# Dividing head math for large # of  gear teeth



## LEEQ (Oct 29, 2020)

I am brainstorming on making a 127 tooth gear. I'm looking at the destructions for the dividing head. The example there is for a 17 tooth gear with a 17 hole dividing circle and a 90 to 1 gear ratio tool. They show you to divide 90 by 17, coming up with an answer of 5 with a remainder of 5/17ths. That would be 5 turns plus 5 holes on the sector arms of the 17 hole dividing circle. That's all hunky dory, but my number is bigger than the 90 from the gear ratio. My thoughts are to mill a 127 hole dividing plate, mount it up to the head and do the math to make it work. When I divide 90 by 127, I come up with .7086614.  I'm just not sure how to turn that number into a holes in the sector arm number. Remember the gear ratio of your tool is pretty irrelevant. If it is 40 to 1, show me the math for say a 47 tooth gear with a 47 hole dividing circle with your 40 to 1 tool. I hope that makes sense. If someone can illuminate this for me with one example, I can bend it to my will from there. Thanks ahead, Lee.


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## jwmelvin (Oct 29, 2020)

127 is a prime number. So you would need a dividing plate with 127 holes because you want to turn 90/127 turns on the input shaft to get the output to rotate 1/127. I think that’s right? Any time your ratio and desired count don’t have a common factor, you’d need the dividing plate to have the same number of holes as the intended gear has teeth?


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## LEEQ (Oct 29, 2020)

yes as stated above I would be using a 127 hole dividing circle.


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## LEEQ (Oct 29, 2020)

It looks to me like you are suggesting 90 holes in the sector arms, and to turn the crank that far every time you index a tooth. Do I understand you right? That sounds pretty good, but I'm not sure how to check it short of doing that 127 or128 times and see if I land where I started.  I'm hoping to find a mathematical way to figure this problem. I'm sure that not every gear I want to cut (such as a 100 tooth gear) is going to be a prime number and demand a hole plate with exactly the same number of holes as teeth on the gear. If I can see how it's done by math, I can then fairly easily extrapolate the difference for using say a 20 hole circle for the 100 tooth gear.  I'm here asking for help because I clearly can't see the forest for the trees.


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## LEEQ (Oct 29, 2020)

Ok, so I'm testing your approach with a gear that's in my destruction manual chart. One that's bigger than 90. I went with 100.  I put 90 over 100. I then reduced both numbers, dividing by 5, to come up with 18 over 20. I do have a 20 hole circle. That gives us an answer of 18/20ths of a turn, or 18 holes in the sector arms. This checks out with the chart. Outstanding. It would stand to reason (to me anyway) that 90 holes on a 100 hole circle would accomplish the same thing. So on my 127 that can't be reduced 90/127th's ( or 90 holes on the 127 circle plate) looks golden.


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## tcarrington (Oct 29, 2020)

Another option is something you don't have - a compound dividing head. Let's let that one alone.
Everything stated above is correct. Rest assured that if you set if up correctly and step 90 holes for each tooth on a 127 hole plate, you will arrive on target, as long as you don't get caught by back lash. I would stick with this, carefully preparing the 127 hole plate. That will have some challenges. 

Another option might be a list of angles down to the resolution of a rotary table - no one really likes doing that.
Finally, an electronic dividing head - more gearing and a high resolution of steps plus the automation of stepping perhaps.


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## LEEQ (Oct 29, 2020)

By golly, I think you found the forest for me. I'd love other input. I'll bet there are other ways to look at this


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## LEEQ (Oct 29, 2020)

I'm working with what I have, the 90 to 1 dividing function of my rotary table. I don't have the plate, but the dro on my mill will let me make the plate I want. I can always go back and add other hole patterns to it as needed too.


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## LEEQ (Oct 30, 2020)

I'm also considering making the plate such that it can be mounted to the face of my rotary table with a large enough hole to accommodate my collet chuck sliding into the table's center. I'm sure I could rig up a gizmo to hold the pin/handle assembly out of the rotary table. I could direct index then. I wanted to solve the problem of simple dividing though. Any thoughts as to what's potentially more accurate? Direct indexing or simple indexing?


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## tcarrington (Oct 30, 2020)

indeed - beware the spacing. you probably have up to 50 something hole plates. Some folks make a larger diameter and arm. Another path is a small pin. 
Direct indexing is also good. Consider the hole placement error is divided by 90 on your dividing head. Your plate might be off a little and it won't show up after dividing by 90.


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## LEEQ (Oct 30, 2020)

I'm not scared of making a good plate. I won't divide that. I will drill a hole where the dro says to 127 times. I played with hole size, hole spacing, and circumference. Im pretty sure I can get it on a standard size plate. I am curious what method of indexing when I cut the gear will be more accurate. Do you think direct indexing without the gear train (plate mounted to rotable face) will be any less (or more) accurate than simple indexing with the gear train in tact (plate mounted to dividing set up)? tcarrington, what do you mean about 50 something hole plates?


