Like I said, I was doing my write-up from memory. After seeing your post, that is how I did it. With the tangent lines. Not the radials directly. I still used a spline however. I will replicate this construction in Alibre and see how much difference there is.
Size each radial with the "radius + circumference length", from the center of the spool to the end of the radial. Using, of course, the length of the circumference for that radial.
That's where I was wrong. You must draw a tangent line from each radial and then size that tangent line to the circumference segment length.
Oh well, I at least remembered what on involute is.
After having drawn the gear with the pitch dia. and pitch correct, I would think that even just the first arc would be more than long enough. Trouble is that if you are really going to use that tooth profile, an arc will not mesh and unmesh like an involute will. It's probably a mute point anyway in machining since gear tooth cutters are a common tool. However if you are going to be *printing* gears, then it might make a difference. I will still compare the arc to the spline versions.
If you look close at the "arc involute" you will see what I mean by being different. The involute is "bumpy", as it should be with arcs. As I also said, it may not make a bit of difference. Will depend entirely on the application. The above is by no means a critique of "jrheinlaender's" work. Just an observation. Please take it in the spirit given. "There's more than one way to skin a cat."
I always liked the vector approach to solving an involute. It was an easy way to visualize (for me anyway) the taught string solution. If you unwinding the string from the base circle you know the length of the unrolled portion of string = R * Theta..
You know the vector position of R into terms of theta.. By definition the taught line is perpendicular to the base circle so you can get the direction vector by |R X k| (or is it the other way around) Do some simple vector math and you got the solution... Just plot your points small enough and who cares if you've drawn point to point.
No. I was just trying to show the difference between arcs and a spline for the novices. I constructed the arc version, keeping the true involute dimensions. That required *not* making the arcs tangent to one another. If had made them tangent, then the curve would have progressively moved toward the center, destroying any involute profile. And I believe the resulting profile would play havoc with gears. And we are, after all, primarily talking about gears and involutes secondarily.
I constructed the arc version, keeping the true involute dimensions. That required *not* making the arcs tangent to one another. If had made them tangent, then the curve would have progressively moved toward the center, destroying any involute profile.
Mmh. This is getting technical. In the sketch I submitted, I constrained the both end points of the arcs to the true involute point. Plus I constrained the curvature by applying a tangent constraint between one arc and the next. I did not constraint the center point of the arcs.
So in what way do you think this method does not keep to the true involute dimensions?
I did not constraint the center point of the arcs.
And that is because they had to move. An arc can be considered to be (for lack of the correct term) constant, in that if you extend an arc, it will meet it's own tail. Not so with an involute. It will have an ever increasing radius at any point along it's length. Never the same in 2 places, as in a closed arc(aka circle). If you are trying to mesh 2 gears, then the involute form has been found to be the best. And that is very different than an arc for the tooth profile. I constrained the centers of the arcs and didn't constrain the tangents. That's the difference. (I don't want to get into gear tooth profile engineering here)
And really, this all has to do with arcs and splines. Not an involute curve. The involute is simply an example of a graphical depiction of a formula using a spline as opposed to arcs.
Obviously a succession of arcs is not identical with a true involute profile. Neither is a spline, nor a succession of straight lines. But they will become more and more similar to the true involute the more points are constructed.
My idea with the arcs was to get an approximation as good as a spline (because of Sketcher not allowing splines yet) and requiring less constructed points than straight segments. For this purpose I think not constraining the centers is better. This also gives you optical feedback for the accuracy of the approximation: For a true involute, the curvature centers (aka arc centers) should be on the base circle. The nearer the are in the approximation, the better the approximation.
oldestfox wrote:After having drawn the gear with the pitch dia. and pitch correct, I would think that even just the first arc would be more than long enough. Trouble is that if you are really going to use that tooth profile, an arc will not mesh and unmesh like an involute will. It's probably a mute point anyway in machining since gear tooth cutters are a common tool. However if you are going to be *printing* gears, then it might make a difference. I will still compare the arc to the spline versions.
I'm not sure if modeling a gear tooth as an arc would make that big of a difference with a 3D printed line will make that big of a difference since the firmware's such as Marlin don't support G02 G03 http://reprap.org/wiki/G-code I'm thinking that fine spaced point to point would would do just fine. Oth.. I suppose if you cad a wire EDM/ or a 3D printer that supports G02 G03, perhaps you would want something else... ( I suppose in the ultimate freecad gear tooth plug in, there could be a toggle option for what type of curve fitting one would want.
