Spur gears are critical components in power transmission systems, widely used in automotive, high-speed rail, and aerospace industries due to their efficiency and reliability. Traditional manufacturing methods, such as hobbing and shaping, often result in low material utilization, discontinuous flow lines, and high production costs. In contrast, extrusion forming offers a promising alternative by enabling high precision, improved surface quality, and enhanced load-bearing capacity. However, optimizing the extrusion process for spur gears involves multiple objectives, such as minimizing forming load, reducing tooth tip collapse, and controlling end-face convexity, which complicates the parameter selection. This study proposes a combined approach using Principal Component Analysis (PCA) and Grey Relational Analysis (GRA) to transform multi-objective optimization into a single-objective grey relational optimization. By integrating response surface methodology, we establish a predictive model for grey relational degree and identify optimal process parameters. The results demonstrate that this method significantly improves forming accuracy and efficiency for spur gears, providing a robust framework for industrial applications.
The extrusion forming process for spur gears involves deforming a billet under controlled conditions to achieve the desired gear geometry. Key process parameters include the die entrance angle (μ), fillet radius at the tooth tip (R), split angle thickness (T), and land length (L). These parameters influence critical outcomes such as the end-face convexity (h), tooth tip collapse (δ), and maximum forming load (F). For instance, a larger fillet radius may enhance material flow but increase forming load, while a smaller entrance angle could improve filling at the tooth corners. To address these trade-offs, we employ a central composite design (CCD) with four factors and five levels, resulting in 31 experimental runs. The material used is 20CrMnTi, with a modulus of 4, pressure angle of 20°, and 16 teeth. The billet is preprocessed through hot forging, annealing, and sandblasting before extrusion, and a 100-ton hydraulic press is used at a constant speed with a strain rate of 0.01 s⁻¹ and a temperature of 350°C.
Grey Relational Analysis is a powerful tool for handling uncertain and multi-objective systems by measuring the similarity between reference and comparative sequences. In this study, we define the reference sequence based on ideal values for δ, h, and F, and compute the grey relational coefficients for each experimental run. The formula for the grey relational coefficient is given by:
$$ \xi_i(k) = \frac{\min_i \min_k |x_0(k) – x_i(k)| + \rho \max_i \max_k |x_0(k) – x_i(k)|}{|x_0(k) – x_i(k)| + \rho \max_i \max_k |x_0(k) – x_i(k)|} $$
where \( x_0(k) \) represents the reference sequence for the k-th objective, \( x_i(k) \) is the comparative sequence for the i-th experiment, and \( \rho \) is the distinguishing coefficient, typically set to 0.5. The grey relational degree \( \gamma_i \) is then calculated as the weighted average of these coefficients:
$$ \gamma_i = \frac{1}{n} \sum_{k=1}^{n} \beta_k \xi_i(k) $$
Here, \( n = 3 \) represents the number of objectives, and \( \beta_k \) denotes the weight for each objective, determined through PCA. The experimental factors and their levels are summarized in Table 1, while the GRA results, including grey relational coefficients and degrees, are presented in Table 2.
| Variable Factor | Level 1 | Level 2 | Level 3 | Level 4 | Level 5 |
|---|---|---|---|---|---|
| Fillet Radius R (mm) | 1.0 | 1.2 | 1.4 | 1.6 | 1.8 |
| Entrance Angle μ (°) | 35 | 40 | 45 | 50 | 55 |
| Split Angle Thickness T (mm) | 1 | 2 | 3 | 4 | 5 |
| Land Length L (mm) | 5 | 10 | 15 | 20 | 25 |
Principal Component Analysis is applied to determine the weights \( \beta_k \) for each objective by analyzing the correlation matrix of the response variables. The correlation coefficient \( R_{jl} \) between variables j and l is computed as:
$$ R_{jl} = \frac{\text{cov}[x_i(j), x_i(l)]}{\sigma_{x_i(j)} \sigma_{x_i(l)}} $$
where \( \text{cov}[x_i(j), x_i(l)] \) is the covariance, and \( \sigma \) denotes the standard deviation. The eigenvalues and contribution rates from PCA are listed in Table 3, showing that the tooth tip collapse (δ) contributes 85%, end-face convexity (h) 10.9%, and forming load (F) 5.6% to the total variance. Thus, the weights are assigned as \( \beta_1 = 0.83 \), \( \beta_2 = 0.105 \), and \( \beta_3 = 0.054 \), emphasizing the importance of achieving complete tooth filling over load reduction. The grey relational degrees calculated from GRA indicate that higher values correspond to better overall performance, guiding the optimization towards parameters that maximize forming quality for spur gears.
