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Paul Alciatore
08-22-2009, 04:29 PM
Years ago I observed an exhibit in a showcase that was placed in the main lobby of the New Orleans International Airport by a local shop that made gears. It contained almost every type of gear from simple spurs on up. They were all meshed together and a motor drove the exhibit to demonstrate the actions of the various types. The pair that I was most fascinated by was two that were cut on blanks that were probably elipses instead of circles. They were the same tooth count and were meshed with the high spots on one matched to the low spots on the other. The driver was turning at a constant speed and the driven gear would cycle from faster to slower twice each revolution. But they had the same average RPM.

I saw an ad in the latest HSM with a drawing of such a pair of gears and it brought back the fascination.

What I wonder is; how are such gears cut? I would think that each tooth is a different shape from the ones next to it. Probably even the two flanks of a single tooth (or space) are different. I am not even sure that the shape of the pitch line is even an ellipse. Would two ellipses rotate together in contact with fixed centers? If not, what curves would do so? Ovals? The question is complicated by the fact that the poing of contact between two such curves may not necessairly be on the line between their centers at all points in the rotation.

I doubt that a form cutter would work. The best idea I can think of is to use a hob and while the blank rotates, it also moves back and forth toward and away from the hob. This would tend to generate teeth that would mesh with a rack that also moved in relation to the gear in the same manner. At least I think it would.

Anybody know how such gears are actually cut?

JCHannum
08-22-2009, 05:00 PM
You can buy them at the non-circular gear store;

http://www.gearshub.com/non-circular-gears.html

Clarence Myers has a trailer with a collection of full sized sream engines, one of which has square gears, it is hypnotic to watch it run.

One set of elliptical gears has non involute teeth, but I would guess you could use standard involute cutters, changing them to correspond the diameter of the arc of the particular section of the ellipse. Since there is a range of tooth count for a given cutter, probably only two or at most, three different cutters would be needed depending on the shape of the ellipse. Each space would need to be cut individually to get the proper depth.

Black_Moons
08-22-2009, 09:25 PM
One word: CNC

wierdscience
08-22-2009, 10:14 PM
Paul,would that have been Prager?

http://www.pragerservice.com/Services.htm

Bmyers
08-22-2009, 10:21 PM
Many years ago I saw a file cutting machine that had that same type of gears. This machine was well over 100 years old, so no CNC. The gears were cast ( I saw the patterns) and most likely hand finished

jdunmyer
08-22-2009, 10:29 PM
I am presently at the Portland engine show, which runs next week, Wednesday through Saturday. There is usually a party set up across from my exhibit who has a pump jack that has this sort of gearing. The idea is that the engine be able to provide lots of lifting power on the pump's upstroke, yet cycle the pump piston back down quickly, thus increasing the average pumping speed (strokes/minute) over what it would otherwise be. Gives the same effect as upshifting your car and speeding downhill, and downshifting for the uphill run. Much faster overall speed than having to keep it in the lower gear.

This pump jack was made well before CNC. :-)

Paul Alciatore
08-23-2009, 02:25 AM
Paul,would that have been Prager?

http://www.pragerservice.com/Services.htm

As I said, I really can't remember. The name Prager does sound somewhat familiar, but I just don't know. Since New Orleans is a major port city and also close to the oil business, it has many shops capable of the kind of work you would need to repair or even build large ships and that would include some serious gears. Probably more than one gear shop capable of making them. Likely more than a dozen.

A.K. Boomer
08-23-2009, 09:58 AM
When you mentioned oval and stuff it makes me cringe on the math involved,
In the books ingenious mechanisms they have about every strange gear design you could imagine.

lakeside53
08-23-2009, 11:07 AM
One word: CNC


Isn't that three?:D

lazlo
08-23-2009, 11:07 AM
Anybody know how such gears are actually cut?

"Attention Walmart Shoppers: Sir John or Les to the gear desk" :)

Seriously, I'm intrigued by this question. It doesn't seem possible to manually index these gears on a dividing head. For a square gear, you could do the flats, but the corners would be another operation, and that approach wouldn't work for oval shapes.

Just a guess, but were these non-circular gears done on a Fellows shaper, with a matching cam that offset the shaper head?

Evan
08-23-2009, 12:20 PM
There is no particular reason they must be made with involute teeth. Prior to the use of involute gearing there were pretty good compromises that were a lot easier to lay out.

