Gears and Gear-Tooth Kinematics

Introduction and context

Gears are toothed machine elements that transmit rotary motion and torque between shafts. Unlike friction drives, gears rely on positive engagement, so their velocity ratio is determined primarily by tooth geometry rather than by friction or belt slip. They are used in vehicle transmissions, machine tools, robotics, wind turbines, pumps, clocks, aerospace actuators and industrial reducers.

The central kinematic requirement is that two gears must rotate with a constant angular-velocity ratio while their teeth remain in contact. This requirement is expressed by the law of gearing. The involute tooth profile is especially important because it satisfies this law, is relatively easy to manufacture, and tolerates small centre-distance errors without changing the fundamental velocity ratio.

This chapter develops the geometry and kinematics of gear pairs, then examines spur, helical, bevel and worm gears. It also considers contact ratio, interference, undercutting, manufacturing, noise, efficiency and practical selection.

Learning objectives

After studying this chapter, a student should be able to:

1. State and apply the law of gearing.
2. Explain involute geometry and conjugate action.
3. Calculate gear ratio, pitch-line velocity, contact ratio and basic gear dimensions.
4. Explain interference, undercutting and profile modification.
5. Distinguish spur, helical, bevel and worm gears.
6. Assess the effects of geometry, manufacturing accuracy, lubrication and operating conditions.
7. Select an appropriate gear arrangement for a practical transmission problem.

1. Fundamental terminology and gear geometry

A gear pair consists of a driving gear, or pinion, and a driven gear. The following terms are standard for involute gears.

For a gear with N teeth and pitch diameter d, the module is

where m and d have the same length units, commonly millimetres. The circular pitch is

These relations assume a standard cylindrical gear measured in the transverse plane. In SI practice, module is usually specified in millimetres and tooth number is dimensionless.

For standard full-depth metric involute spur gears, commonly used nominal proportions are addendum a = m and dedendum approximately b = 1.25m, although actual values depend on the standard and profile system. Consequently, the working depth, clearance and whole depth must always be taken from the applicable gear standard rather than assumed universally.

For two external gears with pitch diameters d₁ and d₂, the centre distance is

and the ideal speed ratio is

where ω₁ and ω₂ are angular speeds and subscripts 1 and 2 denote driver and driven gears. Direction reverses for an external gear pair. For internal gearing, the gears rotate in the same direction and the pitch relationship uses the difference between the internal and external pitch radii.

2. The law of gearing and conjugate action

Suppose two tooth profiles contact at point P. The common normal at P intersects the line of centres at point Q. Constant velocity ratio requires that Q remain fixed as the contact point moves. This is the law of gearing:

The common normal to the mating tooth profiles at every point of contact must pass through a fixed point on the line of centres.

That fixed point is the pitch point. If the normal does not pass through a fixed pitch point, the instantaneous velocity ratio changes, producing transmission error, vibration and possible dynamic loading.

The law can be derived from relative motion. Let the centres be O₁ and O₂, and let the angular velocities be ω₁ and ω₂. At the contact point, the components of velocity along the common normal must be equal because the surfaces do not separate or penetrate. If the normal intersects O₁O₂ at Q, similar triangles and velocity compatibility give

If Q is fixed, the ratio is constant. At the pitch point, the pitch radii r₁ and r₂ satisfy

The condition of conjugate action means that a pair of profiles produces the prescribed relative motion. Involute profiles are conjugate when generated from compatible base circles and used with the same pressure angle. Other conjugate profiles can be designed, but involutes dominate industrial practice because they combine kinematic reliability with manufacturing simplicity.

3. Involute geometry

An involute is the curve traced by the end of a taut string as it is unwound from a base circle. In gear manufacture, the tooth flank is generated by a rack cutter or pinion-shaped cutter whose working face is a straight line. The corresponding involute has a straight line of action.

For an involute gear with pitch radius r and pressure angle φ, the base-circle radius is

The normal force between involute teeth acts along the common tangent to the two base circles. Since this common normal is fixed, the velocity ratio remains constant throughout contact.

