Spur gears are among the most widely used transmission components in automotive and machinery applications, leading to immense market demand. Traditional manufacturing processes predominantly rely on machining methods such as “gear hobbing + gear shaving” or “gear shaping + gear shaving.” While these techniques offer high precision and stable quality, they suffer from significant drawbacks including low material utilization, interrupted metal flow lines, high production costs, and limited efficiency. In contrast, the advancement of precision plastic forming technologies and mold manufacturing has enabled the successful development and gradual adoption of cold extrusion for producing spur gears of various modules and tooth counts in advanced manufacturing industries. Gears formed by cold extrusion exhibit high dimensional accuracy, excellent surface quality, and enhanced load-bearing capacity, presenting vast application prospects.
Despite the advantages, the cold extrusion of spur gears faces challenges related to forming quality. Common defects include insufficient filling at tooth corners, large machining allowances on end faces, and significant collapse angles at tooth tips, which reduce material yield and increase scrap rates. This study focuses on optimizing the through-type cold extrusion process for a specific spur gear to minimize the bottom end-face protrusion and the tooth tip collapse angle at the gear’s exit side, while also considering the forming load.
We employ numerical simulation based on the FORGE software to analyze the metal flow during the process. A central composite experimental design is adopted to investigate the influence of key die geometrical parameters. Subsequently, a multi-objective optimization problem is transformed into a single-objective optimization of grey relational grade using Grey Relational Analysis (GRA) combined with Principal Component Analysis (PCA) for weighting. A second-order response surface model is established to map the relationship between process parameters and the grey relational grade. The optimal parameter combination is then determined and validated through actual forming experiments.
1. Process Analysis and Numerical Modeling
The subject of this study is a spur gear with a module \(m = 4\), number of teeth \(Z = 16\), pressure angle \(\alpha = 20^\circ\), and material 20CrMnTi. To enhance material utilization and production efficiency, a through-type cold extrusion process is adopted. The process schematic is shown in Figure 1. A pre-formed ring blank is placed in the die cavity. The upper punch moves downward, forcing the metal to flow through the tooth-shaped die insert (female die), thereby forming the gear teeth. The final part is ejected from the bottom.
The critical stage of this forming process is the initial phase where the blank first enters the deformation zone. Inhomogeneous metal flow velocity between the inner and outer layers, caused by friction and geometrical constraints, leads to the primary defects: protrusion of the bottom end-face and collapse at the bottom tooth tips (exit side).
2. Finite Element Simulation Setup
A 3D model of the through-type cold extrusion process was created using UG 8.0. To save computational time while maintaining accuracy, a segment model representing a single tooth with periodic boundary conditions was employed for simulation in FORGE 2011. The blank was defined as a plastic body with the material properties of 20CrMnTi. The simulation parameters are summarized in Table 1.
| Parameter | Value |
|---|---|
| Minimum Mesh Size | 1.21 mm |
| Friction Type | Coulomb |
| Friction Factor | 0.1 |
| Temperature | 20 °C |
| Extrusion Speed | 20 mm/s |
| Blank Inner Diameter | Φ39.25 mm |
| Blank Outer Diameter | Φ74.1 mm |
| Blank Height | 43.3 mm |
The simulation results clearly depicted the three stages of forming. In the initial stage, the metal begins to flow into the tooth cavity. The outer layer metal flows slower than the inner layer due to greater frictional contact with the die wall, causing the inner layer to pull the outer layer and resulting in bottom end-face protrusion and tooth tip collapse. In the middle stage, as the metal enters the sizing land, the flow becomes more uniform and primarily axial. In the final stage, the metal exits the die and undergoes elastic recovery.
A comparison between the simulated and a physically formed spur gear showed good agreement, confirming the validity of the finite element model for analyzing the process. The measured defects from the initial trial were a bottom end-face protrusion \(h\) of approximately 3 mm and a bottom tooth tip collapse \(\delta\) of about 0.6 mm over a length of 5.5 mm.
