Spur Gear Calculator: Geometry and Load Capacity
Calculate the complete geometry of an external cylindrical gear pair (spur and helical, basic rack DIN 867) including profile shift, contact ratio and undercut check – plus a load capacity quick check with the safety factors SF (tooth root) and SH (flank).
Cylindrical gear pair calculator
Gear geometry (αn = 20°, basic rack DIN 867)
Gear pair
- Centre distance a
- 120 mm
- Reference centre distance ad
- 120 mm
- Transverse module mt
- 3 mm
- Working pressure angle αwt
- 20°
- Transverse contact ratio εα
- 1.6708
- Overlap ratio εβ
- 0
- Total contact ratio εγ
- 1.6708
- Gear ratio i
- 3
- Tip shortening factor k (info)
- 0
- Tip clearance c
- 0.75 mm
Gear 1
- Pitch circle d
- 60 mm
- Base circle db
- 56.3816 mm
- Tip circle da
- 66 mm
- Root circle df
- 52.5 mm
- Working pitch circle dw
- 60 mm
- Profile shift x
- 0
- Virtual number of teeth zn
- 20
- Tip tooth thickness san
- 2.085 mm
Gear 2
- Pitch circle d
- 180 mm
- Base circle db
- 169.1447 mm
- Tip circle da
- 186 mm
- Root circle df
- 172.5 mm
- Working pitch circle dw
- 180 mm
- Profile shift x
- 0
- Virtual number of teeth zn
- 60
- Tip tooth thickness san
- 2.357 mm
Sketch: pitch circles (solid), tip circles (dashed) and centre distance
Load capacity quick check (based on DIN 3990 / ISO 6336, endurance limit)
Simplified method: YFa/YSa via 30° tangent with load applied at the tooth tip and Yε, KV = 1.2 fixed, KHβ/KFβ from quality table, KHα = KFα = 1.
Material values are estimates from secondary literature, not curve values per ISO 6336-5.
Tooth forces
- Torque T1
- 49.39 Nm
- Tangential force Ft
- 1,646.4 N
- Radial force Fr
- 599.3 N
- Axial force Fa
- 0 N
- Pitch line velocity v
- 4.56 m/s
Tooth root
- YFa (Gear 1 / Gear 2)
- 2.8 / 2.2863
- YSa (Gear 1 / Gear 2)
- 1.5525 / 1.7286
- Yε
- 0.6989
- Yβ
- 1
- σF0 (1 / 2)
- 41.68 / 37.9 N/mm²
- σF (1 / 2)
- 78.16 / 71.05 N/mm²
Flank
- ZH
- 2.4946
- ZE
- 189.8 √(N/mm²)
- Zε
- 0.8811
- Zβ
- 1
- σH0
- 399.02 N/mm²
- σH
- 598.54 N/mm²
Force factors
KA = 1.25 · KV = 1.2 · KHβ = 1.5 · KFβ = 1.25 · KHα = KFα = 1
Formulas and fundamentals
The geometry follows the involute chain: from normal module mn, tooth numbers z1/z2 and helix angle β one obtains the transverse module mt = mn/cos β, the pitch circles d = z·mt and the base circles db = d·cos αt. The profile shift sum defines the working pressure angle αwt via the involute function inv α = tan α − α, which yields the working centre distance a = ad·cos αt/cos αwt. In the inverse mode the calculator determines the required profile shift sum x1 + x2 from a given centre distance.
The transverse contact ratio εα is the ratio of the length of the path of contact to the base pitch and describes how many tooth pairs are in mesh on average. It must exceed 1; typical spur gear values are 1.1 to 1.9. For helical gears the overlap ratio εβ = b·sin β/(π·mn) is added. The calculator also checks undercut (limit tooth number, minimum profile shift), the pointed tooth limit of the tip thickness and the tip clearance. Tip circles are output without tip shortening; the tip shortening factor k is reported separately.
The tooth root quick check follows DIN 3990-3 / ISO 6336-3: the nominal tooth root stress σF0 = Ft/(b·mn)·YFa·YSa·Yε·Yβ uses the form factor YFa and the stress correction factor YSa from the 30° tangent procedure of Method B, applied at the tooth tip (Method C variant with contact ratio factor Yε). With the force factors KA, KV, KFβ and KFα the safety factor is SF = σFlim·YST/σF with YST = 2.
