Spiral bevel gears are used where shafts intersect and where smooth transmission, load capacity, and ratio conformity matter. The pinion is the difficult member to form. Its small pitch cone angle makes ejection from a closed forging die difficult, and conventional cutting severs the metal flow lines that can benefit fatigue performance. Warm rolling offers a near-net-shape route before heat treatment: a localized induction field heats a thin surface layer to about 650 °C, below the hot-rolling range, with limited scale formation and less thermal distortion than bulk hot working. The technical problem is threefold: the die must reproduce the conjugate pinion flank while leaving grinding stock, the blank must flow without folding or excessive stress, and the process window must be stable enough for dimensional control.
Conjugate Tool Surface and Grinding Allowance
The die tooth surface is generated as the envelope of the pinion surface in the relative motion of the gear pair. In the meshing coordinate system, S1 and S2 rotate with the pinion and wheel, Sh is fixed to the housing, and Sd is an auxiliary frame at the crossing point of the axes. The angles φ1 and φ2 are the meshing rotations, E is the pinion offset, and Γ is the shaft angle, 90° in this case. The pinion surface r1(up, θp) and unit normal n1(up, θp) map to the wheel coordinates as
$$
\mathbf{r}_2(u_p,\theta_p)=\mathbf{M}_{2d}\mathbf{M}_{dh}\mathbf{M}_{h1}\mathbf{r}_1(u_p,\theta_p),\qquad
\mathbf{n}_2(u_p,\theta_p)=\mathbf{L}_{2d}\mathbf{L}_{dh}\mathbf{L}_{h1}\mathbf{n}_1(u_p,\theta_p),
$$
where M2d, Mdh, and Mh1 are the homogeneous transformation matrices between the frames, and L2d, Ldh, and Lh1 are their rotation parts. The same mapping defines the die cavity, because the die is the conjugate wheel flank rather than a simple copy of a production wheel. Discrete die points are obtained by solving the rotary projection equations from the pinion reference point.
A rolled pinion is not a finished gear. The die flank must be rotated outward to leave stock for post-heat-treatment grinding. If Δh is the single-flank allowance and r2 is the mid-face radius of the die, the required rotation is
$$
\Delta\delta=\frac{\Delta h}{r_2}.
$$
This produces a thicker tooth than the final design. The example gear pair used in the process study has the geometry given in Table 1. The blank material is AISI-8620 carburizing steel.
| Parameter | Pinion | Wheel |
|---|---|---|
| Module (mm) | 3 | 3 |
| Number of teeth | 15 | 44 |
| Spiral angle (°) | 35 | 35 |
| Pressure angle (°) | 20 | 20 |
| Face width (mm) | 20.2 | 20.2 |
| Addendum coefficient | 0.85 | 0.85 |
| Clearance coefficient | 0.188 | 0.188 |
| Outer cone distance (mm) | 69.73 | 69.73 |
| Pitch cone angle (°) | 18.82 | 71.17 |
| Hand | Left | Right |
Finite Element Model and Parameter Matrix
The process model is built in deform-3D. The pinion blank is treated as a plastic body; the tool wheel and intermediate shaft are rigid. The blank is meshed with about 50,000 tetrahedral elements, the step size is one-third of the minimum element size, and the solution uses direct iteration. The die rotates at 1.071 rad/s, the spline at 3.141 59 rad/s, and the initial die temperature is 20 °C. The friction factor between the intermediate shaft and the blank is fixed at 0.8. The die–blank interface friction factor, axial feed speed, and initial billet temperature are varied in single-factor runs: friction factor 0.2, 0.3, and 0.4; feed speed 0.1, 0.2, and 0.3 mm/s; initial billet temperature 600, 650, and 700 °C. The friction levels represent average values for different lubrication states, from effective lubrication to dry contact. Three tracking points are used: P1 on the tooth profile, P2 at the root, and P3 at the tip. A three-factor, three-level L9 orthogonal array then ranks the parameters by a stress response.
Stress and Strain Response to Process Parameters
Friction factor. At 600 °C and 0.1 mm/s, the peak equivalent stresses at P2, P3, and P1 were 460, 396, and 452 MPa at friction factor 0.2; 424, 434, and 409 MPa at 0.3; and 464, 410, and 346 MPa at 0.4. The local maxima did not move monotonically, which already indicates that friction is not the dominant factor in this window. The accumulated strain at the three points, however, fell as the friction factor increased. The mechanical interpretation is that a low-friction interface supplies insufficient traction for uniform surface flow, so local deformation concentrates. A high-friction interface constrains surface flow and raises resistance to metal movement, but the model showed smaller accumulated strain at the higher friction levels. Lubrication therefore should not be selected only to reduce friction; it also changes the flow pattern.
Feed speed. At friction factor 0.4 and 600 °C, feed speeds of 0.1, 0.2, and 0.3 mm/s gave P2/P3/P1 peak stresses of 464/410/346, 365/425/382, and 435/455/436 MPa, respectively. The lowest feed produced the smallest strain and the most stable strain history. Lower feed means more rolling passes per tooth and a smaller incremental penetration; higher feed means fewer, deeper bites, more springback, and a larger deformation force. For this small-modulus pinion, 0.1 mm/s therefore favors flow control, at the cost of cycle time.
