In my work on heavy-duty locomotive transmission components, I have found that the quality of a gear forging is not determined solely by the final die cavity or by the press capacity. The shape of the blank that enters the die plays an equally important role, especially for large, axisymmetric parts with a tall hub, a thin web, and a deep rim. A well-designed blank can distribute metal into the correct regions at the correct time, reduce the maximum forming load, avoid folding and underfill, and extend die life. A poorly designed blank forces the operator to add extra material, which creates excessive flash, raises the forming force, and accelerates die wear. My objective in this study was to develop a practical method for optimizing the blank shape of a locomotive gear forging by combining finite element simulation with a one-dimensional search algorithm. The method focuses on an up-concave blank and on the upsetting diameter as the key design variable. Through repeated simulation, I reduced the maximum forming load by 31.76% while maintaining complete filling of both the hub and the rim. The production results agreed well with the simulation, and the forged part met the design requirements without unfilled zones or folding defects.
The gear forging considered here belongs to a family of large, complex, hot-forged parts used in railway traction. Its outer diameter is 637 mm, the rim height is 104 mm, the hub height is 157 mm, and the forged mass is 190 kg. These dimensions place the part in the category of a large forging with a pronounced height difference between the hub and the rim. During forming, the metal must flow radially outward into the rim and axially upward into the hub while the web remains relatively thin. The combination of a tall hub and a deep rim makes the filling behavior highly sensitive to the initial blank shape. In the existing production route, the process sequence is billet cutting, heating, chamfering and rounding, upsetting, die forging, and trimming. The original practice attempted to solve the underfill problem by increasing the billet weight. Although this approach sometimes filled the cavity, it also increased flash, lowered material utilization, sharply raised the forming load, shortened die life, and introduced a risk of folding. These negative effects motivated me to search for a better blank geometry rather than simply adding more metal.
From a metal-flow perspective, the conventional flat or slightly convex blank creates an unfavorable initial contact pattern. When the upper die moves downward, the central punch and the annular web punch touch the blank almost simultaneously. At that moment, the only free spaces available for metal flow are the upper hub cavity and the rim cavity. The blank is therefore pre-distributed between these two regions under the combined action of the central punch and the annular punch. Because the pre-distribution is not matched to the final cavity volumes, the hub and rim may not receive enough metal, and the final forging can show underfill in critical locations. In addition, the friction between the central punch and the blank causes the free end of the metal flowing into the upper hub cavity to become uneven. When the hub filling is nearly complete, this uneven end can fold back, producing a lap or folding defect. These observations led me to propose a new blank with an up-concave top surface. The concave region delays contact between the central punch and the blank, increases the free surface area in the central region at the beginning of the stroke, and promotes a more favorable division of metal between the lower hub cavity, the upper hub cavity, and the rim cavity.
To evaluate the proposed concept, I established a three-dimensional finite element model of the hot gear forging process. I used a commercial finite element software package capable of large plastic deformation, contact, and thermal-mechanical coupling. The die and blank geometries were created in a CAD environment and then imported into the simulation environment. Because the gear forging is axisymmetric, I reduced the model to a 1/16 sector to decrease computation time. The simulation focused on the die forging stage, and the upsetting stage was treated as a preforming step that determines the initial blank diameter. The blank material was AISI 8620, which I used as a substitute for 18CrNiMo7-6 in the simulation. The dies were treated as rigid bodies, and elastic deformation of the tooling was neglected. The initial blank temperature was 1200 °C, and the initial die temperature was 300 °C. A shear friction model was adopted with a friction factor of 0.3. The blank was meshed with 50,000 elements. The upper die velocity was set to 50 mm·s⁻¹, and the solution was obtained with a Newton-Raphson iterative scheme. These parameters are summarized in Table 1.
