In my last post I finished cleaning up my dividing head. Now, I’m going to discuss using it. A dividing head is a tool which divides a circle into equally sized portions. If you want to turn a round piece into a hex shape you can use a dividing head. A dividing head is also useful when making gears. Gears are made by taking a round short cylinder and cutting out the spaces between the teeth. They also need equally spaced teeth to run smoothly.
Some dividing heads let you directly move the spindle in a process called “Direct Indexing”. All dividing heads I’m aware of let you do “Simple (or Plain) Indexing” where plates with holes are used to increase the range of possible indexes. My dividing head only does “Simple (or Plain) Indexing”.
The dividing head lets you divide a circle by placing a pin into holes on the index plate. The index plate has a series of rings with different numbers of holes on it. Some dividing heads have multiple index plates and some plates have different numbers of holes on each side of the plate. The holes are there to provide a positive location for the index pin. The pin is at the end of the crank arm. When this arm is moved it turns a shaft that runs through the center of the index plate. This shaft then goes through a reduction gear (usually a worm gear) and turns the spindle. The chuck or collet holder is located on the spindle which holds your work. (Technically, it’s not the holes we care about. It’s actually the spaces between the holes. Holes work to count by though because each space ends in a hole.)
My dividing head has a 40:1 gear ratio between the shaft and the spindle. So, 40 complete rotations of the crank arm results in one rotation of the spindle. If you wanted to divide a circle into 4 pieces you’d need to do 10 compete turns of the crank arm between milling operations. Simple enough but what if you wanted to do 23 divisions? Usually manufacturers would provide literature which included a table which tells what you need to do. The table lists the number of divisions, number of index ring holes, number of full turns, and number of partial turns. Though there’s copies of this literature out there I couldn’t find one that had lines for every set of hole rings on my dividing head.
Here’s a pic of my dividing head in use on a project I haven’t posted about yet. This is to say sorry because I’m sure you came to see cool pictures and instead you ended up with math.
To solve this problem I turned to math and wrote a script to create this table. First, you need to know the worm gear ratio of your dividing head. This is found easily enough by counting the number of full rotations of the crank arm to get one complete revolution of the spindle. As mentioned earlier, my dividing head has a 40:1 ratio. This seems to be a standard on dividing heads but check yours to be sure.
We know that for how many degrees the spindle rotates the crank arm will rotate 40 times as much. The equation for crank rotations is:
crankdeg = 360/divs*wg where
crankdeg is the angle the crank arm will move for each division
360 is the number of degrees in a circle
divs is the number of divisions of the circle you want
wg is the worm gear ratio
The number of full crank turns can be found by:
wholeturns = floor(crankdeg/360)
where floor is a function that rounds down the value inside to the nearest whole number
The number of partial turns is then:
partialturns = (crankdeg-wholeturns*360)/360*indexval where
indexval is the number of holes in the index plate ring
For a given number of divisions (divs) you try different indexval values until partialturns is a whole number.
Example time. Let’s say we want to get 23 divisions.
crankdeg = 360/divs*wg = 360/23*40 = 15.652*40 = 626.09 deg
This tells us the crank arm needs to move 626.09 degrees between every division. Since this is over 360 deg and under 720 deg we’ll be cranking somewhere over 1.5 turns. The number of whole turns can be found by:
wholeturns = floor(crankdeg/360) = floor(626.09/360) = floor(1.7391) = 1
The number of partial turns can be found by trying out different numbers of holes on the index plate. My index plate has the following number of divisions: 66 62 58 54 49 47 46 43 42 41 39 38 37 34 30 28 24 18. Lets try 34.
partialturns = (crankdeg-wholeturns*360)/360*indexval = (626.09 -1*360)/360*34 = 25.131
Unfortunately, 25.131 is not a whole number so this is not an acceptable index val. Now lets try 46.
partialturns = (crankdeg-wholeturns*360)/360*indexval = (626.09 -1*360)/360*46 = 34
Yay! 34 is a whole number. So, we find that, using the the index ring with 46 holes on it, we need to turn 1 whole turn and then place the index pin in the 34th hole from where the index pin currently is. Don’t count the hole the pin is currently in.
We can check our answer using the following process. Since we want 23 divisions of a circle the spindle needs to turn 360/23 = 15.652 degrees. The crank arm is going to turn (1+34/46)*360 = 626.09 deg. When we account for the worm gear reduction we get 626.09/40 = 15.652 deg. Since the answers match this result checks out.
I want to point out that partialturns really needs to be a whole number. Something like 34.05 is not good enough. Why? We can use the math about to show why. In the example above we found the crank turns 626.09 deg. If we hadtried an index val of 54 we would have ended up with partialturns = 39.914. Say we decided to make that 40. We’d get (1+40/54)*360 = 626.67 deg. The spindle would rotate 626.67/40 = 15.667 deg. After 23 dividions the spindle would have rotated 23*15.662 = 360.34 deg. That additional 0.34 deg is probably going to cause you problems.
To create a new table for a dividing head we need to evaluate a range of divisions. For each division we would try each indexval until partialturns is equal to a whole number. For some divisions there may be multiple indexvals which are good. For other divisions there are no good answers.
Running my script, I found that my dividing head can have 206 different settings between 1 and 2640 divisions.
If you have Octave or MatLab this script should work for you. Sorry, it won’t let me indent it here.
#Calculate table for Dividing Head
wg=40; %Worm gear reduction
indexvals=[66 62 58 54 49 47 46 43 42 41 39 38 37 34 30 28 24 18]; %Available index plate values
divrange=1:4000; %Number of divisions
crankdeg=360*wg/divrange(i); %Total deg that needs to be covered
wholeturns=floor(crankdeg/360); %Whole turns on the index plate
partialturns=(crankdeg-wholeturns*360)/360*indexvals(j); %Partial turns on the index plate
if abs(round(partialturns)-partialturns)<1e-10 %Deal with roundoff error. If partial turns is an integer…
table(count,:)=[divrange(i) indexvals(j) wholeturns round(partialturns)];
This script will create a variable called table which is a matrix with four columns. The columns are, from left to right: number of divisions, indexval, whole turns, and partial turns.
If you can program I’m sure you can adapt it to your language.
Sometimes you might want to rotate a part by a specific number of degrees. This is called “Angular Indexing”. To do this, the above math can be used with a specific value of divrange. This is found by:
divrange = 360/desireddeg where
desireddeg is the number of degrees to turn
For example, if we wished to turn 35 deg then
divrange = 360/35 = 10.286
crankdeg = 360/divrange*wg = 360/10.286*40 = 1400 deg
wholeturns = floor(crankdeg/360) = floor(1400/360) = 3
partialturns = (crankdeg-wholeturns*360)/360*indexval = (1400-3*360)/360*18= 16
To turn 35 deg we’d need to turn 3 whole turns and rotate 16 holes on the 18-hole ring.
The is a slightly easier way to determine this as well. With our 40:1 gear ratio we know that each turn of the crank arm will turn the spindle 360/40 = 9 deg. To turn the spindle 35 deg we’ll need to turn the crank arm 35/9 = 3-8/9 times. Since we don’t have an 8-hole ring we’ll need to convert the denominator (9) in 8/9 to something we do have. 18=9*2 so we can expand the fraction 8/9 to 16/18. We’ll need to turn the crank arm 3 and 16/18 turns which is the same result as we got above.