A gear calculation error that misidentifies the module by one standard size can make an entire gearbox impossible to manufacture — understanding module, pressure angle, and tooth geometry from first principles prevents this and much worse.
Gears are the workhorses of mechanical power transmission. From a simple two-gear speed reducer to a multi-stage planetary gearbox, the geometry of the tooth profile determines whether gears mesh smoothly, carry the required load, and achieve the desired ratio. This guide covers the fundamental parameters every mechanical engineer must understand: module, pressure angle, tooth geometry, center distance, and the basics of strength calculation per ISO 6336 and AGMA 2101.
- Module: The Fundamental Size Parameter
- Pressure Angle
- Standard Tooth Geometry (Full-Depth Involute Profile)
- Center Distance and Gear Ratio
- Minimum Tooth Count and Undercutting
- Spur vs Helical Gears
- Contact Ratio
- Gear Tooth Strength: Bending and Contact Stress Overview
- Worked Example: Gear Pair Design
- Conclusion
Module: The Fundamental Size Parameter
Module m is the basic size parameter of a gear tooth. It is defined as the ratio of the pitch circle diameter to the number of teeth:
m = d / z
Where d = pitch circle diameter (mm) and z = number of teeth. Equivalently, m = p / π where p is the circular pitch (arc length between adjacent teeth along the pitch circle). Module is a dimensioned quantity — m = 1 means 1 mm of pitch circle diameter per tooth. A gear with m = 4 and z = 25 has d = 100 mm.
Gears can only mesh if they have the same module and the same pressure angle. This constraint is what makes standardized module series (JIS B 1701 / ISO 54) critical in practice. Standard preferred module values in JIS/ISO:
| Series | Module values |
|---|---|
| 1st preference (use these first) | 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10, 12, 16, 20 |
| 2nd preference | 1.125, 1.375, 1.75, 2.25, 2.75, 3.5, 4.5, 5.5, 7, 9, 11, 14, 18 |
In North America, the equivalent traditional parameter is diametral pitch Pd = z/d (teeth per inch), which is inversely related to module: m = 25.4/Pd. AGMA standards use diametral pitch, while ISO/JIS standards use module. When working with international suppliers, confirming which system is in use prevents costly errors.
Pressure Angle
The pressure angle φ is the angle between the common normal to the tooth surfaces at the pitch point and the common tangent to the pitch circles. The most common standard values are 20° (modern standard) and 14.5° (legacy standard, still encountered on older machinery).
The 20° pressure angle standard (ISO 1328, JIS B 1701, AGMA 2001) produces a stronger tooth than 14.5° because it creates a broader tooth base and better contact geometry. However, the higher pressure angle also generates higher bearing loads (the separating force component is Fr = Ft tan φ, where Ft is the tangential (transmitted) force). At 20°, the separating force is 36% of the tangential force; at 14.5°, it is only 26%.
Some specialized applications use 25° pressure angle (higher load capacity per tooth, but even higher bearing loads and reduced number of teeth before undercutting becomes an issue).
Standard Tooth Geometry (Full-Depth Involute Profile)
For a standard full-depth spur gear with module m:
| Parameter | Formula |
|---|---|
| Pitch circle diameter | d = m × z |
| Addendum (tooth height above pitch circle) | a = 1.0 × m |
| Dedendum (tooth depth below pitch circle) | b = 1.25 × m |
| Whole depth | h = 2.25 × m |
| Addendum circle diameter (tip) | da = m(z + 2) |
| Dedendum circle diameter (root) | df = m(z − 2.5) |
| Circular tooth thickness (at pitch circle) | t = π × m / 2 |
| Base circle diameter | db = m × z × cos φ |
The base circle is the circle from which the involute tooth profile is generated. The involute profile ensures that the pressure angle remains constant throughout tooth engagement, which is why involute gears transmit smooth, constant velocity. This property — the fundamental law of gearing — requires that the common normal to the contacting tooth surfaces always passes through the pitch point.
Center Distance and Gear Ratio
For a pair of standard external spur or helical gears:
a = m(z1 + z2) / 2
Where z1 and z2 are the tooth counts of the pinion and gear respectively. The gear ratio (speed ratio) is:
i = z2 / z1 = n1 / n2
For internal (ring) gears, the center distance formula becomes: a = m(z2 − z1) / 2. Achieving a non-standard center distance while maintaining correct mesh is possible through profile shift (addendum modification), which adjusts the tooth thickness without changing the module or pressure angle. Profile shift coefficient x specifies the radial distance by which the gear cutting tool is displaced from the standard position, in units of module.
