Gear transmission is a critical method for transmitting speed and torque in mechanical systems, characterized by reliability, stability, and compact structure. It finds extensive applications in aerospace, marine, and automotive industries. Among various gear manufacturing techniques, through-feed rolling has emerged as a novel forming method, offering advantages such as high production efficiency, minimal material waste, and superior mechanical properties compared to traditional cutting processes. The rolling die, as a core component in gear rolling equipment, directly influences the quality of the formed gear teeth, with its tooth profile deformation and machining errors mapping onto the workpiece surface. Additionally, process parameters play a pivotal role in determining forming loads and gear quality. Thus, research on the geometric design of beveloid rolling dies and process optimization holds significant industrial value. In this study, we propose a geometric design method for beveloid rolling dies in cylindrical gear through-feed rolling processes, analyze the effects of varying modification coefficients on forming performance, and optimize process parameters through orthogonal experiments.
The design of rolling dies is crucial for achieving high-quality gear formation. Traditional rolling dies often face challenges such as difficulty in tooth tip penetration, inadequate material flow, and adverse stress conditions during the through-feed rolling process. To address these issues, we introduce a beveloid rolling die with varying tooth thickness along the axial direction. This design incorporates a cutting section, finishing section, and exiting section, each with specific modification coefficients to facilitate material flow and reduce forming loads. The geometric design is based on gear meshing principles and the rack tool machining method, where the tooth profile equations are derived by varying the center distance between the rack tool and the workpiece. By adjusting the modification coefficients, we obtain tooth profiles for different sections of the rolling die, enabling controlled deformation and improved forming accuracy.
To derive the tooth profile equations, we establish coordinate systems for the rack tool, workpiece, and rolling die. The transformation matrices between these coordinate systems are defined to describe the relative motion during the rolling process. The rack tool tooth profile consists of three segments: the tooth tip straight line, tooth tip arc, and tooth flank straight line. For each segment, the equations are expressed in the rack tool coordinate system and then transformed into the workpiece and rolling die coordinate systems using the derived transformation matrices. The general form of the transformation matrix from the rack tool moving coordinate system to the workpiece moving coordinate system is given by:
$$ M_{31} = M_{32} \cdot M_{20} \cdot M_{01} = \begin{bmatrix} \cos\varphi & -\sin\varphi & r_1 \cos\varphi + T \sin\varphi \\ \sin\varphi & \cos\varphi & r_1 \sin\varphi – T \cos\varphi \\ 0 & 0 & 1 \end{bmatrix} $$
where $$ \varphi $$ is the rotation angle of the workpiece, $$ r_1 $$ is the pitch radius of the workpiece, and $$ T $$ is the horizontal displacement of the rack tool. Similarly, the transformation matrix to the rolling die coordinate system is:
$$ M_{51} = M_{54} \cdot M_{42} \cdot M_{20} \cdot M_{01} = \begin{bmatrix} \cos\varepsilon & \sin\varepsilon & -r_2 (\cos\varepsilon + \varepsilon \sin\varepsilon) \\ -\sin\varepsilon & \cos\varepsilon & r_2 (\sin\varepsilon – \varepsilon \cos\varepsilon) \\ 0 & 0 & 1 \end{bmatrix} $$
