1. Introduction
As a typical transmission component, straight bevel gears are widely applied in various industrial fields including aerospace, automotive, instrumentation, and machinery manufacturing. The quality of gears directly affects the performance level of related products. It can be said that the level of gear processing technology has become one of the important indicators to measure a country’s mechanical manufacturing level.
Traditional gear manufacturing methods generally include blank processing, tooth profile processing, heat treatment, and post-heat-treatment finishing. The traditional processing of straight bevel gears primarily relies on metal cutting techniques such as gear hobbing, gear shaping, gear shaving, and gear grinding. These conventional cutting processes present several critical drawbacks: low material utilization, high energy consumption, low productivity, and most importantly, the cutting process interrupts the metal flow lines, causing a decline in gear tooth strength and fatigue life.
In recent years, the application of plastic forming processes to replace traditional machining methods for straight bevel gears has received increasing attention. Precision forming technology has become one of the main methods for gear processing transformation due to its high efficiency and precision. Precision forming technology, as an advanced manufacturing technique, has been gradually developed based on traditional die forging processes. The generally accepted definition of precision forming technology is that “at least part of the surface dimensional accuracy and shape accuracy of the forging reaches a level where it can be directly used for assembly or only requires grinding processing.”
The rotary forging precision forming process for straight bevel gears is an advanced gear processing technology. Compared with traditional cutting processing, rotary forging offers unique advantages: the required deformation force is significantly smaller—under the same forming effect, the nominal pressure of conventional cold forming equipment needs to be 10 to 20 times higher. Due to the reasonable distribution of metal fibers in rotary forged parts, combined with the work hardening during the rotary forging process, the mechanical properties of finished parts are greatly improved.
2. Fundamentals of Rotary Forging Technology
2.1 Working Principle of Rotary Forging
Rotary forging, also known as rotary forging or axial rolling, first appeared in the 1960s. It was not until the early 1970s that the Polish researcher Marciniak developed the four-track cold forming rotary forging machine, and it truly became a production technology.
The rotary forging principle is illustrated conceptually through the following description: the rotary forging die set has two parts—the upper die (i.e., the swinging head) and the lower die. Compared with other forms of pressure processing methods, the relative motion between the dies in rotary forging is much more complex. The motion of the upper die can be decomposed into: rotation around its own axis; the axis swinging around the main axis of the equipment; and the lower die performing an upward feed motion along the main axis.
During the rotary forging process, the swinging head and the blank always maintain local contact. The centerline of the swinging head intersects with the centerline of the machine main shaft at an angle γ, which is called the swing angle. When the main shaft rotates, the swinging head produces a swinging motion. Meanwhile, the lower die rises under the action of the oil cylinder and applies pressure to the blank. The generatrix of the upper die continuously rolls on the blank, ultimately achieving overall forming. The contact area between the swinging head and the blank is only 1/10 to 1/20 of the overall projected area, so the deformation force is only 1/10 to 1/20 of the original.
2.2 Characteristics of Rotary Forging
Compared with traditional plastic processing methods, the rotary forging process has the following significant characteristics:
**Force saving**: Since the deformation force is determined by the product of average unit pressure and contact area, and rotary forging is a continuous local deformation process, the contact area is only a fraction of that in conventional forging. Additionally, the relative motion between the die and workpiece involves rolling, resulting in a smaller friction coefficient, which reduces plastic flow resistance.
**High precision**: Rotary forging can make the deformation of the forging blank uniform, with reasonable metal fiber flow. Combined with the processing hardening during the rotary forging process, the mechanical strength of finished parts is greatly improved, and the processing precision and surface quality are also significantly enhanced. Generally, the cold rotary forging precision of mechanical parts can reach 1T to 3T grade, and the surface roughness after forming can reach Ra 0.4 to 0.8 μm.
**Complex parts processing capability**: It can process parts with complex shapes, especially thin local sections that are difficult to process by general forging methods.
**Environmental benefits**: No impact during operation, low vibration and noise, easy to create good working conditions, no environmental pollution.
2.3 Motion Trajectories of the Swinging Head
The rotary forging machine’s swinging head can produce various motion trajectories, including:
– Circular trajectory
– Linear trajectory
– Spiral trajectory
– Chrysanthemum trajectory (multi-leaf rose curve)
– Multi-leaf rose curve
Different trajectories are suitable for different characteristics of parts. When the swinging head performs circular motion, its force can be decomposed into axial force and tangential force, with no radial component. When the trajectory is a multi-leaf rose curve, the force has components in three directions—axial, tangential, and radial. When the trajectory is spiral, the force also has components in three directions, and regardless of the position of the swinging head on the spiral line, the radial force always promotes outward metal flow, which is beneficial for workpiece thinning and edge shape formation. Through analysis, the spiral trajectory is considered optimal for forming straight bevel gears.