3. Multi-Objective Optimization Based on Grey Relational Analysis
To improve the forming quality of the spur gears, four key geometrical parameters of the tooth die insert were selected as design variables (See Figure 2a): Tooth Tip Fillet Radius (\(R\)), Entry Angle (\(\mu\)), Split Angle Thickness (\(T\)), and Sizing Land Length (\(L\)). The optimization objectives were to minimize the Bottom Tooth Tip Collapse Amount (\(\delta\)), the Bottom End-face Protrusion (\(h\)), and the Maximum Forming Load (\(F\)) (See Figure 2b).
3.1 Experimental Design and Grey Relational Analysis
A four-factor, five-level Central Composite Design (CCD) was employed, resulting in 31 experimental runs. The factors and their levels are listed in Table 2.
| Factor | Level 1 | Level 2 | Level 3 | Level 4 | Level 5 |
|---|---|---|---|---|---|
| Tooth Tip Radius \(R\) (mm) | 1.0 | 1.2 | 1.4 | 1.6 | 1.8 |
| Entry Angle \(\mu\) (°) | 35 | 40 | 45 | 50 | 55 |
| Split Angle Thickness \(T\) (mm) | 1 | 2 | 3 | 4 | 5 |
| Sizing Land Length \(L\) (mm) | 5 | 10 | 15 | 20 | 25 |
For each run in the design, a numerical simulation was performed, and the responses \(\delta\), \(h\), and \(F\) were extracted. Grey Relational Analysis (GRA) was then used to convert this multi-response problem into a single-response problem. First, the experimental data sequences for each response were normalized. Since the goal is to minimize all three objectives, the “smaller-is-better” characteristic was used for normalization. The reference sequence \(X_0\) was defined as the ideal minimum values observed from all experiments: \(X_0 = \{\delta_{min}, h_{min}, F_{min}\}\).
The grey relational coefficient \(\xi_i(k)\) for the \(k\)-th response in the \(i\)-th experiment, which expresses the relationship between the comparative sequence and the reference sequence, was calculated using the following formula:
$$
\xi_i(k) = \frac{\min_i \min_k |x_i(k) – x_0(k)| + \rho \max_i \max_k |x_i(k) – x_0(k)|}{|x_i(k) – x_0(k)| + \rho \max_i \max_k |x_i(k) – x_0(k)|}
$$
where \(\rho\) is the distinguishing coefficient, set to 0.5. A higher grey relational coefficient indicates that the corresponding response is closer to the ideal value.
3.2 Determination of Weighting via Principal Component Analysis
The overall grey relational grade \(\gamma_i\) for the \(i\)-th experiment is a weighted sum of the coefficients for each response. Simply averaging them assumes equal importance, which is not accurate. Principal Component Analysis (PCA) was applied to the matrix of grey relational coefficients (\(\xi_1, \xi_2, \xi_3\)) to determine the objective weights based on their variance and contribution. PCA transforms correlated variables into uncorrelated principal components. The first principal component explains the largest proportion of variance in the data.
The covariance matrix \(R_{jl}\) was computed from the standardized grey relational coefficients:
$$
R_{jl} = \frac{\text{cov}[x_i(j), x_i(l)]}{\sigma_{x_i(j)} \sigma_{x_i(l)}}
$$
The eigenvalues \(\lambda_k\) and eigenvectors were obtained by solving the characteristic equation \(|\lambda I – R| = 0\). The contribution rate \(\alpha_k\) of each principal component was calculated as:
$$
\alpha_k = \frac{\lambda_k}{\sum_{i=1}^{n} \lambda_i}
$$
The results of the PCA are shown in Table 3. The first principal component, dominated by the tooth tip collapse (\(\delta\)), accounts for 84% of the total variance. The second principal component (end-face protrusion \(h\)) accounts for 10.7%, and the third (forming load \(F\)) for 5.3%. This indicates that improving tooth filling is the primary concern in optimizing the cold extrusion of these spur gears, rather than merely reducing the load. Therefore, the weights for the grey relational grade were assigned as \(\beta_1 = 0.84\) for \(\delta\), \(\beta_2 = 0.107\) for \(h\), and \(\beta_3 = 0.053\) for \(F\).
