Ferguson's Paradox

[cheng076]

I became fascinated with the infernal machine as displayed in adds for the fall issue of DM. I know how it works but can not figure out what the tooth form might be to engage the 3 dissimilar size gears to the one 'idler' gear. Clayton Boyer has plans for a wooden version so I suppose he uses the original method of lots of play.
I also ran some simulations showing the thing can be made using a belt system with 2 axies and 4 pulleys but it would be much less eligant than brass gears.
Anyone have thoughts on the gear tooth form?

PJH

[John Stevenson]

Any links or pics ?
Signed up for DM but not got the copy yet.

.

[cheng076]

About a week ago I received a solicitation to subscribe to Digital Machinist. The issue used as a 'comeon' is the comming fall issue and features a construction article on the Paradox machine.
I went to the web and found more info which prompted me to ask about the thread form.
So if you wish to see a pic then go to the Digital Machinist magazine website and look at their subscription offer or it may be a prominent pic on the site.
PJ

[cheng076]

Sorry that didn't work.
Can't figure out how to put in a pic. Just search for Fergusons Paradox and you'll find it
I hate computers. ^$&^(&$(%^*)^)^*^ things!
PJ

[Weston Bye]

I got my author's copies earlier this week, but haven't read that article in depth. A quick scan of some of the drawings and text describes the three gears running concentric but with different tooth counts, some meshing the teeth inside or outside the typical pitch diameter.

Looks like a nice project I'd like to do.

[Don Young]

Seems to be another example of how "wrong" things can work within their limits. I have an very old Jaeger cement mixer on which someone replaced the tub drive bevel gear with a roller chain sprocket. It doesn't mesh very good but it turns the tub just fine!

I would imagine that the best tooth form might be rather narrow and long for the device in question.

[TGTool]

I just read it and it's a very interesting article, both about Ferguson, his mechanism, and the current reproduction. IIRC the author says he made some prototype gears and changed to a different cutter in the series to get a better fit for one of the final meshing gears. He did the "fudge to fit", but using excellent judgement didn't also do the "paint it to match."

[John Stevenson]

OK done a web search and found an article that states he did 3 gears on a common shaft, one correct, one with one tooth less and one with one tooth more.

Obviously a unique concept at that time but done everyday nowadays.

If you look inside an automotive gearbox and check the gear specs, chances are none of them will match to a standard because it's now possible to alter specs to suit an application better, like thickening the teeth on a small pinion and making the teeth match on it's mate.

http://homepage.ntlworld.com/stevenson.engineers/lsteve/files/hob%20indexer8.jpg

Of the three gears above only the right hand one, the 22 teeth is correct. The 20 and 21 have 20 and 21 teeth cut on the same circle as the 22.
So putting these together will have the same function as Ferguson's Paradox.
All that was done to cut these, which were done on a hobbing machine, was to alter the gearing so they cut 20, 21 or 22 teeth, everything else, cutter, depth of cut etc, was the same.

All these 3 gears will run with one another at the same centres and can be swapped in any ratio or order but note the difference in gear tooth shape.

.

[Don Young]

Excellent illustration, John!

Don Young

[cheng076]

A couple of you mentioned having already read the article. And thanks, John, for the 'modern' gears explanation. Did the author mention any particular steps he took to modify the tooth form or how he cut them. I should probably wait for the article but 'curosity k.....'
I have been tossing around the idea of using the form as illustrated in Ferguson's drawing and constructing the gear teeth from flat stock and soldering them into a slotted blank. Seems a little more true to the original.
Thanks for the comments, PJH

[George Bulliss]

According to the author, James Donnelly, the original orrey was constructed using wooden gears. Mr. Donnelly used brass for his and cut them with 32-pitch gear cutters. He did mention that he used a No. 2 cutter for the 44-tooth gear and a No. 3 cutter for the 39- and 37-tooth gears.

