Process Route and Technical Requirements
Spiral bevel gears are used where a single pair must combine a large reduction ratio, high contact ratio, and high load capacity: automotive and construction-machinery drive axles, helicopter transmissions, and marine and port machinery. Their reliability depends not only on nominal tooth geometry but also on whether the tooth flanks retain continuous metal flow lines and on the residual stress state at the surface. Milling, still the dominant production route, cuts through those flow lines and can shorten fatigue life even when dimensional accuracy is acceptable.
Precision forging keeps the flow lines continuous. Hot forging at ≥1,000 °C, however, produces a scaled surface and insufficient flank accuracy, so teeth normally require cutting before heat treatment; this adds cost and interrupts the flow. Warm forging at ≤750 °C avoids the heavy oxidation scale associated with hot forging. The flank is formed by hard pressing, which favors compressive surface stress, and no milling is needed before heat treatment. Because finish grinding or lapping after heat treatment is still required, the forged tooth must be thicker than the finished tooth and carry a controlled stock allowance.
For small and medium modules up to 4 mm, the driven wheel is a practical target: tooling and press capacity are manageable, while the fatigue requirement remains high. The case considered here is a left-hand 44-tooth driven wheel with module 3 mm, pitch diameter 132 mm, 20° mean pressure angle, and 35° spiral angle at the reference point; it meshes with a 15-tooth pinion.
Die Surface Generation from the Gear Flank
The die cavity is reverse-modeled from the required gear flank. For the formate cutting method used for the driven wheel, the inner and outer cutting edges are straight. As the cutter head rotates, each edge sweeps a cone. In the cutter coordinate system \(S_c\), the cutting-edge equation 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 the plus sign denotes the outer cutter and the minus sign the inner cutter; \(r_o\) is the nominal cutter radius, \(\alpha_c\) is the cutter angle, and \(u_c,\theta_c\) are surface coordinates on the cutting edge. Transformation from the cutter system to the workpiece system gives the gear flank:
$$
\mathbf r_2=\mathbf M_{2c}(S_1,q_1,\gamma_1,X_{G20})\,\mathbf r_c
$$
Here \(S_1\) and \(q_1\) are the radial and angular cutter positions, \(\gamma_1\) is the workpiece mounting angle, \(X_{G20}\) is a wheel-position correction, and \(\mathbf M_{2c}\) is the homogeneous transformation matrix from cutter to workpiece coordinates.
With the nominal flank known, the stock allowance is introduced by rotating the flank. For a per-flank allowance \(\Delta h\) and a mid-face radius \(r_1\), the required rotation is approximately
$$
\Delta\delta=\frac{\Delta h}{r_1}
$$
and the stock-included surface is obtained in homogeneous coordinates by
$$
\mathbf r_{\mathrm{stock}}=
\begin{bmatrix}
1&0&0&0\\
0&\cos\Delta\delta&-\sin\Delta\delta&0\\
0&\sin\Delta\delta&\cos\Delta\delta&0\\
0&0&0&1
\end{bmatrix}
\mathbf r_{\mathrm{nominal}}
$$
The matrix rotates the surface about the x-axis of the tooth-surface frame; reversing the resulting stock-included flank yields the die cavity. The basic geometry of the driven wheel is listed in Table 1.
| Parameter | Value |
|---|---|
| Number of teeth (driven / mating pinion) | 44 / 15 |
| Module | 3 mm |
| Mean pressure angle | 20° |
| Spiral angle at reference point | 35° |
| Hand | Left-hand |
| Pitch diameter | 132 mm |
| Addendum | 1.536 mm |
| Dedendum | 4.128 mm |
| Face cone angle | 72.903° |
| Pitch cone angle | 71.17529° |
| Root cone angle | 67.78734° |
| Outer cone distance | 68.83016 mm |
Finite-Element Model
Simulating the entire ring is unnecessary. A 1/12 sector of the preform represents the repeating tooth geometry, with about 120,000 tetrahedral elements and local refinement at the die corners. The dies are treated as rigid and the workpiece as plastic. The workpiece material is 18CrNi4A. The displacement step is one-third of the smallest element: a minimum mesh size of 0.41636 mm gives a step of 0.1388 mm. The interface heat-transfer coefficient is 5,000 W/(m²·K), the convection coefficient is 20 W/(m²·K), and the ambient temperature is 20 °C.
Twelve reference points—four each at the heel, mid-face, and toe—cover the addendum and dedendum regions. Equivalent strain and equivalent stress at these points are used to compare process settings.
Process-Parameter Response
The four main inputs are ram speed, die preheating temperature, friction coefficient, and workpiece heating temperature. Each was varied while the others were held constant. The pointwise trends at the reference points are summarized below; the summed indices used for optimization are reported separately in Table 3.
Forging speed. Three ram speeds—15, 20, and 25 mm/s—represent the electric screw press range. Equivalent strain first rises and then falls with increasing speed. The heel consistently shows much higher strain than the mid-face and toe because a flash forms at the outer edge; the flash constrains radial flow and forces more deformation at the root. Within a tooth, root strain exceeds tip strain. At 20 mm/s, the addendum at the heel shows an abrupt strain increase, a local condition that should be avoided. Pointwise equivalent stress trends downward as speed increases. At the heel and mid-face, root stress is higher than at the tip and flank; at the toe, the order reverses, with tip stress above root stress.
Die preheating. Three die temperatures—200, 250, and 300 °C—cover the usual preheating window. The local strain at the heel rises with die temperature. A hotter die receives less heat from the workpiece, and deformation heating further raises the surface temperature; the resulting lower flow stress allows more deformation. The heel is the most sensitive region, while the mid-face and toe change little. At a fixed die temperature, the toe has the lowest strain. Because the response differs among reference points, the summed index does not follow a simple monotonic trend.
