Hi Daniel:
Well, there are basically three ways to tooth a curve. First you have DSS ,
which is Direct Simulatio
n Subtracti
on ( near as I can tell), in this method
you basically simulate a rack moving around a gear. You focus on one tooth
of that rack and subtract it from the centroid as you go. This is quite complex in
practice, though I may try it myself next iteration
.. Various rules exist for various
instaneou
s radius's found as you progress thorugh the curve. This IS though, the best method
for achieving perfectio
n. I would level this as EXPERT mode programmi
ng.
Secondly, you have formulaic represent
ation. In this method you calculate the root,
pitch and tip curves of the centroid, basically offset curves of the orginal centroid,
then calculate each tooth using the formula for that tooth derived from various statistic
s
about its curve locations from centroid center. This is difficult to do as the centroid center
often creates large swings in motion from the tooth locations center point and normal on its curve.
This too requires EXPERT level programmi
ng and math skills.
The easiest, ( and therefore the one I implement
ed for elliptica
ls since elliptica
ls were my first gear
type) is the Osculatin
g Circle approach. In this you calcuate the instantan
eous radius at a tooth
location. You do this by taking the derivativ
e of the curve on both side of tooth center and fitting
it to the equasion for a circle radius. This gives you the equivalen
t circle size that such a tooth
woudl be fitting on. ( Look for Wiki on Osculatin
g Circle, youll see some good graphics on this method).
Once you have the derived radius of the equivalen
t circle, you simply place that tooth as if it were
on that circle. This means each tooth is different as are my elliptica
l teeth but they will mesh, though
keeping pressure angles low is required as higher pressure angles push the limits of the theory that
an osculated circel will fit on an arbitrary curve. This method works quite well on an ellipse but as you
go higher and higher in eccentric
ity, the osculatio
n matches less and less to the centroid. This requires GOOD
levcel programmi
ng skills to my mind. Its not rocket science, but it isnt easy either.
It sounds like your making centroid that dont match very well at all to circles ( as does an ellipse), and
that likely means the efficicen
cy of the osculated methos may not match well to your designs. If so, then
items one or two are your best bet. Complex, but once you wrap your head around the job it isnt impossibl
e.
Those nice video's you see from China come from such a software written in China (using method 1)and seemingly well
hidden from us mortals.
. Id like to try that one soon as it sounds challengi
ng and would make Gearotic worth
many times what it sells for. ( No, I wouldnt raise the price due to such things, GM is already worth
several times what we charge as we see it.).
So its hard to give you a great answer here, what most people think is easy, is very much not. Toothing a
random centroid is quite hard unless your quite a mathmatic
s wizard, and quite a programmi
ng one as well.
( I am neither, I just dont give up easily.) Just calculati
ng the length of an ellipse to fit teeth on it
is quite hard itself as there is no real answer, just a calculus approxima
tion, the same is true of
random centroids
. If I were trying to do as you are, Id use osculatin
g circle. Approxima
te the circumfer
ance of
the curve, then divide it to section that will take each tooth. Consider each section on its own and figure out the circle
it would fit on closest to.. this gives you the imaginary circles radius, root and pitch circles, then tooth is as if it
actually were a circle. This will work, but you may have to tweak the pressure angles as a function of the distance from the
osculated center to the actual centroid center in order to allow for a clean mesh.
Item one above would be best. It allows for any curve that isnt impossibl
e to be toothed, but requires quite a bit of
computati
on of subtracte
d curves, inside/outside decisions, and derivativ
e calculus to figire out what the subtracti
on
actually gives you. Ill be trying it in the new year, and if it works Ill likely have to rewrite much of GM to take
advantage of it as if gives realistic trochoids and tips on any curve profile.
Yell if you need anything explained as you go, Ill answer anything I can for you. Its an essoteric interest, you wont likely
find much support on the web, I didnt as I went, so if I can help get you over rough pathces Im happy to assist.
Art
