I approached the precision forging of forged gear blanks as a coupled problem of material flow, cavity filling, equivalent strain distribution, flash formation, and press load. The central purpose was not merely to obtain a numerically filled die, but to identify a billet geometry that allows the upper and lower die cavities to fill in a controlled sequence, suppresses internal backflow and folding, keeps the final equivalent strain field as uniform as possible, and avoids unnecessary press tonnage. I therefore constructed a three-dimensional finite element model of a gear blank, varied the initial cylindrical billet diameter and height under a constant-volume constraint, and compared the resulting forging behavior. The study focused on forged gear blanks produced from 45 steel at an elevated temperature, with the die treated as a heated tool and with friction represented by a constant shear friction factor. The numerical experiments were carried out with DEFORM-3D, and the comparisons were organized around measurable or observable indicators: fill sequence, fill-time difference between hub and rim, evidence of metal backflow, tendency toward folding, amount of flash, uniformity of equivalent strain, and maximum upper-die stroke load.
Geometric basis and volume control. I first represented the forged gear blank as an axisymmetric body with a toothed outer rim, a central hub, a bore, and a thin punching web. The toothed rim was approximated in the finite element model by the equivalent outer envelope required for forging simulation, while the final tooth form was retained through the die cavity. The principal dimensions used in my model are summarized in Table 1. The finished outer envelope was taken slightly larger than the nominal toothed tip diameter in order to leave a controlled machining allowance. The bore was not punched fully through in the first forging operation; a thin web was retained to stabilize the hub and to be removed in a subsequent piercing step.
| Feature | Value used in the model |
|---|---|
| Number of teeth | 29 |
| Tip diameter of toothed form | 155 mm |
| Tooth thickness | 8 mm |
| Hub height | 40 mm |
| Hub diameter | 50 mm |
| Bore diameter | 20 mm |
| Punching web thickness | 5 mm |
| Outer blank diameter including allowance | 158 mm |
The target forging volume was first calculated from the finished geometry, and I then added a small allowance for machining, flash, and process variation. For a cylindrical billet of diameter \(D_b\) and height \(H_b\), the nominal volume is
$$ V_b = \frac{\pi}{4}D_b^2 H_b. $$
For the finished forged gear blank, the calculated volume was approximately \(2.08689717\times 10^{5}\ \mathrm{mm^3}\). After adding the allowance for machining and flash, I used a billet volume of approximately \(2.14413699\times 10^{5}\ \mathrm{mm^3}\). The volume-conservation condition used throughout the study was therefore
$$ V_b \ge V_{\mathrm{part}} + V_{\mathrm{flash}} + V_{\mathrm{allowance}}, $$
and for each candidate billet diameter the billet height followed from
$$ H_b = \frac{4V_b}{\pi D_b^2}. $$
The height-to-diameter ratio of the billet was tracked because it influences the initial aspect ratio of the deformation zone and therefore the order in which the hub and rim fill:
$$ r_h = \frac{H_b}{D_b}. $$
This ratio is a convenient first-order indicator for forged gear blanks, although it is not sufficient by itself because the local die geometry and friction conditions also control the filling sequence. I used it to design the initial set of numerical experiments and to ensure that all candidate billets had the same nominal volume.
Thermal, frictional, and kinematic inputs. I treated the process as a hot forging operation. The workpiece material was 45 steel, the initial gear-blank temperature was set to \(1200^{\circ}\mathrm{C}\), and the die temperature was set to \(200^{\circ}\mathrm{C}\). I used a constant shear friction factor of \(0.5\), an upper-die speed of \(5\ \mathrm{mm/s}\), and a die fillet radius of \(3\ \mathrm{mm}\). These values are collected in Table 2. The die speed was kept moderate so that the simulation remained stable while still representing a practical press stroke. The friction factor was deliberately chosen on the high side of the usual hot-forging range because the contact between the hot workpiece and the relatively cooler die can produce strong shear resistance, especially in the hub and web regions.
| Parameter | Value |
|---|---|
| Workpiece material | 45 steel |
| Initial gear-blank temperature | \(1200^{\circ}\mathrm{C}\) |
| Die temperature | \(200^{\circ}\mathrm{C}\) |
| Shear friction factor | \(0.5\) |
| Upper-die speed | \(5\ \mathrm{mm/s}\) |
| Die fillet radius | \(3\ \mathrm{mm}\) |
| Billet volume used in all candidates | \(2.14413699\times 10^{5}\ \mathrm{mm^3}\) |
Finite element discretization and solution controls. I used a Lagrangian incremental formulation with automatic remeshing. The initial mesh for each candidate contained approximately \(50\,000\) elements. The simulation step size was set to approximately one third of the minimum element edge length in the active deformation zone. This choice is a compromise between numerical stability and computational cost. If \(\ell_{\min}\) is the minimum element length, then the step size was selected as
$$ \Delta s \approx \frac{\ell_{\min}}{3}. $$
For the larger billets I used a nominal step size of \(0.4\ \mathrm{mm}\), while for the smallest billet I reduced it to \(0.35\ \mathrm{mm}\) to resolve the more severe local deformation near the hub and web. The remeshing criterion was based on element distortion, interference with the die surface, and strain-gradient intensity. I monitored the upper-die load, the volume of material in the hub and rim, the contact area, and the maximum equivalent strain at each step. The main discretization and solution controls are listed in Table 3.
