Non-Circular Gear Theory

[Mooselake]

Being 40+ years out of engineering school, I remember that I took a lot of math classes but not what was in them.  I googled some pages on cardiod equations to try to pry some of it back.  Try not to laugh too loud if the following doesn't make any sense...

I did a bit of playing around with cardioid equations, and got some interesting effects.  Using Nate's equation and generalizing it a bit to t2=a+b*cos(c*t)

For 4 knots 1.5+cos(t) looked like a cardiod.  When I upped the knot count the results looked a bit weird, like this example at 50 - I'd expected it would smooth the shape out a bit.  I left this one HD to show selected options.

1.5+cos(2*t) with 4 knots gave a 2 leaf rose, but with 50 knots was again strange.  I also got a "Warning, ratios not as demanded" for both so I'm probably  doing something improper.

Aren't you supposed to be on vacation?

Kirk

[Nate]

...
For 4 knots 1.5+cos(t) looked like a cardiod.  When I upped the knot count the results looked a bit weird, like this example at 50 - I'd expected it would smooth the shape out a bit.  I left this one HD to show selected options.
...

That's a really neat shape for gears.  Not really a cardioid, but visually interesting just the same.

I don't have a copy of gearotic, and have no idea what's going on 'under the hood' so to speak.

Based on Art's comments, I think the interface on gearotic is for the 'realtive turn' rather than the radius as a function of theta.  You might get smoother results if you try something like "t+cos(t)/4" in the formula slot.

Gearotic the formula you enter is not used for the shape of the gear, its the first derivative of the formula that defines shape.

If I understand correctly, the 'gear ratio' should be the first derivative of the formula that is entered.  That's useful for creating a particular motion, but not so great for making a particular gear shape.  (Of course, you'd have to be a little loony to want a particular gear shape...)

[ArtF]

Nate:

>>If I understand correctly, the 'gear ratio' should be the first derivative of the formula that is entered.  That's useful for creating a particular motion, but not so great for making a particular gear shape.  (Of course, you'd have to be a little loony to want a particula r gear shape...)

  Correct. The formula uses t as a descriptor for angle, it is specifically asking for a t2 ( angle of the second gear) in relation to t1 ( angle of the first gear.) I designed it for many that wrote me asking for a formulaic motion designer, as you said youd have to be looney to want to use a formula for shape..but then Im frequently descibed as looney... so Ill give some thought as to allowing that as an option on the designer. Im never against the artistic merits of any shape.

Art

[JustinO]

"Noncircular Gears, Design and Generation" Faydor L. Litvin.

This subject is discussed.

[Mooselake]

Nate:

>>If I understand correctly, the 'gear ratio' should be the first derivative of the formula that is entered.  That's useful for creating a particular motion, but not so great for making a particular gear shape.  (Of course, you'd have to be a little loony to want a particula r gear shape...)

  Correct. The formula uses t as a descriptor for angle, it is specifically asking for a t2 ( angle of the second gear) in relation to t1 ( angle of the first gear.) I designed it for many that wrote me asking for a formulaic motion designer, as you said youd have to be looney to want to use a formula for shape..but then Im frequently descibed as looney... so Ill give some thought as to allowing that as an option on the designer. Im never against the artistic merits of any shape.

Art

I must have forgotten that, along with the contents of my calc and diffeq classes.  Strange that it comes close to what I was expecting.  Also explains why using sin(t) instead of cos(t) did a lot more than a shift from x to y.  Maybe it's time to hit one of those online intro to calc classes and see if some of it'll come back; didn't have those the first time around.

t+cos(t)/4 gives a slightly out of round circle.

Kirk

[Nate]

t+cos(t)/4 gives a slightly out of round circle.

Yeah, I don't have the 'right' formula handy.  You could try tweaking the coefficient on the cosine - t+cos(t)/1.414 should be a bit more out of round.

Edit:
Based on the numbers I generated, you could try 1.1t+sin(t)

Re-Edit:
That should be t+0.8 sin(t)

[ArtF]

>>"Noncircul ar Gears, Design and Generatio n" Faydor L. Litvin.

Litvin is the master. Ive read most of what he's written, and lately Ive found things I must go back and revisit
as my math skills have improved over the last couple years.

Art

[Nate]

  On gears such as you have pictured, the issue often isnt the pressure angle. Its the toothing, depending on the teeth they can be hard to place
in concavities.

It doesn't seem *that* bad, though I haven't tested them.

[ArtF]

doesnt look too bad..

Art

[Nate]

...
Correct. The formula uses t as a descriptor for angle, it is specifically asking for a t2 ( angle of the second gear) in relation to t1 ( angle of the first gear.) I designed it for many that wrote me asking for a formulaic motion designer, as you said youd have to be looney to want to use a formula for shape..but then Im frequently descibed as looney... so Ill give some thought as to allowing that as an option on the designer. Im never against the artistic merits of any shape.
...

As long as you're volunteering to do silly things... how do you feel about non-circular bevel gears?

[ArtF]

Strickly speaking, wouldnt that be impossible? Well, I mean I guess one could have a movable shaft version.. but in trying to imagine some
other configuration in my head makes it hurt.. :) .. Though I guess one could design one something liek a crown gear which has varying height as opposed to haveing the shaft move.... Ill have to think about that one..

Art

[Nate]

Strickly speaking, wouldnt that be impossible? Well, I mean I guess one could have a movable shaft version.. but in trying to imagine some
other configuration in my head makes it hurt.. :) .

It seems like it should work the same way that planar non-circular gears do, just spherical geometry instead.  I'll see about modeling and making a pair...

[JustinO]

I shaved an almost correct pair of spherical NCGs, but I need another axis. I've mostly built a machine that should make a theoretically correct pair. The more axes I add, the simpler the math gets. I need another parallel port and sixth Gecko 203v.

Yes, I'm completely mad. Mwahahahahah!

[Nate]

I shaved an almost correct pair of spherical NCGs, but I need another axis. I've mostly built a machine that should make a theoretically correct pair. The more axes I add, the simpler the math gets. I need another parallel port and sixth Gecko 203v.

Yes, I'm completely mad. Mwahahahahah!

I don't understand why you'd want or need more than a 5th axis.

[Nate]

Strickly speaking, wouldnt that be impossible?...

I think I've modeled a matching pair of gear profiles.  Can anyone suggest good tools for closing up the STL to print them?