Friction. Friction coefficients of 0.12, 0.25, and 0.30 represent different lubricant regimes. Strain first decreases and then increases with friction. At low friction, the surface layer flows more easily because interfacial shear is low. At high friction, friction-generated heat softens the surface and again promotes flow. The lowest friction coefficient, 0.12, gives the highest equivalent stress of the three: easier surface flow accelerates flash formation at the heel, which increases constraint and the stress required to fill the tooth.
Workpiece temperature. Workpiece temperatures of 650, 700, and 750 °C lie within the 600–800 °C warm-forging range of 18CrNi4A. Strain is almost independent of temperature, but the spatial pattern remains: heel > mid-face > toe, and within a tooth root > flank > tip. The heel effect is again linked to flash. The root/flank/tip order reflects the filling sequence: the convex die surface is contacted first, and the concave flank becomes the main forming surface later as metal fills the cavity, so the concave side accumulates more strain than the convex side. Equivalent stress falls markedly with temperature; at 750 °C, the heel stress is clearly below the mid-face and toe values because the material is softer and flows more readily.
Orthogonal Optimization
Single-factor curves do not define a unique optimum: a setting that lowers stress may reduce strain or move the maximum to an unfavorable location. A four-factor, three-level orthogonal array was therefore used. Table 2 lists the levels; Table 3 gives the nine runs and the resulting summed response indices.
| Factor | Symbol | Level 1 | Level 2 | Level 3 |
|---|---|---|---|---|
| Ram speed | A | 15 mm/s | 20 mm/s | 25 mm/s |
| Die preheating temperature | B | 200 °C | 250 °C | 300 °C |
| Friction coefficient | C | 0.12 | 0.25 | 0.30 |
| Workpiece temperature | D | 650 °C | 700 °C | 750 °C |
| Run | Combination | Summed equivalent strain | Summed 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 entries are sums over the reference points and are used as comparative indices, not as local stress magnitudes. Normalized influence weights were calculated from grouped strain and stress sums. For strain, the highest-weight combinations were A2B3C3D2 (3.854), A2B1C3D2 (3.833), A2B3C3D1 (3.601), A2B1C3D1 (3.582), and A2B3C3D3 (3.283). For stress, the lowest-weight combinations were A3B2C2D3 (1.345), A3B3C2D3 (1.355), A3B2C3D3 (1.364), A1B2C2D3 (1.365), and A3B3C3D3 (1.373). No combination appeared in both groups, so the minimum-stress criterion was used for final selection.
The selected setting is A3B2C2D3: 25 mm/s ram speed, 250 °C die preheating, 0.25 friction coefficient, and 750 °C workpiece temperature. Because this exact run was not in the array, it was simulated separately. The additional run gave a total equivalent-stress index of 4,433 MPa, below the 4,502 MPa obtained for A1B2C2D3, confirming the expected reduction.
Warm-Forging Trial and Dimensional Validation
The optimized setting was transferred to a 2,000 t hydraulic press. A warm-forging trial produced the driven wheel. After forging, the teeth were used for location; the mounting face, bore, outer diameter, and front face were turned on a vertical lathe to establish inspection datums. The gear was then measured on a Gleason 650GMS gear measuring center.
Maximum topology error was 13.7 µm on the left flank and 13.9 µm on the right flank. The concave flank had an upper deviation of 5.1 µm, a 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, respectively. Cumulative pitch deviations were 60.7 and 86.8 µm, corresponding to grades 8 and 9 of GB/T 11365—2019. The results are summarized in Table 4.
| Measurement | Result |
|---|---|
| Maximum topology error, left / right flank | 13.7 / 13.9 µm |
| Concave flank deviation, upper / lower / total | 5.1 / 13.7 / 18.7 µm |
| Convex flank deviation, upper / lower / total | 13.9 / 13.8 / 27.7 µm |
| Cumulative pitch deviation, two flanks | 60.7 / 86.8 µm (GB/T 11365—2019 grades 8 / 9) |
These values are within the allowance left for grinding after heat treatment. They establish that the die-surface inversion, stock rotation, and stress-minimized process window produce a tooth form with sufficient margin for finishing.
Engineering Significance and Limits
Warm forging at ≤750 °C changes the manufacturing chain in two ways. It removes pre-heat-treatment milling, so the tooth flank is not cut through and the metal flow lines can follow the forged shape; it also uses a hard-pressed flank that is intended to leave compressive surface stress. The dimensional trial supports the first part of this route. Fatigue life itself was not measured, so the fatigue benefit remains a process-based expectation rather than a demonstrated service result. Residual stress and microstructure were not part of the dimensional validation.
The model covers the main forming parameters but not heating rate, ambient temperature, the full temperature history, or the coupling between process variables and microstructure. Those factors matter for grain size, phase state, and final distortion. The present result is therefore a process window and a dimensional proof for a 3 mm module driven gear, not a general fatigue qualification. Extending the approach to other modules or to the pinion changes die load, flash geometry, and heat balance, so the window cannot be transferred without new tooling and re-optimization.
Within that scope, the combination of a reverse-modeled die surface, a stock-compensated flank, a 1/12-sector finite-element model, and a stress-minimized orthogonal setting provides a coherent route for small- and medium-module spiral bevel gears. The measured pitch grades and flank deviations show that the forged tooth can carry the grinding allowance needed for finishing, which is the practical requirement for replacing pre-heat-treatment milling.