How can one gearset give different speeds?

Follow one planet through spin and orbit, then change the fixed member to compare speed, direction and ideal torque.

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Preparing the experiment…

0:00Four parts, one gearset
Gold in the middle: the sun gear. Three planets surround it, inside a ring with inward-facing teeth. The carrier holds their axles. Slide the parts together before turning the input. Each planet must mesh with both neighbors at once.
0:20One planet, two motions
Follow the planet with the gold mark. Its axle travels around the sun, while the gear itself spins. The stationary ring gives it an inner track to roll along. The teal carrier follows the orbiting axle, not the planet’s own spin. Those are two different motions.
0:41Count one output turn
Start counting from the marks. With the ring fixed, the sun makes three and a half turns while the carrier completes just one. The two counters keep that proportion throughout. Our twenty-four-tooth sun and sixty-tooth ring give a three-point-five reduction, with the output turning in the same direction.
1:03Slower, with more turning force
A slower output can push harder. Ignore friction for this ideal comparison: one unit of input torque becomes three and a half at the carrier. But output angle is only one part in three and a half. Torque times angle still gives the same work. Real gears also lose some energy.
1:27Hold a different member
Stop and choose another fixed member. This time the sun stays still and the ring supplies the input. Both gear contacts remain engaged, but the constraint has changed. The carrier follows in the same direction at a different reduction: one point four ring turns now give one carrier turn.
1:48Reverse the output
Hold the carrier. The axles stay put, and the planets spin in place. The outside mesh reverses rotation once; the inside mesh keeps that direction. The ring therefore turns opposite to the sun. With these teeth, two and a half input turns give one reverse output turn.
2:10Turn together, without relative motion
Now couple the sun and ring so that they turn together. The carrier must match their speed. Our marked planet still moves with the assembly, but it no longer spins relative to the carrier. One input turn gives one output turn: the gearset behaves as a single rotating unit.
2:32The constraint selects the result
Compare one input turn in each arrangement. Fix the ring, the sun or the carrier, and the input and output roles change with it. Tooth counts matter, but so do those roles and constraints. Choose a fixed member yourself, then turn the input slowly and follow the marked planet.

A gear can turn while its axle travels

The center gear is the sun, surrounded by smaller planet gears. A planet spins around its own axle while its carrier moves that axle around the sun.

An outer ring has teeth facing inward. Hold the ring, sun or carrier still and drive another member: the remaining parts respond in different ways.

Follow the planet with the gold mark. Separate its own spin from its axle’s orbit, then change which part is held still and compare the output speed and direction.

Change the constraint, keeping the same teeth

Follow the gold mark. Separate a planet’s own spin from its axle’s trip around the center.
Hold a different part still. Give the same input turn and compare output speed and direction.

Tooth counts and constraints select the speed

Look closer: the full process and its parts

Start with the gold sun gear. Three planet gears surround it, inside a ring whose teeth face inward. Their axles attach to a single carrier. When that carrier turns, all three axles travel around the sun.

Follow the green planet with the gold mark. The gear spins around its own axle while its axle orbits with the carrier. Tracking only one tooth makes these two motions easy to confuse. With the ring held stationary and the sun driven forward, the carrier advances slowly while the planets spin backward.

The gearset does not have one inherent reduction ratio. First choose the stationary member, input and output. Holding the ring, sun or carrier selects different speeds—and can reverse the output direction.

Detailed controls

Hold the ring and turn the sun through 1260°, or 3.5 revolutions. The carrier completes exactly one revolution.
Hold the sun and turn the ring through 504°, or 1.4 revolutions. The carrier completes one revolution.
Hold the carrier and turn the sun through 900°, or 2.5 revolutions. The ring completes one revolution backward; the planet axles no longer orbit.
Lock the set together. All central members turn at the same speed. The marks still travel around the center, but there is no relative rotation between gears.

Choose Explore to select a constraint and move the input-angle slider, including with the keyboard. Each constraint starts from the same assembly phase. The 2D cutaway retains the same tooth profiles, angles and controls.

Why do these gears fit?

The sun has 24 teeth, each planet has 18 and the ring has 60. Equal module makes pitch radii proportional to tooth counts. Concentric assembly requires:

Nr = Ns + 2Np = 24 + 2 × 18 = 60

For three equally spaced planets, (Ns + Nr) / 3 must also be an integer: here it is 28. Neighboring planet tip circles must clear each other. The displayed teeth use 20° involute flanks. Internal teeth extend inward; external teeth extend outward. The internal profile is not simply an external gear with its colors reversed.

A small tooth-thickness clearance is included and root transitions are simplified. Motion follows ideal rolling constraints; finite-backlash impacts and reversal lost motion are not simulated. This geometry is not a manufacturing drawing.

How do spin and orbit enter the same equation?

Let ωs, ωr and ωc be the angular velocities of the sun, ring and carrier. In the carrier’s frame, the planet axles are stationary. The two meshes give:

Ns(ωs − ωc) + Nr(ωr − ωc) = 0

Holding the ring gives ωc/ωs = 24/84 = 2/7. Holding the sun gives ωc/ωr = 60/84 = 5/7. Holding the carrier gives ωr/ωs = −24/60 = −0.4; the minus sign means reversal. Constrain any two central members to turn together and the third must follow at the same speed.

A planet’s absolute spin is ωp = ωc − (Ns/Np)(ωs − ωc). Positions, marks and revolution counts share these constraints, allowing any playback position to be reconstructed directly.

Why can a slower output deliver more torque?

For ideal steady transmission with one input, one output and no losses, input and output power are equal: τin ωin = τout ωout. With the ring stationary, output speed is 1/3.5 of input speed, so output torque magnitude can be 3.5 times input torque. The output travels through a smaller angle; no energy is created.

The stationary member carries a reaction torque, but does no work because its angular velocity is zero. A real assembly also has friction, inertia, bearing losses, elastic deformation and uneven load sharing. This model omits those effects and does not simulate a gearbox clutch changing state. The opening axial separation identifies the components; transmission begins only after the teeth return to their assembled positions.

A gear can turn while its axle travels

Change the constraint, keeping the same teeth

Tooth counts and constraints select the speed

Detailed controls

Further reading & sources

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