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## Bi11Hudson (Oct 30, 2020)

My preference is a little more "brute force", I don't like electronic gizmos doing my thinking for me. I used a "spin indexer", is that the proper name? In any case, I used the indexer and mounted a 127 tooth gear from a Grizzly lathe. The Grizzly was Modulus 1 which equates to 25.4 DP. With this rig, I cut a 16 DP (plastic) gear for an Atlas lathe. Simple, boring as hell, slower than Christmas, but everything came out first try. Do I *need* a 16 DP 127 tooth gear? No, there are several combinations for an Atlas that get within a partial percentage. But I proved to myself that I *could* do it with what I had and that's what matters. 

For what it's worth, a fellow here recently ran a 127 point circle on a DRO and came up with a 126 point circle on the mill. Something about that particular DRO, I don't recall what actually happened but I simply don't trust an electronic gizmo. I worked 50 plus years building such machines. For my own use, I prefer to do everything manually. I use AutoCAD for convenience, it can divide a circle 127 times. But that little gear from the Grizzly wasn't electronic. It just worked. . .
.


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## addertooth (Oct 30, 2020)

I think the DRO I got would have a neurotic breakdown if I asked it to do a 100+ hole bolt circle.  
But then, you get what you paid for


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## Lo-Fi (Oct 30, 2020)

Bi11Hudson said:


> For what it's worth, a fellow here recently ran a 127 point circle on a DRO and came up with a 126 point circle on the mill. Something about that particular DRO,




This is a classic DRO gotcha. The problem arises when you ask it for N number of holes and set the start _and end_ at the same angle. Let's say you ask for 4 and enter a start and end angle of zero. It'll give you a three hole pattern because you've asked for the last hole to be at the same angle and position as the first. You can either ask for 4 holes and manually calc the ending angle of 270, or you can just ask it for 5 holes, set zero start and end and get the result you're actually wanting with no calc needed. It's an operator issue, rather than an electronic one (meaning no disrespect to whoever was involved) 

It's actually because if it's flexibility that you get this issue. If the function was simply "circular pattern", you'd expect to enter the number of holes, start angle and let her rip. As it let's you spread the pattern over an arc shorter than 360 - lets say 90 -you'll see that it works exactly as intended: 4 holes, 0, 30, 60, 90. You need that "ah ha" moment to realise that entering zero as the end angle isn't going to end well!!!

@LEEQ I've successfully 3D printed plates for odd jobs like this. They won't last forever, but for one offs like this it's well worth it, cheap and quick


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## Bi11Hudson (Oct 30, 2020)

Lo-Fi said:


> This is a classic DRO gotcha. The problem arises when you ask it for N number of holes and set the start _and end_ at the same angle. Let's say you ask for 4 and enter a start and end angle of zero. It'll give you a three hole pattern because you've asked for the last hole to be at the same angle and position as the first. You can either ask for 4 holes and manually calc the ending angle of 270, or you can just ask it for 5 holes, set zero start and end and get the result you're actually wanting with no calc needed. It's an operator issue, rather than an electronic one (meaning no disrespect to whoever was involved)



As a point of reference I would ask if the DRO systems have the capibility of doing a *reverse* or negative angle. With AutoCAD I want to rotate an angle to 270 degrees, I can enter* -90* and get the same result. It's no big deal for the basic compass points, but an angle of negative 13.25* saves me from doing the math in my head to find the inverse angle.
.


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## LEEQ (Oct 30, 2020)

Sounds like Bill votes for direct indexing. Funny thing ,Bill. I'm looking to cut a 127 tooth gear for my grizzly like you cut for your atlas with the grizzly gear. It's also a can I kind of thing.


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## LEEQ (Oct 30, 2020)

I'm sure I can chuck up the old sharpie and mark the positions to check the dro.


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## LEEQ (Oct 30, 2020)

As far as printing plates, that seems to be a good easy way out. I don't have the capability though. I think letting the dro think for me is a plenty good cheat for me, and my plate can have other dividing circles added to it as needed with no worries of longevity.


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## LEEQ (Oct 30, 2020)

Well I dug through the manual for my el400 dro. It looks like it wants me to input center in x and y, radius of pitch circle, number of holes, and start angle. after that I can mark the first hole, then advance to hole 127 and mark it to make sure its not on top of the first hole. This dro doesn't ask me for finish angle, and it lets me scroll through or select by number the position I want to go to next.


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## Lo-Fi (Oct 30, 2020)

I'm sure you'll get a great result from the DRO method. That being said, there are plenty of people here who would be happy to help out printing a plate for you, I'm sure.

