In heavy machinery such as tunnel boring machines, large internal gears are critical components that often face uneven load distribution due to shaft misalignments. This misalignment, typically caused by operational environmental factors, leads to localized stress concentrations, premature wear, and potential tooth surface failure. As an internal gear manufacturer, addressing these issues is paramount to enhancing the durability and performance of internal gears in demanding applications. Traditional modification methods, including profile and lead crowning, have been employed to mitigate such problems, but they often treat these modifications independently, limiting their effectiveness. In this study, we propose a comprehensive tooth surface modification approach that integrates neural network optimization with orthogonal experiments to simultaneously adjust profile and lead parameters, transforming eccentric loads into uniformly distributed stresses across the tooth surface. This method not only reduces localized stresses but also minimizes transmission errors, offering a cost-effective solution for large internal gears without the need for extensive physical testing.
The tooth surface modification begins with a mathematical model that defines the modified surface using a set of spatial points. For a spur gear, the modified tooth surface can be represented in a cylindrical coordinate system, where each point \( P_{ki} \) on the surface is described by the following equations:
$$ r_{ki} = \sqrt{r_a^2 + (l – l_{ki})^2 – 2r_a (l – l_{ki}) \sin \alpha_a} $$
$$ \theta_{ki} = (\theta_{1ki} + \theta_{2ki} – \theta_{3ki}) \frac{180}{\pi} + \theta_k $$
$$ z_{ki} = z_k $$
Here, \( r_a \) is the tip radius, \( l \) is the modification length, \( l_{ki} \) is the modification length at point \( P_{ki} \), \( \alpha_a \) is the tip pressure angle, and \( z_k \) is the axial coordinate. The modification length \( l \) is derived from the gear geometry as \( l = (\epsilon – 1) m_t \cos \alpha_t \pi \), where \( \epsilon \) is the contact ratio, \( m_t \) is the transverse module, and \( \alpha_t \) is the transverse pressure angle. The angular components \( \theta_{1ki} \), \( \theta_{2ki} \), and \( \theta_{3ki} \) are given by:
$$ \theta_{1ki} = \tan \alpha_{ki} – \alpha_{ki} $$
$$ \theta_{2ki} = \frac{s_{ki}}{2r_{ki}} $$
$$ \theta_{3ki} = \arctan \left( \frac{r_{ki} \sin \theta_{2ki} – \Delta_{ki}}{l_{ki}} \right) $$
$$ \theta_k = \frac{c_k}{r_a} \frac{180}{\pi} $$
In these equations, \( \alpha_{ki} \) is the pressure angle at point \( P_{ki} \), \( s_{ki} \) is the tooth thickness, \( \Delta_{ki} \) represents the profile modification value, and \( c_k \) is the lead modification amount at section \( k \). The pressure angle \( \alpha_{ki} \) is calculated as \( \alpha_{ki} = \alpha_a – \arcsin \left( \frac{l – l_{ki}}{r_{ki}} \cos \alpha_a \right) \). The profile modification \( \Delta_{ki} \) and lead modification \( c_k \) are controlled by parameters such as superposition coefficients \( \sigma \) and \( w \), and exponential coefficients \( \gamma \), \( \beta \), \( \mu \), and \( \phi \), as shown below:
$$ \Delta_{ki} = \Delta_{\text{max}} \left[ \sigma \left( \frac{l_{ki}}{l} \right)^\gamma + (1 – \sigma) \left( \frac{l_{ki}}{l} \right)^\beta \right] $$
$$ c_k = c_c \left[ w \left( \frac{z_k}{b_c} \right)^\mu + w \left( \frac{z_k}{b_c} \right)^\phi \right] $$
Here, \( \Delta_{\text{max}} \) is the maximum profile modification, \( c_c \) is the axial modification amount, and \( b_c \) is the distance from the lead modification center to the gear face. The parameters \( \sigma, w \in [0,1] \) and \( \gamma, \beta, \mu, \phi \in [1,3] \) allow for flexible adjustment of the tooth surface shape, enabling a comprehensive modification that addresses both profile and lead aspects simultaneously. This approach is particularly beneficial for internal gears, where traditional methods may fall short in handling complex load distributions. By generating a point cloud from these equations and fitting a surface in CAD software, we create a 3D model of the modified gear, which is then used for finite element analysis (FEA) to evaluate performance under misalignment conditions.
