Spur gears are fundamental machine elements widely used in power transmission systems. Conventional machining processes for gear teeth inevitably interrupt the continuity of metal fibers, leading to stress concentration sensitivity and reduced fatigue life. Precision forging offers significant advantages in preserving metal flow lines, improving mechanical properties, and enhancing service performance. This study focuses on a large-module spur gear made of 18CrNiMo7-6 steel. A warm-cold composite forging process is proposed, combining warm preforming and cold sizing to achieve high dimensional accuracy and good surface quality. Finite element simulations using DEFORM-3D are performed to evaluate different forming schemes, optimize process parameters, and determine the optimal cold sizing allowance. Furthermore, a virtual reality simulation of the forging process is implemented on the Open Inventor platform by converting the finite element geometrical data into a readable format. This work demonstrates the feasibility of the composite forming approach and provides a practical reference for industrial production of spur gears.
Spur gears are among the most critical components in transmission systems. Their manufacturing quality directly affects the performance and durability of mechanical equipment. Traditional cutting processes, such as hobbing and shaping, break the continuous metal fiber structure along the tooth profile, making the gear surface susceptible to fatigue cracks and reducing the service life. Precision forging, in contrast, deforms the material plastically, allowing the metal flow lines to follow the tooth contour. This results in higher strength, better wear resistance, and improved fatigue life. Precision forging of spur gears has therefore become an active research field, especially for large-module gears where the material flow is complex and the forming load is high.
There are several precision forging variants: hot forging, cold forging, and warm forging. Hot forging achieves good formability but suffers from oxidation and poor dimensional accuracy. Cold forging provides excellent accuracy but requires high forming loads and offers limited formability. Warm forging, conducted at temperatures below the recrystallization point but above room temperature, balances formability and accuracy. However, a single warm forging step may not meet all precision requirements. A combination of warm and cold forging, often referred to as warm-cold composite forging, leverages the advantages of both processes: warm forging for bulk deformation to obtain the approximate shape, and cold sizing to refine the dimensions and surface quality. This approach has gained popularity in the automotive and machinery industries.
This study addresses a large-module spur gear with a module of 5 mm, 30 teeth, and a face width of 26 mm. The gear material is 18CrNiMo7-6, a high-strength low-alloy steel commonly used in heavy-duty gearboxes. The objective is to design a warm-cold composite forming process, optimize the process parameters through finite element simulation, and implement a virtual reality simulation of the forging process. The results provide theoretical guidance and numerical support for the practical production of spur gears.
This thesis is organized as follows: Section 2 presents the fundamental finite element theory for metal forming, including rigid-plastic and elastic-plastic formulations, friction models, and wear models. Section 3 describes the gear design and the proposed composite forming route, including the workpiece geometry, die design, and process parameters. Section 4 presents the finite element simulation results and optimization of the warm forging process, including the study of preform schemes, die structure improvements, and process parameter optimization. Section 5 addresses the cold sizing simulation and the selection of the optimal sizing allowance. Section 6 explains the virtual reality simulation implementation. Finally, Section 7 summarizes the main conclusions.
Finite Element Theoretical Foundation
Finite element analysis of metal forming processes can be classified into two main categories: rigid-plastic finite element method and elastic-plastic finite element method. For bulk forming processes with large plastic deformation, the rigid-plastic formulation is often employed because elastic deformation is negligible compared to plastic strain. For small deformation processes such as cold sizing, elastic-plastic formulation is necessary to capture the unloading and springback behavior.
Rigid-Plastic Finite Element Method
The rigid-plastic finite element method assumes the material is rigid-plastic, neglecting elastic effects. The basic assumptions are as follows:
- Elastic deformation is ignored.
- The material is homogeneous and isotropic.
- The volume is incompressible.
- The material obeys the Levy-Mises flow rule.
- The yield condition follows the von Mises criterion.