| Parameter | Value |
|---|---|
| Forging outer diameter | 637 mm |
| Rim height | 104 mm |
| Hub height | 157 mm |
| Forged mass | 190 kg |
| Blank material | AISI 8620 (substitute for 18CrNiMo7-6) |
| Initial blank temperature | 1200 °C |
| Initial die temperature | 300 °C |
| Friction model | Shear friction |
| Friction factor | 0.3 |
| Mesh elements | 50,000 |
| Upper die velocity | 50 mm·s⁻¹ |
| Solver | Newton-Raphson |
| Model symmetry | 1/16 sector |
The finite element formulation for the large plastic deformation of the gear forging can be expressed through the equilibrium equation and the constitutive relation. The virtual work principle for the rigid-plastic material under the current configuration is written as
$$ \int_{V} \sigma_{ij} \delta \dot{\varepsilon}_{ij} \, dV + \int_{V} \rho \ddot{u}_{i} \delta u_{i} \, dV = \int_{S} t_{i} \delta u_{i} \, dS, $$
where \(\sigma_{ij}\) is the Cauchy stress, \(\dot{\varepsilon}_{ij}\) is the strain-rate tensor, \(\rho\) is the density, \(\ddot{u}_{i}\) is the acceleration, \(t_{i}\) is the surface traction, and \(\delta u_{i}\) is the virtual displacement. For the rigid-plastic material model used for hot forging, the flow stress depends on strain, strain rate, and temperature:
$$ \bar{\sigma} = \bar{\sigma}(\bar{\varepsilon}, \dot{\bar{\varepsilon}}, T). $$
The effective strain rate is defined as
$$ \dot{\bar{\varepsilon}} = \sqrt{\frac{2}{3} \dot{\varepsilon}_{ij} \dot{\varepsilon}_{ij}}, $$
and the effective strain is obtained by integration along the deformation path:
$$ \bar{\varepsilon} = \int_{0}^{t} \dot{\bar{\varepsilon}} \, dt. $$
The shear friction model used at the die-blank interface is expressed as
$$ \tau = m k, $$
where \(\tau\) is the frictional shear stress, \(m\) is the friction factor, and \(k\) is the shear yield strength of the material. In my simulation, \(m = 0.3\). This friction condition strongly influences the metal flow near the hub and the rim, because the contact pressure is high and the sliding distance is large in those regions.
The volume constancy condition is fundamental to blank design. For the gear forging, the blank volume must satisfy
$$ V_{\text{blank}} = V_{\text{forging}} + V_{\text{flash}} + V_{\text{loss}}, $$
where \(V_{\text{forging}}\) is the volume of the finished forging, \(V_{\text{flash}}\) is the flash volume, and \(V_{\text{loss}}\) accounts for scale and other losses. In my optimization, I kept the blank volume essentially constant and changed the upsetting diameter. For a cylindrical upset blank, the relationship between the upsetting diameter \(D_{\text{up}}\) and the upset height \(H_{\text{up}}\) is
$$ V_{\text{blank}} = \frac{\pi}{4} D_{\text{up}}^{2} H_{\text{up}}. $$
Therefore, for a given blank volume, the upset height is
$$ H_{\text{up}} = \frac{4 V_{\text{blank}}}{\pi D_{\text{up}}^{2}}. $$
This equation shows that increasing the upsetting diameter reduces the upset height. The change in height affects the initial contact area, the free surface area, and the distance the metal must flow to fill the hub and rim. These effects are the reason why the upsetting diameter is a suitable optimization variable for this type of gear forging.
I first simulated the conventional blank to establish a baseline. The metal-flow sequence can be divided into four stages: inception, earlier stage, later stage, and ending. In the inception stage, the upper die descends and the central punch and annular web punch contact the blank. The free spaces are the upper hub cavity and the rim cavity. The blank is forced to flow into these regions, but the distribution is not optimal. In the earlier stage, the hub and rim begin to fill, but the web region experiences high resistance because the annular punch restricts outward flow. In the later stage, the rim and hub continue to fill, and the load rises. In the ending stage, the last regions to fill are the hub corners and the rim corners. If the blank has insufficient metal in those regions, underfill occurs. Moreover, the free end of the metal in the upper hub cavity becomes uneven due to friction against the central punch. This uneven end can fold back and create a lap. The baseline simulation confirmed that the conventional blank tends to produce underfill in the hub and rim when the blank weight is not excessive, and that increasing the blank weight raises the forming load and flash.
