What an involute is
Almost every modern gear has involute gearing. The involute of a circle is the curve formed when a taut string is unwound from the base circle and the endpoint of the string is traced. Each tooth flank follows this curve exactly.
This leads to the fundamental law of gearing: The common normal at the point of contact between two meshing tooth flanks must always pass through the rolling point on the line connecting the centers of the gears. In the case of an involute, this normal is the common tangent to both base circles. Therefore, the gear ratio remains constant—even if the center distance changes slightly due to assembly tolerances (in which case only the rolling point and the operating mesh angle shift).
It is precisely this insensitivity to tolerances, combined with cost-effective manufacturing using gear hobbing and gear shaping processes (a single tool module covers the entire number of teeth for the same module), makes involute gearing according to DIN 3960/DIN 867 the standard in mechanical engineering.
Description and values of the figure
The base circle has a radius of r_b= 40 mm. The involute is formed when a taut string is unwound from the base circle: its endpoint traces the curve x(θ) =r_b·(cos θ + θ·sin θ), y(θ) =r_b·(sin θ − θ·cos θ). As the unwinding angle θ increases, the curve moves away from the base circle at an ever-faster rate. The dashed operating pitch circle with r ≈42.57 mm marks the radius at which the pressure angle α = 20° is measured.
| θ (rad) | x (mm) | y (mm) |
|---|---|---|
| 0,0 | 40,00 | 0,00 |
| 0,1 | 40,20 | 0,01 |
| 0,2 | 40,79 | 0,11 |
| 0,3 | 41,76 | 0,36 |
| 0,4 | 43,07 | 0,84 |
| 0,5 | 44,69 | 1,63 |
| 0,6 | 46,56 | 2,78 |
The pressure angle α
The pressure angle α is the angle between the line of engagement (the line of action of the force between the meshing tooth flanks) and the tangent to the operating pitch circle at the rolling point. In involute gearing, it is also the flank angle of the tool reference profile.
| Pressure angle | Load capacity | Bearing force | Typical Applications |
|---|---|---|---|
| 14.5° | gering | niedrig | Older standards, precision engineering |
| 20° (standard) | mittel | mittel | General Mechanical Engineering (DIN 867) |
| 25° | hoch | higher | High-performance gearboxes, small numbers of teeth |
20° is the standard angle because it combines sufficient tooth root thickness, moderate radial force on the bearings, and good manufacturability. A larger angle increases the load-carrying capacity and lowers the undercut limit—but also generates higher bearing forces.
Reference circle and operating pitch circle: two different things
The reference circle (d = m·z) is a fixed property of the individual gear—it does not change. The operating pitch circle, on the other hand, is a parameter that depends on the gear pair: It is created only through the interaction of two gears, at the point where the circumferential speeds of both gears are equal (the rolling point).
For standard gearing with a center distance that complies with standards, the reference circle and the operating pitch circle coincide. If the center distance is increased or a profile shift is used, the operating pitch circle shifts outward, and the operating mesh angle deviates from the reference value of 20°.
Profile Shift: Avoid Undercut
Below a number of teeth of z ≈ 17 (at a 20° pressure angle), undercut occurs: During gear hobbing, the tool reference profile cuts away material at the tooth root because it falls below the involute curve at that point. This weakens the tooth and reduces the tooth overlap.
This can be remedied by a positive profile shift x·m: The tool reference profile is shifted radially outward, so that it no longer undercuts the involute. The following applies as a guideline for the required minimum profile shift:
x_min ≈ (17 − z) / 17
x_min = Minimum profile displacement factor · z = actual number of teeth (at a 20° pressure angle)
Mnemonic
Worked example
Given
A small pinion with a module of m = 2 mm must have only z = 14 teeth due to space constraints—which is below the limit of 17 teeth for a 20° pressure angle.
Calculation
- Minimum profile shift: x_min ≈ (17 − 14) / 17 = 3 / 17 ≈ 0.18
- Absolute displacement: x_min · m ≈ 0.18 · 2 mm ≈ 0.35 mm
With this positive profile shift of approximately 0.35 mm, the tooth root remains free of undercut even with only 14 teeth—the final design is based on DIN 3960.
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