Planetary Gear Simulator & Ratio Calculator

Epicyclic gear train lab • hold, drive or free the sun, carrier and ring • real machines — Simulate • Machines • Explore • Practice • Quiz

Mode

Gear Trains

Gearbox

Gear Strength 📖 User Guide

Display Controls

Labels & values Speed arrows Power flow Pitch circles Planet path trace

📖 Understand what you see

Σ Live equations — your numbers substituted

▦ Tabular method — the textbook table for this set-up

💡 What-if coach — predictions to test

User Guide — Planetary Gear Simulator

1 What this planetary gear simulator does

This is a free planetary gear simulator and planetary gear ratio calculator. It runs in any browser, with no download or login. The canvas shows a bench planetary gear trainer in two views of one machine:

View A, front cover removed: the real involute sun, planets and ring, with the planet carrier and its pins.
Section: the three concentric members on one axis. Each one ends in a station that shows its job: a motor (driving), a clamped brake with a red lamp (held), a load dynamometer (output) or nothing (free). Each station has its own tachometer.

Five modes: Simulate (the trainer), Machines (four real epicyclic machines on the same engine), Explore (16 concept cards), Practice (12 problem types) and Quiz (5 questions from a pool of 20).

2 How to use it in four steps

Pick what is held. The tiles above the canvas are the six one-member-held arrangements and two specials: Two inputs (a differential) and Direct drive (two members clutched).
Change the gears. Set the sun and planet teeth; the ring follows from the coaxial condition Nr = Ns + 2Np. Choose the number of planets, the module, the input speed and torque, and the two mesh efficiencies.
Press Run. The motor starts and every member turns at its true speed ratio. Real speeds are far too fast to follow, so the drawing is slowed down by a factor printed on the canvas.
Switch to Diagram to see the same state as a schematic, a velocity diagram, a lever analogy and the tabular method. Show me How walks through the table step by step.

Keyboard: Space run/stop, D Real/Diagram, H Show me How, R reset.

3 The readings and the planets-fit check

Under the canvas, a one-line status names what is held, what drives and what turns, and in which direction. The tiles show:

the ratio (input speed over output speed; negative means reverse);
the output speed and torque;
the efficiency;
whether the planets fit.

The fit check is the assembly condition: (Ns + Nr)/n must be a whole number. If it is not, the simulator still builds the set, but places the planets at the nearest angles where they can mesh, so they are visibly unequally spaced. If adjacent planets would touch, the neighbour condition fails and the planets that clash are outlined in red.

4 Diagram view: velocity diagram, lever analogy and tabular method

Schematic and velocity diagram: the textbook stick diagram, with ground hatching on the held member. Beside it, the speeds of the sun pitch point, planet centre and ring pitch point lie on one straight line. Where that line crosses zero is the planet's instantaneous centre.
Lever analogy: a vertical lever with the sun, carrier and ring as nodes. The distances are proportional to Nr and Ns. Speeds are drawn as offsets and always lie on a straight line; torques are drawn as forces that balance.
Tabular method: lock everything and turn it +x, then hold the carrier and turn the sun +y, then add. Every number comes from the live engine.

5 Machines: Prius, automatic transmission, bicycle hub, wind turbine

Toyota Prius power split: engine on the carrier, MG1 on the sun, wheels and MG2 on the ring. Set the road speed and the engine speed; MG1's speed and whether it is generating or motoring follow.
Simpson 3-speed automatic: two sets sharing one sun. Pick P, R, N, 1, 2 or 3 to see which clutches and bands apply, and the 2.45 / 1.45 / 1.00 / −2.22 ratios.
Bicycle 3-speed hub: the sun is fixed to the axle; the gears give 0.75 / 1 / 1.333 and a road speed from your cadence.
Wind turbine step-up: rotor on the carrier, ring held, sun to the next stage. It shows the torque each planet carries.