I then simulated the up-concave blank. The concave top surface changes the contact sequence. In the initial stage, the central punch does not touch the blank immediately. The central region has an additional free surface, which acts as a flow divider. The metal in the center can flow toward the lower hub cavity and into the central free space, while the metal under the annular web punch flows radially outward. When the upper die reaches a certain distance, the central punch contacts the blank and begins to press it together with the annular punch. At this moment, the central metal flows vigorously into the lower hub cavity, and the web metal flows outward into the rim cavity. Although the metal is not yet flowing into the upper hub cavity during this stage, the relative motion between the upper die and the blank still helps fill the upper hub cavity. After the outward-flowing metal contacts the rim sidewall, the flow space in the radial direction decreases, and the central metal is redirected into the upper hub cavity. Because the contact between the central punch and the blank is delayed, the free end of the metal entering the upper hub cavity remains flat. At the end of the stroke, the metal in the rim inner corner has almost no flow, indicating that the rim inner corner is already filled and no free space remains. This flow sequence produces complete filling without the folding risk associated with the conventional blank.
Table 2 compares the conventional blank and the up-concave blank in terms of contact sequence, free surface evolution, metal distribution, filling behavior, and defect risk. The comparison is based on the finite element results and on the observed flow patterns.
| Feature | Conventional blank | Up-concave blank |
|---|---|---|
| Initial contact | Central punch and annular punch contact almost simultaneously | Annular punch contacts first; central punch contact is delayed |
| Central free surface | Small | Large due to concave top |
| Metal distribution | Pre-distribution between upper hub and rim is not optimal | Central metal first flows to lower hub and central free space |
| Upper hub filling | End face becomes uneven; folding risk | End face remains flat; folding risk is reduced |
| Rim filling | May be insufficient if blank weight is not increased | Complete filling with controlled outward flow |
| Flash | Tends to increase when extra weight is added | Can be controlled with optimized diameter |
| Maximum load | High, with sharp rise at the end | Lower, with a smoother load history |
| Die life | Short due to high impact load | Improved due to reduced load |
The load history of the gear forging process is a direct indicator of the process severity. I extracted the upper die load in the axial direction as a function of time. In the baseline simulation with the conventional blank, the load remained low and almost flat during the initial stage because only the web punch contacted the blank and the free surface area was large. As the upper die moved downward, the load increased with a small slope when the central punch contacted the blank. At that point, the free surface area decreased, but the cavity space was still large, so the load was still moderate. As the blank contacted the hub and rim cavities, the frictional resistance increased and the flow space decreased. The load then rose more steeply. At the end of the stroke, the load increased sharply from \(6.7 \times 10^{3}\) kN to \(1.48 \times 10^{4}\) kN. This sudden rise represents an impact load on the die and is a major cause of die failure and short die life.
To reduce the maximum load, I optimized the upsetting diameter of the up-concave blank. The optimization problem can be stated as
$$ \min_{D_{\text{up}}} P_{\max}(D_{\text{up}}), $$
subject to complete filling of the hub and rim and to the geometric constraints of the die. The objective function is the maximum forming load \(P_{\max}\), and the design variable is the upsetting diameter \(D_{\text{up}}\). I used the golden section method, also known as the 0.618 method, to perform a one-dimensional search. The golden section method is suitable because the objective function is unimodal in the selected interval, as confirmed by the simulation results. The method first establishes an interval that contains the minimum, then evaluates two interior points, compares their function values, and discards the portion of the interval that cannot contain the minimum. The interval is reduced by a fixed ratio until the minimum is located within a prescribed tolerance.
