In modern manufacturing, the rolling forming technology for bevel gears plays a critical role in enhancing metal utilization and improving fatigue strength. Unlike traditional machining methods that cut through metal fibers, rolling forming preserves the integrity of the material flow lines, which is essential for high-performance applications. In this study, we focus on the numerical simulation of the rolling forming process for bevel gears, specifically addressing the formation and control of lug defects. Bevel gears are widely used in power transmission systems due to their ability to transmit motion between intersecting shafts, and optimizing their forming process is vital for industrial efficiency. We begin by deriving the tooth surface equations for bevel gears, which allows for precise calculation of tooth profile points. Using DEFORM software, we simulate the hot rolling forming process, analyzing key parameters such as equivalent stress fields, equivalent strain fields, and rolling forces. Our investigation reveals that maximum strain typically occurs near the root of the tooth profile midpoint, while stress peaks are concentrated at the contact points between the mold wheel and the blank. Furthermore, we explore the mechanisms behind lug defect formation, examining how factors like friction coefficient influence these imperfections. By optimizing process parameters, particularly the friction coefficient, we aim to suppress lug defects and improve the overall quality of bevel gears. This research provides a foundational framework for advancing the rolling forming technology of bevel gears, contributing to more sustainable and efficient manufacturing practices.
The rolling forming process for bevel gears can be broadly divided into three stages: initial tooth splitting, tooth forming, and tooth finishing. During the splitting stage, the mold wheel begins to impress a preliminary tooth profile onto the blank. The forming stage involves deeper penetration, where plastic deformation allows the metal to flow and take the shape of the target bevel gear teeth. Finally, the finishing stage refines the tooth profile through alternating rotations of the mold wheel, ensuring dimensional accuracy and surface quality. To model this process mathematically, we derive the tooth surface equations for bevel gears. Unlike spur gears, bevel gears have tapered teeth, and their geometry is more complex. The tooth surface of a straight bevel gear can be represented using parametric equations based on the gear’s pitch cone angle and other parameters. For a bevel gear with a pitch cone angle of $\delta$, the position vector of a point on the tooth surface can be expressed as:
$$ \mathbf{r}(u, \theta) = \begin{bmatrix} x(u, \theta) \\ y(u, \theta) \\ z(u, \theta) \end{bmatrix} = \begin{bmatrix} (R – u \sin \delta) \cos \theta \\ (R – u \sin \delta) \sin \theta \\ u \cos \delta \end{bmatrix} $$
where $u$ is the parameter along the tooth height, $\theta$ is the angular parameter around the gear axis, and $R$ is the pitch cone radius. This equation accounts for the conical nature of bevel gears, which is essential for accurate simulation. By discretizing these equations, we can generate point clouds that define the tooth profile, which are then used to create 3D models in software like UG. For instance, the coordinates for the mold wheel teeth are computed iteratively, and the resulting model is assembled to represent the rolling process. The following table summarizes the basic parameters used for the target bevel gear and the mold wheel in our simulation:
| Parameter | Target Gear | Mold Wheel |
|---|---|---|
| Module (mm) | 2 | 2 |
| Pressure Angle (°) | 20 | 20 |
| Number of Teeth | 19 | 53 |
| Pitch Cone Angle (°) | 30 | 30 |
Using these parameters, we calculate the tooth surface points and construct the rolling process model. The initial blank diameter is determined through the volume constancy principle, assuming no change in axial dimensions during plastic deformation. For a bevel gear, the volume calculation must consider the tapered geometry, leading to a more complex formulation. The cross-sectional area equivalence method is applied, where the area before and after deformation is equated to find the blank’s initial dimensions. This approach ensures that the metal flow during rolling is accurately represented in the simulation.
To set up the numerical simulation in DEFORM, we make several assumptions to simplify the model. The mold wheel is treated as a rigid body, neglecting elastic deformations, while the blank is modeled as a homogeneous, rigid-plastic material. The contact friction between the mold wheel and the blank is assumed constant, and the environmental temperature is maintained steady during the hot rolling process. The simulation model includes a half-section of the blank to reduce computational load, with localized mesh refinement in the deformation zones. The mesh consists of approximately 101,536 elements, with a refinement ratio of 0.01 in the outer 4 mm thickness of the blank where plastic deformation is most significant. Boundary conditions are applied to prevent slippage; for example, the inner surface of the blank is fixed in all directions, and symmetric constraints are imposed on the cut plane. The mold wheel is assigned a radial feed velocity of 0.1 mm/s and a rotational speed of 2.2525 rad/s around its axis, while also rotating at 6.2832 rad/s around the blank’s center (Y-axis). These settings mimic real-world rolling conditions for bevel gears, enabling a realistic analysis of the forming process.
