Non-Circular Gear Theory

[ArtF]

Nate:

The problem with measuring it from that tangent is that it would still seem reasonable even when ( in relation to the focus) its approaching 90 degrees and can fall out of mesh. Since its importance is in meshing, the referance should be to the focal point. ( Ill give this a check in the
morning, this is from memory , but I know the formula is derived from the computation of "u" which is the angle variation from angle of generation to the current tangent. ( This is also the reason the addendum  ellipse is not the same shape as the pitch line ellipse.)

Art

[Nate]

The problem with measuring it from that tangent is that it would still seem reasonable even when ( in relation to the focus) its approaching 90 degrees and can fall out of mesh. Since its importance is in meshing, the reference should be to the focal point. ( Ill give this a check in the morning, this is from memory , but I know the formula is derived from the computation of "u" which is the angle variation from angle of generation to the current tangent. ( This is also the reason the addendum  ellipse is not the same shape as the pitch line ellipse.)

I agree that that angle is important.  I'm just not sure what it's called.

[ArtF]

According to one source::

>>According to the law of gearing, a pressure angle can
be defined as the angle between the transmission force
and speed of any point in the pitch curve. As shown in Fig.
2, the speed of point P is perpendicular to the radius vector,
and therefore, the pressure angle can be expressed as

  TRUEPA = PI/2 - u + PA;

where u is the angle between tangency and radial vector.

[Nate]

     Its an interesting topic, there are a lot of things to consider, so feel free to ask
when your curious as to how they work in theory and practice... I dont pretend to be
an expert, but I have spent quite a number of hours on ellipticals over the past few years. :)

Is there a simple closed form for the n-lobe versions of the elliptical gears?

[ArtF]

Hi Nate:

A Simple form..you mean of the elliptical equations?

Radius =  a( 1 - e^2))    /  (1-e* cos( order *angle)))

will work for a general elliptical formula for higher orders. Gearotics is  bit more complex
to take into account what I think is a nonlinearity of the eccentricity term of that
formula, but the one above is the one most texts use for illustration and easy to implement.

  There is no standard, so ellipticals can be made many ways and with many formulas..


Art

[Nate]

...Gearotics is  bit more complex  to take into account what I think is a nonlinearity of the eccentricity term of that
formula, but the one above is the one most texts use for illustration and easy to implement. ...

Yeah, the arc length will grow superlinearly with the order if the exentricity is greater than 0.  That's the part I was wondering about.  I guess it's just easier to do things numerically.

[Nate]

Yeah, the numeric profile generator seems much simpler than the analysis.  The cardioid is 1.5+cos(theta), the egg shape is generated to match it.  I haven't done pressure angle calculations yet.  Are there any other easy checks I can do before making the gears to test them?

[ArtF]

HI 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. I notice my functional generator works with  the formula theta = 1.5+cos(t) and provides the expected image only with a low number of 4 points..
(Ill investigate to see why I made such a limitation, I suspect the numbers are beiong clipped in a safety with that formula, however, even with 4
points it does show the sape you have and it reports 56 degrees max pressure angle.

Art

[ArtF]

Nate:

I was curious so I anaylysed why the functional doesnt generate that gear pair form that formula, Gearotic see's it
as degenerate, it runs under the radius of zero, so it limits the inflection to .2 , this makes it effectively a circular curve in the inflection
zone. But with only 4 points selected, it never see's the inflection, so you get a close approximation to the cos( theta) gear without the
inflected concavity.  The pressure angle min/max is approximate, but correct. About  56 degree's , That seems quite reasonable and if you make them
it shouldnt be an issue. (Ill have to consider if I should auto rescale such formulas to allow for the same sort of generation you show..)

  In gearotic of course, the scalar would have to be removed from the formula, the 1.5 additive term just shifts the curve to one side and the radius
is then clipped. Its that that I think perhaps I shoudl readjust my thinking on..

Art

[Nate]

...
  In gearotic of course, the scalar would have to be removed from the formula, the 1.5 additive term just shifts the curve to one side and the radius is then clipped. Its that that I think perhaps I shoudl readjust my thinking on..

Maybe I'm not following, but you can't just remove fixed radius offset like that without changing the shape.  For example, that the pressure angle due to the roll line (i.e. the 'u' in the formula above) is atan((dr/dt)/r), and adding a fixed offset to the radius will always move that closer to zero.  Similarly if you remove the 1.5+ form that cardioid, you get r=cos(theta) which is degenerate as a gear shape in multiple ways.

[ArtF]

Nate:

Your right of course, I spoke too quicky, you should get a .5 radius min to 2.5radius max in this case... I was thinking G2 was autoscaling it.. Ill dig in the code to see wht with more than 4 points it does what it does..

Art

[ArtF]

Nate:

  lol, you know , Summer breaks really hurt your head. I went back and looked and it became obvious. Your formula would indeed give
that shape,but in Gearotic the formula you enter is not used for the shape of the gear, its the first derivative of the formula that defines shape.
This makes the gear ratio implied by the formula to hold true. The formula is meant to specify
the ratio of the two gears. Its why t2 = t gives a circle, it implies the two gears run at a 1:1 relationship throughout 1 period.
 
  I wonder if perhaps an option to enter the shape by formula with the resultant ratios ( whatever they may be ) is worth adding at some point?

Art

[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