where $$ \varepsilon $$ is the rotation angle of the rolling die, and $$ r_2 $$ is the corresponding pitch radius of the rolling die. The tooth profile equations for the rack tool segments are as follows. For the tooth tip straight line segment (ab):
$$ R^1_{ab} = \begin{bmatrix} R^1_{ab,x} \\ R^1_{ab,y} \\ 1 \end{bmatrix} = \begin{bmatrix} \pm (-\pi m – 4X_1 m \tan\alpha + l_{1ab}) \\ -h_a^* m – X_1 m + r_{bc} \sin\alpha – r_{bc} \\ 1 \end{bmatrix} $$
For the tooth tip arc segment (bc):
$$ R^1_{bc} = \begin{bmatrix} R^1_{bc,x} \\ R^1_{bc,y} \\ 1 \end{bmatrix} = \begin{bmatrix} \pm \left( -\frac{\pi}{2} m + 2X_1 m \tan\alpha – h_a^* m \tan\alpha – X_1 m \tan\alpha – r_{bc} \cos\alpha + r_{bc} \sin\theta_{2bc} \right) \\ -h_a^* m – X_1 m + r_{bc} \sin\alpha – r_{bc} \cos\theta_{2bc} \\ 1 \end{bmatrix} $$
For the tooth flank straight line segment (cd):
$$ R^1_{cd} = \begin{bmatrix} R^1_{cd,x} \\ R^1_{cd,y} \\ 1 \end{bmatrix} = \begin{bmatrix} \pm \left( -\frac{\pi}{2} m + 2X_1 m \tan\alpha – h_a^* m \tan\alpha – X_1 m \tan\alpha + l_{cd} \sin\alpha \right) \\ l_{cd} \cos\alpha – h_a^* m – X_1 m \\ 1 \end{bmatrix} $$
In these equations, $$ m $$ is the module, $$ \alpha $$ is the pressure angle, $$ h_a^* $$ is the addendum coefficient, $$ X_1 $$ is the modification coefficient, $$ r_{bc} $$ is the radius of the tooth tip arc, and $$ l_{1ab} $$, $$ \theta_{2bc} $$, $$ l_{cd} $$ are parameters defining points on the segments. The ± sign applies to left and right tooth flanks, respectively. Using the transformation matrices, the workpiece tooth profile equations are derived by applying the coordinate transformations to the rack tool profiles. For the cutting section, the workpiece tooth profile equations become:
$$ W^1_{ab} = \begin{bmatrix} W^1_{ab,x} \\ W^1_{ab,y} \\ 1 \end{bmatrix} = \begin{bmatrix} R^1_{ab,x} \cos\varphi – R^1_{ab,y} \sin\varphi + r_1 \cos\varphi + T \sin\varphi \\ R^1_{ab,x} \sin\varphi + R^1_{ab,y} \cos\varphi + r_1 \sin\varphi – T \cos\varphi \\ 1 \end{bmatrix} $$
Similarly, for the rolling die, the tooth profile equations in the cutting section are:
$$ B^1_{ab} = \begin{bmatrix} B^1_{ab,x} \\ B^1_{ab,y} \\ 1 \end{bmatrix} = \begin{bmatrix} R^1_{ab,x} \cos\varepsilon + R^1_{ab,y} \sin\varepsilon – r_2 (\cos\varepsilon + \varepsilon \sin\varepsilon) \\ -R^1_{ab,x} \sin\varepsilon + R^1_{ab,y} \cos\varepsilon + r_2 (\sin\varepsilon – \varepsilon \cos\varepsilon) \\ 1 \end{bmatrix} $$
The same approach is applied to derive equations for the finishing and exiting sections of the rolling die. In the finishing section, the modification coefficient is held constant at zero, resulting in a standard tooth profile without variation. The exiting section is designed with a gradually increasing clearance to prevent scratching of the formed tooth surface, using modification coefficients similar to the cutting section but with a decreasing trend.
To validate the design method, we consider a specific gear shaft example with parameters listed in Table 1. The gear shaft has a module of 1.75 mm, pressure angle of 20°, and 46 teeth. The rolling die has 128 teeth, and the face widths for the cutting, finishing, and exiting sections are 40 mm, 25 mm, and 5 mm, respectively. The modification coefficients for the cutting section range from -2 to 0, for the finishing section is 0, and for the exiting section from 0 to -0.5. Using these parameters, we generate discrete points for the tooth profiles in MATLAB and reconstruct the 3D model in SolidWorks. The finite element model is built in Deform software to simulate the rolling process, with the workpiece material set as AL-6061-T6-cold at 20°C. The model includes axial and side baffles to constrain material flow, and mesh refinement is applied near the tooth formation zone to capture deformation accurately.