6 Practice, Quiz and Explore

Practice deals unlimited problems from 12 types: ratios, output speed, ring teeth, reverse, two-input carrier speed, planet spin, assembly, output torque, efficiency, held-member torque, Prius MG1 speed and Simpson ratios. Answers within 2 % count, and a worked solution follows. Quiz draws 5 questions from 20 and shuffles the options every time. Explore has 16 concept cards in five groups.

7 SI and Imperial units

The SI / Imperial switch sits in the toolbar after Reset. Speeds stay in rpm in both systems. Torque switches N·m ↔ lbf·ft, power kW ↔ hp, force N ↔ lbf, length mm ↔ in, and road speed km/h ↔ mph. In Imperial the gear size is given as a diametral pitch P = 25.4/m (teeth per inch of pitch diameter), as US gear catalogues do. Your choice is remembered across the site.

Planetary Gear Ratio Calculator and Epicyclic Gear Train Simulator

A planetary gear set (also called an epicyclic gear train) has three members on one axis:

a central sun gear;
an internal-toothed ring gear (annulus);
a carrier holding two or more planet gears that mesh with both.

The set has two degrees of freedom, so it has no ratio until one member is held (or two are driven). Hold a different member and the same gears give a different ratio, a reverse or an overdrive. That is why planetary gears are found in automatic transmissions, hybrid cars, bicycle hubs and wind turbines. The simulator above lets you hold, drive or free each member of a working set and read every speed, torque and power.

The Willis equation

Seen from the carrier, the planets spin on fixed pins and the set is an ordinary gear train with ratio −Nr/Ns. That observation is the Willis equation:

(ω_s − ω_c) / (ω_r − ω_c) = −N_r/N_s ⇔ N_sω_s + N_rω_r = (N_s + N_r)ω_c

Any two speeds fix the third. With standard (unshifted) gears the ring must also satisfy the coaxial condition N_r = N_s + 2N_p, so that the sun-to-planet and planet-to-ring centre distances are equal.

Planetary gear ratios for every arrangement

For the simulator's default set (sun 24, planets 18, ring 60 teeth), with input speed ÷ output speed as the ratio:

Held	Input	Output	Ratio formula	24 / 60 set	Use
Ring	Sun	Carrier	1 + N_r/N_s	3.50	Low reduction
Sun	Ring	Carrier	1 + N_s/N_r	1.40	Mild reduction
Carrier	Sun	Ring	−N_r/N_s	−2.50	Reverse
Sun	Carrier	Ring	N_r/(N_s + N_r)	0.714	Overdrive
Ring	Carrier	Sun	N_s/(N_s + N_r)	0.286	High overdrive
Carrier	Ring	Sun	−N_s/N_r	−0.40	Reverse overdrive
Two members clutched	Any	Any	1	1.00	Direct drive

The ring-held arrangement gives the largest reduction from one set. That is why it is the standard industrial planetary gearbox stage, with typical single-stage ratios of 3 to 10.

Worked example: the tabular method

Ring held, sun driven at 1200 rpm, sun 24 and ring 60 teeth. Lock the whole set and turn it +x, then hold the carrier and turn the sun +y:

Step	Sun	Carrier	Ring
All locked, turn +x	x	x	x
Carrier held, sun +y	+y	0	−y·24/60
Total	x + y	x	x − 0.4y

The ring is held, so x = 0.4y. The sun turns at 1200 rpm, so x + y = 1.4y = 1200, giving y = 857.1 and x = 342.9 rpm. The carrier turns at 342.9 rpm, a ratio of 3.5, in the same direction as the sun. Each planet spins at x − y·24/18 = −800 rpm absolute, against the sun.

Torque, power and planetary gearbox efficiency

At steady speed the three external torques must balance: T_s : T_r : T_c = N_s : N_r : −(N_s + N_r). The carrier always carries the largest torque, and the held member carries a reaction torque but does no work. Mesh losses depend on the power flowing relative to the carrier, not on the power through the set. This basic-ratio method gives a well-known result: with a basic-train efficiency of η₀ = 0.98 × 0.99 = 0.970, the ring-held reduction runs at (1 + 2.5×0.970)/3.5 = 97.9 %. That is better than the 97.0 % of the same gears used as a fixed-carrier reverse train, because part of the power passes through the set without sliding over teeth. The Two inputs set-up shows that a set can also split or merge power, which is how a hybrid car and a vehicle differential work.

