Precision forming of gears represents a significant near-net-shape manufacturing technology. Compared to traditional machining processes, precision forming offers distinct advantages, including significantly improved material utilization and the preservation of continuous internal metal flow lines. Consequently, precision-formed helical gear components exhibit superior mechanical properties compared to their machined counterparts. Since the 1980s, extensive research has been conducted on the precision forming of helical gears, yielding considerable progress. However, much of the focus has been on macroscopic deformation behavior and die structure design. Investigations into the microstructural evolution during the hot forming of helical gears remain relatively limited.
The floating die process, proposed as an improvement over traditional closed-die forging where the die is stationary, addresses issues related to material flow resistance caused by friction between the workpiece and the die. This technique utilizes the “effective friction” between the die and the billet to promote downward material flow, thereby enhancing tooth cavity filling. During the hot forming process of a helical gear, complex thermal interactions occur: heat is exchanged between the material, the dies, and the environment, while simultaneous deformation generates additional heat within the billet. This leads to a transient temperature field within the workpiece. The material’s formability is highly dependent on this temperature field, and the characteristics and evolution of the internal microstructure, which ultimately govern the mechanical properties of the finished helical gear, are directly influenced by the thermomechanical processing conditions.
This study focuses on a cylindrical helical gear with the following specifications: number of teeth \( z = 18 \), module \( m_n = 2 \) mm, helix angle \( \beta = 16^\circ \), pressure angle \( \alpha_n = 20^\circ \), and face width \( B = 15 \) mm. Based on the floating die principle, a coupled finite element model integrating deformation, heat transfer, and microstructure evolution was established using the commercial software Deform-3D. The model simulates the hot precision forming process under varying die speeds to investigate the influence of different upper punch velocities and floating die velocities on the microstructural evolution of the helical gear. The results indicate that under the same kinematic mode, as the upper punch speed increases, the region experiencing dynamic recrystallization and grain refinement gradually expands from the tooth root towards the tooth flank. Furthermore, with increasing upper punch speed, the average volume fraction of dynamic recrystallization increases, and the average grain size decreases.
Geometric Model and Material Characterization
The initial billet design is critical for successful forming. A preform with a central分流孔 (flow-diverging hole) is often employed to facilitate material distribution and reduce forming load. For this helical gear, the hole diameter was set to 5 mm, and the outer diameter of the billet was designed to be equal to the gear’s root circle diameter. The initial height \( H_0 \) of the billet was determined based on the principle of volume constancy, calculated from the volume of the final helical gear and the preform geometry. The three-dimensional geometries of the billet and the die assembly (upper punch, lower punch, and floating die) were created using CAD software and imported into Deform-3D.
The workpiece material selected for this study is 20CrMnTiH steel, a common carburizing steel used for high-strength gears. Its flow stress and microstructural evolution during hot deformation are highly sensitive to temperature, strain, and strain rate. The material’s constitutive behavior and microstructure models were incorporated into the simulation.
The flow stress \( \sigma \) is typically described by a constitutive equation that accounts for work hardening, dynamic recovery, and dynamic recrystallization. A common form is the Arrhenius-type equation, often modified to include the effects of dynamic recrystallization (DRX):
$$ \sigma = f(\varepsilon, \dot{\varepsilon}, T, Z, X_{DRX}) $$
where \( \varepsilon \) is the true strain, \( \dot{\varepsilon} \) is the strain rate, \( T \) is the absolute temperature, \( Z \) is the Zener-Hollomon parameter (\( Z = \dot{\varepsilon} \exp(Q/(RT)) \)), and \( X_{DRX} \) is the dynamic recrystallization volume fraction.
The kinetics of dynamic recrystallization are frequently modeled using an Avrami-type equation:
$$ X_{DRX} = 1 – \exp\left[-k_d \left( \frac{\varepsilon – \varepsilon_c}{\varepsilon_p} \right)^{n_d}\right] \quad \text{for} \quad \varepsilon > \varepsilon_c $$
where \( \varepsilon_c \) is the critical strain for the onset of DRX, \( \varepsilon_p \) is the strain corresponding to the peak stress, and \( k_d \) and \( n_d \) are material-dependent constants. The critical strain \( \varepsilon_c \) is often related to the peak strain \( \varepsilon_p \) and the Zener-Hollomon parameter.
The evolution of grain size is another crucial aspect. The dynamically recrystallized grain size \( d_{DRX} \) is primarily influenced by the Zener-Hollomon parameter:
$$ d_{DRX} = A Z^{-m} $$
where \( A \) and \( m \) are material constants. The average grain size \( d_{avg} \) in the material is a mixture of the original grains and the new recrystallized grains, evolving according to the recrystallization fraction.