The simulation results for the hot rolling of bevel gears show distinct stages of tooth formation. At a feed depth of 12% of the total tooth depth, initial tooth impressions are visible on the blank. As the feed increases to 30%, the tooth profiles become more defined, and by 60% feed, the teeth are nearly fully formed. At 100% feed, the teeth achieve their final shape, and the process enters the finishing stage, where alternating rotations of the mold wheel correct any deviations and enhance surface quality. The equivalent strain distribution during these stages indicates that the maximum strain occurs at the midpoint of the tooth profile,偏向 the root region, due to the concentrated deformation from the mold wheel’s tooth tips. For example, at 30% feed, the maximum equivalent strain is around 0.8, increasing to 1.2 at 60% feed, and reaching 1.5 at full feed. This pattern is consistent with the plastic flow characteristics of bevel gears, where the root area experiences higher deformation because of the tapered geometry. The equivalent stress fields follow a similar trend, with stress concentrations at the contact points between the mold wheel and the blank. Initially, at 30% feed, the maximum stress is approximately 309 MPa, rising to 342 MPa at 60% feed, and peaking at 347 MPa at full feed. During the finishing stage, the stress stabilizes around 250 MPa as the material undergoes elastic recovery and minor adjustments. The following equation illustrates the relationship between stress and strain in the plastic deformation zone, using the von Mises criterion for bevel gears:
$$ \sigma_{eq} = \sqrt{\frac{3}{2} \mathbf{s} \cdot \mathbf{s}} $$
where $\sigma_{eq}$ is the equivalent stress, and $\mathbf{s}$ is the deviatoric stress tensor. This formulation helps in understanding the material behavior under rolling conditions, particularly for bevel gears where stress distribution is non-uniform due to the conical shape.
The rolling forces during the forming process are dynamic and vary with time, impacting the stability and precision of bevel gear production. We analyze the forces in the X, Y, and Z directions, as shown in the simulation outputs. In the X-direction, the rolling force starts low and increases steadily with feed depth, exhibiting fluctuations during the mid-stage due to intense plastic deformation. Similarly, the Z-direction force follows an upward trend, reflecting the resistance from metal flow. The Y-direction force, however, is negligible compared to the others, as it aligns with the minor axis of deformation for bevel gears. Overall, the rolling force curves demonstrate a progressive increase, with peak forces occurring at high feed depths. This behavior is analogous to observations in spiral bevel gears and other gear types, validating our simulation approach for bevel gears. The table below summarizes the maximum rolling forces recorded at different feed percentages:
| Feed Percentage | Max X-Force (N) | Max Y-Force (N) | Max Z-Force (N) |
|---|---|---|---|
| 30% | 1,200 | 50 | 1,100 |
| 60% | 2,500 | 80 | 2,300 |
| 100% | 3,000 | 100 | 2,800 |
Lug defects are a common issue in the rolling forming of bevel gears, characterized by protrusions at the tooth tips that exceed the intended profile. These defects arise from mismatches in process parameters, such as mold wheel speed, feed rate, and blank rotation. Mechanically, lugs form due to the frictional forces acting on the tooth surfaces during rolling. When the mold wheel teeth engage with the blank, normal pressures and frictional forces drive metal flow toward the tooth tips. As one mold tooth disengages, the frictional force direction shifts upward, pulling material and creating elevated regions. For bevel gears, this effect is exacerbated by the tapered geometry, which concentrates stress at the tips. To quantify lug defects, we define an evaluation index based on the volume ratio of the lug to the entire tooth at 55% feed depth, where lugs are most prominent. The formula is:
$$ F_0 = \frac{\sum_{i=1}^{n} \left( \frac{V_i^0}{V_i} \times 100 \right)}{n} \% $$
where $V_i^0$ is the lug volume of the $i$-th tooth, $V_i$ is the total volume of the $i$-th tooth, and $n$ is the number of teeth. This metric allows for consistent comparison across different process conditions for bevel gears.
Friction coefficient is a key factor influencing lug formation in bevel gears. Higher friction coefficients increase the frictional forces that drive material toward the tooth tips, thereby enlarging lug volumes. We investigate this by simulating the rolling process with friction coefficients of 0.1, 0.2, and 0.3, corresponding to common lubricants like spindle oil with additives, graphite, and heavy oil. The results show a clear trend: as friction coefficient increases, so does the lug volume percentage. For instance, at a friction coefficient of 0.1, the average lug volume is 16.63%, rising to 17.01% at 0.2, and reaching 18.51% at 0.3. This aligns with findings in other gear types, such as spiral bevel gears, where friction plays a similar role. The following equation models the frictional force $f$ in terms of the normal pressure $N$ and friction coefficient $\mu$:
$$ f = \mu N $$
In the context of bevel gears, optimizing the friction coefficient through lubricant selection can significantly reduce lug defects. For example, using low-friction lubricants like spindle oil with additives (μ ≈ 0.1) minimizes material pull-up, whereas higher-friction conditions worsen lugs. This insight is crucial for process control in bevel gear manufacturing, as it directly impacts product quality and reduces the need for post-processing.
In conclusion, our numerical simulation of the rolling forming process for bevel gears provides detailed insights into strain-stress distributions, rolling forces, and defect mechanisms. The derivation of tooth surface equations enables accurate modeling, while the analysis of lug defects highlights the importance of friction control. By optimizing parameters such as friction coefficient, manufacturers can enhance the forming quality of bevel gears, leading to improved performance and longevity. Future work could explore additional factors like temperature variations and material properties for further refinement. This study establishes a theoretical foundation for the rolling forming of bevel gears, supporting advancements in gear manufacturing technology.