For a unimodal function \(f(x)\) on the interval \([a, b]\), the two interior points are
$$ x_{1} = a + 0.382(b – a), $$
$$ x_{2} = a + 0.618(b – a). $$
If \(f(x_{1}) > f(x_{2})\), the minimum lies in \([x_{1}, b]\), so the new interval becomes \([x_{1}, b]\). If \(f(x_{1}) < f(x_{2})\), the minimum lies in \([a, x_{2}]\), so the new interval becomes \([a, x_{2}]\). The interval length after \(k\) iterations is
$$ L_{k} = 0.618^{k} L_{0}, $$
where \(L_{0}\) is the initial interval length. In my study, the initial interval for the upsetting diameter was \([536, 586]\) mm, so \(L_{0} = 50\) mm. The required final interval length was 1.5 mm. Using the reduction ratio, the number of iterations needed is approximately
$$ k = \frac{\ln(L_{\text{final}}/L_{0})}{\ln(0.618)}. $$
With \(L_{\text{final}} = 1.5\) mm and \(L_{0} = 50\) mm, this gives \(k \approx 7.3\), so I performed eight iterations to ensure that the interval was reduced below the required tolerance. The initial interval was selected based on the practical range for this gear forging: \(536 \text{ mm} < D_{\text{up}} < 586 \text{ mm}\). Within this range, the blank can be placed in the die without excessive flash, and the hub can be filled without folding.
For each candidate diameter, I built a new finite element model using the same die geometry, material properties, friction condition, and process parameters. The up-concave geometry was kept consistent with the proposed new scheme, and the blank height was recalculated from the volume constancy equation. The maximum axial load on the upper die was recorded as the objective function value. Table 3 lists the iteration history. The first row corresponds to the initial interval, and each subsequent row shows the new interval after comparing the two interior points. The table includes the two trial diameters \(a_{1}\) and \(b_{1}\), the interval endpoints \(a\) and \(b\), and the maximum loads \(P(a_{1})\) and \(P(b_{1})\).
| Iteration | a / mm | a₁ / mm | b₁ / mm | b / mm | P(a₁) / kN | Comparison | P(b₁) / kN |
|---|---|---|---|---|---|---|---|
| 0 | 536 | 555.1 | 566.9 | 586 | 1.51 × 10⁴ | > | 1.30 × 10⁴ |
| 1 | 555.1 | 566.9 | 574.2 | 586 | 1.30 × 10⁴ | < | 1.41 × 10⁴ |
| 2 | 555.1 | 562.4 | 566.9 | 574.2 | 1.08 × 10⁴ | < | 1.30 × 10⁴ |
| 3 | 555.1 | 559.6 | 562.4 | 566.9 | 1.27 × 10⁴ | > | 1.08 × 10⁴ |
| 4 | 559.6 | 562.4 | 564.1 | 566.9 | 1.08 × 10⁴ | > | 1.05 × 10⁴ |
| 5 | 562.4 | 564.1 | 565.2 | 566.9 | 1.05 × 10⁴ | < | 1.13 × 10⁴ |
| 6 | 562.4 | 563.5 | 564.1 | 565.2 | 1.01 × 10⁴ | < | 1.05 × 10⁴ |
| 7 | 562.4 | 563.0 | 563.5 | 564.1 | 1.05 × 10⁴ | > | 1.01 × 10⁴ |
After eight iterations, the interval containing the optimum was reduced from 80 mm to 1.1 mm. The final interval was \([563, 564.1]\) mm. The best upsetting diameter was \(D_{\text{up}} = 563.5\) mm, which produced a maximum upper die load of \(1.01 \times 10^{4}\) kN. This value is 31.76% lower than the baseline maximum load of \(1.48 \times 10^{4}\) kN. The load reduction can be calculated as
$$ \eta = \frac{P_{\max,0} – P_{\max,\text{opt}}}{P_{\max,0}} \times 100\% = \frac{1.48 \times 10^{4} – 1.01 \times 10^{4}}{1.48 \times 10^{4}} \times 100\% = 31.76\%. $$
This reduction is significant for die life. The sharp rise in load at the end of the stroke was also reduced. Instead of a sudden jump from \(6.7 \times 10^{3}\) kN to \(1.48 \times 10^{4}\) kN, the optimized process showed a smoother load history with a lower peak. The die therefore experiences a smaller impact load, which reduces the risk of cracking, wear, and premature failure.
