In our study, we investigate the influence of cold extrusion process parameters on the forming accuracy of straight spur gear using finite element numerical simulation. The aim is to fully leverage the advantages of precision plastic forming technology, such as high material utilization, ease of automation, and excellent product quality. We focus on a straight spur gear manufactured via cold extrusion, and we systematically analyze the effects of six key parameters: tooth root fillet radius coefficient, number of teeth, module, die entry angle, blank diameter coefficient, and extrusion speed. The results demonstrate that the die entry angle and blank diameter are the most critical factors affecting the forming accuracy of straight spur gear. Through orthogonal experimental design and numerical simulation, we identify optimal parameter combinations that achieve full die filling and high dimensional precision. Our findings provide valuable guidance for the design and production of high-quality straight spur gear through cold extrusion.
Cold extrusion is a well-established manufacturing process for producing straight spur gear with high efficiency and precision. The technology offers significant benefits, including reduced material waste, improved mechanical properties, and the ability to mass-produce components with consistent quality. However, achieving the desired forming accuracy in straight spur gear requires careful control of process parameters. Improper selection can lead to defects such as incomplete die filling, excessive forming loads, or geometric deviations. In this work, we employ the finite element method (FEM) using DEFORM-3D software to simulate the cold extrusion of straight spur gear. We design a six-factor, five-level orthogonal experiment to evaluate the influence of each parameter on the forming accuracy. The results are analyzed through range analysis and validated by a confirmatory simulation. Our objective is to provide a comprehensive understanding of how these parameters affect the final geometry of straight spur gear and to propose an optimized process window for industrial application.
Numerical Simulation Setup
We establish the finite element model of the cold extrusion process for straight spur gear using DEFORM-3D. The geometric models of the punch, die, and billet are created in SolidWorks and exported as STL files. To reduce computational cost, we take advantage of the symmetry of the straight spur gear. Only a wedge-shaped section corresponding to one tooth (central angle π/n, where n is the number of teeth) is modeled. The billet material is AISI-1015 steel, with properties defined for cold working at 20 °C. The mesh consists of 100,000 tetrahedral elements, with local refinement in the tooth-forming region (mesh size ratio 0.01) and coarser mesh elsewhere (size ratio 10). Volume compensation is applied to account for mesh-induced volume loss. Two symmetry planes are defined on the wedge-shaped billet. The friction between the billet and the die is modeled using the constant shear friction law. The Lagrangian incremental algorithm is used for the simulation, with a time step set to one-third of the minimum element edge length. The total simulation steps equal the extrusion stroke divided by the time step. The extrusion stroke is 80% of the billet length L, where L = 0.8D (D is the billet diameter). The billet diameter is defined as d = k·m·z, where k is the blank diameter coefficient, m is the module, and z is the number of teeth. The tooth root fillet radius is given by Rg = Kg·m, with Kg ≤ 0.45 according to reference standards. The pressure angle is fixed at 20°, and the profile shift coefficient is zero. The die entry angle (α) is varied from 60° to 140° to cover a wide range of conditions. The extrusion speed is also varied from 10 mm/s to 6000 mm/s to study its effect. Table 1 lists the six factors and their five levels used in the orthogonal experiment.
| Level | Kg | Number of teeth (z) | Module m (mm) | Die entry angle α (°) | Blank diameter coefficient k | Extrusion speed v (mm/s) |
|---|---|---|---|---|---|---|
| 1 | 0.1 | 13 | 0.8 | 60 | 1.25 | 10 |
| 2 | 0.1 | 17 | 1.5 | 80 | 1.30 | 20 |
| 3 | 0.1 | 21 | 2.5 | 100 | 1.35 | 133 |
| 4 | 0.1 | 25 | 3.5 | 120 | 1.40 | 266 |
| 5 | 0.1 | 29 | 4.5 | 140 | 1.45 | 6000 |
| 6 | 0.2 | 13 | 1.5 | 100 | 1.40 | 6000 |
| 7 | 0.2 | 17 | 2.5 | 120 | 1.45 | 10 |