The key material parameters for 20CrMnTiH used in the simulation are summarized in the table below.
| Material Property / Constant | Symbol | Value / Expression | Notes |
|---|---|---|---|
| Activation Energy for Deformation | \( Q \) | ~350 kJ/mol | Typical value for low-alloy steels |
| Initial Grain Size | \( d_0 \) | 55.67 μm | Defined for the simulation |
| Critical Strain Coefficient | \( \varepsilon_c / \varepsilon_p \) | ~0.6 – 0.8 | Material dependent relationship |
| DRX Kinetics Constant | \( k_d \) | Material specific | From published data for 20CrMnTiH |
| DRX Kinetics Exponent | \( n_d \) | Material specific | From published data for 20CrMnTiH |
| DRX Grain Size Constant | \( A \) | Material specific | From published data for 20CrMnTiH |
| DRX Grain Size Exponent | \( m \) | Material specific | From published data for 20CrMnTiH |
The Floating Die Process and Finite Element Modeling
Principles of the Floating Die Process for Helical Gears
The conventional closed-die forging of a helical gear involves a stationary die cavity. Frictional forces at the die-workpiece interface resist material flow, making it difficult to completely fill the intricate tooth profiles, especially at the corners. The floating die concept introduces controlled axial movement to the die. In the setup for this helical gear, both the upper and lower punches are designed with the conjugate tooth profile to maintain meshing and guide the die movement. During operation, the upper punch moves axially downward at a velocity \( v_{up} \) while simultaneously rotating about its axis with an angular velocity \( \omega \). The floating die moves axially downward at a prescribed velocity \( v_{die} \). The lower punch remains axially stationary but rotates with the same angular velocity \( \omega \) as the upper punch to ensure synchronized rotation of the forming billet.
The key benefit is that the downward motion of the die imparts a additional downward frictional force on the billet surface in contact with it. This “effective friction” aids in pulling material into the tooth cavities, improving fillability compared to a stationary die where friction only hinders flow. The rotation of the punches is essential to form the helix angle of the gear. The required rotation angle \( \alpha \) and angular velocity \( \omega \) are geometrically determined by the helix angle, the axial displacement, and the base circle radius. The relationships can be derived as follows, where \( L \) is the axial stroke, \( \beta \) is the helix angle, \( \alpha_t \) is the transverse pressure angle, and \( r_b \) is the base circle radius:
$$ \alpha = \frac{180 \cdot L \cdot \tan[\arctan(\tan \beta \cos \alpha_t)]}{\pi r_b} $$
$$ t = L / v_{up} $$
$$ \omega = \alpha / t = \frac{v_{up} \cdot 180 \cdot \tan[\arctan(\tan \beta \cos \alpha_t)]}{\pi r_b} $$
This ensures the material is correctly twisted into the helical form during axial compression.
Coupled Deformation-Thermal-Microstructure FE Model
A fully coupled thermomechanical-metallurgical finite element model was developed. The workpiece (billet) was defined as a plastic body with the material properties of 20CrMnTiH, including its flow stress model and microstructural evolution laws for dynamic recrystallization and grain growth. All dies (upper punch, lower punch, floating die) were modeled as rigid bodies. The material for the dies was set as H13 hot-work tool steel.
The initial temperatures were defined: the billet was preheated to 950°C, and all dies were preheated to 200°C. The thermal boundary conditions included:
- Heat transfer coefficient between the workpiece and dies: 11 N/(s·mm·°C).
- Convection coefficient from free surfaces to the environment: 0.02 N/(s·mm·°C).
- Environment temperature: 20°C.
- Heat generation due to plastic deformation: 90% of plastic work converted to heat.
The friction at the tool-workpiece interfaces was modeled using the shear friction model with a friction factor \( m \) set to 0.25.
To investigate the effects of process kinematics, three distinct motion schemes for the floating die were defined relative to the upper punch speed \( v_{up} \):
- Scheme A: \( v_{die} = 0.5 \cdot v_{up} \)
- Scheme B: \( v_{die} = 1.0 \cdot v_{up} \)
- Scheme C: \( v_{die} = 2.0 \cdot v_{up} \)
For each scheme, the upper punch speed \( v_{up} \) was varied across a range: 2, 6, 10, 14, and 18 mm/s. The corresponding angular velocity \( \omega \) was calculated for each case using the formulas above. The lower punch rotational speed was set equal to \( \omega \), and its axial velocity was zero. The floating die moved axially downward at \( v_{die} \) with no rotation.
