Gear Contact Stress (Hertz)
Estimate the gear tooth contact fatigue stress σ_H with a simplified Hertz formula using elastic coefficient Z_E and combined load factor K.
Inputs
Formula
Fundamentals
Estimate the gear tooth contact fatigue stress σ_H with a simplified Hertz formula using elastic coefficient Z_E and combined load factor K.
This tool computes Gear Contact Stress (Hertz) from the formula: σH = ZE·√[ Ft·(u+1)/(b·d1·u) · K ], u = Z2/Z1
History
Design theory of gears, threads and bearings matured with machine tools and standardisation (AGMA, ISO) since the Industrial Revolution.
Engineering applications
Used for transmission design, joint strength checks and component selection.
Glossary
| Tangential force Ft | Input parameter |
| Face width b | Input parameter |
| Pinion pitch dia. d1 | Input parameter |
| Teeth ratio u | Output result |
| Contact stress σH | Output result |
How to use
- Fill in Tangential force Ft, Face width b, Pinion pitch dia. d1, Gear teeth Z₂, Pinion teeth Z₁, Elastic coeff. ZE in the Inputs section (watch the unit on each field).
- Click Calculate; the tool evaluates the formula shown above.
- Read Teeth ratio u, Contact stress σH in the results area.
Formula · Worked Example · Knowledge
Formula
核心计算关系：σH = ZE·√[ Ft·(u+1)/(b·d1·u) · K ], u = Z2/Z1
输入变量：
- Tangential force Ft
Ft— N - Face width b
b— mm - Pinion pitch dia. d1
d1— mm - Gear teeth Z₂
Z2 - Pinion teeth Z₁
Z1 - Elastic coeff. ZE
ZE— √MPa - Load factor K
K
输出结果：
- Teeth ratio u
u - Contact stress σH
sigmaH— MPa
假设/适用：Gearing/thread formulas assume standard, zero-backlash mounting; include error & load factors K_A.
Worked Example
根据公式 σH = ZE·√[ Ft·(u+1)/(b·d1·u) · K ], u = Z2/Z1，取 Tangential force Ft = 2,000 N；Face width b = 30 mm；Pinion pitch dia. d1 = 60 mm；Gear teeth Z₂ = 60；Pinion teeth Z₁ = 20；Elastic coeff. ZE = 189.8 √MPa；Load factor K = 1.5。
代入计算得：Teeth ratio u = 3；Contact stress σH = 282.9 MPa。
Key Points
- Gearing/thread formulas assume standard, zero-backlash mounting; include error & load factors K_A.
- Compare stress with allowable stress and keep a safety factor (≥1.5, ≥2 for critical parts).
- For high speed/load also check lubrication, heat and fatigue life, not just static strength.
Parameters
Inputs
| Parameter | Symbol | Unit | Default |
|---|---|---|---|
| Tangential force Ft (N) | Ft | N | 2000 |
| Face width b (mm) | b | mm | 30 |
| Pinion pitch dia. d1 (mm) | d1 | mm | 60 |
| Gear teeth Z₂ | Z2 | 60 | |
| Pinion teeth Z₁ | Z1 | 20 | |
| Elastic coeff. ZE (√MPa) | ZE | √MPa | 189.8 |
| Load factor K | K | 1.5 |
Outputs
| Result | Symbol | Unit |
|---|---|---|
| Teeth ratio u | u | |
| Contact stress σH | sigmaH | MPa |
Applications
- Common engineering use cases