| Control item | Setting |
|---|---|
| Formulation | Lagrangian incremental with remeshing |
| Initial element count | Approximately \(50\,000\) |
| Step size | Minimum element edge divided by 3 |
| Nominal step for larger billets | \(0.4\ \mathrm{mm}\) |
| Nominal step for the smallest billet | \(0.35\ \mathrm{mm}\) |
| Remeshing trigger | Element distortion, die interference, strain gradient |
| Die treatment | Rigid heated tool |
| Workpiece treatment | Thermo-viscoplastic deformable body |
Governing mechanical quantities. For the stress and strain comparisons, I used the standard von Mises equivalent stress, equivalent strain, and equivalent strain rate. The equivalent stress is
$$ \sigma_{\mathrm{eq}} = \sqrt{\frac{1}{2}\left[(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2\right]}, $$
where \(\sigma_1\), \(\sigma_2\), and \(\sigma_3\) are the principal stresses. The equivalent strain is
$$ \varepsilon_{\mathrm{eq}} = \sqrt{\frac{2}{3}\left(\varepsilon_1^2+\varepsilon_2^2+\varepsilon_3^2\right)}, $$
and the equivalent strain rate is
$$ \dot{\varepsilon}_{\mathrm{eq}} = \sqrt{\frac{2}{3}\left(\dot{\varepsilon}_1^2+\dot{\varepsilon}_2^2+\dot{\varepsilon}_3^2\right)}. $$
For the hot deformation of 45 steel, I represented the flow stress in the general form
$$ \bar{\sigma} = K \bar{\varepsilon}^{\,n}\dot{\bar{\varepsilon}}^{\,m}, $$
where \(K\) is a strength coefficient, \(n\) is the strain-hardening exponent, and \(m\) is the strain-rate sensitivity. At the temperatures used here, the strain-hardening contribution is partially offset by dynamic recovery and recrystallization, so the exact constants are temperature-dependent. For the purpose of comparing billet geometries under identical thermal and kinematic conditions, the relative trends in forged gear blanks are governed more strongly by die filling and strain path than by small changes in the flow-stress constants.
The frictional shear stress was represented with the shear friction model
$$ \tau_f = m_f k_s, $$
where \(m_f\) is the friction factor and \(k_s\) is the shear yield strength of the workpiece. With the von Mises yield condition, \(k_s = \sigma_s/\sqrt{3}\), so the friction stress can also be written as
$$ \tau_f = m_f \frac{\sigma_s}{\sqrt{3}}. $$
The upper-die load was obtained by integrating the normal pressure and the friction stress over the contact surface:
$$ F = \int_{A_c} p_n\,dA + \int_{A_c} \tau_f\,dA, $$
where \(A_c\) is the instantaneous contact area. The stroke work was evaluated from the load-displacement history:
$$ W = \int_{0}^{s_{\max}} F(s)\,ds. $$
These quantities allowed me to compare candidate billets not only by whether they filled the die, but also by the mechanical cost of filling it. For forged gear blanks, a geometry that fills the die with a lower load and a more uniform strain field is preferred, provided that no filling defects are introduced.
Filling criteria used in the comparison. I defined the quality of a candidate billet using the criteria in Table 4. The hub should fill before the rim, or at least the two regions should finish filling within a short time interval. If the rim fills first and the hub remains open, subsequent die motion forces material to flow backward from the rim toward the hub. That backflow is undesirable because it can break the continuity of the forged flow lines and produce a less favorable mechanical texture in the finished forged gear blank. I also rejected candidates that showed folding, severe flash extrusion, or a large time gap between hub filling and rim filling.
| Criterion | Definition used in this study | Preferred condition |
|---|---|---|
| Filling sequence | Order in which hub and rim cavities become full | Hub fills first, rim fills shortly afterward |
| Fill-time difference | Difference between rim fill time and hub fill time | As small as possible |
| Backflow | Local flow directed from rim toward hub after rim filling | No visible backflow |
| Folding | Surface or internal overlap caused by reversed flow | No folding |
| Flash | Material extruded into the die clearance | Minimal and distributed |
| Strain uniformity | Variation of equivalent strain in hub and rim | High uniformity |
| Press load | Maximum upper-die stroke load | Low enough for the selected press |
I quantified the fill-time difference as
$$ \Delta t_f = t_{\mathrm{rim}} – t_{\mathrm{hub}}, $$
where \(t_{\mathrm{hub}}\) and \(t_{\mathrm{rim}}\) are the times at which the hub and rim cavities reach full contact with the die. A positive \(\Delta t_f\) means that the hub filled first; a negative value means that the rim filled first. In my comparison, a small positive \(\Delta t_f\) was considered favorable, while a negative \(\Delta t_f\) was a warning sign for backflow in forged gear blanks. I also used a normalized flash indicator
$$ \Phi_b = \frac{A_{\mathrm{flash}}}{A_{\mathrm{projected}}}, $$
where \(A_{\mathrm{flash}}\) is the projected area of material extruded into the die clearance and \(A_{\mathrm{projected}}\) is the projected area of the forged gear blank. This indicator is not a substitute for a detailed flash analysis, but it is useful for ranking candidate billets under otherwise identical conditions.