Assembly and neighbour conditions

Equally spaced planets can be fitted only when (N_s + N_r)/n is a whole number. Sun 24, ring 60 takes 2, 3, 4 or 6 planets but not 5. Adjacent planets must also clear each other at their tips: (N_s + N_p)·sin(180°/n) > N_p + 2. Increase the planet count in the simulator until the planets clash and it outlines them in red. Choose a tooth sum that does not divide and the planets move to unequal spacing, which some real gearboxes use deliberately.

Planetary gears in real machines

Hybrid power split: the Toyota Prius uses a single planetary set (sun 30, ring 78) as an electronically controlled CVT. The generator MG1 on the sun sets the engine speed independently of road speed. Its real pinions have 23 teeth and are profile-shifted, which is why 30 + 2×23 is not 78.
Automatic transmission: the Simpson gear set uses two planetary sets sharing one sun. Two clutches and two bands give 2.45, 1.45, 1.00 and a 2.22 reverse.
Bicycle hub gear: a three-speed hub fixes the sun to the axle and gives 0.75, 1 and 1.333.
Wind turbine: a planetary first stage (about 5.7:1) shares the enormous rotor torque between three or more planets.

Who uses this simulator?

Mechanical and automotive engineering students studying epicyclic gear trains, the Willis equation and the tabular method for exams; diploma and vocational learners on automatic transmissions; teachers who want a projector demonstration of which member is held; and designers who need a quick planetary gear ratio, tooth-count and assembly check.

Frequently asked questions

How do you calculate the gear ratio of a planetary gear set?

Hold one member, drive a second and take the output from the third, then use the Willis equation Ns·ωs + Nr·ωr = (Ns + Nr)·ωc. With the ring held, sun in and carrier out the ratio is 1 + Nr/Ns. With the sun held and the ring driving the carrier it is 1 + Ns/Nr. With the carrier held, sun in and ring out it is −Nr/Ns, a reverse. The other three arrangements are the inverses of these.

What is the Willis equation for an epicyclic gear train?

It states that, seen from the carrier, the set is an ordinary gear train: (ωs − ωc)/(ωr − ωc) = −Nr/Ns. Rearranged it reads Ns·ωs + Nr·ωr = (Ns + Nr)·ωc. Any two speeds fix the third, which is why a planetary set needs one member held (or two driven) before it has a definite ratio.

How many planet gears can a planetary gear set have?

Equally spaced planets fit only when (Ns + Nr) divided by the number of planets is a whole number; that is the assembly condition. The planets must also not touch each other: (Ns + Np)·sin(180°/n) must be greater than Np + 2. The ring must also satisfy the coaxial condition Nr = Ns + 2Np for standard gears.

How does a planetary gear set give reverse?

Hold the carrier. The planets then spin on fixed pins and act as idlers between the sun and the ring, so the ring turns the opposite way to the sun at a ratio of −Nr/Ns. Automatic transmissions get reverse this way, by braking a carrier.

How does the Toyota Prius power split device work?

It is one planetary set: the engine drives the carrier, the generator MG1 is on the sun, and the ring is geared to the wheels and to motor MG2. With 30 sun and 78 ring teeth, 72 % of the engine torque reaches the ring and 28 % is held by MG1. Because the Willis equation leaves one degree of freedom, MG1's speed sets the engine speed independently of road speed, which makes it an electronically controlled CVT.

Explore Related Simulators

For spur, idler, compound and worm trains, use the Gear Train Calculator. For a layshaft manual transmission, open the Gearbox Simulator. To check that the teeth of your planetary set will survive the load, run the planet mesh through Gear Strength (Lewis and AGMA). The Belt & Chain Drive simulator covers the other common way of changing speed.

Updated 10 October 2026