An Epicyclic Train form 5 is a planetary gear arrangement where the ring gear (annulus) is held fixed, the planet carrier is the input, and the sun gear is the output. The carrier — the rotating arm that holds the planets — is what defines this form, because it forces the planets to walk around the stationary ring while spinning the sun faster than itself. We use form 5 when you need a compact step-up from a slow shaft, like a hand crank or a low-RPM motor driving a tachometer, flywheel, or generator. Typical ratios run from about 1:3 to 1:11 in a single stage.
Epicyclic Train Form 5 Interactive Calculator
Vary the fixed ring teeth, sun teeth, and carrier speed factor to see the step-up ratio, sun output speed, required planet tooth count, and ideal torque tradeoff.
Equation Used
This calculator applies the fixed-ring Willis relation for an epicyclic train form 5. The ring tooth count Zr and sun tooth count Zs set the speed multiplication; the carrier speed is then multiplied by that ratio to estimate the sun output speed.
- Ring gear is fixed and the carrier is the input.
- Sun gear is the output and rotates in the same direction as the carrier.
- Planet gears are idlers for the speed ratio.
- Planet tooth output uses Z_planet = (Z_ring - Z_sun) / 2 for standard mounting.
- Torque factor is ideal and neglects friction and bearing losses.
Operating Principle of the Epicyclic Train (form 5)
Form 5 inverts what most people picture when they hear "planetary gearbox." In a normal reduction, the sun is the input and the carrier is the output. In form 5 we flip that — the carrier drives, the sun is driven, and the ring stays bolted to the housing. Because the planets are forced to roll around a fixed annulus, every degree the carrier turns, the planets spin on their own pins, and that spin gets multiplied through the sun. The result is a step-up. Slow input, fast output.
The ratio comes straight from the Willis equation. With the ring fixed, the speed of the sun relative to the carrier is set by the tooth counts: ωsun / ωcarrier = 1 + Zring / Zsun. So a 60-tooth ring and a 12-tooth sun gives you 1:6 step-up. Simple. The planet tooth count never appears in the ratio — the planets are just idlers in the speed equation, though they absolutely matter for load sharing and meshing geometry.
Get the geometry wrong and the train binds or hammers. The mounting equation Zring = Zsun + 2 × Zplanet must be exact — not approximate. If Zring is one tooth off, the planets won't sit on a common pitch circle and you'll feel a tight spot once per carrier rev. Backlash also stacks: in form 5, sun-side backlash gets multiplied by the ratio when reflected to the input. A 0.05 mm backlash at the sun feels like 0.3 mm of lash at the carrier on a 1:6 step-up. That's why a carrier-driven planetary used as a tachometer drive will read jittery if the planet bearings have any radial play above about 20 µm.
Key Components
- Sun Gear (output): Central spur gear on the output shaft. In form 5 this is the fastest-spinning member. Tooth counts typically run 10-20 teeth — go below 10 and undercut starts wrecking the contact ratio. Module is usually 0.5-1.5 for desktop builds, 1.5-3 for industrial.
- Planet Gears: Three or four idler gears riding on pins fixed to the carrier. They mesh simultaneously with the sun and the ring. They share the load — three planets gives ~85% load sharing in practice because of manufacturing tolerance, four gives ~75%. Equal spacing of 120° (or 90° for four) must hold within ±0.05° or load distribution collapses.
- Planet Carrier (input): The rotating frame that holds the planet pins. In form 5 this is what you drive. The carrier must be stiff — any flex of the planet pins under load shifts mesh contact and causes audible whine above 2000 RPM output. Pin parallelism within 0.02 mm over a 50 mm span is the spec we hold.
- Ring Gear / Annulus (fixed): Internal-tooth gear bolted to the housing. Because it's stationary, it carries reaction torque equal to (ratio − 1) times input torque. On a 1:6 step-up driven at 5 N·m input, the ring sees 25 N·m reaction — your housing bolts and pilot fit must take that without flexing.
- Bearings on Carrier and Sun: Carrier bearings see input speed. Sun bearings see output speed — meaning at a 1:8 step-up they spin 8× faster than the input shaft. We size sun bearings for the actual output RPM, not the input RPM, which catches a lot of first-time designers out.
Where the Epicyclic Train (form 5) Is Used
Form 5 shows up anywhere you need to step up rotation in a small package. Hand-cranks, anemometers, tachometer drives, small generators, centrifuges — any time the input shaft is slow and the output needs to spin fast without a belt or a chain. It's also common as the second stage of a compound planetary where the first stage reduces and the second stage steps part of it back up to drive an auxiliary like a cooling fan or oil pump.
- Renewable Energy: The auxiliary tachometer drive inside a Bergey Excel 10 small wind turbine nacelle, stepping rotor RPM up about 1:8 to feed the speed sensor.