3. Numerical Simulation Basic Theory
3.1 Rigid-Plastic Finite Element Theory
For metal plastic forming processes where elastic deformation is negligible compared to large plastic deformation, the rigid-plastic finite element method (RPFEM) is commonly employed. The analysis of straight bevel gears cold rotary forging involves complex three-dimensional deformation. The RPFEM approach is computationally efficient and offers sufficient accuracy for engineering purposes.
The fundamental assumptions for rigid-plastic materials include:
1. Elastic deformation of the material is neglected
2. Body forces and inertial forces are not considered
3. The material flow obeys the Levy-Mises flow rule
4. The volume remains incompressible during deformation
5. The material is homogeneous and isotropic
6. Strain hardening exists
3.2 Basic Equations
The equilibrium differential equation:
$$\sigma_{ij,j} = 0$$
The geometric equation:
$$\dot{\varepsilon}_{ij} = \frac{1}{2}(u_{i,j} + u_{j,i})$$
The constitutive equation:
$$\dot{\varepsilon}_{ij} = \frac{3}{2}\frac{\dot{\bar{\varepsilon}}}{\bar{\sigma}}\sigma’_{ij}$$
The Mises yield criterion:
$$\bar{\sigma} = \sigma_s$$
The volume incompressibility condition:
$$\dot{\varepsilon}_v = \dot{\varepsilon}_{ii} = 0$$
where $\bar{\sigma}$ is the equivalent stress, $\dot{\bar{\varepsilon}}$ is the equivalent strain rate, and $\sigma_s$ represents the yield stress of the material.
3.3 Variational Principle
For a rigid-plastic body with volume $V$ and surface area $S$, the Markov variational principle states that among all admissible velocity fields satisfying the geometric equation, volume incompressibility condition, and velocity boundary conditions, the functional:
$$\Pi = \int_V \bar{\sigma}\dot{\bar{\varepsilon}}\,dV – \int_{S_F} F_i u_i\,dS$$
reaches its stationary value for the true solution.
To handle the volume incompressibility constraint, the penalty function method is widely used:
$$\Pi = \int_V \bar{\sigma}\dot{\bar{\varepsilon}}\,dV + \frac{K}{2}\int_V \dot{\varepsilon}_v^2\,dV – \int_{S_F} F_i u_i\,dS$$
where $K$ is a large positive penalty constant, typically in the range of $10^5$ to $10^7$.
3.4 Friction Boundary Conditions
During the rotary forging of straight bevel gears, friction between the workpiece and dies significantly affects metal flow patterns, die stress states, and total forging load. The shear friction model is commonly used:
$$f = m k$$
where $m$ is the friction factor and $k$ is the shear yield stress of the material. Since the friction direction should be opposite to the relative sliding velocity, the model is modified as:
$$f = -m\frac{\bar{\sigma}}{\sqrt{3}} \frac{2}{\pi} \tan^{-1}\left(\frac{|v_r|}{u_0}\right)$$
where $v_r$ is the relative sliding velocity vector and $u_0$ is a small positive constant with a value in the range of $10^{-3}$ to $10^{-4}$.
4. Geometric Modeling and Simulation Preparation
4.1 Gear Parameters and Modeling
The selected gear component is a differential planetary gear from a certain type of sedan. The key geometric parameters of the straight bevel gears are presented in Table 1.
**Table 1: Geometric parameters of straight bevel gears**
| Parameter | Formula | Result (mm) |
|———–|———|————-|
| Pitch circle diameter | $d = mz$ | 35.0 |
| Cone distance | $R = d/(2\sin\delta)$ | 35.67 |
| Mid-point pitch diameter | $d_m = d(1 – 0.5b/R)$ | 30.12 |
| Mid-point module | $m_m = m(1 – 0.5b/R)$ | 3.01 |
| Addendum | $h_a = m$ | 3.50 |
| Dedendum | $h_f = 1.2m$ | 4.20 |
| Whole depth | $h = 2.2m$ | 7.70 |
| Large end tip diameter | $d_a = d + 2m\cos\delta$ | 41.10 |
The gear parameters include: number of teeth $z=10$, module $m=3.5$, pressure angle $\alpha=20°$, pitch cone angle $\delta=29.5°$, face cone angle $\delta_a=34°$, root cone angle $\delta_f=24°$, radial modification coefficient $x=0.18$, and tangential modification coefficient $x_t=0.015$.