The involute has useful geometric properties:

• It is normal to the generating string at every point.
• The line of action is straight.
• The pressure angle varies along the tooth flank if contact extends beyond the pitch point, but the operating pressure angle for standard gears is normally fixed by the common tangent geometry.
• A small change in centre distance changes the operating pressure angle and working pitch radii, but not the fundamental tooth velocity ratio, provided the teeth remain conjugate and contact is maintained.

This last property explains why involute gears are comparatively tolerant of centre-distance errors. It does not mean that centre-distance errors are harmless: backlash, contact ratio, load distribution, noise and bearing loads can still deteriorate.

4. Contact ratio and smooth transmission

Contact ratio measures the average number of tooth pairs in contact. For spur gears, the transverse contact ratio is

where L_a is the length of the path of contact and p_b is the base pitch. The base pitch is

for a standard spur gear with transverse circular pitch p and pressure angle φ. The path of contact consists of the approach and recess portions. A value ε_α ≥ 1 is required for continuous contact in an ideal spur pair. In practice, values above about 1.2 are preferred because manufacturing errors, deflection and dynamic effects reduce the effective margin.

For a gear pair with pitch radii r₁, r₂, addendum radii r_a1, r_a2 and base radii r_b1, r_b2, the approximate path of contact for external involute gears is

This expression assumes standard external involute geometry, no profile modification, and a common operating pressure angle. It must be used with consistent length units.

When 1 < ε_α < 2, one pair of teeth is in contact during part of the mesh and two pairs during the remainder. Higher contact ratio generally improves smoothness and load sharing, but may increase sliding, tooth dimensions or manufacturing cost. Helical gears add an overlap component, discussed later.

Worked example 1: speed ratio and contact ratio

A 20-tooth pinion meshes with a 60-tooth gear. The module is 3 mm and the pressure angle is 20°. Assume standard full-depth addenda a = m. Find the speed ratio, pitch diameters, centre distance and approximate transverse contact ratio.

The speed ratio is

Thus the driven gear rotates at one-third of the pinion speed and in the opposite direction. The pitch diameters are

The centre distance is

The pitch radii are 30 mm and 90 mm. The addendum radii are 33 mm and 93 mm. The base radii are

The path of contact is approximately

which gives approximately L_a = 27.3 mm. The base pitch is

Therefore,

This unusually high value indicates substantial theoretical overlap of successive tooth pairs; in a practical design, the actual value should be checked against the selected standard, addendum proportions and contact geometry. The calculation illustrates the method, but a designer should verify the result with a gear-design standard or software because unusually high results can signal that a differing definition or geometry has been used.

5. Interference and undercutting

Interference occurs when a portion of one tooth profile contacts a non-conjugate portion of the mating tooth. For involute gears, the involute exists only outside the base circle. If contact extends inside the base circle of the mating gear, the tip of one tooth may strike the non-involute root region of the other.

Undercutting is the removal of material near the tooth root during generation, usually by a rack cutter or shaping tool. It weakens the tooth, reduces the contact ratio and can increase backlash. It is especially likely for pinions with few teeth and standard pressure angles.

For a standard 20° full-depth involute system, the theoretical minimum tooth number for avoiding undercutting by a standard rack cutter is approximately

At φ = 20°,

Thus 18 teeth is often cited as a practical minimum for an unmodified standard pinion, although the exact limit depends on profile shift, tooth system, clearance, and the definition of undercutting. At 14.5° pressure angle the minimum is much larger; at 25° it is smaller, but tooth forces and radial loads increase.

Undercutting can be reduced by:

• increasing the pinion tooth number;
• using a larger pressure angle, with an associated increase in radial force;
• applying positive profile shift to the pinion;
• using stub teeth or a nonstandard tooth system;
• reducing the required ratio per stage;
• using helical teeth, where the virtual or formative tooth number is higher.

