Planetary (Epicyclic) Gearset

kinematicstorque-power

Verified build 7 relations · 6 identities proven · 0 modeling steps · 12 parity samples

A planetary (epicyclic) gearset packs three coaxial connection points into one compact, concentric stage: a central sun gear, an internally-toothed ring gear, and a carrier that holds two or more planet gears meshing with both. You have one in every automatic transmission, most EV reduction gearboxes, wind-turbine speed increasers, cordless-drill gearboxes, and bicycle hub gears.

What makes it worth studying first: it is the simplest mechanism that has no single gear ratio. The speed relation (the Willis equation) links three speeds, so the gearset has two kinematic degrees of freedom — you must pick two speeds before the third is determined. Fix the ring and it’s a ~3.5:1 reduction; fix the sun and the same hardware gives a different ratio; fix the carrier and the output reverses; drive two members at once and it adds speeds like a differential. The widget below stores the relation, not a one-way formula — switching configurations just changes which variables you’re allowed to turn.

Two practical notes baked into the relations: the planets are idlers (their tooth count cancels out of the speed law, though it sets the ring size via $N_r = N_s + 2N_p$ ), and an assembled set also needs $(N_s + N_r)$ divisible by the number of equally-spaced planets — with three planets, the default 24/60 set passes: $(24+60)/3 = 28$ .

Try it

Governing relations

\frac{\omega_s - \omega_c}{\omega_r - \omega_c} = -\frac{N_r}{N_s}

Assumes: rigid gears, no backlash or slip; all axes parallel; planets are idlers (their count and speed drop out of the speed relation)

Source: Norton, R. L., Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines, 5th ed., McGraw-Hill, 2012 — ch. 9 (gear trains; epicyclic/Willis analysis).

N_r = N_s + 2 N_p

Assumes: sun, planets, and ring share one module (tooth size) and standard external/internal meshing

Source: Norton, R. L., Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines, 5th ed., McGraw-Hill, 2012 — ch. 9 (gear trains; epicyclic/Willis analysis).

T_s + T_r + T_c = 0

Assumes: static equilibrium of the gearset as a free body (externally applied shaft torques)

Source: Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §13-13 (planetary gear trains: force, torque, and power flow).

T_s\,\omega_s + T_r\,\omega_r + T_c\,\omega_c = 0

Assumes: ideal gearset — no friction losses (real planetary stages run ~96–99% efficient per mesh)

Source: Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §13-13 (planetary gear trains: force, torque, and power flow).

T_r = \frac{N_r}{N_s} T_s

Assumes: follows from tangential force balance at the sun–planet and planet–ring meshes

Source: Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §13-13 (planetary gear trains: force, torque, and power flow).

T_c = -\frac{N_s + N_r}{N_s} T_s

Assumes: combination of the torque balance and the mesh force balance

Source: Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §13-13 (planetary gear trains: force, torque, and power flow).

T_{out} = -T_c

Assumes: action–reaction across the output shaft interface — T_c is the torque the carrier shaft exerts ON the gear set; the gear set exerts −T_c back on the shaft, and that is what the shaft delivers to the load

Source: Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §13-13 (planetary gear trains: force, torque, and power flow).

Derivation

Steps marked modeling step are where physics enters by citation — every other line is machine-proven to follow from them, and the cited models are independently re-derived in the test pipeline where possible. See what is and isn't machine-verified.

N_{s} \left(- \omega_{c} + \omega_{s}\right) = N_{r} \left(\omega_{c} - \omega_{r}\right)

1. Ride on the carrier. In that rotating frame the gearset is an ordinary two-gear train from sun to ring (through the planet idlers), so the relative speeds are locked by the tooth ratio, with a sign flip because an internal mesh reverses once. This is the Willis equation, cross-multiplied. — change to the carrier frame

N_{s} \omega_{s} = \omega_{c} \left(N_{r} + N_{s}\right)

2. Hold the ring still (ω_r = 0) and collect the carrier terms on one side. — apply the constraint and rearrange

\omega_{c} = \frac{N_{s} \omega_{s}}{N_{r} + N_{s}}

3. Solve for the carrier. The carrier always turns the same way as the sun, slower by the factor 1 + N_r/N_s — this configuration is a torque-multiplying reduction. — solve

\frac{\omega_{s}}{\omega_{c}} = \frac{N_{r} + N_{s}}{N_{s}}

4. The speed ratio of the stage. With the default teeth (N_s = 24, N_r = 60) the ratio is 3.5:1 — and notice it depends only on tooth counts you can read off the hardware. — form the ratio

T_{r} = \frac{N_{r} T_{s}}{N_{s}}

5. The same tangential tooth force acts at the sun radius and (through each planet) at the ring radius, and radii are proportional to tooth counts — so the ring reaction torque scales by N_r/N_s. — mesh force balance

T_{c} = - \frac{T_{s} \left(N_{r} + N_{s}\right)}{N_{s}}

6. Equilibrium of the whole gearset: the carrier shaft reacts the sum. Check the ideal power books balance: T_s ω_s + T_c ω_c = 0 exactly when the ring is stationary. — torque balance

How it fails

Planetary stages are compact because three or more planets share the load — and most of their failure modes come from that sharing being imperfect.

Unequal load sharing. Manufacturing errors (planet pin position, tooth spacing) make one planet carry more than its share; design standards apply a mesh load factor rather than assuming a perfect split. Floating suns or flexible pins are common fixes.
Gear tooth bending fatigue and pitting. The same Lewis-bending and Hertzian-contact failure modes as any gear pair (see AGMA 2001 methodology), but planet teeth see more load cycles per revolution — each planet tooth meshes with both sun and ring.
Assembly constraint violations. If $(N_s + N_r)$ isn’t divisible by the planet count, the set physically cannot assemble with equal spacing; forcing unequal spacing changes phasing and vibration behavior.
Carrier and pin deflection. At high torque the carrier windup misaligns planets, concentrating load at tooth edges.
Lubrication starvation in the planet bearings. Planet bearings orbit at carrier speed while spinning on their pins — a hard duty cycle that often limits stage life before the teeth do.

The torque relations in this entry assume an ideal, lossless, rigid gearset — real stages run roughly 96–99% efficient per mesh, and the lost few percent become heat in exactly the components listed above.

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Chains with

Outputs whose SI dimension and quantity kind match another THING's input — the only wires the planner's connectionLegal accepts (invariant 2, computed at build time, not hand-listed). Wire these on the chaining demo.

Fixed-Fixed Torsion Shaft (Interior Torque)
- T_c T
- T_out T
- T_r T
Rectangular Shaft in Torsion (Saint-Venant)
- T_c T
- T_out T
- T_r T
Shaft in Torsion (Solid, Circular)
- omega_c omega
- omega_r omega
- T_c T
- T_out T
- T_r T
Shaft under Combined Bending + Torsion
- T_c T
- T_out T
- T_r T
Thin-Walled Tube in Torsion (Bredt)
- T_c T
- T_out T
- T_r T
Rotating Disk with a Central Bore
- omega_c omega
- omega_r omega
Power Screw (Square Thread)
- T_c T_R
- T_out T_R
- T_r T_R
Spur Gear Pair (Lewis Bending)
- N_r N_g
- N_r N_p
- omega_c omega_p
- omega_r omega_p
- T_c T
- T_out T
- T_r T

+ 8 more THINGs its outputs can legally feed (showing the first 8 in course order).

Sources

Norton, R. L., Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines, 5th ed., McGraw-Hill, 2012 — ch. 9 (gear trains; epicyclic/Willis analysis).
Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §13-13 (planetary gear trains: force, torque, and power flow).