| Parameter | Value |
|---|---|
| Module (mm) | 1.75 |
| Pressure Angle (°) | 20 |
| Number of Workpiece Teeth | 46 |
| Addendum Coefficient | 1 |
| Number of Rolling Die Teeth | 128 |
| Bottom Clearance Coefficient | 0.25 |
| Modification Coefficient (Cutting Section) | -2 to 0 |
| Face Width (Cutting Section, mm) | 40 |
| Modification Coefficient (Finishing Section) | 0 |
| Face Width (Finishing Section, mm) | 25 |
| Modification Coefficient (Exiting Section) | 0 to -0.5 |
| Face Width (Exiting Section, mm) | 5 |
We analyze three variants of beveloid rolling dies with different modification coefficient ranges: Die I (cutting: -2 to 0, finishing: 0, exiting: 0 to -0.5), Die II (cutting: -1 to 1, finishing: 1, exiting: 1 to 0.5), and Die III (cutting: 0 to 2, finishing: 2, exiting: 2 to 1.5). The effects on forming load, material flow velocity, and tooth profile error are evaluated. The forming load during the rolling process exhibits a single-peak trend, increasing initially as the die penetrates the workpiece and decreasing as the exiting section disengages. The maximum forming loads for Dies I, II, and III are 40,500 N, 40,700 N, and 41,300 N, respectively, indicating that lower modification coefficients in the cutting section reduce the load. This reduction is attributed to decreased radial forces and improved material flow conditions.
Material flow velocity is assessed at points located at different depths from the workpiece surface: P1-P10 on the surface, P11-P20 at 1 mm depth, and P21-P30 at 2 mm depth. The velocity fluctuations are most significant on the surface, with Die I and Die II showing maximum surface velocities of 25 mm/s, while Die III reaches 21.5 mm/s. At 1 mm depth, Die I has a maximum velocity of 5.8 mm/s, compared to 8.5 mm/s for Die II and 8.1 mm/s for Die III. At 2 mm depth, the velocities are 5.4 mm/s, 8.5 mm/s, and 2.8 mm/s for Dies I, II, and III, respectively. These results suggest that Die I promotes more uniform material flow with lower velocity gradients, reducing the risk of defects such as ear formation on the gear shaft.
Tooth profile error is evaluated by comparing the formed tooth profiles with the standard involute profile. The error distribution typically shows a “concave” shape, with significant deviations in the transition arc region. For Die I, the maximum error is 0.18 mm in the transition arc, while for Die II and Die III, the errors are 0.19 mm and 0.23 mm, respectively. Die III also exhibits insufficient tooth tip height due to excessive positive modification. Overall, Die I demonstrates superior performance in terms of forming load, material flow, and profile accuracy, making it the preferred design for the gear shaft rolling process.
Next, we optimize the process parameters using orthogonal experiments. The factors considered are friction factor (A), feed speed (B), and rolling die rotation speed (C), each at three levels as shown in Table 2. The response variables include effective tooth height, forming load, and tooth profile error. The orthogonal array L9 is employed, with nine experimental runs. The results are analyzed using range analysis to determine the optimal levels and factor significance.
| Experiment | A: Friction Factor | B: Feed Speed (mm/s) | C: Rotation Speed (rad/s) |
|---|---|---|---|
| 1 | 0.15 | 0.5 | 1.34 |
| 2 | 0.15 | 0.75 | 4.94 |
| 3 | 0.15 | 1.00 | 3.14 |
| 4 | 0.30 | 0.5 | 4.94 |
| 5 | 0.30 | 0.75 | 3.14 |
| 6 | 0.30 | 1.00 | 1.34 |
| 7 | 0.45 | 0.5 | 3.14 |
| 8 | 0.45 | 0.75 | 1.34 |
| 9 | 0.45 | 1.00 | 4.94 |
The effective tooth height is measured against the standard value of 42 mm. Experiments 1, 3, 5, and 9 achieve the full tooth height, while others fall short due to inadequate parameter matching. Range analysis for effective tooth height (Table 3) shows that rotation speed has the greatest influence (R = 0.0899), followed by friction factor (R = 0.0634) and feed speed (R = 0.05). The optimal levels are A1 (0.15), B1 (0.5 mm/s), and C3 (4.94 rad/s).
