## Dividing heads –  Dividing the circle

As one might expect from the name, the dividing head is all about dividing. Dividing is not about moving through a certain angle but is more about dividing a whole circle into a specific number of parts. The most obvious use of this is when gear cutting. The circle has to be divided by the number of teeth needed on the gear to be cut. (Of course, in reality, teeth are not cut – it is the space between them that is cut.) One tooth is cut then the workpiece is rotated by the required amount and then the next tooth is cut and so on till all of the teeth have been cut.

Whereas a rotary table is always calibrated in degrees round its edge a dividing head never is.

#### A – Simple indexing

Most dividing heads have a built in indexing ring. This is fixed to the spindle. It has holes spaced round it and a pin mechanism to lock the spindle on any one of these holes. This form of indexing does not have to use the worm and worm wheel. These can be disengaged. The workpiece is free to rotate except where the indexing pin locks with the indexing plate. Of course the only numbers that can be divided by are numbers that divide into the number of holes on the indexing plate. But there are many times when this is useful.

The advantage of this is that it is much quicker than having to rotate the spindle using the worm and worm wheel. It is also possible to have the worm and wormwheel engaged and to turn it using the handle and then to lock the spindle using the indexing plate.

Whenever the spindle is locked using the pin and the indexing plate, the spindle does not need to be clamped.

The holes for indexing are those that can be seen on this side of the wormwheel. The indexing pin it the black knob on the top of the dividing head.

#### B – Use of dividing plates

The most common way that a dividing head is used is with dividing plates. A dividing plate is covered with concentric rings of holes.

Dividing plates can either have rings of holes on one side only. In this case the holes are drilled all the way through. Alternatively they can have rings of holes on both sides. In this can the holes are drilled just less than halfway through. Only the rings of holes on one side can be used at any one time. If one of the rings on the other side is needed the dividing plate has to be taken off, turn round and screwed back on.

On a small dividing head there might two or three dividing plates. Together these will cover rings of holes up to about 50 holes. The numbers selected in the rings of holes will be arranged so the division by any number up to 50 is possible. Of course, it is possible to divide by numbers way beyond fifty but not numbers whose factors include one or more primes larger than the ring with the largest number of holes that are prime. For example, on a small dividing head it might be 51. This is the largest number of holes that can be fitted in one circle on the plate – given its size.

The dividing plate fits onto the body of the dividing head and is concentric with the crank handle. On more complex heads this plate can rotate but on plain dividing heads it is fixed.

On a simple crank handle there is just one handle that contains a retractable pin. This handle can be moved so the pin can engage the holes in any one of the rings of holes. On more complicated , and therefore, larger, dividing heads there are two handles. One contains the pin and can be moved as before. The other is fixed and does not contain a pin. It is as far out as possible so is easier to use for turning the worm that turns the spindle.

The other feature that goes with this are the sector arms. These arms can be adjusted to encompass any number of holes in any ring of holes. This can be set so the space between the two arms goes from one hole through any number of spaces between the holes ending up at another hole. Between one arm and the other there will be n+1 holes but only n spaces. We are only really interested in the number of spaces between the sector arms. In practice the arms are used so that the handle can be moved through so many spaces between the holes in one ring of holes.

The number of holes on a dividing plate depends on the manufacturer of the plate. For example a Browne and Sharp dividing head uses a set of three dividing plates. These plates contain rings with the following numbers of holes:

Plate 1     15, 16, 17, 18, 19, 20

Plate 2     21, 23, 27, 29, 31, 33

Plate 3     37, 39, 41, 43, 47, 49.

Cincinnati dividing heads use one dividing plate with rings of holes on both sides. These are:

First side  24, 25, 28, 30, 34, 37, 38, 39, 41, 42, 43

Second side 46, 47, 49, 51, 53, 54, 57, 58, 59, 62, 66

We have to remember that one turn of the handle turns the worm by 1/40 of a circle. To go through a whole circle, therefore, requires 40 turns. Using just whole turns of the handle we can divide by any permutation of the factors of 40, for example, 2, 4, 5, 8, 10, 20.

Many amateurs use homemade dividing head with a ratio of 90. In this case, the factors become 2, 3, 3, 5

In the simplest way of using these, the dividing plate is fixed in position.

If we can turn the handle by a fraction of a turn then the choices are much greater. Suppose we pick the ring of holes with 15 holes in it. It will also have 15 spaces in it. To turn the spindle through a whole circle involves the handle going past the number of spaces in the currently selected circle, say, 15 times 40 turns. That is 600 holes.

