a rogue method?
|Andrew Johnston||11/10/2021 11:04:15|
6316 forum posts
Unfortunately that is mathematically incorrect. A facet is a feature of a polyhedron, generally of one dimension less than the original object. Assuming that we're discussing 3D Euclidian geometry then a facet will be 2D. By definition that means it exists on a plane and so will be planar, ie, flat.
Here's a thought experiment. Let's assume we hob a spur gear of zero width, so the hob teeth only cut when they cross the plane of the blank. As the hob enters the plane of the blank, and exits the other side, it will cut a reproduction of its shape, ie, straight sided. Since the blank is of zero thickness the cut time is also zero, so there is no rotation of the blank during the cut to consider. What will the resultant shape of the cut space be; a series of straight lines, or a smooth curve?
|Martin Kyte||11/10/2021 11:16:22|
2607 forum posts
Please don't call me a pedant. Turning marks on a lathe turned cylinder would be referred to as surface roughness. Taper would be the approximation to the correct geometery. Unless you produce a form tool for each tooth count some of the gears are going to be approximations of the curve. Thats geometry. Hobbs will generate the right geometery but create machining facets which is surface finish. Lumping surface finish and geometrical errors into the same bucket makes the discussion woolly.
|duncan webster||11/10/2021 11:41:33|
|3588 forum posts|
Taking Andrews thought experiment, his zero thickness gear will be a series of flats, but if there is another one alongside the flats will have an angular offset, so for a real case there will be a series of spirals. The repeat rate is the same as the cut per tooth, ie a few thou.
|Dave S||11/10/2021 12:53:17|
|257 forum posts|
Firstly I was mostly calling myself a pedant - for insisting that the hobbed profile is actually a series of facets.
Apologies if that came across as insulting to anyone.
In the case of a Lathe turned cylinder, assuming the lathe is in good order the cross section of that cylinder will be circular. (the intended form). There is a helical groove running down the cylinder, which does affect the cylindrical size - crests are larger than troughs. but it does not make the cylinder none circular.* Hence the helix can be described as surface roughness - now close to size (not form) is the cylinder.
The hobbed gear has both surface roughness issues (its a cutting process and will leave tooth marks after all) AND form issues. The faceting is a departure from the ideal surface form, and hence is more appropriately termed an approximation, just as using a circular form tool cutter is a departure from the idea from, and hence an approximation.
Gear grinding has 2 benefits. Firstly the cutting tools are capable of taking a much smaller cut - so can leave the surface roughness at a lower level, and secondly there are millions of cutting tools operating all the time - equivalent to a hob with millions of gashes - so the form produced is much more perfect than it would be with the limited number of cuts a hob makes.
*actually it does if you cross section you get an oval with the DOC as you go peak to trough.
|Martin Kyte||11/10/2021 13:05:38|
2607 forum posts
I do understand what you are getting at. Any deviation from the ideal surface could be termed an approximation to form but as I say lumping everything into one term is not helpfull to any discussion. I think you have answered the question really when you equate the grinding wheel to a hobb with millions of gashes. All that changes is the surface finish.
|Michael Gilligan||11/10/2021 13:11:08|
19285 forum posts
He didn’t … he called you a pendant
[ perhaps hanging on your every word ]
|Dave S||11/10/2021 13:15:36|
|257 forum posts|
As a thought experiment imagine a hob with a single gash.
If you visualise the hob as having a single gash then the top of the picture shows all the places that you are cutting on a single hob rotation.
Further down the page Tony illustrates how by 4 cuts per tooth (in other words a hob with 4 gashes) gets much closer to the correct form - to the point that although the tooth must have flats on it - it can be no other way with a straight sided cutter - the approximation to a curve is such that it would be easy to miss, and for all practical purposes is an involute.
|Martin Kyte||11/10/2021 13:43:19|
2607 forum posts
But there is always one point on the rack where the cutter is exactly on the involute. With three cuts there will be three points on each flank where the cut and the involute touch. So the rack exactly follows the path of the involute. Increasing the number of cuts per degree of rotation is just the same as dialing in a finer feed on the lathe.
7686 forum posts
Duncan Webster helped me out last month by writing a program in Python3 to draw involute gears. (I was and still am struggling with the maths!) Duncan's code, which requires the easygraphics module, is here on Dropbox.
Duncan's teeth aren't drawn with curves, instead the involute is approximated by drawing a series of straight lines, i.e. facets. By default, the teeth are drawn with 12 facets, so a 20 tooth Mod 1 gear looks like this:
I bodged Duncan's code to draw only one tooth with 1,2,3,4 and 8 facets, all scaled up so the eye can see the facets.
Single facet teeth are obviously notchy:
Two facets are a considerable improvement:
Four facets are quite good:
And at this scale, eight facets are almost indistinguishable from a true curve.
All of these 'involutes' are approximations because of the way they are generated by drawing straight lines, but there are more issues to come.
Duncan's program could be modified to produce G-code to make subtractive gears with a CNC mill or additive gears with a 3D printer. In both cases his mathematical involute could be an excellent approximation, say 24 facets or more, but the resulting gear will be degraded by the production process. Although both gears can be improved by cleaning up if necessary, the plastic and metal versions differ in strength while the metal version could also be greatly improved by polishing and hardening. The properties of a Duncan gear depend on the maths, the material, the basic production process, and the finishing. It's possible for Duncan gears to be either cheaply or better made as required. Most engineering objects are like this. They can be made up to an advanced specification or down to a price. Both useful, but don't waste money on uneccessarily expensive gear or on on cheap junk. It's the engineer's job to make cost effective choices.
My general point was lathe change-gears ain't anything special, and that it doesn't matter as long as they do the job.
|Dave S||11/10/2021 17:05:08|
|257 forum posts|
Indeed more cuts is a bit like a finer lathe feed, but the difference is that the lathe will still turn round (form) with either.
The hob with “courser feed” (1 gash in extremity) creates a different, more approximate form.
The other difference is that the hobs “feed” is set at manufacture but the number of gashes.
The thing to remember is that even a couple of gashes gets you a very close approximation of the correct form, and by the time you have a typical number of gashes in a commercial hob the form is essentially perfect, and other factors in machining are likely to have at least as big an effect on the perfection of form.
|John Haine||11/10/2021 17:06:32|
|4272 forum posts|
Nice! And thanks for the link to the code, and Duncan thanks for generating it. I am wondering about writing a Python wizard for gears, this will be very useful to get me started I think.
|Michael Gilligan||11/10/2021 17:09:04|
19285 forum posts
Very useful, Dave [and Duncan]
P.S. __ Your final point is valid, Dave ... but it does depend a little on 'the job'
... See my earlier link to the Bryant-Symons lathe
... Some people use[d] lathes to do very serious screw-cutting
|Pete Rimmer||11/10/2021 18:39:07|
|1096 forum posts|
I don't do theroetical debate very well I see it as wasted energy. I take your point about the use of the term 'facet' but it's the most fitting terminology I could come up with.
Basically my hobber when it's feeding has very little infeed in fact the feed handle moves like clock's minute hand. Almost all of the 'movement' is the hob teeth passing the blank whilst the blank is constantly rolling so whilst a zero-thickness blank might theoretically receive a flat facet cut ANY thickness of blank will be cut via a simultaeneous action of helical-motion teeth passing through the part whilst it's rolling. It will generate a curve, not a flat.
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