Gear terminology, involute profile and interference

Spur gear terminology, the law of gearing, why the involute works, pressure angle, path of contact and contact ratio, and interference with the minimum-teeth formulas.

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Why it matters

Gears transmit motion at an exact speed ratio, which is why servo gearheads, robot joints, machine-tool feeds and vehicle gearboxes use them. Whether a pair runs quietly, carries load smoothly and avoids gouging its own teeth depends on the tooth geometry: module, pressure angle, addendum and number of teeth. These checks are made before any strength calculation.

Key ideas

Basic terminology (spur gears).

  • Pitch circle: the imaginary circle that rolls without slipping on the mating gear's pitch circle; the pitch point P is where the two touch on the line of centres.
  • Module m = d/T (mm per tooth), the standard size unit in SI. Circular pitch p = πd/T = πm. Diametral pitch P_d = T/d (per inch in older practice).
  • Addendum: radial height of the tooth above the pitch circle (standard full-depth: 1m). Dedendum: depth below the pitch circle (1.25m), leaving a clearance of 0.25m.
  • Pinion: the smaller gear; gear (wheel): the larger; rack: a gear of infinite radius.
  • Backlash: small gap between non-driving flanks, needed for lubrication and thermal growth, but a source of lost motion in servo positioning.

Law of gearing. For a constant angular velocity ratio, the common normal at the point of contact must always pass through the fixed pitch point. Then ω₁/ω₂ = r₂/r₁ = T₂/T₁. Profiles that satisfy this are conjugate; the involute and the cycloid are the practical ones.

Involute profile. The curve traced by the end of a taut string unwound from a base circle. Its normal at any point is tangent to the base circle, so for two involute gears the common normal is the fixed common tangent to the two base circles: the line of action. Consequences:

  • the force direction (pressure angle) stays constant throughout mesh;
  • the velocity ratio is set by base-circle radii, so a small change of centre distance does not change the ratio (it only changes backlash and the operating pressure angle);
  • a single rack cutter or hob generates all tooth counts of a given module, so involute gears are cheap to make. Cycloidal teeth do not have these properties and survive mainly in clocks and some pumps.

Pressure angle (φ). The angle between the line of action and the common tangent to the pitch circles at P. Standards: 20° (most common), 14.5° (older), 25° (higher load). A larger φ gives thicker-based, stronger teeth and fewer teeth before interference, but higher radial (separating) bearing load and a lower contact ratio.

Path, arc and contact ratio. Contact starts where the line of action meets the driven gear's addendum circle and ends where it meets the driver's addendum circle. The length between is the path of contact. The contact ratio (number of tooth pairs in contact on average) must exceed 1, and is usually 1.4–1.8 for quiet running.

Interference and undercutting. The involute exists only outside the base circle. The interference points are where the line of action touches the two base circles. If a gear's addendum circle extends beyond the interference point of its mating gear, its tip digs into the non-involute flank of the other gear (interference). In generation by a rack cutter, the cutter removes that flank material instead (undercutting), weakening the tooth root. It is most likely with few teeth on the pinion and a small pressure angle. Remedies: more pinion teeth, larger φ, stub teeth (smaller addendum), or profile shift (positive correction on the pinion).

Formulas

m = d / T ; p = π·m ; d = m·T (d = pitch diameter, mm; T = number of teeth) C = m(T₁ + T₂) / 2 (centre distance, mm) r_b = r·cos φ (base circle radius) ; p_b = p·cos φ (base pitch) G = T / t = ω_pinion / ω_gear (gear ratio)

Path of contact (pinion radii r, r_a; gear radii R, R_a; addendum radii = pitch radius + addendum):

  • path of approach = √(R_a² − R_b²) − R·sin φ
  • path of recess = √(r_a² − r_b²) − r·sin φ
  • arc of contact = path of contact / cos φ
  • contact ratio = arc of contact / p = path of contact / (p·cos φ)

No interference if R_a ≤ √(R_b² + ((r + R)·sin φ)²) and r_a ≤ √(r_b² + ((r + R)·sin φ)²).

