Spiral bevel and hypoid gears carry the final reduction in vehicle drivelines, differentials and many industrial transmissions because their curved tooth traces give gradual, overlapping contact, low noise and high torque capacity per unit volume. The weakest link in that chain is usually the pinion: it has fewer teeth, a smaller pitch cone angle and a longer active flank than its mating wheel, so it accumulates more load cycles and bends further under the same torque. How the pinion teeth are made therefore governs the fatigue life of the whole pair.
The pinion is where near-net-shape forming breaks down
Conventional face milling and hobbing cut the tooth space out of a solid blank. The cut interrupts the grain flow that would otherwise run continuously from the root fillet into the web, and the resulting tooth root is measurably weaker in bending fatigue than a formed tooth of the same geometry. Near-net-shape forming, which displaces material instead of removing it, keeps the flow lines continuous and is the main reason the process is attractive for highly loaded gears.
Forming has been applied successfully to the wheel member of the pair. With a pitch cone angle of about 71° (71.17° in the case examined here), the wheel flank is shallow and a precision-forged tooth can be ejected axially from the die without gouging. The pinion is the opposite case: an 18.82° pitch cone angle means the flank wraps tightly around the cone apex, and no rigid punch can be withdrawn from the formed tooth space. Rotary rolling with a gear-like tool in mesh with the blank avoids the ejection problem entirely, because the tool leaves the tooth space by continued rotation rather than by axial withdrawal.
Hot versus warm: why the process window is narrow
Rolling a gear tooth can be done hot or warm. Hot rolling works above roughly 1 000 °C, in the austenitic range where flow stress is low and large reductions are possible in one pass. The penalties are equally large: a visible oxide scale forms on the heated surface and is rolled into the flank, the through-thickness thermal gradient is steep, and the combination of thermal contraction after rolling and elastic die deflection makes tooth accuracy difficult to hold without a finishing cut.
Warm rolling heats only a thin surface layer of the tooth region by high-frequency induction, bringing roughly the outer skin to about 650 °C while the core stays considerably cooler. Three consequences matter for accuracy: no oxide layer forms at this temperature, the smaller thermal gradient limits differential contraction after the tool disengages, and the softened surface layer still flows plastically at die loads that a tool steel insert can survive. The process therefore occupies a narrow band — hot enough for the flank to flow, cool enough to stay clean and dimensionally stable — and that band has to be mapped experimentally and numerically before the process can be specified.
Designing the rolling tool as the conjugate of the pinion flank
The tool is not a standard generated gear. Its tooth surface must be the envelope, or conjugate, of the pinion flank it is meant to produce, so that the pinion is formed as the mirror image of the meshing action. The pinion surface is defined parametrically as r1(up, θp) with unit normal n1(up, θp), where up and θp are the surface coordinates. Transforming into the wheel coordinate system through the housing frame gives the tool surface and its normal:
$$ \mathbf{r}_2(u_p,\theta_p)=\mathbf{M}_{2d}\,\mathbf{M}_{dh}\,\mathbf{M}_{h1}\,\mathbf{r}_1(u_p,\theta_p), \qquad \mathbf{n}_2(u_p,\theta_p)=\mathbf{L}_{2d}\,\mathbf{L}_{dh}\,\mathbf{L}_{h1}\,\mathbf{n}_1(u_p,\theta_p), $$
where φ1 and φ2 are the rotational angles of pinion and wheel, E is the offset and Γ the shaft angle (90° in this case), and the L matrices are the rotation parts of the corresponding M matrices:
$$ \mathbf{M}_{2d}=\begin{bmatrix}1&0&0&0\\0&-\cos\varphi_2&-\sin\varphi_2&0\\0&\sin\varphi_2&-\cos\varphi_2&0\\0&0&0&1\end{bmatrix},\quad \mathbf{M}_{dh}=\begin{bmatrix}\cos\Gamma&0&-\sin\Gamma&0\\0&1&0&-E\\ \sin\Gamma&0&\cos\Gamma&0\\0&0&0&1\end{bmatrix},\quad \mathbf{M}_{h1}=\begin{bmatrix}1&0&0&0\\0&\cos\varphi_1&\sin\varphi_1&0\\0&-\sin\varphi_1&\cos\varphi_1&0\\0&0&0&1\end{bmatrix}. $$
With the wheel geometry known, the conjugate points are obtained by rotary projection: the pinion surface coordinates at the reference point serve as the initial guess, and the nonlinear system formed by the projection plane and the surface is iterated until the discrete conjugate points converge. Those points are the tool tooth surface.