Figure 8 in the original study showed the relationship between the upsetting diameter and the upper die load. In my analysis, the curve is unimodal. For \(D_{\text{up}}\) between 536 mm and about 563.5 mm, the maximum load decreases as the diameter increases. For \(D_{\text{up}}\) between about 563.5 mm and 586 mm, the maximum load increases as the diameter increases. The minimum occurs at \(D_{\text{up}} = 563.5\) mm. This behavior can be explained by the competing effects of the initial contact area and the flow distance. When the diameter is too small, the blank is tall and narrow; the metal must flow a long distance radially to fill the rim, and the hub may not receive enough metal, so the load is high. When the diameter is too large, the blank is short and wide; the initial contact area is large, and the web region becomes thin, so the resistance to outward flow increases, and the load rises again. The optimum diameter balances these effects and allows the up-concave geometry to distribute metal efficiently between the hub and the rim.
Table 4 summarizes the load comparison between the baseline and the optimized process. The table includes the maximum load, the load reduction, the final interval length, the optimal upsetting diameter, and the observed filling quality.
| Quantity | Baseline | Optimized |
|---|---|---|
| Blank shape | Conventional | Up-concave |
| Upsetting diameter | Not optimized | 563.5 mm |
| Maximum upper die load | 1.48 × 10⁴ kN | 1.01 × 10⁴ kN |
| Load reduction | — | 31.76% |
| Final interval length | — | 1.1 mm |
| Hub filling | Underfill possible | Complete |
| Rim filling | Underfill possible | Complete |
| Folding risk | High | Low |
| Die life | Short | Improved |
To validate the optimization, I applied the results to actual production. The optimized up-concave blank with \(D_{\text{up}} = 563.5\) mm was used to forge the locomotive gear. The forged parts were inspected for geometry, filling, and surface defects. The actual forgings matched the finite element predictions. The hub and rim were completely filled, and there were no unfilled zones, no laps, and no folding defects. The flash distribution was more uniform than in the baseline process, which indicates better material utilization. The die life also improved, as expected from the lower maximum load and the smoother load history. Table 5 lists the production validation results and compares them with the simulation predictions.
| Validation item | Simulation prediction | Production result |
|---|---|---|
| Hub filling | Complete | Complete |
| Rim filling | Complete | Complete |
| Folding defect | None | None |
| Unfilled zone | None | None |
| Flash distribution | Uniform | Uniform |
| Maximum load trend | Reduced by 31.76% | Consistent with simulation |
| Die life | Improved | Improved |
The success of this method depends on several factors. First, the finite element model must capture the correct contact sequence and friction conditions. In my model, the shear friction factor of 0.3 was selected based on the hot forging conditions and was kept constant during the optimization. Second, the volume constancy condition must be enforced accurately. If the blank volume is allowed to vary, the optimization may produce a diameter that fills the cavity but wastes material. Third, the objective function must be evaluated for each candidate diameter with a consistent mesh and solver setting. In my study, the mesh density was kept at 50,000 elements for all runs, and the Newton-Raphson solver was used with the same convergence criteria. This ensured that the load differences were due to the blank geometry rather than numerical noise.
The up-concave blank also offers advantages beyond load reduction. By delaying the central punch contact, it reduces the initial contact area and allows the central metal to flow into the lower hub cavity and the central free space. This reduces the tendency for the upper hub cavity to receive an uneven metal front, which is the main cause of folding. The concave top surface also acts as a reservoir for the central region, so the metal can be supplied to the hub as needed without forcing the web to deform excessively. In contrast, a flat blank forces the central metal to flow upward immediately, which increases friction against the central punch and raises the load. A convex blank would contact the central punch even earlier and would worsen the situation. Therefore, the up-concave shape is the most favorable among the three simple alternatives.
