In the field of mechanical transmission, straight spur gears are fundamental components widely used due to their simplicity and efficiency. Traditional machining methods for these gears, such as hobbing and shaping, suffer from low material utilization, high production cost, and poor mechanical performance due to the cutting of fiber flow lines in the tooth profile. To overcome these limitations, cold forging has emerged as a promising alternative. Compared to hot forming, cold forging offers superior dimensional accuracy, excellent surface finish, enhanced strength, and elimination of subsequent machining steps. In this study, we investigate the cold forging of straight spur gears using a combined process of central hole axial分流 (axis分流) and floating die technology. Numerical simulations are performed using the finite element method (FEM) to analyze the deformation behavior, stress and strain distributions, metal flow, and forming load. The goal is to provide theoretical guidance for the practical production of high-quality straight spur gears.
Straight spur gears are typically characterized by their module, number of teeth, face width, pressure angle, and profile shift coefficient. In our work, the gear parameters are as follows: module m = 2 mm, number of teeth Z = 18, face width h = 10 mm, normal pressure angle α = 20°, and profile shift coefficient x = 0.0. The billet material is AISI-4140 (cold), and the dies are considered rigid. The friction between the billet and dies is modeled as shear friction with a coefficient of 0.12. The initial billet dimensions are determined based on the principle of volume constancy before and after cold forging, with an allowance for subsequent machining. The billet is a cylinder with diameter 30 mm and height 15 mm. The central hole diameter in the upper and lower dies is 10 mm. To reduce computational cost while maintaining accuracy, we exploit the geometric symmetry of the straight spur gear and model only one quarter of the entire assembly. The geometric model is created using CAD software, and tetrahedral elements are used to mesh the billet. The upper die and floating die move downward at a constant velocity of 1 mm/s. The simulation is carried out using DEFORM-3D, a specialized FEM software for metal forming.
The FEM model is illustrated in the above figure, which shows the quarter-symmetry setup including the upper die, floating die, lower die, and billet. The mesh of the billet consists of tetrahedral elements, refined in regions where large deformation is expected. The simulation is performed in incremental steps, and the results are extracted at different stages to study the evolution of equivalent strain, equivalent stress, velocity field, and forming load.
The volume constancy condition is fundamental in determining the initial billet size. For cold forging of straight spur gears, the relationship between the billet volume \( V_b \) and the final gear volume \( V_g \) plus machining allowance \( V_a \) can be expressed as:
$$
V_b = V_g + V_a
$$
Given the gear geometry, the volume of the straight spur gear (including the central hole) is calculated as:
$$
V_g = \frac{\pi}{4} \left( d_a^2 – d_i^2 \right) h – (\text{tooth space volume})
$$
where \( d_a \) is the addendum circle diameter, \( d_i \) is the inner diameter of the central hole, and the tooth space volume is computed from the tooth profile. The billet dimensions are then adjusted to satisfy the volume condition.
The material behavior is modeled using the flow stress curve of AISI-4140 (cold). A typical constitutive equation for cold forming is:
$$
\sigma = K \bar{\varepsilon}^n \dot{\bar{\varepsilon}}^m
$$
where \( \sigma \) is the flow stress, \( \bar{\varepsilon} \) the equivalent plastic strain, \( \dot{\bar{\varepsilon}} \) the strain rate, \( K \) the strength coefficient, \( n \) the strain hardening exponent, and \( m \) the strain rate sensitivity. For simplicity, in our simulation we assume the material to be rigid-plastic with a given flow stress data.
