This revision replaces arbitrary tooth combinations and estimated torque multiplication with one fully declared, assemblable 20/20/60-tooth fixed-ring gearset whose sun, carrier and planet speeds close exactly.
Fixed-Ring Planetary Kinematics Explorer
Enter signed sun speed and elapsed time for a declared 20-tooth sun, four 20-tooth planets and fixed 60-tooth ring. The worksheet reports exact carrier speed, planet spin and accumulated carrier revolutions.
Equation Used
- The declared tooth counts are Ns=20, Np=20 and Nr=60 with equal module.
- The ring is fixed and the sun is the input; the carrier is the output.
- Four identical planets are equally spaced at 90 degrees.
- The assembly condition (Ns+Nr)/4=20 is an integer and the declared planets have circumferential clearance.
- All members are rigid and the animation uses ideal pitch-circle rolling without backlash.
- Torque, efficiency, power loss, load sharing, tooth form, strength, bearings, friction, lubrication, heat, noise, wear, life and safety are excluded.
Replaces arbitrary tooth combinations and unsupported torque multiplication with one assemblable 20/20/60 fixed-ring set and exact Willis speed closure.
Sun, planets, carrier and fixed ring share one exact motion
KHK defines a planetary drive as a sun gear, planet gears, an internal ring gear and a carrier. For equal module, its first geometry condition is Nr=Ns+2Np. The declared 20/20/60 set satisfies that identity exactly.
The animation uses four equally spaced planets. KHK's assembly condition gives (Ns+Nr)/4=20, an integer, so the repeated planet positions are compatible with the tooth pattern.
This page isolates kinematics from capacity
A fixed-ring, sun-input planetary stage provides coaxial speed reduction through the carrier. Actual reducers require tooth-form, material, bearing, carrier, lubrication, tolerance, thermal and load-sharing design that this ideal diagram does not supply.
Use the carrier-relative Willis relation
With the ring fixed, omega_r=0 and omega_c=omega_s Ns/(Ns+Nr). The planet's absolute spin follows the external sun-planet mesh: omega_p=omega_c-(Ns/Np)(omega_s-omega_c).
A declared exact speed example
At 1200 rpm sun speed, the 20/20/60 set has a 4.00:1 sun-to-carrier ratio. Carrier speed is 300 rpm, each planet spins at -600 rpm in the fixed frame, and the carrier completes 50 revolutions in 10 seconds.
Speed closure is not torque or service capacity
The prior worksheet multiplied input torque by an entered efficiency without defining losses, member reactions, tooth loads or hardware. Those torque and performance results are removed.
Unsupported universal ratio ranges, efficiencies, tolerances, backlash bands, temperatures, failure timelines, brand comparisons and product-installation claims are also removed. Use manufacturer data and a complete gear design for real hardware.
Planetary-motion questions
Why is the ring count fixed at 60?
Equal-module geometry requires Nr=Ns+2Np, so 20+2(20)=60.
Why do the planets spin opposite the sun?
The external sun-planet mesh reverses their relative rotation; the carrier motion is included in the reported absolute speed.
Does 4.00:1 imply four times usable torque?
No. This worksheet reports kinematics only.
Does the animation include backlash?
No.
Planetary-gear references
- KHK Technical Information, Internal Gears: planetary tooth-count geometry, equal-spacing assembly and planet-clearance conditions.
- RoyMech, Epicyclic Gear Systems: carrier-relative speed method and fixed-ring sun-to-carrier ratio.
- KHK Steel Internal Gears and Planetary System Examples: fixed internal gear, sun input, carrier output and documented stock tooth-count combinations.
Building or designing a mechanism like this?
Explore the precision-engineered motion control hardware used by mechanical engineers, makers, and product designers.