- Laboratory Equipment: Step-up stage in a Hettich EBA 200 benchtop centrifuge — slow motor input, 6000 RPM rotor output, single planetary stage handling part of the ratio.
- Hand-Powered Generators: Crank-to-alternator step-up in a BioLite charger-style emergency generator, taking ~60 RPM crank input to ~480 RPM alternator input.
- Marine Instruments: Anemometer cup-shaft to encoder shaft step-up inside a Davis Vantage Pro2 wind sensor, where the cups rotate slowly but the encoder needs higher resolution per unit time.
- Robotics: Auxiliary fan drive on a custom FRC robot gearbox where the main reduction drives the wheels and a form-5 stage off the carrier spins a 12,000 RPM cooling fan.
- Watchmaking & Instruments: Seconds-hand step-up in a complicated chronograph movement where the going train delivers slow rotation and a small epicyclic stage multiplies it for the sweep hand.
The Formula Behind the Epicyclic Train (form 5)
The ratio formula tells you how fast the sun spins relative to the carrier when the ring is locked. At the low end of practical tooth counts — say a 36-tooth ring with a 12-tooth sun — you get a modest 1:4 step-up that's easy to manufacture and runs quietly. At the high end, a 96-tooth ring on a 10-tooth sun gives nearly 1:11, but the small sun is now the weak link: tooth root stress climbs and the sun bearing sees output speeds where lubrication regime matters. The sweet spot for general use sits around 1:5 to 1:7, which keeps the sun above 12 teeth and the ring under 80 teeth.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| ωsun | Angular velocity of the sun gear (output) | rad/s or RPM | RPM |
| ωcarrier | Angular velocity of the planet carrier (input) | rad/s or RPM | RPM |
| Zring | Tooth count of the fixed ring gear (annulus) | teeth (count) | teeth (count) |
| Zsun | Tooth count of the sun gear | teeth (count) | teeth (count) |
Worked Example: Epicyclic Train (form 5) in a benchtop yarn-twist tester step-up drive
Sizing the form-5 step-up between the hand-crank shaft and the twist counter spindle on a benchtop yarn-twist tester similar to a James Heal Tortion 200. The operator turns the crank at roughly 80 RPM, and the spindle needs to read out twist at around 480 RPM to match the marked dial scale. We've picked a 60-tooth ring and a 12-tooth sun. Three 24-tooth planets fit the mounting equation. We need to confirm the actual output RPM at the operator's typical cranking range of 40-120 RPM and figure out whether the sun bearing is sized for the high end.
Given
- Zring = 60 teeth
- Zsun = 12 teeth
- Zplanet = 24 teeth
- ωcarrier,nom = 80 RPM
- Operating range = 40 to 120 RPM
Solution
Step 1 — confirm the planets actually fit. The mounting equation must hold exactly:
Step 2 — compute the step-up ratio from the Willis equation:
Step 3 — at the nominal cranking speed of 80 RPM, the spindle output is:
That matches the dial scale. The operator gets the right reading at a comfortable cranking pace.
Step 4 — at the low end of the operator range, 40 RPM crank, the spindle runs at:
Slow, but the test still works — twist counts integrate over time so a slower test just takes longer. No mechanical issue, just operator patience.
Step 5 — at the high end, 120 RPM crank:
Now we need to check the sun bearing. A typical 8 mm bore deep-groove ball bearing (e.g. 608ZZ) is rated for grease-lubricated continuous use up to about 24,000 RPM, so 720 RPM is nowhere near a problem. But if the operator hammers the crank above about 150 RPM, you'll start to hear planet-mesh whine because the sun is now spinning at 900+ RPM and the 12-tooth sun has a meshing frequency of 180 Hz — right where small housings resonate.
Result
At nominal 80 RPM crank input, the spindle reads 480 RPM — exactly what the dial expects. Across the operator's natural range of 40 to 120 RPM, the spindle covers 240 to 720 RPM, and the whole range stays inside safe bearing speed for a standard 608ZZ. The sweet spot sits around 70-90 RPM crank input, fast enough to finish a test in reasonable time, slow enough that mesh whine stays inaudible. If the measured spindle RPM comes back low — say 450 instead of 480 at 80 RPM crank — check three things in this order: (1) the ring gear is actually clamped to the housing and not slipping under torque, which makes the ratio drop toward 1:1, (2) carrier pin parallelism is within 0.02 mm because skewed pins let planets back-drive on the ring tooth flanks, and (3) the sun shaft pinned-to-output coupling isn't slipping under the higher output torque seen at low input speed.