To optimize the modification parameters, we employ an orthogonal experimental design combined with a neural network. The orthogonal array \( L_{25}(5^6) \) is used to systematically vary the six key parameters: \( \sigma \), \( \gamma \), \( \beta \), \( w \), \( \mu \), and \( \phi \). This design reduces the number of experiments while covering a wide range of parameter combinations. Each combination is evaluated using FEA to determine the maximum Hertzian contact stress, pinion root stress, and internal gear root stress. The results are then used to train a neural network based on the Levenberg-Marquardt (L-M) algorithm, which enhances the prediction accuracy for optimal parameter sets. The L-M algorithm combines the fast convergence of Gauss-Newton methods with the global search capability of gradient descent, making it suitable for small-sample problems. The weight update in the neural network is given by:
$$ \Delta x = -\left[ J^T(x) J(x) + \mu_1 I \right]^{-1} J(x) e(x) $$
where \( J(x) \) is the Jacobian matrix, \( e(x) \) is the error vector, \( I \) is the identity matrix, and \( \mu_1 \) is a damping parameter that adjusts during training. This network takes the three stress outputs as inputs and predicts the six modification parameters, allowing us to identify optimal values that minimize stresses and improve load distribution. For large internal gears, this method eliminates the need for costly physical trials, which is a significant advantage for internal gear manufacturers seeking to enhance product reliability.
The finite element modeling process involves creating a detailed mesh of the gear pair, with emphasis on refining the contact regions to ensure accurate stress analysis. Based on the contact ratio, we include multiple tooth pairs in the model to capture a full meshing cycle. The mesh is generated using hypermesh software, with hexahedral elements to improve computational efficiency. The gear parameters used in this study are summarized in the table below:
| Parameter | Value |
|---|---|
| Pinion teeth number \( z_1 \) | 15 |
| Internal gear teeth number \( z_2 \) | 105 |
| Module \( m \) (mm) | 22 |
| Pinion width \( b_1 \) (mm) | 250 |
| Internal gear width \( b_2 \) (mm) | 240 |
| Pressure angle \( \alpha \) (°) | 20 |
| Elastic modulus \( E \) (GPa) | 211 |
| Poisson’s ratio \( \nu \) | 0.33 |
| Shaft misalignment angle \( \theta \) (°) | 0.1833 |
In the orthogonal experiments, 25 parameter combinations are tested, and the resulting stresses are recorded. The table below shows a subset of the results, highlighting the variation in stresses with different parameter sets:
| Experiment | \( \sigma \) | \( \gamma \) | \( \beta \) | \( w \) | \( \mu \) | \( \phi \) | Max Contact Stress (MPa) | Pinion Root Stress (MPa) | Internal Gear Root Stress (MPa) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 0.1 | 1 | 1 | 0.1 | 1 | 1 | 1689.28 | 209.35 | 286.57 |
| 2 | 0.1 | 1.5 | 1.5 | 0.3 | 1.5 | 1.5 | 1442.01 | 249.64 | 275.99 |
| 3 | 0.1 | 2 | 2 | 0.5 | 2 | 2 | 1462.42 | 256.85 | 270.60 |
| 4 | 0.1 | 2.5 | 2.5 | 0.7 | 2.5 | 2.5 | 1429.32 | 282.05 | 267.99 |
| 5 | 0.1 | 3 | 3 | 0.9 | 3 | 3 | 1423.07 | 275.99 | 269.50 |
From these results, we observe that certain parameter combinations, such as Experiment 5, yield lower contact stresses, indicating better load distribution. However, the orthogonal experiment alone may not capture the optimal trade-offs between stresses. Therefore, we use the neural network to predict parameters that target specific stress values, such as 1200 MPa for contact stress, 270 MPa for pinion root stress, and 260 MPa for internal gear root stress. The neural network, divided into two sub-networks for profile and lead parameters, achieves training accuracies of over 95% for profile parameters and 87% for lead parameters. The predicted parameters are \( \sigma = 0.2403 \), \( \gamma = 2.9387 \), \( \beta = 2.8492 \), \( w = 0.4595 \), \( \mu = 1.8027 \), and \( \phi = 2.4967 \).
Finite element analysis of the gear model with these optimized parameters shows a significant improvement in stress distribution. The maximum contact stress is reduced to 1277.66 MPa, which is close to the target, with errors of 6.47% for contact stress, 4.09% for pinion root stress, and 4.74% for internal gear root stress. The load distribution becomes more uniform across the tooth surface, avoiding the edge loading seen in unmodified gears. Additionally, the transmission error, which is a key indicator of vibration and noise, is analyzed over three meshing cycles. The peak-to-peak transmission error decreases from 0.02794 in the unmodified case to 0.0042 after optimization, demonstrating smoother operation and reduced dynamic loads.
This integrated approach offers substantial benefits for internal gear manufacturers, as it enables the design of high-performance internal gears without relying on iterative physical testing. By leveraging neural networks and orthogonal experiments, we can efficiently identify modification parameters that enhance load capacity and longevity. Future work could extend this method to helical internal gears or incorporate dynamic effects for even more robust designs. In conclusion, the combination of comprehensive tooth surface modification, orthogonal testing, and neural network optimization provides a powerful tool for addressing the challenges faced by internal gears in heavy machinery applications.