| Factor | Level 1 | Level 2 | Level 3 | R | Optimal Level |
|---|---|---|---|---|---|
| A: Friction Factor | 42.0367 | 41.9733 | 41.9733 | 0.0634 | 1 |
| B: Feed Speed (mm/s) | 42.0167 | 41.9667 | 42.0000 | 0.0500 | 1 |
| C: Rotation Speed (rad/s) | 41.9367 | 42.0200 | 42.0267 | 0.0899 | 3 |
Forming load analysis reveals that all experiments exhibit a single-peak trend, with the peak load occurring during the stable forming phase. The maximum loads range from 40,500 N to 45,000 N, depending on the parameter combination. Range analysis for maximum forming load (Table 4) indicates that rotation speed is the most influential factor (R = 1608.36), followed by friction factor (R = 1313.14) and feed speed (R = 977.30). The optimal levels are A1 (0.15), B1 (0.5 mm/s), and C2 (3.14 rad/s).
| Factor | Level 1 | Level 2 | Level 3 | R | Optimal Level |
|---|---|---|---|---|---|
| A: Friction Factor | 43700.03 | 44004.80 | 45013.17 | 1313.14 | 1 |
| B: Feed Speed (mm/s) | 43831.67 | 44077.36 | 44808.97 | 977.30 | 1 |
| C: Rotation Speed (rad/s) | 44078.95 | 43515.34 | 45123.71 | 1608.36 | 2 |
Tooth profile error is evaluated by measuring the deviation from the standard involute profile. The maximum errors occur in the transition arc or tooth tip regions, with values ranging from 0.13 mm to 0.23 mm. Range analysis for maximum profile error (Table 5) shows that rotation speed has the highest influence (R = 0.1424), followed by friction factor (R = 0.1040) and feed speed (R = 0.0483). The optimal levels are A1 (0.15), B1 (0.5 mm/s), and C3 (4.94 rad/s).
| Factor | Level 1 | Level 2 | Level 3 | R | Optimal Level |
|---|---|---|---|---|---|
| A: Friction Factor | 0.1284 | 0.1854 | 0.2325 | 0.1040 | 1 |
| B: Feed Speed (mm/s) | 0.1501 | 0.1984 | 0.1978 | 0.0483 | 1 |
| C: Rotation Speed (rad/s) | 0.2682 | 0.1523 | 0.1258 | 0.1424 | 3 |
Considering all response variables, the optimal process parameters are determined as friction factor 0.15, feed speed 0.5 mm/s, and rotation speed 4.94 rad/s. This combination minimizes forming load, reduces tooth profile error, and achieves the full effective tooth height for the gear shaft. To validate the findings, we conduct rolling experiments using Die I and the optimized parameters. The workpiece material is AL-1100 with an initial diameter of 80.75–80.85 mm and thickness of 25 mm. The formed gear shaft exhibits uniform tooth distribution, complete tooth profiles, and dimensions meeting the requirements, with an average addendum radius of 42.09 mm and dedendum radius of 38.11 mm. The results align with the finite element simulations, confirming the effectiveness of the beveloid rolling die design and process optimization.
In conclusion, we have developed a geometric design method for beveloid rolling dies in cylindrical gear through-feed rolling processes. The design incorporates varying modification coefficients in the cutting, finishing, and exiting sections to enhance material flow and reduce forming loads. Through finite element analysis and orthogonal experiments, we identified Die I (modification coefficients: cutting -2 to 0, finishing 0, exiting 0 to -0.5) as the optimal design, and determined the best process parameters as friction factor 0.15, feed speed 0.5 mm/s, and rotation speed 4.94 rad/s. The experimental results demonstrate that this approach produces high-quality gear shafts with minimal defects, offering a viable solution for industrial applications. Future work could explore the application of this method to helical gears or other complex profiles, and investigate the effects of material properties on forming behavior.