Profile shift changes the radial position of the generating rack relative to the gear blank. Positive shift generally thickens the pinion tooth at the root and moves the working geometry outward; negative shift has the opposite tendency. Profile shift must be allocated to the pair, not selected independently without checking centre distance, contact ratio, tooth thickness and interference.

6. Forces, sliding and efficiency

If the transmitted torque on a gear is T and its pitch radius is r, the nominal tangential force is

For a spur gear with pressure angle φ, the separating or radial force is

The resultant tooth force is

These equations neglect dynamic amplification, load concentration, friction and tooth flexibility. They are useful for preliminary bearing-load and shaft calculations. Units must be consistent: if T is in N·m and r in m, force is in newtons.

At the pitch point, pure rolling occurs ideally. Away from the pitch point, the contacting tooth surfaces slide relative to each other. Sliding is greatest near the start and end of contact and contributes to friction, wear, scuffing and heat. Gear efficiency depends on tooth friction, bearing losses, seals, windage and lubrication. Spur and helical gear pairs can achieve high efficiency; worm gears may have substantially lower efficiency because sliding dominates.

Worked example 2: tooth forces

A gear transmits 1.2 kW at 600 r/min through a pitch diameter of 120 mm. The pressure angle is 20°. Find torque, tangential force and radial force, neglecting losses.

Angular speed is

Torque is

With r = 0.060 m,

The radial force is

The bearing and shaft designer must also consider dynamic factors, helix forces if applicable, misalignment and transient torque.

7. Spur gears

Spur gears have straight teeth parallel to the shaft axis. They are simple, economical and efficient. Their principal advantages are easy manufacture, low axial thrust and suitability for moderate speeds and parallel shafts. Their main disadvantages are abrupt engagement, relatively high noise at speed, and sensitivity to pitch and profile errors.

Spur gear design must address:

• tooth bending strength;
• pitting or contact fatigue;
• wear and scuffing;
• dynamic loads due to transmission error;
• backlash and thermal expansion;
• shaft, bearing and housing stiffness.

A basic bending-stress estimate often begins with the Lewis concept, in which tooth strength is related to module, face width, allowable stress and a geometry factor. Modern design normally uses standards such as ISO 6336 or AGMA methods rather than relying on the simplified Lewis equation alone. Standards include factors for dynamic effects, size, load distribution, surface finish, reliability and material.

8. Helical gears

Helical gear teeth are inclined at helix angle β to the shaft axis. They engage progressively, so helical gears are quieter and generally carry more load than comparable spur gears. Their total contact ratio includes transverse and overlap contributions:

where ε_α is transverse contact ratio and ε_β is overlap ratio. For a uniform face width b, a common approximation is

where p_x is axial pitch. Since axial pitch relates to normal pitch p_n by p_x = p_n/sin β, an equivalent form is

The exact convention depends on whether module and pitch are specified in the normal or transverse system; the designer must not mix these systems.

Helical gears produce axial thrust. For a gear with tangential force F_t, transverse pressure angle φ_t and helix angle β, approximate force components are

where F_a is axial force. Opposite-hand helices are required for external parallel-shaft pairs. Herringbone gears cancel axial thrust using opposed helices, but are more difficult and expensive to manufacture.

Helical gears are preferred for high-speed transmissions, automotive gearboxes and compact power reducers when increased axial bearing capacity and manufacturing cost are acceptable. They require careful alignment, suitable thrust bearings and often better lubrication than spur gears.

9. Bevel gears

Bevel gears transmit motion between intersecting shafts, commonly at 90°. Their pitch surfaces are cones rather than cylinders. Straight bevel gears are relatively simple but can be noisy at high speed because tooth engagement is less progressive. Spiral bevel gears provide smoother engagement and greater load capacity but require more complex manufacture and adjustment.