Even if involute teeth are used laying it out isn't difficult, just tedious. Back when everything was done by draughtsmen on paper a it was normal practise to actually draw the teeth accurately to scale. A method called Unwin's Construction was commonly used and is still used internally by CAD programs to calculate involute tooth shape.

To lay out the teeth on a non-circular gear one would simply determine the pitch circle for each tooth on the gear and proceed normally using Unwin's Construction as if that tooth were on a circular gear.

The same can be done with some difficulty using a CAD program to lay out the teeth for a series of gears all stacked on top of each other with a common centre. Then the teeth that are required for each portion of the shape are used from each of the developed gears and the rest discarded.

The two most common non-circular gear shapes are the ellipse and the trochoid. The trochoid is particularly interesting because it will roll between two plane surfaces just as well as a circle will and yet is not round.

aostling
08-23-2009, 12:53 PM
The trochoid is particularly interesting because it will roll between two plane surfaces just as well as a circle will and yet is not round.

A trochoid is not a closed curve like an ellipse. A bicycle pedal follows the path of a trochoid.

Were you thinking of a Reuleaux Triangle? This is a closed curve of constant width. http://en.wikipedia.org/wiki/Reuleaux_Triangle.

JCHannum
08-23-2009, 01:25 PM
The most common non-circular gear shape is the rack.

lazlo
08-23-2009, 01:31 PM
There is no particular reason they must be made with involute teeth. Prior to the use of involute gearing there were pretty good compromises that were a lot easier to lay out.

Sure, but the point is that they are involute teeth. So how did they cut involute teeth on gears with complex curves, pre-cnc?

oil mac
08-23-2009, 02:32 PM
Many years ago, in Scotland, the old machine tool firms of Thomas shanks & co & also Loudon brothers, used to make large slotting machines, (for our American readers, vertical shapers) In these machines the ram was driven by a adjustable crank pin, set for the length of stroke by adjusting its position from the centre axis, still the same systems are used yet in more modern shapers of the 20th century,
This crank system was driven, by a large epicycloidal gear which was cast on the back of it, This gear had two tracks at different radii, driven by a smaller matching gear which was similar When the two gears tracked on the large ratio, (slow cutting speed) before it reached the end of its track, the larger of the elliptical track on the small primary gear, picked up on the matching smaller gear on the matching ram crank, thus throwing the ram upwards on its return, or non cutting part of the cycle at a ratio of double the cutting speed.
This was a most fascinating and smooth system to see in operation, The last of these big machines, i observed in production work, was 15 years ago, The teeth on the gears were cast,

JCHannum
08-23-2009, 03:28 PM
Sure, but the point is that they are involute teeth. So how did they cut involute teeth on gears with complex curves, pre-cnc?

I'm trying to get my head around indexing, but the individual tooth can be cut by touching off on the blank and infeeding the required depth.

As I mentioned, the ellipse can probably be treated as two circles, the large arc being one and the small radius on the ends another. Use the cutter corresponding to gears of those dimensions, say a #1 on the arc and a #8 on the ends. You will have to fake the transition. It won't be exact, but then, gear cutting seldom is.

dfw5914
08-23-2009, 03:45 PM
The most common non-circular gear shape is the rack.

Nope, a rack is a circular gear of an infinite radius.

JCHannum
08-23-2009, 03:52 PM
Nope, a rack is a circular gear of an infinite radius.

Technically maybe, but let's be realistic here.

Evan
08-23-2009, 03:59 PM
Were you thinking of a Reuleaux Triangle? This is a closed curve of constant width

Yes, that is what I was thinking of. The two are closely related.


As mentioned, the rack is the limit case of the circle and the teeth are the involute of an infinite diameter circle.


Sure, but the point is that they are involute teeth. So how did they cut involute teeth on gears with complex curves, pre-cnc?

Use a copy attachment on a shaper. The variety of attachments available for shapers was large. The could copy complex curves as well as anything that was an object of revolution.

John Stevenson
08-23-2009, 06:11 PM
Even if involute teeth are used laying it out isn't difficult, just tedious. Back when everything was done by draughtsmen on paper a it was normal practise to actually draw the teeth accurately to scale. A method called Unwin's Construction was commonly used and is still used internally by CAD programs to calculate involute tooth shape.

.

Nearly, the calculations were done by Grant, Unwin, who is still alive, brought the idea up to date.
Grants Ondontograph is published in all the early copies of machinery handbook.

dfw5914
08-23-2009, 06:33 PM
Technically maybe, but let's be realistic here.

