Planetary gearbox design: tooth count rules, gear ratio calculation and 3D printing tips (with valid tooth combinations)

How a planetary gear set works and its ratio

A planetary set has a central sun gear (Zs), N planet gears (Zp) around it, an outer ring gear (Zr) and a carrier holding the planets. Fix one member, drive another and take output from the third to get different ratios; the usual layout fixes the ring, drives the sun and takes output from the carrier.

Willis equation: (ns − nc) / (nr − nc) = −Zr / Zs　→　ring fixed (nr = 0): i = ns / nc = 1 + Zr / Zs

Torque (ideal): Tc = Ts · i　　ring reaction torque Tr = Ts · (i − 1)

The load is shared by N planets at once, which is why planetary gearboxes are compact and torque-dense. The ring, however, reacts nearly the full output torque, so the housing and ring mounting must be stiff. Stages multiply: two 6:1 stages give 36:1. With the ring fixed i = 2 + 2Zp/Zs, so a single stage is always above 2:1.

TIPInput and output are coaxial and turn in the same direction, which is why planetary gearboxes bolt so neatly onto the front of a stepper or servo motor.

Tooth count rules: coaxial, assembly and adjacency conditions

Planetary tooth counts can't be chosen freely. With standard (unshifted) gears, all of these must hold:

Coaxial condition: Zr = Zs + 2·Zp　　centre distance a = m(Zs + Zp)/2 = m(Zr − Zp)/2

Assembly condition: (Zs + Zr) / N = integer

Adjacency condition: (Zs + Zp) · sin(180° / N) > Zp + 2

• Coaxial: the sun–planet and planet–ring meshes must have the same centre distance so each planet meshes with both.
• Assembly: for planets spaced evenly at 360°/N, every planet must drop into a tooth space of both the sun and the ring. If the condition fails, planets can only sit at multiples of 360°/(Zs + Zr), i.e. unevenly spaced.
• Adjacency: neighbouring planet centres are 2a·sin(180°/N) apart and must clear the planet tip diameter m(Zp + 2); for printing leave another 0.5–1 mm.
• Internal interference: keep the ring–planet tooth difference Zr − Zp (= Zs + Zp) at 12 or more at 20° — practical sets satisfy this automatically.
• Undercut: the sun is usually the smallest gear and undercuts below 17 teeth at 20°; 12–16 teeth still work for printing, with weaker roots.

Adjacency caps the single-stage ratio: bigger planets give a higher ratio, but more planets crowd each other. With a 12-tooth sun (adjacency only):

Above about 6:1 you are practically limited to three planets; four to six planets for load sharing usually means 3:1–5:1 per stage. Three planets share load most evenly and are the most common choice.

Valid tooth combinations (ratios 3–10)

Standard sets for m = 1 that satisfy the coaxial, assembly and adjacency conditions — multiply lengths by the module. "Valid N" checks assembly and adjacency (N = 2 omitted); the last column is the gap between neighbouring planet tip circles at the largest valid N.

• 12 / 24 / 60 is the internal gear generator's "Planetary m1 · 60T" preset (72 mm outer diameter): ratio 6 with three planets.
• 15 / 15 / 45 accepts five planets, but the gap is only 0.63 mm at m1 — print at most four.
• 20 / 20 / 60 and 16 / 16 / 48 have Zs + Zr not divisible by 3, so they take four (or five) planets, not three.
• From 7:1 up the sun gets small and the ring large (12 / 48 / 108 has a ⌀108 mm ring pitch circle); two stages are usually more compact.

TIPTo estimate output speed and torque, enter an equivalent single stage in the gear ratio calculator (e.g. z1 = 12, z2 = 72 for i = 6) — just remember a planetary stage keeps the input direction.

Non-standard tooth counts: profile shift for ratios like 5.18:1

When Zr − Zs is odd, Zp = (Zr − Zs)/2 is not an integer and standard gears can't mesh with both sides. Drop the planet by half a tooth and use positive profile shift to stretch the sun–planet centre distance until it matches the planet–ring centre distance. For the common 5.18:1 ratio, one feasible set is:

Zs = 11, Zp = 17, Zr = 46　→　i = 1 + 46/11 ≈ 5.18　　(11 + 46) / 3 = 19 → three planets

Planet–ring a = m(46 − 17)/2 = 14.5 mm　　standard sun–planet a₀ = m(11 + 17)/2 = 14.0 mm (m = 1)

cos αw = a₀ · cos α / a　→　αw ≈ 24.87°　　Σx = (z₁ + z₂)(inv αw − inv α) / (2 tan α) ≈ +0.56

So the sun and planet need a combined shift of about +0.56 (centre distance modification y = 0.5, tip shortening about 0.06m). Putting it all on the 11-tooth sun (the profile shift calculator's "All on pinion" split) keeps the planets and ring standard and also cures the sun's undercut (an 11-tooth gear needs at least about 0.36). The price is a sun tip thickness of only about 0.30·m — fine for SLS or SLA, too pointed for FDM at m1, so use m1.5 or a standard set.

• The internal gear generator makes standard (unshifted) ring gears, so put any shift on the sun or planets. Shifting the planets also changes the planet–ring centre distance, so recalculate it.
• Profile shift doesn't change the assembly condition: (Zs + Zr)/N must still be an integer.
• Enter m, z₁ = 11, z₂ = 17 and the 14.5 mm target centre distance in the profile shift calculator to get Σx, then enter the shift in the spur gear generator to make the sun.

TIPIf an approximate ratio is enough, pick a standard set from the table. Use profile shift only when you need an exact ratio (for example to match an existing specification) — design and checking are much simpler.

3D-printed planetary gearboxes: module, backlash and layout

• Add another 0.05–0.1 mm to the centre distance to absorb print error and thermal growth.
• Double helical (herringbone) planets self-centre, can't slide out axially and run quieter.
• Run the planets on ⌀3 mm pins or small bearings (e.g. 623: 3 × 10 × 4 mm) and clamp them between a two-plate carrier — much stiffer than cantilevered pins.
• Let the sun float radially (no bearing of its own) so three planets share the load more evenly.
• The ring can be printed into the housing; its mounting must react (i − 1) times the input torque.
• Use PTFE or silicone grease; petroleum oils attack some plastics.

Example: a 6:1 planetary gearbox for a NEMA 17 stepper

1. Input: NEMA 17 stepper with about 0.4 N·m at low speed; target ratio about 6.
2. Tooth counts: Zs 12 / Zp 24 / Zr 60 from the table, m = 1, three planets, i = 1 + 60/12 = 6.
3. Dimensions: sun–planet centre distance a = 1 × (12 + 24)/2 = 18 mm; ring pitch circle ⌀60 mm — use the internal gear generator's "Planetary m1 · 60T" preset (72 mm outer diameter).
4. Output torque: T ≈ 0.4 × 6 × 0.95 ≈ 2.28 N·m (95% efficiency).
5. Tooth load: tangential force per planet mesh Ft = 2·Ts / (N·ds) = 2 × 400 N·mm / (3 × 12 mm) ≈ 22 N, or about 29 N with a 1.3 load-sharing factor.
6. For more reduction stack two stages: 6 × 6 = 36, with about 0.95² ≈ 90% efficiency.

Make the sun and planets with the spur gear generator (or the spur gear pair generator to create the sun–planet pair and check its centre distance in one go), the ring with the internal gear generator, and confirm speed and torque with the gear ratio calculator.