With the rapid expansion of urban areas and increasing reliance on public transportation, subway systems have become a critical component of urban mobility. The safety and reliability of subway vehicles are paramount, and the transmission system, particularly the helical gears within the gearbox, plays a vital role in ensuring smooth operation. Helical gears are widely used due to their ability to transmit power efficiently with reduced noise and vibration. However, bending fatigue failure, often originating at the tooth root, is a common issue that can lead to catastrophic failures if not properly addressed. In this study, we focus on predicting the bending fatigue life of helical gears used in subway vehicle transmissions and optimizing their design parameters, specifically the profile shift coefficients, to enhance fatigue performance. We employ finite element analysis (FEA), experimental testing, and theoretical modeling to develop a comprehensive approach for life prediction and optimization.
The importance of helical gears in subway transmissions cannot be overstated, as they directly influence the vehicle’s operational safety and maintenance costs. Traditional design methods often rely on standard calculations, but these may not fully capture the complex stress states under dynamic operating conditions. Therefore, we aim to integrate advanced simulation techniques with empirical data to create a more accurate fatigue life prediction model. This research not only addresses the specific needs of subway systems but also contributes to the broader field of gear design by providing insights into the effects of profile shift coefficients on fatigue behavior. Throughout this work, we emphasize the role of helical gears in transmitting torque and their susceptibility to bending fatigue, which is a key concern in high-cycle fatigue scenarios.
To begin, we analyze the operational characteristics of subway vehicles, which experience varying loads and speeds during service. The transmission helical gears are subjected to cyclic stresses that can lead to crack initiation and propagation at the tooth root. We developed a finite element model of the helical gear pair based on actual subway gearbox parameters. The model was constructed using SolidWorks with the GearTrax plugin and processed in HyperMesh for meshing. Transient dynamic simulations were performed in ABAQUS to simulate the gear meshing process under different operating conditions. The results revealed that the maximum root bending stress occurs during two-tooth contact phases, which account for approximately 15% of the meshing cycle. Importantly, only specific speed ranges (0–50 km/h) produced stresses exceeding the material’s fatigue limit, indicating that these conditions dominate fatigue damage accumulation. The stress distribution and variation patterns were documented, providing a foundation for subsequent fatigue life analysis.
In terms of fatigue life prediction, we conducted single-tooth bending fatigue tests on equivalent spur gears made of 18CrNiMo7-6 steel, as helical gears pose challenges for direct testing. The tests were performed on a PLG-200 high-frequency fatigue testing machine, and strain data were collected to monitor crack initiation. We observed that the crack propagation phase contributes less than 10% to the total fatigue life, allowing us to focus on crack initiation life as the primary indicator. Based on the test results, we developed a bending fatigue life prediction model for cylindrical gears that combines Miner’s linear cumulative damage theory with energy accumulation curves. The energy dissipated during fatigue loading was calculated using the formula: $$E = \sum_{i=1}^{n} \sigma_m \Delta l_i S$$ where \(\sigma_m\) is the mean stress amplitude, \(\Delta l_i\) is the displacement increment, and \(S\) is the area of the crack initiation zone. The cumulative energy \(E\) follows a quadratic relationship with the number of cycles \(n\): $$E = C_1 n^2 + C_2 n + C_3$$ where \(C_1\), \(C_2\), and \(C_3\) are material-dependent constants derived from experimental data. The fatigue life \(N\) is then predicted by equating the cumulative energy to the fracture energy \(E^*\), which is expressed as: $$E^* = A_0 e^{-t_0 \sigma} + B_0$$ Here, \(A_0\), \(B_0\), and \(t_0\) are parameters obtained through regression analysis. This model was validated against experimental data, showing that predicted lives fall within a factor of three of the measured values, confirming its applicability for helical gears in subway transmissions.
