting table and draw vertical lines up from each end.
Now pick a length on the meter stick for each division of your wheel. For a 127 tooth I would go with 5mm. 10 would be simpler but you would need a ruler 1.27 meters long.
127*5=635mm
Lay the meter stick on the table at an angle with 0 on one end of the paper strip and 635 intersecting with the vertical line drawn at the opposite end of the strip.
Now put a tick mark on the table every 5 mm on the meter stick.
Using the T square and triangle drop a vertical from every tick to the strip and make a line.
Wrap this around the rotary table and tap.
Make something to act as a point and rotate the table from line to line and drill.
If this is not clear, buy the book it is all clearly photographed and explained. Plus there is lots of other great stuff in there.

 

[RJSakowski]

In the village press book "The Shop Wisdom of Rudy K", He describes a method of making dividing head wheels from scratch.
He gave 3 options depending on the machinery in your shop. 1. Rotary table, 2. Drill Press with circular table, 3. Plywood cut round to sit on drill press.
You also need some old fashioned drafting equipment like T square and triangles. and a long meter ruler.
You get a roll of adding machine paper and wrap it around your rotary table and cut it at the over lap so it fits perfect.
Lay that out on your drafting table and draw vertical lines up from each end.
Now pick a length on the meter stick for each division of your wheel. For a 127 tooth I would go with 5mm. 10 would be simpler but you would need a ruler 1.27 meters long.
127*5=635mm
Lay the meter stick on the table at an angle with 0 on one end of the paper strip and 635 intersecting with the vertical line drawn at the opposite end of the strip.
Now put a tick mark on the table every 5 mm on the meter stick.
Using the T square and triangle drop a vertical from every tick to the strip and make a line.
Wrap this around the rotary table and tap.
Make something to act as a point and rotate the table from line to line and drill.
If this is not clear, buy the book it is all clearly photographed and explained. Plus there is lots of other great stuff in there.

A dividing plate would normally be made to much greater precision than the method you describe. Rudy may be much more adept then I am but I would find it difficult to scribe a line to better than .01" accuracy. Additionally, there will be error associated properly intersecting a vertical line with the scribe line and with lining the pointer up with the vertical lines. If he is marking with a pencil, add another .02" for the line width. It isn't unrealistic to have the stacked error be as much as .05". On a 10" disk, this would be .6º. My RT will resolve to 5 seconds of arc which is .0014º and better than 400 x better.

This method might work for making something visual but I definitely wouldn't use the plate made by this method for making a gear.

 

[Bob Korves]

There are dividing heads and Dividing Heads. If you need to make a hex bolt head or a nut, there is plenty of latitude for slight and even more than slight errors. To make gears and other precision tooling, good accuracy is critical, and makeshift methods will not be able to do the job. In the beginning, of course, there was no accuracy, it needed to be created from crude resources. The study of how tools and tooling became more accurate over time is an interesting study. Sorry, I do not have a link to a single book or article that chronicles that evolution, but maybe someone else here does.
https://www.amazon.com/Foundations-mechanical-accuracy-Wayne-Moore/dp/B0006CAKT8 This is a good one that mostly explores the higher end of it, Wayne's father Richard F. Moore also wrote books on the subject. Moore tools is in the stratosphere of making precision tools.

 

[RJSakowski]

A process come to mind where a person could make a custom dividing plate without precision measuring equipment. Make a plate as described in post #33. It won't be accurate but if used on a 40:1 dividing head, it can produce a second plate with 40 x better accuracy. At that point, other errors associated with machining will probably outweigh the errors in the plate. The only precision tool required is the dividing head.

 

[British Steel]

A process come to mind where a person could make a custom dividing plate without precision measuring equipment. Make a plate as described in post #33. It won't be accurate but if used on a 40:1 dividing head, it can produce a second plate with 40 x better accuracy. At that point, other errors associated with machining will probably outweigh the errors in the plate. The only precision tool required is the dividing head.

I've always heard that is how the first proto-dividing-plates were made, by layout on a circle with actual ( > ) dividers - as long as the worm and wheel are accurate, a 1-degree error will be reduced to an (e.g.) 1/40th or 1/90th (or on my big rotary table 1/120th) of a degree on the first generation, second generation 1/1600th, 1/8100th (or 1/14400th, a quarter-second of arc - this is probably beyond the accuracy of most worms and wheels...). A third generation would probably be unnecessary.

I may just print a 127-hole circle from TurboCAD, stick it to an old freebie CD-ROM and use it as a plate with a pointer, locking the table each set of steps, and generate a more accurate plate in steel (with tiny-tiny holes!) - after the reduction in the little 6" RT I'll be looking at (assuming a 1-degree printed error - which would be obvious on the print-out) still better accuracy, it should theoretically produce work with 4/9ths of a second error or less...?

Then I'll have to work out how I can cut this 127-tooth 48 DP internal gear

(I'm thinking at the moment that I'll make an indexing plate to attach to the lathe chuck, and a slotter attachment to sit on the topslide - at least I won't have hefty cuts to make, just lots of 'em)

Dave H. (the other one)

 

[mathsquid]

Tell ya what. If somebody will give all the holes in an X#.### Y#.### format, one hole per line, with the center of the plate at X0.000 Y0.000 I will run it.
karl

That's trivial using trigonometry. If you want it let me know the radius you'd want for the circle.

 

[Wreck™Wreck]

That's trivial using trigonometry. If you want it let me know the radius you'd want for the circle.

What he said, simple trig easily done with a calculator.

 

[EPAIII]

I have been preaching this on other boards for some time. A dividing head (a worm gear) is an accuracy amplifier. So if you use a 40:1 worm to make a plate with the same number of holes as your original plate, then the errors on that second generation plate will be just 1/40th of those in the original. This can be shown with a rigorous mathematical proof. It is not just an assumption or a rough approximation. It is mathematically exact. The error is reduced (divided) by the worm ratio.

If you need a 127 hole plate, you can make a first generation plate using any method you want to lay it out. Just for Ss & Gs, lets say your first plate has the holes located +/- 1 degree. Then, when you make a second generation plate with it, your error will be just 1/40 degree or about 1.5 seconds. If you use that second generation plate then your work with it will be 1/40th of that or about 2.25 seconds. That is down to 25% of the error spec of my RT. So, if you are going to use the plate with your dividing head or rotary table, then a second generation plate will probably give you greater accuracy than the specs of that DH or RT. You can stop there. If you want to use the plate for direct indexing, then you may want to make a third generation plate which would have that +/-2.25 second accuracy.

This method does not require any math. Your first plate can have a lot of error in it. All you have to do is make two or three generations of hole plates. It will work for ANY number of divisions and produce work that is as accurate as your DH or RT is capable of.

I have read about compound and differential indexing and can only shake my head. I am sure those techniques do work. Sometimes they are exact and sometimes they are only approximate so if you go around a second time the error increases. A third time around and you have three times the original error, etc. I just shake my head. All that math and you still have error. For 127 divisions I would just make the plate. And using my, multiple generation plate method, the error does not increase when you go around a second, or a third, or even a millionth time. You will land on the same point every time around.

A process come to mind where a person could make a custom dividing plate without precision measuring equipment. Make a plate as described in post #33. It won't be accurate but if used on a 40:1 dividing head, it can produce a second plate with 40 x better accuracy. At that point, other errors associated with machining will probably outweigh the errors in the plate. The only precision tool required is the dividing head.