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Table of Contents Key input parameters Geometry and basic formulas Pitch diameter equals module multiplied by tooth count d = m × z Pitch radius equals half the pitch diameter r = d / 2 = m × z / 2 For ideal concentric geometry the ring tooth count equals sun plus two times planet Zr = Zs + 2 × Zp Kinematic core relation The angular speeds of sun, carrier and ring satisfy the Willis equation (ωs − ωc) / (ωr − ωc) = − Zr / Zs From this relation the calculator derives closed form expressions for three operating modes Planetary gearbox modes Sun drives carrier, ring fixed When the ring does not rotate the carrier to sun speed ratio is ωs / ωc = 1 + Zr / Zs Expressed in revolutions per minute the numeric ratio used by the tool is i_sc = 1 + Zr / Zs Sun drives ring, carrier fixed With carrier fixed the ring speed is proportional to sun speed with opposite sign ωr = − Zs / Zr × ωs Magnitude of the gear ratio reported is i_sr = Zr / Zs Carrier drives ring, sun fixed When the sun is held stationary the ring follows the carrier with a scale factor ωr = ωc × (1 + Zs / Zr) The calculator reports the useful inverse form when needed i_cr = ωc / ωr = 1 / (1 + Zs / Zr) Practical kinematic and power formulas Output speed given input speed and numeric ratio equals input speed divided by ratio Angular velocity in radians per second equals two pi times rpm divided by 60 Output torque approximated with efficiency equals Tin times numeric ratio times eta Power equals torque multiplied by angular velocity so Pin = Tin × ωin and Pout ≈ eta × Pin Worked examples Sun drives carrier Input setup : Zs = 17, Zp = 21, Zr = 59, N = 3, m = 2.5 millimetres, input speed 1200 rpm, input torque 40 N·m, efficiency 0.95 Check geometry: Zr equals Zs plus two times Zp, 59 equals 17 plus 42, geometry is compatible Numeric ratio i equals 1 plus Zr divided by Zs, i = 1 + 59 / 17 = 4.4706 Computed results Carrier speed equals 1200 divided by 4.4706 equals 268.48 rpm Output torque equals 40 times 4.4706 times 0.95 equals 169.84 N·m Pitch radii in millimetres, sun 21.25, planet 26.25, ring 73.75 Tangential force on sun equals input torque divided by sun radius in metres, 40 divided by 0.02125 equals 1882.35 newtons Load per planet equals 1882.35 divided by 3 equals 627.45 newtons Sun drives ring Input setup : Zs = 12, Zp = 14, Zr = 40, input speed 2000 rpm, input torque 25 N·m, efficiency 0.94 Numeric ratio equals Zr divided by Zs, 40 divided by 12 equals 3.3333 Ring speed equals 2000 divided by 3.3333 equals 600 rpm Output torque equals 25 times 3.3333 times 0.94 equals 78.33 N·m Carrier drives ring Input setup : Zs = 10, Zp = 15, Zr = 40, carrier speed 80 rpm, input torque 30 N·m, efficiency 0.96 Relative factor equals one minus Zs divided by Zr, one minus 10 divided by 40 equals 0.75 Ring speed equals 80 times 0.75 equals 60 rpm Numeric inverse ratio equals 1 divided by 0.75 equals 1.3333 Output torque approximated equals 30 times 1.3333 times 0.96 equals 38.40 N·m Geometry notes and practical cautions Approximate centre distance between two wheels equals module times the sum of teeth divided by two a ≈ m × (Z1 + Z2) / 2 Adding more planets reduces load per planet nearly in proportion to the planet count when load distribution is even 📝 Calculator outputs are engineering level estimates
collection: Accessories
Pedal Pins:Titanium Gold
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machined aluminum clamp