| Principal Component | Eigenvalue | Contribution Rate (%) |
|---|---|---|
| PC1 (Dominant: \(\delta\)) | 2.5185 | 84.0 |
| PC2 (Dominant: \(h\)) | 0.3217 | 10.7 |
| PC3 (Dominant: \(F\)) | 0.1598 | 5.3 |
The grey relational grade \(\gamma_i\) for each experimental run was then calculated using the weighted formula:
$$
\gamma_i = \sum_{k=1}^{n} \beta_k \cdot \xi_i(k) = 0.84\xi_i(1) + 0.107\xi_i(2) + 0.053\xi_i(3)
$$
A higher \(\gamma\) value indicates that the corresponding combination of process parameters yields responses closer to the ideal values. The calculated grades for all 31 runs are part of the full experimental data. The mean grey relational grade for each level of the four factors was computed to identify preliminary trends, as shown in Table 4. The range (max-min) of these mean grades indicates the factor’s influence: a larger range means the factor has a stronger effect on the multi-response outcome.
| Process Parameter | Level 1 | Level 2 | Level 3 | Level 4 | Level 5 | Range |
|---|---|---|---|---|---|---|
| Tooth Tip Radius \(R\) | 0.1769 | 0.2161 | 0.2252 | 0.2369 | 0.2508 | 0.0740 |
| Entry Angle \(\mu\) | 0.2991 | 0.2497 | 0.2232 | 0.2033 | 0.1547 | 0.1444 |
| Split Angle Thickness \(T\) | 0.3206 | 0.2875 | 0.2255 | 0.1655 | 0.1030 | 0.2175 |
| Sizing Land Length \(L\) | 0.2872 | 0.2397 | 0.2197 | 0.2133 | 0.2116 | 0.0757 |
From Table 4, it is evident that the Split Angle Thickness (\(T\)) has the most significant influence on the forming quality of the spur gears, followed by the Entry Angle (\(\mu\)). The Tooth Tip Radius (\(R\)) and Sizing Land Length (\(L\)) have relatively smaller effects. The preliminary optimal level combination based on the highest mean grade for each factor is: \(R=1.8\) mm, \(\mu=35^\circ\), \(T=1\) mm, \(L=5\) mm.
3.3 Development of the Response Surface Model
To precisely locate the optimum within the design space and understand the interaction effects, a second-order Response Surface Methodology (RSM) model was developed. The grey relational grade (\(\gamma\)) was taken as the response, and the four process parameters (\(R, \mu, T, L\)) were the input variables. Using statistical software, a quadratic polynomial regression model was fitted to the data. The resulting model in coded units is presented below:
$$
\begin{aligned}
\gamma = & 0.22156 + 0.01395R – 0.02498\mu – 0.05797T – 0.01595L \\
& – 0.00197R^2 + 0.0038\mu^2 – 0.00248T^2 + 0.00692L^2 \\
& + 0.00384R\mu – 0.00273\mu T + 0.00722TL
\end{aligned}
$$
The adequacy of the model was rigorously checked. Figure 3 shows a comparison between the predicted values from the model and the actual simulated values. The points lie close to the 45-degree line, indicating a good fit. The analysis of variance (ANOVA) for the model is summarized in Table 5. The extremely low p-value (0.0001) for the model, well below the significance level of 0.05, confirms that the model is statistically significant and that the terms in the model have a significant effect on the grey relational grade.
