Gears and gearbox

Geometry, contact ratio, pitting and tooth-root safety of a spur or helical gear pair, with the gears drawn from their real involute tooth forms and running — then a complete one- to three-stage gearbox, sized and shown in 3D.

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INPUTmm · N · min⁻¹

1 · Load

Set by Power [kW]

Pinion speed [min⁻¹]

Application factor KA 1.0 uniform … 1.75 heavy shocks Required life [h]

2 · Gears

Pinion teeth z1 Wheel teeth z2

Normal module [mm] Face width [mm]

Helix angle [°] 0 for spur gears Pressure angle [°]

Double helical (the face width is both halves) Centre distance from

Profile shift x1 Profile shift x2

Hob, pinion Hob, wheel

3 · Materials and finish Pinion Wheel

Accuracy grade ISO 1328-1 Oil viscosity ν40 [mm²/s]

Oil temperature [°C] Flank roughness Rz [µm] ground about 3–4

Root roughness Rz [µm]

4 · Shafts and helix Pinion arrangement Bild 22.3/7

Helix modification Contact pattern

Idler gear: pinion loaded both ways (YA = 0.7)

5 · Required safeties Tafel 22.3/10

Pitting SH Tooth root SF

Example values, not a real gearbox. The listed steels carry the endurance limits used in Niemann & Winter's worked examples; for your own material choose “your material”.

PITTING SAFETY SH1.73

MESH · LIVE

PINION TOOTH

SAFETIESDIN 3990

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HOW IT'S CALCULATED · WITH YOUR NUMBERS

Step by step, with your numbers

01
Equilibrium████ ██ ███████ · ██████ = ██.██
02
Reactions████ ██ ███████ · ██████ = ██.██
03
Shear force████ ██ ███████ · ██████ = ██.██
04
Bending moment████ ██ ███████ · ██████ = ██.██
05
Deflection████ ██ ███████ · ██████ = ██.██
06
Stress████ ██ ███████ · ██████ = ██.██

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References: DIN 3990 Parts 1–3 (the basis of ISO 6336:1996) as set out in Niemann & Winter, Maschinenelemente Band II (Springer), Abschnitte 21–22, with its six worked examples of spur and helical gearboxes; ISO 1328-1 for the tolerances; DIN 780 module series; ISO 281 for the bearings.

ASSUMPTIONS & LIMITS

Load capacity by DIN 3990, Parts 1 to 3 (the basis of ISO 6336:1996), as set out by Niemann & Winter, Maschinenelemente Band II, Abschnitte 21–22: methods B and C, tooth-root stress for load at the tip.
Solid steel gears (Fc = 1, CR = 1), external gears, parallel shafts; the nominal torque times the application factor; no load spectrum. Scuffing, micropitting, wear and the bearings are not checked.
Tolerances fpe and fHβ from the ISO 1328-1 equations for the grade given (unrounded). The face load factor uses the deflection constants of Tafel 22.3/6 and Bild 22.3/7; for a precise KHβ the tooth contact must be measured.
Endurance limits σHlim and σFE are those of the material chosen (for the listed steels: the values used in Niemann & Winter's worked examples, quality MQ). For through-hardened steel the proof stress in Yδ is estimated from the hardness.
Life factors: the static value up to 10⁵ (pitting) or 10³–10⁴ (root) load cycles, the endurance value from the knee (5·10⁷ or 3·10⁶) on, log-linear between; 1.0 beyond the knee (no further decline is taken).

DISCLAIMER · For education and preliminary sizing only. Results must be reviewed by a qualified engineer before use in design.

BEYOND THE STANDARD

Tooth contact under load, housing and shaft deflection, gear whine and transmission error, failure investigations — we analyse gearboxes with FEA and multibody dynamics.

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