For a pair of bevel gears with pitch-cone angles δ₁ and δ₂, intersecting shaft angle Σ and external gearing,

For a right-angle pair, Σ = 90°. The tooth ratio remains approximately

but dimensions vary along the face width. Bevel gears require accurate control of mounting distance, tooth contact pattern, shaft intersection and backlash. Incorrect adjustment can concentrate load at the heel or toe of the tooth, causing noise and premature failure.

10. Worm gears

A worm gear set consists of a screw-like worm and a worm wheel. It is used for large speed reduction between generally perpendicular, non-intersecting shafts. If the worm has z₁ starts and the wheel has N₂ teeth, the ideal ratio is

Worm drives offer compactness, high reduction in one stage and, in some designs, resistance to back-driving. However, sliding is substantial, so efficiency and heat generation are major concerns. Self-locking is not guaranteed merely by a large ratio; it depends on lead angle, friction, lubrication, surface finish, load and vibration.

For a worm with lead angle γ and coefficient of friction μ, a simplified efficiency estimate for a worm driving a wheel is

This expression is an idealized sliding model. It neglects bearing and seal losses and should not be used as a final rating equation. It shows that increasing lead angle generally improves efficiency but reduces the tendency toward self-locking.

Worm wheels are often made from bronze or other compatible materials against hardened steel worms to control adhesive wear. Lubricant selection, heat rejection and surface finish are critical. Worm gears are useful in lifts, conveyors, rotary tables and steering mechanisms, but they are inefficient for high-power continuous transmission unless thermal performance is carefully designed.

11. Manufacturing and operational implications

Gear performance depends as much on manufacturing and assembly as on theoretical geometry. Important manufacturing routes include hobbing, shaping, milling, broaching, grinding, shaving and additive or hybrid processes. Hobbing is productive for many external gears; shaping is useful for internal gears and shoulders; grinding provides high accuracy and surface quality after heat treatment.

Common errors include pitch variation, profile deviation, helix deviation, runout, eccentricity and surface waviness. These errors create transmission error, often a primary source of gear noise. Heat treatment improves bending and contact strength but may introduce distortion. Grinding or honing can correct some distortion, at increased cost.

Operational best practices include:

• specifying the correct accuracy grade for speed and load;
• ensuring adequate housing and shaft stiffness;
• controlling centre distance and alignment;
• selecting lubricant viscosity and additives for load and temperature;
• filtering lubricant and preventing contamination;
• maintaining designed backlash;
• checking tooth contact pattern during assembly;
• monitoring temperature, vibration and acoustic emissions;
• avoiding overloads and abrupt torque reversals.

Typical failure modes include bending fatigue, pitting, micropitting, scuffing, abrasive wear, tooth fracture, plastic flow and fretting. Failure analysis should examine tooth appearance, lubricant condition, loading history, alignment, hardness and manufacturing records rather than attributing failure automatically to material weakness.

12. Applied case: selecting a reduction drive for a conveyor

Background

This illustrative case concerns a packaging conveyor requiring a 12:1 speed reduction from a 1,450 r/min motor. The output must be continuous, quiet and reliable, with limited maintenance access.

Problem

A single-stage spur gear cannot normally provide the required ratio without an impractically small pinion or a very large gear. A worm drive is compact but may produce excessive heat at the required power. The design team must choose an arrangement that balances efficiency, noise, size and cost.

Analysis

A two-stage helical gearbox can use ratios of approximately 3.5:1 and 3.43:1, giving an overall ratio near 12:1. Helical teeth improve smoothness and distribute load over several tooth pairs. The design must provide thrust bearings for axial forces and must verify thermal capacity.

A single worm stage could achieve 12:1 easily and would be compact. However, its sliding losses may be unacceptable for a conveyor operating continuously. The resulting heat could require a larger housing or forced cooling. Back-drivability may also be undesirable if the conveyor must coast or be manually positioned.

Approach and solution

The preferred preliminary solution is a two-stage helical reducer. The pinions should have sufficient tooth numbers to avoid undercutting, and profile shift can be used if packaging requires a small first-stage pinion. The housing should be rigid, gears accurately finished, and bearings selected for combined radial and axial loads. Lubrication should be specified according to speed, load, temperature and manufacturer guidance.