Minimum teeth to avoid interference (A_w = addendum of the wheel in modules, 1 for full depth):

  • wheel: T_min = 2A_w / (√(1 + (1/G)(1/G + 2)·sin² φ) − 1); pinion t_min = T_min / G
  • pinion with a rack: t_min = 2A_r / sin² φ
  • equal gears (G = 1): T_min = 2A / (√(1 + 3 sin² φ) − 1)

v_sliding = (ω₁ + ω₂)·x (x = distance of the contact point from the pitch point; zero at P)

Worked examples

Example 1 (standard): contact ratio of a spur pair. Given: pinion t = 20, gear T = 40, m = 5 mm, φ = 20°, full-depth addendum 1m = 5 mm.

  1. r = 5 × 20 / 2 = 50 mm, R = 100 mm; r_a = 55 mm, R_a = 105 mm.
  2. r_b = 50 cos 20° = 46.98 mm, R_b = 93.97 mm.
  3. Approach: √(105² − 93.97²) − 100 sin 20° = 46.85 − 34.20 = 12.65 mm
  4. Recess: √(55² − 46.98²) − 50 sin 20° = 28.600 − 17.101 = 11.49 mm
  5. Path of contact = 24.14 mm; arc = 24.14 / cos 20° = 25.69 mm; p = π × 5 = 15.71 mm.
  6. contact ratio = 25.69 / 15.71 = 1.64
  7. Interference check: R_a,max = √(93.97² + (150 sin 20°)²) = 107.1 mm > 105 mm; r_a,max = 69.6 mm > 55 mm. No interference. Answer: path of contact ≈ 24.1 mm, contact ratio ≈ 1.64, no interference.

Example 2 (GATE level): minimum pinion teeth. Given: φ = 20°, full-depth teeth (A = 1 module), gear ratio G = 3. Find the minimum pinion teeth; compare with a rack and with equal gears.

  1. sin² 20° = 0.11698.
  2. Wheel: T_min = 2 / (√(1 + (1/3)(1/3 + 2) × 0.11698) − 1) = 2 / (√1.09098 − 1) = 2 / 0.04450 = 44.9
  3. Pinion: t_min = 44.9 / 3 = 14.98 → 15 teeth (wheel 45).
  4. Rack: t_min = 2 / 0.11698 = 17.1 → 18 teeth.
  5. Equal gears: T_min = 2 / (√(1 + 3 × 0.11698) − 1) = 2 / 0.1625 = 12.3 → 13 teeth.
  6. For 14.5° with a rack: 2 / sin² 14.5° = 31.9 → 32 teeth, which is why 20° replaced 14.5°. Answer: 15 teeth for G = 3; 18 with a rack; 13 for equal gears (20°, full depth). Always round up.

Common mistakes

  • Using pitch-circle radius in place of base-circle radius in the path-of-contact formula, or forgetting to add the addendum to get r_a.
  • Dividing the path of contact by p instead of by p·cos φ (base pitch) for contact ratio.
  • Rounding the minimum tooth count down.
  • Thinking a change in centre distance changes the velocity ratio of involute gears; it does not.
  • Treating circular pitch as module; p = πm.
  • Mixing up interference (a geometry fault in mesh) with undercutting (the material the cutter removes because of it).

For GATE ME

Typical questions: module, pitch diameter and centre distance; base circle radius; length of path of contact and contact ratio; minimum number of teeth to avoid interference (often with a rack); properties of the involute and the law of gearing. Practise the path-of-contact calculation end to end and the three minimum-teeth formulas.

Quick check

  1. A gear has 36 teeth and a pitch diameter of 108 mm. What is the module?
  2. What is the base circle radius of a 100 mm pitch-diameter gear with φ = 20°?
  3. Why does a change in centre distance not alter the speed ratio of involute gears?
  4. What is the minimum tooth count for a 20° full-depth pinion meshing with a rack?
  5. Is contact ratio 0.9 acceptable?

Answers: 1. 3 mm. 2. 50 cos 20° = 46.98 mm. 3. The ratio depends on base-circle radii, which are fixed. 4. 18 (2/sin² 20° = 17.1). 5. No; it must exceed 1 for continuous contact.