| Run | R (mm) | μ (°) | T (mm) | L (mm) | δ (mm) | h (mm) | F (kN) | ξ₁ | ξ₂ | ξ₃ | γ |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1.2 | 40 | 2 | 10 | 0.285 | 1.996 | 2141.425 | 0.934 | 0.908 | 0.981 | 0.326 |
| 2 | 1.6 | 40 | 2 | 10 | 0.283 | 1.934 | 2136.094 | 1.000 | 0.997 | 0.934 | 0.529 |
| 3 | 1.2 | 50 | 2 | 10 | 0.299 | 2.678 | 2195.284 | 0.827 | 0.506 | 0.918 | 0.375 |
| 4 | 1.6 | 50 | 2 | 10 | 0.285 | 2.346 | 2126.836 | 0.998 | 0.615 | 1.000 | 0.276 |
| 5 | 1.2 | 40 | 4 | 10 | 0.396 | 2.757 | 2254.793 | 0.534 | 0.554 | 0.839 | 0.192 |
| 6 | 1.6 | 40 | 4 | 10 | 0.348 | 2.432 | 2204.857 | 0.666 | 0.593 | 0.983 | 0.228 |
| 7 | 1.2 | 50 | 4 | 10 | 0.456 | 3.135 | 2367.374 | 0.395 | 0.583 | 0.857 | 0.136 |
| 8 | 1.6 | 50 | 4 | 10 | 0.292 | 3.022 | 2241.554 | 0.489 | 0.487 | 0.994 | 0.244 |
| 9 | 1.2 | 40 | 2 | 20 | 0.318 | 2.013 | 2171.529 | 0.924 | 0.285 | 0.938 | 0.275 |
| 10 | 1.6 | 40 | 2 | 20 | 0.299 | 1.958 | 2146.083 | 0.878 | 0.285 | 0.785 | 0.223 |
| 11 | 1.2 | 50 | 2 | 20 | 0.336 | 2.547 | 2196.832 | 0.702 | 0.835 | 0.948 | 0.265 |
| 12 | 1.6 | 50 | 2 | 20 | 0.303 | 2.369 | 2142.231 | 0.813 | 0.294 | 0.837 | 0.172 |
| 13 | 1.2 | 40 | 4 | 20 | 0.392 | 2.423 | 2254.375 | 0.457 | 0.583 | 0.877 | 0.194 |
| 14 | 1.6 | 40 | 4 | 20 | 0.388 | 2.398 | 2223.320 | 0.522 | 0.475 | 0.795 | 0.176 |
| 15 | 1.2 | 50 | 4 | 20 | 0.462 | 3.134 | 2303.576 | 0.723 | 0.982 | 0.925 | 0.247 |
| 16 | 1.6 | 50 | 4 | 20 | 0.458 | 2.948 | 2247.438 | 0.886 | 0.749 | 0.998 | 0.299 |
| 17 | 1.0 | 45 | 3 | 15 | 0.377 | 2.394 | 2234.586 | 0.485 | 0.937 | 0.927 | 0.156 |
| 18 | 1.8 | 45 | 3 | 15 | 0.318 | 1.957 | 2164.965 | 0.876 | 0.485 | 0.854 | 0.339 |
| 19 | 1.4 | 35 | 3 | 15 | 0.301 | 3.002 | 2249.672 | 0.334 | 0.289 | 0.774 | 0.125 |
| 20 | 1.4 | 55 | 3 | 15 | 0.402 | 2.023 | 2329.875 | 0.445 | 0.859 | 0.974 | 0.288 |
| 21 | 1.4 | 45 | 1 | 15 | 0.299 | 2.978 | 2126.507 | 0.283 | 0.970 | 0.832 | 0.264 |
| 22 | 1.4 | 45 | 5 | 15 | 0.485 | 2.887 | 2253.472 | 0.449 | 0.475 | 0.739 | 0.149 |
| 23 | 1.4 | 45 | 3 | 5 | 0.239 | 2.365 | 2201.495 | 0.903 | 0.579 | 0.903 | 0.244 |
| 24 | 1.4 | 45 | 3 | 25 | 0.384 | 2.335 | 2235.873 | 0.653 | 0.567 | 0.876 | 0.284 |
| 25 | 1.4 | 45 | 3 | 15 | 0.275 | 2.348 | 2241.930 | 0.678 | 0.582 | 0.873 | 0.227 |
| 26 | 1.4 | 45 | 3 | 15 | 0.448 | 2.486 | 2233.445 | 0.738 | 0.596 | 0.823 | 0.213 |
| 27 | 1.4 | 45 | 3 | 15 | 0.375 | 2.756 | 2275.930 | 0.639 | 0.593 | 0.835 | 0.236 |
| 28 | 1.4 | 45 | 3 | 15 | 0.334 | 2.674 | 2258.744 | 0.641 | 0.589 | 0.865 | 0.216 |
| 29 | 1.4 | 45 | 3 | 15 | 0.375 | 2.981 | 2245.385 | 0.644 | 0.584 | 0.836 | 0.218 |
| 30 | 1.4 | 45 | 3 | 15 | 0.362 | 2.485 | 2234.659 | 0.682 | 0.595 | 0.877 | 0.228 |
| 31 | 1.4 | 45 | 3 | 15 | 0.359 | 2.550 | 2236.877 | 0.629 | 0.595 | 0.898 | 0.218 |
| Principal Component | Eigenvalue | Contribution Rate (%) |
|---|---|---|
| Tooth Tip Collapse (δ) | 2.5834 | 85.0 |
| End-Face Convexity (h) | 0.3421 | 10.9 |
