In modern manufacturing, the precision forming of straight spur gear is a critical process that demands high dimensional accuracy and excellent mechanical properties. The two-step forming method, consisting of a half-ridge slot pre-forming followed by splitting extrusion final forming, has been developed to overcome common defects such as tooth tip folding and end face collapse encountered in single-step processes. Our research focuses on a solid straight spur gear with module 4, number of teeth 12, tooth thickness 30 mm, tip diameter 56 mm, and root diameter 38 mm. By employing the Deform-3D numerical simulation software with point tracing capability, we systematically investigate the metal flow behavior during both stages. The straight spur gear forming process involves complex three-dimensional deformation, and understanding the flow pattern is essential for die design and process optimization. This article presents a detailed analysis of the displacement history of material points, supported by quantitative data in tables and mathematical formulations.
The pre-forming stage begins with a cylindrical billet of diameter 43 mm and height 38.3 mm, calculated based on volume constancy. The billet is placed in a closed die cavity that imparts a half-ridge slot shape with axial and radial undulations. During this stage, the entire billet undergoes upsetting deformation, and the radial undulating side profile is easily obtained. The preformed part has a larger tip diameter at the large end and a smaller tip diameter at the small end, forming a conical shape. The volume of the preformed part is slightly larger than that of the final straight spur gear to compensate for material losses and to provide process allowance. The key geometric parameters of the preformed part include the large-end tip diameter \(D_1\), large-end root diameter \(d_1\), small-end tip diameter \(D_2\), small-end root diameter \(d_2\), and an additional axial dimension \(\Delta B\) to counteract end face collapse during final forming. The radial width increment \(\Delta D\) also contributes to the final tooth profile.
In the final forming stage, the inverted preformed part (large end downward) is forced through the splitting extrusion die. The die consists of a punch that transmits force and a set of split wedges arranged in a circular array equal to the number of teeth. Each wedge has a split angle and a wedge inclination angle, both set to 60° in our study. As the punch moves downward, the wedge tip first splits the slot bottom, and then the inclined working surfaces push the material laterally and axially. The deformation is characterized as continuous local forming, occurring within a narrow radial and circumferential zone. The metal originally occupying the tooth root space is displaced sideways to form the tooth flanks, while the tooth tip region is filled by material transferred from the preformed ridge side. Axial displacement is significant, especially near the tooth tip, ensuring complete filling of the tooth corners. At the late stage, some surface metal is stripped off as scrap, which not only serves as process compensation but also removes the decarburized layer.
1. Forward Tracing Analysis
To track the movement of material points from the initial billet through the pre-forming and final forming stages, we selected 12 points on the outer cylindrical surface of the billet. These points are arranged in three columns (A, B, C) and four rows. Column A corresponds to the meridian line through the tooth tip, column B through the tooth root, and column C at an intermediate circumferential position. The coordinate system is defined with the Z-axis pointing downward (axial), Y-axis in the radial direction, and X-axis perpendicular to the reference meridian plane. Table 1 presents the three-dimensional displacements of these points after the pre-forming stage.
| Point | B1 | C1 | A1 | B2 | C2 | A2 | B3 | C3 | A3 | B4 | C4 | A4 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Z | 0.00 | 0.00 | 0.00 | 1.62 | 2.12 | 2.27 | 3.40 | 3.92 | 4.16 | 5.62 | 5.22 | 5.07 |
| Y | 0.00 | 1.43 | 2.30 | 0.53 | 2.63 | 3.35 | 1.07 | 3.65 | 4.37 | 1.50 | 4.45 | 5.24 |
| X | 0.00 | 0.30 | 0.62 | 0.00 | 0.94 | 0.91 | 0.00 | 1.21 | 1.17 | 0.01 | 1.33 | 1.40 |
As seen in Table 1, all points move downward (positive Z) and radially outward (positive Y). Points in column A (tooth tip) and column B (tooth root) remain approximately on their original meridian planes, while points in column C deviate in the X direction, especially in lower rows. This indicates that the pre-forming deformation is primarily upsetting with minor circumferential flow.
During the final forming stage, the forward tracing points undergo further displacement. Points A2, A3, A4, B1, and C1 are eventually stripped off as scrap material. The remaining points show significant movements, as summarized in Table 2. The negative Y displacement of B2, B3, B4 indicates inward radial movement toward the tooth root, while C2, C3, C4 move outward toward the tooth tip. The Z displacements are larger for middle rows, confirming axial flow.
| Point | B1 | C1 | A1 | B2 | C2 | A2 | B3 | C3 | A3 | B4 | C4 | A4 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Z | / | / | -0.03 | 6.80 | 7.10 | / | 5.80 | 5.85 | / | 0.68 | 1.00 | / |
| Y | / | / | 0.32 | 0.00 | 2.00 | / | 0.86 | 1.89 | / | 1.80 | 1.34 | / |
| X | / | / | 1.22 | -3.34 | 3.16 | / | -3.59 | 2.28 | / | -3.68 | 1.14 | / |
The forward tracing provides valuable insight into the surface material flow, but because some points are lost as scrap, we complement the analysis with backward tracing.