| 8 | 0.2 | 21 | 3.5 | 140 | 1.25 | 20 |
| 9 | 0.2 | 29 | 4.5 | 60 | 1.30 | 133 |
| 10 | 0.2 | 29 | 0.8 | 80 | 1.35 | 266 |
| 11 | 0.3 | 13 | 2.5 | 140 | 1.30 | 266 |
| 12 | 0.3 | 17 | 3.5 | 60 | 1.35 | 6000 |
| 13 | 0.3 | 21 | 4.5 | 80 | 1.40 | 10 |
| 14 | 0.3 | 25 | 0.8 | 100 | 1.45 | 20 |
| 15 | 0.3 | 29 | 1.5 | 120 | 1.25 | 133 |
| 16 | 0.4 | 13 | 3.5 | 80 | 1.45 | 133 |
| 17 | 0.4 | 17 | 4.5 | 100 | 1.25 | 266 |
| 18 | 0.4 | 21 | 0.8 | 120 | 1.30 | 6000 |
| 19 | 0.4 | 25 | 1.5 | 140 | 1.35 | 10 |
| 20 | 0.4 | 29 | 2.5 | 60 | 1.40 | 20 |
| 21 | 0.45 | 13 | 4.5 | 120 | 1.35 | 20 |
| 22 | 0.45 | 17 | 0.8 | 140 | 1.40 | 133 |
| 23 | 0.45 | 21 | 1.5 | 60 | 1.45 | 266 |
| 24 | 0.45 | 25 | 2.5 | 80 | 1.25 | 6000 |
| 25 | 0.45 | 29 | 3.5 | 100 | 1.30 | 10 |
Definition of Forming Accuracy
To quantify the forming accuracy of straight spur gear, we define the filling ratio P as follows:
$$ P = \frac{S_{\text{simulated}}}{S_{\text{design}}} \times 100\% $$
where \(S_{\text{simulated}}\) is the cross-sectional area of the formed tooth measured from the simulation, and \(S_{\text{design}}\) is the theoretical cross-sectional area of the tooth profile. The measurement procedure involves: (1) obtaining a cross-section of the extruded part along the Z-direction using the Slicing function in DEFORM-3D; (2) importing the image into AutoCAD and tracing the tooth profile contour with a polyline; (3) measuring the straight portion length \(L_1\) in the traced contour, knowing the actual design length \(L_2\); and (4) calculating \(S_{\text{simulated}} = S_{\text{AutoCAD}} / (L_1 / L_2)^2\). For the example of the 8th experiment, we obtained \(P = 118 / 121.3 = 97.5\%\). In addition, we use the tooth root fillet radius \(r\) (in mm) as an alternative indicator of filling quality. Table 2 shows the relationship between \(r\) and \(P\) derived from the 8th gear specimen.
| \(r\) (mm) | \(P\) (%) |
|---|---|
| 0.0 | 100.00 |
| 0.5 | 99.99 |
| 1.0 | 99.95 |
| 1.5 | 99.88 |
| 2.0 | 99.77 |
| 2.5 | 99.62 |
| 3.0 | 99.41 |
| 3.5 | 99.15 |
Orthogonal Experiment Results and Range Analysis
We conducted 25 simulation runs according to the L25 orthogonal array (6 factors, 5 levels). The measured forming accuracy \(P\) for each run is listed in Table 3. The range analysis (direct observation) is performed to determine the influence of each factor. The mean value of \(P\) for each factor at each level is computed, and the range (maximum mean minus minimum mean) indicates the sensitivity of the factor.
| Exp. No. | Kg | z | m (mm) | α (°) | k | v (mm/s) | P (%) |
|---|---|---|---|---|---|---|---|
| 1 | 0.1 | 13 | 0.8 | 60 | 1.25 | 10 | 96.2 |
| 2 | 0.1 | 17 | 1.5 | 80 | 1.30 | 20 | 95.1 |
| 3 | 0.1 | 21 | 2.5 | 100 | 1.35 | 133 | 98.6 |
| 4 | 0.1 | 25 | 3.5 | 120 | 1.40 | 266 | 99.1 |
| 5 | 0.1 | 29 | 4.5 | 140 | 1.45 | 6000 | 99.9 |
| 6 | 0.2 | 13 | 1.5 | 100 | 1.40 | 6000 | 98.4 |
| 7 | 0.2 | 17 | 2.5 | 120 | 1.45 | 10 | 99.9 |
| 8 | 0.2 | 21 | 3.5 | 140 | 1.25 | 20 | 100.0 |
| 9 | 0.2 | 29 | 4.5 | 60 | 1.30 | 133 | 94.1 |
| 10 | 0.2 | 29 | 0.8 | 80 | 1.35 | 266 | 98.4 |
| 11 | 0.3 | 13 | 2.5 | 140 | 1.30 | 266 | 97.1 |
| 12 | 0.3 | 17 | 3.5 | 60 | 1.35 | 6000 | 93.6 |
| 13 | 0.3 | 21 | 4.5 | 80 | 1.40 | 10 | 95.3 |
| 14 | 0.3 | 25 | 0.8 | 100 | 1.45 | 20 | 99.9 |
| 15 | 0.3 | 29 | 1.5 | 120 | 1.25 | 133 | 100.0 |
| 16 | 0.4 | 13 | 3.5 | 80 | 1.45 | 133 | 97.5 |
| 17 | 0.4 | 17 | 4.5 | 100 | 1.25 | 266 | 99.3 |
| 18 | 0.4 | 21 | 0.8 | 120 | 1.30 | 6000 | 100.0 |
| 19 | 0.4 | 25 | 1.5 | 140 | 1.35 | 10 | 99.4 |
| 20 | 0.4 | 29 | 2.5 | 60 | 1.40 | 20 | 96.2 |
| 21 | 0.45 | 13 | 4.5 | 120 | 1.35 | 20 | 100.0 |
| 22 | 0.45 | 17 | 0.8 | 140 | 1.40 | 133 | 100.0 |
| 23 | 0.45 | 21 | 1.5 | 60 | 1.45 | 266 | 95.6 |
| 24 | 0.45 | 25 | 2.5 | 80 | 1.25 | 6000 | 97.3 |
| 25 | 0.45 | 29 | 3.5 | 100 | 1.30 | 10 | 98.7 |
Table 4 presents the mean accuracy \(P\) for each level of each factor and the range (R). The results show that the die entry angle α has the largest range (4.66%), making it the most influential parameter on the forming accuracy of straight spur gear. The blank diameter coefficient k exhibits the second-largest range (1.56%), followed by Kg (1.30%), module m (1.20%), number of teeth z (1.06%), and extrusion speed v (1.02%). The optimal levels for maximizing accuracy are: Kg = 0.4, z = 29, m = 0.8 mm, α = 120°, k = 1.25, and v = 20 mm/s. At these settings, the predicted mean accuracy is 100%.
