Gearing: A Complete Guide to Types, Parameters, and Calculations

Introduction

Whether you are a design office technician, a student in Industrial Product Design, or an engineer in charge of a mechanical transmission, the sizing of a cylindrical spur gear remains one of the fundamental skills in industrial mechanics. However, French-language resources that cover the entire chain—from specifications to metrological control in the workshop—are rare.

This comprehensive technical guide accompanies you through two real numerical case studies: an industrial mine conveyor gearbox (45 kW) and a standard academic pinion (15 kW, m = 3 mm). For each case, you will find the formulas, detailed step-by-step calculations, mechanical validation criteria (Lewis, Hertz), shaft and bearing sizing, as well as the quality control protocol to apply after machining.

By the end of this article, you will be able to size, validate, and control a cylindrical spur gear from A to Z, using the same tools as a professional design office.

1. Main Types of Cylindrical Gears

• Spur Gears: Teeth parallel to the axis. Simple but noisy at high speeds.
• Helical Gears: Inclined teeth. Smoother and quieter transmission.
• Bevel Gears: Transmission between non-parallel axes (often at 90°).
• Worm Gear System: High speed reduction and self-locking in reverse.
• Rack and Pinion System: Conversion of rotation → linear motion.

2. Key Parameter: The Module (m) of the Gear

Two gears can only mesh if they have the same module (standardized unit in mm). The module defines the size of the teeth and determines the entire geometry.

d = m × Z

3. Risks Associated with Poor Gear Design

• Pitting: Surface fatigue due to excessive contact pressure (Hertz).
• Root Bending Fatigue Failure: The most destructive failure mode.
• Scuffing: Breakdown of the oil film → instantaneous welding of surfaces.
• Abrasive Wear: Particles or lack of lubricant reducing tooth thickness.

4. Pitfalls to Avoid During Sizing

• Undercutting: Z < 17 for α = 20° weakens the tooth root.
• Center Distance Too Rigid or Too Loose: Jamming or load concentration.
• Neglected Face Width Factor: b > 25·m causes misalignment defects.
• Overlooked Dynamic Shocks: Incorporate service factor K_A (crushers, conveyors).

5. Case A — Step-by-Step Calculation of a Spur Gear (45 kW Conveyor)

Data: P = 45 kW | N₁ = 1500 rpm | u = 3 | Carburized steel (R_pe = 300 MPa) | α = 20°

Step 1 — Calculation of Torque on Pinion

ω₁ = (2 × π × N₁) / 60 ≈ 157.08 rad/s
C₁ = P / ω₁ = 45,000 / 157.08 ≈ 286.48 N·m

Step 2 — Number of Pinion and Gear Teeth

Z₁ = 20 | Z₂ = Z₁ × u = 20 × 3 = 60

Step 3 — Calculation of Module by Lewis Formula

m_theoretical = 3.10 mm → m_standardized = 4 mm

Step 4 — Final Gear Geometry

d₁ = 4 × 20 = 80 mm | d₂ = 4 × 60 = 240 mm
b = 10 × 4 = 40 mm | a = (80 + 240) / 2 = 160 mm

6. Manufacturing and Quality Control in the Workshop

Machining Processes

• Gear Hobbing: Hob cutter or Fellow's process.
• Heat Treatment: Carburizing + quenching.
• Flank Grinding: ISO 5 to 7 precision.

Dimensional Control

• Dimensional: Measurement over pins/balls, plate micrometer.
• Profile and Helix: Gear-specific CMM.
• NDT: Magnetic particle inspection or dye penetrant inspection.

7. Recalculation for Helical Gearing (β = 12°)

Step 1 — Apparent Module

m_t = m_n / cos(β) = 4 / cos(12°) ≈ 4.0895 mm

Step 2 — Pitch Diameters

d₁ = 4.0895 × 20 = 81.79 mm | d₂ = 4.0895 × 60 = 245.37 mm

Step 3 — Nominal Center Distance

a = (81.79 + 245.37) / 2 = 163.58 mm

Step 4 — Minimum Face Width

b_min = (π × 4) / sin(12°) ≈ 60.4 mm → b chosen = 65 mm

8. Center Distance Tolerances According to ISO 286 (ISO Class 7)

163.58 ^+0.040_+0.010 mm

Pitfall: A negative deviation causes tooth jamming due to thermal expansion and immediate scuffing.

9. Calculation of Bearing Loads

F_t = 2·C₁ / d₁ ≈ 7,005 N

F_r = F_t × tan(α_n) / cos(β) ≈ 2,606 N

F_a = F_t × tan(β) ≈ 1,489 N

10. Bearing Selection — Case A

Selected: Tapered roller bearings in opposition — SKF 32208
40 × 80 × 24.75 mm | C ≈ 76.5 kN | Life > 20,000 h

11. Functional Clearance (Backlash) According to ISO 1328

According to ISO 1328 / AGMA, for m_n = 4 mm, a = 163.58 mm, ISO class 7:

j_t = 0.15 to 0.25 mm

• Dial Indicator: Lock the gear, oscillate the pinion by hand.
• Lead Wire: Thickness of crushed wire measured with micrometer.

12. Final Technical Summary — Case A

13. Keyway Sizing (Case A)

Parallel key, form A — ISO 773 / NF E 22-177 — for d = 40 mm, C₁ = 286.48 N·m.

• Section: b = 12 mm | h = 8 mm | t₁ = 5 mm | t₂ = 3.3 mm

Steel 35NCD16 (100 MPa): l ≥ 43.4 mm ✓

Final choice: Key 12 × 8 × 50 mm in treated 35NCD16 steel.

