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ArtF
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« Reply #30 on: September 24, 2015, 06:34:13 AM »

Nate:

  Smiley, you guys hurt my head, your math backgroun d is far advanced to mine. I struggle to do such things
as tooth a noncircul ar, and generalis ing it is something Ive spent many attempts at, including generatio n
point by point. While Ive gotten close, the virtual hob seems to be the only solution I can come up with
so far.
       Its a good discussio n, and I agree with its direction, Ive always felt there is a formulaic generaliz ation
of involutio n for any surface. Ive tried and tried to derive it, but its just over my head. When this happens
I just try to keep studying the subject till I understan d. So your comments are helpful for what I can glean.
       
   
Art
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Mooselake
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« Reply #31 on: September 24, 2015, 07:24:20 AM »

Hi, Michael!  Welcome to the Gearotic forum.

Guess it's time to cough up a few bucks and get rid of the wiggling in the Gearotic display Smiley   They're forecasti ng rain in Moosevill e, so the outside project I got talked into (running antenna wiring in an old firehall) might get cancelled, which will free up the day for dinking around with the laser and Gearify/Gearotic.

I didn't see any provision for shafts, either sizes or locations, in Gearify gears, did I miss them?  Also, does the postmaste r address on your site still work?

Kirk
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Nate
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« Reply #32 on: September 24, 2015, 10:12:21 AM »

Nate:

  Smiley, you guys hurt my head, your math backgroun d is far advanced to mine. I struggle to do such things
as tooth a noncircul ar, and generalis ing it is something Ive spent many attempts at, including generatio n
point by point. While Ive gotten close, the virtual hob seems to be the only solution I can come up with
so far.
...

Since Micheal was asking about it, I've been thinking about the best way to explain it.  So here's an attempt:

I'm going to assume that you can work out how to generate circular involute tooth profiles point-by point starting with the theory of involute gears.  So you know, for example, that there are two contact points on tooth flanks that correspon d to every point on the pitch circle.   If that doesn't make sense then starting with a foundatio n of involute gear theory might be more productiv e.

The non-circular analogue of the pitch circle is called a roll line.  (I tend to call it a pitch line, or pitch profile, and there may be other terms, but I'll call it a roll line here.)  Similarly, let's call the analogue of the pitch point the roll point.

So let's say we want to make a set of involute non-circular gears, and let's suppose that we've produced two "nice" roll lines so that they'll roll against each other with fixed centers of rotation and with the point of contact - i.e. the roll point - always on the line between centers.

Then it's relativel y straightf orward to model the rolling action of one roll line against the other.  (For example, that's something that gearify already does.)

Now, to build involute tooth profiles from this action we need to pick some way to determine tooth phase, and a pressure line.  There are natural ways to do both of those:

The pressure line can be effective ly the same as it would be for circular involutes:  It's the line that intersect s the line between centers at the roll point, and is off perpendic ular by the pressure angle.  (There are two of these pressure lines, one for the rising flank, and one for the falling flank that correspon d to the two direction s of the perpendic ular.)

Similarly, the tooth phase is a linear sawtooth function of the arc length of the roll line.

So for every point on the roll line, we have a pressure line and a tooth phase, so we can work out the correspon ding contact points, and this lets us generate the tooth flanks point-by-point.
« Last Edit: September 24, 2015, 02:28:19 PM by Nate » Logged
ArtF
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« Reply #33 on: September 24, 2015, 01:21:58 PM »

Nate:

 Very well explained, the problem Ive found is in the convexiti es.. that does work for me till the point where convexity causes a problem, been so long I cant say exactly what the problem was.. .. but you know, the one point you mentioned I hadnt tried was using a sawtooth phase on the arc length for flank position,...thats brilliant .. I may have to revisit
that code..

Art
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Gearify
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« Reply #34 on: September 25, 2015, 10:03:45 AM »

Nate,

"Now, to build involute tooth profiles from this action we need to pick some way to determine tooth phase, and a pressure line."

Wow. This is essential ly what I had in mind, only I was of the opinion that "picking" my own pressure line and tooth phase would not constitut e something I could advertise as legitimat e involute teeth, but rather my own personal bastardiz ation of a precisely defined concept.  Cheesy I have not had time to extensive ly review and understan d in-volute tooth theory to know how much is good enough (I am involved in too many projects. .. )

This discussio n is very motivatin g to me! If we can come up with a feasible definitio n I can definitel y implement it in Gearify! Smiley

Here is my question for you though, Nate. By whatever definitio n you're working with, what is the pressure angle measured from on a non-circular gear? In circular gears, the line tangent to the roll lines at the point of contact is always perpendic ular to the line through the centers of rotation (which gives us some nice propertie s). On non-circular gears (especiall y the more eccentric ones) the tangent line can be quite far from perpendic ular to the line through the center. See the diagram below:



If measured from (p1-p2) I feel that there would be severe disortion s where t1-t2 differs significa ntly. But if from t1-t2, the gears may not "push" on eachother properly and the entire benefit of involute teeth is compromis ed.