The governing equations for a rigid-plastic body are:
Equilibrium equation:
$$\sigma_{ij,j}=0$$
Strain rate–velocity relation:
$$\dot{\varepsilon}_{ij}=\frac{1}{2}\left(v_{i,j}+v_{j,i}\right)$$
Constitutive relation:
$$\dot{\varepsilon}_{ij}=\lambda \sigma’_{ij}$$
where $\sigma’_{ij}$ is the deviatoric stress tensor and $\lambda$ is a scalar factor derived from the equivalent stress and equivalent strain rate.
Yield criterion (Mises):
$$\bar{\sigma}=Y$$
Volume incompressibility:
$$\dot{\varepsilon}_{v}=\dot{\varepsilon}_{ii}=0$$
Boundary conditions:
$$\sigma_{ij} n_j = P_i \quad \text{on } S_p$$
and velocity boundary conditions on $S_v$.
The variational principle of Markov states that among all kinematically admissible velocity fields $v_i^*$, the actual solution minimizes the functional:
$$\Pi = \int_V \bar{\sigma} \dot{\bar{\varepsilon}} \, dV – \int_{S_p} P_i v_i^* \, dS$$
To enforce the incompressibility constraint, a Lagrange multiplier $\lambda$ is introduced, leading to a generalized functional:
$$\Pi_1 = \int_V \bar{\sigma} \dot{\bar{\varepsilon}} \, dV + \int_V \lambda \dot{\varepsilon}_v \, dV – \int_{S_p} P_i v_i^* \, dS$$
The stationary condition of this functional yields the velocity field and the mean stress.
Elastic-Plastic Finite Element Method
For cold sizing, the deformation is small and the elastic recovery is significant. The elastic-plastic formulation is based on the incremental theory of plasticity. The total strain increment is decomposed into elastic and plastic parts:
$$d\varepsilon = d\varepsilon^e + d\varepsilon^p$$
The elastic part follows Hooke’s law:
$$d\sigma = D^e \, d\varepsilon^e$$
The plastic part is derived from the yield criterion and flow rule. The von Mises yield criterion is commonly used:
$$\sigma_{von} = \sqrt{\frac{3}{2} \sigma’_{ij} \sigma’_{ij}} = \sigma_s$$
The hardening behavior is assumed to be isotropic, and the plastic strain increment is given by:
$$d\varepsilon^p = d\lambda \frac{\partial f}{\partial \sigma}$$
where $f$ is the yield function and $d\lambda$ is a positive scalar.
Friction Model
In bulk forming, the shear friction model is widely used. The friction stress is expressed as:
$$\tau_f = m \, k$$
where $m$ is the friction factor ($0 \leq m \leq 1$) and $k$ is the shear flow stress of the material. When the contact pressure is low, Coulomb friction may apply:
$$\tau_f = \mu \, p$$
DEFORM-3D uses a hybrid model that switches between Coulomb and shear friction depending on the normal pressure. For warm and cold forging, the shear friction model is more accurate.
Wear Model
Die wear is a critical factor affecting tool life and product quality. The Archard wear model is commonly used for forging dies. The wear depth $W$ is given by:
$$W = K \int \frac{p^a \, v^b}{H^c} \, dt$$
where $p$ is the contact pressure, $v$ is the sliding velocity, $H$ is the die hardness, and $K$, $a$, $b$, $c$ are empirical constants. For steel dies, typical values are $a=1$, $b=1$, $c=2$, and $K$ is calibrated from experiments.
Composite Forging Process Design for Spur Gears
The spur gear under investigation has the following geometrical parameters:
| Parameter | Value |
|---|---|
| Number of teeth | 30 |
| Module | 5 mm |
| Pressure angle | 20° |
| Pitch diameter | 150 mm |
| Addendum diameter | 160 mm |
| Tooth height | 11.25 mm |
| Face width | 26 mm |
| Bore diameter (after machining) | 57 mm |
The material is 18CrNiMo7-6 steel. Its chemical composition (measured) is:
| Element | C | S | Si | Mn | P | Cr | Ni | Mo |
|---|---|---|---|---|---|---|---|---|
| Content (wt%) | 0.19 | 0.007 | 0.29 | 0.73 | 0.020 | 1.57 | 1.52 | 0.32 |
The processing route for the composite forming is:
Precision cutting of the billet → Medium-frequency induction heating → Warm preforming (closed-die forging) → Punching of the web → Annealing → Surface treatment → Cold sizing → Subsequent machining → Heat treatment.