Outcome

The helical arrangement is expected to provide higher continuous efficiency and lower operating noise than a comparable worm reducer, at the cost of more components and a larger axial-bearing requirement. Final selection requires rating calculations using an accepted standard, thermal analysis, service factor, safety assessment and prototype or field validation.

Learning points

1. Ratio alone does not determine gear selection.
2. Efficiency and heat rejection are decisive in continuous service.
3. Helical smoothness creates axial thrust that must be designed for.
4. Manufacturing accuracy and alignment strongly affect noise and life.
5. A gearbox should be evaluated as a complete system of gears, shafts, bearings, housing, lubricant and controls.

13. Contemporary engineering considerations

Computer-aided gear design now integrates involute geometry, tolerance analysis, finite-element tooth-stress analysis, dynamic simulation and condition monitoring. Digital manufacturing enables highly accurate profile modification and lightweight designs. Additive manufacturing is useful for prototypes, polymer gears and complex low-load components, but material anisotropy, surface finish, dimensional stability and wear must be validated before safety-critical use.

Sensors can detect vibration, temperature, acoustic emissions and lubricant debris. Data-driven maintenance can identify changes in mesh frequency and sidebands, but algorithms require representative data and engineering interpretation. Artificial intelligence should support—not replace—failure physics, inspection and safety controls. Traceability of design assumptions, training data and alarm thresholds is important in regulated industries.

Environmental considerations include lubricant toxicity, energy loss, material selection, recyclability and service life. Efficient gearing reduces operating energy, while durable surface treatments may reduce replacement waste. Safety-critical applications require documented calculations, verification, quality assurance and compliance with relevant national and international standards.

Chapter Summary

Gears transmit rotary motion through conjugate tooth profiles. The law of gearing requires the common normal at every contact point to pass through a fixed pitch point, thereby ensuring a constant velocity ratio. Involute profiles satisfy this condition with a straight line of action and are practical to manufacture.

Module, pitch diameter, tooth number, pressure angle and base circle define essential gear geometry. Contact ratio measures the average number of tooth pairs sharing load; adequate contact ratio promotes continuous and smooth transmission. Interference and undercutting are important limitations, particularly for small pinions, and may be controlled through tooth number, pressure angle, profile shift or modified tooth systems.

Spur gears are simple and efficient but can be noisy at high speed. Helical gears engage progressively and carry load smoothly but generate axial thrust. Bevel gears transmit motion between intersecting shafts and require careful adjustment. Worm gears provide high reduction and compactness but usually have greater sliding loss and thermal demands.

Practical gear engineering requires more than kinematic calculations. Strength, pitting, wear, lubrication, manufacturing accuracy, alignment, backlash, thermal behaviour and safety must be considered together. Standards, inspection, condition monitoring and responsible use of computational tools support reliable professional decisions.

Key Terms

Law of gearing: Requirement that the common normal at contact pass through a fixed pitch point.

Conjugate action: Tooth-profile behaviour that produces a prescribed relative motion.

Involute: Curve generated by unwinding a taut string from a base circle.

Pitch circle: Imaginary rolling circle used to define gear ratio.

Base circle: Circle from which an involute tooth profile is generated.

Pressure angle: Angle between the line of action and the tangent to the pitch circle.

Module: Pitch diameter divided by tooth number.

Circular pitch: Arc distance between corresponding points of adjacent teeth.

Line of action: Common normal along which tooth force acts.

Contact ratio: Average number of tooth pairs simultaneously in contact.

Backlash: Difference between tooth-space width and mating tooth thickness.

Interference: Contact between non-conjugate portions of mating tooth profiles.

Undercutting: Removal of tooth-root material during generation.

Profile shift: Controlled displacement of the generating tool relative to the gear blank.

Spur gear: Gear with teeth parallel to the shaft axis.