Minimum Tooth Count and Undercutting
When a gear has too few teeth, the tip of the mating gear cuts into the base of the tooth during manufacture (undercutting), weakening the tooth root and degrading the involute profile. The minimum tooth count to avoid undercutting for a standard gear at pressure angle φ is:
zmin = 2 / sin²φ
At φ = 20°: zmin = 2 / sin²(20°) = 2 / 0.117 ≈ 17 teeth. At φ = 14.5°: zmin ≈ 32 teeth. This is one reason 20° has replaced 14.5° as the standard — it allows pinions with as few as 17 teeth without undercutting, enabling more compact designs. For fewer than 17 teeth (e.g., small pinions in high-ratio gearboxes), positive profile shift is applied to avoid undercutting.
Spur vs Helical Gears
Spur gears have teeth parallel to the shaft axis. Their advantages are simplicity, no axial thrust load, easy manufacturing, and straightforward calculation. Their disadvantage is that the full tooth width comes into contact simultaneously at each mesh cycle, causing noise and vibration — particularly noticeable at higher speeds.
Helical gears have teeth cut at an angle (helix angle β, typically 15–30°) to the shaft axis. Multiple teeth are in contact simultaneously, the load is distributed gradually, and operation is much smoother and quieter. This comes at the cost of an axial thrust force Fa = Ft × tan β that must be absorbed by the bearings. Double-helical (herringbone) gears cancel the axial thrust by having opposing helix angles on the same gear.
For helical gears, the normal module mn is the standard module in the normal plane (perpendicular to the tooth). The transverse module mt = mn / cos β is used for pitch circle diameter calculation: d = mt × z = mn × z / cos β. The normal pressure angle φn = 20° is standard; the transverse pressure angle φt = arctan(tan φn / cos β) is slightly larger.
Contact Ratio
The contact ratio εα is the average number of teeth in contact at any instant. For smooth, continuous transmission of motion, εα must be greater than 1. Typical spur gears have εα = 1.3–1.8. A contact ratio of 1.3 means that for 30% of the mesh cycle, two pairs of teeth are in contact; for the remaining 70%, only one pair is in contact. Higher contact ratio reduces noise, distributes load better, and generally improves gear performance.
The transverse contact ratio is calculated from the addendum circle radii, base circle radii, and center distance. For helical gears, the total contact ratio εγ = εα + εβ where εβ = b × sin β / (π × mn) is the overlap ratio due to the helix. This is why helical gears inherently have better load distribution than spur gears of the same size.
Gear Tooth Strength: Bending and Contact Stress Overview
ISO 6336 and AGMA 2101 provide comprehensive methods for calculating gear tooth bending stress (σF) and contact stress (σH, Hertzian pitting stress). The fundamental bending stress formula (Lewis equation, simplified) is:
σF = Ft × KA × Kv × KHβ × KHα / (b × mn × YJ)
Where Ft = tangential load, KA = application factor (overload), Kv = dynamic factor (speed effect), KHβ = face load distribution factor, KHα = transverse load distribution factor, b = face width, and YJ = geometry factor for bending strength. The allowable bending stress depends on material grade, heat treatment, and surface finish per ISO 6336-5. For standard case-hardened steel (material grade MQ), allowable root stress σFP is typically 430–530 MPa.
Worked Example: Gear Pair Design
Design a spur gear pair for a 5:1 speed reduction. Input: 1450 rpm, 7.5 kW. Target center distance approximately 150 mm. Material: Grade 8.8 steel (not applicable — let us say ISO grade MQ carburized). Pinion tooth count z1 = 20 (above zmin = 17). Gear tooth count z2 = 5 × 20 = 100. Module m: a = m(z1 + z2)/2 = m × 60. For a = 150 mm: m = 150/60 = 2.5. Select standard m = 2.5 mm. Check: d1 = 2.5 × 20 = 50 mm, d2 = 2.5 × 100 = 250 mm, a = (50+250)/2 = 150 mm. Tangential force: Ft = (2 × 1000 × P) / (d1 × n1 × π / 30) = (2 × 7500 × 30) / (0.050 × π × 1450) = 1991 N.
Conclusion
Gear design begins with a thorough understanding of module, pressure angle, and tooth geometry. Module is the fundamental size parameter that must match between mating gears. The 20° pressure angle is the modern standard, offering a good balance between tooth strength and bearing loads. The involute profile ensures constant velocity transmission, while contact ratio determines load sharing and noise characteristics. For any real gearbox design, full stress analysis per ISO 6336 or AGMA 2101 is required to verify bending and contact strength against the applied loads and required service life. Mastering the geometric fundamentals covered here is the essential prerequisite for that analysis.
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