Cold Forging Drawing Design
The final cold forging drawing is designed based on the product requirements. Machining allowances are assigned to the critical surfaces: 1.5 mm per side for the bore, 1 mm per side for the end faces, and 0.1 mm per side for the tooth profile. The center hole is not pierced during forging; a web is retained for subsequent punching.
Warm Forging Drawing Design
The warm forging drawing is derived from the cold forging drawing by adding the cold sizing allowance and considering thermal expansion. The cold sizing allowance (per side) is determined as follows.
The minimum cold sizing allowance must exceed the elastic recovery of the forged part. According to Hooke’s law:
$$\varepsilon = \frac{\sigma}{E}$$
For a solid cylinder of diameter $D$ and elastic recovery $\Delta D$, the circumferential strain is:
$$\varepsilon = \ln\left(\frac{D+\Delta D}{D}\right) \approx \frac{\Delta D}{D}$$
Thus:
$$\Delta D = D \frac{\sigma}{E}$$
For 18CrNiMo7-6, the yield strength is approximately 1016 MPa and the elastic modulus is 210 GPa. Taking $D=160$ mm (addendum diameter), the single-side elastic recovery is:
$$\Delta r = \frac{D}{2} \frac{\sigma}{E} = \frac{160}{2} \frac{1016}{210000} \approx 0.19 \text{ mm}$$
Therefore, a cold sizing allowance of 0.2 mm per side is selected.
The web thickness is calculated using empirical formula:
$$S = 0.45 d – 0.25 h – 0.6 h + 5 \quad (\text{mm})$$
where $d$ is the bore diameter (57 mm) and $h$ is the gear height (26 mm). Substituting:
$$S = 0.45 \times 57 – 0.25 \times 26 – 0.6 \times 26 + 5 \approx 6.1 \text{ mm}$$
Thus the web thickness is chosen as 6 mm.
Thermal expansion is accounted for in the warm forging die design. The linear expansion of the workpiece at 850 °C with a coefficient of $1.17 \times 10^{-5}$ °C-1 gives:
$$D_{warm} = D_0 (1 + \alpha \Delta T)$$
For the addendum circle diameter $D_0 = 160$ mm and $\Delta T = 830$ °C (from 20 °C to 850 °C), the hot dimension is approximately 161.5 mm. The die cavity dimensions are further adjusted considering die preheating and elastic expansion.
Billet Size Determination
A solid cylindrical billet is selected for simplicity and easy placement in the die. The billet diameter is chosen slightly smaller than the root circle diameter of the warm forging die (about 138.8 mm). Taking the billet diameter as 138 mm, the height is calculated based on volume constancy, including allowances for scale loss (1%), the web volume, and the cold sizing allowance. The total volume is:
$$V_{billet} = (V_{cold forging} + V_{sizing} + V_{web}) \times (1 + \delta)$$
Using CAD volume calculations, $V_{cold forging}+V_{sizing} \approx 426528 \text{ mm}^3$ and $V_{web} \approx 16578 \text{ mm}^3$. With $\delta = 0.01$, the billet volume is about 447,000 mm3, leading to a height of 31 mm.
Warm Forging Schemes
Three warm forging die configurations are proposed, all based on the float die concept:
- Closed-die backward extrusion: The upper punch has a boss, the lower die is flat. The upper punch moves downward while the lower die is stationary.
- Closed-die unidirectional upsetting-extrusion: The upper die consists of an upsetting punch and an extrusion punch. They move together initially, then the extrusion punch continues after the upsetting punch stops.