Helical gear: Gear with teeth inclined at a helix angle.

Bevel gear: Gear with teeth arranged on a conical pitch surface.

Worm gear: Screw-like worm meshing with a toothed wheel.

Transmission error: Deviation of actual motion from ideal conjugate motion.

Pitting: Surface-fatigue damage producing small cavities on tooth flanks.

Review Questions

Short Answer Questions

1. State the law of gearing.
2. Define module and give its relation to pitch diameter and tooth number.
3. Why are involute profiles widely used in industrial gears?
4. What is meant by contact ratio?
5. State one major advantage and one disadvantage of helical gears.

Long Answer Questions

1. Explain involute generation and show how it produces constant velocity ratio.
2. Discuss interference and undercutting, including methods for avoiding them.
3. Compare spur, helical, bevel and worm gears in terms of geometry, application and limitations.
4. Explain the effects of manufacturing errors, alignment and lubrication on gear performance.
5. Describe the forces acting on a spur-gear tooth and explain their significance for shafts and bearings.

Analytical / Application Questions

1. A 24-tooth pinion drives a 72-tooth gear at 1,200 r/min. Determine the ideal speed of the gear, direction of rotation and ratio of pitch diameters.
2. A gear transmits 2 kW at 750 r/min through a pitch radius of 80 mm. For a 20° pressure angle, calculate the ideal tangential and radial tooth forces.
3. A continuously operating conveyor requires a compact 15:1 reduction. Compare a two-stage helical gearbox with a single worm stage, considering efficiency, heat rejection, noise, axial or thrust loads and maintenance.

Multiple Choice Questions with Answers

Question 1

For an external gear pair, the ideal speed ratio is expressed as:

A. ω₁/ω₂ = N₁/N₂

B. ω₁/ω₂ = N₂/N₁

C. ω₁/ω₂ = N₁ + N₂

D. ω₁/ω₂ = N₁ − N₂

Correct Answer: B. ω₁/ω₂ = N₂/N₁

Question 2

According to the law of gearing, the common normal at the contact point must:

A. Be tangent to both addendum circles

B. Pass through a fixed point on the line of centres

C. Remain perpendicular to the line of centres

D. Pass through the midpoint of the face width

Correct Answer: B. Pass through a fixed point on the line of centres

Question 3

The base-circle radius of an involute gear is:

A. r sin φ

B. r tan φ

C. r cos φ

D. r/cos φ

Correct Answer: C. r cos φ

Question 4

For a standard spur gear, the base pitch is:

A. πm sin φ

B. πm cos φ

C. m tan φ

D. πm/cos φ

Correct Answer: B. πm cos φ

Question 5

A 20-tooth gear with module 2.5 mm has a pitch diameter of:

A. 8 mm

B. 22.5 mm

C. 50 mm

D. 80 mm

Correct Answer: C. 50 mm

Question 6

For a 20° standard full-depth involute system, the approximate minimum tooth number to avoid ordinary undercutting is closest to:

A. 8

B. 12

C. 18

D. thirty

Correct Answer: C. 18

Question 7

Compared with spur gears, helical gears generally:

A. Produce no axial force

B. Engage less progressively

C. Produce axial thrust

D. Cannot be used on parallel shafts

Correct Answer: C. Produce axial thrust

Question 8

A worm with 2 starts drives a 40-tooth worm wheel. The ideal speed reduction is:

A. 2:1

B. 10:1

C. 20:1

D. 80:1

Correct Answer: C. 20:1

Question 9

For a spur gear with tangential force Fₜ and pressure angle φ, the radial force is:

A. Fₜ sin φ

B. Fₜ cos φ

C. Fₜ tan φ

D. Fₜ/cos φ

Correct Answer: C. Fₜ tan φ

Question 10

Which feature most directly measures the average number of tooth pairs sharing contact?

A. Backlash

B. Contact ratio

C. Module

D. Helix angle

Correct Answer: B. Contact ratio