Spiral bevel gears transmit motion between intersecting shafts through teeth whose traces curve along the pitch cone. Compared with straight bevel gears, the curved tooth trace gives more gradual contact, higher load capacity, and smoother meshing. These advantages come at a manufacturing cost: the tooth space is narrow, the tooth depth changes along the face width, and the root fillets are small. Precision forging can produce the tooth form with little or no subsequent machining, but the die must fill these confined regions without generating excessive forming load or die stress. The process window is therefore controlled by billet temperature, deformation speed, and interface friction.
For spur gears and straight bevel gears, closed-die precision forging has already been developed around controlled material flow and split cavities. Spiral bevel gears are more difficult because the curved tooth space changes direction along the face width. A small increase in flow resistance can produce a sharp rise in load near the end of the stroke, while incomplete filling leaves a tooth form that cannot be corrected by simple trimming. The relevant process question is not only whether the cavity can be filled, but how the forming load responds to the main process variables.
Closed-die pre-forging and split-flow final forging
The process route considered here is closed-die pre-forging followed by split-flow final forging. In the pre-forging stage, the upper and lower tooling contain matching boss features, so the preform cavity is filled progressively from the top and bottom. This distributes the initial deformation and avoids concentrating the entire reduction in the tooth region.
In the final stage, the die includes a split cavity in addition to the tooth cavity. As the punch advances, material fills the tooth impressions and also flows into the split cavity. The billet volume is deliberately controlled so that the split cavity is not completely filled when the tooth form is complete. The remaining volume acts as a relief space: excess material can continue to flow instead of causing a steep pressure rise at the end of the stroke. This is important for spiral bevel gears because the curved tooth ends and the narrow root region are sensitive to overload, and an excessive end-of-stroke load can damage the die or cause elastic die deflection that worsens tooth accuracy.
Finite element model and baseline parameters
A three-dimensional finite element model of the closed-die forging process was built in DEFORM-3D. The upper die, upper punch, tooth die, and lower punch were treated as rigid bodies, while the 20Cr workpiece was treated as a plastic body. Elastic deformation of the workpiece was neglected because it is small compared with the plastic deformation in bulk forging. The workpiece mesh contained 52,936 elements. The baseline gear and process parameters are listed in Table 1.
| Category | Parameter | Value |
|---|---|---|
| Gear geometry | Number of teeth, \(Z\) | 18 |
| Gear geometry | Module, \(m\) | 4 mm |
| Gear geometry | Spiral angle, \(\beta\) | 31° |
| Gear geometry | Pressure angle, \(\alpha\) | 20° |
| Gear geometry | Addendum coefficient, \(h_a^*\) | 1 |
| Gear geometry | Normal clearance coefficient, \(c\) | 0.2 |
| Gear geometry | Hand | Left |
| Materials | Billet | 20Cr |
| Materials | Die | H13 |
| Thermal conditions | Billet temperature, baseline | 800 °C |
| Thermal conditions | Die preheat temperature | 200 °C |
| Interface | Friction factor, baseline | 0.25 |
| Kinematics | Deformation speed, baseline | 10 mm/s |
| Discretization | Workpiece mesh | 52,936 elements |
Flow stress and friction representation
In the rigid-plastic formulation, the workpiece flow stress depends on equivalent plastic strain, equivalent strain rate, and temperature:
\[
\bar{\sigma}=\bar{\sigma}\left(\bar{\varepsilon},\dot{\bar{\varepsilon}},T\right)
\]
where \(\bar{\sigma}\) is the flow stress, \(\bar{\varepsilon}\) is the equivalent plastic strain, \(\dot{\bar{\varepsilon}}\) is the equivalent strain rate, and \(T\) is the temperature. This relationship captures the competition between strain hardening, strain-rate hardening, and thermal softening. For a forging operation with heated dies, the temperature term is not constant: heat transfer between the workpiece and the tooling changes the local flow stress during the stroke.
Interface friction was represented with a constant shear friction model:
\[
f_s = m k
\]
where \(f_s\) is the friction stress, \(k\) is the shear yield stress of the workpiece material, and \(m\) is the friction factor. In this formulation, \(m\) is not the Coulomb friction coefficient; it scales the shear resistance at the interface relative to the shear yield stress. The model is convenient for bulk forming because the normal pressure at the tooth surfaces is high and the real contact condition is difficult to describe with a single Coulomb coefficient.
Effect of billet temperature
Billet temperature has a direct effect on flow stress, ductility, die filling, and the final microstructure. Simulations were compared at billet temperatures of 600 °C, 700 °C, and 800 °C, with the other baseline conditions held constant. The load–stroke curves show that the forming load decreases as the billet temperature increases. At 800 °C, the maximum forming load was 39.8% lower than at 600 °C.