Initial billet candidates. I began with four cylindrical billets of the same nominal volume but different diameters and heights. The initial candidates are listed in Table 5. The largest diameter produced a short, squat billet, while the smallest diameter produced a taller billet with a larger initial height-to-diameter ratio. The purpose of this first sweep was to locate the diameter range in which the filling sequence changes from rim-first to hub-first behavior.
| Initial scheme | Billet diameter \(D_b\) (mm) | Billet height \(H_b\) (mm) | Height-to-diameter ratio \(r_h\) | Nominal step (mm) |
|---|---|---|---|---|
| 1 | 120 | 19.0 | 0.158 | 0.40 |
| 2 | 110 | 22.6 | 0.205 | 0.40 |
| 3 | 100 | 27.3 | 0.273 | 0.40 |
| 4 | 90 | 33.7 | 0.374 | 0.35 |
For the 120 mm and 110 mm billets, I observed that the rim region reached the die surface first, while the hub region remained difficult to fill until late in the stroke. Once the rim was full, continued downward motion of the upper die forced material from the rim toward the hub. This produced a clear backflow pattern. The rim also became locally thicker than the intended final dimension before the hub was completely filled, which is exactly the condition that should be avoided in precision forging of forged gear blanks. The resulting flow-line disruption is likely to reduce the mechanical performance of the final part, even if the die is eventually filled.
For the 100 mm billet, the behavior was different. The hub region filled first, and the rim region filled shortly afterward. The time difference between hub filling and rim filling was small, and the amount of material extruded into the die clearance was limited. The metal flow direction remained broadly consistent throughout the stroke, and I did not observe a pronounced reversal from rim to hub. This candidate therefore satisfied the main filling criteria for forged gear blanks. For the 90 mm billet, the hub also filled before the rim, and the flow direction was generally acceptable, but the time gap between hub filling and rim filling was larger than for the 100 mm billet. That larger gap is unfavorable because the hub and rim experience different strain histories for a longer portion of the stroke, which can increase strain nonuniformity.
| Initial scheme | Observed filling sequence | Backflow tendency | Flash tendency | Preliminary assessment |
|---|---|---|---|---|
| 1, \(D_b=120\ \mathrm{mm}\) | Rim first, hub late | Strong | Pronounced rim thickening before hub fill | Rejected |
| 2, \(D_b=110\ \mathrm{mm}\) | Rim first, hub late | Strong | Excess material near rim | Rejected |
| 3, \(D_b=100\ \mathrm{mm}\) | Hub first, rim shortly after | Not significant | Limited | Promising |
| 4, \(D_b=90\ \mathrm{mm}\) | Hub first, rim after a longer delay | Not significant | Limited | Acceptable but requires further study |
Refined search around the promising range. The initial sweep showed that the transition between unfavorable rim-first filling and favorable hub-first filling occurs between 100 mm and 110 mm, while the lower bound for acceptable flow is near 90 mm. I therefore refined the search with three additional billets: 108 mm, 105 mm, and 102 mm. Their heights were again calculated from the constant-volume condition and are listed in Table 7. The intention was to determine whether a diameter slightly larger than 100 mm could improve the filling sequence without introducing backflow, and whether a diameter slightly smaller than 100 mm could reduce the fill-time difference without increasing the load excessively.
| Refined scheme | Billet diameter \(D_b\) (mm) | Billet height \(H_b\) (mm) | Height-to-diameter ratio \(r_h\) | Nominal step (mm) |
|---|---|---|---|---|
| 5 | 108 | 23.4 | 0.217 | 0.40 |
| 6 | 105 | 24.8 | 0.236 | 0.40 |
| 7 | 102 | 26.2 | 0.257 | 0.40 |
The 108 mm billet behaved similarly to the larger initial candidates. The rim filled first, the hub filled later, and metal backflow occurred. I rejected this scheme for the same reason as the 120 mm and 110 mm billets: the flow-line continuity in a precision-forged gear blank is more important than a small change in initial aspect ratio. The 105 mm billet did not show instability, folding, or pronounced backflow, which was an improvement over the 108 mm case. However, it produced more flash from the die clearance than the 100 mm billet. That flash is not merely a material-loss issue; it can create local stress concentrations and can make the final trimming operation more difficult. The 102 mm billet performed well. It avoided instability, folding, and backflow, and it produced only a small amount of flash immediately before the final step. From a filling standpoint, the 102 mm billet was therefore a credible alternative to the 100 mm billet.
