Spur gears modules and calculation
Spur gears are the simplest type of gear; they transmit motion between parallel shafts using straight teeth oriented radially. Contact between involute profiles ensures a constant transmission ratio. The absence of axial thrust makes them suitable for moderate-speed applications. Modern standard pressure angles are 20° and 25°, while 14.5° is found in older equipment.
The relationship between the module m and the number of teeth z defines all main dimensions. For gears with 17 or more teeth, the pitch diameter = m × z. In gears with fewer than 17 teeth, a profile shift is required, resulting in a corrected pitch diameter = m × z + m. Interference and undercutting are avoided by this shift.
Nomenclature and Standard Dimensions
Section titled “Nomenclature and Standard Dimensions”Standard modules according to ISO 54 series are used universally. The conversion between metric module m (mm) and diametral pitch DP (in⁻¹) is DP = 25.4 / m. The following table lists preferred modules and the resulting pitch diameter for a 20-tooth pinion.
| Module m (mm) / Equivalent DP (in⁻¹) | Pitch diameter for z=20 (mm / in) |
|---|---|
| 0.5 / 50.8 | 10 / 0.394 |
| 0.8 / 31.75 | 16 / 0.630 |
| 1 / 25.4 | 20 / 0.787 |
| 1.25 / 20.32 | 25 / 0.984 |
| 1.5 / 16.933 | 30 / 1.181 |
| 2 / 12.7 | 40 / 1.575 |
| 2.5 / 10.16 | 50 / 1.969 |
| 3 / 8.467 | 60 / 2.362 |
| 4 / 6.35 | 80 / 3.150 |
| 5 / 5.08 | 100 / 3.937 |
| 6 / 4.233 | 120 / 4.724 |
| 8 / 3.175 | 160 / 6.299 |
| 10 / 2.54 | 200 / 7.874 |
The face width b is typically between 8 and 12 times the module. For commercial modules from 1 to 6 mm, the width usually ranges from 10 to 60 mm (0.39 to 2.36 in). The total tooth depth h is 2.25 × m and the circular thickness at the pitch circle is π × m / 2.
Load Capacity
Section titled “Load Capacity”Although load capacity for bending and pitting follows methodologies such as AGMA or ISO 6336, a first estimate of the transmission is based on the following kinematic relationships. For a gear pair with efficiency η, the output speed n₂ and output torque M₂ are given by:
n₂ = n₁ / i
M₂ = M₁ × i × η
where i = z₂/z₁ is the transmission ratio. The mechanical power is maintained except for losses: P₂ = P₁ × η.
Calculation Example
Section titled “Calculation Example”- Input torque: M₁ = 500 N·m / 369 lbf·ft
- Transmission ratio: i = 3.8
- Efficiency: η = 0.90
- Input speed: n₁ = 2000 rpm
Results:
M₂ = 500 N·m × 3.8 × 0.90 = 1710 N·m (1261 lbf·ft)
n₂ = 2000 rpm / 3.8 = 526 rpm
Input power P₁ = 300 kW / 402 hp → P₂ = 270 kW / 362 hp
To safely transmit these loads, tooth fatigue failure must be verified. As an initial guideline, allowable stress values for case-hardened steel of 200 to 400 MPa (29 000 to 58 000 psi) at the tooth root are used.
Assembly and Adjustments
Section titled “Assembly and Adjustments”Spur gears require precise mounting on parallel shafts. The center distance a = m(z₁+z₂)/2 must be maintained with tolerances of ±0.05 mm (±0.002 in) for small modules and up to ±0.2 mm (±0.008 in) for large modules. Shaft and hub fits follow the ISO system; a fit of H7/g6 or H7/js6 prevents excessive clearance. Axial fixation is achieved with retaining rings or lock nuts.