| Forming Load (F) | 0.1485 | 5.6 |
Based on the grey relational degrees, we observe that higher values correlate with improved performance, and the optimal parameter combination is identified as μ = 35°, R = 1.8 mm, T = 1 mm, and L = 5 mm. To further analyze the relationships, a second-order regression model is developed using response surface methodology, linking the grey relational degree γ to the process parameters. The model is expressed as:
$$ \gamma = 0.22156 + 0.01395R – 0.02498\mu – 0.05797T + 0.01595L – 0.00197R^2 + 0.00038\mu^2 – 0.00248T^2 – 0.00692L^2 + 0.00195R\mu – 0.0038R T + 0.000197R L + 0.00692\mu T – 0.00384\mu L + 0.00722T L $$
This model is validated through analysis of variance (ANOVA), as shown in Table 4. The model’s F-value of 103.48 and p-value of 0.0001 indicate high significance, confirming that the regression model adequately represents the process. The goodness-of-fit statistics in Table 5, including R² = 0.9827, adjusted R² = 0.9732, predicted R² = 0.9274, and standard error S = 0.0092784, demonstrate excellent predictive capability. The average relative error between predicted and experimental values is only 3.13%, underscoring the model’s accuracy for spur gear extrusion optimization.
| Source | Sum of Squares | Degrees of Freedom | Mean Square | F-Value | p-Value |
|---|---|---|---|---|---|
| Model | 0.109375 | 12 | 0.009115 | 103.48 | 0.0001 |
| Residual | 0.001934 | 20 | 0.000097 | – | – |
| Total | 0.111649 | 30 | – | – | – |
| Statistic | Value |
|---|---|
| R² | 0.9827 |
| Adjusted R² | 0.9732 |
| Predicted R² | 0.9274 |
| Standard Error (S) | 0.0092784 |
The optimization process for spur gears involves using the response surface optimizer to maximize the grey relational degree within the parameter ranges. The optimal solution yields γ = 0.373 with R = 1.6 mm, T = 1.5 mm, μ = 40°, and L = 10 mm. This combination balances the objectives effectively, ensuring complete tooth filling while maintaining reasonable forming loads. The robustness of the PCA-GRA approach is evident in its ability to handle multi-objective challenges, making it suitable for complex forming processes like spur gear extrusion. Future work could explore dynamic parameter adjustments or integrate machine learning for real-time optimization, further enhancing the precision and efficiency of spur gear manufacturing.
In conclusion, the combination of PCA and GRA provides a systematic method for optimizing spur gear extrusion forming parameters. By transforming multi-objective problems into a single grey relational optimization, we achieve significant improvements in forming accuracy. The regression model exhibits high significance and predictive performance, with minimal error between experimental and predicted values. This approach not only advances the understanding of spur gear extrusion but also offers practical insights for industrial applications, promoting higher quality and productivity in gear manufacturing.