Spiral bevel gears carry the final reduction in vehicle and tractor drive axles, helicopter transmissions, and ship and port machinery. A single pair combines a large reduction ratio, a high contact ratio, and high load capacity, so the reliability of the whole drivetrain is set by the tooth flanks and roots of that pair. The driven member — the large wheel — is the more difficult of the two to manufacture, because its teeth are wide, strongly tapered, and must be produced to a flank tolerance tight enough to leave a uniform grinding allowance after carburizing.
Why warm forging is preferred over cut teeth
Milling remains the standard route for spiral bevel gears. Its limitation is metallurgical rather than geometric: the cutter severs the grain flow, so the flow lines that would otherwise follow the tooth profile terminate at the flank surface. Fatigue cracks initiate and propagate along those interrupted lines, and the load-bearing capacity of the finished gear is reduced accordingly. Forging avoids the problem because the flow lines are continuous through the root fillet, and the hard pressing of the blank against the die cavity leaves a surface layer in compression, which further retards crack initiation.
Conventional hot forging of these gears takes place at or above 1 000 °C and brings its own penalty. Oxide scale forms on the hot surfaces, and the dimensional scatter of a hot-forged tooth is too large to be ground directly, so a cutting step is inserted before heat treatment. That step removes the very surface layer the forging was meant to create and adds cost and lead time. Warm forging, held at or below 750 °C, stays clear of the scaling regime. The tooth surface is formed by hard squeezing against the die, the compressive surface stress is retained, and no tooth cutting is needed before carburizing; only a grinding or lapping allowance has to be left on the flanks.
The process route considered here therefore targets the driven wheel of small-modulus gears (module ≤ 4 mm): form the tooth in the warm state, leave a controlled allowance, carburize, and grind. What follows examines how the die cavity is derived, how the four principal process variables move the equivalent stress and strain in the workpiece, how an orthogonal screening narrows the parameter window, and how a forging trial on a 2 000 t press compared with the metrology predicted by the simulation.
From the gear tooth surface to the die cavity
The die cavity is obtained by inverting the gear tooth surface, since the mapping between the two is single-valued regardless of how the cavity is finally machined — by EDM, electrode, or ball-end milling. The gear surface itself is computed for the generating method, in which the inner and outer straight cutting edges of the cutter head sweep two cones as the head rotates. In the cutter coordinate system Sc, with nominal radius ro, blade angle αc, and surface coordinates uc and θc, the cutting edge is
$$\mathbf{r}_c(u_c,\theta_c)=\begin{bmatrix}\left(r_o \pm u_c\sin\alpha_c\right)\cos\theta_c\\ \left(r_o \pm u_c\sin\alpha_c\right)\sin\theta_c\\ -u_c\cos\alpha_c\end{bmatrix}$$
where the upper sign refers to the outer blade and the lower sign to the inner blade. Mapping the edge into the workpiece system through the machine settings — radial setting S1, angular setting q1, workpiece mounting angle γ1, and the horizontal/axial setting XG20 — gives the generating wheel vector
$$\mathbf{r}_2=\mathbf{M}_{2c}\left(S_1,q_1,\gamma_1,X_{G20}\right)\mathbf{r}_c(u_c,\theta_c)$$
with M2c the homogeneous transformation from the cutter system to the workpiece system. The resulting surface describes the finished tooth. Because a warm-forged gear is ground after heat treatment, each flank must be thickened by a single-side allowance Δh. This is done by rotating the surface about the pitch axis through an angle Δδ = Δh/r1, where r1 is the radius at mid-face width:
$$\mathbf{r}_e=\mathbf{M}_{es}\,\mathbf{r}_s,\qquad
\mathbf{M}_{es}=\begin{bmatrix}1&0&0&0\\0&\cos\Delta\delta&-\sin\Delta\delta&0\\0&\sin\Delta\delta&\cos\Delta\delta&0\\0&0&0&1\end{bmatrix}$$
Discrete points of the allowance-bearing surface were generated by a dedicated program, imported into a three-dimensional modelling package, and used to build both the workpiece and the die model. The gear used for the demonstration is a left-hand 44-tooth wheel of 3 mm module, paired with a 15-tooth pinion; its principal data are listed below.