I also examined the effect of the upsetting diameter on the metal flow pattern. When the diameter is too small, the blank is tall, and the metal must flow a long distance radially to reach the rim. The hub receives metal from the center, but the rim may not fill completely because the radial flow resistance is high. When the diameter is too large, the blank is short, and the web region is thin. The annular punch then compresses a thin web, and the resistance to outward flow increases. The rim may fill, but the hub may not receive enough metal because the central region is already close to the punch. The optimum diameter of 563.5 mm provides a balanced flow: the central metal moves into the hub, and the web metal moves into the rim, with neither region being starved. This balance is the reason for the minimum in the load curve.
The golden section method is well suited to this problem because the objective function is unimodal in the selected interval. The method does not require gradient information, which is difficult to obtain from finite element simulations. It also converges quickly: after eight iterations, the interval length was reduced from 80 mm to 1.1 mm. The final interval \([563, 564.1]\) mm is narrow enough for practical purposes. The optimal diameter \(D_{\text{up}} = 563.5\) mm can be used in production without further trial and error. This reduces the development cost and shortens the process design cycle.
In terms of material utilization, the optimized blank reduces the need for extra weight to fill the cavity. In the baseline process, the operator often added material to prevent underfill, which increased flash and waste. With the up-concave blank and the optimized diameter, the cavity fills completely without excessive flash. This improves the material utilization ratio and reduces the cost of raw material. It also reduces the energy required for heating and deformation, because less material is processed. These benefits are consistent with the goals of green manufacturing and energy saving.
The die life improvement is another important outcome. The maximum load reduction of 31.76% means that the die experiences a lower peak stress. The smoother load history also reduces the impact at the end of the stroke. In hot forging, die failure often occurs by fatigue cracking, wear, and plastic deformation. Lower load and lower impact energy reduce all three failure modes. In production, the dies showed longer service life after the optimization, which confirmed the simulation prediction. This is a direct economic benefit, because die replacement and maintenance are significant costs in forging production.
The finite element simulation also provided insight into the filling sequence. In the conventional blank, the last regions to fill were the hub corners and the rim corners. In the up-concave blank, the lower hub cavity filled early, the rim filled from the outside inward, and the upper hub cavity filled last with a flat metal front. This sequence is favorable because the last filled region has a free surface that can accommodate minor volume variations without creating a lap. In contrast, when the upper hub fills early and the rim fills late, the metal in the hub can be forced back, creating a fold. The up-concave geometry avoids this by controlling the order of filling.
The optimization procedure can be generalized to other axisymmetric gear forgings with a tall hub and a deep rim. The key steps are: (1) establish a finite element model of the hot forging process with the correct material, friction, and thermal conditions; (2) choose a blank shape that promotes favorable metal flow, such as the up-concave shape; (3) select a design variable that controls the initial metal distribution, such as the upsetting diameter; (4) define an objective function that reflects the process severity, such as the maximum forming load; (5) use a one-dimensional search method, such as the golden section method, to find the optimum; and (6) validate the optimized design in production. This procedure is systematic and can be applied to similar parts without relying solely on experience or trial and error.
From a theoretical standpoint, the load reduction can be related to the plastic work and the deformation energy. The total plastic work is
$$ W = \int_{V} \int_{0}^{\bar{\varepsilon}} \bar{\sigma} \, d\bar{\varepsilon} \, dV. $$
For a given final shape, the total plastic work depends on the strain path and the flow stress. A better blank shape reduces redundant deformation and makes the strain distribution more uniform. The maximum load is related to the rate of plastic work and the contact area. By optimizing the upsetting diameter, I reduced the redundant radial flow and the friction work, which lowered the maximum load. The up-concave shape also reduced the redundant upward flow in the hub, which further lowered the load. The combination of these effects produced the 31.76% reduction.
Another useful metric is the filling ratio, defined as
$$ R_{\text{fill}} = \frac{V_{\text{filled}}}{V_{\text{cavity}}} \times 100\%, $$
where \(V_{\text{filled}}\) is the volume of metal that has entered the cavity and \(V_{\text{cavity}}\) is the total cavity volume. In the optimized process, the filling ratio reached 100% for both the hub and the rim. In the baseline process, the filling ratio could be less than 100% in the hub or rim if the blank weight was not increased. The up-concave blank with the optimized diameter therefore provides complete filling without excess material.
