Drivetrain
Spur Gear Design Calculator
Design and validate spur gears: involute geometry, contact ratio, bending and contact stress per ISO 6336, matched model kits.
Parameters
Results
- Pitch diameter (pinion)(d₁)
- 40.00 mm
- Pitch diameter (gear)(d₂)
- 80.00 mm
- addendumPinion
- 44.00 mm
- addendumGear
- 84.00 mm
- Root diameter (pinion)(d_f1)
- 35.00 mm
- dedendumGear
- 75.00 mm
- baseDiameterPinion
- 37.59 mm
- baseDiameterGear
- 75.18 mm
- Center distance(a)
- 60.00 mm
- Gear ratio(i)
- 2.000
- Contact ratio(ε)
- 1.635
- Circumferential speed(v)
- 3.02 m_s
- Tangential force(F_t)
- 3125.0 N
- Radial force(F_r)
- 1137.4 N
- Total force(F)
- 3325.6 N
- Gear torque(T₂)
- 100.0 Nm
- Gear speed(n₂)
- 720 rpm
- Bending stress (pinion)(σ_F1)
- 156.3 MPa
- Bending stress (gear)(σ_F2)
- 118.8 MPa
- Contact stress(σ_H)
- 259.1 MPa
Design checks
- [PASS] Bending σ_F max 156.25 MPa ≤ allowable 210 MPabending_safety
- [PASS] Contact σ_H 259.14 MPa within allowable 550 MPacontact_safety
- [PASS] Contact ratio ε = 1.64 ≥ 1.2contact_ratio_check
- [PASS] z₁ = 20 ≥ 17, no undercutundercut_check
- [PASS] b/m = {faceWidth/module} within recommended 8–20face_width_ratio
- [PASS] Pitch-line velocity 3.02 m/s within limit for grade Q7speed_check
Formulas
d₁ = m × z₁= 40.0000
Expression: module * teethPinion
d₂ = m × z₂= 80.0000
Expression: module * teethGear
dₐ₁ = m(z₁ + 2)= 44.0000
Expression: module * (teethPinion + 2)
dₐ₂ = m(z₂ + 2)= 84.0000
Expression: module * (teethGear + 2)
d_f₁ = m(z₁ − 2.5)= 35.0000
Expression: module * (teethPinion - 2.5)
d_f₂ = m(z₂ − 2.5)= 75.0000
Expression: module * (teethGear - 2.5)
d_b₁ = d₁ × cos α= 37.5877
Expression: module * teethPinion * cos(pi / 180 * pressureAngle)
d_b₂ = d₂ × cos α= 75.1754
Expression: module * teethGear * cos(pi / 180 * pressureAngle)
a = m(z₁+z₂)/2= 60.0000
Expression: module * (teethPinion + teethGear) / 2
i = z₂ / z₁= 2.00000
Expression: teethGear / teethPinion
ε = [√(rₐ₁²−r_b₁²) + √(rₐ₂²−r_b₂²) − a·sin α] / (π·m·cos α)= 1.63519
Expression: (sqrt((addendumPinion/2)^2-(baseDiameterPinion/2)^2) + sqrt((addendumGear/2)^2-(baseDiameterGear/2)^2) - centerDistance*sin(pi/180*pressureAngle)) / (pi*module*cos(pi/180*pressureAngle))
v = π · d₁ · n₁ / 60000= 3.01593
Expression: pi * pitchDiameterPinion * pinionSpeed / 60000
60000 — π·d[mm]·n[rpm] → m/s: π·d·n gives mm/min, ÷1000 for m and ÷60 for s
F_t = 2T₁·Kₐ / d₁= 3125.00
Expression: 2000 * pinionTorque * serviceFactor / pitchDiameterPinion
2000 — 2 × 1000: converts N·m ÷ mm into N
F_r = F_t × tan α= 1137.41
Expression: tangentialForce * tan(pi / 180 * pressureAngle)
F = √(F_t² + F_r²)= 3325.56
Expression: sqrt(tangentialForce^2 + radialForce^2)
T₂ = T₁ × i= 100.000
Expression: pinionTorque * gearRatio
n₂ = n₁ / i= 720.000
Expression: pinionSpeed / gearRatio
σ_F1 = F_t / (b·m) · Y_F₁= 156.250
Expression: tangentialForce / (faceWidth * module) * lewisFormFactorPinion
σ_F2 = F_t / (b·m) · Y_F₂= 118.750
Expression: tangentialForce / (faceWidth * module) * lewisFormFactorGear
σ_H = Z_E · √(F_t·(u+1) / (b·d₁·u)) — ISO 6336-2 simplified= 259.136
Expression: 0.564 * sqrt(tangentialForce / (faceWidth * pitchDiameterPinion) * (gearRatio + 1) / gearRatio) * elasticCoefficient
0.564 — Simplified ISO 6336-2 consolidation of Z_H (node region factor) and load-distribution factors, calibrated to the reference hand calculation. Z_E itself is computed from E/ν, never hardcoded.
Results are for preliminary design. Verify before manufacturing.
Export
STEP export runs entirely in your browser; the geometry kernel downloads once on first export.
Download a sample STEP (m2, z20)A real file to open in your CAD right away, before waiting on the engine.
DXF: closed 2D tooth profile, extrude it in your CAD. STL: watertight mesh, axis = Z, 1 unit = 1 mm. STEP: single solid, involute flanks as B-spline surfaces. Root fillet is a circular approximation, not a trochoidal root — for fatigue-critical gears use a dedicated gear tool.
How the spur gear calculator works
This calculator implements the ISO 6336 series for cylindrical gears, together with the ISO 53 tooth profile and the ISO 54 module series. The geometry block follows the standard involute relationships: pitch diameter d₁ = m·z₁, tip diameter dₐ = m(z+2), root diameter d_f = m(z−2.5), and base diameter d_b = d·cos α. The transverse contact ratio ε is computed from the path of contact: ε = [√(rₐ₁²−r_b₁²) + √(rₐ₂²−r_b₂²) − a·sin α] / (π·m·cos α). A ratio of at least 1.2 is recommended; below 1.0 the gears cannot run continuously.
The load block converts the pinion torque into the tangential force F_t = 2·T₁·Kₐ/d₁, where Kₐ is the application (service) factor per ISO 6336-1. Bending stress is checked per ISO 6336-3 using a tooth form factor Y_F interpolated from the standard’s chart trend (normalised to Y_F = 2.0 at z = 20); this means the bending stress genuinely changes with tooth count instead of using a textbook constant. Contact stress follows the simplified ISO 6336-2 relation σ_H = Z_E·√(F_t·(u+1)/(b·d₁·u)), with the elastic coefficient Z_E = √(1/(π·((1−ν₁²)/E₁ + (1−ν₂²)/E₂))) computed from the actual material pair — switch from steel to PA6 nylon and the contact stress drops by roughly a factor of eight, exactly as it should. Allowable stresses σ_FP and σ_HP come from the built-in material database (C45E, 42CrMo4, 16MnCr5, 304, PA6, GG25) per the ISO 6336-5 basis.
Six design checks run on every change: bending strength, contact strength, contact ratio, undercut prevention (z₁ ≥ 17 at 20° pressure angle per ISO 53), the face-width ratio b/m ∈ [8, 20], and a pitch-line velocity limit tied to the ISO 1328 quality grade. Use the module suggestions to stay on the ISO 54 series, and share any configuration by copying the URL — all parameters are encoded in the query string. Everything you type stays in your browser; there is no server.
Need the geometry too? The Spur Gear Pack (module 1–4) matches this calculator’s parameter space.