Spur Gear Contact Fatigue Life Simulation Under Multi-Condition Operations

By /

Stress Analysis and Theoretical Basis

The contact fatigue life of spur gears depends on stress distribution during meshing. Using Hertz contact theory, the maximum contact stress at the pitch point is calculated as:

$$ \sigma_H = 2.5Z_EZ_\varepsilon\sqrt{\frac{2K_HT_1}{Bd_1^2} \cdot \frac{u+1}{u}} $$

Where $Z_E$=189.8 (elastic coefficient), $K_H$=1.25×1.2×1.0×1.299=1.948 (load factor), and $T_1$=20,619 Nmm (input torque). For spur gears with parameters shown in Table 1, theoretical stress calculations align with finite element (FE) results within 5% error.

Table 1: Spur Gear Specifications
Parameter Driven Gear Driver Gear
Teeth Count 24 16
Module (mm) 4.5 4.5
Face Width (mm) 20 20
Transmission Efficiency 0.97

Transient Dynamics and Load Spectrum

The time-dependent stress history for spur gears is derived through transient dynamics. The load spectrum scaling factor is defined as:

$$ S(step) = P(step) \times ScaleFactor \times S_{FE} $$

Where $S_{FE}$ represents FE-solved stress. A typical stress-time history for spur gear meshing under 30,000 Nmm torque shows peak stress at the pitch line:

Table 2: FE vs Theoretical Stress Comparison
Torque (Nmm) FE Stress (MPa) Theoretical (MPa) Error (%)
30,000 498.43 485.34 2.63
60,000 698.58 686.38 1.74
90,000 866.56 840.64 2.99

Elastohydrodynamic Lubrication (EHL) Effects

Surface roughness significantly impacts spur gear friction coefficients under EHL conditions:

$$ \mu_{rms} = \frac{1}{N}\sqrt{\sum_{i=1}^{N}(\mu_i – \bar{\mu})^2} $$

Measured friction coefficients for different roughness levels are:

Table 3: EHL Friction Coefficients
Roughness (μm) Friction Coefficient
0.2 0.014
0.4 0.021
0.6 0.042

Fatigue Life Prediction Model

The S-N curve for Cr-alloy steel (UTS=1,010 MPa) is estimated using:

$$ \frac{\sigma_a}{\sigma_{UTS}} + \frac{\sigma_m}{\sigma_{UTS}} = 1 $$

Fatigue life calculations employ Palmgren-Miner rule:

$$ D = \sum_{i=1}^{k}\frac{n_i}{N_i} $$

Where $n_i$=cycles at stress level $σ_i$, $N_i$=failure cycles from S-N curve. Critical nodes predominantly appear at gear tooth ends due to edge loading.

Multi-Condition Simulation Results

Rotational Speed Effects

Table 4: Speed Influence on Spur Gear Life
Speed (rpm) Cycles Service Life (hrs)
1,200 2.97×10⁹ 41,222
1,500 1.37×10⁹ 15,222
1,800 6.63×10⁸ 6,152

Torque Loading Effects

Table 5: Torque Impact on Fatigue Life
Torque (Nmm) Cycles Service Life (hrs)
30,000 4.61×10⁸ 3,650
60,000 7.13×10⁶ 57
90,000 5.77×10⁵ 5

Surface Roughness Effects

Table 6: Surface Finish Impact
Friction Coefficient Cycles Life Reduction (%)
0.042 2.97×10⁹ Base
0.021 2.80×10⁹ 5.7
0.014 2.64×10⁹ 11.0

Conclusion

This study establishes a comprehensive spur gear contact fatigue model considering multiple operational parameters. Key findings include:

  1. Critical contact fatigue nodes concentrate at tooth ends along face width
  2. Load torque exhibits greatest influence on life reduction (99.8% decrease from 30kNmm to 90kNmm)
  3. Surface roughness reduction improves fatigue life through EHL friction modulation

The methodology provides valuable insights for spur gear design optimization in aerospace and power transmission applications.

Scroll to Top