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Spline Curvature and Geartrax

Posted in GearTrax, Splines by Administrator on the May 28th, 2008

As promised, more [yawn] detail about spline curvature, involutes, and Geartrax. Part 1 of this subject is here.

As I wrote before, there are some errors in the involute spline generated by Geartrax. While the spline passes through all of the defining points, it meanders between these points, introducing minute error as it goes. This is due to the fact that the spline is only defined by its points, with no attention paid to tangency or curvature at any point.

screenshot of Geartrax sample part screenshot of Geartrax curvature comb
Screen shot of spur gear sample from Geartrax’ web site. Curvature comb of Geartrax tooth profile. (Click image to view full size)

Curvature of an involute

The curvature profile of an involute with respect to its length is asymptotic. It is infinite (zero radius) at the root and approaches (but never reaches) zero as the length increases. The formula is c=1/sqrt(s), where c is curvature and s is length from the root. For details, see 2dcurves.com.

The problem with the Geartrax “involute” is that it does not follow this curvature profile. The curvature is zero at each end. Also, there are inflecctions in the curvature comb. The curvature should always get smaller as the length from the root increases. The Geartrax spline has regions where the curvature increases with distance. the image below shows the Geartrax spline with curvature comb along with a curve showing what the idealized curvature comb should look like.

Geartrax curvature comb vs. ideal
Curvature comb of Geartrax tooth profile (purple) vs. ideal (red line)
A: Ideal curvature is infinite at root, Geartrax spline curvature abruptly reverts to zero
B: Geartrax spline curvature has region where curvature increases as distance from root increases
C: Geartrax spline curvature is zero at endpoint. Ideal curvature approaches but never reaches zero
(Click image to view full size)

Is this a problem?

Probably not. It depends on how much detail you need in your involutes. If you are cutting gear teeth right from CAD data, you are copying the errors. If you are cutting gear teeth with hobs, the error would not be carried through. For most common uses, performance will not be noticeably affected in either case.

I did have one application where this could have been an issue. I was tasked to model a very large gear for a very large press. The gear was to be wire EDM cut right from CAD geometry. The gear was large enough that CAD geometry errors could possibly be detected. Being a former submariner, I appreciate large, quiet, smooth gears.

Still, I am disappointed. Involutes are nothing new, and the mathematics behind them are clear and well-established. Geartrax’ results are a bit ham-fisted when placed next to an ideal involute’s simple elegance.

What you can learn

The big lesson is that there is more to drawing splines than connecting the dots. Many spline control problems are only made worse by adding more points. Controlling tangency and curvature at key points will go a long way toward creating a spline that suits your needs.

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