| Refined scheme | Observed behavior | Backflow | Folding | Flash | Assessment |
|---|---|---|---|---|---|
| 5, \(D_b=108\ \mathrm{mm}\) | Rim first, hub later | Present | Not dominant | Moderate | Rejected |
| 6, \(D_b=105\ \mathrm{mm}\) | Acceptable sequence | Not significant | None observed | Excessive compared with 100 mm | Possible but not preferred |
| 7, \(D_b=102\ \mathrm{mm}\) | Hub first, rim shortly after | Not significant | None observed | Slight near final step | Good alternative |
Equivalent strain distribution. I compared the final equivalent strain fields for the 90 mm, 100 mm, 102 mm, and 105 mm billets. In all four cases, the hub region had a relatively low equivalent strain, while the rim and the punching web had higher equivalent strain. This pattern is expected because the hub deforms under a more constrained stress state, whereas the rim and web experience larger shape changes and more severe shear. The important point for forged gear blanks is not simply the absolute strain level, but whether the strain is distributed smoothly and whether the hub and rim reach their final shapes at compatible times.
The 90 mm billet showed the least uniform strain distribution in the rim. I attribute this to the relatively long interval between hub filling and rim filling. During that interval, the hub continues to densify and deform while the rim is still open, so the two regions follow different strain paths. The 100 mm and 102 mm billets produced the most uniform overall equivalent strain fields. In those cases, the hub filled first and the rim filled soon afterward, so the strain accumulated in a more coordinated manner across the forged gear blank. The 105 mm billet was acceptable in terms of strain uniformity, but its greater flash tendency and higher load made it less attractive than the 100 mm and 102 mm candidates.
To rank the strain fields in a consistent way, I used the coefficient of variation of equivalent strain over the hub and rim control volumes:
$$ C_v = \frac{\mathrm{std}(\bar{\varepsilon})}{\mathrm{mean}(\bar{\varepsilon})+\epsilon}, $$
where \(\epsilon\) is a small positive number introduced to avoid division by zero. A smaller \(C_v\) indicates a more uniform strain field. I also defined a normalized uniformity index
$$ U = 1 – \frac{C_v}{C_{v,\mathrm{ref}}}, $$
where \(C_{v,\mathrm{ref}}\) is the coefficient of variation for a reference case. This index is useful for comparing candidate billets within the same study; it is not intended as an absolute material property. The qualitative ranking from the finite element results is given in Table 9. The 100 mm and 102 mm billets were the best in terms of strain uniformity, while the 90 mm billet was the poorest among the candidates that otherwise satisfied the basic filling requirement.
| Billet diameter (mm) | Hub strain level | Rim strain level | Web strain level | Observed uniformity | Relative ranking |
|---|---|---|---|---|---|
| 90 | Low | High | High | Poorest rim uniformity | 4 |
| 100 | Low to moderate | Moderate to high | High | Good and balanced | 1 |
| 102 | Low to moderate | Moderate to high | High | Good and balanced | 2 |
| 105 | Low to moderate | Moderate to high | High | Acceptable | 3 |
Upper-die stroke load. I recorded the upper-die load history for the 90 mm, 100 mm, 102 mm, and 105 mm billets. In every case the load increased with stroke, with the steepest rise occurring near the end of filling when the contact area became large and the material in the hub and web was highly constrained. The maximum loads are listed in Table 10. The 100 mm billet required approximately \(2.46\times 10^{7}\ \mathrm{N}\), essentially the same as the 90 mm billet at \(2.47\times 10^{7}\ \mathrm{N}\). The 102 mm billet required approximately \(2.61\times 10^{7}\ \mathrm{N}\), and the 105 mm billet required approximately \(2.98\times 10^{7}\ \mathrm{N}\). The load increase from 100 mm to 105 mm is about 21 percent, which is significant for press selection and die stress.
| Billet diameter (mm) | Maximum upper-die load \(F_{\max}\) (N) | Relative load \(F_{\max}/F_{\max,100}\) | Load assessment |
|---|---|---|---|
| 90 | \(2.47\times 10^{7}\) | 1.004 | Lowest, close to 100 mm |
| 100 | \(2.46\times 10^{7}\) | 1.000 | Lowest |
| 102 | \(2.61\times 10^{7}\) | 1.061 | Moderately higher |
| 105 | \(2.98\times 10^{7}\) | 1.211 | Highest among the compared candidates |
The relative load was calculated as
$$ \bar{F} = \frac{F_{\max}}{F_{\max,100}}, $$
and the increase relative to the 100 mm case was
$$ \Delta \bar{F} = \bar{F} – 1. $$
For the 105 mm billet, \(\Delta \bar{F} \approx 0.211\), meaning that the maximum load was about 21 percent higher. That penalty is not justified when the 100 mm billet already satisfies the filling and strain-uniformity requirements. The 90 mm billet had a similarly low load, but its larger fill-time difference and poorer strain uniformity made it less attractive for forged gear blanks. The 102 mm billet is an intermediate case: it is close to the 100 mm billet in forging quality but requires about 6 percent more load.
