Presets
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Gear set
A planetary set has three coaxial members, sun, ring and carrier. Hold one, drive another, and the third is the output. The ring has Zr = Zs + 2 Zp teeth, so changing the tooth counts changes every ratio.
Held member and drive
Model checks
The science
What is modelled
One sun gear (Zs teeth), several planets (Zp teeth) on a carrier and an internal ring gear with Zr = Zs + 2 Zp teeth. Meshing forces the surface speeds at the sun-planet and planet-ring contacts to match, which gives one kinematic constraint between the three coaxial speeds.
Equations
- Willis:
Zs (ωs - ωc) + Zr (ωr - ωc) = 0. - Planet spin:
ωp - ωc = -(Zs/Zp)(ωs - ωc). - Ideal torques:
Ts : Tr : Tc = Zs : Zr : -(Zs + Zr). - Assembly:
(Zs + Zr)/Nmust be an integer for N equally spaced planets.
Checks run in the page
- Ratios 3:1, 1.5:1 and -2:1 for 24/12/48 teeth.
- The Willis relation for several tooth combinations and held members.
- Net power is zero for the ideal torque split.
- Integrated tooth counts passing the mesh.
Honest simplifications
Ideal, lossless meshing with rigid teeth: no efficiency loss, backlash, load sharing between planets or tooth strength. Gear teeth are drawn schematically and are not true involute profiles.
What to try
- Press the three presets to compare 3:1, 1.5:1 and the reversing -2:1 set.
- Press Assembly fails: with 24 + 48 teeth, 72 / 5 is not an integer, so five equally spaced planets cannot mesh; three, four or six can.
- Increase sun teeth with ring held: the reduction ratio falls towards 1.