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Elliptical gear calculations
*To*: mathgroup at smc.vnet.net
*Subject*: [mg110626] Elliptical gear calculations
*From*: dean k <deaken.psu at gmail.com>
*Date*: Tue, 29 Jun 2010 06:59:06 -0400 (EDT)
Through two identical elliptical gears rotating about their centers, I
would like to calculate the torque and angular velocity ratios as a
function of the rotation angle of one of the gears. I'm not a
mathematician, but an engineer working on a personal project I thought
would be interesting. It is quickly becoming more interesting than I
expected
My first attempt to do this went like this:
Calculate the radius of 1 gear for a given angle theta; call this R1
Assume the other radius (R2) must be a+b-R1
Calculate ratios using R1 and R2 as if at that instant these were 2
circular gears of radii R1 and R2
The above solution created strange problems including details where
2pi radians did not appear to take me around 1 revolution of an
elliptical gear. I was also concerned that torque might not actually
be equal to Force*radius since the torque vector might not be
perpendicular to the radius.
So I started looking around wikipedia and quickly got bogged down in
elliptical integrals and material frankly above my head. I was hoping
to use Mathematica to do the difficult work for me as I saw
Mathematica has elliptical integral functions, but alas Mathematica is
designed to help people who know what they are doing, aka not me.
Certainly the documentation makes that assumption.
So my questions are:
Am I even on the right track, or is there a simpler way to do
this? An approximation with 99% accuracy would be perfectly
acceptable for my needs.
Can I use Mathematica to calculate the solution, and if so, how?
Thank you for any help,
Dean
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