Interview questions

All Theory of Machines and Vibrations interview questions

Try answering each one aloud before you open it.

  1. 1.What is the involute profile in gear design?Concept

    The involute profile is a curve that is generated by unwinding a taut string from a base circle. In gear design, the involute profile is used for the teeth of gears because it ensures smooth and constant velocity transmission between gears. This profile allows for a consistent pressure angle, which is crucial for efficient power transmission.

  2. 2.Explain the term 'interference' in the context of gears.Concept

    Interference in gears occurs when the non-working portions of the gear teeth come into contact during meshing. This typically happens when the addendum of one gear enters the base circle of the mating gear, leading to undesirable contact and potential damage. Interference is more likely in gears with fewer teeth or when the pressure angle is too low.

  3. 3.Why is the involute profile preferred over other profiles in gear design?Application

    With involute teeth the common normal is always the common tangent to the two base circles, so the line of action and pressure angle stay fixed and the law of gearing is satisfied throughout mesh. The velocity ratio depends only on the base-circle radii, so a small change in centre distance from manufacturing or bearing tolerances changes only backlash and operating pressure angle, not the ratio. A single straight-sided rack cutter or hob generates every tooth count of a given module, which makes involute gears cheap and interchangeable.

  4. 4.What happens if interference occurs in a gear system?Application

    If interference occurs in a gear system, it can lead to increased friction, noise, and wear, potentially causing damage to the gear teeth. This can result in inefficient power transmission and may require the gears to be redesigned or modified to prevent further issues. In severe cases, interference can lead to gear failure.

  5. 5.How can interference be avoided in gear design?Application

    Interference can be avoided by using a larger number of teeth on the gears, increasing the pressure angle, or using profile shifting techniques. These methods help ensure that the addendum of one gear does not enter the base circle of the mating gear, thus preventing unwanted contact.

  6. 6.Explain the term 'pressure angle' in gears and its significance.Concept

    The pressure angle is the angle between the line of action (the direction of the tooth force) and the common tangent to the pitch circles at the pitch point. Standard values are 20°, the most common, 14.5° in older practice and 25° for heavy duty. A larger pressure angle gives wider-based, stronger teeth and allows fewer pinion teeth before interference, but raises the radial separating force on the bearings and lowers the contact ratio, which can make the pair noisier.

  7. 7.What is the base circle in the context of involute gears?Concept

    The base circle is the circle from which the involute profile is generated. It is smaller than the pitch circle and serves as the starting point for the involute curve. The base circle is crucial because it determines the shape of the involute profile and, consequently, the gear's performance.

  8. 8.What is the minimum number of teeth on a 20° full-depth involute pinion to avoid interference with a rack?Numerical

    For a pinion meshing with a rack, t_min = 2·A_r / sin²φ, where A_r is the rack addendum in modules (1 for full depth). With φ = 20°, sin²φ = 0.117, so t_min = 2 / 0.117 = 17.1, which is rounded up to 18 teeth. For two equal 20° gears the limit is lower, 2 / (√(1 + 3 sin²φ) − 1) = 12.3, so 13 teeth; with 14.5° teeth the rack limit rises to 32.

  9. 9.What is the effect of increasing the pressure angle on gear performance?Application

    A larger pressure angle makes the tooth wider at the root, so bending strength rises, and it reduces the minimum number of teeth needed to avoid interference and undercutting. The drawbacks are a larger radial (separating) component of the tooth force, so heavier bearing loads, and a shorter path of contact, so a lower contact ratio and usually more noise. That is why 20° is the common compromise and 25° is used for heavily loaded gears.

  10. 10.A gear with 20 teeth is meshing with a gear with 40 teeth. If the module is 5 mm, calculate the pitch circle diameter of both gears.Numerical

    The pitch circle diameter (PCD) of a gear is calculated using the formula: PCD = number of teeth × module. For the gear with 20 teeth, PCD = 20 × 5 mm = 100 mm. For the gear with 40 teeth, PCD = 40 × 5 mm = 200 mm.

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