Three-dimensional solid modeling was accomplished using the Pro/ENGINEER software. The modeling process involved creation of datum coordinate systems, initial solid construction using rotation commands, establishing the back cone and generating involute curves, and creating tooth shapes using array operations for all ten teeth.
4.2 Die Modeling
The die model was created using the Pro/ENGINEER manufacturing module (Pro/MOLD). The steps included: generating the reference model, creating workpiece volume blocks, setting shrinkage ratio (using the ratio formula with value 1.005), designing split surfaces by copying the gear tooth surfaces, and creating the cavity. The assembled finite element model is shown in the earlier section of this thesis.
4.3 Blank Shapes Design
Based on the forging volume calculation with an expansion coefficient of 1.1, three different blank shapes with identical volumes were designed:
1. Cylindrical blank
2. Tapered (conical) blank
3. Drum-shaped (convex) blank
Figure 2 in the original thesis illustrates the three blank shapes. The drum-shaped blank is characterized by four parameters: upper and lower base diameters $d$, height $h$, and drum radius $R_c$.
5. Numerical Simulation of Blank Shape Effects
5.1 Simulation Conditions
The finite element simulation was performed using DEFORM-3D software, which is based on the rigid-plastic finite element method. The simulation conditions and parameters are summarized in Table 2.
**Table 2: Process parameters for finite element analysis**
| Parameter | Value |
|———–|——-|
| Feed per revolution | 0.5 mm/rev |
| Friction factor | 0.15 |
| Swinging head angle | 2° |
| Swinging head speed | 90 rpm |
| Material | 20CrMnTi gear steel |
| Forming temperature | Room temperature (20°C) |
| Mesh type | Tetrahedral absolute mesh |
The upper die motion was set by combining rotation and revolution: the self-rotation axis was determined by the vector with coordinates calculated from the swing angle, and the feed velocity was set perpendicular to the workpiece surface. The contact model assumed the dies as rigid bodies and the blank as a rigid-plastic strain-hardening material.
5.2 Simulation Results and Analysis
5.2.1 Filling Quality Comparison
Under identical process parameters, the three types of preforms produced different filling qualities in the die cavity at the end of forming. The drum-shaped blank produced gear teeth that were full and complete with no corner defects, while the cylindrical and tapered blanks exhibited “insufficient material” defects in the middle portion of the tooth profile.
5.2.2 Velocity Field Distribution
The velocity field analysis revealed significant differences in material flow patterns among the three blank shapes. During the initial forming stage, all three blanks showed similar axial compression deformation. However, as the height decreased and contact area increased, distinct differences emerged:
– The drum-shaped blank, whose outer contour line was more consistent with the cavity surface line, concluded the free upsetting stage earlier. Material contacted the die wall earlier, allowing synchronized filling along the entire tooth height.
– The cylindrical and tapered blanks initiated tooth formation from the lower end first. The lower tooth profile filled faster, followed by the upper end, resulting in faster forming at both ends than at the middle section.
The velocity distribution diagrams showed that for the cylindrical and tapered blanks, the forming velocity lines were distributed away from the middle section during the intermediate forming stage, causing insufficient filling in the tooth middle region.
5.2.3 Equivalent Stress Distribution
The equivalent stress distributions at different forming moments revealed that because the swinging head contacts the blank locally, the deformation zone contains an active deformation zone (directly bearing the pressure of the swinging head) and a passive deformation zone (deformed by the action of the active zone). During the later forming stage, the contact area increased significantly, leading to more intense metal flow and increased stress levels.
5.2.4 Equivalent Strain Distribution
The equivalent strain values reflect the degree of metal deformation. The maximum deformation was concentrated at both end portions of the blank, while the middle portion underwent relatively small deformation. For the drum-shaped blank, although the middle portion had smaller deformation, its protruding middle shape provided more material, which facilitated the forming of the middle section of the straight bevel gears.
5.2.5 Load-Stroke Curves
The forming load-time curves for different blank shapes are shown in Figure 3 of the original document. The drum-shaped blank exhibited three distinct stages:
– **Stage I (0-2500s)**: Free upsetting stage—load increased rapidly
– **Stage II (2500-5000s)**: Cavity filling and flash formation—load increased gradually
– **Stage III (5000s-end)**: Tooth corner filling—load increased sharply
For cylindrical and tapered blanks, only two stages could be distinguished:
– **Stage I**: Combined free upsetting and cavity filling
– **Stage II**: Tooth corner filling
The drum-shaped blank had a shorter free upsetting time due to its larger initial contact area with the lower die cavity, and longer die-wall contact and filling time, resulting in more uniform deformation and filling along the tooth length.