George

[gordon]

I became fascinated with the infernal machine as displayed in adds for the fall issue of DM. I know how it works but can not figure out what the tooth form might be to engage the 3 dissimilar size gears to the one 'idler' gear. Clayton Boyer has plans for a wooden version so I suppose he uses the original method of lots of play.
I also ran some simulations showing the thing can be made using a belt system with 2 axies and 4 pulleys but it would be much less eligant than brass gears.
Anyone have thoughts on the gear tooth form?

PJH
Re : Fergusons Paradox
Gear tooth cutters for involute gears come in sets, each cutter in the set is engraved with the range of nos of teeth that can be produced with that cutter. The cutter for the largest no of teeth will be marked for example 100 – rack.
OK so far: A rack is a straight bar with tooth forms equi-spaced along its length such that a gear can run along. The teeth on the rack have sides that are straight. As the nos of teeth engraved on the cutter decrease the form produced become more curved. ( this is the involute curve ) See the rack under a lathe bed that controls the saddle travel, the teeth have straight sides.
Years ago back in the 60’s I made a version of the Fergusons Paradox using 3 gears on common spindles all of the same diameter. They had 59, 60 & 61 teeth. Using a gear tooth vernier to measure the tooth thickness on the pitch line, I cut the 60 tooth gear to the correct size. Then adjusted the depth of cut on the 59 & 61 gears to achieve the same thickness of tooth on their pitch diameters.
NOTE the tooth thickness was measured at the same pitch circle dia on all 3 gears.
This method is not strictly correct but for this application worked perfectly.
My version of the paradox had a fixed central gear mounted on a base. A plate revolved around the central spindle, 2 spindles mounted on that plate carried a thick idler gear. Outside of the idler spindles another 2 spindles carried the 2 sets of 3 gears cut from Perspex. The 59 & 61 tooth gears had arrows pointing opposite directions.The 60 tooth mounted between the other 2 had a large dot engraved . A top plate secured all spindles to maintain parallel and a central knob mounted on the top enabled the whole assembly to be spun around. After a little time you could see the top and bottom arrows moving relative to each other and the dot on the middle gear staying in the middle.
The outside dia of the 3 gears being identical and the small difference is the number of teeth leads the observer to assume the gears are the same.
I hope this explains the paradox.
Regards
Gordon.

[gordon]

Re : Fergusons Paradox
Gear tooth cutters for involute gears come in sets, each cutter in the set is engraved with the range of nos of teeth that can be produced with that cutter. The cutter for the largest no of teeth will be marked for example 100 – rack.
OK so far: A rack is a straight bar with tooth forms equi-spaced along its length such that a gear can run along. The teeth on the rack have sides that are straight. As the nos of teeth engraved on the cutter decrease the form produced become more curved. ( this is the involute curve ) See the rack under a lathe bed that controls the saddle travel, the teeth have straight sides.
Years ago back in the 60’s I made a version of the Fergusons Paradox using 3 gears on common spindles all of the same diameter. They had 59, 60 & 61 teeth. Using a gear tooth vernier to measure the tooth thickness on the pitch line, I cut the 60 tooth gear to the correct size. Then adjusted the depth of cut on the 59 & 61 gears to achieve the same thickness of tooth on their pitch diameters.
NOTE the tooth thickness was measured at the same pitch circle dia on all 3 gears.
This method is not strictly correct but for this application worked perfectly.
My version of the paradox had a fixed central gear mounted on a base. A plate revolved around the central spindle, 2 spindles mounted on that plate carried a thick idler gear. Outside of the idler spindles another 2 spindles carried the 2 sets of 3 gears cut from Perspex. The 59 & 61 tooth gears had arrows pointing opposite directions.The 60 tooth mounted between the other 2 had a large dot engraved . A top plate secured all spindles to maintain parallel and a central knob mounted on the top enabled the whole assembly to be spun around. After a little time you could see the top and bottom arrows moving relative to each other and the dot on the middle gear staying in the middle.
The outside dia of the 3 gears being identical and the small difference in the number of teeth lead
the observer to assume the gears are the same.
I hope this explains the paradox.
Regards
Gordon.