Gear terminology, involute profile and interference

Why it matters

A gearbox, a differential and a timing drive all depend on teeth that pass motion at a perfectly steady speed ratio. The involute tooth form makes that possible and tolerates small errors in centre distance, which is why almost every automotive gear uses it. Knowing the terminology, how long the teeth stay in contact and when they interfere lets you size a gear pair, check its smoothness and avoid undercut pinions.

Key ideas

Terminology (spur gears):

• Pitch circle: the imaginary circle that rolls without slipping on the mating gear's pitch circle. Its diameter d is the pitch circle diameter (PCD).
• Module m = d / z (mm): the size of the teeth in the SI system. Mating gears must have the same module.
• Circular pitch p = π·m: the distance between corresponding points on adjacent teeth along the pitch circle. Diametral pitch P = z/d (teeth per inch in older practice) is the inverse idea.
• Addendum: radial height of the tooth above the pitch circle. Dedendum: depth below it. For standard 20° full-depth teeth, addendum = 1m and dedendum = 1.25m, so the whole depth is 2.25m and the clearance is 0.25m.
• Pressure angle φ: the angle between the line of action (common normal at the contact point) and the common tangent to the pitch circles at the pitch point. Standard values are 20° (most common) and 14.5°.
• Base circle: the circle from which the involute is generated; its radius is r·cos φ.
• Pitch point P: where the line of centres crosses the pitch circles.
• Backlash: the gap between the non-driving faces of mating teeth, needed for lubrication and thermal expansion.

Law of gearing. For a constant velocity ratio, the common normal at the point of contact must always pass through the fixed pitch point. Then ω₁/ω₂ = r₂/r₁ = z₂/z₁.

Involute profile. The curve traced by a point on a taut string as it unwinds from the base circle. Its normal at every point is tangent to the base circle, so for two involute gears the common normal is always the common tangent to the two base circles: a fixed straight line through the pitch point. Consequences:

• The velocity ratio is constant (law of gearing satisfied automatically).
• The pressure angle is constant during meshing, so the bearing load direction is steady.
• If the centre distance changes slightly, the velocity ratio is still the base circle ratio; only the working pressure angle and backlash change. Cycloidal teeth do not have this property.
• One cutter (rack or hob) can cut all gears of a given module and pressure angle.

Path of contact, arc of contact and contact ratio. Contact starts where the driven gear's addendum circle crosses the line of action and ends where the driver's addendum circle crosses it. The length between is the path of contact (path of approach + path of recess). The contact ratio is the average number of tooth pairs in contact; it must exceed 1, and is usually 1.4 to 1.8 for spur gears. Higher contact ratio means smoother, quieter running.

Interference. The involute exists only outside the base circle. If the tip of one gear contacts the other gear below its base circle (the non-involute flank), the tip digs into the flank: interference. In generating processes the cutter removes that flank material instead, which is undercutting and weakens the tooth root. Interference happens when the addendum circle of the mating gear goes beyond the interference points, the points where the line of action touches the base circles.

Avoiding interference. Use more teeth on the pinion; a larger pressure angle (20° or 25° instead of 14.5°); stub teeth (smaller addendum, 0.8m); profile shift (positive correction on the pinion: the cutter is withdrawn so the pinion teeth are thicker at the root and the active profile moves outward); or limit the mating gear's addendum. Module size alone does not change the minimum number of teeth.

Formulas

m = d / z, p = π·m, p_b = p·cos φ

• m: module (mm); d: pitch circle diameter (mm); z: number of teeth; p: circular pitch (mm); p_b: base pitch (mm); φ: pressure angle.

r_b = r·cos φ

• r_b: base circle radius (mm); r: pitch circle radius (mm).

C = m·(z₁ + z₂) / 2

• C: standard centre distance of external spur gears (mm).

path of approach = √(R_a² − R_b²) − R·sin φ and path of recess = √(r_a² − r_b²) − r·sin φ

• R, R_a, R_b: pitch, addendum and base circle radii of the gear (wheel) (mm); r, r_a, r_b: the same for the pinion (mm). Pinion drives.

ε = path of contact / p_b and arc of contact = path of contact / cos φ

• ε: contact ratio (dimensionless).