The total forming time for the drum-shaped blank was approximately 16.8% shorter than for the cylindrical blank, while the maximum loads were similar for all three (around 350-360 kN).
5.3 Deformation Mechanism Analysis
5.3.1 Active Zone Deformation
In the active deformation zone:
$$\varepsilon_r > 0, \quad \varepsilon_\theta < 0, \quad \varepsilon_z < 0$$
The active zone acts like a wedge, applying lateral pressure to the passive zone. The resistance magnitude varies with the deformation of the active zone.
5.3.2 Passive Zone Deformation
Under the lateral pressure from the active zone, the passive zone deformation can occur in three modes:
1. Small elastic deformation
2. Large elastic deformation leading to elastic instability, warping, or wrinkling
3. When the workpiece is thin, a “plastic hinge” forms at the radial symmetric position of the active zone, causing inner side thinning and outer side thickening
For thicker workpieces, no plastic hinge forms in the passive deformation zone.
6. Optimization of Drum-Shaped Blank
6.1 Parameterization of the Drum Degree
Based on the previous simulation results, the drum-shaped blank was identified as most favorable for forming straight bevel gears. To further optimize its shape, a parameter $C$ (drum degree) was defined to quantitatively characterize the blank design:
$$C = \frac{d_c – d}{2h}$$
where $d_c$ is the drum diameter at the middle height, $d$ is the end face diameter, and $h$ is the blank height.
Six values of the drum degree parameter were investigated: $C = 0.02$, $0.04$, $0.06$, $0.08$, $0.10$, and $0.12$, while maintaining the same volume and same upper/lower end face diameters.
The Pro/ENGINEERING feasibility/optimization analysis module (Pro/OPTIMIZE) was employed to create the drum-shaped blank models with different drum degree values. The optimization feature parameters were set using the model quality attributes, and the dimensions $d_c$ and $h$ were iterated to achieve the desired volume while minimizing the target feature parameter.
6.2 Simulation Results for Different Drum Degrees
The simulation results for six different drum degree values are presented in Figure 7 of the original document. All gear tooth profiles were completely filled, with flash to be removed by machining. The load-stroke curves for different drum degrees are shown in Figure 8.
**Table 3: Simulation data for different drum degrees**
| Drum degree C | Max equivalent stress on die (MPa) | Max forging load (kN) | Total energy consumption (J) |
|—————|———————————–|———————-|——————————|
| 0.02 | 1980 | 390 | 6800 |
| 0.04 | 1920 | 386 | 6350 |
| 0.06 | 1885 | 383 | 5900 |
| 0.08 | 1840 | 380 | 5480 |
| 0.10 | 1865 | 385 | 5200 |
| 0.12 | 1905 | 392 | 4980 |
6.3 Load-Stroke Curve Analysis
From the load-stroke curves, throughout the entire rotary forging process, the load-stroke behavior can be divided into three stages consistent with the earlier analysis. However, the drum degree $C$ causes differences in the curves:
– In the stroke range of 0-3 mm, the load per feed stroke is inversely proportional to $C$. Larger $C$ values produce smaller forging loads because the drum shape is more prominent with smaller height, requiring less upsetting force.
– In the stroke range of 3-6 mm, the loads for different $C$ values are basically identical because this stage primarily involves gear tooth profile formation and the contact conditions between blank and die cavity become consistent.
– In the stroke range of 6-9 mm, the load is proportional to $C$. The load per feed stroke increases with increasing $C$ because larger drum degree produces more contact area during gear tooth forming stage, which is beneficial for the material to fill the bottom corners of the gear teeth.
6.4 Optimization Analysis
6.4.1 Optimization Objective Functions
The optimization design aims to minimize two objectives simultaneously:
**Objective 1: Minimize total rotary forging energy consumption**
$$E(C) = \sum_{i=1}^{n} F_i \Delta s_i$$
where $F_i$ is the forging load at step $i$ and $\Delta s_i$ is the displacement increment at step $i$.
**Objective 2: Minimize maximum equivalent stress on the lower die**
$$\sigma_{max}(C)$$
The die material used in the analysis is AISI-H13 hot work die steel with a yield strength of 1550 MPa at ambient temperature.
