Epicyclic Gear Train

Take a planetary set and identify sun, planets, carrier and ring.

Count teeth on the sun (Z_s) and the ring (Z_r). Check the geometry holds: Z_r = Z_s + 2 × Z_planet.

If that identity fails, the parts do not belong together — the planets cannot sit between sun and ring.

Clamp the ring. Drive the sun, take output from the carrier.

Predict first: ratio = 1 + Z_r/Z_s. With Z_s = 24 and Z_r = 72, that is 4:1.

Now turn the sun four times and confirm the carrier turns once. Same direction as input.

Now clamp the carrier instead. Drive the sun, take output from the ring.

Predict: ratio = −Z_r/Z_s = −3:1 for the same set.

Confirm the magnitude, and note the minus sign is real — the ring turns backwards. The planets are now simple idlers reversing direction, and you have obtained a reverse gear by changing which member is held, not by adding any parts.

That is exactly how an automatic gearbox produces reverse.

Clamp any two members to each other — sun to carrier, say — and drive the assembly.

Expect the whole train to rotate as one lump: 1:1, direct drive.

Three configurations, three ratios including a reverse and a direct, from one set of castings. Stack two such sets and you have the four to six ratios of a conventional automatic gearbox.

Load the output and note how many planets are transmitting: all of them, simultaneously.

Three planets means roughly a third of the tooth force on each mesh, so the same torque needs teeth about a third the width of a single pair.

Then check the shaft: with planets equally spaced, their radial reactions cancel, so there is almost no side load on the bearings. A single spur pair pushes its shafts apart hard and needs bearings sized for it. This is why planetary gearboxes are so compact — it is not clever teeth, it is symmetry.

Older than the industrial revolution's machinery, and used to dodge a patent. James Watt's sun-and-planet gear (1781) converted his beam engine's reciprocating motion to rotation using epicyclic gearing. The straightforward solution was a crank — patented by James Pickard — so Watt's engineer William Murdoch worked around it. One of the clearest cases in engineering history of a patent forcing an alternative into existence, and it is generally credited to Murdoch rather than Watt.

The Antikythera mechanism. The 2nd-century BC Greek device is now widely held to contain epicyclic gearing used to model the Moon's variable motion. That interpretation rests on reconstruction from a heavily corroded object and is not universally settled — worth stating as the well-supported reading it is, not as plain fact.

Why automatics are built this way. A planetary set changes ratio by restraining a member, which a brake band or clutch can do smoothly while everything is turning. A conventional gearbox changes ratio by sliding gears in and out of mesh, which cannot be done under load. That difference — not the torque converter — is why automatic gearboxes are planetary.

Where else it hides. Hub gears in bicycles, cordless drill reduction stages, wind-turbine gearboxes, aircraft turbofan reduction drives. Anywhere torque density and coaxial shafts matter more than efficiency.

The honest limitation. More meshes than a simple pair means more losses — a planetary stage typically runs 95–98% against 97–99% for a good spur pair. And it demands accurate concentricity: if the ring, sun and carrier are not truly coaxial the planets share load unequally, and the whole advantage in step 5 evaporates.