Weighted comparison and selection. I combined the observations into a weighted decision matrix. The weights reflect the priorities of precision forging of forged gear blanks: metal flow and defect avoidance are more important than a small reduction in load, while strain uniformity is important for mechanical performance. I used the following objective function for candidate \(i\):
$$ S_i = w_1 s_{i,\mathrm{flow}} + w_2 s_{i,\mathrm{strain}} + w_3 s_{i,\mathrm{defect}} + w_4 s_{i,\mathrm{load}}, $$
with
$$ w_1 + w_2 + w_3 + w_4 = 1. $$
I selected \(w_1=0.30\), \(w_2=0.25\), \(w_3=0.25\), and \(w_4=0.20\). The flow score represents the filling sequence and backflow behavior; the strain score represents the uniformity of equivalent strain; the defect score represents folding and flash tendency; and the load score represents press-load economy. The scores are relative and were assigned from the finite element observations, with 5 being the best and 1 being the worst. The weighted comparison is given in Table 11.
| Candidate diameter (mm) | Flow score \(s_{\mathrm{flow}}\) | Strain score \(s_{\mathrm{strain}}\) | Defect score \(s_{\mathrm{defect}}\) | Load score \(s_{\mathrm{load}}\) | Weighted score \(S_i\) |
|---|---|---|---|---|---|
| 90 | 3.0 | 2.5 | 4.0 | 5.0 | 3.53 |
| 100 | 5.0 | 5.0 | 5.0 | 5.0 | 5.00 |
| 102 | 4.5 | 4.5 | 4.5 | 4.0 | 4.40 |
| 105 | 3.5 | 3.5 | 3.0 | 3.0 | 3.28 |
The 100 mm billet received the highest weighted score. It filled the hub first and the rim shortly afterward, showed no significant backflow or folding, produced only limited flash, gave a relatively uniform equivalent strain field, and required the lowest maximum load among the compared candidates. The 102 mm billet was the closest alternative, but its slightly higher load and slight flash near the final step made it less desirable. The 105 mm billet was acceptable but not superior. The 90 mm billet had low load but a larger fill-time difference and poorer strain uniformity. I therefore selected a billet diameter of \(100\ \mathrm{mm}\) and a corresponding height of \(27.3\ \mathrm{mm}\) as the preferred initial geometry for the precision forging of these forged gear blanks.
Process window and practical interpretation. The selected geometry should not be treated as an isolated point. I evaluated the robustness of the decision by considering small variations in billet diameter, billet height, friction, and die speed. The recommended process window is summarized in Table 12. A small increase in diameter above 100 mm moves the process toward rim-first filling and backflow, so the upper side of the window should be controlled more tightly than the lower side. A small decrease in diameter toward 90 mm is safer with respect to backflow but increases the fill-time difference and can reduce strain uniformity. The friction factor is also important: a higher friction factor increases the resistance to radial flow and can shift the filling sequence. The die speed affects the strain-rate distribution and the load history, but under the conditions studied here its influence on the filling sequence was secondary to billet geometry.
| Parameter | Recommended value or range | Reason |
|---|---|---|
| Billet diameter | \(100\ \mathrm{mm}\), preferably \(99\) to \(101\ \mathrm{mm}\) | Best filling sequence and strain uniformity |
| Billet height | \(27.3\ \mathrm{mm}\) at the nominal volume | Volume-conserving height for \(D_b=100\ \mathrm{mm}\) |
| Billet volume | \(2.14413699\times 10^{5}\ \mathrm{mm^3}\) | Includes part, flash, and allowance |
| Workpiece temperature | \(1200^{\circ}\mathrm{C}\) | Hot-forging condition used in the study |
| Die temperature | \(200^{\circ}\mathrm{C}\) | Reduces thermal shock and controls heat loss |
| Shear friction factor | \(0.5\) | Representative of the studied hot-forging contact |
| Upper-die speed | \(5\ \mathrm{mm/s}\) | Stable simulation and practical press motion |
| Die fillet radius | \(3\ \mathrm{mm}\) | Reduces local stress concentration and aids flow |
| Maximum upper-die load | Approximately \(2.46\times 10^{7}\ \mathrm{N}\) | Press-selection basis for the selected billet |
| Press capacity recommendation | At least \(30\ \mathrm{MN}\) nominal capacity | Provides margin for friction, temperature, and variation |
The maximum load of approximately \(2.46\times 10^{7}\ \mathrm{N}\) corresponds to about \(24.6\ \mathrm{MN}\). In practice, I would not specify a press whose nominal capacity is only marginally above the predicted load, because the actual flow stress, friction, billet temperature, and die temperature can all vary. A nominal capacity of at least \(30\ \mathrm{MN}\) provides a reasonable margin for forged gear blanks of this size. The exact margin should be confirmed with a production-specific flow-stress curve and a die-temperature survey.