- Closed-die bidirectional upsetting-extrusion: Both upper and lower dies have upsetting and extrusion punches. The upper extruding punch and lower extruding punch move toward each other during the final stage.
Schematic diagrams of these schemes are described, but for brevity we refer to them as Scheme A, B, and C in the following sections.
Cold Sizing Scheme
Cold sizing is performed by pushing the warm-forged preform through a sizing die. The die has an entrance angle of 5–10° and a working zone length of 5–8 mm. The punch is flat without tooth profile to avoid entrapment of material. The process provides the final dimensional accuracy and surface finish.
Finite Element Simulation and Optimization of Warm Forging
Finite element simulations are carried out using DEFORM-3D. The workpiece is modeled as a plastic body, and the dies as rigid bodies. The material model for 18CrNiMo7-6 is created using JMatPro software based on the measured chemical composition. The flow stress is a function of strain, strain rate, and temperature:
$$\sigma = f(\varepsilon, \dot{\varepsilon}, T)$$
The simulation parameters are listed in Table 4.1.
| Parameter | Value |
|---|---|
| Workpiece initial temperature | 850 °C |
| Die preheating temperature | 250 °C |
| Environment temperature | 20 °C |
| Heat transfer coefficient | 11 kW/(m²·K) |
| Emissivity | 0.7 |
| Friction model | Shear friction, m = 0.25 |
| Punch velocity | 20 mm/s |
| Step size | 0.2 mm |
| Data saving interval | Every 10 steps |
| Wear model | Archard |
To reduce computational time, a segment with one tooth is modeled and symmetric boundary conditions are applied.
Simulation Results and Discussion
Velocity field
For Scheme A, the metal initially flows axially upward because the contact area with the upper punch is small. A slight upsetting occurs until the material fills the upper die cavity. The tooth corners are the last to fill. This behavior leads to non-uniform metal flow and a higher risk of folding.
In Scheme B, the upsetting stage causes the metal to flow radially into the tooth cavity, followed by extrusion from the bottom punch. The metal flow is more uniform, but the web is located at the bottom, increasing the distance for the upper corner filling and causing a higher load.
In Scheme C, the symmetric arrangement of the upper and lower extrusion punches produces a balanced flow pattern. The web is located at the middle, which shortens the flow distance for both upper and lower tooth cavities. The tooth corners are filled simultaneously, and the deformation is more homogeneous.
Forming load
The load-stroke curves for the three schemes are shown in Figure 4.5 (not depicted here). In all cases, the load increases mildly at the beginning and rises sharply at the final stage of die filling. The maximum loads are:
| Scheme | Maximum Load (kN) |
|---|---|
| A | 1080 |
| B | 911 |
| C | 890 |
Scheme C yields the lowest load because the material flow path is shortest and the frictional resistance is reduced by the symmetric movement.
Die wear
Die wear is calculated using the Archard model. The wear distribution indicates that Scheme A produces the most severe wear due to high sliding velocities and pressure. Scheme C shows the least wear because of the balanced flow and shorter sliding distances.
Effective stress
The effective stress distribution in Scheme C is more uniform compared with Scheme B. In Scheme B, high stress concentrations appear at the tooth root near the end faces, which may initiate micro-cracks. Scheme C reduces the stress gradient by promoting simultaneous radial and axial flow.
Based on the above analysis, Scheme C (floating die bidirectional upsetting-extrusion) is selected as the optimal warm forging scheme.
Process Optimization
Although Scheme C shows better performance, the load still increases sharply during the final filling of the tooth corners. To reduce the forming load and improve die life, a flow relief groove is added to the end face of the forging (in the region that will be machined later). This groove provides an additional free surface, allowing excess metal to flow out rather than being forced into the sharp corners. The modified scheme is shown in Figure 4.10 (not reproduced here). The simulation results indicate that the maximum forming load is reduced from 890 kN to 632 kN, a reduction of about 30%. Die wear is slightly increased due to the larger metal flow, but this can be mitigated by using an effective lubricant.