The numerical simulation yields critical information about the deformation process. Table 1 summarizes the key process parameters used in the simulation.
| Parameter | Value |
|---|---|
| Module m | 2 mm |
| Number of teeth Z | 18 |
| Face width h | 10 mm |
| Pressure angle α | 20° |
| Profile shift coefficient x | 0.0 |
| Billet diameter | 30 mm |
| Billet height | 15 mm |
| Central hole diameter | 10 mm |
| Billet material | AISI-4140 (cold) |
| Friction coefficient (shear) | 0.12 |
| Upper die velocity | 1 mm/s |
| Floating die velocity | 1 mm/s |
| Mesh type | Tetrahedral |
| Symmetry | 1/4 model |
During the simulation, we extract the equivalent strain distribution at different incremental steps. At step 60, the material begins to fill the tooth cavity. At step 75, the tooth profile is partially formed, and at step 89, the die cavity is nearly completely filled. The equivalent strain distribution indicates that the tooth region experiences significantly higher strain compared to the central hole region. The maximum equivalent strain occurs at the tooth root and the corner regions of the die, where material undergoes severe plastic deformation. This large deformation helps to refine the grain structure and improve the mechanical properties of the straight spur gears, especially in the tooth profile which is critical for load transmission.
The equivalent stress fields follow a similar trend. At the early stage, the stress is relatively low and uniformly distributed. As the billet flows into the die cavity, stress concentrates in the tooth area due to the constraint of the die walls. At the final stage, the tooth cavity is fully filled, and the central hole acts as a relief channel, allowing material to flow axially. The equivalent stress in the tooth region is much higher than that in the central hole, which is beneficial for healing internal defects and damage. The maximum stress is observed when the material in the tooth corners is under nearly three-dimensional compressive stress state.
The velocity field provides insight into the material flow pattern. At early steps, the material flows preferentially toward the tooth cavities because the flow resistance there is lower than toward the central hole. The floating die exerts axial force on the tooth cavity, enhancing the filling of the lower tooth profile. Since the upper die moves downward while the lower die remains stationary, the upper part of the billet flows faster than the lower part. As deformation progresses, the tooth cavity becomes nearly filled, and the material flow direction changes. At the final stage, the central hole offers a free surface with lower resistance, so the material flows predominantly toward the central hole. This combined flow pattern ensures complete filling of the tooth profile while preventing excessive pressure buildup. The velocity distribution confirms that the axis分流 (central hole分流) and floating die coupling strategy effectively balances material flow and reduces forming load.
The forming load versus punch displacement curve (stroke-load curve) is shown in the simulation results. Initially, the load increases linearly as the upper die contacts the billet. Then, a period of upsetting occurs where the load increases slowly because the billet is being compressed without significant lateral flow. Once the material begins to enter the tooth cavity, the load rises more steeply. As the tooth cavity fills progressively, the load increases gradually, while a small amount of material still flows into the central hole. Near the end of the stroke, the tooth cavity is almost completely filled, and only the corner regions remain to be filled. At this stage, the material experiences triaxial compressive stress, leading to a sharp rise in the forming load. The maximum load occurs at the final stroke. The curve can be roughly divided into three stages: (I) linear elastic/upsetting, (II) steady filling, and (III) final sharp rise. Understanding this load characteristic is essential for die design and press capacity selection.
Table 2 lists the values of equivalent strain, equivalent stress, and velocity at selected stages for quantitative reference.
| Increment step | Max. equivalent strain | Max. equivalent stress (MPa) | Max. velocity (mm/s) |
|---|---|---|---|
| 60 | 1.8 | 720 | 1.3 |
| 75 | 2.9 | 890 | 1.8 |
| 89 | 4.2 | 1050 | 2.4 |
From the above analysis, it is clear that the combined process of central hole分流 (axis分流) and floating die is effective for forming straight spur gears with good tooth profile quality. The tooth region undergoes large strain and high stress, which improves the material’s mechanical properties. The central hole serves as a material reservoir during the final stage, preventing a drastic increase in forming load and allowing complete filling of the tooth corners. The floating die provides axial support to enhance lower tooth filling.
One of the key advantages of cold forging for straight spur gears is the preservation of continuous metal flow lines along the tooth profile. Unlike conventional machining, which cuts the fibers, cold forging deforms the material in a way that follows the gear shape. This results in improved strength and load-bearing capacity. The high strain in the tooth region further contributes to grain refinement and work hardening, making the gear more resistant to wear and fatigue.
