Acme thread? New Arbor?

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Looks like it could be 5/8 x 13 tpi
BUT thats assuming the 5/8 is accurate. It could also be a 16mm with 2mm pitch, the two sizes only differ by about 18 thousandths between them.
 
Looks like it could be 5/8 x 13 tpi
BUT thats assuming the 5/8 is accurate. It could also be a 16mm with 2mm pitch, the two sizes only differ by about 18 thousandths between them.
The difference between 5/8" & 16mm is in fact only .0049"

I had considered that this might be a 'Trapizoudal' thread and therefore 16mm OD x 2mm pitch but having re-visited the latest image I'm not convinced. Recognising that the OD could well be undersize I made the image exact as to 20mm over 10 threads. This showed that the OD would in fact be about 15.36mm - which I consider somewhat too much at variance - though not impossible.

All of these figures are of course garnered from a quite unreliable source so are mainly speculation.
 
Clean it up really well.

Buy some of this:

https://www.amazon.co.uk/Morphits-Mouldable-ThermoPlastic-BioPlastic-Polymorph/dp/B0C4Z7P778/
Mould around the male thread. Let it set. Unscrew the 'nut'. Cut it exactly in half and photograph very close up, very square to the cut surface*. Measure the outside diameter with a pair of calipers and report that number.

That would give the photo-reconnaissance people here the best chance of telling you what it is.

* If you cut it well, you could paint it with ink and make a stamping on a piece of paper. Scanning that piece of paper guarantees squareness.
 
You might try contacting e bay seller Reltub34. He makes square and acme thread feed nuts for a variety of lathes. I believe he is the same chap who used to offer a service whereby you could send him your screw and old nut and he would clean up the thread and make a new nut to suit. I dare say he would be able to do this for you. I assume the retainer is a fairly simple threaded collar or wheel, do you have a picture or drawing of it?
 
Looks like 2mm pitch.

In inches, that’s 0.0787”, which equates to about 12.7 tpi, which looks plausible from the photo.

There is a trapezoidal metric thread in the MH, of 2mm pitch, but with a nominal diameter of 16.5mm.

So, non-standard as far as I can see. A decent machinist could probably grind a small threading tool to fit the threads, then do a dry run in the lathe to see if it syncs up at 2mm pitch. If so, carry on and make the nut. Depending on how complicated the arbour is, it might be easier to just make a new one.
 
I would have thought of it is Rexon it would be metric. Metric or Imperial I very much doubt they would use some odd pitch. Of course they are not obliged to use a thread pitch that is normally associated with that particular diameter. They could cut a 2mm pitch thread on any diameter. If I were doing it I would probably buy or grind a suitable tool, then make a female thread in something like Delrin to start with. If it matches up at 2mm pitch then you can go ahead and make a real one. Of course if you have it in your hand and can measure it accurately then you would know if it is 2mm, or something else. It looks as though the threads at the base are in fairly good shape, as you might expect, so that is where I would try and measure. Or as Guineafowl says, just get a new one made from scratch then it will be no problem to match up the nut.
 
Looks like 2mm pitch.

In inches, that’s 0.0787”, which equates to about 12.7 tpi, which looks plausible from the photo.

There is a trapezoidal metric thread in the MH, of 2mm pitch, but with a nominal diameter of 16.5mm.

So, non-standard as far as I can see. A decent machinist could probably grind a small threading tool to fit the threads, then do a dry run in the lathe to see if it syncs up at 2mm pitch. If so, carry on and make the nut. Depending on how complicated the arbour is, it might be easier to just make a new one.
It's not 'About' 12.7 tpi - it's exactly 12.7tpi - nor is it 'non-standard' - DIN 103-4 lists 16mm x 2mm Trapizoidal as in the 'Extra Fine' Series.

From the information provided to date, it appears to be 16 x 2 but with a reduced OD which may well be acceptable since the OD is not the important dimension - the Pitch (or Effective) dia. is the most important and that should be between 14.924 & 13.98 mm (7e tolerance). The difficulty is the to measure that requires some specialist equipment that I doubt is available to the OP. Measuring the root will be easier and that should be 13.5mm.

The correct depth of thread should be 1.25mm but if the OD is 15.36 (as gleaned from the photo's) and the root is correct at 13.5mm - suggesting that the pitch dia. is also accurate - then the DoT will be only 0.93mm -- effectively a Stub Trapizoidal being ~75% DoT.
 
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Fastenright Ltd have lots of Acme thread rod, and nuts. I have bought from them in the past. No idea if they do this size as most places that do the bearing grade acetal and similar drive nuts for machinery seem to list a 3mm pitch with a 16mm diameter. Metric threads generally have a number of pitch options for any given diameter. You will have a coarse or "standard" pitch, the one that is most typically used, then one or more progressively finer pitches. As JG suggests it looks like you have a fine pitch option for that diameter. I suspect very few if any suppliers will have one on the shelf, especially in a left hand thread, pretty niche I would have thought. But worth a try, if they haven't got what you want they might be able to suggest someone else.

Still think the guy on e bay might be your best bet as he will be very familiar with this type of thread, and will have all the necessary tooling. I suspect you would have to send him the part, so he can accurately identify the thread. I would be willing to bet JG is right, he knows a thing or two about threads, and it is a metric acme in 16x2. If all you need is a flanged nut or similar I wouldn't have thought it would be too outrageously expensive, you can only ask.
 
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