Orthogonal experiment for process parameters
Three key parameters are selected for optimization: initial workpiece temperature (A), die preheating temperature (B), and friction coefficient (C). Each parameter has three levels as listed in Table 4.2.
| Level | A (°C) | B (°C) | C |
|---|---|---|---|
| 1 | 750 | 200 | 0.1 |
| 2 | 800 | 250 | 0.2 |
| 3 | 850 | 300 | 0.25 |
A orthogonal array $L_9(3^4)$ is used. The nine combinations are simulated, and the forming load and die wear are recorded in Table 4.3.
| No. | A | B | C | Load (kN) | Wear (10-6 mm) |
|---|---|---|---|---|---|
| 1 | 750 | 200 | 0.1 | 904 | 2.65 |
| 2 | 750 | 250 | 0.2 | 854 | 2.69 |
| 3 | 750 | 300 | 0.25 | 848 | 2.70 |
| 4 | 800 | 200 | 0.2 | 885 | 2.52 |
| 5 | 800 | 250 | 0.25 | 878 | 2.58 |
| 6 | 800 | 300 | 0.1 | 826 | 2.51 |
| 7 | 850 | 200 | 0.25 | 855 | 2.51 |
| 8 | 850 | 250 | 0.1 | 815 | 2.33 |
| 9 | 850 | 300 | 0.2 | 828 | 2.37 |
Range analysis is performed. For forming load, the factors in order of influence are B (die preheat temperature) > A (workpiece temperature) > C (friction coefficient). The optimal combination is A3B3C1, i.e., workpiece at 850 °C, die preheated to 300 °C, and friction coefficient 0.1. For die wear, the order is A > C > B, and the same optimal combination is obtained. Therefore, the optimum process parameters for warm forging are: workpiece temperature 850 °C, die preheat temperature 300 °C, and friction coefficient 0.1.
Cold Sizing Simulation
Cold sizing is simulated as an elastic-plastic process. The preform is modeled as an elasto-plastic body, and the die as rigid. The friction coefficient is 0.1, and the punch speed is 20 mm/s. Four different sizing allowances are considered: 0.2, 0.25, 0.3, and 0.35 mm per side. The mesh is refined near the tooth profile to capture the small deformation accurately.
The simulation results for die wear and forming load are summarized in Table 4.4.
| Sizing allowance (mm) | Extrusion load (kN) | Die wear (relative) |
|---|---|---|
| 0.20 | 695 | Lowest |
| 0.25 | 791 | Increase |
| 0.30 | 1050 | Increase |
| 0.35 | 1450 | Rapid increase |
As the sizing allowance increases, both the forming load and die wear increase significantly. Since the elastic recovery is about 0.19 mm, a 0.2 mm allowance is sufficient to achieve the desired accuracy. Therefore, the optimal cold sizing allowance is 0.2 mm per side.
Virtual Forging Simulation on Open Inventor
Virtual reality (VR) technology provides an immersive and interactive environment for visualizing manufacturing processes. However, VR development platforms such as Open Inventor lack the ability to accurately model the plastic deformation of metals. On the other hand, finite element software like DEFORM-3D can simulate the deformation accurately but offers limited post-processing interaction. To combine the strengths of both, a data bridge is established to convert the finite element results into a format readable by Open Inventor.
During the finite element simulation, the transient geometry of the workpiece is saved in the DEFORM-3D database at intervals of ten steps. Each geometry is exported in the STL (ASCII) format. Open Inventor, however, uses the IV format. A conversion program is written in C++ to extract the triangular facet data from the STL file and generate a corresponding IV file.
The STL format records each triangular facet with its normal vector and three vertices. The IV format stores a list of all vertices with indices and then defines each face by a list of vertex indices. A simple example of the conversion process is illustrated. The conversion algorithm reads the STL file, assigns a sequential index to each unique vertex, and writes the vertex coordinates and face indices into a new IV file.
