Dividing by difficult ratios

[John HaineParticipant @johnhaine32865]

There was a question here about how to get a difficult number of divisions on an RT (or, equally, a dividing head) that didn't correspond to any of the available plates. Usually this arises because you want a prime number or something. A similar problem is finding a pair of gears for a peculiar ratio (such as imperial/metric conversion). If you have CNC or DROs with a circle function, or an electronic divider, it's easy because you just dial in the number you want and let the technology do the rest.

Another way is to use CAD of some sort to plot a circle of dots with the right number, print it out, and glue it to a plate and drill to make a plate. This might be accurate enough, or you can use it in a 2-stage process. Suppose you have a 90 tooth wheel on your DH or RT and want 133 divisions – plot and drill 133 holes, mount the plate on the DH, and use it to drill 133 holes in another plate stepping 90 holes for each. This produces a second plate where the errors in the first are reduced by a factor of 90 (or whatever the number of teeth on the DH gear is). I've read this many times, and to satisfy myself I just made a spreadsheet to try it, and it really does work.

There is another way though where you find a ratio of whole numbers smaller than the one that you want but which give approximately the right answer. In the thread linked above two alternate ratios, each possible with the available plates, were given: every 23 holes in a 34 hole plate, and every 67 holes in a 99 hole plate. The first gave a maximum error on the last hole of 0.117 degrees, the second 0.04 degrees. Both pretty good and might be fine for some applications; and those errors can be reduced by a further factor of 90 by using the 2-stage method.

I did some digging into how to calculate the best ratios and found a useful on-line calculator. You basically take the fraction you want to approximate and expand it as a "continued fraction" (CF). Then various numbers of terms in the CF are expanded to find the "convergents" which are a series of integer ratios that are better and better approximations to the starting ratio. The beauty of the method is that there are no better approximations than the ones it gives.  If you are lucky, one of the ratios will correspond to one of the dividing plates you have for your DH or RT.

Edited By John Haine on 29/04/2018 15:22:27

[John HaineParticipant @johnhaine32865]

[John HaineParticipant @johnhaine32865]

The calculator is here. Here are some pics of it in action.

This the ratio above. You need to clear the input, then type the numbers you want in. If there is no square root part leave it blank, it doesn't like zeros. Then you click the right arrow to get the CF.

The CF is just given as a list of numbers – 0; 1,2,10,1,2,1 in this case. You don't need to worry what these mean but if interested it's explained in a long article on an accompanying web page. There are two forms of the fraction which you can get by clicking the "alternate ending" button – the one you want has its last digit as 1 and the longest list. Then click "convergents" to get the approximate ratios you want, given in a third part of the page

This shows the list of possible ratios, with the 23/34 and 67/99 appearing.

[John HaineParticipant @johnhaine32865]

It's interesting to play around with other ratios. Putting in 50/127 for example gives a list of gear pairs that are alternatives to having a big 127 tooth change wheel for metric thread cutting. Or put pi x 10^9 / 10^9 and get a list of approximate ratios for pi.

Hope that this might be useful for someone.

[Howard LewisParticipant @howardlewis46836]

Thank You John,

I have used this calculator to suggest making an extra division plate which could then be used to produce the OP's 133 divisions. Have also downloaded it against any future need.

Howard

[JasonBModerator @jasonb]

But his existing plates give less error that what you suggested.

[WearyParticipant @weary]

And, the 'calculator' doesn't seem to rank the results/options by accuracy nor give any indication of which might be give the least 'error'…. Or does it & I have simply not understood???

Phil

[John HaineParticipant @johnhaine32865]

Usually the trick is to find a ratio that corresponds to a set of holes in a plate you already have – so for the 90/133 example the OP had plates with 34 (so 23/34 gives 0.117 degrees error) and 99 (the best, 67/99 gives 0.04 degrees.

Accuracy improves as you go down the table as you can see from the results pane. Goes 2/3, 21/31, 23/34, 67/99, then 90/133 which of course is the original ratio.

[NealebParticipant @nealeb]

I had a look at this problem a little while ago when I was musing on making prime-number divisions. Looks to me as if most of the usual approximations assume the usual approach of "same number of holes each time". If you are prepared to drop this, then I think it isn't too difficult to get better approximations, using that modern workshop tool of a spreadsheet!

Start with the concept of "total number of holes for one rotation of workpiece". That is, number of holes in chosen circle times division ratio of worm (typically 40, 60, or 90). Say, biggest circle is 57 holes, for the sake of an example. So total number of holes (for my dividing head) is 57*60 = 3420. Using the OP's number, how can we divide into 133 divisions? For each successive cut, we need to move by 3420/133 holes, that is, 25.7142857. Can't do non-integer number of holes, so for the first move the nearest "whole number" (pardon the pun) is 26 holes. Next move would be 2*25.71etc, giving 51. Note that you must do the multiplication first (step number * exact number of holes complete with decimal places), then round to the nearest integer – which is a trivial job to do if you use a spreadsheet to do the sums for you and give a list of hole movements to use. What you find in this case is that about 2 out of 3 steps, you move 26 holes, and the rest 25. However, you will be minimising the error at each step, and errors will be small and scattered through the entire workpiece. In fact, the very worst error possible with this method is equal to a half-hole step, which would be 360/114deg on the plate,. and therefore 360/114min (60:1 division ratio) at the workpiece. It is possible to do a bit of analysis and maybe find a hole circle which could do better – and again, very easily done with a spreadsheet.