Next, we investigated the influence of profile shift coefficients on the bending fatigue life of helical gears. Profile shift coefficients alter the tooth geometry, affecting stress distribution and fatigue resistance. Using the allowable range derived from standard gear design charts, we selected multiple combinations of profile shift coefficients for the pinion and gear. For each combination, we created a single-tooth meshing finite element model in ABAQUS to compute the root bending stress under static loading. The results were compared with calculations based on GB/T 3480 standards, and the agreement was within acceptable limits (e.g., errors below 5.387%), validating the model’s accuracy. The table below summarizes the root stress values for different profile shift coefficients:
| Pinion Profile Shift Coefficient (\(x_1\)) | Gear Profile Shift Coefficient (\(x_2\)) | Pinion Root Stress (MPa) | Gear Root Stress (MPa) |
|---|---|---|---|
| 0.35 | -0.65 | 649.0 | 785.8 |
| 0.40 | -0.35 | 645.7 | 710.0 |
| 0.45 | -0.05 | 646.2 | 628.8 |
| 0.50 | 0.25 | 648.8 | 676.9 |
| 0.55 | 0.55 | 590.3 | 618.3 |
| 0.60 | 0.85 | 648.8 | 516.8 |
| 0.65 | 1.15 | 662.5 | 508.8 |
The fatigue life for each configuration was analyzed using FE-Safe software, incorporating the stress results from ABAQUS. The Smith-Watson-Topper method was applied for stress correction in high-cycle fatigue regimes. The analysis showed that the fatigue life of helical gears varies significantly with profile shift coefficients. For instance, the pinion’s fatigue life changed by up to 3.15 times, while the gear’s life varied by 22.30 times across the range. To visualize these trends, we plotted the root stress and fatigue life against the profile shift coefficients, enabling the identification of optimal values that balance the stress and life between the pinion and gear. The line graph method demonstrated high accuracy, with root stress prediction errors as low as 0.02% and fatigue life errors within 18.76%, making it a reliable tool for optimization.
Building on these findings, we optimized the helical gear pair for subway vehicle transmissions by applying the equal-strength principle through the line graph method. The goal was to minimize the root stress difference between the pinion and gear while maximizing fatigue life. The optimal profile shift coefficients were determined to be \(x_1 = 0.559\) for the pinion and \(x_2 = 0.603\) for the gear, resulting in a predicted root stress of 601 MPa and a fatigue life of 8.199e6 cycles. This represents a 3.29-fold improvement in fatigue life compared to the original design. We also evaluated the transmission quality of the optimized helical gears by calculating the contact ratio, which includes the transverse contact ratio \(\varepsilon_\alpha\) and the overlap ratio \(\varepsilon_\beta\). The total contact ratio \(\varepsilon_\gamma\) is given by: $$\varepsilon_\gamma = \varepsilon_\alpha + \varepsilon_\beta$$ where \(\varepsilon_\alpha\) is computed as: $$\varepsilon_\alpha = \frac{1}{2\pi} \left[ z_1 (\tan \alpha_{a1} – \tan \alpha’) + z_2 (\tan \alpha_{a2} – \tan \alpha’) \right]$$ and \(\varepsilon_\beta\) is: $$\varepsilon_\beta = \frac{b \tan \beta}{\pi m_n}$$ For the optimized gears, \(\varepsilon_\gamma = 2.4353\), ensuring smooth meshing and low noise, which are critical for subway applications. The table below compares the fatigue performance of the original and optimized gear pairs:
| Gear Pair Configuration | Pinion Fatigue Life (Cycles) | Gear Fatigue Life (Cycles) | Overall Fatigue Life (Cycles) |
|---|---|---|---|
| Original (\(x_1=0.488, x_2=0.257\)) | 4.039e6 | 2.492e6 | 2.492e6 |
| Optimized (Equal-Strength Principle) | 10e7 | 8.199e6 | 8.199e6 |
| Optimized (Equal-Life Principle) | 7.624e6 | 10e7 | 7.624e6 |
In conclusion, this study presents a comprehensive framework for predicting and optimizing the bending fatigue life of helical gears in subway vehicle transmissions. The integration of finite element analysis, experimental testing, and theoretical modeling provides a robust approach to address fatigue-related issues. The optimization of profile shift coefficients based on the equal-strength principle significantly enhances the fatigue performance of helical gears, ensuring longer service life and improved reliability. Future work should focus on extending this methodology to helical gears with different tooth counts and conducting direct fatigue tests on helical gears to refine the prediction model. Ultimately, these advancements contribute to safer and more efficient subway operations, highlighting the critical role of helical gears in urban transportation systems.