The forming load in the final stage increases sharply because the remaining free surface area is very small. This is a typical characteristic of closed-die forging. The load can be estimated by the following empirical relationship:
$$
F_{\text{max}} = A_{\text{proj}} \cdot \sigma_{\text{flow}} \cdot K_f
$$
where \( A_{\text{proj}} \) is the projected area of the gear in the forging direction, \( \sigma_{\text{flow}} \) is the flow stress of the material, and \( K_f \) is a factor accounting for friction and geometry complexity. For our straight spur gear, the projected area is approximately the area of the gear blank minus the central hole area. Using the simulation results, the load at the end of stroke is found to be around 800 kN for the quarter model, which corresponds to a full-scale load of about 3200 kN.
In summary, the cold forging process of straight spur gears using axis分流 and floating die technology is numerically simulated and analyzed. The following conclusions are drawn:
1. The combined process can successfully produce straight spur gears with fully filled tooth profiles and good dimensional accuracy.
2. The equivalent strain and stress are highest in the tooth region, which is beneficial for improving gear strength and durability.
3. The velocity field shows that material first flows predominantly into the tooth cavity, and later into the central hole, ensuring a balanced filling sequence.
4. The forming load history consists of three stages: initial linear increase, steady filling, and sharp rise at the end. The maximum load occurs when the tooth corners are filled under triaxial compression.
5. The simulation results provide valuable guidance for die design, process optimization, and selection of forming equipment for industrial production of straight spur gears.
The methodology presented here can be extended to other gear geometries and materials. Future work may include experimental validation of the simulation results and investigation of the effects of process parameters such as friction coefficient, die velocity, and billet temperature (if warm forging is adopted). The cold forging of straight spur gears offers a cost-effective and high-performance manufacturing route, especially for medium to large batch production.
Another important aspect is the optimization of the central hole diameter and the floating die pressure. The axis分流 hole diameter of 10 mm used in this study was effective, but further parametric studies could identify the optimal size to minimize forming load while ensuring complete filling. The floating die velocity relative to the upper die also plays a role; a mismatch could cause uneven filling. Our simulation assumed equal velocities, which worked well.
To further quantify the process, we can define the filling ratio of the tooth cavity as:
$$
R_f = \frac{\text{Volume of material in tooth cavity}}{\text{Volume of tooth cavity}} \times 100\%
$$
At the final step, the simulation shows that \( R_f = 100\% \) for the straight spur gear, indicating complete filling. The central hole also absorbs some material, reducing the final load. The relationship between the volume of material flowing into the central hole \( V_h \) and the total billet volume \( V_b \) is expressed as:
$$
V_h = V_b – V_{\text{gear}} – V_{\text{flash}}
$$
where \( V_{\text{flash}} \) is negligible in this closed-die process. By adjusting the billet volume, the amount of material flowing into the central hole can be controlled.
In the context of sustainable manufacturing, cold forging of straight spur gears significantly reduces material waste compared to conventional machining. The material utilization rate can exceed 95% in cold forging, while traditional cutting methods often leave 30-50% of the material as chips. Additionally, cold forging eliminates the need for heating, saving energy and reducing greenhouse gas emissions. The higher strength of cold-forged straight spur gears allows for lighter designs, further contributing to environmental benefits.
Although the current study focuses on a specific gear size and material, the underlying principles are applicable to a wide range of straight spur gear dimensions. Scaling laws for cold forging can be derived using geometric similarity. For example, the forming load scales with the square of the gear diameter, and the strain distribution is primarily influenced by the tooth geometry and friction conditions. The numerical simulation approach adopted here can be readily adapted to other cases, providing a powerful tool for process design without costly trial-and-error.
Finally, we emphasize that the successful application of cold forging to straight spur gears requires careful control of die design, lubrication, and process parameters. The use of floating die and central hole分流 effectively addresses the common issues of incomplete filling and excessive load. The results presented here form a solid foundation for industrial implementation.
In conclusion, this research demonstrates that cold forging, augmented by axis分流 and floating die coupling, is a viable and advantageous method for manufacturing high-quality straight spur gears. The finite element analysis provides deep insights into the deformation mechanics and guides practical production. By adopting this technology, manufacturers can achieve better product performance, higher material efficiency, and lower costs, meeting the increasing demands of modern mechanical transmission systems.