Planetary Gear Calculator — Ratio, Contact Stress & Load Sharing (ISO 6336 / AGMA 6123)

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 all three standard configurations simultaneously and highlights the active one: 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 equal-spacing assembly condition (z_s + z_r) / n_planets = integer and the tooth-mesh condition z_r = z_s + 2·z_p are both checked before any stress calculation — a wrong tooth count is reported immediately rather than producing a silently incorrect stress number.

Pitting durability is evaluated at both mesh pairs — the external sun-planet mesh and the internal ring-planet mesh — using the ISO 6336-2 surface-durability method. The nominal contact stress σH0 combines the zone factor ZH (accounting for the curvature of the tooth flanks at the pitch point), the elasticity factor ZE (material pair), the contact-ratio factor Zε and the helix-angle factor Zβ. Load factors KA (application), KV (dynamic) and KH (distribution) amplify the tangential force before the stress formula is applied. The internal ring-planet mesh uses the (u−1)/u curvature term rather than the external (u+1)/u, correctly reducing the Hertzian stress for the concave-convex contact geometry.

The safety factor for each mesh is computed as the ratio of the material's allowable contact stress σH,lim (from the material database) to the calculated contact stress. A minimum safety factor of 1.2 is the acceptance threshold, consistent with ISO 6336 guidance for enclosed industrial gear drives.

In a real planetary set, manufacturing tolerances — pitch errors, carrier runout, bearing clearances — mean the planets do not share the load perfectly equally. ISO 6336-1 accounts for this with the mesh load factor K_gamma, which increases the effective tangential force per planet above the ideal 1/n_planets share. More planets reduce the per-planet load but increase K_gamma; the optimum is typically 3–4 planets for precision industrial gearboxes.

The planet-pin diameter is sized from the bending moment on the cantilever pin (Shigley method), using the total pin reaction of 2·F_t per planet — both the sun-side and ring-side tangential components act in the same circumferential direction, so they add; the opposing radial (separating) components cancel.