# The common or garden Dividing Head - how does it work?

#### sophiecentaur

Gold Member
I have looked all over the net and can only find descriptions of how to use a dividing head, followed by countless numerical examples. There must be someone on PF who can give a short algebraic description of the way they work. Machinists as a whole, don't seem to want to present the topic in a way that's approachable for me
I realise this is totally the opposite way round for most PF questions that ask for "a Physical explanation" etc.

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#### berkeman

Mentor
What's a garden dividing head? Like a garden tool for digging?

#### berkeman

Mentor
Ah. So "garden" must mean "garden variety" in this case, not dirt and flowers...

#### sophiecentaur

Gold Member
What's a garden dividing head? Like a garden tool for digging?
Use of good old English here. "Common or Garden" is a general term which means ordinary, run of the mill, expect to find one anywhere. I think it is probably a Horticultural origin, as in Common Ragwort, or perhaps in Ornithology, as with Common or Garden Woodpidgeon. It shows my age and culture I think because I thought everyone would be familiar with it.
Any way, is the actual device known well enough by anyone to give a PF-like description of its function. @gmax137 that Wiki article doesn't seems to describe the principle. It just describes a typical arrangement and a numerical example, afaics, which is like all the other hits I found.
The device seems to be based on approximating 'difficult' ratios to simpler ratios and multiple turns of the wheel seem to improve the accuracy. Using different number bases, perhaps?
I guess what I really need is for a nerdy Mathematician randomly to spot this question and to dish up an answer that reduces the thing to an equivalent to the description of the old Slide Rule, which "adds logarithms together to produce the product of two numbers".

#### AZFIREBALL

Here is some math,
Lets say we want 11 holes equally spaced around a circle.
Using a 40:1 table we have 9 degrees for each turn of the handle (360/40=9)
The angle between each of the 11 holes is 32.7272 degrees, (360/11=32.7272)
That is 3.6363 turns of the handle. (32.7272/9=3.6363)
Turning it 3 times is easy but, the 0.6363 portion needs a solution.
We need to pick an index plate with the right number of holes to give us the o.6363.
Since one turn of the handle is 9 degrees three turns will equal 27 degrees.
This leaves 5.7272 degrees. (32.7272-27=5.7272) more to go.
What index plate will get us close to this figure?
We find an index plate with 33 holes will work just fine since each turn of the handle is 9 degrees, therefore each hole in a 33 hole index plate moves us 0.2727 degrees.(9/33=0.2727)
Therefore, if we move 21 holes forward we add 5.7272 degrees to our last sitting.(27+5.7272=32.7272 degrees) So. we go 3 complete turns of the handle plus 21 holes in the 33 hole plate for each of the 11 holes.

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#### sophiecentaur

Gold Member
Here is some math,
Thanks for the effort but that is Arithmetic and not 'Maths'. There are dozens of numerical examples (some of them are excellent instructional videos) that we can all find but they don't do what I am needing. Even Wiki used numbers and commercial names but no algebra afaics. I have watched probably a dozen YouTube videos and they all discuss how to use the various forms of head / indexer and they start by showing you pictures of hardware. Not a single algebraic symbol in sight. I'm afraid the shutter comes down when I try to read a description that relies of six digit numbers; a, b and c are much easier to follow through in an explanation.
Just imagine you wanted to launch a space rocket to reach the Moon. Would a load of numerical examples to be as good as some basic orbital mechanics equations? All I am after is an equivalent description of the dividing process. I guess that the rationale behind the system is along the same lines as a Vernier scale but I'd just like an explanation that's couched in reliable and generally applicable terms.

#### Bystander

Homework Helper
Gold Member
I guess that the rationale behind the system is along the same lines as a Vernier scale but
Your "intuition" agrees with my recollection, but I fear that "my recollection" has faded beyond "formal" recall. @jim hardy , @Baluncore ?

#### Spinnor

Gold Member
θ=9(a/b) where a and b are integers. Using AZFIREBALL's example

θ=360/11 = 9(a/b)

a/b = 3.6363636363636363636363...

.636363636363... = 63/99 = 7/11

If we had a plate with 11 or 99 equally spaced holes we could divide 360 by 11 exactly?

#### AZFIREBALL

Spinnor: A hole plate with any multiple of 11 will work to get the 7/11's.

#### gmax137

Sophie, I think you are making this harder than it needs to be. The indexing head just allows you to turn the held workpiece by a given angle. The holes in the plates let you turn the crank a rational fraction of a rotation. The accuracy comes from the 40:1 ratio. The sector arms you see in the photos are not vernier devices; they simply help you keep track when counting holes (tedious to count out 24 of the 39 holes (or whatever) over & over again as you are cutting a 65 tooth gear).

