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