Thoughts?

« Last Edit: September 25, 2015, 10:23:20 AM by Gearify » Logged
Nate
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« Reply #35 on: September 25, 2015, 11:14:18 AM »

Here is my question for you though, Nate. By whatever definitio n you're working with, what is the pressure angle measured from on a non-circular gear? In circular gears, the line tangent to the roll lines at the point of contact is always perpendic ular to the line through the centers of rotation (which gives us some nice propertie s). On non-circular gears (especiall y the more eccentric ones) the tangent line can be quite far from perpendic ular to the line through the center.

I don't know what the official definitio n is or even if there is one for non-circular gears.  Art and I discussed the same question in the other thread.   As far as I'm concerned, the preferred usage is the angle off p1-p2.


Quote
If measured from (p1-p2) I feel that there would be severe disortion s where t1-t2 differs significa ntly. But if from t1-t2, the gears may not "push" on eachother properly and the entire benefit of involute teeth is compromis ed.

Thoughts?

Mechanica lly speaking you probably want to measure the angle from p1-p2 and live with the distortio ns.   That's what I was trying to describe, and, mechanica lly, you want the action to be close to that line.  There's a lot of freedom in tooth profiles, so the roll line tangent line can somtimes also work.

In the illustrat ion, if we imagine that we're turning the red gear clockwise as the master and the yellow gear is the slave and the gear flanks are roughly perpendic ular to the t1-t2 line, then it's likely that the gears would just separate.

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Nate
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« Reply #36 on: September 25, 2015, 11:16:01 AM »

Here is my question for you though, Nate. By whatever definitio n you're working with, what is the pressure angle measured from on a non-circular gear? In circular gears, the line tangent to the roll lines at the point of contact is always perpendic ular to the line through the centers of rotation (which gives us some nice propertie s). On non-circular gears (especiall y the more eccentric ones) the tangent line can be quite far from perpendic ular to the line through the center.

I don't know what the official definitio n is or even if there is one for non-circular gears.  Art and I discussed the same question in the other thread.   As far as I'm concerned, the preferred usage is the angle off p1-p2, but I worked this stuff out for myself.  Other people will have other notions.


Quote
If measured from (p1-p2) I feel that there would be severe disortion s where t1-t2 differs significa ntly. But if from t1-t2, the gears may not "push" on eachother properly and the entire benefit of involute teeth is compromis ed.

Thoughts?

Mechanica lly speaking you probably want to measure the angle from p1-p2 and live with the distortio ns.   That's what I was trying to describe, and, mechanica lly, you want the action to be close to that line.  There's a lot of freedom in tooth profiles, so the roll line tangent line can somtimes also work.

In the illustrat ion, if we imagine that we're turning the red gear clockwise as the master and the yellow gear is the slave and the gear flanks are roughly perpendic ular to the t1-t2 line, then it's likely that the gears would just separate.


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Gearify
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« Reply #37 on: September 25, 2015, 12:10:52 PM »

"so the roll line tangent line can somtimes also work."

Sounds like this could potential ly be a user option!  Cheesy

I do wonder how the distortio ns would appear when the roll-tangent line is significa ntly off-normal to the center-to-center line...

Here is another question.

We can define the pressure lines as a series of "snap shots" where the line is chosen to pass through the contact point at a particula r point in time. So there is in a sense a "jump" from one pressure line to the next as the position of the line switches for each tooth, to pass through the varying point of contact of the roll lines. This jump may produce some abrupt changes in stress and pressure that may be undesirab le (am I wrong?).

One could, however, define a MOVING pressure line.  Such a pressure line would always be intersect ing the point of contact between the roll lines, and the point of contact would linearly travel along this "floating" pressure line.

This is similar to how a Cubic Bezier curve is generated . Imagine the green line below is our "floating" pressure line. The orientati on of the line is defined by either of the two "pressure angle" notions we defined (this is constant if measured from (p2-p1) as described earlier, and continuou s if measured from (t2-t1)), and the position of the line would be continuou sly defined by the point of contact on the roll lines...

What do you guys think?  

« Last Edit: September 25, 2015, 12:16:25 PM by Gearify » Logged
ArtF
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« Reply #38 on: September 25, 2015, 01:54:22 PM »

I think youll find the pressure angle is measured not from the pitch circle, but from the base circle, so if we shrink the ellipse by the base circle amount,
then compute a tangent between the two offset base ellipses, that would be the pressure angle Id use ( and do currently) in computing the involutes .