Sources like "Machinery's Handbook" have tables to help you find the closest match (how many holes to count on which circle). I don't know if these tables were developed by brute force or by some kind of algebra but I suspect it was brute force. There are a finite number of hole circles in the standard plates. Maybe you can turn up the patent (Brown & Sharpe Indexing Head) -- it may explain the rationale for the standard set of hole circles.

You can see that if you want a gear with a prime number of teeth (say, 127 teeth), then you will need a circle with that number of holes (or twice as many...). Otherwise the result will not be exact. On the other hand, machining is never "exact" rather it is "to given tolerance" so you might be satisfied with a gear that has a fraction of a thousandths inch error.

#### sophiecentaur

Gold Member
Sophie, I think you are making this harder than it needs to be.
Yep. Probably but PF is more than just a "do it this way" Forum. I take your point about 127 tooth gear and that it may not matter if one tooth is fractionally wider or narrower than the rest. Except that it could result in a periodicity in experimental results and the experimenter may not be aware why.

I now feel empathy with people who thing the opposite way round to me - i.e. people who are kinaesthetic learners. I have followed through the demonstrated methods on a few videos and it all makes sense but the description without numbers has been missing in all of them - and in all these posts. It puts me in mind of the ready reckoner books that people used to use before digital calculators were available and when they didn't want to get involved with what we called "mechanical arithmetic". They would need to learn the whole book off by heart in order to know the answer to any calculation they were presented with.
The system seems to be based on finding the least error by choosing the best available hole circle. A ratio is expressed as a mixed fraction - so many whole turns and a fraction of a turn. I think I am happy now.
But, boy, it looks like a big opportunity for getting angles wrong if you lose concentration at all. I think I will be sticking with two, three, four and six sides for all my work.
Cheers chaps, for indulging me.

#### AZFIREBALL

sophiecentaur: You said, “The system seems to be based on finding the least error by choosing the best available hole circle. A ratio is expressed as a mixed fraction - so many whole turns and a fraction of a turn.”

Not true! The system is based on finding the right tools to make exact cuts to met specified requirements (within tolerances of the machine being used).

If you select the proper series of holes in a hole plate you get the ‘exact’ location of the fractional part of the mixed fraction.

BTW – a fraction (7/11) is an exact expression of a value only approximated by a decimal equivalent. 7/11 means exactly 7 parts of 11.

The decimal equivalent is a never ending string of digits. (0.636363.......) that never quite reaches the exact value of 7 parts of 11.

#### Tom.G

Slightly off-topic but I am curious:
How are the locations of the original 11 holes determined?

#### sophiecentaur

Gold Member
Slightly off-topic but I am curious:
How are the locations of the original 11 holes determined?
Yes. That's very relevant. We always assume that our measuring equipment has been made to a certain tolerance. There are many divisions of a circle that can't be achieved geometrically ('exactly'). You can't even produce an exact 1/3 division with geometric construction.
The thing that gets you off the hook in Engineering is the acceptance of tolerances. But systematic errors show up in experiments - even when cutting threads.
If you select the proper series of holes in a hole plate you get the ‘exact’ location of the fractional part of the mixed fraction.
That's only true if the hole plate has a suitable set of holes. As has been mentioned previously, there can be an error in spacing for the teeth of a prime number gear if there's not a suitable set of holes, f you try to achieve it with integer numbers of turns plus integer numbers of extra holes. There is an example above, iirc, that shows the errors for two different choices of hole number.
All this could be clear if someone could express this algebraically. I don't have the skills to avoid falling down in producing something like that but there will always be a 'remainder' in some cases.

Decimals are a complete red herring here, of course.

#### sophiecentaur

Gold Member
The accuracy that's required of n high precision mechanical linkages these days may not be as great these days because it's always (usually) possible to correct for regular of even irregular variations when using gears etc. to perform tracking. At one time, telescopes relied on highly accurate clock drives to keep them tracking objects for long exposure times. Nowadays it's possible to reduce the effects of an imperfect tracking mechanism by using a guide scope, which keeps the main scope pointing at a faint object by locking onto a single 'guide star' or, better still, using correlation of the whole field. People claim to be quite sloppy about setting up their equipment and getting everything back by shifting and stacking many hundreds of images. As long as your pointing is good enough to avoid star trails, over each exposure the results can be fine.

#### gmax137

The system seems to be based on finding the least error by choosing the best available hole circle. A ratio is expressed as a mixed fraction - so many whole turns and a fraction of a turn.
Yes. The least error may be exactly zero for some gears but not all.

When simple indexing doesn't work out closely enough, the tables show "compound" indexing where you have K whole turns, plus L holes on circle with N holes, plus M holes on circle with P holes. So the angle moved (in degrees) is (K + L/N + M/P)*(360/40) Where 40 is the worm ratio. Note that the standard B&S setup has three plates, each with six circles.

