This form 4 calculator uses the carrier as input, the sun as output and a fixed ring. It increases speed by 1+ring/sun while reducing ideal output torque. The diagram generates compatible integer tooth counts from the ratio control.
Epicyclic Train Form 4 Interactive Calculator
Set the ring-to-sun ratio and carrier speed. Watch the carrier drive two planets around a fixed ring and turn the sun faster in the same direction.
Equation Used
- Common module and fixed ring.
- Two opposed planets.
- Ideal rigid gearing without losses.
- Illustrative tooth outlines with simplified roots.
Two opposed planets; illustrative 25° stub tooth outlines. Playback normalized for inspection.
Drive the carrier to speed up the sun
The orange arm is the input carrier. Its blue planets roll inside the stationary ring while meshing with the gold sun. The sun turns faster than the carrier in the same direction.
The ratio slider generates an illustrative 20-tooth sun. Ring teeth are 20 times the selected ratio, and planet teeth are half the difference between ring and sun. The slider’s 0.1 increments therefore produce integer tooth counts for this example.
The illustration has two opposed planets with correct relative tooth phases. Member speed ratios remain correct when high playback settings are slowed for inspection.
Compare speed multiplication
Use the ring-to-sun ratio to explore the ideal trade between speed and torque. Carrier speed is expressed as a multiple of a common reference, and the sun output uses that same reference.
The fixed ring supplies reaction torque. It does not supply power because it does not rotate. Increasing sun speed does not create power; the ideal available output torque falls by the reciprocal factor.
The form 5 calculator describes the same power-flow arrangement using separate sun and ring tooth-count controls.
Calculate the step-up
For ratio r=Nr/Ns, the fixed-ring relation gives ωs=(1+r)ωc. The step-up factor is 1+r. Ideal power balance gives Tout/Tin=1/(1+r).
Input and output turn in the same direction. Their opposing applied torques leave a ring reaction with magnitude Tin−Tout=Tin×r/(1+r). It is not r times the carrier input torque.
The illustrative tooth set uses Ns=20, Nr=20 r and Np=(Nr−Ns)/2. The 25° stub outlines have 0.75-module addendum and simplified roots; they are not a production profile selection.
Ring-to-sun ratio 4:1
A ring/sun ratio of 4 generates 20 sun teeth, 80 ring teeth and 30 teeth on each planet. A carrier speed of 1× reference produces a sun speed of 5× reference. Ideal sun torque is 0.2 times carrier input torque.
At 3× carrier speed, the same geometry produces 15× sun speed without changing the torque fraction. At ring/sun ratio 10, the speed multiplication is 11 and the ideal torque fraction is 1/11.
Keep kinematics separate from production design
Correct centre distances and speed ratios do not establish tooth strength, acceptable backlash, bearing life or manufacturability. The example uses two opposed planets and illustrative stub involutes; low-count cases outside its profile checks use pitch circles.
Operating limits, tolerances and wear depend on the specified gears and application. A detailed gearbox requires its own tooth geometry, interference, material, load and bearing calculations.
Form 4 questions
Does the sun rotate backwards?
No. It turns in the same direction as the carrier in this fixed-ring arrangement.
Why does the diagram show 20 sun teeth?
That chosen example lets every ratio-slider step produce compatible integer counts.
What does the fixed ring do?
It provides the reaction that constrains the planet motion. Its speed and mechanical power are zero.
How does form 5 differ?
On these pages it uses the same power flow, but accepts individual tooth counts instead of the ring/sun ratio.
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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