epicyclic gear calculator

What is an epicyclic gear train?

An epicyclic (planetary) gear train is a compact transmission architecture where one or more planet gears rotate around a central sun gear while meshing with an internal ring gear. Because all three major members can rotate, planetary systems give many speed and torque combinations in a small package.

You will find epicyclic gearing in automatic transmissions, EV reducers, robotics joints, industrial drives, aerospace mechanisms, and precision actuators. Their key advantage is flexibility: by holding one member, driving another, and taking output from the third, you can create reduction, overdrive, or reverse behavior.

Core relationship used by this calculator

For a simple planetary set with sun teeth Ns and ring teeth Nr, the fundamental speed equation is:

(ωs - ωc) / (ωr - ωc) = -Nr / Ns

Where:

• ωs = sun angular speed
• ωr = ring angular speed
• ωc = carrier angular speed
• Ns = sun gear tooth count
• Nr = ring gear tooth count

This calculator rearranges that equation depending on the unknown member:

• ωc = (Ns·ωs + Nr·ωr) / (Ns + Nr)
• ωs = ((Ns + Nr)·ωc - Nr·ωr) / Ns
• ωr = ((Ns + Nr)·ωc - Ns·ωs) / Nr

How to use the calculator

Step 1: Enter tooth counts

Provide sun and ring tooth counts. In most practical designs, ring teeth are greater than sun teeth. The calculator also estimates planet teeth as (Nr - Ns)/2, which should typically be an integer for a standard layout.

Step 2: Pick the unknown speed

Select whether you want to solve for carrier speed, sun speed, or ring speed. The corresponding input is disabled and auto-computed.

Step 3: Enter known speeds and calculate

Enter known speeds in rpm, including sign for direction. Click Calculate to see the complete state of the gear set and a consistency check.

Practical engineering notes

• Direction matters: negative values are physically meaningful and indicate reverse rotation relative to your reference.
• Carrier as weighted average: ωc lies between sun and ring speeds weighted by tooth counts.
• Fixed ring case: very common for reduction drives. If ωr = 0, output at carrier gives a reduction from sun.
• Fixed carrier case: behaves like a simple gear pair between sun and ring, with opposite directions.
• Always verify load capacity: speed math alone does not guarantee acceptable stress, lubrication, or thermal behavior.

Worked example

Suppose Ns = 30, Nr = 70, sun speed ωs = 1200 rpm, and ring fixed ωr = 0. Then:

ωc = (30·1200 + 70·0) / (30 + 70) = 360 rpm

So the carrier turns in the same direction as the sun at 360 rpm, giving a reduction ratio of roughly 3.33:1 from sun input to carrier output.

Common mistakes to avoid

• Mixing sign conventions between design notes and test data.
• Using diameters instead of tooth counts in the kinematic formula.
• Ignoring that some tooth combinations are geometrically infeasible for the chosen planet count and spacing.
• Assuming torque scaling without considering efficiency, bearing losses, and load sharing error.

Final thoughts

Epicyclic gear sets are elegant because one compact mechanism can produce multiple motion profiles. Use this calculator for quick kinematic checks, concept selection, and design reviews—then follow up with full mechanical validation for production decisions.