Initial billet temperature. At friction factor 0.4 and 0.1 mm/s, the peak stresses at P1/P2/P3 were 442/405/373 MPa at 600 °C, 370/400/391 MPa at 650 °C, and 359/386/320 MPa at 700 °C. The overall stress level fell with temperature, and the 700 °C case had the lowest peak at all three points. Strain moved the other way: the 600 °C run ended with the smallest accumulated strain, and the 700 °C run with the largest. Higher temperature lowers flow stress and promotes ductility, so the blank deforms more uniformly and with less force; lower temperature raises the rolling load and the risk of forming defects. The stress optimum is therefore not the strain optimum, and the process window has to balance the two.
Orthogonal Optimization
Table 2 gives the L9 array and the stress response. The level averages are 1 348.5, 1 318.5, and 1 288.5 MPa for friction factor; 1 272.5, 1 318.5, and 1 364.5 MPa for feed speed; and 1 396.0, 1 318.5, and 1 241.0 MPa for initial temperature. The corresponding ranges are 60, 92, and 155 MPa, so initial temperature has the largest effect, feed speed the next, and friction factor the smallest. The minimum average stress occurs at A3B1C3: friction factor 0.4, feed speed 0.1 mm/s, and initial billet temperature 700 °C. Trial 4 gave the lowest single response, but the range analysis selects the combination with the best level averages; the two criteria answer different questions.
| Trial | A: friction factor | B: feed speed (mm/s) | C: initial temperature (°C) | Stress response (MPa) |
|---|---|---|---|---|
| 1 | 0.2 | 0.1 | 600 | 1 380.0 |
| 2 | 0.2 | 0.2 | 650 | 1 348.5 |
| 3 | 0.2 | 0.3 | 700 | 1 317.0 |
| 4 | 0.3 | 0.1 | 700 | 1 195.0 |
| 5 | 0.3 | 0.2 | 600 | 1 396.0 |
| 6 | 0.3 | 0.3 | 650 | 1 364.5 |
| 7 | 0.4 | 0.1 | 650 | 1 242.5 |
| 8 | 0.4 | 0.2 | 700 | 1 211.0 |
| 9 | 0.4 | 0.3 | 600 | 1 412.0 |
Warm-Rolling Trial and Measured Accuracy
Using the A3B1C3 combination, a pinion was warm rolled on a bench with a tool wheel, a balance wheel, and an induction coil. A KGPS medium-frequency induction unit heated the blank. The coil geometry followed the tooth region, and the voltage setting was calibrated to the 600, 650, and 700 °C levels. The conjugate die cavity was machined by CNC engraving because the allowance-modified conjugate flank cannot be produced by the conventional face-milling method used for production wheels. After rolling, the blank was turned to the reference surfaces and inspected on a Gleason 650GMS gear measuring center.
| Flank | Maximum upper deviation (µm) | Maximum lower deviation (µm) | Total deviation (µm) | Cumulative pitch deviation (µm) |
|---|---|---|---|---|
| Convex | 9.9 | 4.5 | 14.4 | 57.0 |
| Concave | 10.1 | 4.0 | 14.1 | 56.3 |
The convex flank had a maximum upper deviation of 9.9 µm, a maximum lower deviation of 4.5 µm, and a total deviation of 14.4 µm. The concave flank had 10.1 µm, 4.0 µm, and 14.1 µm. The cumulative pitch deviations were 57.0 µm and 56.3 µm. The teeth reached GB accuracy grade 8. This is not a finished-ground gear; the result is a pre-heat-treatment body with enough flank and pitch accuracy to leave the intended grinding stock.
Interpretation, Limits, and Next Steps
The result supports a route in which warm rolling replaces rough cutting for the pinion, and final accuracy is generated after heat treatment by grinding. The dominant role of temperature is consistent with the thermal–mechanical nature of the process: flow stress, contact pressure, and strain partitioning all change with temperature. The secondary role of feed speed reflects the number of loading cycles per tooth and the size of each incremental bite. Friction is the least influential of the three under the tested range, but it still changes the strain distribution and should be controlled through lubrication.
Several boundaries remain. The parameter set covers one pinion geometry and one carburizing steel, AISI-8620. It does not include preform shape, motion mismatch between tool and blank, residual stress, microstructure, grain-flow continuity, or fatigue life. The validation is dimensional, not metallurgical. The 700 °C optimum is a stress-based choice within the 600–700 °C window; it does not establish a general thermal limit for warm rolling. These gaps define the next technical work: active tooth-flank topology control, preform optimization, and fatigue and grain-flow assessment on rolled-and-ground pinions. For spiral bevel gears, the pinion remains the limiting member, and its warm-rolling window must be evaluated together with the downstream grinding allowance rather than as a standalone forming step.