| Source | Degrees of Freedom | Sum of Squares | Mean Square | F-Value | p-Value |
|---|---|---|---|---|---|
| Regression Model | 11 | 0.109802 | 0.009982 | 102.45 | 0.0001 |
| Residual Error | 19 | 0.001851 | 0.0000974 | – | – |
| Total | 30 | 0.111653 | – | – | – |
Further model statistics, shown in Table 6, validate its predictive capability. The high values of the coefficient of determination (\(R^2 = 98.34\%\)) and the adjusted \(R^2\) (\(R^2_{adj} = 97.38\%\)) indicate that the model explains over 97% of the variability in the grey relational grade. The predicted \(R^2\) (\(R^2_{pred} = 93.93\%\)) is in reasonable agreement with the adjusted \(R^2\), suggesting good predictive power for new observations. The low standard deviation (S) value also points to a precise model.
| Statistic | \(R^2\) | \(R^2_{adj}\) | \(R^2_{pred}\) | S |
|---|---|---|---|---|
| Value | 98.34% | 97.38% | 93.93% | 0.00987 |
4. Optimization Results and Experimental Verification
Using the numerical optimization function based on the validated response surface model, the goal was set to maximize the grey relational grade \(\gamma\) within the defined variable ranges. The optimization algorithm found a maximum \(\gamma\) of 0.374. The corresponding optimal process parameter combination for manufacturing the spur gears was:
- Tooth Tip Fillet Radius (\(R\)): 1.6 mm
- Entry Angle (\(\mu\)): 40°
- Split Angle Thickness (\(T\)): 1.5 mm
- Sizing Land Length (\(L\)): 10 mm
This optimal combination differs from the preliminary one derived from the main effects plot, highlighting the importance of considering interaction effects captured by the RSM model. A final simulation run with these optimal parameters was conducted. Furthermore, a physical cold extrusion experiment was performed on a 4 MN hydraulic press. The billets were subjected to spheroidizing annealing and phosphating-soaping treatment for lubrication prior to extrusion. The manufactured die insert with the optimized geometry and the resulting extruded spur gear are shown in Figure 4.
The formed spur gear exhibited a significant improvement in quality. Compared to the initial trial (baseline parameters), the bottom tooth tip collapse \(\delta\) was reduced by 51.7%, and the bottom end-face protrusion \(h\) was reduced by 36.3%. The tooth profile was fully filled, and the surface quality was excellent. This successful verification confirms the effectiveness and reliability of the integrated optimization approach combining finite element simulation, Grey Relational Analysis, Principal Component Analysis, and Response Surface Methodology for the cold extrusion process of spur gears.
5. Conclusions
This study presents a systematic methodology for optimizing the through-type cold extrusion process of spur gears, focusing on mitigating forming defects. The following key conclusions are drawn:
- Process Analysis: The finite element simulation accurately captured the metal flow during the cold extrusion of spur gears, identifying the root cause of bottom end-face protrusion and tooth tip collapse as the velocity gradient between the inner and outer metal layers during the initial forming stage.
- Integrated Optimization Method: A robust multi-objective optimization framework was established. Grey Relational Analysis (GRA) effectively converted the multi-response problem (minimize \(\delta\), \(h\), \(F\)) into a single-response problem (maximize grey relational grade \(\gamma\)). Principal Component Analysis (PCA) provided a rational, data-driven method for determining objective weights, revealing that improving tooth filling is the primary concern (84% contribution) over reducing forming load for these spur gears.
- Factor Influence: Among the four die geometry parameters studied, the Split Angle Thickness (\(T\)) has the most pronounced effect on the forming quality of the spur gears, followed by the Entry Angle (\(\mu\)). The Tooth Tip Radius (\(R\)) and Sizing Land Length (\(L\)) have comparatively lesser influence.
- Optimal Solution and Validation: The response surface model built upon the GRA-PCA results successfully mapped the complex relationships between process parameters and the overall performance index (\(\gamma\)). The optimal parameter combination (\(R=1.6\) mm, \(\mu=40^\circ\), \(T=1.5\) mm, \(L=10\) mm) was determined through numerical optimization. Experimental validation confirmed that this combination significantly improves the quality of the cold extruded spur gears, reducing the target defects by over 36%, thereby enhancing material utilization and product consistency.
The proposed methodology provides a valuable reference for optimizing forming processes for similar gear-like components and other complex multi-response manufacturing operations.