Spur Gearboxes: Fundamentals and Design
A spur gearbox transmits torque between parallel shafts via involute-toothed gears — achieving efficiencies up to 99%, making it the most efficient standard gearbox type in mechanical engineering. Its simplicity, reliability, and broad availability make it the default choice for most industrial torque-transmission tasks, offering an excellent balance of cost, performance, and maintainability.
This guide explains the fundamentals of spur gear geometry, the differences between straight and helical teeth, and the steps required for correct sizing per DIN 3990 and 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 | 95–98% | 97–99% (slightly higher) |
| Typical helix angle | 0° | 15°–25° |
| 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 (16MnCr5, 42CrMo4): Standard for heavily loaded gears; excellent strength and machinability. For non-standard requirements TEA manufactures spur gears to drawing — with any module and tooth-count combination.
- 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 (case hardening): 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 spur 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
- Tooth type: straight (spur gearing)
- Material: 16MnCr5, case-hardened
Selection result:
- Module m = 2 mm, z1 = 20, z2 = 80
- Center distance a = m × (z1 + z2) / 2 = 100 mm
- Face width b = 40 mm
- SH > 1.5 → requirements met (root strength σF verified separately)
Contact-stress check (ISO 6336)
The decisive criterion for service life is the Hertzian contact stress σH on the tooth flank. It starts from the tangential force at the pinion reference circle (d1 = m × z1 = 40 mm):
Ft = 2 × M1 / d1 = 2 × 47.8 Nm / 0.040 m ≈ 2,390 N
σH0 = ZH × ZE × Zε × √( Ft/(d1×b) × (u+1)/u )
With ZH ≈ 2.5 (pressure angle α = 20°), ZE = 190 √(N/mm²) for steel/steel, Zε ≈ 0.9, u = 80/20 = 4 and b = 40 mm: σH0 = 2.5 × 190 × 0.9 × √( 2,390/(40×40) × 5/4 ) ≈ 580 N/mm².
The permissible value for case-hardened 16MnCr5 is σH lim ≈ 1,300–1,500 N/mm². Even after applying the application and load factors (KA, Kv, KHβ) per ISO 6336, the contact stress stays well below it — the pitting safety SH = σH lim/σH exceeds 1.5, above the usual minimum (SH ≥ 1.0–1.3). The root-strength proof (σF per ISO 6336-3) is verified separately. All Z/K factors and σH lim are reference values; binding figures come from the standard and the material data sheet.
TEA Recommendation
Practical Tip from TEA:
The most common design mistake we see in consultations is calculating with the nominal torque without an application factor: for shock-loaded drives (conveyors, presses), KA per ISO 6336 quickly reaches 1.5–1.75 — if it is omitted, the gearbox is safe on paper but undersized in practice. With helical gears, also make sure that the axial force arising from the helix angle is accounted for in the bearing concept (angular-contact or tapered-roller bearings instead of pure radial bearings). And always state the quality grade per DIN 3961/3962 in your enquiry — without this figure, noise, smoothness, and price vary considerably, because every supplier assumes a different tolerance.
Request Gear ConsultationFrom design to enquiry: procurement notes
- Cost drivers: Module, quality grade (DIN 3961), and heat treatment determine price — case carburizing or nitriding adds a noticeable premium over through-hardening.
- Standard vs. custom: Catalogue spur gears with standard modules (DIN 780) cover the majority of standard applications. Non-standard modules or unusual tooth-count combinations require production to drawing — worthwhile when installation space is constrained or special material properties are needed.
- Enquiry checklist: Gear ratio, input torque and speed, centre distance (if fixed), required module, tooth type (straight/helical), material and hardness requirement, target quality grade per DIN 3961, operating hours/design life.
- TCO note: Higher manufacturing quality (tighter tolerances, finer tooth-flank grinding) reduces noise and wear — for continuous-duty applications the additional investment pays off through lower maintenance costs.
- Further reading: Custom spur gears to drawing: Custom gears at TEA — or contact us directly.
Frequently 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 15°–25°) 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.
About the Author
Senior Sales & Application Engineer · Technical Sales
Specializes in gear technology and drive system design for industrial applications.