One correction is required before the tool can be cut. A rolled pinion is not the finished part; it is the pre-grind state, and the rolled tooth must be thicker than the finished tooth to leave grinding allowance after heat treatment. A single-flank allowance Δh is converted into a rotation of the flank about the mid-face radius r2:
$$ \Delta\delta=\frac{\Delta h}{r_2}, \qquad \mathbf{r}_e=\mathbf{M}_{es}\,\mathbf{r}_s, $$
where rs and re are the flank points before and after rotation and Mes is the corresponding rotation matrix. The resulting geometry for the case studied is listed in Table 1. The discrete flank points were imported into a CAD environment to build solid models of blank and tool, and the assembly was exported as STL for the forming simulation.
| Parameter | Pinion | Wheel |
|---|---|---|
| Module (mm) | 3 | 3 |
| Spiral angle (°) | 35 | 35 |
| Number of teeth | 15 | 44 |
| 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 |
| Hand of spiral | Left | Right |
| Height modification coefficient | +0.34 | −0.34 |
| Tangential modification coefficient | +0.075 | −0.075 |
| Addendum (mm) | 3.57 | 1.53 |
| Dedendum (mm) | 2.094 | 4.134 |
| Pitch cone angle (°) | 18.82 | 71.17 |
| Face cone angle (°) | 21.09 | 72.89 |
| Root cone angle (°) | 17.10 | 67.78 |
A practical consequence of deriving the tool from the conjugate condition is that the tool flank is not a standard generated surface and cannot be cut on a conventional gear machine with a face-milling cutter. In the work described here the tool was engraved by milling on a machining centre, which is slow but geometrically unrestricted — an economic constraint that any production route based on this process has to absorb.
Finite-element representation of the rolling process
The forming simulation uses a rigid–plastic formulation. The blank is plastic, assigned AISI-8620 case-hardening steel; the tool wheel and the intermediate shaft that drives the blank are rigid. The blank is meshed with roughly 50 000 tetrahedral elements with local remeshing at the flank, since accuracy improves with mesh density but solution time rises steeply. The friction factor at the spline interface between the intermediate shaft and the blank is set to 0.8, the tool rotates at 1.071 rad/s, the corresponding spline speed is 3.141 59 rad/s, the tool starts at 20 °C, the step size is one third of the smallest element dimension, and the equations are solved by direct iteration.
Three locations are tracked through the process: a point on the flank (P1), a point in the tooth root (P2) and a point at the tooth tip (P3). Each of the three process variables — friction factor, feed speed and initial rolling temperature — is varied on its own while the other two are held fixed, so the response of stress and strain can be attributed without confounding. The friction levels 0.2, 0.3 and 0.4 correspond to the average values associated with oil-lubricated, graphite-lubricated and dry (unlubricated) hot rolling respectively, which makes the parameter sweep physically anchored rather than purely numerical.
How friction, feed and starting temperature shift stress and strain
The single-factor results are collected in Table 2. Two features hold across all nine runs. First, equivalent stress at every tracked point follows a rise–fall–rise or rise–fall pattern rather than a monotonic curve, because the instantaneous contact area between tool and blank changes as the tooth space is progressively filled. Second, the strain readings increase monotonically with rolling time and never fully saturate within the simulated stroke, which means the process ends while the material is still being worked.
| Series | Level | σP1 (MPa) | σP2 (MPa) | σP3 (MPa) | Strain P1 | Strain P2 | Strain P3 |
|---|---|---|---|---|---|---|---|
| Friction factor (600 °C, 0.1 mm/s) |
0.2 | 452 | 460 | 396 | 10.70 | 9.20 | 5.05 |
| 0.3 | 409 | 424 | 434 | 8.25 | 6.58 | 1.99 | |
| 0.4 | 346 | 464 | 410 | 4.41 | 5.66 | 3.25 | |
| Feed speed (0.4, 600 °C) |
0.1 mm/s | 346 | 464 | 410 | 4.41 | 5.66 | 3.25 |
| 0.2 mm/s | 382 | 365 | 425 | 8.87 | 8.97 | 6.78 | |
| 0.3 mm/s | 436 | 435 | 455 | 6.05 | 8.90 | 4.45 | |
| Initial temperature (0.4, 0.1 mm/s) |
600 °C | 442 | 405 | 373 | 5.70 | 5.41 | 3.71 |
| 650 °C | 370 | 400 | 391 | 7.00 | 10.10 | 6.37 | |
| 700 °C | 359 | 386 | 320 | 10.30 | 10.30 | 6.04 |
Friction factor: lower friction means larger deformation, not lower stress
At a friction factor of 0.2 the three locations reach their maxima at essentially the same instant, 7.80 s, with 460 MPa at the root, 452 MPa at the flank and 396 MPa at the tip. Raising the factor to 0.3 separates the events: the tip peaks early at about 4.20 s and then unloads, the flank peaks at 6.53 s and the root last at 9.79 s, with peak values of 434, 409 and 424 MPa. At 0.4 the separation is more pronounced — flank and tip peak at roughly 1.00 s and the root only at 8.94 s, reaching 464 MPa, the highest root stress of the whole sweep.