With this ring of holes we can divide the rotation of the spindle into any number of equal parts where the number is a factor of 600.

Suppose we want to divide a circle by 30. In this case the number of spaces will be 600/30 which is 20. Since this is greater than 15 this means each step will be one complete turn, that is, 15 spaces, plus another 5 to make 20 spaces altogether. The sector arms would be set to cover 5 spaces between holes. Between the sector arms would be 5 spaces but it would start with a hole and end with a hole so 5 spaces would have the arms encompassing 6 holes.

As we have just seen, dividing by any number less than 40 will need one or more complete turns of the handle plus a certain number of spaces. Dividing by 40 will require one whole turn of the handle. Dividing by any number greater than 40 will involve less than one rotation of the handle, i.e., just so many spaces.

If we look at the numbers of holes in the rings on the Brown and Sharpe dividing plate above we can see that the rings of holes give us the following factors:

Holes       factors

In

ring

15          3     5

16          2     2     2     2

17          17

18          2     3     3

19          19

21          3     7

23          23

27          3     3     3

29          29

31          31

33          3     11

37          37

39          3     13

41          41

43          43

47          47

49          7     7

It can be seen that with this dividing plate we can divide by any number up to 49. What we often cannot do is divide a circle where 40 times the number of spaces in a circle does not contain the factors required for the division we want to do. For example we cannot divide a circle by 51, whose factors are 3 and 17, because we cannot get a ring with both 3 and 17 at the same time and these numbers are not factor of 40. Similarly we cannot divide by 81 because it has 4 three’s as factors and we can only get a ring with 3 three’s. Apart from these sorts of problems we can divide by most numbers up to 49 x 40, nearly 2000.

Remember that, at any time, we also have the factors that divide into 40, that is, 5, 2, 2, and 2.

#### Dividing by degrees

The reader might have noticed that dividing heads seldom use degrees. But if it is necessary it can be done. Since there are 360º in a circle, one degree is simply a whole circle divided by 360.

To do this requires that the number of spaces on the dividing plate times the number of turns to do 360°, ie, 40,  must be divisible by 360.

Find the factors of 360. They are 5, 3, 3, 2, 2, 2. All of these factors have to be in the product (multiplication) of 40 and the number of spaces in the ring.

Find the factors of 40 are 5, 2, 2, 2.

The difference in the factors of 360 and 400 is 3 and 3.

The smallest ring of holes that includes 3 and 3 as factors is 18.

Using this ring gives 40 x 18 as the number of spaces in a whole circle, that is, 720.

Divide this by 360. The result is 2.

This means one degree is equivalent to three spaces on the 18 hole ring.

For n degrees the number of holes is n x 3.

One space is equivalent to 20 minutes of one degree.

#### The choice of ring

With a small dividing head with only one handle, if there is more than one ring that contains the required factors it is best to choose the larger ring. It is easier to turn the handle.

In the above example the 18 ring was chosen for having the factors required. But the 27 ring also has the required factors and would, therefore, be easier to use.

With a dividing head with two handles it does not make much difference.

#### Differential indexing

A more complex type of dividing head, which is sometimes referred to as being universal, has an auxiliary input at the back. This input can be used to rotate the dividing plate. The dividing plate is on a shaft that contains the shaft with the worm on it. This worm rotates the wormwheel etc.

Usually the dividing plate is locked in position. The worm is rotated by the handle and is locked to the dividing plate by means of a pin.

In differential indexing the dividing plate is not locked. If the auxiliary shaft rotates it rotates. In differential indexing the main shaft of the dividing head is connected via a gear train to the auxiliary input.

If we turn the handle from one hole on the dividing plate to another the workpiece turns. But doing that feeds through to the auxiliary input which, in turn, rotates the dividing plate.

This enables us to divide by numbers we could not do using just the dividing plates. In this method of indexing we cannot rotate the workpiece from the leadscrew – we cannot do helical milling but then that is not why we are using this set-up. Also it is not possible to have the dividing head tilted at an angle.

The merit of this is that it makes it easy to divide a circle by unpleasant numbers. Suppose we want to make a gear with 127 teeth. The number 127 is of great interest because it is the factor that converts from imperial to metric but it also happens to be a prime number.

The method is to set things up to produce a gear with something near the number of teeth we require.

This number is a number that is easy to handle. In the case of wanting 127 teeth a good number would be 120. The closer this number is to the required number the smaller the gear ratio needed to correct it.