| Parameter | Value |
|---|---|
| Number of teeth | 44 |
| Module | 3 mm |
| Mean pressure angle | 20° |
| Mean spiral angle | 35° |
| Hand of spiral | Left |
| Pitch diameter | 132 mm |
| Addendum | 1.536 mm |
| Dedendum | 4.128 mm |
| Face cone angle | 72.903° |
| Pitch cone angle | 71.175 29° |
| Root cone angle | 67.787 34° |
| Outer cone distance | 68.830 16 mm |
| Mating pinion teeth | 15 |
Simulation model and the quantities tracked
Simulating a complete ring blank is expensive and unnecessary for the purpose of parameter comparison, so a one-twelfth sector of the preform was modelled. The sector was meshed with tetrahedral elements, 120 000 in total, with local refinement in the region that contacts the die edges — the corner fillets where the tooth root and tip are formed, and where gradients in the temperature, strain, and stress fields are steepest. Reducing the sector and refining only where needed keeps resolution high without inflating the solve time.
Upper and lower dies were treated as rigid bodies and the preform as plastic. The workpiece material was 18CrNi4A, a carburizing steel used for heavily loaded gears. The integration step was set to one third of the smallest element edge, 0.416 36 mm, giving 0.138 8 mm. The interface heat transfer coefficient was 5 000 W/(m²·K), the convection coefficient 20 W/(m²·K), and the ambient temperature 20 °C. Twelve reference points were distributed over the big end, the mid-face, and the small end, covering both the tooth tip and the root fillet, so that equivalent stress and strain could be read at comparable locations in every run rather than from a single peak value.
How the four process variables move stress and strain
Each variable was varied on its own while the others were held at a baseline, giving a clean picture of the direction and the magnitude of its influence before any combination was attempted.
Forging speed
Three ram speeds were compared — 15, 20, and 25 mm/s — chosen to match the characteristics of an electric screw press as used industrially for straight bevel and hot-forged spiral bevel gears. Equivalent strain at the reference points first rises and then falls with increasing speed. The big end consistently shows far higher strain than the mid-face and the small end, because flash forms there during the stroke; the flash restricts radial metal flow and forces the root region to accept a larger share of the deformation. Within a single tooth, the root strain exceeds the tip strain. One anomaly matters for practice: at 20 mm/s the big-end tip strain jumps sharply, a local flow instability that should be avoided when the press schedule is set. Equivalent stress falls monotonically as speed rises from 15 to 25 mm/s. At the big end and mid-face the root carries higher stress than the tip and flank; at the small end the order is reversed, with the tip stressed more than the root.
Die preheating temperature
Die preheating was sampled at 200, 250, and 300 °C, inside the 150–300 °C range customary for warm forging. Strain increases with preheating temperature across all reference points. The mechanism is thermal: a hotter die extracts less heat from the workpiece, and the deformation energy added during the stroke raises the surface temperature further, so the material flows more easily and deforms more before the cavity fills. The response is strongly non-uniform. The big end is the most sensitive region, while the mid-face and small end change comparatively little, and the small end always shows the lowest strain of the three.
Friction coefficient
Three friction coefficients were examined — 0.12, 0.25, and 0.30 — spanning the range associated with lubricants used in warm forging. Strain first decreases and then increases with rising friction. Two competing effects explain the shape: at high friction, shearing between die and workpiece generates additional heat, softening the surface layer and improving flow; at low friction, the surface layer slides easily and deforms readily for that reason. Equivalent stress is highest at the lowest friction coefficient of the set. The interpretation offered is that easy surface flow promotes flash formation at the big end, and the additional flash constraint raises the effective stress in the part rather than lowering it.
Workpiece heating temperature
For 18CrNi4A the warm forging window is 600–800 °C, and 650, 700, and 750 °C were simulated. Equivalent strain is almost insensitive to workpiece temperature; the spatial pattern is unchanged, with the big end still far above the mid-face and small end because of the flash, and within a tooth the sequence root > flank > tip. The root–flank–tip ordering reflects the filling sequence: the convex side of the die contacts the blank first, and the concave side becomes the main forming surface as the cavity fills, so the concave flank accumulates more strain than the convex one. Equivalent stress, by contrast, falls markedly as temperature rises, because the softer, more fluid material requires less pressure to fill the cavity. At 750 °C the big end carries noticeably lower stress than the mid-face and the small end.
