Planetary Gear Calculator — Sun, Planet & Ring Flank and Tooth-Root Rating (ISO 6336)

Governing standard: ISO 6336-based· ISO 6336-based flank (ISO 6336-2) and tooth-root (ISO 6336-3 Method B) rating of the sun, planet and ring on the shared /gears engine, each member with its own load cycles; planet root fully reversed · K_γ load sharing from IEC 61400-4:2012 by planet count unless measured · K_Hα / K_Fα by ISO 6336-1 Method B and K_Hβ by Method C (no shaft deflection) · kinematics from the Willis equation

Page last updated 23 October 2026

How ISO 6336 works — the method explained

This tool provides an engineering estimate — it uses an accepted simplified model rather than a single citable governing standard. Use it for preliminary sizing and verify the final design against manufacturer data or a licensed engineer.

Video walkthrough (3:51)

Read the transcript

This planetary stage is SAFE. Four planets share the load. Contact safety 1.57.

Add a fifth planet, and it fails. The planets would collide.

This is the MechanixCalc planetary gear calculator. Ratio, assembly, contact stress to ISO 6336-2, and the planet pin. Let's go through it, step by step.

A planetary stage has a sun in the middle. Planets roll around it, inside a ring. A carrier holds the planets.

Before any stress, the set must be buildable. The ring must equal the sun plus two planets. The planets must space evenly. And they must not touch.

Step one: the arrangement. The sun drives, the ring is held, and the carrier is the output. Four planets share the load.

The table shows all six ways to connect the same gears. Ours gives a ratio of 5.

Step two: the teeth. Sun 20, planets 30, ring 80. 20 plus two times 30 is 80. So the mesh condition holds.

And 20 plus 80, over four planets, is 25. A whole number, so the planets space evenly.

Step three: module 3, a 45 mm face, and 42CrMo4 steel, quenched and tempered.

Step four: the load. 15 kW at 1500 rpm. The ratio is 5, so the carrier turns at 300 rpm. With 471 N·m.

Each planet carries 995 N. That includes a load-sharing factor of 1.25, because four planets never share exactly evenly.

Step five: the checks. Contact safety 1.57 at the sun, 3.55 at the ring. Against 1.2. Contact governs. SAFE.

The tooth root is only an estimate: a Lewis formula with the load factors. 4.82. For a full ISO root rating, try our gear calculator, at mechanixcalc.com/gears.

Planet tip clearance 10.1 mm. Efficiency 98.7 %. No undercut.

Step six sizes the planet pin. 2941 N, with the application and dynamic factors. A 15 mm pin. It never changes the verdict.

Now try five planets. Their tips overlap by 7.8 mm. They cannot be assembled. FAIL, whatever the stress says.

Three planets: 100 over 3 is not a whole number. They cannot space evenly, so it reads CHECK.

Make it 22, 29 and 80. Now 102 over 3 is 34. SAFE, with a ratio of 4.64.

Back to four planets, and raise the load to 26 kW. Contact safety drops to 1.19. CHECK.

Widen the face to 55 mm: 1.30. SAFE again.

Or keep 45 mm, and use case-hardened 16MnCr5. Contact safety 2.17.

What the tool does not check is listed: the ring tooth root, the planet bearings, the carrier and scuffing. So silence never reads as a pass.

Every input, every factor and every check goes into a PDF report. A report your reviewer can follow.

That's a planetary stage, checked. Try it on your own gearbox, on MechanixCalc.com.

The MechanixCalc planetary gear calculator rates a single epicyclic stage the way a gearbox engineer does: both meshes are run through the same ISO 6336 engine as the /gears tool, and every member — sun, planet and ring — gets its own flank (pitting, ISO 6336-2) and tooth-root (ISO 6336-3 Method B) safety factor with its own load cycles. Enter the tooth counts, module, face width, power and speed; add K_A, the accuracy grade, the required life, the sun profile shift and the ring's cutter and tip diameter when you have them. The tool returns the ratio, output speed and torque, the per-planet force with the load-sharing factor K_γ, the safety of each member, the input power at which the stage reaches its limits and the planet bearing load. All three assembly conditions are carried into the verdict, so a set that cannot be built is never shown as passing.