| Level | Kg | z | m (mm) | α (°) | k | v (mm/s) |
|---|---|---|---|---|---|---|
| Mean at level 1 | 97.78 | 97.84 | 98.90 | 95.14 | 98.56 | 97.90 |
| Mean at level 2 | 98.16 | 97.58 | 97.70 | 96.72 | 97.00 | 98.75 |
| Mean at level 3 | 97.18 | 97.90 | 97.82 | 98.98 | 98.00 | 97.73 |
| Mean at level 4 | 98.48 | 97.96 | 97.78 | 99.80 | 97.80 | 97.90 |
| Mean at level 5 | 98.32 | 98.64 | 97.72 | 99.28 | 98.56 | 97.84 |
| Range R | 1.30 | 1.06 | 1.20 | 4.66 | 1.56 | 1.02 |
Discussion
The observed strong influence of the die entry angle on the forming accuracy of straight spur gear can be explained by the material flow behavior during extrusion. A larger die entry angle (e.g., 120° or 140°) reduces the resistance to metal flow into the tooth cavity, promoting better filling of the die corners. However, excessively large angles (above 140°) may lead to flow dead zones and increased friction, which can slightly degrade accuracy. The blank diameter also plays a significant role: a smaller blank diameter coefficient (k = 1.25) results in less redundant material and more uniform deformation, which helps achieve near-net shape. On the other hand, a larger blank diameter increases the required extrusion load and may cause uneven flow, leading to underfill or flash. The module and number of teeth affect the geometry of the tooth profile, but their influence is relatively moderate within the tested range. Extrusion speed has the smallest effect, indicating that the process is relatively insensitive to speed variations under cold extrusion conditions for straight spur gear. The tooth root fillet radius coefficient Kg, while not the most dominant, still influences the stress concentration at the root and the ability to fill the root region completely. Our results suggest that a moderate value of Kg (0.4) provides a good balance.
Optimization and Validation
Based on the range analysis, we select the optimal combination: Kg = 0.4, z = 29, m = 0.8 mm, α = 120°, k = 1.25, and v = 20 mm/s. We conduct a confirmatory simulation under these conditions. The formed straight spur gear exhibits a fully filled tooth profile with no visible defects. The measured forming accuracy P is 100%, which agrees with the predicted value. The tooth root fillet radius is effectively zero, indicating perfect die replication. This validation confirms that the identified parameter set yields the highest possible forming accuracy for straight spur gear in our study.
Conclusions
From our numerical investigation on the cold extrusion of straight spur gear, we draw the following conclusions:
- The die entry angle is the most critical factor affecting the forming accuracy of straight spur gear. A die entry angle of 120° provides the best filling performance.
- The blank diameter coefficient is the second most influential parameter. A coefficient of 1.25 is recommended to achieve high accuracy.
- The tooth root fillet radius coefficient, number of teeth, module, and extrusion speed have relatively minor effects, but optimal values (Kg = 0.4, z = 29, m = 0.8 mm, v = 20 mm/s) further enhance the accuracy.
- The combination of these optimal parameters yields a forming accuracy of 100%, demonstrating the feasibility of producing high-precision straight spur gear through cold extrusion with proper process control.
- Our study provides a systematic methodology for selecting process parameters to minimize defects and maximize dimensional fidelity in cold extrusion of straight spur gear.
Future work may extend the analysis to include other gear types and investigate the influence of lubrication, die coating, and preform design. The results presented here form a solid foundation for the practical implementation of cold extrusion in the manufacturing of straight spur gear components.