Involute Gear ... Profile approximated

I came across this article on the forum’s archive:

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The technique looks refreshingly simple ... but how good an approximation can two angles be to an involute curve ?

Just wondering ... Has anyone tried this ?

and are the referenced articles by Messers Hobbs And Cleeve also available ?

MichaelG.

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Ref. **LINK**

https://www.model-engineer.co.uk/sites/7/documents/me-3503-2.pdf

It will depend upon the involute, which in turn depends upon the number of teeth on the gear. Looking at the extremes the fit will be perfect for a rack but awful for, say, a gear with 10 teeth than needs undercutting. Another problem is how to assess the quality of fit. Least squares may be approproiate?

Andrew

I have a spreadsheet that calculates the error between a circular approximation to an involute (such as the button method) and the true involute. It is easy to then optimise the parameters of the circular fit to minimise the maximum error (which for this sort of thing is probably better than LS). It could easily be modified to do the same for a straight line approximation. I can't help thinking that there will be a horrible glitch as the corners of the straight lines "roll" past each other. Could you post some more details please Michael? I don't have access to the digital archives.

The linked article is the design for the cutter-holder, for its grinding.

We really need know what Mr. Hobbs and Mr. Cleeves (weren't they polite in those days?) described, but I wonder if it is based on the industrial process of gear-shaping?

Here, the cutter is of involute-rack form, so with straight-sided teeth, reciprocated across the blank at the same time as the blank is rotated and traversed with respect to the rack, under the control of appropriate gearing. After each pass, the blank is indexed to cut the next space - the adjoining cutter teeth trimming the previous and starting the next.

This generates involute teeth of the same DP and pressure-angle as the rack; approximated mainly by the traverse / rotation increments; and I think the finished wheels are sometimes lapped together. Its advantage is of not limiting teeth-numbers by cutter.

In another version, the shaper-cutter is a formed as a gear with the appropriate clearances; and it and the blank (on its dividing-plate) are rotated with each other by suitable change-wheels. I don't know if each of such cutters can generate only a set range of teeth numbers.

It can be done on a shaper with a single-tooth, straight-sided cutter, but very much more slowly; and potentially less accurately. However, I gather a few model-engineers have obtained good results by it. It might be feasible to make a sort of insert-tooth rack in which the individual teeth are gauge-plate bars set side-by-side in a holder.

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It look as if Messrs. Hobbs, Cleeve and Walsome used the same basic principle but with a revolving single-acting cutter, and it would indeed be interesting to see if their technique is worth re-visiting, though the set-up is half-way to being a gear-hobbing machine.

Mr. Walsome suggests a surface-grinder for forming the cutter, but one could equally use a tool-&-cutter grinder.

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Or one could do as many engineering companies do nowadays.. buy standard "blank" gears and machine the bores and faces to suit. One might not gain Gold Medal though...

Yet another job for this wonderful machine! Load te gears then set the gear centres and wind the handle. Tooth to tooth deviations are recorded on the pressure paper by the stylus. Came out of a skip!

Will do both external and internal gears.

I wonder if the Mr W.K Walsome worked for Parkson, which were I believe an associate company of Sunderland Gear?

Martin

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No need, John ...

The link is to a file hosted on the forum, not to the digital subscribers’ archives.

... Sorry, I can’t remember how I originally found it.

MichaelG.

Edited By Michael Gilligan on 06/03/2021 15:26:41

Update :

I’ve just rediscovered this previous thread: **LINK**

https://www.model-engineer.co.uk/forums/postings.asp?th=116802

MichaelG.

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Start with a quick look at this :

https://www.model-engineer.co.uk/albums/member_photo.asp?a=40710&p=668592

Edited By Michael Gilligan on 06/03/2021 19:01:33

CAD Jockeys might be interested in this: **LINK**

http://web.mit.edu/harishm/www/papers/involuteEWC.pdf

... I confess it’s beyond me

MichaelG.

Thanks for the ME link Michael, so he seems to have one angle above the pitch line and another below, with a "corner" on the pitch circle itself. I guess that would pretty quickly wear to a curve! Given that the circular arc approximation is so easy to make I wonder why one would bother with this? Anyway I'll add it to the todo list for when I pick up that gear project again...

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Nice summary, John

... although it would certainly be interesting to see your analysis.

MichaelG.

The maths seems fairly straightfoward, at least in principle. While I'd agree with the statement that no algebraic equation exists for the involute I also agree with John, why? Equations for the involute are simple (albeit transcendental) and can be calculated to a precision far in excess of any practical need. That is what I did when designing an internal gear. Just calculate a series of points and let CAD join the dots. I can't remember how many points I used but it was significantly more than that shown in the example Bezier splines.

I suppose it would be quicker for CAD to calculate a Bezier spline than an involute. However, I think most gear manufacturers will have their own set of tweaks and adjustments anyway so won't be using just a basic involute. In summary an interesting article but I won't be using the results in anger.

Andrew

Ken Walsome was my apprentice instructor, during my first year of work in 1968. He worked for several years for W E Sykes, a gear cutting machine manufacturer at Staines, Middx. He was a good instructor and made some very nice models, and a plate camera during my first year at the company. He may have worked previously at HEGTA (Hounslow Engineering Group Training Association), if anyone here is from that place and time. After his death some years ago, items he had made were sold at auction. He encouraged me to build an enlarger when he dicovered I was interested in photography.

As Andrew says, this grinding jig approximates the involute to two straight lines; it would be interesting to see how the form works on the Parkson gear rolling tester in Martin's photo, above. That brings back memories of my time in the Inspection Department at the above firm

Bill

Thanks for sharing the interesting [auto]biographical note, Bill

MichaelG.

Gears cut with a rack form cutter and just one pass at each tooth position (and hence a small number of facets on each tooth) work surprisingly well. The 'skinny' teeth are dues to compensating for non-standard centre distances. They look odd but work fine.

Neil

I seem to recall that clock gears can work with only the dedendum of the wheel in play with only the addendum of the pinion which is why straight sided wheels can work adequately with lantern pinions. The curved top of the wheel being for appearance not function.