It is built for engineers working on industrial reducers, wind-turbine and slewing drives, winches and robotic joints who need to see WHY a stage passes or fails. Speeds are taken relative to the carrier, the planet root is rated as fully reversed (it is an idler, loaded by the sun on one flank and the ring on the other), the sun sees one load cycle per planet per revolution. The page opens with K_A 1.25, accuracy grade 7, a 20 000 h life, flank roughness by material (6 µm for the through-hardened default pair, stated as assumed), ISO VG 220 oil and the 0.85 critical-service long-life line filled in — change them to your duty; an optional input left empty (K_γ, the tooth friction) takes a stated default that is listed in the result and the report; an empty ring rim is rated solid, and when a thinner rim the standard allows would fail the ring root the verdict reads CHECK and names the minimum rim. Read the scope notes before signing off: K_Hβ is ISO 6336-1 Method C with no shaft deflection (carrier twist and sun float are not in it — enter an Annex E or FE value as the override), K_Hα / K_Fα are Method B, and carrier deflection is otherwise represented only through K_γ.

What this calculator does

Gear ratio for 3K-I (ring fixed) and 3K-II (sun fixed), plus a Wolfrom option that is an explicitly badged single-stage STAND-IN, not a Wolfrom rating
Read-only kinematic table covering three input/output/fixed arrangements, including the carrier-fixed reversal
Flank (ISO 6336-2) and tooth-root (ISO 6336-3 Method B) safety of the sun, the planet (both flanks) and the ring, on the shared /gears engine
Load cycles per member from the carrier-relative speeds: sun one per planet per revolution, planet one per flank, ring one per planet per revolution
Real contact path with undercut and form circles; tip-to-fillet interference of the internal mesh, rated on the ring's shaper cutter when entered
Sun profile shift with the planet shift solved for coaxial centre distances
Planet load sharing K_γ from IEC 61400-4:2012 for 3–7 planets, or your measured value
Separate ring material and rim thickness; K_A, accuracy grade, life, roughness and oil as inputs, each default listed
Inverse job: the input power at which every member reaches S_H 1.2 and S_F 1.4
Planet bearing load (pin force plus centrifugal force) for the bearing selection
All three assembly conditions carried into the headline verdict: the mesh condition and actual planet interference as hard failures, equal spacing and tip clearance as warnings
Mesh efficiency from the tooth friction coefficient, swept over input power, with a cooling-threshold rule of thumb
Branded PDF engineering report with every input, every member's safety, the assumed defaults and the scope notes

Method & formulas

Kinematic analysis (Willis equation)

The gear ratio of a planetary set follows from the Willis equation, which relates the speeds of the three members — sun (S), planet carrier (C) and ring (R) — through the tooth counts z_s and z_r. The configuration is defined by which member is held fixed. MechanixCalc evaluates every input/output/fixed permutation simultaneously in a read-only kinematic table and highlights the active one, while the configuration you select for the stress calculation is 3K-I or 3K-II. 3K-I (ring fixed, sun input, carrier output) is the most common industrial choice because it gives the largest ratio for a given package size.

The tooth-mesh condition z_r = z_s + 2·z_p, the equal-spacing assembly condition (z_s + z_r) / n_planets = integer, and the neighbour condition (adjacent planet tip circles must not overlap) are all checked, reported, and carried into the headline verdict: a set that cannot be assembled is never shown as passing, however large its contact safety factor is. The stresses are still computed and shown for reference, so you can see how far the design is from working once the geometry is corrected.

Gear ratio — 3K-I (ring fixed)

i = 1 + z_r / z_s

where i = gear ratio (dimensionless); z_r = ring tooth count; z_s = sun tooth count

Tooth-mesh assembly condition

z_r = z_s + 2 · z_p

where z_p = planet tooth count. Deviations > 1 tooth are flagged as a mesh error.