Sensitivity and design implications. I found that the maximum load increases with billet diameter in the range that remains free of severe backflow. A simple local sensitivity can be written as
$$ \frac{\partial F_{\max}}{\partial D_b} > 0 \quad \text{for the selected process window}. $$
This does not mean that a larger diameter always increases load in a strictly monotonic way, because the filling sequence changes when the diameter becomes too large and backflow develops. In the range from 100 mm to 105 mm, however, the trend is clear: the larger billet requires a higher load, and the increase from \(2.46\times 10^{7}\ \mathrm{N}\) to \(2.98\times 10^{7}\ \mathrm{N}\) is large enough to matter for press selection. I also examined the sensitivity of the fill-time difference to diameter:
$$ \frac{\partial (\Delta t_f)}{\partial D_b} < 0 \quad \text{when moving from the hub-first regime toward the rim-first regime}. $$
In other words, increasing the diameter reduces the time advantage of the hub and eventually causes the rim to fill first. That transition is the key design constraint for forged gear blanks of this geometry. The selected 100 mm billet lies on the favorable side of the transition, with enough margin to avoid rim-first filling but not so much margin that the hub-to-rim fill-time difference becomes excessive.
I also considered the effect of friction. A higher friction factor increases the resistance to radial material flow, which tends to make the hub fill more slowly relative to the rim. If the friction factor increases beyond the assumed value, the 100 mm billet may move toward a larger fill-time difference or even toward a different filling sequence. A lower friction factor has the opposite effect. The recommended process should therefore control lubrication and die-surface condition within a narrow range. For forged gear blanks produced at elevated temperature, die coating and scale control are important because they affect both friction and heat transfer.
Numerical accuracy and limitations. The finite element model used a Lagrangian formulation with automatic remeshing, which is appropriate for large deformation in closed-die forging. The mesh size of approximately \(50\,000\) elements was sufficient to capture the global filling sequence, the load history, and the main trends in equivalent strain. A finer mesh would improve the local resolution near the die fillets and the punching web, but it would also increase computational cost. For the purpose of ranking billet geometries, the mesh density was adequate because the differences among the candidates were large and consistent. The model treated the die as rigid and used a constant shear friction factor. In reality, the die undergoes elastic deflection, and the friction condition varies with contact pressure, temperature, and lubrication. These effects should be considered in a final die-design verification.
The thermal model also involved simplifications. I used a fixed initial die temperature and a fixed workpiece temperature, but in production the die temperature rises during continuous operation. That rise reduces the heat loss from the workpiece and can change the flow stress and the friction condition. The selected billet geometry is robust with respect to moderate thermal variation, but the process window should be rechecked with a coupled thermal-mechanical analysis that includes the full die assembly and the press cycle. For forged gear blanks, the most important thermal effect is usually the local cooling near the die contact, which can increase the load and reduce the fillability of thin sections.
Design summary for the selected forged gear blanks. The selected process uses a cylindrical billet of \(100\ \mathrm{mm}\) diameter and \(27.3\ \mathrm{mm}\) height. The billet volume is \(2.14413699\times 10^{5}\ \mathrm{mm^3}\), which includes the forged part, a controlled flash allowance, and a machining allowance. The hub fills first, the rim fills shortly afterward, and the fill-time difference is small. No significant backflow or folding is observed. The equivalent strain is relatively uniform across the hub and rim, and the maximum upper-die load is approximately \(2.46\times 10^{7}\ \mathrm{N}\). The recommended press capacity is at least \(30\ \mathrm{MN}\). The process window should maintain the billet diameter close to \(100\ \mathrm{mm}\), with tighter control on the upper side to avoid rim-first filling. The die fillet radius should be maintained at approximately \(3\ \mathrm{mm}\), and the friction condition should be controlled to avoid shifting the filling sequence.