If the 3 minute or so error above is too great, then as mentioned the "two-step" approach will significantly reduce this, down to the point that the errors in your dividing worm wheel are probably the limiting factor.

Advantage of this method is that you can very quickly try all the hole circle numbers in whatever plates you have and see which gives the minimum error. But as I haven't actually needed to make a prime-number-divided circle yet, this is a rather theoretical approach at the moment!

Edited By Nealeb on 29/04/2018 20:35:57

[Neil WyattModerator @neilwyatt]

Posted by Nealeb on 29/04/2018 20:33:54:

I had a look at this problem a little while ago when I was musing on making prime-number divisions. Looks to me as if most of the usual approximations assume the usual approach of "same number of holes each time". If you are prepared to drop this, then I think it isn't too difficult to get better approximations, using that modern workshop tool of a spreadsheet!

Start with the concept of "total number of holes for one rotation of workpiece". That is, number of holes in chosen circle times division ratio of worm (typically 40, 60, or 90). Say, biggest circle is 57 holes, for the sake of an example. So total number of holes (for my dividing head) is 57*60 = 3420. Using the OP's number, how can we divide into 133 divisions? For each successive cut, we need to move by 3420/133 holes, that is, 25.7142857. Can't do non-integer number of holes, so for the first move the nearest "whole number" (pardon the pun) is 26 holes. Next move would be 2*25.71etc, giving 51. Note that you must do the multiplication first (step number * exact number of holes complete with decimal places), then round to the nearest integer – which is a trivial job to do if you use a spreadsheet to do the sums for you and give a list of hole movements to use. What you find in this case is that about 2 out of 3 steps, you move 26 holes, and the rest 25. However, you will be minimising the error at each step, and errors will be small and scattered through the entire workpiece. In fact, the very worst error possible with this method is equal to a half-hole step, which would be 360/114deg on the plate,. and therefore 360/114min (60:1 division ratio) at the workpiece. It is possible to do a bit of analysis and maybe find a hole circle which could do better – and again, very easily done with a spreadsheet.

If the 3 minute or so error above is too great, then as mentioned the "two-step" approach will significantly reduce this, down to the point that the errors in your dividing worm wheel are probably the limiting factor.

Advantage of this method is that you can very quickly try all the hole circle numbers in whatever plates you have and see which gives the minimum error. But as I haven't actually needed to make a prime-number-divided circle yet, this is a rather theoretical approach at the moment!

Edited By Nealeb on 29/04/2018 20:35:57

A refreshingly realistic take on the issue.

Another is to 3D print a disc (or just make a paper one, it need not be very accurate) with the required number of holes, you then move same number of holes as the worm ratio. Any error in the disc will be reduced by the worm ratio, so a hand-drawn disc with the points marked to two-degree accuracy (we can all do better than that with a protractor) will be accurate to about 2 arc-minutes.

I usually just use a 60:1 worm with a 60 division graduated collar and a spreadsheet to tell me where to turn the wheel to. This gives a precision of 6 arc minutes. Rounding to the nearest division for a 2" diameter gear gives a fundamental precision of +/- 0.0008" which is fine for the sort of gears I make!

Edited By Neil Wyatt on 01/05/2018 15:48:20

[Martin KyteParticipant @martinkyte99762]

My George Thomas versatile dividing head has an additional worm wheel that allows the division plate to be rotated by a mesured amount. This allows for rotation of the primary spindle by a whole number of diviion plate holes plus a fraction of a hole. Essentially it allows you to make any division plate you require by direct angular rotation. In fact this is the way that the tool generates its own 'standard' plates the first time round.

regards Martin

[Michael GilliganParticipant @michaelgilligan61133]

Nealb has a good concept, and describes it well.

As quite an 'early adopter' of Tony Jeffree's DivisionMaster, I have assumed that to use very similar logic:

[quote] DivisionMaster always calculates moves to the nearest motor half-step. This means that the theoretical positioning accuracy is always within a quarter of a motor step of the desired position; i.e., within 1/200th of a degree with a 90:1 worm drive ratio. Actual positioning accuracy will ultimately depend upon the mechanical accuracy of the rotary device. Positioning errors are not cumulative, but are spread evenly over a full rotation of the rotary axis. [/quote]

MichaelG.

.

Source: http://divisionmaster.co.uk/divisionmaster.html

Edited By Michael Gilligan on 01/05/2018 19:12:50

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