On
29 November 2024 at 05:31 garryh Said:
Something is puzzling me, Why does this math work:
Assuming I have a 20t on either the spindle or leadscrew and a 55t on the other with an 8tpi leadscrew, then 20×55/50=22tpi
I’m too fuzzy headed to attempt that, and the numbers I offer next could be wrong too! Check them yourself.
Manually calculating the gears needed to generate threads on a lathe is labour intensive. Continued fractions if anyone wants to look it up. Same method used to find fractional approximations of pi, 3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, etc. Don’t bother with fractions myself – 3.142 is easier.
An alternative is to program a computer to generate all the permutations available from a given set of gears. Though the maths is much simpler, it’s rarely done by humans because the number of permutations is enormous, doing the sums takes forever and the work is extremely tedious. Though it can take a long time to print the results, a computer whizzes through the same problem in microseconds: doesn’t get bored, and does sums in giga-floating point operations per second. Ages ago I wrote the necessary program, partly to find out what unadvertised threads are possible on my WM280. Loads of them, mostly useless!
I fed garrys list of gears into the program [ 20,20,40,45,60,70,80 ] and told it he has a 1/8 leadscrew. Short list of useful TPI follows:
List is filtered because, for example, there are 16 different combinations of garry’s gears that all produce 8TPI. I just printed the first.
Garryh’s gears aren’t metric friendly, though there are a few near misses, for instance 5 ways of producing close to a 1.0 pitch. 1.021 is 24.89 turns per banana which garryh can do with 6 different gear combinations:
Teeth………. Ratio. Metric. .. TPI
60 80 45 70 9/224 1.0205 24.89
40 80 45 70 9/224 1.0205 24.89
40 70 45 80 9/224 1.0205 24.89
20 70 45 80 9/224 1.0205 24.89
60 70 45 80 9/224 1.0205 24.89
20 80 45 70 9/224 1.0205 24.89
As to which gears to buy without doing sums, many imperial lathes have 1/8 ratio lead-screws, so it’s a good bet that gears supplied with them will deliver useful ratios on garryh’s lathe. A potential minor gotcha is modern lathes may not deliver the required ratios in obvious ways.
- A 127 toothed gear works straightforwardly by converting inches into millimetres. Direct and accurate but requires a large diameter gear that may not fit on the banjo. Check whether or not a big gear will fit on your lathe before buying one!
- 63 toothed gears are a practical size, but the accuracy of the metric conversion drops. 63 tooth approximations may not be as inaccurate as might be expected, because the maths is different – not a direct conversion as with 127 teeth.
- The “different maths” approach to ratios is often taken by modern lathes. They come with gears allowing close approximations of both metric and imperial threads, not necessarily obvious! The advantage is fewer gears are needed.
Quite interesting to compare metric and imperial versions of the same modern lathe. A metric WM280 comes with 11 gears (5472 permutations), whilst the Imperial version has 13 (12540 combinations). Oddly, it appears that a metric WM280 can do a few more of the less useful TPI threads than it’s Imperial sibling. However, not checked, I expect some of the metric lathe’s TPI approximations are mildly inferior to the imperial lathe, whilst the Imperial machine does a few metric pitches a tad inaccurately compared with its metric sister. For the difference to matter, the operator has to be doing something special, like cutting a very long high accuracy thread, for which a general purpose lathe isn’t the right tool anyway.
Dave