| Output quantity | Selected result | Interpretation |
|---|---|---|
| Preferred billet diameter | \(100\ \mathrm{mm}\) | Best overall filling and strain behavior |
| Preferred billet height | \(27.3\ \mathrm{mm}\) | Volume-conserving height |
| Preferred billet volume | \(2.14413699\times 10^{5}\ \mathrm{mm^3}\) | Part plus allowance and flash |
| Filling sequence | Hub first, rim shortly after | Favorable for forged gear blanks |
| Backflow | Not significant | Flow-line continuity preserved |
| Folding | Not observed | Reduced risk of internal defects |
| Flash | Limited | Lower trimming burden and stress concentration risk |
| Equivalent strain uniformity | Good | More balanced mechanical properties |
| Maximum upper-die load | \(2.46\times 10^{7}\ \mathrm{N}\) | Basis for press selection |
Interpretation of the rejected candidates. The 120 mm and 110 mm billets failed because the rim filled before the hub. Once the rim was full, continued die motion forced material backward toward the hub, which disrupted the flow lines and produced local thickening in the rim. For forged gear blanks intended for precision applications, that flow reversal is a serious defect because it can create a nonuniform grain-flow pattern and reduce fatigue resistance. The 108 mm billet showed the same problem, which confirmed that the transition from acceptable to unacceptable behavior is close to the 105 mm to 108 mm range. The 105 mm billet did not show severe backflow, but it produced more flash than the 100 mm billet and required a higher load. The 90 mm billet avoided backflow but had a larger hub-to-rim fill-time difference, which reduced strain uniformity. The 102 mm billet was a good alternative but did not surpass the 100 mm billet in the weighted comparison.
Relationship between flow sequence and strain uniformity. The results show a clear coupling between filling sequence and equivalent strain uniformity. When the hub fills first and the rim fills soon afterward, the two regions deform under a more coherent strain path. When the rim fills first, the hub continues to deform while the rim is already constrained, and the subsequent backflow creates a different strain history in the rim. This is why the 120 mm, 110 mm, and 108 mm billets were rejected even though they eventually filled the die. A filled die is not necessarily a good forging. For forged gear blanks, the quality criterion must include the path by which the die fills, not only the final shape. The 100 mm billet satisfied both the final-shape requirement and the path requirement.
I also noted that the equivalent strain in the hub was lower than that in the rim and web for all candidates. This is a natural consequence of the geometry and the constrained deformation in the hub. The objective is not to make the strain perfectly uniform, which is impractical in a complex forging, but to avoid unnecessary extremes and to prevent the rim from accumulating excessive strain before the hub is filled. The 100 mm and 102 mm billets achieved the best balance. The 90 mm billet produced a less uniform rim strain field because the rim remained open for a longer time while the hub continued to deform.
Press-load implications. The load history is important for die design as well as press selection. The maximum load occurs near the end of the stroke, when the contact area is large and the material in the hub and web is highly constrained. A higher billet diameter increases the contact area and the radial resistance, which explains the higher load for the 105 mm billet. The 100 mm billet is attractive because it achieves good filling at the lowest load among the compared candidates. The 90 mm billet has a similar maximum load, but its poorer strain uniformity makes it less desirable. The 102 mm billet requires about 6 percent more load than the 100 mm billet, which is acceptable but not advantageous. The 105 mm billet requires about 21 percent more load, which is a significant penalty. For forged gear blanks, this load difference can determine whether an existing press can be used or whether a larger press is required.
The stroke work can be approximated from the load-displacement curve. If the load increases approximately linearly in the final stage, then
$$ W \approx \frac{1}{2}F_{\max}s_{\mathrm{final}}, $$
where \(s_{\mathrm{final}}\) is the final stroke increment over which the load rises to its maximum. This approximation is not used to replace the integrated load history, but it illustrates why a higher maximum load also tends to increase the energy demand. The 100 mm billet is therefore favorable from both the load and energy perspectives. The 90 mm billet is comparable in load but less favorable in strain uniformity. The 102 mm billet is a compromise. The 105 mm billet is the least favorable among the compared candidates from the load and energy perspective.
Practical recommendations for forged gear blanks. Based on the numerical experiments, I recommend the following. First, use a cylindrical billet of \(100\ \mathrm{mm}\) diameter and \(27.3\ \mathrm{mm}\) height for the initial forging operation. Second, control the billet volume carefully because a small excess volume increases the flash and the load, while a small deficit volume can leave the rim underfilled. Third, maintain the hub-first filling sequence by keeping the billet diameter on the lower side of the transition range. Fourth, preserve the die fillet radius at approximately \(3\ \mathrm{mm}\) to avoid local stress concentration and to promote smooth flow into the rim. Fifth, verify the friction condition with a production trial because the filling sequence is sensitive to friction. Sixth, use a press with at least \(30\ \mathrm{MN}\) nominal capacity to provide margin above the predicted \(24.6\ \mathrm{MN}\) maximum load. Seventh, inspect the final forged gear blanks for flash distribution and flow-line continuity, especially near the rim-to-hub transition.