Ahh, but that would be infinetly less fun.:D

Dragons_fire
08-23-2009, 08:51 PM
i think i saw this on this forum before, but check this out of you like crazy gearing..

http://blog.makezine.com/archive/2008/11/gear_heart.html

and its all made from paper

lazlo
08-23-2009, 09:20 PM
Nearly, the calculations were done by Grant, Unwin, who is still alive, brought the idea up to date.

It's way older than Grant. The excersize of drawing the involute curves is described in the Brown & Sharpe and Fellows gear handbooks from the late 1800's, early 1900's.

Here's a great tutorial on drawing the involute curve from the 1910 American Machinist:

http://books.google.com/books?id=bGYpAAAAYAAJ&pg=PA9&lpg=PA9&dq=fellows+involute+gear+book&source=bl&ots=YsGWrPM-cE&sig=BqVwZvk8C1z4-dMRXxt3TWWfcT8&hl=en&ei=QuR-SomLJ4SItgfD5I3zAQ&sa=X&oi=book_result&ct=result&resnum=1

As TexasTurnado pointed out, if you scan back a couple of pages to "The Origin of the Involute Gear", it talks about a French book published in 1694, "An Essay on the Teeth of Wheels", which describes the involute gear in great detail, including the math necessary to generate the tooth form:

http://i164.photobucket.com/albums/u15/rtgeorge_album/Involute1.png
http://i164.photobucket.com/albums/u15/rtgeorge_album/Involute2.png

davidfe
08-24-2009, 10:10 AM
A little earlier publication -
Machine design, construction and drawing: a textbook for the use of young engineer

By Henry John Spooner pub. in 1908 is available for download as a pdf file
at http://books.google.com/books?id=GVdDAAAAIAAJ&dq=Ondontograph&source=gbs_navlinks_s

Starting on page 293 is a description of Willis' Ondontograph from Principles of mechanism. A treatise on the modification of motion by means of the elementary combinations of mechanism, or of the parts of machines (1896)'.

This pdf is available from http://www.archive.org/details/principlesofmech00robirich

This thread about Non Circular Gears leads to some very interesting and old texts.

Thanks so very much for starting it.

OldToolmaker
08-24-2009, 12:00 PM
The theory behind the involute shape is if you take a string and pull it off the perimeter of a wheel the pattern generated would be an involute. If you put Prussian Blue on the gears and rotate them you will see constant contact if the gears are cut right. There are also indicator contraptions sold that will check gear contact and record it on paper.
Many years ago I worked in a large gear shop with customers all over US and Canada. One job was to duplicate oval gears for some sort of antique machinery. After much profanity coming from the office we finally wound up beating them out on a shaper with a rotary table on it. This was 1968, before CNC. Each X-Y location had to be calculated separately. No $5 calculators either, in those days "pocket calculator" was yellow and had an eraser on the end. :)

Evan
08-24-2009, 02:14 PM
I'm still trying to figure out where the assumption that the teeth are involute came from. There is another shape that looks almost the same but is much better suited as it will easily make gears that will mesh universally with any other size of gear as long as the same radius is used to generate the curve of the teeth and the pitch is similar. It is the predecessor of the involute form and is called the cycloidal gear. They are much less sensitive to variations in separation and even tooth pitch at the expense of greater sliding friction. They are also a lot easier to make because the shape of the sides of the teeth is a simple curve. They would be well suited to this sort of application.

aostling
08-24-2009, 03:41 PM
It is the predecessor of the involute form and is called the cycloidal gear. .... They would be well suited to this sort of application.

Evan,

Your comment caused me to dig up my old 1959 edition of Kinematic Analysis of Mechanisms, by Shigley. This has a section describing cycloidal gears. Here are two paragraphs from that section:

http://i168.photobucket.com/albums/u183/aostling/cycloidalteeth1.jpg

http://i168.photobucket.com/albums/u183/aostling/cycloidalteeth2.jpg

Evan
08-24-2009, 07:16 PM
The main thing I recall about cycloidal gearing is that they can tolerate variations in the tooth pitch that involute cannot. That would be very uncommon except in non-circular gearing.

LES A W HARRIS
08-28-2009, 01:02 AM
Just got in last night from 16 day vacation.


Paul,
Cam on hobber, very possible, tough getting a budget to try it these days?


Laslo,
Various Fellows shapers were modified for cam operation, elliptical, square, logrithmic spiral, etc. CNC coversions were also done.