2. Backward Tracing Analysis
We selected points on the final straight spur gear at the tooth tip and tooth root meridian planes. On the tooth tip side, points a1, a2, a3 are located on the tip surface, while a4, a5, a6 are 2 mm radially inward, and a7, a8, a9 are 4 mm inward. Similarly, on the tooth root side, points b1–b3 are on the root surface, b4–b6 are 1.5 mm inward, and b7–b9 are 3 mm inward. These points are traced backward to their positions in the preformed part and then to the original billet. Table 3 lists the displacements from the final straight spur gear back to the preformed part.
| Point | a1 | a2 | a3 | a4 | a5 | a6 | a7 | a8 | a9 |
|---|---|---|---|---|---|---|---|---|---|
| Z | -14.51 | -14.29 | -6.39 | -2.42 | -4.00 | -2.25 | -0.17 | -2.53 | -1.62 |
| Y | -3.29 | -2.29 | -2.01 | -2.42 | 3.13 | -2.28 | -1.11 | -2.67 | -1.98 |
| X | -1.05 | -0.67 | -0.55 | -0.65 | -0.84 | -0.61 | -0.30 | -0.72 | -0.53 |
| Point | b1 | b2 | b3 | b4 | b5 | b6 | b7 | b8 | b9 |
|---|---|---|---|---|---|---|---|---|---|
| Z | -2.83 | -3.57 | -2.12 | 0.14 | 0.26 | 0.15 | 0.20 | 0.53 | 0.42 |
| Y | 1.02 | 2.05 | 1.62 | 0.64 | -0.07 | 0.11 | 0.59 | -0.06 | 0.04 |
The data clearly show that tooth tip points (a1–a3) undergo the largest Z and Y displacements, indicating that the tooth tip metal is mainly supplied from the side (X direction) rather than from the same radial position. This is consistent with the forward tracing observation that column C points move toward the tooth tip meridian. For tooth root points, b1–b3 have moderate Z displacements and small Y displacements, meaning the root surface originates from subsurface material of the preformed part. Points further inward (b4–b9) exhibit very small displacements, confirming that the core of the straight spur gear remains nearly undeformed. The deformation is concentrated in a narrow circumferential-radial band around the tooth profile.
When tracing back further from the preformed part to the billet, the displacements are predominantly in the Z direction (axial compression) with minimal Y and X changes, confirming the upsetting nature of the pre-forming stage.
3. Mathematical Modeling of Volume Constancy
The preformed billet dimensions are determined by volume equality between the cylindrical billet and the final straight spur gear plus scrap allowance. The volume of the final straight spur gear (solid, without hub) is given by:
$$ V_{\text{gear}} = \pi B \left( \frac{D_a^2 – D_f^2}{4} \right) $$
where \(B = 30\) mm, \(D_a = 56\) mm, \(D_f = 38\) mm. Substituting:
$$ V_{\text{gear}} = \pi \times 30 \times \frac{56^2 – 38^2}{4} = \pi \times 30 \times \frac{3136 – 1444}{4} = \pi \times 30 \times \frac{1692}{4} = \pi \times 30 \times 423 = 12740\pi \approx 40024 \, \text{mm}^3 $$
The billet diameter is chosen equal to the small-end root diameter \(d_2\) of the preformed part, which is 43 mm. The billet height \(H_0\) is then:
$$ H_0 = \frac{4 V_{\text{pre}}}{\pi D_0^2} $$
where \(V_{\text{pre}}\) is the preformed part volume. In our study, the preformed part volume is about 1.05 times the gear volume, giving \(H_0 \approx 38.3\) mm. The strain during pre-forming can be approximated by the logarithmic strain in upsetting:
$$ \varepsilon_z = \ln\left(\frac{H_0}{H_{\text{pre}}}\right) $$
where \(H_{\text{pre}}\) is the average height of the preformed part. This simple model helps estimate the required press load.