The strain response runs the other way. Peak flank strain falls from 10.70 at a friction factor of 0.2 to 8.25 at 0.3 and 4.41 at 0.4, and root strain falls from 9.20 to 6.58 to 5.66. The mechanism is straightforward: with little interfacial friction the tool slides against the flank, the contact pressure is distributed over a small area and the surface material is dragged and stretched; with high friction the contact area grows as the feed proceeds but the material adheres to the tool, metal flow becomes difficult and the deformation per unit time drops. High friction therefore protects the flank from excessive stretching while concentrating load at the root — an unfavourable trade for a part that fails in bending fatigue at the root fillet.
Feed speed: fewer, deeper bites versus many shallow ones
Feed speed controls how many times each tooth space is worked. At 0.1 mm/s the tooth is rolled repeatedly in shallow increments: the flank and tip peak early (about 1.00 s, 346 and 410 MPa), the root peaks at 8.94 s at 464 MPa, and the peak strains stay low (4.41, 5.66, 3.25). At 0.2 mm/s the flank and tip rise together to a maximum at 7.20 s (382 and 425 MPa) while the root keeps loading until 12 s (365 MPa); strains rise to 8.87, 8.97 and 6.78. At 0.3 mm/s each location follows a rise-and-fall curve with the tip peaking first at 4.50 s (455 MPa), the flank at 6.70 s (436 MPa) and the root at about 11.00 s (435 MPa), with strains of 6.05, 8.90 and 4.45.
The interpretation is that a low feed rate multiplies the number of rolling passes and keeps each increment small enough to be accommodated by local flow; the deformation is stable and the required forming force is modest. A high feed rate reduces the pass count and increases the depth of each indentation, so the material stores more elastic energy and springs back between passes, the total deformation increases and the forming force rises. The 0.2 mm/s case is intermediate but shows the largest strain at the root, which suggests that intermediate feed rates can be worse than either extreme when the pass pattern interacts with the tooth index.
Initial temperature: stress down, strain up
Temperature acts on flow stress directly. Raising the initial rolling temperature from 600 °C to 700 °C lowers the peak stress at every tracked location: the flank falls from 442 to 370 to 359 MPa, the root from 405 to 400 to 386 MPa and the tip from 373 to 391 to 320 MPa, with the non-monotonic tip value at 650 °C reflecting the change in when the tip peak occurs (1.10 s at 600 °C, 15.00 s at 650 °C, 3.28 s at 700 °C). Peak strain moves the other way, rising from 5.70/5.41/3.71 at 600 °C to 7.00/10.10/6.37 at 650 °C and 10.30/10.30/6.04 at 700 °C. Hotter material is softer and more ductile, so it deforms further for the same tool motion; colder material resists flow, raises the rolling force and, if the force exceeds what the flank can accommodate, produces forming defects rather than useful deformation.
Ranking the parameters by orthogonal screening
Because the strain response varies little between levels while the stress response varies considerably, the optimisation is driven by stress. A three-factor, three-level orthogonal array covers the space in nine runs instead of the twenty-seven a full factorial would require. The trial response is a stress index obtained by summing the peak equivalent stresses recorded over the tracked positions, and lower is better. Range analysis of the level means gives the influence ranking and the preferred level combination (Table 3).