Early on, 1800's multiple cutters were used, different form on each flank, calculate at each pitch radius for the same DP but varying N.O.T. like this.

Note angle varies from each space!

http://i37.photobucket.com/albums/e97/CURVIC9/09%20GEARS/21T3DP20PAELIPTICAL08.jpg


Cheers,

Paul Alciatore
08-29-2009, 04:06 PM
Thanks Les. Finally a real clue as to how it was done before CNC.

Several have said CNC, but is that for real or just a quick way to dismiss the question? Would anyone use a small diameter cutter to do the profile of the CAD drawn teeth? I guess it could be done, but is it done?

I know gear teeth can be drawn to any degree of accuracy you want with CAD. I have done so. But would this ensure proper mesh with involute or any other tooth form or would modifications need to be made to each individual tooth? Les' drawing suggests that this is the case.

This may be the true apex of gear design and construction.

LES A W HARRIS
08-29-2009, 11:04 PM
Thanks Les. Finally a real clue as to how it was done before CNC.

Several have said CNC, but is that for real or just a quick way to dismiss the question? Would anyone use a small diameter cutter to do the profile of the CAD drawn teeth? I guess it could be done, but is it done?

I know gear teeth can be drawn to any degree of accuracy you want with CAD. I have done so. But would this ensure proper mesh with involute or any other tooth form or would modifications need to be made to each individual tooth? Les' drawing suggests that this is the case.

This may be the true apex of gear design and construction.
Here is an extract from page 421 of machinists & draughtsmens handbook, by Peder Lobben, 1900.

He states depending on the degree of elipse multiple cutter's may be used. I have another reference somewhere that stated the form changes for each flank, and multiple cutters were used. My plottings were to understand the principles, have not cut one, my example is even teeth which as extract shows should be odd.

http://i37.photobucket.com/albums/e97/CURVIC9/09%20GEARS/Machinists__and_Draftsmen_s_Handboo.jpg


Cheers,

lazlo
08-29-2009, 11:14 PM
Laslo,
Various Fellows shapers were modified for cam operation, elliptical, square, logrithmic spiral, etc.

Thanks Les -- that's what I had guessed earlier. Oil Mac describes how one setup worked, which sounds simply brilliant:


the ram was driven by a adjustable crank pin, set for the length of stroke by adjusting its position from the centre axis,

This crank system was driven, by a large epicycloidal gear which was cast on the back of it, This gear had two tracks at different radii, driven by a smaller matching gear which was similar When the two gears tracked on the large ratio, (slow cutting speed) before it reached the end of its track, the larger of the elliptical track on the small primary gear, picked up on the matching smaller gear on the matching ram crank, thus throwing the ram upwards on its return

I also found this great scan of an ancient Popular Mechanic's article, showing how to draw eliptical gears. He uses the same method you describe in your second post:

Drawing Gear Wheels. Part 4 (http://chestofbooks.com/crafts/popular-mechanics/Mechanical-Drawing-Self-Taught/Drawing-Gear-Wheels-Part-4.html)


http://chestofbooks.com/crafts/popular-mechanics/Mechanical-Drawing-Self-Taught/images/Fig-254-td.jpg

But these circular arcs may be rectified and subdivided with great facility and accuracy by a very simple process, which we take from Prof. Rankine's "Machinery and Mill Work," and is illustrated in Figure 252. Let O B be tangent at O to the arc O D, of which C is the centre. Draw the chord D O, bisect it in E, and produce it to A, making O A=O E; with centre A and radius A D describe an arc cutting the tangent in B; then O B will be very nearly equal in length to the arc O D, which, however, should not exceed about 60 degrees; if it be 60 degrees, the error is theoretically about 1/900 of the length of the arc, O B being so much too short; but this error varies with the fourth power of the angle subtended by the arc, so that for 30 degrees it is reduced to 1/16 of that amount, that is, to 1/14400. Conversely, let O B be a tangent of given length; make O F=1/4 O B; then with centre F and radius F B describe an arc cutting the circle O D G (tangent to O B at O) in the point D; then O D will be approximately equal to O B, the error being the same as in the other construction and following the same law.

LES A W HARRIS
08-30-2009, 12:02 AM
Thanks for the link, hadn't seen that.
"Gear Design & Application" (Chironis), McGraw-Hill 1967 has a good section on NCG, pg 158-165 & elipticals pg 166-168, also eccentrics, pg 169-173.

Cheers,