Flank and tooth-root rating of every member — ISO 6336-2 and ISO 6336-3

A planetary stage is two gear meshes sharing one member. The tool rates the sun–planet mesh as an external gear pair and the planet–ring mesh as an internal mesh on the same engine as the /gears tool — the zone, elasticity, contact-ratio and single-pair factors (Z_H, Z_E, Z_ε, Z_B / Z_D), the load factors K_A, K_V (DIN 3990-1 Method C at the entered accuracy grade, on the pitch-line velocity RELATIVE TO THE CARRIER), K_Hβ by ISO 6336-1 Method C (unmodified flanks as mounted unless you set crowning or assembly adjustment) and K_Hα, and the ISO 6336-2 permissible stress with the life, lubricant, velocity and roughness factors. The contact ratio comes from the real contact path, so an undercut sun is no longer credited with involute it does not have. The internal mesh uses the (u−1)/u curvature term and Z_D = 1.

Each member has its own load cycles, because each one sees a different number of meshes per revolution relative to the carrier: the sun meshes with every planet, so it sees N cycles per relative revolution; a planet sees one cycle on each flank; a ring tooth sees N. The tooth root of every member is rated by ISO 6336-3 Method B (form and stress-correction factors on the real load point). The planet is an idler — loaded by the sun on one flank and by the ring on the other — so its root is fully reversed (Y_M = 0.7). The ring root is rated on its exact shaper-cut geometry when the cutter is entered, otherwise on the conservative rack limit, with the rim factor for the entered rim thickness. Minimum safety factors are S_H 1.2 and S_F 1.4 (DIN 3990-11 industrial practice; ISO 6336 leaves the minimum to agreement).

ISO 6336-2 contact stress

σH = Z_B,D · Z_H · Z_E · Z_ε · Z_β · √( F_t · K_A · K_V · K_Hβ · K_Hα / (b · d₁) · (u ± 1) / u )

where F_t = tangential force per planet incl. K_γ (N); b = face width (mm); d₁ = pinion reference diameter (mm); u = tooth ratio of the pair; '+' external (sun–planet), '−' internal (planet–ring); Z_D = 1 on the internal mesh

ISO 6336-3 tooth-root stress (Method B)

σF = F_t / (b · m) · Y_F · Y_S · Y_β · Y_B · Y_DT · K_A · K_V · K_Fβ · K_Fα

where Y_F, Y_S = form and stress-correction factors at the real load point; Y_B = rim factor (ring); K_Fβ = K_Hβ^N_F. Safety S_F = σFlim · Y_ST · Y_NT · Y_δrelT · Y_RrelT · Y_X · Y_M / σF, with Y_M = 0.7 for the fully reversed planet

Load cycles per member

N_sun = 60 · n_s,rel · L_h · N_p ; N_planet = 60 · n_p,rel · L_h ; N_ring = 60 · n_r,rel · L_h · N_p

where n_rel = speed relative to the carrier, |n − n_c| (rpm) — for the ring |n_r − n_c|, which is n_c in 3K-I (ring fixed) and n_r − n_c in 3K-II; L_h = life (h); N_p = number of planets; the planet count is per flank

Planet load sharing (K_γ, IEC 61400-4)

Manufacturing tolerances — pitch errors, carrier and pin position errors, bearing clearances — mean the planets do not share the load equally. The mesh load factor K_γ raises the tangential force per planet above the ideal 1/N share. Without a measured value the tool uses the IEC 61400-4:2012 table by planet count: 1.10 for 3 planets, 1.25 for 4, 1.35 for 5, 1.44 for 6 and 1.47 for 7. The standard permits lower values only when they are demonstrated by measurement, for example with a floating sun or flexible pins — enter the measured value and the result states that it was entered.