Since the ratio built into the dividing head is 40 this means each tooth will need only one third of a turn of the handle. Therefore we have to use a ring on the dividing plate that contains a factor of 3. If we use a ring with 27 holes then each tooth will be 9 spaces.

This arrangement is called differential because the ratio is determined by a difference, i.e., two numbers added or subtracted. It is easy to see that this technique is a very powerful way of dealing with ratios with intractable factors.

Formula for differential indexing

D     the ratio between the handle and the spindle

A     a convenient number near to   the  number required

N     the required number

R     the required gear ratio

R = (A – N ) x D /A

in the above example

R = ( 120 – 127 ) x 40/120

R = – 7 x 40 /1 20

R = – 7/3

If top and bottom are multiplied by 8 this gives a ratio of 56/24 which are gears in the standard Browne and Sharpe gear set.

If the number chosen is too big then the ratio will be negative. If it is too small the ratio will be positive. This means that the dividing plate will have to turn one way rather than the other. This can be changed by adding an idler gear in the gear train to change the direction of rotation.

see Machinery’s Handbook 17th ed p1308

see MEW 153 p26

#### Compound indexing

For the sake of completeness some mention will be made here of compound indexing. This can be used to divide by odd numbers. The trick here is to rotate the main shaft by a certain number of holes on one ring of holes on a dividing plate and to add or subtract from this a different number of holes from another ring. It is necessary that both of the rings being used are on the same plate. In “Practical Treatise on Milling and the Milling Machine” published by Brown and Sharpe in 1927 refers to this as being of “little practical accuracy”.

1. samwel kibet maina said: kindly send me indexing chart showing number of divisions, turns, n holes in plate having. 37,38,39,49, 42 etc.

• johnf said: I dont have such a chart. I am not sure I know what you are trying to do. In most cases, when dividing, the problem is given a number that you are dividing by has to be turned in what plate do you need, which ring of holes do you need, whats the spacing. I think I go through this in some detail in the page called “dividing the circle”. Let me know if this is what you are doing and what number you are dividing by. I would also need to know what numbers you have got on your division plates and the ratio on your dividing head.
john f

2. Mark Jones said: FAO John F. Hello I hope you can help me here I am a welding tutor and need to start milling 30° bevels on mild steel 10 mm plate as my company are on the look out for a milling machine currently. I wonder if you could provide some advice as to a cost effective and efficient machine that should be considered. I would like something automatic that will do 600 to 700mm lengths at a time any ideas would be great. Markj

• johnf said: Generally a vertical machine is far more flexible than a horizontal one. Given a vertical mill for a small workpiece that could be held in a vice I would use a tilting vice to hold it and use a large endmill in the vertical drive. For a larger workpiece as is what you probably have I would mount the workpiece horizontally on parallels and clamped so the edge being machined was along the x axis.I would then use a slot drill ground to give the required angle.
in the simple case the length that can be cut is the travel of the milling table along the x axis. If a longer edge is needed then table stops are fitted to the milling table so the workpiece can be slid left/right so the whole edge can be milled accurately but just a piece at a time.
hope this helps
john f

3. Mark Jones said: Thanks for the advice. I will work through all you have stated, but what I need advice additionally about is what automatic cross slide table Millers are out there because we have not bought one just yet and need you to help me make the right choice , so the requirement is firstly something automatic and secondly the longest movement action I can get, so any help you can give John will be apreciated. Thanks mark

• johnf said: I buy most of my stuff on ebay including my milling machine. I’d look on ebay. There is a lot of info about individual milling machines on http://www.lathes.co.uk. You can always ask the seller a question on ebay. If you are doing a lot of this work it will save you a lot of trouble is there is a power feed on the x direction .
Watch out some of these machines are much larger and heavier than you might think. You need to move it, not have it sink in the floor or knock against the ceiling. Try and see if you can see the machine working.
john f hi
i have a MS Plate with 800 MM dia and i want 162 holes in it with triangle shape so i need the formula or any software which calculate no’s of tub and distance between tube to tube or Pitch size because i want every hole equal distance in shell plate. so guide me please.
Best wishes from