 I think Nates idea is a really good one. Ill probably play with it in the coming months to see if it improves on hobbing, but I have read papers
on the hobbing being superiour simply because it deals with degenerat e solutions to Nates idea. Its like the tips of the teeth on high K areas
of the curve, the hob finds it naturally, the involute equations start to produce some nasty overlap that needs to be dealt with.. at least thats
what Ive experienc ed so far.

Art

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ArtF
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« Reply #39 on: September 25, 2015, 01:58:39 PM »

Michael:

 As an example, in your drawing, Id imagine the proper line of action is if you moved T1 to a 1/4 inch inside the red on a line from
c1 to p1, and then places t2 at a line crossing pitchpoin t to a point 1/4" inside the yellow... that woudl be the proper line for that gear.

Art
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Nate
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« Reply #40 on: September 25, 2015, 02:43:04 PM »

...
 I think Nates idea is a really good one. Ill probably play with it in the coming months to see if it improves on hobbing, but I have read papers
on the hobbing being superiour simply because it deals with degenerat e solutions to Nates idea. Its like the tips of the teeth on high K areas
of the curve, the hob finds it naturally, the involute equations start to produce some nasty overlap that needs to be dealt with.. at least thats
what Ive experienc ed so far.
...

Right, the hob is good for finding clearance s, but if the profiles are degenerat e, then 'virtual hobbing' will produce bad gears.  (I.e. gears that lose mesh or with rotationa l ratios that are inconsist ent with the roll lines.)
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Nate
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« Reply #41 on: September 25, 2015, 03:03:36 PM »

...
We can define the pressure lines as a series of "snap shots" where the line is chosen to pass through the contact point at a particula r point in time. So there is in a sense a "jump" from one pressure line to the next as the position of the line switches for each tooth, to pass through the varying point of contact of the roll lines. This jump may produce some abrupt changes in stress and pressure that may be undesirab le (am I wrong?).

One could, however, define a MOVING pressure line.  Such a pressure line would always be intersect ing the point of contact between the roll lines, and the point of contact would linearly travel along this "floating" pressure line.
...

The pressure line is stationar y in the reference frame of the pitch point. In any reference frame where the pitch point is moving the pressure line will be moving as well.

In this video the pressure line is the red line, and the red dots are points of contact.  Can you explain what time in the cycle the "jump" you're concerned about occurs?

https://www.youtube.com/watch?v=14yMFdgWM-A
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Gearify
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« Reply #42 on: September 25, 2015, 03:16:48 PM »

Let's say that P(t) is the point of contact between the roll lines at time t. The line of action (as you defined it) passes through P(t), and is oriented based on the pressure angle.

Suppose the pitch point begins contact at time t0 and moves smoothly along the line of action until the next tooth is engaged at time t1. Now a new line of action is engaged and will pass through point P(t1).

For a circular gear, P(t0) = P(t1). for a non-circular gear, they are most likely not equal.

So imagine in the video you posted, imagine that the height of the red line would instantan eously jump to a different vertical height every time a new tooth engaged.
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Nate
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« Reply #43 on: September 25, 2015, 03:48:28 PM »

...
So imagine in the video you posted, imagine that the height of the red line would instantan eously jump to a different vertical height every time a new tooth engaged.

The pitch point is moving continuou sly along a continuou s path.  There's also a continuou s rotation.  What does your math education tell you about the compositi on of continuou s things?  Also, does continuou s motion 'instantan eously jump'?

The "dots" do jump back to the start of the line of action - that's the sawtooth phase I described - but the motion of the line of action as a whole is going to be continuou s in any setting where the motion of the gears is continuou s.

...

BTW:  If we imagine that the gear in the video is rolling along a stationar y rack, then the only way that P(t0) = P(t1) is if t0=t1.  You're not thinking in terms of the reference frame of the pitch point.
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Gearify
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« Reply #44 on: September 25, 2015, 04:31:16 PM »

I think I'm not communica ting what I am trying to say properly. I do understan d compositi on of continuou s functions is continuou s. I think we're talking about two different things but I'm not sure where our disconnec t is yet.



In the above diagram, the circular gear has a line of action who's position (I'm talking about the entire line segment's position) remains constant relative to (let's call it) the observer. But the non-circular gear has a line of action that is at a different horizonta l location. So, either the horizonta l position (relative to the observer) of the line of action would need to jump to new horizonta l position after each tooth engagemen t, OR, the location of the line of action is allowed to vary continuou sly over a single tooth (in which case the "dots", which are interpola ting across a moving line, would actually follow a continuou s curve).

Do what I' m saying make more sense now?

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