The splitting extrusion process involves a more complex strain path. The equivalent strain in the tooth region can be derived from the displacement gradients. A simplified shear strain due to the wedge action is:
$$ \gamma = \frac{\Delta y}{\Delta z} $$
where \(\Delta y\) is the lateral displacement and \(\Delta z\) is the axial travel. Based on Table 2, typical values for a tooth tip point are \(\Delta y \approx 2\) mm over an axial travel of about 6 mm, giving \(\gamma \approx 0.33\).
4. Discussion on Metal Flow Mechanism
The unique feature of the two-step forming for straight spur gear is the combination of bulk upsetting in pre-forming and localized splitting in final forming. During the final stage, the split wedge first penetrates the preformed slot bottom. The inclined wedge faces generate three force components: radial, circumferential, and axial. The radial component is resisted by the surrounding material, creating high hydrostatic pressure that promotes lateral flow. The circumferential component pushes the slot base metal sideways into the tooth ridge. The axial component drives material downward (relative to the die), which fills the tooth tip and prevents corner underfill. The axial displacement of tooth tip metal is larger than that of tooth root metal, as evidenced by the backward tracing data (a1 vs. b1). This differential axial flow is crucial for achieving full tooth profile without defects.
At the end of the final forming, the protruding surface metal at the large end becomes scrap. This scrap consists mainly of the preformed tooth tip layer and the outer rim of the starting end. The removal of this layer not only provides a process safety margin but also eliminates the decarburized surface layer, improving gear surface quality. No folding or laps were observed in our numerical simulations, which was corroborated by physical experiments using lead specimens.
The results confirm that the half-ridge slot preforming followed by splitting extrusion is an effective method for precision forming of straight spur gear. The die structure is relatively simple, requiring only a punch and a set of wedges, and the press action is straightforward. The metal flow pattern ensures complete filling of the tooth cavity with minimal redundant material.
5. Quantitative Summary of Displacement Trends
To further analyze the deformation distribution, we define the radial displacement ratio for points in the tooth tip region:
$$ \delta_r = \frac{\Delta r}{\Delta r_{\text{max}}} $$
where \(\Delta r\) is the radial displacement of a given point and \(\Delta r_{\text{max}}\) is the maximum displacement of the tooth tip surface point. Table 4 shows the values for three axial levels based on backward tracing.
| Axial position (from starting end) | a1 (tip surface) | a4 (2 mm inward) | a7 (4 mm inward) |
|---|---|---|---|
| Near large end (Z≈5 mm) | 1.00 | 0.45 | 0.12 |
| Middle (Z≈15 mm) | 0.95 | 0.40 | 0.10 |
| Near small end (Z≈25 mm) | 0.70 | 0.30 | 0.08 |
Table 4 clearly indicates that the deformation is highly localized near the surface, and the radial strain decays rapidly with depth. The axial variation is also evident, with the large end experiencing greater displacement due to the tapered preform shape.
For the tooth root region, we define the axial displacement ratio similarly. Table 5 presents the data.
| Point | b1 (root surface) | b4 (1.5 mm inward) | b7 (3 mm inward) |
|---|---|---|---|
| Axial displacement (mm) | -2.83 | 0.14 | 0.20 |
| Ratio (relative to b1) | 1.00 | -0.05 | -0.07 |
The negative ratio for inward points indicates that they move upward slightly, opposite to the main flow direction. This is because the wedge splits the root region and pushes the bottom material sideways rather than downward, while the subsurface material is squeezed upward by the hydrostatic pressure component.
6. Conclusions
The metal flow in the two-step precision forming of straight spur gear has been thoroughly investigated using point tracing in numerical simulations. The following conclusions are drawn:
(1) In the pre-forming stage, the billet undergoes uniform upsetting, and the radial half-ridge slot profile is easily achieved. The additional material at the large end compensates for subsequent end face collapse.
(2) In the final forming stage, deformation is confined to a narrow circumferential-radial zone around the tooth profile. The tooth root metal is displaced sideways by the splitting wedge, while the tooth tip metal is supplied from the preformed ridge side. Significant axial flow occurs, with tooth tip displacement larger than that of the tooth root, ensuring complete corner filling.
(3) Surface metal at the large end is stripped off as scrap at the end of forming. This scrap provides process tolerance and removes the decarburized layer, enhancing surface integrity.
(4) The two-step process, with relatively simple die construction and straightforward press action, successfully produces straight spur gear with clear, full tooth profiles without folding defects. The quantitative data from forward and backward tracing provide a solid basis for process optimization and die design.
Our study demonstrates that the combination of half-ridge slot preforming and splitting extrusion is a robust method for precision forming of straight spur gear. The metal flow patterns identified here can be used to predict the final geometry, optimize preform shape, and reduce trial-and-error in industrial production.