The planet bearing load is reported as the pin force (both mesh forces act in the same circumferential direction and add, with the meshes' K_A·K_V) plus the centrifugal force of the planet on the carrier radius. The planet-pin diameter is a bending sizing rule against a fixed 200 MPa allowable — no shear, fatigue or fit check — so size the pin with your carrier design.

Tangential force per planet

F_planet = ( F_t,sun · K_γ ) / N_p

where F_t,sun = tangential force at the sun reference circle (N); N_p = number of planets; K_γ = IEC 61400-4:2012 default or the measured value

Worked example

A 15 kW, 1 500 rpm motor drives a 3K-I planetary reducer (ring fixed, sun input, carrier output): z_s = 20, z_p = 30, z_r = 80, module 3 mm, face width 45 mm, 4 planets, case-hardened 18CrNiMo7-6, K_A 1.25, grade 7, 20 000 h. Check the assembly, find the ratio, the per-planet force, the load cycles of each member, and read the governing member.

Given

Input power P15 kW
Input speed n_in1 500 rpm
Tooth counts z_s / z_p / z_r20 / 30 / 80
Module m, face width b3 mm, 45 mm
Number of planets N_p4
Material, K_A, grade, life18CrNiMo7-6 (case-hardened), 1.25, 7, 20 000 h
Long-life linecritical service (0.85 at 10¹⁰ cycles)

Result

Gear ratio i5.000
Force per planet (incl. K_γ 1.25)994.8 N
Sun flank S_H (governing)1.43 — SAFE (minimum 1.2)
Output torque (after 98.71 % mesh efficiency)471.3 N·m

Assembly: z_s + 2·z_p = 20 + 60 = 80 = z_r ✓; equal spacing (z_s + z_r)/N_p = 100/4 = 25, a whole number ✓.
Ratio: i = 1 + z_r/z_s = 1 + 80/20 = 5; carrier speed n_c = 1 500/5 = 300 rpm.
Sun torque T = 9 550 · 15/1 500 = 95.5 N·m; F_t,sun = 2 000 · 95.5/60 = 3 183.3 N; K_γ(4) = 1.25 → F_planet = 3 183.3 · 1.25/4 = 994.8 N.
Speeds relative to the carrier: sun 1 500 − 300 = 1 200 rpm; planet 1 200 · 20/30 = 800 rpm; ring 300 rpm.
Load cycles over 20 000 h: sun 60 · 1 200 · 20 000 · 4 = 5.76·10⁹; planet 60 · 800 · 20 000 = 9.6·10⁸ per flank; ring 60 · 300 · 20 000 · 4 = 1.44·10⁹.
Life factors: the sun's 5.76·10⁹ cycles sit far past the contact knee (5·10⁷); on the critical-service line Z_NT = 0.85^(ln(5.76·10⁹/5·10⁷) / ln(10¹⁰/5·10⁷)) = 0.8645 — the flat optimum line would have credited 1.0, 16 % more margin than the material is shown to have.
Rating on the shared engine: the sun flank governs at S_H 1.43 — the most cycles and the smallest curvature radius, with the transverse load factor K_Hα 1.25 on the sun–planet mesh by ISO 6336-1 Method B, the face-load factor K_Hβ 2.96 by ISO 6336-1 Method C (unmodified flanks as mounted: at this light unit load the helix tolerance and the sun's torsional wind-up under the full sun torque put the load on part of the face) and the flank roughness taken by material (3 µm for ground case-hardened gears, assumed). The stage reads SAFE and carries about 27.6 kW input before a check falls below its minimum. Every root is far above S_F 1.4, the planet's lowest because it is fully reversed. With a through-hardened 42CrMo4 set the same stage reads FAIL (sun S_H 0.77) — the reason industrial planetary gears are case-hardened.

The page's opening example. The ring here is cut by a 25-tooth shaper with tip radius 0.25·m and has a 234.6 mm tip diameter; without the ring's cutter and tip the tool reads a possible tip-to-fillet interference CHECK, because it cannot prove the ring clears the planet's fillet.

