Spiral bevel gears are the default choice when torque must be transmitted between intersecting shafts with a high ratio and smooth motion. Axle drives in cars and construction machinery, helicopter transmissions, and marine and port machinery all rely on them. The conventional manufacturing route is milling: a cutter removes material from a solid blank until the flank is formed. Cutting interrupts the metal flow lines that would otherwise run continuously along the tooth, and the fatigue life of the part suffers as a result.
Precision forging avoids that interruption. Hot forging above 1 000 ℃ can produce the overall shape, but flank accuracy is poor and a thick oxide scale forms; the tooth still has to be cut before heat treatment, which adds a machining step and cost. Warm forging at or below 750 ℃ is a different compromise. The scale is much thinner, the flank is coined under high pressure, and the resulting tooth can be carburized without prior milling. A grinding allowance is left on the flank and removed after heat treatment. For small-module driven gears (module ≤ 4 mm), the allowance and the die deflection are small enough for this route to be practical.
The die cavity is obtained by reverse modeling the stock-including tooth flank, so the flank must be computed first. The process window then has to be narrowed: forging speed, die preheating, friction, and workpiece temperature all change the strain and stress state. Finally, the selected parameters must survive a real forging and a gear inspection. The case considered here is a 44-tooth driven wheel, module 3 mm, left-hand, pitch diameter 132 mm, made of 18CrNi4A and meshing with a 15-tooth pinion.
Die Tooth Surface from the Gear Flank
The die surface that forms the flank is geometrically identical to the stock-including gear flank; reverse modeling of that surface gives the die cavity. The flank was calculated with the formate (fixed-setting) method. The inner and outer cutting edges are straight, and as the cutter head rotates each edge sweeps a cone. In the cutter coordinate system \(S_c\), a point on the cutting edge is
$$
\mathbf{r}_c(u_c,\theta_c)=
\begin{bmatrix}
(r_o\pm u_c\sin\alpha_c)\cos\theta_c\\
(r_o\pm u_c\sin\alpha_c)\sin\theta_c\\
-u_c\cos\alpha_c
\end{bmatrix}
$$
where \(r_o\) is the nominal cutter radius, \(u_c\) and \(\theta_c\) are surface coordinates, and \(\alpha_c\) is the blade angle. The upper sign refers to the outer blade and the lower sign to the inner blade. The cutter surface is carried into the workpiece coordinate system by the machine settings:
$$
\mathbf{r}_2=\mathbf{M}_{2c}(S_1,q_1,\gamma_1,X_{G20})\,\mathbf{r}_c(u_c,\theta_c)
$$
Here \(S_1\) is the radial cutter position, \(q_1\) the angular cutter position, \(\gamma_1\) the workpiece mounting angle, \(X_{G20}\) the axial wheel position correction, and \(\mathbf{M}_{2c}\) the homogeneous transformation matrix from cutter to workpiece.
After forming, the tooth must be thicker than the finished tooth because grinding stock is required. With a single-flank allowance \(\Delta h\) and a radius \(r_1\) at the middle of the face width, the flank is rotated by
$$
\Delta\delta=\frac{\Delta h}{r_1}
$$
The stock-including surface is obtained in homogeneous coordinates as
$$
\mathbf{r}_s=\mathbf{R}_x(\Delta\delta)\,\mathbf{r}_e,\qquad
\mathbf{R}_x(\Delta\delta)=
\begin{bmatrix}
1&0&0&0\\
0&\cos\Delta\delta&-\sin\Delta\delta&0\\
0&\sin\Delta\delta&\cos\Delta\delta&0\\
0&0&0&1
\end{bmatrix}
$$
where \(\mathbf{r}_e\) is the finished flank point and \(\mathbf{r}_s\) the stock-including point. The rotation adds material to the flank while preserving the cutting geometry. Reverse modeling of this surface gives the die tooth surface. Table 1 lists the gear parameters used for the calculation.
| Parameter | Value |
|---|---|
| Number of teeth | 44 |
| Module | 3 mm |
| Mean pressure angle | 20° |
| Mean spiral angle | 35° |
| Hand | Left |
| Pitch diameter | 132 mm |
| Addendum | 1.536 mm |
| Dedendum | 4.128 mm |
| Face angle | 72.903° |
| Pitch angle | 71.175 29° |
| Root angle | 67.787 34° |
| Outer cone distance | 68.830 16 mm |
| Mating pinion teeth | 15 |
Finite Element Model
The full blank would require a very large mesh to resolve the tooth root, so a 1/12 sector was taken from the preform. The sector was meshed with 120 000 tetrahedral elements. Local refinement was applied near the die corners, where the root fillet and crest are formed. This keeps the element count manageable while resolving the temperature, strain, and stress gradients in the tooth.