The finite element mesh for the billet was locally refined in regions expecting high strain gradients, such as the tooth forming zones. The model simulated the entire process until the final forged helical gear shape was achieved.
| Parameter | Setting / Value |
|---|---|
| Workpiece Material | 20CrMnTiH Steel (Plastic) |
| Die Material | H13 Steel (Rigid) |
| Workpiece Initial Temperature | 950 °C |
| Die Initial Temperature | 200 °C |
| Friction Model | Shear, Factor m = 0.25 |
| Workpiece-Die Heat Transfer Coefficient | 11 N/(s·mm·°C) |
| Convection Coefficient (to environment) | 0.02 N/(s·mm·°C) |
| Upper Punch Speed, \( v_{up} \) | 2, 6, 10, 14, 18 mm/s |
| Floating Die Speed Schemes | \( v_{die} = 0.5v_{up}, 1.0v_{up}, 2.0v_{up} \) |
| Initial Grain Size | 55.67 μm |
Simulation Results and Analysis
Forming Load Analysis
The forming load-stroke curves for different motion schemes and upper punch speeds provide insight into the process mechanics. The general trend observed across all schemes is that the load increases gradually during the initial stages as the material fills the central cavity and begins to flow into the tooth spaces. A sharp increase in load occurs during the final filling of the tooth corners. The maximum forming load is a critical parameter for die and press selection.
The analysis reveals that for a given upper punch speed \( v_{up} \), the maximum forming load decreases as the floating die speed \( v_{die} \) increases. Scheme C (\( v_{die} = 2v_{up} \)) consistently yields the lowest maximum load, while Scheme A (\( v_{die} = 0.5v_{up} \)) results in the highest. This is a direct consequence of the floating die principle: a faster downward-moving die applies a greater downward frictional pull on the billet, actively assisting material flow into the tooth cavities and reducing the resistance that must be overcome by the upper punch. Furthermore, at any fixed \( v_{die}/v_{up} \) ratio (i.e., within a specific scheme), increasing the absolute speed \( v_{up} \) generally leads to a higher maximum load due to the increased strain rate sensitivity of the material’s flow stress at high temperatures.
| \( v_{up} \) (mm/s) | Scheme A: \( v_{die}=0.5v_{up} \) | Scheme B: \( v_{die}=1.0v_{up} \) | Scheme C: \( v_{die}=2.0v_{up} \) |
|---|---|---|---|
| 2 | ~1250 | ~1200 | ~1150 |
| 6 | ~1650 | ~1550 | ~1450 |
| 10 | ~1900 | ~1800 | ~1680 |
| 14 | ~2100 | ~1980 | ~1850 |
| 18 | ~2250 | ~2120 | ~2000 |
Temperature Field Distribution in the Formed Helical Gear
The temperature distribution within the forged helical gear at the end of the process is non-uniform and significantly influences the microstructural outcome. The simulation results show a consistent pattern across all motion schemes and punch speeds, though the absolute temperature values vary.
- Highest Temperature Regions: The core (center) of the gear and the regions near the tooth roots exhibit the highest temperatures, often exceeding 900°C. This is due to two factors: limited heat transfer from the interior to the dies, and the significant amount of plastic deformation (and associated heat generation) occurring in these zones as material is forced into the root fillets.
- Lowest Temperature Regions: The tooth tips (addendum) and the top/bottom end surfaces of the gear show the lowest temperatures, sometimes dropping below 600°C. These areas are in direct and prolonged contact with the cooler dies (200°C), leading to rapid heat extraction (quenching effect). The material here also undergoes less severe deformation.
The effect of increasing the upper punch speed \( v_{up} \) is also evident. A higher punch speed reduces the total process time, thereby decreasing the time available for heat conduction to the dies. Consequently, the temperature drop in the cooler regions (tooth tips, surfaces) is less severe. The core and tooth root temperatures also remain slightly higher due to the adiabatic heating effect becoming more pronounced at higher strain rates, where a larger proportion of the deformation work is converted to heat before it can be conducted away.
Dynamic Recrystallization (DRX) Behavior
Dynamic recrystallization is a crucial grain refinement mechanism during hot forming. The volume fraction of DRX (\( X_{DRX} \)) determines the extent of this refinement. The simulation maps of \( X_{DRX} \) reveal a characteristic distribution pattern for the formed helical gear.