The flank quick check per DIN 3990-2 / ISO 6336-2 models the Hertzian contact pressure at the pitch point: σH0 = ZH·ZE·Zε·Zβ·√(Ft/(d1·b)·(u+1)/u) with zone factor ZH, elasticity factor ZE (computed from Young's modulus and Poisson's ratio), contact ratio factor Zε and helix factor Zβ. The safety factor is SH = σHlim/σH; the softer material governs. The standard switch accounts for the definition difference Zβ = √cos β (DIN 3990) versus 1/√cos β (ISO 6336).
Deliberate simplifications of the quick check: dynamic factor fixed at KV = 1.2 (valid up to about v = 10 m/s), face load factors KHβ/KFβ from a quality grade table, KHα = KFα = 1, single contact factor ZB/D = 1 and endurance limit without life, lubricant and size factors. The material values σHlim/σFlim are estimates from published scatter bands, not curve values per ISO 6336-5. For a documented proof per full Method B the standard itself is required.
Worked example
Reference example: a spur gear pair with mn = 3 mm, z1 = 20, z2 = 60 (u = 3), b = 40 mm transmits P = 7.5 kW at n1 = 1450 rpm. This gives T1 = 49.4 Nm, Ft = 1646 N and v = 4.56 m/s. The geometry yields εα = 1.67; for gear 1 the factors are YFa = 2.80 and YSa = 1.55.
With KA = 1.25 (moderate shocks), KV = 1.2 and quality 7–8 (KHβ = 1.5 / KFβ = 1.25) and case hardened 16MnCr5 on both gears (σHlim = 1470, σFlim = 430 N/mm²) the results are: σF = 78 N/mm² and SF = 11.0 at the pinion, σH = 598 N/mm² and SH = 2.46 at the flank. As is typical for case hardened gears the flank is the governing criterion while the tooth root has large reserves – both safety factors are well above the required minimum values.
Frequently asked questions
What does profile shift do?
The profile shift x moves the basic rack radially outwards (x > 0) or inwards (x < 0). Positive values avoid undercut at small tooth numbers, thicken the tooth root and allow a given centre distance to be matched exactly. In the mode "x from centre distance" the calculator determines the required sum x1 + x2 automatically.
When does undercut occur and how do I avoid it?
For the DIN 867 basic rack with αn = 20° the theoretical limit is about 17 teeth (practically 14). Below that, the cutting tool digs into the tooth root and weakens it. The remedy is a positive profile shift of at least xmin = 1 − z·sin²αn/2; the calculator warns and states the required value.
What does the contact ratio tell me?
The transverse contact ratio εα states how many tooth pairs are in mesh on average. Values below 1 mean an interrupted mesh and are inadmissible; below 1.1 it becomes critical (noise, shocks). For helical gears the overlap ratio εβ further improves smoothness; the total contact ratio is εγ = εα + εβ.
How accurate is the load capacity quick check?
The geometry factors YFa/YSa are calculated exactly via the 30° tangent, while the force factors are deliberately simplified (KV = 1.2 fixed, KHβ/KFβ from a quality table, KHα = KFα = 1) and endurance limit conditions are assumed. The result is suitable for preliminary sizing and plausibility checks. A reliable, documented proof requires the full Method B per DIN 3990 / ISO 6336 and material values per ISO 6336-5.
How do DIN 3990 and ISO 6336 differ?
Both standards share the same core concept. Practically relevant in this calculator is the flank helix factor: DIN 3990 uses Zβ = √cos β, ISO 6336 (since 2006) the reciprocal 1/√cos β – at β = 15° a difference of about 3.5 % on σH0. Further differences (YB, YDT, YNT curve) concern special cases outside this quick check.
Where do the material values σHlim and σFlim come from?
The stored values are estimates from published scatter bands in secondary literature (e.g. case hardened 16MnCr5: σHlim ≈ 1470, σFlim ≈ 430 N/mm²). The binding hardness and quality dependent curves are given in ISO 6336-5 and DIN 3990-5 and govern any final design.
Related tools
Shaft strength DIN 743
Calculate the safety verification for shafts and axles per DIN 743.
Cam design per VDI 2143
Design motion laws, check characteristic values and calculate cam profiles per VDI 2143.
Fit calculator ISO 286
Determine limit deviations, clearance and interference for hole and shaft per ISO 286.