Spur Gearboxes: Fundamentals and Design
Spur gearboxes (also called parallel-shaft gearboxes) are among the most common and versatile types in mechanical engineering. They transmit power between parallel shafts with high efficiency of up to 99% per stage and are available in a wide range of sizes and gear ratios.
This guide explains the fundamentals of spur gear technology, compares spur and helical gears, and describes the design procedure per DIN 3990 / ISO 6336.
Operating Principle
Spur gears transmit force and torque through the involute tooth profile. The involute is a mathematically defined curve that ensures constant velocity transmission between meshing gears regardless of manufacturing tolerances in center distance. The standard pressure angle is 20° (DIN 3960).
The force in the tooth contact acts along the line of action (pressure line) at the pressure angle to the common tangent at the pitch point. This results in a normal force Fn and its components: tangential force Ft (torque transmission) and radial force Fr (bearing loads).
Ft = 2M / d | Fr = Ft × tan(α) | Fn = Ft / cos(α)
M = torque [Nm], d = pitch circle diameter [m], α = pressure angle (standard 20°)
Spur vs. Helical Gears
| Criterion | Straight (Spur) | Helical |
|---|---|---|
| Noise level | Higher (abrupt engagement) | Low (gradual engagement) |
| Load capacity | Standard | 15–30% higher |
| Axial force | None | Yes (thrust bearings required) |
| Speed suitability | Medium | High |
| Manufacturing cost | Lower | Higher |
| Efficiency | Up to 99% | Up to 99% (slightly lower) |
| Typical helix angle | 0° | 8°–20° |
| Applications | Low-speed, simple | High-speed, precision |
Key Design Parameters
- Module m: m = d / z. Defines tooth size. Both gears must have the same module. Standard modules per DIN 780: 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10 mm.
- Tooth count z: Determines gear ratio i = z2 / z1. Minimum tooth count without undercutting: zmin = 17 for spur gears (lower with profile shift).
- Center distance a: a = m × (z1 + z2) / 2 for standard gears. Must be precisely maintained.
- Face width b: b = ψd × d1, where ψd = 0.3–1.2 depending on application. Larger face width increases load capacity but also shaft deflection.
- Profile shift x: Addendum modification factor. Positive shift increases tooth root strength and prevents undercutting.
- Pressure angle α: Standard is 20°. Higher pressure angles (25°) increase load capacity but also radial force on bearings.
Materials and Heat Treatment
Common gear materials:
- Gray cast iron (GG): Low-cost, good vibration damping, suitable for low loads and speeds
- Steel (C45, 42CrMo4): Standard for industrial gears; good strength and machinability
- Plastics (PA, POM, PEEK): For light loads, quiet operation, chemical resistance, or food applications
Common heat treatments:
- Through-hardened: Uniform hardness HRC 30–45; good for moderate loads
- Case carburized (Einsatzhärten): Carbon-enriched surface layer, hardened to HRC 58–63 with tough core; highest load capacity for high-performance gears
- Nitrided: Nitrogen diffusion into surface, achieving >700 HV surface hardness; lower distortion than case carburizing, suitable for precision gears
- Induction-hardened: Selective hardening of tooth surface; cost-effective alternative to case carburizing
Design per DIN 3990 / ISO 6336
The standardized strength verification procedure per DIN 3990 / ISO 6336 follows 5 steps:
- Determine load spectrum (torques, speeds, duty cycles)
- Select material and heat treatment, determine allowable stress values
- Calculate pitting resistance (surface durability): σH ≤ σHP
- Calculate tooth root strength (bending): σF ≤ σFP
- Verify safety factors: SH ≥ 1.3 (pitting), SF ≥ 1.4 (bending)
Practical Example
Application: Single-stage helical gearbox
- Input power: P = 7.5 kW at n1 = 1,500 rpm
- Input torque: M1 = 9550 × P / n = 9550 × 7.5 / 1500 = 47.8 Nm
- Required gear ratio: i = 4:1 → n2 = 375 rpm, M2 ≈ 191 Nm
- Material: 42CrMo4, case carburized
- Helix angle: β = 15°
Selection result:
- Module m = 3 mm, z1 = 18, z2 = 72
- Center distance a = 135 mm
- Face width b = 50 mm (ψd = 0.93)
- SH = 1.45, SF = 1.65 → requirements met
TEA Recommendation
TEA offers standard spur and helical gear pairs as well as complete gearbox units. For custom applications, we work with certified manufacturers to deliver gear pairs designed exactly to your specifications.
Request Gear ConsultationFrequently Asked Questions about Spur Gearboxes
The module m = d / z describes the ratio of pitch circle diameter d [mm] to number of teeth z. It is the fundamental parameter for gear sizing. Two meshing gears must have the same module. Larger modules mean larger teeth with higher load capacity but lower speed (larger pitch circle). Standard modules are: 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10 (DIN 780).
Helical gears run significantly more quietly and smoothly because multiple teeth are always in simultaneous contact – the load is distributed. The helix angle (typically 8°–20°) creates an axial force component that must be supported by bearings. Helical gears transmit higher torques and achieve higher speeds with the same module. The disadvantage is the axial force requiring thrust bearing capacity.
Center distance a = m × (z1 + z2) / 2 for standard gears without profile shift. With profile shift x1, x2: a = m × (z1 + z2) / 2 + m × (x1 + x2). The center distance determines the installation distance between the two shaft axes and must be maintained precisely to ensure the correct mesh.
Profile shift (addendum modification) shifts the generation rack relative to the gear center. A positive shift (x > 0) increases tooth thickness and tip circle, improves load capacity, and prevents undercutting for small tooth counts. A negative shift reduces these values. Profile shift allows optimization of the gear pair without changing the center distance.
DIN 3990 / ISO 6336 requires minimum safety factors of: SF_H ≥ 1.2–1.5 (surface durability / pitting) and SF_F ≥ 1.4–1.7 (tooth root strength). Higher values are required for shock loads, limited inspection access, or safety-critical applications. The exact required values depend on the reliability target and operating conditions.
Über den Autor
Alexander Olenberger
Sales & Application Engineer · Technical Sales
Specializes in gear technology and drive system design for industrial applications.