The dies were rigid and the workpiece was plastic. The workpiece material was 18CrNi4A. The step size was one third of the smallest element: 0.138 8 mm for a minimum element size of 0.416 36 mm. The interface heat-transfer coefficient was 5 000 W/(m²·K), the convection coefficient 20 W/(m²·K), and the ambient temperature 20 ℃. The simulation was run in DEFORM-3D.
Twelve reference points were placed at the heel, middle, and toe of the tooth, four per section, covering crest and root. Equivalent stress and equivalent strain at these points were extracted for each run.
Single-Parameter Effects on Strain and Stress
The four factors were varied one at a time around the baseline A1B2C2D1: 15 mm/s, 250 ℃ die preheating, 0.25 friction, and 650 ℃ workpiece heating. The reference-point response was then compared across the three levels.
Forging Speed
Three ram speeds were tested: 15, 20, and 25 mm/s, spanning the working range of an electric screw press. Equivalent strain first increased and then decreased with speed. The heel strain was much larger than the middle and toe strain. A flash forms at the heel, and the flash resists outward flow, so the root region needs a higher forging force. At 20 mm/s the heel crest showed a sharp local jump in strain, a condition that should be avoided in production. Within a tooth, root strain exceeded crest strain. Equivalent stress tended to fall as speed increased; the 20 mm/s case was an exception, consistent with the local strain jump at the heel crest. At the heel and middle sections, root stress was higher than crest and flank stress; at the toe, crest stress was the highest.
Die Preheating Temperature
Die preheating was varied over 200, 250, and 300 ℃, inside the usual 150–300 ℃ range. Strain increased with die temperature. A hotter die reduces heat loss from the workpiece; deformation heating remains near the surface, the local temperature rises, flow stress falls, and the metal deforms more readily. The heel was the most sensitive section, while the middle and toe changed little. The toe had the lowest strain at all three die temperatures.
Friction Coefficient
Friction coefficients of 0.12, 0.25, and 0.30 represented different lubricant regimes. Strain was lowest at 0.25 and higher at both 0.12 and 0.30. At high friction, frictional heating raises the local temperature and improves flow; at low friction, the surface layer is less constrained and deforms more readily. Equivalent stress was highest at 0.12. With low friction the metal flows outward more freely, the heel flash grows faster, and the forging stress rises.
Workpiece Heating Temperature
Workpiece heating was set at 650, 700, and 750 ℃, within the 600–800 ℃ warm range quoted for 18CrNi4A. Strain was nearly independent of workpiece temperature. The heel remained the highest-strain section because of the flash. Within a tooth, root strain was highest, flank strain intermediate, and crest strain lowest. The convex flank contacts the die first and forms earlier; the concave flank then becomes the main filling surface, so it accumulates more strain. Stress fell strongly with temperature as the alloy softened. At 750 ℃, the heel stress was clearly lower than the middle and toe stress.
Parameter Optimization
Table 2 gives the four factors and three levels, and Table 3 gives the nine runs and the two computed responses. Equivalent strain ranged from 39.96 to 64.88, and equivalent stress from 4 502 to 5 507 MPa.
| Factor | Level 1 | Level 2 | Level 3 |
|---|---|---|---|
| A: forging speed (mm/s) | 15 | 20 | 25 |
| B: die preheating temperature (℃) | 200 | 250 | 300 |
| C: friction coefficient | 0.12 | 0.25 | 0.30 |
| D: workpiece heating temperature (℃) | 650 | 700 | 750 |
| Run | Combination | Equivalent strain | Equivalent stress (MPa) |
|---|---|---|---|
| 1 | A1B2C2D1 | 43.84 | 5 422 |
| 2 | A2B2C2D1 | 52.75 | 5 497 |
| 3 | A3B2C2D1 | 42.25 | 5 345 |
| 4 | A1B1C2D1 | 52.32 | 5 503 |
| 5 | A1B3C2D1 | 51.59 | 5 463 |
| 6 | A1B2C1D1 | 56.84 | 5 507 |
| 7 | A1B2C3D1 | 64.88 | 5 498 |
| 8 | A1B2C2D2 | 46.92 | 4 922 |
| 9 | A1B2C2D3 | 39.96 | 4 502 |
The nine runs were not a full factorial. To rank the levels, the response values at each level of a factor were aggregated, normalized against the smallest value of that factor, and multiplied by a within-level proportion; the product is a dimensionless global weight. This is a relative ranking of levels, not an analysis of variance. The weights are listed in Table 4.