R_a,max = √(R_b² + (C·sin φ)²)

• Largest addendum circle radius of the gear that avoids interference with the pinion (mm); similarly for the pinion with R and r swapped.

z_p,min = 2A / [G·(√(1 + (1/G)·(1/G + 2)·sin²φ) − 1)]

• z_p,min: minimum teeth on the pinion to avoid interference; A: addendum of the gear in modules (1 for standard teeth); G = z_gear / z_pinion (gear ratio).

z_min = 2A / sin²φ (pinion with a rack)

• For φ = 20°, A = 1: z_min = 17.1, so 18 teeth (17 is usually accepted in practice); for φ = 14.5°: 31.9, so 32.

Worked examples

Example 1 (standard: geometry and contact ratio). A 20-tooth pinion drives a 40-tooth gear; m = 5 mm, φ = 20°, standard addendum 1m. Find the centre distance, the path of contact and the contact ratio.

1. r = 5 × 20 / 2 = 50 mm; R = 5 × 40 / 2 = 100 mm. C = 50 + 100 = 150 mm.
2. Addendum radii: r_a = 55 mm, R_a = 105 mm. Base radii: r_b = 50 cos 20° = 46.98 mm, R_b = 100 cos 20° = 93.97 mm.
3. Path of approach = √(105² − 93.97²) − 100 sin 20° = √(11025 − 8830.2) − 34.20 = 46.85 − 34.20 = 12.65 mm.
4. Path of recess = √(55² − 46.98²) − 50 sin 20° = √(3025 − 2207.5) − 17.10 = 28.59 − 17.10 = 11.49 mm.
5. Path of contact = 12.65 + 11.49 = 24.14 mm.
6. Base pitch p_b = π × 5 × cos 20° = 14.76 mm.
7. Contact ratio ε = 24.14 / 14.76 = 1.64: between one and two pairs of teeth are in contact, on average 1.64.

Example 2 (GATE level: interference). For the same pair, check whether the gear's 5 mm addendum causes interference, and find the minimum pinion teeth for a 3 : 1 ratio with standard 20° teeth.

1. R_a,max = √(R_b² + (C sin φ)²) = √(93.97² + (150 × 0.3420)²) = √(8830.2 + 2632.2) = √11462.4 = 107.06 mm.
2. Maximum gear addendum = 107.06 − 100 = 7.06 mm > 5 mm, so no interference.
3. For G = 3: (1/G)(1/G + 2)·sin²φ = (0.3333)(2.3333)(0.11698) = 0.09098. √1.09098 − 1 = 0.04450.
4. z_p,min = 2 × 1 / (3 × 0.04450) = 2 / 0.1335 = 14.98, so 15 teeth minimum.
5. Comparison: for G = 1, the same formula gives 12.3 (13 teeth); for a rack, 17.1 (18 teeth). The larger the mating gear, the more pinion teeth are needed.

Common mistakes

• Using a diametral-pitch formula with SI module, or mixing radius and diameter in m = d/z.
• Taking the dedendum equal to the addendum. Standard full-depth teeth have dedendum 1.25m.
• Dividing the path of contact by the circular pitch. Divide by the base pitch, or divide the arc of contact by the circular pitch.
• Swapping which addendum goes with which path: approach uses the driven gear's addendum, recess uses the driver's.
• Believing a smaller module reduces interference. Minimum tooth count depends on pressure angle, addendum (in modules) and gear ratio only.
• Rounding the minimum teeth down. 14.98 means 15.

For GATE ME

Expect calculations of module, centre distance, base circle, path or arc of contact and contact ratio; checks on interference or the minimum number of pinion teeth for a given pressure angle and ratio; and concept items on the involute's properties and the law of gearing. Practise example 1 with different addenda until the path-of-contact formula is automatic.

Quick check

1. Module of a 30-tooth gear with PCD 120 mm?
2. Whole depth of a standard 20° full-depth tooth of module 3 mm?
3. Base circle diameter of a 100 mm PCD gear with φ = 20°?
4. Why does a slight change in centre distance not change the velocity ratio of involute gears?
5. Minimum teeth on a 20° full-depth pinion meshing with a rack?

Answers: 1. 4 mm. 2. 2.25 × 3 = 6.75 mm. 3. 100 cos 20° = 93.97 mm. 4. The ratio equals the ratio of base circle radii, which are fixed. 5. 2 / sin²20° = 17.1, so 18 (17 commonly accepted).