15 16 17 18 19 20 holes;
21 23 27 29 31 33 holes:
37 39 41 43 47 49 holes.

So these are the integers "available" for N and P. The tables tell you what to pick for L and M, and how far off that would be from the desired number of divisions.

This set allow you to make gears (or bolt hole circles, or whatever divisions) up to 50 exactly; and alot of higher numbers exactly as well. The Machinery's Handbook tables show the error; here's a portion of the table. Look (for example) at the 127 tooth choices -- 2 turns plus 23 on the 39 hole circle plus 12 on the 49 hole circle gives 127.00018 divisions.

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#### sophiecentaur

Gold Member
Thanks - and there was even a pukka formula in there too!

#### 256bits

Gold Member
2 turns plus 23 on the 39 hole circle plus 12 on the 49 hole circle gives 127.00018 divisions.
Interesting this.
360 degrees / 127 divisions = 2.83465 degrees per division
with an error of 0.00018 divisions
or 0.0000005 degrees for the full ctrcle
or again 0.00018 seconds of degree

#### sophiecentaur

Gold Member
Interesting this.
360 degrees / 127 divisions = 2.83465 degrees per division
with an error of 0.00018 divisions
or 0.0000005 degrees for the full ctrcle
or again 0.00018 seconds of degree
Very small - until you are doing interferometry and that has a habit of magnifying errors. If that angular error is translated into a translational motion then fringes could shift a tad. But @Tom.G makes a very relevant comment about how accurately you can actually divide up a circle. I remember reading about methods for improving the accuracy of a helical thread on a lead screw by using a long 'pith half nut' to filter out periodic variations of pitch on a source screw when it's used to cut a 'better' one.

I am accepting that the available sets of hole plates will give you a very acceptable result in nearly all cases. Thanks for the many comments.

#### Baluncore

It all comes down to accurate frequency synthesis using the integer ratios of selected modulus dividers.

As the operator of a Zeatz geared dividing head, I know the head is rotated by a worm gear with a ratio of 40, which has factors of 23 x 5. If I want n divisions of the circle, I first factorise n. I satisfy the factors I can from the worm factors, the product of the remaining factors must come from one of the index plate circles. I skip around that index plate circle in steps of the product of the unused worm factors. Simple!

The Zeatz universal dividing head can also be driven with selected gears from the lead screw on my Huron milling machine when 'spiral milling', helical lines between centres. That always makes it more fun.

If I needed an unavailable plate hole count, I could always make another index plate for the head using the OMT optical dividing table (with projected optical micrometer), flat on the bed of the Hauser Jig Borer. My computer program generates the angles in deg, min & sec for the hole positions required on the index plate. I compute those from i*360/n rather than by accumulating steps with errors. I replaced the scale projector light globe with an LED so it is cooler and brighter.

120 and 127 tooth gears are needed for mixing metric and imperial thread cutting in a lathe. For that reason there will usually be a 127 tooth gear hidden somewhere in the workshop if you need to borrow one for use as an index plate.

As you can see, the solution I prefer uses old and trusted international tools. Zeatz is from Spain. OMT is Optical Measuring Tools Ltd, once part of Newall in the UK. Hauser precision jig borers were made in Switzerland. Huron milling machines are from Paris, France. And all that gear ended up on an Australian Island.

#### sophiecentaur

Gold Member
I know the head is rotated by a worm gear with a ratio of 40, which has factors of 23 x 5.
That's another thing that was nagging me. If the worm were 36:1, there would be more factors 2,3,4,6,9,12 which I imagined would be more useful / flexible. But that would only apply if the user weren't limited to the integer hole positions, I think.
You are discussing more advanced stuff and I imagine that the essentially practical YouTubers are unlikely to be discussing it. Sounds like a combination of nice equipment and healthy mental exercise!!

#### Baluncore

If the worm were 36:1, there would be more factors 2,3,4,6,9,12
I believe it is a compromise and a tradition. Both have 4 factors; 36 = 2x2 x 3x3, while 40 = 2x2x2 x 5.

The number of holes in the plate circles tend to be prime. A two and a five may be needed more often than two threes.
The Zeatz has plates with circles of 15,17,18,19,20,21,23,24,27,29,31,33,37,39,41,43,47 and 49 holes.
That gives all divisions of a circle up to 50. It then misses 51,53,57,59,61,63,67,69,71...
There are many useful divisions beyond 50, listed up to 490, or computed to 1960.
With that set there are 7 different ways to divide by 15, 30, 60 or 120 which seems to me to be inefficient.

Maybe there is a better way? 48 = 2x2x2x2 x 3, while 60 = 2x2 x 3 x 5.

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#### gmax137

2x2 x 3 x 5
Machinery's Handbook has tables for the Cincinnati Milling Co indexing head, which apparently uses a 60:1 worm.

"The common or garden Dividing Head - how does it work?"

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