Frequently asked questions

Which standard does this planetary gear calculator use?

ISO 6336: the flank of every member to ISO 6336-2 and the tooth root of every member to ISO 6336-3 Method B, on the same engine as the /gears tool, with load cycles per member from the speeds relative to the carrier. Load sharing uses the IEC 61400-4:2012 K_γ table unless you enter a measured value. The enclosed-unit K_Hβ is the /gears form rather than ISO 6336-1 Method B (K_Hα / K_Fα follow Method B) — which is why the result is labelled ISO 6336-based. Kinematic ratios come from the Willis equation; AGMA 6123 is background reading, not an implemented basis.

How does the calculator handle the internal ring-planet mesh differently from the sun-planet mesh?

The ring-planet mesh is a concave-convex (internal) contact, which has lower curvature stress than the equivalent external pair at the same load. ISO 6336-2 handles this by replacing the (u+1)/u curvature term used for external gears with (u−1)/u for the internal mesh, and by computing the contact-ratio factor Zε from the internal tooth geometry. The calculator applies both formulae automatically based on the mesh type.

What is the K_gamma factor and why does it matter?

K_γ is the mesh load factor for imperfect load sharing among the planets, from pitch, carrier and pin-position errors. Without it a 4-planet set would be assumed to put exactly a quarter of the load on each planet, which is optimistic. Without a measured value the tool uses the IEC 61400-4:2012 defaults — 1.10 for 3 planets, 1.25 for 4, 1.35 for 5, 1.44 for 6, 1.47 for 7. A floating sun or flexible pins can bring it lower, but the standard accepts lower values only from measurement — enter yours and the report says it was entered. K_γ/N still falls with planet count, so the tip-clearance check, not load sharing, bounds the number of planets.

Can it analyse configurations other than ring-fixed (3K-I)?

Yes, for viewing. The kinematic table evaluates three input/output/fixed arrangements at once — including the carrier-fixed arrangement, which reverses the output shaft — and shows the ratio for each. The configurations you can SELECT for the stress calculation are 3K-I (ring fixed, sun input, carrier output) and 3K-II (sun fixed, ring input, carrier output); the carrier-fixed reversal is shown for comparison but is not a selectable stress case. A third selector option, Wolfrom, is offered but is a single-stage STAND-IN and NOT a Wolfrom rating: a real Wolfrom (3K) drive needs a SECOND ring gear, and its very high ratios come from two rings differing by one or two teeth. This tool models one ring, so within its own assembly rule the stand-in ratio tends to 2·z_p/z_s — at typical tooth counts it returns a ratio LOWER than the plain 3K-I value for the same gears. It is badged as illustrative in the tool; use 3K-I or 3K-II for a defensible number.

Does it check tooth-root bending as well as contact stress?

Yes — for every member. The tooth root of the sun, the planet and the ring is rated by ISO 6336-3 Method B, with the form and stress-correction factors at the real load point. The planet is an idler, loaded by the sun on one flank and by the ring on the other, so its root carries the fully-reversed factor Y_M = 0.7. The ring root is rated on its exact shaper-cut geometry when you enter the cutter's teeth and tip radius, otherwise on the conservative rack limit, and a thin ring rim adds the rim factor Y_B. The governing member is named in the result.

What are the limits of the ISO 6336-2 contact-stress result?

The geometry factors, the life, lubricant, velocity and roughness factors, and the load factors K_A and K_V (at your accuracy grade) are computed per member. One load factor is simplified: K_Hβ is ISO 6336-1 Method C with no shaft deflection — carrier twist and sun float are not in it (enter an Annex E / FE value as the override); K_Hα / K_Fα follow Method B, with the grade's pitch tolerance; the size factors Y_X / Z_X are applied, the work-hardening factor Z_W is 1.0. Carrier deflection is represented only through K_γ. Scuffing and micropitting are not rated. Treat a design close to S_H 1.2 as needing a full gearbox analysis with the real mesh misalignment.

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