Abstract: This article focuses on the simulation study of spur gear contact fatigue life. It begins with an introduction to the importance of spur gear transmission systems and the need for fatigue life research. The spur gear’s basic parameters, contact strength, and friction factors considering surface roughness are analyzed. Then, finite element simulation models for static and transient dynamics are established and verified. The time sub-step load spectrum is compiled, and the material S – N curve is estimated. Based on these, a contact fatigue life prediction model is developed. The simulation results under different input speeds, load torques, and friction factors are presented and analyzed, revealing the influence laws and magnitudes of these factors on the contact fatigue life.
1. Introduction
Gear transmission systems play a crucial role in various mechanical devices such as helicopter main reducers, advanced turbofan engines, and gas turbine main reducers. These systems are required to operate continuously in complex and harsh environments, demanding high reliability. To evaluate the reliability of gears under high-speed and heavy-load conditions, research on the fatigue damage life of spur gear components is essential.
1.1 Research Methods
- Fatigue Damage Theory Numerical Simulation: Some studies have focused on numerical simulation based on fatigue damage theories. For example, in , stress analysis was conducted under elastohydrodynamic lubrication (EHL) conditions, and spur gear contact fatigue life was analyzed using Matake and DangVan criteria. In, a multi-axis fatigue life model was established considering the influence of surface roughness on the subsurface stress field.
- Traditional Contact Fatigue Testing: Contact fatigue tests using testing machines have been carried out. In , spur gear contact fatigue tests were performed to fit the gear contact fatigue life curve.
- Finite Element Simulation Modeling: Finite element analysis has also been widely used. In , static and transient dynamic finite element analyses were conducted on simplified gear models, and fatigue life was predicted.
1.2 Limitations of Existing Research
- Traditional contact fatigue tests have high data reliability but are time-consuming and resource-intensive, especially for aviation spur gear.
- Numerical simulations often have limitations in considering three-dimensional contact fatigue changes and lack intuitive explanations for fatigue damage regions.
- In finite element simulation studies of contact fatigue life, there are issues such as inaccurate load spectrum compilation and lack of consideration for spur gear working environment variables and boundary conditions.
2. Spur Gear Meshing Process Analysis
2.1 Spur Gear Basic Parameters
The parameters of the spur gear used in a certain contact fatigue testing machine are shown in Table 1.
| Gear Parameter | Driven Gear | Driving Gear |
|---|---|---|
| Number of Teeth | 24 | 16 |
| Module (mm) | 4.5 | 4.5 |
| Pitch Circle Diameter (mm) | 108 | 72 |
| Tooth Width (mm) | 20 | 20 |
| Transmission Efficiency | 0.97 | 0.97 |
2.2 Tooth Surface Contact Strength
The Hertz contact stress is used for calculating the tooth surface contact strength. The formula for calculating the tooth surface contact strength is given by: where is the load coefficient, is the contact ratio coefficient, is the elastic coefficient, is the input torque, is the tooth width, is the driving gear pitch circle diameter, and is the gear ratio. The values of relevant parameters for a certain aviation gear are provided.
2.3 Tooth Surface Friction Factor Considering Surface Roughness
Based on experimental formulas for time-varying friction factors under EHL conditions, the tooth surface friction factors for different surface roughness values (0.2 μm, 0.4 μm, 0.6 μm) are calculated, and their root mean square values are 0.014, 0.021, and 0.042 respectively.
3. Establishment of Spur Gear Contact Fatigue Finite Element Simulation Model
3.1 Spur Gear Static Analysis Model
- Modeling and Meshing: The spur gear is modeled according to the parameters in Table 1. In Hypermesh, the wheel tooth contact model is divided using hexahedral elements. The driving gear single-tooth model has 239,646 nodes and 215,700 elements, and the driving gear base circle single-tooth model has 4,900 nodes and 4,000 elements.
- Boundary Conditions and Load Application: The input-output relationship of the spur gear meshing is considered. If the load torque is 30,000 Nmm, the input torque of the driving gear is 20,619 Nmm. The driving gear is set with a single-degree-of-freedom rotation joint using joint – revolute, and the driven gear is set as a fixed rotation joint using joint – fixed.
- Result Verification: The contact stress results obtained by Workbench are compared with the theoretical solutions. The errors are within 5% and are generally higher than the theoretical solutions, indicating that the simulation results are relatively reliable for conservative life prediction. The comparison data are shown in Table 2.
| Load Torque (Nmm) | Finite Element Solution (MPa) | Theoretical Solution (MPa) | Relative Error (%) |
|---|---|---|---|
| 30000 | 498.43 | 485.34 | 2.63 |
| 60000 | 698.58 | 686.38 | 1.74 |
| 90000 | 866.56 | 840.64 | 2.99 |
3.2 Spur Gear Transient Dynamic Analysis Model
- Modeling and Meshing: The spur gear is modeled according to Table 1. In Hypermesh, the whole gear model has 103,840 nodes and 86,080 elements to control the mesh quantity considering computational resources and efficiency.
- Boundary Conditions and Load Application: The driving gear speed is set using joint – revolute, the driven gear load torque is set using joint – moment, and the tooth surface friction factor is set according to its root mean square value calculated in 1.3.
3.3 Time Sub-step Load Spectrum Compilation
The fatigue assessment stress time sub-step load spectrum used in fatigue analysis is defined as , where is the time sub-step history stress, is the finite element solution stress, is the load multiplier of the time sub-step load channel, and is the user-defined load scaling factor.