- High DRX Fraction Zones: The central regions of the tooth roots consistently show the highest volume fraction of dynamic recrystallization, often approaching 1.0 (fully recrystallized). This correlates directly with the regions of highest strain, strain rate, and temperature, which are the driving forces for DRX nucleation and growth.
- Moderate DRX Zones: The areas extending from the tooth root towards the middle of the tooth flank show moderate levels of DRX. The fraction decreases as one moves away from the root.
- Low/No DRX Zones: The tooth tips and the gear’s end surfaces (away from the central hole) show little to no dynamic recrystallization. The low temperature and lower effective strain in these regions suppress the DRX process. The material here may only undergo dynamic recovery.
A critical observation is the effect of punch speed \( v_{up} \). For a given motion scheme (e.g., Scheme B), as \( v_{up} \) increases from 2 to 18 mm/s, the region of high DRX fraction expands from being concentrated mainly in the tooth root to covering a larger portion of the tooth flank. This is because higher strain rates, while potentially increasing the critical strain for DRX, also lead to greater heat generation (adiabatic heating), raising the local temperature and accelerating the recrystallization kinetics. The combination can result in a larger volume of material meeting the conditions for DRX within the shorter process time.
The average DRX volume fraction across the entire helical gear tooth volume was calculated for each case. The results show a clear trend: for each motion scheme, the average \( \bar{X}_{DRX} \) increases monotonically with increasing \( v_{up} \). Furthermore, at higher punch speeds (e.g., >6 mm/s), Scheme C (\( v_{die}=2v_{up} \)) tends to produce helical gears with a slightly higher average DRX fraction compared to the other schemes under the same \( v_{up} \). This is likely due to the more homogeneous and assisted material flow reducing localized dead zones and promoting more widespread deformation conducive to DRX.
| \( v_{up} \) (mm/s) | Scheme A (\( v_{die}=0.5v_{up} \)) | Scheme B (\( v_{die}=1.0v_{up} \)) | Scheme C (\( v_{die}=2.0v_{up} \)) |
|---|---|---|---|
| 2 | ~22 | ~24 | ~25 |
| 6 | ~35 | ~38 | ~42 |
| 10 | ~48 | ~52 | ~56 |
| 14 | ~58 | ~62 | ~66 |
| 18 | ~65 | ~68 | ~72 |
Grain Size Evolution and Distribution
The final grain size distribution is the most direct indicator of microstructural quality. Finer grains generally lead to improved strength, toughness, and fatigue resistance in the helical gear. The simulated average grain size maps show a strong inverse correlation with the DRX maps.
- Finest Grains: The tooth root areas and the region around the central分流孔 show the smallest grain sizes, often refined to below 5-10 μm from an initial 55.67 μm. This is precisely where DRX is most complete (\( X_{DRX} \approx 1 \)).
- Moderately Refined Grains: The tooth flanks show a gradient in grain size, increasing from the root towards the tip. This follows the gradient in DRX fraction.
- Coarse/Unrefined Grains: The tooth tips and peripheral end surfaces retain grain sizes close to or even larger than the initial size due to the absence of DRX and possible grain growth in the high-temperature environment before cooling.
The influence of process parameters is significant. Under the same motion scheme, increasing the upper punch speed \( v_{up} \) leads to a reduction in the average grain size \( \bar{d} \) across the tooth region. This is a direct result of the increased average DRX fraction discussed earlier. Higher strain rates promote a greater nucleation rate for new grains during DRX, leading to a finer recrystallized grain size as per the relationship \( d_{DRX} \propto Z^{-m} \). Although higher speeds might reduce process time, the dominant effect is the enhanced DRX kinetics and finer intrinsic DRX grain size.
Comparing the different floating die schemes, at sufficiently high punch speeds (e.g., \( v_{up} \geq 10 \) mm/s), Scheme C (\( v_{die}=2v_{up} \)) consistently results in the smallest average grain size in the tooth-forming region. This can be attributed to its ability to achieve slightly higher and more uniform deformation (and thus DRX) due to the beneficial material flow assistance, coupled with the high strain rate effects.