Scene Graph Establishment
In Open Inventor, a scene graph defines the objects, their properties, and the camera/lighting. The scene graph for the forging process includes:
- Root node
- Camera node
- Light node
- Group nodes for the workpiece and dies
- Transform nodes to control the motion of dies
- Draw style nodes to control visibility of the workpiece geometry
The gear workpiece undergoes plastic deformation, so its geometry must be updated at each time step. This is achieved by using sensor and alarm mechanisms to switch the visibility of successive IV models. The upper and lower dies are moved by using engines and translation nodes. A time-based sensor triggers the motion and visibility changes.
Implementation of the Virtual Forging Process
The converted IV files are read into the scene graph. The spatial positions are already defined relative to the fixed die and the moving punches. The upper upsetting punch moves downward at a speed of 1 mm/s for a duration corresponding to the simulated stroke. The extrusion punches are activated at appropriate times. The workpiece geometry is switched at intervals of 1 second, representing the saved simulation steps.
The following pseudocode shows the control logic for the upper punch motion:
if (time >= start_time && time <= end_time)
topmove->on = TRUE;
else
topmove->on = FALSE;
The translation is driven by an engine that converts time to displacement:
topcalcXZ->expression.setValue(0, "ta=-a*1"); topcalcXZ->expression.setValue(1, "oA=vec3f(0,0,-ta)"); topmoveTranslation->translation.connectFrom(&topcalcXZ->oA);
The visibility of successive workpiece models is controlled by alarm sensors that change the draw style from FILLED to INVISIBLE and vice versa.
Simulation Results
The virtual forging simulation runs smoothly in a Windows VC++ environment. The user can rotate, pan, and zoom the scene using the mouse. The figure above shows a snapshot of the virtual forging process at different stages. To observe the deformation inside the die, the die is made semi-transparent. The animation clearly demonstrates the plastic flow and die filling.
This approach is not limited to spur gears. It can be easily extended to other plastic forming processes. For example, a cross-shaft forging process (used in automotive couplings) was also simulated using the same method. The converted geometry is accurate, and the interactive visualization provides a powerful tool for engineers and designers to understand the forming process and validate the simulation results.
Conclusion
In this study, the composite warm-cold precision forging process for large-module spur gears was investigated by means of finite element simulation and virtual reality simulation. The main conclusions are as follows:
- Three warm forging schemes were compared using DEFORM-3D. The floating die bidirectional upsetting-extrusion scheme (Scheme C) exhibited the most uniform metal flow, the lowest forming load (890 kN), and the least die wear. It was identified as the optimal warm forging scheme.
- Adding a flow relief groove on the gear end face reduced the maximum forming load by about 30%, from 890 kN to 632 kN, while ensuring complete filling of the tooth profile and improving die loading conditions.
- An orthogonal experiment revealed that the die preheating temperature has the greatest influence on forming load, while the workpiece temperature has the greatest influence on die wear. The optimal warm forging parameters are: workpiece temperature 850 °C, die preheating temperature 300 °C, and friction coefficient 0.1.
- The minimum cold sizing allowance was calculated from Hooke’s law as about 0.19 mm. Simulation of cold sizing with allowances of 0.2, 0.25, 0.3, and 0.35 mm showed that the forming load and die wear increase rapidly with increasing allowance. Therefore, 0.2 mm per side is the optimal cold sizing allowance for this spur gear.
- A data conversion method between STL and IV formats was developed, enabling the finite element geometry data to be imported into Open Inventor. A virtual forging simulation of the spur gear was successfully implemented, providing an interactive and visually realistic representation of the forming process. The method is general and can be applied to other metal forming processes.
These results provide a solid theoretical and numerical foundation for the practical production of high-quality spur gears using warm-cold composite forging. Future work could extend the virtual simulation to include temperature and stress fields, and to incorporate real-time user interaction for process optimization.