Choosing the Epicyclic Train (form 5): Pros and Cons
Form 5 is one option among several when you need a compact step-up. The honest comparison is against a parallel-shaft spur gear pair — the most common alternative — and a belt-and-pulley step-up, which is what most prototype builders reach for first.
| Property | Epicyclic Train Form 5 | External Spur Gear Pair | Timing Belt Step-Up |
|---|---|---|---|
| Single-stage ratio range | 1:3 to 1:11 | 1:1 to 1:6 practical | 1:1 to 1:5 practical |
| Input/output shaft alignment | Coaxial (shafts on same centreline) | Parallel offset | Parallel offset, larger centre distance |
| Volumetric envelope for given ratio | Smallest — typically 40-60% of spur pair volume | Medium | Largest — needs centre-distance space |
| Load capacity at given size | High — load shared across 3-4 planets | Low to medium — single mesh | Low — belt tension limit |
| Backlash reflected to input | Moderate — multiplied by ratio | Low — single mesh | Low to negligible (toothed belt) |
| Manufacturing cost (one-off) | High — needs internal ring gear | Low | Lowest |
| Noise at output >2000 RPM | Quiet if planets are matched within 0.01 mm runout | Moderate | Quiet |
| Tolerance to misalignment | Low — needs precise carrier | Medium | High |
Frequently Asked Questions About Epicyclic Train (form 5)
Because in form 5 the ratio works against you on backlash. Any clearance at the sun mesh gets multiplied by the ratio when you reflect it back to the carrier input. A 0.05 mm circumferential backlash at the sun teeth shows up as roughly 0.3 mm of carrier lash on a 1:6 step-up.
On top of that, planet pin clearance and carrier flex add directly. If you measured 0.5° of input lash and the gear-only prediction is 0.15°, the missing 0.35° is almost always pin-to-carrier-bore slop. Press-fit those pins or use a shoulder-bolt design with a clamped fit, not a slip fit.
Three planets self-balance better. With three, the planet positions form a triangle that geometrically averages out manufacturing variation, so you typically get 80-85% load sharing in real hardware. Four planets only achieve 70-75% load sharing because two opposing planets carry most of the load and the other two ride along.
Pick four planets only when you need the extra load capacity AND you can afford ground gears with sub-5 µm tooth-to-tooth variation. For a hand-cranked or low-torque step-up, three planets is almost always the right answer.
You're probably measuring under load and seeing torsional wind-up plus bearing drag, not a kinematic error. The Willis equation gives you the kinematic ratio assuming rigid bodies. Real systems have shaft twist, mesh deflection, and bearing preload drag that subtract a few percent of effective output.
Spin it unloaded with the input driven at constant RPM and measure the output — that should match the calculation within 0.5%. If it does, your loaded 5% loss is normal stiffness and friction, not a gearing fault. If the unloaded test still reads low, then the ring gear is rotating slightly and you have a housing-clamping problem.
Technically yes, practically rarely a good idea. Steppers already lose torque rapidly above their corner speed, and a form-5 step-up multiplies the output speed but divides the available torque. A 1:6 step-up on a NEMA 17 stepper running at 600 RPM gives you 3600 RPM output but with about 1/6 the already-falling torque.
Use form 5 when the input is mechanical and slow — hand cranks, wind cups, low-RPM motors with high stall torque. For steppers, drive the load directly or use a step-down to get more torque, not less.
You almost certainly have a mounting-equation violation that's only obvious after the planets walk around once. Re-check Zring = Zsun + 2 × Zplanet with the actual tooth counts on the cut gears, not the nominal CAD numbers. A common build error is grabbing a 25-tooth planet instead of a 24, which makes the equation 60 vs 62 — close enough to assemble at one position but binds when the planets index forward.
The other cause is planet phasing. If your three planets aren't all at the same tooth-engagement phase relative to the sun, they'll fight each other. With three equally-spaced planets on a 60-tooth ring, the ring tooth count must be divisible by 3 for clean phasing — 60 works, 61 wouldn't.
The sun bearing in a form-5 step-up runs at the output speed, which is the fastest shaft in the assembly. On a 1:8 ratio, 200 RPM input means 1600 RPM at the sun bearing — and bearing power loss scales roughly with speed squared at constant load.
Most first-time builders size the sun bearing for input speed and find it heats up under continuous duty. Use a higher-speed-rated bearing on the sun (a precision class deep-groove or a small angular contact), and consider grease with a higher base oil viscosity index. Drop the preload by 20-30% from what you'd use on the input side.
Pin shear is the usual failure mode in compact form-5 builds, not gear teeth. With three planets sharing load at ~85% efficiency, each pin carries roughly (Tinput / 3 / 0.85) divided by the planet pitch radius as a tangential force. For a 5 mm steel pin in double shear, you're typically good to about 3-4 N·m input torque before pin deflection lets the planets skew.
Above that, step up to 6 mm pins or, better, support both ends of each pin with a closed carrier (a plate on each side rather than a cantilever). Closed carriers roughly double the practical torque limit at the same pin diameter.
References & Further Reading
- Wikipedia contributors. Epicyclic gearing. Wikipedia
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