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:
15 3 5
16 2 2 2 2
18 2 3 3
21 3 7
27 3 3 3
33 3 11
39 3 13
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.
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
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”.