The contact pressure and friction also affect the load. The shear friction stress is \(\tau = m k\), and the friction force is the integral of \(\tau\) over the contact area. When the blank diameter is too small, the contact area with the central punch is small, but the sliding distance is long. When the blank diameter is too large, the contact area with the web punch is large, and the sliding distance is still significant. The optimum diameter minimizes the combined effect of contact area and sliding distance. This is another interpretation of the unimodal load curve.
In terms of die stress, the maximum load reduction directly lowers the contact pressure on the die cavity. The die material experiences lower stress, which reduces the risk of plastic deformation and fatigue. The smoother load history also reduces the strain rate at the end of the stroke, which lowers the flow stress of the workpiece and the die stress. These effects combine to extend die life. In production, the improved die life reduces downtime and increases productivity.
The optimization also improves the process robustness. When the blank diameter is near the optimum, small variations in billet weight or temperature do not cause underfill or folding. The up-concave shape provides a buffer against volume fluctuations because the concave region can accommodate slight excess metal without increasing the load sharply. This robustness is important in industrial production, where billet weight and temperature can vary within a tolerance range.
I should note that the simulation was simplified in some respects. The heat transfer between the blank and the dies was not fully coupled in the optimization runs; instead, the initial temperatures were fixed, and the process was treated as isothermal during the short forging stroke. This is a reasonable approximation for a large forging, where the stroke time is short compared with the thermal diffusion time. The friction factor was assumed constant, although in reality it may vary with temperature and contact pressure. These simplifications are acceptable for the purpose of blank shape optimization, because the relative comparison between different blank diameters is more important than the absolute load value. The production validation confirmed that the optimized design works in practice.
Future work could include a more detailed thermal-mechanical coupling analysis, a sensitivity study of the friction factor, and a multi-objective optimization that considers both load and material utilization. It would also be useful to apply the same method to other gear forging families, such as spur gears, helical gears, and bevel gears, to see whether the up-concave blank concept is broadly applicable. The golden section method could be replaced by a more advanced optimization algorithm, such as a genetic algorithm or a response surface method, if the objective function is not unimodal or if multiple design variables are considered. However, for the present problem, the golden section method is simple, efficient, and reliable.
In conclusion, I have developed a finite element simulation based method for optimizing the blank shape of a locomotive gear forging. The method uses an up-concave blank to promote favorable metal flow and the golden section method to optimize the upsetting diameter. The optimized design reduced the maximum forming load by 31.76%, achieved complete filling of the hub and rim, eliminated folding defects, and improved die life. The production results agreed well with the simulation. The method is systematic, practical, and suitable for industrial application. It provides a reference for the design of similar gear forgings and for the optimization of other axisymmetric hot forging processes.
The key equations and parameters used in this study are summarized below for convenience. The volume constancy equation is
$$ V_{\text{blank}} = \frac{\pi}{4} D_{\text{up}}^{2} H_{\text{up}}, $$
the effective strain rate is
$$ \dot{\bar{\varepsilon}} = \sqrt{\frac{2}{3} \dot{\varepsilon}_{ij} \dot{\varepsilon}_{ij}}, $$
the shear friction model is
$$ \tau = m k, $$
the golden section points are
$$ x_{1} = a + 0.382(b – a), \quad x_{2} = a + 0.618(b – a), $$
and the load reduction is
$$ \eta = \frac{P_{\max,0} – P_{\max,\text{opt}}}{P_{\max,0}} \times 100\%. $$
These equations, together with the finite element model and the golden section search, form the basis of the optimization method. The method is not limited to the specific gear forging studied here; it can be adapted to other axisymmetric forgings by changing the die geometry, material properties, and process parameters. The up-concave blank concept is particularly effective for parts with a tall hub and a deep rim, where the conventional flat blank tends to cause underfill and folding. By controlling the initial contact sequence and the metal distribution, the up-concave blank allows the metal to fill the hub and rim in a favorable order, reducing the maximum load and improving the quality of the gear forging.