Not sure about negative angle, Bill. I suspect it only works "clockwise" and you'll manually calculate the start and end angles. 

That's an interesting way your DRO works, Leeq. Sounds pretty sensible to me! Pics as you progress  please


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## LEEQ (Oct 30, 2020)

I think if the dro were to want start and finish angle and zero was given for both values, a guy might get a 127 hole pattern by asking for a 128 hole pattern. I'm not at all sure about that, but If that's how mine worked I would sharpie it out and verify my theory. I'm curious myself. I only have a nice dro on the mill. Knowing what to expect out of my chinese glass set up wouldn't hurt at all.


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## LEEQ (Oct 30, 2020)

I appreciate the offer/sentiment of help printing things, but if I didn't have to make a thing to make a thing to make things, this hobby probably wouldn't suit me.


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## RJSakowski (Oct 30, 2020)

I just checked my Grizzly DRO and if the start angle and end angle are the same or differ by some multiple of 360º, (360º, 720º, etc.) the number of holes entered will be correct.  However, if the end angle differs by even .001º.  As an example, drilling a four hole bolt circle; center 0,0, diameter 2, start point 0º, end point 360º will put holes at -1,0; 0,-1; 1.0; and 0,1  However, entering 359.999º for the end point, the holes are located at -1,0; -.866,-.5; -.5,-.866; ans 0,-1.

To answer Bill's question about entering negative angles, the negative angle is converted to its positive equivalent.  For example, drilling four holes between 0º and 90º, center 0,0, diameter 2, start point 0º, end point 90º, the holes are located at -1,0; -.5,.866;  .5, .866.  If the end point is -90º, it is internally converted to 270º and the holes are located at -1,0; 0,-1; 1.0; and 0,1.

This is on a 16 y.o. DRO, model nos. H6087 -H6098, running software version 8M-B


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## DAT510 (Oct 30, 2020)

You may want to look into Compound Indexing (which is different than Differential Indexing), which can allow you to increase the indexing combinations available beyond plain indexing.

(You could also look into Differential indexing, which adds a gear train to the compound, so you get indexing plates and a variable gear train.)

In Compound indexing multiple hole circles are used simultaneously on a given indexing plate. This may include counting forward a certain number of holes on one hole circle, and in some instances even backing up a certain number of holes on a different hole circle.  By combining the use of multiple hole circles, its possible to improve the resolution of your indexing.  It may not be exact, but if you can get the resolution small enough, it may not be noticeable.


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## LEEQ (Oct 30, 2020)

Hello SoCal. I'd love to see references for figuring "compound indexing". A how to, kind of thing. It sounds like you are working with one plate on an indexing setup like mine. I am curious about backlash if you are working in two directions.


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## pontiac428 (Oct 30, 2020)

Both of these books will fill in the gaps for anyone new to dividing.  I would also like to give an honorable mention to Kurt Moltrecht for his chapter on dividing- like most of his work, he conveys the concepts clearly.
---
Okay, the B&S manual is 6 MB, too large to attach.  Here is a link:


			http://vintagemachinery.org/pubs/2185/5863.pdf


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## RJSakowski (Oct 30, 2020)

So to drill 127 equally spaced holes, I would either enter 128 for the number of hole with the start and end point the same or I would enter 127 for the number of holes and the end point would be 360 x126/127 = 357.165. 

On a 8" bolt circle, the holes would be .1979" apart.  You would probably want to drill two or three rows with staggered holes depending on your hole and plate size.

Incidentally, this issue came up in another thread some years ago. https://www.hobby-machinist.com/threads/prime-number-division-without-gearing.62486/
Post #33 on this thread has my concept for a 127 hole dividing plate.


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## LEEQ (Oct 30, 2020)

chapter 11 of the big one had the answer to my original question. Thanks for the reading material.


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## LEEQ (Oct 30, 2020)

RJ, I will have to check out that thread, thanks.


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## Optic Eyes (Apr 10, 2021)

pontiac428 said:


> Both of these books will fill in the gaps for anyone new to dividing.  I would also like to give an honorable mention to Kurt Moltrecht for his chapter on dividing- like most of his work, he conveys the concepts clearly.
> ---
> Okay, the B&S manual is 6 MB, too large to attach.  Here is a link:
> 
> ...


I always was in awe of how Brown & Sharpe engineered standards for threads, gears, measurements I preferred B& S over Starrett in many tools


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## Flyinfool (Apr 10, 2021)

Something else to consider.
If your indexing plate has 1/8 inch holes and you leave 1/16 of material between the holes, Your plate will have a bolt circle of Ø7.580 so you will need an 8 inch diameter plate. Does your setup have room for a plate that big?

Another idea, Rather than doing the indexing on the rotory table to make a 127 hole indexing plate, You could jjust as easy set it up so that you each time you index 1/127 instead of drilling a hole, cut a tooth.


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