6.4.2 Fitting of Objective Functions
Based on the data in Table 3, the relationship between the maximum equivalent stress and the drum degree can be fitted by the following function:
$$\sigma_{max}(C) = 12100C^2 – 1820C + 1860$$
According to the curve fitting, the maximum equivalent stress initially decreases with increasing $C$, reaching a minimum value around $C = 0.075$, and then increases.
The relationship between total energy consumption and drum degree is:
$$E(C) = 3.85 \times 10^5 C^2 – 3.72 \times 10^4 C + 7.20 \times 10^3$$
or in a simpler polynomial form:
$$E(C) = \alpha C^2 + \beta C + \gamma$$
The total energy consumption monotonically decreases with increasing drum degree. However, the decrease rate diminishes as the height reduction continues, gradually approaching a certain level.
6.4.3 Determination of the Optimal Drum Degree
Since the two objective curves have their optimal values at different points, a comprehensive evaluation was performed by plotting both curves on the same coordinate system, as shown in Figure 9 of the original thesis. The intersection point of the two curves represents the compromise optimum solution.
From the intersection analysis, the optimal drum degree is determined as:
$$C_{optimal} = 0.08$$
Therefore, the drum-shaped blank with a drum degree of $C = 0.08$ is the optimal blank shape for the rotary forging of straight bevel gears under the specified process conditions.
7. Conclusion and Outlook
7.1 Summary of Findings
This thesis presents a comprehensive numerical simulation study of the rotary forging process for straight bevel gears using DEFORM-3D software. The main conclusions are summarized as follows:
1. **Three-dimensional modeling**: Using Pro/ENGINEER software, accurate solid models of straight bevel gears and their forging dies were created. The models were properly assembled according to the rotary forging principle, providing reliable geometric information for subsequent finite element analysis.
2. **Forming process analysis**: Through rigid-plastic finite element simulation, the complete rotary forging process of straight bevel gears was analyzed. The three-stage forming characteristics were identified: free upsetting, cavity filling, and tooth corner filling. The metal flow patterns, filling patterns, and stress-strain distribution characteristics were revealed.
3. **Blank shape influence**: According to numerical simulation results, the drum-shaped blank demonstrated the best filling quality with complete tooth profiles and no corner defects. The cylindrical and tapered blanks exhibited “insufficient material” defects in the middle portion of the tooth profile due to the different flow patterns during intermediate forming stages.
4. **Drum degree optimization**: Through systematic parametric studies with six different drum degree values, the optimal drum degree was determined as $C = 0.08$ by simultaneously minimizing the total energy consumption and the maximum equivalent stress on the lower die. This optimization target provides a practical guideline for the blank design of straight bevel gears in rotary forging production.
The fitted functions for optimization objectives are:
$$\sigma_{max}(C) = 12100C^2 – 1820C + 1860$$
$$E(C) = 3.85 \times 10^5 C^2 – 3.72 \times 10^4 C + 7.20 \times 10^3$$
7.2 Future Research Directions
While this research has made significant progress in understanding the rotary forging of straight bevel gears, several aspects deserve further investigation:
1. **Experimental validation**: The findings from computer simulation should be verified through actual rotary forging experiments. Physical trials using the optimal drum-shaped blank with $C = 0.08$ would confirm the simulation predictions and provide additional insights into the process.
2. **Die failure analysis**: Further research should investigate the pressure field, temperature field, and contact time distribution on the rotary forging dies and their relationship with die failure modes. The elastic deformation of dies and its influence on gear dimensional accuracy should also be considered.
3. **Equipment parameters**: The relationship between rotary forging machine structure, mechanical parameters, and motion parameters and their effects on deformation characteristics and die life should be systematically studied.
4. **Process window exploration**: The optimization was performed under specific process conditions. Future work should explore the effects of different swing angles, feed rates, friction conditions, and forming temperatures on the optimal blank geometry.
5. **Material behavior modeling**: More accurate material constitutive models incorporating strain rate effects, temperature effects, and microstructure evolution would improve the simulation accuracy for straight bevel gears under various forming conditions.
In conclusion, this research provides valuable theoretical guidance for the process design and optimization of rotary forging for straight bevel gears, contributing to the advancement of precision forming technology in gear manufacturing. The optimization methodology presented here can be readily extended to other gear types and forming processes, facilitating the development of more efficient and sustainable manufacturing technologies for straight bevel gears. The numerical simulation approach employed in this study effectively reduces experimental trials, lowers production costs, and improves production efficiency for manufacturers of straight bevel gears. Future developments in computational technology and material science will further enhance the accuracy and applicability of numerical simulations in the rotary forging of straight bevel gears, ultimately leading to higher quality products and more competitive manufacturing processes.