The main reason is thermal softening. As the temperature rises, the yield limit of 20Cr decreases and the material becomes more ductile, so a smaller stress is required to produce the same local shape change. Better metal flow also helps the tooth cavity fill before the load reaches an excessive level. Within the examined range, a higher billet temperature is therefore favorable for reducing press load and improving fill. The practical upper limit is set by grain growth, surface oxidation, die softening, and the need to maintain a controlled final microstructure; the result should not be read as a recommendation to raise the temperature without metallurgical limits.
Effect of deformation speed
Deformation speed changes both the strain rate and the thermal history. Simulations were run at 1 mm/s, 10 mm/s, and 20 mm/s with all other parameters fixed. The forming load decreased when the speed increased from 1 mm/s to 10 mm/s. Between 10 mm/s and 20 mm/s, the change in final load was small.
Two mechanisms act in opposite directions. A higher deformation speed raises the strain rate, which tends to increase flow stress and deformation resistance. At the same time, the workpiece remains in contact with the die for a shorter period, so less heat is transferred from the billet to the die. The billet stays hotter, and thermal softening partly offsets strain-rate hardening. In the examined range, the thermal effect is sufficient to lower the load at moderate speed, while further speed increase produces only a minor net change. This balance is sensitive to die preheat, contact time, and lubricant condition; outside the tested range, strain-rate hardening or incomplete cavity filling may become dominant.
Effect of friction factor
Friction controls the lateral flow of metal into the tooth impressions. The friction factor was varied at 0.15, 0.25, and 0.35 while the other conditions remained unchanged. The maximum forming load increased with the friction factor.
The trend is consistent with the geometry of a spiral bevel gear die. Metal must flow laterally into a curved, narrowing tooth space, and interface friction resists that flow. Higher friction raises the internal pressure needed to push material toward the tooth tip and root fillet, so both the forming load and the local die pressure increase. In production, the friction factor can be reduced by selecting a suitable lubricant and by controlling die surface condition. Lower friction also reduces the risk of adhesion and die wear, but the lubricant must remain stable at the billet and die temperatures used in the process.
Comparison of parameter effects
Table 2 summarizes the parameter ranges and the observed load trends. The comparison is based on one-factor-at-a-time simulations; it identifies the direction and relative importance of each variable but does not resolve interactions between temperature, speed, and lubrication.
| Parameter | Range examined | Effect on maximum forming load | Interpretation |
|---|---|---|---|
| Billet temperature | 600 °C, 700 °C, 800 °C | Decreases as temperature rises; 800 °C is 39.8% lower than 600 °C | Thermal softening lowers flow stress and improves ductility and cavity filling |
| Deformation speed | 1 mm/s, 10 mm/s, 20 mm/s | Decreases from 1 mm/s to 10 mm/s; only a small change from 10 mm/s to 20 mm/s | Reduced die chilling at higher speed offsets strain-rate hardening |
| Friction factor | 0.15, 0.25, 0.35 | Increases with friction factor | Interface friction impedes lateral flow into the tooth cavity and raises internal pressure |
Process-control implications
For spiral bevel gear precision forging, the forming load should be controlled through the combination of billet temperature, stroke speed, lubrication, and die-cavity relief. Temperature is the strongest direct lever on flow stress: a uniform billet temperature and a controlled die preheat reduce the risk of local chilling and incomplete tooth filling. Speed must be selected with care. A moderate increase in speed can reduce die chilling and lower the load, but an excessive speed can raise strain-rate resistance and make the load sensitive to small changes in contact condition.
Friction control is equally important because the tooth cavity is narrow and curved. Adequate lubrication lowers the load required to fill the tooth ends and reduces the pressure on the die inserts. The split cavity and the controlled billet volume provide a second safeguard: by leaving the relief cavity partially unfilled at the end of the tooth-filling stage, the process avoids the sharp load rise that would otherwise occur when the tooth cavity becomes closed. This design principle links the process parameters to die life and tooth accuracy, not only to press capacity.
The reported trends are numerical predictions from a rigid-plastic finite element model with a constant shear friction law. They are useful for ranking parameters and for defining the direction of process adjustment, but they are not a substitute for forging trials. Die elasticity, thermal distortion, lubricant film breakdown, friction evolution at high temperature, and microstructure evolution can all shift the optimum window. Experimental validation should therefore include load–stroke measurement, tooth-form filling at the toe and heel, die wear inspection, and metallurgical examination of the forged spiral bevel gears.