A simple planetary reduction uses a rotating sun gear, planets carried around the sun, and a fixed internally toothed ring. Taking the output from the carrier gives a slower coaxial rotation. The calculator checks that the selected tooth counts fit before reporting an assembled output speed.
Epicyclic / Planetary Gear Train Interactive Calculator
Enter sun, planet and ring tooth counts. Compatible gears animate together; incompatible combinations stay visible as a parts preview.
Equation Used
- Common module and fixed ring.
- Two opposed identical planets.
- Rigid ideal gearing and no losses.
- Illustrative stub profile with simplified roots.
Two opposed planets; illustrative 25° stub involutes. Input speeds are relative multiples, not rpm.
Sun input and carrier output
The gold sun drives the blue planets. The orange carrier supports their pins and turns more slowly than the sun. The surrounding ring is fixed. A planet therefore spins on its own pin while its centre orbits with the carrier.
Both meshes use the same tooth size, or module. The sun-to-planet centre distance must equal the ring-to-planet centre distance. This requires ring teeth = sun teeth + twice the planet teeth.
Two opposed planets are illustrated. The tooth phases are calculated for both meshes, so both meshes follow the same relative-speed constraints. Valid integer counts meet the phase condition for this opposed arrangement.
Explore tooth counts and reduction
Adjust the three tooth counts independently to see whether they assemble. Sun speed is entered as a multiple of a common reference speed. The carrier result uses that same reference; the control does not imply rpm.
Changing planet count alone generally breaks the radial fit. In that case the animation shows separated parts and identifies the mismatch. An incompatible combination cannot produce an assembled-drive speed result.
Real multi-planet gearboxes also require appropriate tooth profiles, pin supports, bearings and load distribution. This two-planet kinematic example does not predict equal torque sharing or select production components.
Use the relative-speed equation
For sun, ring and carrier speeds ωs, ωr and ωc, the simple-train relation is Ns(ωs−ωc)+Nr(ωr−ωc)=0. With the ring fixed, ωr=0, so ωc=ωs/(1+Nr/Ns). Sun and carrier turn in the same direction.
The reduction is i=1+Nr/Ns, and carrier speed as a percentage of sun speed is 100/i. The planet-tooth mismatch is Np−(Nr−Ns)/2. A zero mismatch is required for the same-module geometry shown.
The absolute planet speed is ωp=ωc−(Ns/Np)(ωs−ωc). This same relationship drives the animation. Tooth outlines use an illustrative 25° pressure angle and 0.75-module addendum stub form, with simplified roots. They are not the standard stock-gear profiles in the reference catalog.
20-tooth sun, 30-tooth planets and 80-tooth ring
The fit condition is 80=20+2×30. Reduction is 1+80/20=5:1. At a sun input of 5× reference speed, the carrier turns at 1× reference speed, or 20% of the sun speed.
If only the planet input changes to 31 teeth, the mismatch is +1 tooth. The same ring and sun cannot enclose that planet at a common module. The display keeps the parts visible and asks for compatible counts.
The animated inspection speed is normalized at high settings; relative member speeds remain correct. Pause and use the cycle slider to inspect engagement.
Assembly is separate from strength
A correct radial fit and a valid speed ratio do not certify a gearbox. Internal gearing can encounter involute, trochoid or assembly interference. Tooth strength, contact ratio, backlash, bearing loads and manufacturing tolerances require more information than tooth counts alone.
The renderer uses pitch circles when the illustrative stub profile falls outside its geometric display checks. The pitch-circle view shows kinematics without specifying a production tooth shape.
The calculation concerns geometry and relative speed. Tolerances, bearing life, wear and application suitability require a complete gearbox specification.
Planetary-gear questions
Why does changing planet teeth remove the speed result?
The chosen sun and ring fix the space available for the planet. An incompatible count cannot form the displayed same-module gearbox.
Does the planet affect the reduction?
For this fixed-ring arrangement, the ratio uses sun and ring counts. The planet count must nevertheless satisfy the fit equation.
Why show only two planets?
Two opposed planets make both engagements visible. Other planet counts require their own spacing and assembly-phase checks.
Are these stock gear profiles?
No. The stub involutes are illustrative outlines with simplified roots. This is not a machining drawing or product selector.
Reference
- KHK Internal Gears — Technical Information — planetary tooth-count conditions, relative rotation and internal-gear interference. The catalog’s standard product profiles differ from the illustrative stub teeth drawn here.
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