14. Maintenance Plan and Operational Monitoring

15. Case B — Academic Pinion Sizing: 20 Teeth, m = 3 mm, α = 20°

15.1 Operating Data (Specifications)

• Nominal Power (P): 15 kW
• Input Speed (N₁): 1,500 rpm
• Pinion Torque (C₁): 95.5 N·m
• Reduction Ratio (r): 0.4 (2.5 reduction)
• Expected Mechanical Efficiency (η): 98.7% (standard value for cut spur gear, oil bath lubricated, ISO class 7–9)

15.2 Geometric Machining Specifications

15.3 Mechanical Validation: Lewis Bending and Hertz Contact Pressure

• Tangential Force (F_t): 3,183 N
• Radial Force (F_r): 1,159 N
• Total Normal Force (F_n): 3,390 N
• Bending Strength (Lewis): σ_f = 93.2 MPa ≤ σ_fp = 150 MPa ➔ VALIDATED ✅
• Contact Pressure Strength (Hertz): σ_H = 747.2 MPa ≤ σ_Hp = 800 MPa ➔ VALIDATED ✅

15.4 Calculation of Torque and Forces on the Tooth

ω = (2π × 1,500) / 60 = 157.08 rad/s | C = 15,000 / 157.08 = 95.49 N·m

F_t = (2 × 95.49) / 0.06 = 3,183 N
F_r = 3,183 × tan(20°) = 1,159 N
F_n = 3,183 / cos(20°) = 3,386 N

15.5 Sizing of the Pinion Shaft in Pure Torsion

R_pg = 120 / 3 = 40 MPa | d_shaft ≥ ∛(16 × 95,500 / π × 40) ≈ 22.99 mm → 25 mm (ISO R40)

Keyway ISO 2491 for d = 25 mm: w = 8 mm | h = 7 mm | groove depth = 4 mm

15.6 Reactions at Bearings A and B

R_A = R_B = √(1,591.5² + 579.5²) ≈ 1,693.7 N ≈ 1,700 N

16. Metrological Control: Measurement over Teeth with Plate Micrometer

16.1 Gear Inspection Instruments

• Plate micrometer (gear caliper): Measures the measurement over teeth W_k.
• Vernier caliper (double vernier): Measures chordal thickness on the pitch diameter.
• Gauge pins (rollers): Measure the measurement over pins.
• Rolling bench (double flank): Detects overall defects.
• CMM (Coordinate Measuring Machine): Probes the exact tooth profile.

16.2 The 4 Major Controls After Machining

Step A — Tooth Thickness Control (Measurement over teeth W_k)

1. Theoretical calculation of the number of teeth k to encompass.
2. Insert the micrometer plates around k teeth.
3. Measure at 3 different locations (at 120°).

Step B — Involute Profile Control (Profile error f_f)

1. The probe scans the tooth flank from the root diameter to the tip.
2. The machine plots a deviation graph. A perfect curve yields a straight line.

Step C — Tooth Alignment Control (Helix error f_β)

1. The probe moves parallel to the shaft axis over width b = 30 mm.
2. Detects runout or taper.

Step D — Double Flank Meshing Control (Dynamic Control)

1. Mount the gear on a movable axis opposite a master gear.
2. A spring keeps the two gears in double flank contact.
3. Rotate 360°. A sensor records variations in center distance.
4. This test highlights the total composite error and the gear runout.

16.3 ISO 1328 Accuracy Classes for Gears

• Classes 4 to 6: High precision (automotive, aeronautics). Grinding mandatory.
• Classes 7 to 9: Standard industrial gears (our case 15 kW). Direct hobbing.

16.4 Calculation of Measurement over Teeth W_k with Micrometer — 20-tooth pinion, m = 3 mm

k = round(20 × 20/180 + 0.5) = 3 teeth
inv(20°) = tan(20°) − α_rad = 0.36397 − 0.34906 = 0.01491

W_k = 3 × 0.93969 × [π × 2.5 + 20 × 0.01491] = 22.981 mm

• Workshop acceptance range: 22.910 mm to 22.960 mm (ISO 1328, class 8)
• If value > 22.98 mm → teeth too thick, jamming.
• If value < 22.80 mm → tool penetration too deep, weakening in bending.

17. Manufacturing and Inspection Dashboard

18. Validation Supplements — Design Office Level

⚠️ This section complements the previous calculations with essential verifications for a complete industrial validation file.

18.1 Energy Balance — Power and Torque at Driven Wheel

The efficiency η = 98.7 % is a standard value for a straight-cut gear, oil-lubricated, in ISO class 7–9 (source: Niemann, Maschinenelemente, vol. 2).

P₂ = 15,000 × 0.987 = 14,805 W ≈ 14.8 kW
C₂ = 95.5 × 2.5 × 0.987 = 235.6 N·m
Thermal losses: ΔP = 195 W → natural cooling sufficient

18.2 Safety Factors S_F and S_H

⚠️ S_H Alert: With K_A = 1.25 (frequent starts), σ_H,eff ≈ 835 MPa > 800 MPa. Correct by increasing b (30 → 35 mm) or improving surface hardness.

18.3 Bearing Selection for Pinion Shaft d = 25 mm

Required C ≥ 1,700 × (20,000 × 60 × 1,500 / 10⁶)^1/3 = 20,672 N ≈ 20.7 kN

✅ Final Choice: SKF 6305 — 25 × 62 × 17 mm | C = 22.5 kN | L_10h ≈ 24,300 h > 20,000 h ✓

⭐ Key Formulas to Remember — Cylindrical Spur Gear