• johnf said: I am not quite sure what you are asking but it seems probable that it involves dividing a circle by 162. One way of doing this might be as follows. I would not drill them on the milling machine. I would simply drill centers on the milling machine and then drill these using a pillar drill. Clearly you will need a space of 400mm between the center drill and the colum of the milling machine.
I would use a rotary table preferably fitted with a dividing attachment. If possible holes would be drilled in the workpiece so it can be clamped onto the rotary table. If this is not possible then a larger plate can be fitted to the rotary table and the workpiece is clamped to this round the edges.
If the workpiece clashes with the dividing attachment then it needs something underneath it to raise it up..
The rotary table is centered relatice to the center drill as explained elsewhere. Then the milling table is moved in the x direction by the rasius of the cirle the holes on.
the factors of 162 are 2, 3, 3, 3, 3 you need to know the gearing for the rotary table. working this out is covered on a page called something like “dividing the circle” on my website.
john

• It may be that the spindle to column clearance could be less than 400mm. Picture an 800mm disc with holes just 50mm in from the edge. With a supplementary outboard table, you can position the centre of the 800mm disc away from the column: Left, Right, or Towards the operator. It could be a pain to setup sufficient support, and it would be trickier to set the correct radius for the drilled holes, but it is possible, and for drilling holes, rigidity in the setup would be a little less critical than for milling.

• johnf said: It is some time since I wrote this. And I’m not sure if this is a reply to something I have written or not. If it relates to the dividing head the usual limit is the height between the table nd the cutter.

Somewhere I do cover the use of an auxiliary table to use the dividing head at a odd angle as needed for making bevel gears.

john f

• Hi John, yes – it’s in reply to a comment you made on 26th April 2018 on the dividing head page of your website. Your comment was in reply to someone (Tanveer Ahmad) talking about using a rotary table to drill holes in an 800mm diameter disk, and you mentioned the limitation of the spindle to column clearance. I was trying (maybe inelegantly) to point out that the disk, and disk centre, can hang off the front of the table, so long as you can reach the holes from the edge of the disk, and adequately support the cantilevered mass. No big deal, I just though the option may be worth mentioning. Cheers, Craig

• johnf said: Thanks. True.

john f

5. John Finfrock said: Unrelated to dividing the circle I was looking for some answers on the Cos-Par divider. I have one similar that has clamping screws on the top and a full cover around auxiliary and plate gears. How does the worm drive get lubricant? The plastic cover on the bottom doesn’t seem like it will keep a bath worth of oil in. What kind of grease fittings are those (new to a lot of this)? Do they just take a needle type grease gun?

Thanks,
John F.

• johnf said: On my dividing head when the body is tipped up there is an opening in the bottom that is covered by a casting.
The worm and wormwheel are covered in grease.
There is no other way to get grease in apart from taking this cover off.
Other parts that need greasing have grease nipples.

john f

6. Hi John. Just working through your excellent and down to earth explanation of dividing. I think I may have found an error: “Using this ring gives 40 x 18 as the number of spaces in a whole circle, that is, 720. Divide this by 360. The result is 3.” and the few lines that follow.

Thanks for the great dividing resource. I’m currently fleshing out my knowledge of dividing and angular maths, as I am planning to build a rotary table with added indexing plates.

• johnf said: You are quite right. I’m surprised no-one else has noticed this. Thanks.

john f

7. Warren Buck said: I have a Cincinnati dividing head and I am trying to make gears with 69 teeth and when I do my math I end up with 129 but on one side of the plate I have 128 and on the other side I have 127. Is it possible there are additional plates that were made for the Cincinnati dividing head (one that would give me a 128 as one of the choices)?

• johnf said: There are no additional plates. For number that cant be done using the plates the solution is differential indexing. I cover this somewhere. I think my example is for making a gear with 127 teeth.
let me know how you get on.
john

• Warren Buck said: John, thank you for your prompt reply. If I understand you correctly, I don’t have everything I need to do this operation. Can you take me by the hand a little? I am tearing out what little hair I have left. I want to make a gear with 69 teeth-so I use the formula 40 divided by 69 and I get 0.58 so when I fractionalized that I get 37/64 or 74/128 and the only choices I have for that denominator are either 127 or 129. Now, I don’t know how to do differential indexing and can it be done with what I currently have? Thanks. Warren

8. johnf said: I thought I had replied to this.
use differential indexing as explained on my website somewhere.
If you cant do this
use a spreadsheet to compute the angle for each tooth in a circle,
machine, mount the workpiece in three/for jaw chuck mounted on a rotary table in
the vertical position with gear cutter in stub arbor on vertical head
cut tooth, rotate rotary table, cut tooth etc.

I think there might be photo of this setup somewhere on my website

john