| Run | A | B (mm/s) | C (°C) | Stress index |
|---|---|---|---|---|
| 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 |
| K1 | 4 045.5 | 3 817.5 | 4 188.0 | |
| K2 | 3 955.5 | 3 955.5 | 3 955.5 | |
| K3 | 3 865.5 | 4 093.5 | 3 723.0 | |
| F1 | 1 348.5 | 1 272.5 | 1 396.0 | |
| F2 | 1 318.5 | 1 318.5 | 1 318.5 | |
| F3 | 1 288.5 | 1 364.5 | 1 241.0 | |
| R | 60 | 92 | 155 | |
| Best level | A3 | B1 | C3 |
The ranges order the variables unambiguously: initial temperature (155) dominates, feed speed (92) is intermediate and friction factor (60) is the weakest of the three, so the influence sequence is temperature > feed speed > friction factor. The lowest level mean in each column gives the combination 700 °C, friction factor 0.4 and 0.1 mm/s. The ranking is consistent with the single-factor results: temperature changes the flow stress of the whole deforming volume, feed speed changes the incremental depth of each pass, and friction acts only on the interface.
It is worth being explicit about what this optimum does and does not mean. The objective was the stress index alone. Since strain rises steeply with temperature, the 700 °C optimum sits at the hot end of the warm-rolling window, where stress is lowest but the flank is worked hardest; that is acceptable only if the strain at 700 °C stays below the ductility limit of the surface layer and does not produce laps or folds on the flank. The friction factor of 0.4, the highest value tested, is also the least favourable choice from a root-loading standpoint, and it was selected because the stress index is dominated by the flank rather than the root. A different weighting of root and flank would shift the optimum.
Rolling trials and tooth measurement
The trial rig carries the toothed tool, a balance wheel that supports the blank and the induction coil around the tooth region. Heating is provided by a medium-frequency induction power supply with a matching transformer; the coil is shaped to the tooth band, an alternating field drives eddy currents into the blank surface, and a voltage control on the panel sets the temperature, which is how the 600, 650 and 700 °C levels are reproduced. After rolling, the blank body is turned to remove the clamping geometry and leave the finished pinion form.
Tooth surface and pitch were inspected on a Gleason 650GMS gear measuring centre. The measured deviations are summarised in Table 4.
| Item | Convex flank | Concave flank |
|---|---|---|
| Maximum upper deviation (μm) | 9.9 | 10.1 |
| Maximum lower deviation (μm) | 4.5 | 4.0 |
| Total flank deviation (μm) | 14.4 | 14.1 |
| Cumulative pitch deviation (μm) | 57.0 | 56.3 |
| Accuracy grade | GB 8 | |
Total flank deviations of 14.4 μm and 14.1 μm, with cumulative pitch deviations of 57.0 μm and 56.3 μm, place the as-rolled pinion at GB grade 8. That grade is not a finished-gear specification; it is the pre-grind condition, and the relevant question is whether the deviation band is small enough for a grinding allowance to clean up the flank without removing the hardened case depth that the heat treatment will produce. The reported result is that it is: the as-rolled flank is regular enough that grinding after heat treatment can finish the tooth without an intermediate cutting step, which is exactly the saving in process chain that motivates warm rolling.
What the result establishes and what it leaves open
The contribution of the work is a demonstrated chain from a conjugate tool surface, through a three-parameter finite-element window and an orthogonal screening step, to a rolled pinion that measures within a defined pre-grind tolerance. Three limits of that chain deserve as much attention as the result.
First, the model treats the tool and the machine as rigid while the actual process passes load through a tool stack, a balance wheel and a workpiece fixture; elastic deflection in that stack is comparable to 14 μm and will appear directly in the flank deviation. Second, the temperature is treated as uniform over the heated layer, whereas induction heating produces a gradient that follows the coil geometry, so the surface flow stress varies around the tooth and the strain distribution is unlikely to be as symmetric as the model suggests. Third, the optimisation covers only three process variables; blank preform shape, the mismatch between the kinematic motion of the tool and the theoretical conjugate motion, cooling rate and the resulting residual stress field were not included, and any of them can dominate the final distortion after heat treatment.
The geometric scope also deserves note: one gear pair with module 3 mm, 15 and 44 teeth, 20.2 mm face width and a 90° shaft angle. The parameter ranking temperature > feed speed > friction is likely to be robust, since it follows from the physics of flow stress and pass depth, but the specific optimum of 700 °C and 0.1 mm/s is tied to this combination of module, face width and steel. Larger modules concentrate more deformation per pass and would be expected to shift the optimum toward higher temperature or lower feed. Finally, the fatigue benefit that motivates forming in the first place — continuous grain flow through the root fillet — is not tested by a dimensional inspection; flank deviation and pitch show that the process holds geometry, not that it delivers longer life, which requires bending fatigue testing of rolled and cut pinions of identical geometry.