Orthogonal screening and the selected parameter set
Single-variable curves give directions, not a recipe, and the four variables do not all point the same way. A nine-run screening plan was therefore built around a baseline combination, varying one factor at a time through three levels each: forging speed A (15, 20, 25 mm/s), die preheating temperature B (200, 250, 300 °C), friction coefficient C (0.12, 0.25, 0.30), and workpiece heating temperature D (650, 700, 750 °C). For each run, the equivalent strain and equivalent stress were aggregated over the twelve reference points.
| Run | Combination | Aggregated equivalent strain | Aggregated 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 |
To rank the factors, each level was expressed relative to the smallest response in its own group, and the resulting ratios were multiplied to give a global weight. The strain ranking was led by A2B3C3D2, A2B1C3D2, A2B3C3D1, A2B1C3D1, and A2B3C3D3, with global weights of 3.854, 3.833, 3.601, 3.582, and 3.283. The stress ranking, ordered from the lowest global weight upward, was A3B2C2D3, A3B3C2D3, A3B2C3D3, A1B2C2D3, and A3B3C3D3, at 1.345, 1.355, 1.364, 1.365, and 1.373.
No combination appears in both lists: nothing simultaneously combines high strain with low stress. Since the forming load and the die stress are the binding constraints in production, equivalent stress was adopted as the optimization criterion. The selected combination is A3B2C2D3 — forging speed 25 mm/s, die preheating 250 °C, friction coefficient 0.25, workpiece heating 750 °C. This set had not been simulated in the screening plan, so it was run as an additional case. The aggregated equivalent stress over the big end, mid-face, and small end came out at 4 433 MPa, below the 4 502 MPa of A1B2C2D3, confirming the prediction and the ranking method itself.
Forging trial and gear measurement
The optimized parameters were then applied on a 2 000 t hydraulic press to produce a warm-forged driven wheel. Before measurement, the forging was located and its mounting face and bore were turned on a vertical lathe, followed by the outer diameter and front face, so that the metrology would reference machined surfaces rather than forging surfaces. Tooth flank topology and pitch were then measured on a Gleason 650GMS gear inspection centre.
The maximum flank topology error was 13.7 µm on the left flank and 13.9 µm on the right, both inside the permitted band. On the concave flank the largest upper deviation was +5.1 µm and the largest lower deviation −13.7 µm, a total spread of 18.7 µm; on the convex flank the corresponding values were +13.9 µm and −13.8 µm, a spread of 27.7 µm. Cumulative pitch deviation totalled 60.7 µm and 86.8 µm on the two flanks, which corresponds to grade 8 and grade 9 of GB/T 11365—2019. That is sufficient accuracy for a forging that will be carburized and ground, and it confirms that the allowance left by the surface-rotation step is of the right size and distribution: enough material for grinding, not so much that grinding time becomes excessive.
| Measured item | Concave flank / left | Convex flank / right |
|---|---|---|
| Maximum flank topology error | 13.7 µm | 13.9 µm |
| Largest upper deviation | +5.1 µm | +13.9 µm |
| Largest lower deviation | −13.7 µm | −13.8 µm |
| Total deviation spread | 18.7 µm | 27.7 µm |
| Cumulative pitch deviation | 60.7 µm (grade 8) | 86.8 µm (grade 9) |
Scope and remaining questions
The result supports a specific, bounded claim: for a small-modulus driven spiral bevel gear of 18CrNi4A, a warm forging route built on a die cavity derived from the allowance-bearing tooth surface, with parameters selected by simulation and orthogonal screening, produces a tooth form accurate enough for post-carburizing grinding without any pre-heat-treatment cutting. The measurable benefit is the uninterrupted grain flow through the root and the compressive surface layer left by hard squeezing — the two features that milling cannot provide, and the reason the route is of interest for fatigue-limited axle and transmission gears.
The scope is equally clear. Only the four principal variables were screened, at one level each per run, so the ranking reflects main effects and not interactions; and warm forging is a coupled thermal–mechanical process in which heating rate, ambient conditions, and the temperature history of the stroke all shift the result. The link between the simulated strain field and the resulting microstructure and hardness after carburizing was not examined, yet it is this link that ultimately governs tooth root fatigue life. Those couplings, rather than further refinement of the die surface, are the next constraint to be resolved before the process can be specified with confidence for series production.