| Recommendation | Target | Verification method |
|---|---|---|
| Billet diameter | \(100\ \mathrm{mm}\) | Dimensional inspection and volume check |
| Billet height | \(27.3\ \mathrm{mm}\) | Height gauge and mass check |
| Billet volume | \(2.14413699\times 10^{5}\ \mathrm{mm^3}\) | Mass divided by density |
| Filling sequence | Hub first, rim shortly after | Short-shot trials and simulation correlation |
| Backflow | Not significant | Flow-line inspection on sectioned parts |
| Flash | Limited and symmetric | Visual inspection and flash-weight measurement |
| Strain uniformity | Good | Microhardness mapping and grain-flow evaluation |
| Maximum load | About \(2.46\times 10^{7}\ \mathrm{N}\) | Press load monitor |
| Press capacity | At least \(30\ \mathrm{MN}\) | Press specification and load calibration |
Further verification. I would verify the numerical prediction with a set of short-shot experiments. In a short-shot trial, the press stroke is stopped before the die is completely filled, and the partially formed forged gear blank is sectioned to reveal the filling sequence. If the hub fills before the rim, the simulation is consistent with the experiment. If the rim fills first, the friction or thermal boundary conditions should be recalibrated. I would also measure the load-stroke curve on the press and compare it with the simulated curve. A close match in both the filling sequence and the load history would give confidence in the selected billet geometry. After the first production trial, I would inspect the flash distribution, the rim fill, and the hub fill. Any systematic underfill at the rim would suggest a slight increase in billet volume or a small adjustment in billet diameter, while any evidence of backflow would suggest a reduction in diameter or a change in friction condition.
The finite element results also suggest that the die design should avoid sharp transitions in the hub-to-rim region. The fillet radius of \(3\ \mathrm{mm}\) used in the model is a reasonable compromise between flow guidance and die strength. A larger fillet would reduce local stress concentration but could change the filling sequence and the final geometry. A smaller fillet would increase the risk of local shear and die wear. For forged gear blanks with a thin web and a relatively tall hub, the fillet radius should be reviewed together with the billet diameter rather than as an independent variable.
Conclusions from my analysis. The numerical study led me to the following conclusions. When the billet diameter is in the range from approximately \(105\ \mathrm{mm}\) to \(120\ \mathrm{mm}\), the rim tends to fill before the hub, and the subsequent deformation produces metal backflow that is unfavorable for the mechanical performance of forged gear blanks. When the billet diameter is less than approximately \(90\ \mathrm{mm}\), the hub fills first and the rim fills later, but the time difference becomes large enough to cause nonuniform equivalent strain. When the billet diameter is in the range from approximately \(90\ \mathrm{mm}\) to \(105\ \mathrm{mm}\), the forging process can avoid significant backflow and folding, and the filling quality is generally good. Among the candidates, the \(100\ \mathrm{mm}\) diameter billet with a height of \(27.3\ \mathrm{mm}\) gives the best overall combination of metal flow, equivalent strain uniformity, flash control, and press-load economy. I therefore recommend this geometry as the preferred billet for the precision forging of these forged gear blanks, with a controlled process window around the selected diameter and with a press capacity of at least \(30\ \mathrm{MN}\) for production safety.
The broader lesson is that the design of forged gear blanks should be based on the coupled response of the billet, die, and process. A billet that fills the die is not necessarily optimal. The filling sequence, the flow-line continuity, the strain path, and the load history all matter. The finite element method is useful because it makes those quantities visible before a die is cut. In this study, the numerical comparison reduced the candidate range from a broad set of possible billet diameters to a narrow window around \(100\ \mathrm{mm}\). That is the practical value of the analysis: it converts a large trial-and-error space into a small set of physically justified trials. For forged gear blanks produced by precision forging, this kind of reduction is essential for controlling cost, improving material utilization, and achieving consistent mechanical performance.
| Final decision item | Selected value | Basis |
|---|---|---|
| Billet diameter | \(100\ \mathrm{mm}\) | Best weighted score and lowest maximum load |
| Billet height | \(27.3\ \mathrm{mm}\) | Constant-volume calculation |
| Billet volume | \(2.14413699\times 10^{5}\ \mathrm{mm^3}\) | Part plus allowance and flash |
| Filling sequence | Hub first, rim shortly after | No significant backflow |
| Equivalent strain | Good uniformity | Balanced hub and rim deformation |
| Maximum load | \(2.46\times 10^{7}\ \mathrm{N}\) | Finite element load history |
| Press capacity | At least \(30\ \mathrm{MN}\) | Production margin |
| Primary defect risk | Rim-first filling if diameter is too large | Process-window control |
| Secondary defect risk | Fill-time difference if diameter is too small | Strain uniformity control |
I regard the \(100\ \mathrm{mm}\) diameter billet as the most defensible choice for the forged gear blanks studied here. It gives the hub-first filling sequence that I wanted, keeps the rim fill close to the hub fill, avoids the backflow that appeared in the larger billets, limits flash, and maintains a relatively uniform equivalent strain field. The \(102\ \mathrm{mm}\) billet is a reasonable backup if production constraints require a slightly larger diameter, but it should be treated as a secondary option because it requires a higher load and shows a slight flash tendency near the final step. The \(105\ \mathrm{mm}\) billet is not preferred. The \(90\ \mathrm{mm}\) billet is acceptable only if the die and press conditions strongly favor a smaller diameter, and even then the fill-time difference should be monitored. For the present geometry and process assumptions, the \(100\ \mathrm{mm}\) billet is the best balance of flow, strain, flash, and load for forged gear blanks.