| Factor | Response | Level 1 | Level 2 | Level 3 |
|---|---|---|---|---|
| A | Strain | 1.102 | 1.326 | 1.062 |
| A | Stress | 1.111 | 1.126 | 1.095 |
| B | Strain | 1.314 | 1.123 | 1.321 |
| B | Stress | 1.120 | 1.104 | 1.112 |
| C | Strain | 1.642 | 1.267 | 1.875 |
| C | Stress | 1.151 | 1.113 | 1.128 |
| D | Strain | 1.097 | 1.174 | 1.000 |
| D | Stress | 1.204 | 1.046 | 1.000 |
For strain, the highest weight at each factor was A2 (1.326), B3 (1.321), C3 (1.875), and D2 (1.174). Their product is 3.854. For stress, the lowest weight at each factor was A3 (1.095), B2 (1.104), C2 (1.113), and D3 (1.000), giving 1.345. No single combination gave both a high strain weight and a low stress weight. Stress was therefore used as the selection criterion.
The selected combination is A3B2C2D3: forging speed 25 mm/s, die preheating 250 ℃, friction coefficient 0.25, and workpiece heating 750 ℃. A separate simulation of this combination, which was not in the nine runs, gave a summed equivalent stress of 4 433 MPa at the heel, middle, and toe. The A1B2C2D3 run, which shares B2C2D3, gave 4 502 MPa. The lower value is consistent with the ranking.
Experimental Validation
The selected parameters were used in a warm-forging trial on a 2 000 t hydraulic press. The forged gear was turned on a vertical lathe: the mounting face and bore were established first, followed by the outer diameter and front face. This gave a stable datum for inspection. The gear was measured on a Gleason 650GMS gear measuring center. Table 5 summarizes the results.
| Characteristic | Concave flank | Convex flank |
|---|---|---|
| Maximum upper deviation (μm) | 5.1 | 13.9 |
| Maximum lower deviation (μm) | 13.7 | 13.8 |
| Total deviation (μm) | 18.7 | 27.7 |
| Maximum topology error (μm) | 13.7 | 13.9 |
| Cumulative pitch deviation (μm) | 60.7 | 86.8 |
| Accuracy grade (GB/T 11365—2019) | 8 | 9 |
Maximum topology error was 13.7 μm on the concave flank and 13.9 μm on the convex flank. The concave flank had a maximum upper deviation of 5.1 μm, a maximum lower deviation of 13.7 μm, and a total deviation of 18.7 μm. The convex flank had 13.9 μm, 13.8 μm, and 27.7 μm. The cumulative pitch deviations were 60.7 μm and 86.8 μm, corresponding to grades 8 and 9 of GB/T 11365—2019.
These values are within the grinding stock. The trial therefore shows that the optimized route can deliver a tooth form that needs only post-heat-treatment grinding, not preliminary milling.
Process Implications and Open Questions
The route has three practical consequences. The tooth is formed with continuous flow lines instead of being cut, which is the intended basis for longer fatigue life. The flank is coined under high pressure, which favors a compressive surface stress state. And the gear can be carburized without a milling step; only grinding after heat treatment is required. These points are relevant to small-module driven gears in axle drives, helicopter transmissions, and marine and port machinery.
The present parameter study covered forging speed, die preheating, friction, and workpiece temperature. Warm forging is a coupled thermal-mechanical process, and the heating rate, ambient temperature, full time-temperature history, and microstructure evolution were not varied. The validation is geometric; fatigue life and residual stress were not measured. Coupling the temperature history to microstructure and properties is the next step.
Even with these open points, the combination of a computed die surface, a refined sector model, and a validated parameter set provides a workable route for warm forging of small-module spiral bevel gears.