Backlash j is recommended between 0.03 and 0.10 mm (0.0012 – 0.004 in) for modules from 1 to 4 mm, increasing proportionally with the module. A practical guide is j ≈ 0.05 × m (in mm). The following table details typical values:
| Module m (mm) | Minimum backlash (mm / in) | Maximum backlash (mm / in) |
|---|---|---|
| 1 | 0.03 / 0.0012 | 0.06 / 0.0024 |
| 2 | 0.05 / 0.0020 | 0.10 / 0.0039 |
| 3 | 0.08 / 0.0031 | 0.15 / 0.0059 |
| 4 | 0.10 / 0.0039 | 0.20 / 0.0079 |
| 5 | 0.13 / 0.0051 | 0.25 / 0.0098 |
| 6 | 0.15 / 0.0059 | 0.30 / 0.0118 |
During assembly, the clearance must be checked with feeler gauges and the contact pattern verified using Prussian blue.
Selection by Application
Section titled “Selection by Application”The following table relates application sectors with typical recommendations for module, pressure angle, and material. The values are indicative and must be confirmed by detailed calculation.
| Application | Typical module m (mm) | Pressure angle (°) | Material | Remarks |
|---|---|---|---|---|
| Office mechanisms, instrumentation | 0.5 – 1.0 / DP 50.8 – 25.4 | 20° | Brass, plastic (POM) | Low noise, loads < 10 N·m / 7.4 lbf·ft |
| Household appliances (mixers, blenders) | 1.0 – 2.0 / DP 25.4 – 12.7 | 20° | Sintered steel, PA6+GF | Medium speeds, torque up to 50 N·m / 37 lbf·ft |
| Machine tools (lathes, milling machines) | 2.5 – 5.0 / DP 10.16 – 5.08 | 20° or 25° | Case-hardened alloy steel 20MnCr5 | High precision, continuous loads of 200 – 800 N·m / 148 – 590 lbf·ft |
| Light vehicle transmissions | 3.0 – 6.0 / DP 8.47 – 4.23 | 20° | Induction hardened steel 42CrMo4 | Fluctuating loads, typical torque 500 – 1500 N·m / 369 – 1106 lbf·ft |
| Mining, heavy conveyors | 8.0 – 12.0 / DP 3.18 – 2.12 | 25° | Cast steel, deep case hardening | Severe impacts, torque > 2000 N·m / 1475 lbf·ft |
Frequently Asked Questions (FAQ)
Section titled “Frequently Asked Questions (FAQ)”What is the formula for the pitch diameter of a pinion with 20 teeth and module 2 mm?
Section titled “What is the formula for the pitch diameter of a pinion with 20 teeth and module 2 mm?”The pitch diameter is 20 × 2 mm = 40 mm (1.575 in). For standard modules this relationship is valid as long as the number of teeth is ≥ 17.
What backlash value is recommended for a module 4 gear?
Section titled “What backlash value is recommended for a module 4 gear?”A minimum backlash of 0.10 mm (0.004 in) and maximum of 0.20 mm (0.008 in) is recommended, following the practical rule of 0.05 × m.
What is the diametral pitch equivalent to module 3?
Section titled “What is the diametral pitch equivalent to module 3?”The equivalent diametral pitch is 25.4 / 3 = 8.467 teeth per inch. A module 3 gear with 30 teeth would have a pitch diameter of 90 mm (3.543 in).
What typical efficiency does a pair of well-lubricated spur gears have?
Section titled “What typical efficiency does a pair of well-lubricated spur gears have?”Efficiency varies between 94% and 98%. For preliminary calculations η = 0.95 is used, meaning that with 100 kW (134 hp) input, 95 kW (127 hp) useful output is obtained.
How many minimum teeth without correction are needed for a 20° pressure angle?
Section titled “How many minimum teeth without correction are needed for a 20° pressure angle?”To avoid interference with a 20° pressure angle, at least 17 teeth on the pinion are required. With 25° the minimum drops to 12 teeth, allowing more compact designs.
What torque can a module 4, case-hardened steel gear transmit for infinite life?
Section titled “What torque can a module 4, case-hardened steel gear transmit for infinite life?”A 20-tooth pinion with face width 40 mm, σ_Flim = 350 MPa (50 800 psi), can transmit approximately 1200 N·m (885 lbf·ft) according to ISO 6336. The exact value depends on quality and service factor.
References
Section titled “References”- engineeringtoolbox.com: https://www.engineeringtoolbox.com/gear-output-torque-speed-horsepower-d_1691.html
- mcmaster.com: https://www.mcmaster.com/products/spur-gears/