| \( v_{up} \) (mm/s) | Scheme A (\( v_{die}=0.5v_{up} \)) | Scheme B (\( v_{die}=1.0v_{up} \)) | Scheme C (\( v_{die}=2.0v_{up} \)) |
|---|---|---|---|
| 2 | ~42 | ~40 | ~39 |
| 6 | ~32 | ~30 | ~27 |
| 10 | ~24 | ~22 | ~19 |
| 14 | ~19 | ~17 | ~15 |
| 18 | ~16 | ~14 | ~12 |
Discussion and Synthesis
The comprehensive simulation study elucidates the complex interplay between process kinematics, thermal history, and microstructural evolution in the hot precision forming of a helical gear using a floating die. The primary findings can be synthesized as follows:
- Floating Die Kinematics: The ratio \( v_{die}/v_{up} \) has a measurable impact on the forming process. A higher relative die speed (Scheme C) effectively reduces the maximum forming load by assisting material flow. While the basic pattern of temperature, DRX, and grain size distribution is similar across schemes, the enhanced flow in Scheme C appears to promote marginally more uniform and extensive DRX at higher punch speeds, leading to a better overall refined microstructure in the critical tooth regions of the helical gear.
- Upper Punch Speed (\( v_{up} \)): This is the most influential parameter on microstructural evolution for the studied helical gear. Increasing \( v_{up} \) has several concurrent effects:
- Increases strain rate, elevating flow stress and forming load.
- Reduces process time, limiting heat loss to dies, thus maintaining higher workpiece temperatures.
- Increases adiabatic heating, particularly in high-strain zones like tooth roots.
- Promotes higher nucleation rates for DRX, leading to a finer DRX grain size.
The net result is a significant expansion of the DRX zone and a reduction in the average grain size within the helical gear teeth as \( v_{up} \) increases.
- Microstructural Gradients: The formed helical gear inevitably possesses microstructural gradients. The tooth root, experiencing the most severe thermomechanical processing, becomes the finest-grained and most recrystallized region, which is beneficial for fatigue resistance as roots are typical failure origins. The tooth tip remains coarse-grained. This gradient must be considered in the context of the gear’s final application and heat treatment.
The relationships can be summarized by the following conceptual equations linking process parameters to microstructural outcomes for the helical gear:
$$ \text{Strain Rate in Tooth Root} \propto v_{up} $$
$$ \text{Adiabatic Heating} \propto \sigma \cdot \dot{\varepsilon} \propto f(v_{up}) $$
$$ T_{local} = T_{initial} + \Delta T_{adiabatic} – \Delta T_{conduction}(t_{process}) $$
$$ \text{where } t_{process} \propto 1/v_{up} $$
$$ \varepsilon_c, \varepsilon_p = g(Z) = g(\dot{\varepsilon} \exp(Q/RT_{local})) $$
$$ X_{DRX} = h(\varepsilon, \varepsilon_c, \varepsilon_p, T_{local}) $$
$$ d_{DRX} = A Z^{-m} $$
$$ \bar{d}_{final} = \int (X_{DRX} \cdot d_{DRX} + (1-X_{DRX}) \cdot d_{0/recov}) \, dV $$
The simulations demonstrate that optimizing \( v_{up} \) and \( v_{die}/v_{up} \) allows for the control of these equations to maximize \( X_{DRX} \) and minimize \( \bar{d}_{final} \) in the load-bearing parts of the helical gear.
Conclusion
This investigation into the hot precision forming of a helical gear using a floating die process, through coupled thermomechanical-metallurgical finite element analysis, provides detailed insights into microstructural control. The key conclusions are:
- The floating die speed relative to the punch speed (\( v_{die}/v_{up} \)) influences the forming load and, to a secondary degree, the uniformity of microstructure. A faster floating die (e.g., \( v_{die} = 2v_{up} \)) reduces load and can enhance dynamic recrystallization at higher strain rates.
- The upper punch speed \( v_{up} \) is the dominant factor governing microstructural evolution in the helical gear. As \( v_{up} \) increases, the region undergoing dynamic recrystallization expands from the tooth root towards the flank, the average volume fraction of DRX increases, and the average grain size decreases significantly.
- For the specific helical gear geometry and material (20CrMnTiH) studied, process windows favoring a refined microstructure can be identified. Specifically, using a high upper punch speed (e.g., 14-18 mm/s) in combination with a floating die speed twice that of the punch (\( v_{die}=2v_{up} \)) yields helical gear components with the highest average DRX fraction and the smallest average grain size in the tooth region, which is promising for obtaining superior mechanical properties.
- The simulations confirm the presence of inevitable microstructural gradients, with the finest grains in the high-strain tooth root areas and coarser grains in the less-deformed tooth tips. This knowledge is essential for designing subsequent heat treatments to homogenize properties if required.
This work establishes a numerical framework for designing and optimizing the hot precision forming process for helical gears, with a focus on achieving tailored microstructures. The findings contribute to the foundational knowledge required for the property-controlled forming (控性控形) of high-performance helical gear components.