3.4 Material S – N Curve
The material used for a certain aviation gear is Cr alloy forged steel with a tensile strength of 1,010 MPa. Since this material is not in the nCode material library, the S – N curve method with UTS correction is used for conservative estimation, and the S – N curve is presented.
3.5 Tooth Surface Contact Fatigue Life Prediction Model
- Analysis of Gear Meshing Region: The stress and strain characteristics of the spur gear meshing region during stable meshing transmission are studied using Ansys Workbench gear transmission static analysis.
- Determination of Load Scaling Factor: Based on the load change during gear transient dynamic loading and comparison, the load scaling factor is determined.
- Calculation of Fatigue Life: In the SN fatigue (stress fatigue) module of ncode, the stress range and average stress are calculated using the critical plane method and rain flow counting method, and combined with Goodman’s life average stress correction. Finally, the Palmgren – Miner linear damage cumulative theory is applied to predict the contact fatigue life, obtaining the number of cycles for failure initiation and the tooth surface area.
4. Simulation Results and Analysis
4.1 Analysis of Contact Fatigue Life under Different Input Speeds
- Simulation Setup: The driving gear speeds of 1,200 r/min, 1,500 r/min, and 1,800 r/min and the driven gear load torque of 30,000 Nmm are set for transient dynamic analysis. The spur gear stress – time sub-step history is obtained and compared with the static analysis results of 30,000 Nmm load torque to get the scaling factor for contact fatigue life calculation.
- Results and Analysis: The fatigue life cloud charts for different speeds are obtained. As shown in Table 3, under the same load torque, the speed is negatively correlated with the number of cycles. This is because an increase in speed leads to an increase in the fluctuation of spur gear stress history and a more rapid damage accumulation process. At the same time, the higher the speed, the shorter the working life because the time for spur gear to experience a complete cycle of stress is shorter. The dangerous nodes mostly appear at the two ends of the tooth surface pitch line.
| Rotational Speed (r/min) | Torque (Nmm) | Friction Factor | Dangerous Nodes | Number of Cycles | Working Life (h) |
|---|---|---|---|---|---|
| 1200 | 30000 | 0.042 | 423925 | 2.968×1041222 | |
| 1500 | 30000 | 0.042 | 4614601.370×1015222 | ||
| 1800 | 30000 | 0.042 | 4211556.630×1036152 |
4.2 Analysis of Contact Fatigue Life under Different Load Torques
- Simulation Setup: The driving gear speed is set to 2,100 r/min, and the load torques are set to 30,000 Nmm, 60,000 Nmm, and 90,000 Nmm for transient dynamic analysis. The static analysis results are superimposed to generate the scaling factor and obtain the fatigue life cloud charts for different load torques.
- Results and Analysis: As shown in Table 4, the larger the load torque, the smaller the number of cycles. This is because an increase in load torque leads to an increase in input torque and contact stress. Compared with the analysis results in 4.1, it can be seen that the influence of load torque on contact fatigue life is much greater than that of input speed. For aviation gears, to maintain a high working life under high-speed and heavy-load conditions, materials with higher yield strength may need to be replaced.
| Rotational Speed (r/min) | Torque (Nmm) | Friction Factor | Dangerous Nodes | Number of Cycles | Working Life (h) |
|---|---|---|---|---|---|
| 2100 | 30000 | 0.042 | 421155 | 4.61×108 | 3650 |
| 2100 | 60000 | 0.042 | 423923 | 7.13×10 | 57 |
| 2100 | 90000 | 0.042 | 4211555.77×10 | 5 |
4.3 Analysis of Contact Fatigue Life under Different Friction Factors
- Simulation Setup: The driving gear speed is set to 1,200 r/min, and the load torque is set to 30,000 Nmm for transient dynamic analysis. Different tooth surface surface roughness values (0.2 μm, 0.4 μm, 0.6 μm) are set to affect the friction factor for static analysis, and the corresponding load scaling factors are obtained to get the fatigue life cloud charts for different tooth surface surface roughness values.
- Results and Analysis: As shown in Table 5, the friction factor is positively correlated with contact fatigue. This is because under EHL conditions, a smaller friction factor indicates a larger oil film thickness between the tooth surfaces. However, an excessive oil film thickness can lead to an increase in the viscosity of the lubricating oil between the tooth surfaces and an increase in the tooth surface contact pressure. The influence of the tooth surface friction factor on the working life is relatively small, and the working life of spur gear under EHL conditions is relatively high.
| Rotational Speed (r/min) | Torque (Nmm) | Friction Factor | Dangerous Nodes | Number of Cycles | Working Life (h) |
|---|---|---|---|---|---|
| 1200 | 30000 | 0.042 | 4239252.968×1041222 | ||
| 1200 | 30000 | 0.021 | 4239252.799×1038875 | ||
| 1200 | 30000 | 0.0144239252.640×1036667 |
5. Conclusion
- Spur gear contact fatigue life simulation model has been established, and the contact fatigue lives and the most vulnerable dangerous nodes under different input speeds, load torques, and friction factors have been obtained.
- The influence laws and magnitudes of the three influencing factors on the tooth surface contact fatigue life have been determined. The prediction of the dangerous node positions shows that the two ends of spur gear along the tooth width direction are more prone to damage than the middle position. The simulation experimental data can provide references for spur gear contact strength evaluation and subsequent fatigue tests.