In modern mechanical transmission systems, high-quality cylindrical gear shafts serve as critical components in aerospace, marine, and automotive applications due to their high power transmission capacity, efficiency, and superior service performance. Traditional manufacturing methods such as hobbing, shaping, and grinding often involve significant material waste and lower production efficiency. In contrast, rolling forming technology, as a near-net-shape processing method, offers advantages like minimal material loss, high forming efficiency, and enhanced fatigue resistance. However, conventional axial rolling tools face challenges such as substantial tooth sliding, difficult material flow, and poor stress conditions during the forming process. To address these issues, this study focuses on the design and application of variable tooth thickness rolling tools for the axial rolling gradient forming of cylindrical gear shafts. The research encompasses the derivation of tooth profile equations, finite element simulation of the forming process, and experimental validation using a custom-built rolling platform.
The design of variable tooth thickness rolling tools involves three distinct sections: the cutting-in section, the finishing section, and the exit section. The tooth thickness, addendum, and dedendum vary axially in the cutting-in and exit sections, enabling gradual engagement and disengagement with the gear shaft workpiece. This design mitigates issues like excessive sliding and uneven material flow. The minimum displacement coefficient for the cutting-in section is crucial to prevent interference between the workpiece and the small end of the rolling tool. For a gear shaft with module m = 1.75 mm, workpiece tooth count z’₁ = 46, workpiece diameter D = Φ80.75 mm, rolling tool tooth count z’₂ = 128, addendum coefficient h*ₐ = 1, and tip clearance coefficient c* = 0.25, the minimum displacement coefficient is calculated as -1.32. To ensure non-interference, a value of -2 is adopted for the cutting-in section. The tooth profile equations for each section are derived based on the rack-gear engagement principle, facilitating the construction of a three-dimensional model of the variable tooth thickness rolling tool.
The tooth profile equations for the cutting-in, finishing, and exit sections are expressed as follows. For the cutting-in section, the profiles for segments AB (tip line), BC (tip arc), and CD (involute) are given by:
$$ B^{1}_{AB} = \begin{bmatrix} B^{1x}_{AB} \\ B^{1y}_{AB} \\ 1 \end{bmatrix} = \begin{bmatrix} R^{1x}_{ab} \cos \epsilon + R^{1y}_{ab} \sin \epsilon – r_2 (\cos \epsilon + \epsilon \sin \epsilon) \\ -R^{1x}_{ab} \sin \epsilon + R^{1y}_{ab} \cos \epsilon + r_2 (\sin \epsilon – \epsilon \cos \epsilon) \\ 1 \end{bmatrix} $$
$$ B^{1}_{BC} = \begin{bmatrix} B^{1x}_{BC} \\ B^{1y}_{BC} \\ 1 \end{bmatrix} = \begin{bmatrix} R^{1x}_{bc} \cos \epsilon + R^{1y}_{bc} \sin \epsilon – r_2 (\cos \epsilon + \epsilon \sin \epsilon) \\ -R^{1x}_{bc} \sin \epsilon + R^{1y}_{bc} \cos \epsilon + r_2 (\sin \epsilon – \epsilon \cos \epsilon) \\ 1 \end{bmatrix} $$
$$ B^{1}_{CD} = \begin{bmatrix} B^{1x}_{CD} \\ B^{1y}_{CD} \\ 1 \end{bmatrix} = \begin{bmatrix} R^{1x}_{cd} \cos \epsilon + R^{1y}_{cd} \sin \epsilon – r_2 (\cos \epsilon + \epsilon \sin \epsilon) \\ -R^{1x}_{cd} \sin \epsilon + R^{1y}_{cd} \cos \epsilon + r_2 (\sin \epsilon – \epsilon \cos \epsilon) \\ 1 \end{bmatrix} $$
For the finishing section, where no displacement occurs, the profiles are:
$$ B^{2}_{AB} = \begin{bmatrix} B^{2x}_{AB} \\ B^{2y}_{AB} \\ 1 \end{bmatrix} = \begin{bmatrix} R^{1x}_{ab} \cos \epsilon + R^{1y}_{ab} \sin \epsilon – r_2 (\cos \epsilon + \epsilon \sin \epsilon) \\ -R^{1x}_{ab} \sin \epsilon + R^{1y}_{ab} \cos \epsilon + r_2 (\sin \epsilon – \epsilon \cos \epsilon) \\ 1 \end{bmatrix} $$
$$ B^{2}_{BC} = \begin{bmatrix} B^{2x}_{BC} \\ B^{2y}_{BC} \\ 1 \end{bmatrix} = \begin{bmatrix} R^{1x}_{bc} \cos \epsilon + R^{1y}_{bc} \sin \epsilon – r_2 (\cos \epsilon + \epsilon \sin \epsilon) \\ -R^{1x}_{bc} \sin \epsilon + R^{1y}_{bc} \cos \epsilon + r_2 (\sin \epsilon – \epsilon \cos \epsilon) \\ 1 \end{bmatrix} $$
$$ B^{2}_{CD} = \begin{bmatrix} B^{2x}_{CD} \\ B^{2y}_{CD} \\ 1 \end{bmatrix} = \begin{bmatrix} R^{1x}_{cd} \cos \epsilon + R^{1y}_{cd} \sin \epsilon – r_2 (\cos \epsilon + \epsilon \sin \epsilon) \\ -R^{1x}_{cd} \sin \epsilon + R^{1y}_{cd} \cos \epsilon + r_2 (\sin \epsilon – \epsilon \cos \epsilon) \\ 1 \end{bmatrix} $$
For the exit section, with increasing negative displacement, the equations are:
$$ B^{3}_{AB} = \begin{bmatrix} B^{3x}_{AB} \\ B^{3y}_{AB} \\ 1 \end{bmatrix} = \begin{bmatrix} R^{1x}_{ab} \cos \epsilon + R^{1y}_{ab} \sin \epsilon – r_2 (\cos \epsilon + \epsilon \sin \epsilon) \\ -R^{1x}_{ab} \sin \epsilon + R^{1y}_{ab} \cos \epsilon + r_2 (\sin \epsilon – \epsilon \cos \epsilon) \\ 1 \end{bmatrix} $$
$$ B^{3}_{BC} = \begin{bmatrix} B^{3x}_{BC} \\ B^{3y}_{BC} \\ 1 \end{bmatrix} = \begin{bmatrix} R^{1x}_{bc} \cos \epsilon + R^{1y}_{bc} \sin \epsilon – r_2 (\cos \epsilon + \epsilon \sin \epsilon) \\ -R^{1x}_{bc} \sin \epsilon + R^{1y}_{bc} \cos \epsilon + r_2 (\sin \epsilon – \epsilon \cos \epsilon) \\ 1 \end{bmatrix} $$
$$ B^{3}_{CD} = \begin{bmatrix} B^{3x}_{CD} \\ B^{3y}_{CD} \\ 1 \end{bmatrix} = \begin{bmatrix} R^{1x}_{cd} \cos \epsilon + R^{1y}_{cd} \sin \epsilon – r_2 (\cos \epsilon + \epsilon \sin \epsilon) \\ -R^{1x}_{cd} \sin \epsilon + R^{1y}_{cd} \cos \epsilon + r_2 (\sin \epsilon – \epsilon \cos \epsilon) \\ 1 \end{bmatrix} $$
These equations are utilized to generate discrete points for the tooth profiles, which are then assembled to form the complete three-dimensional model of the variable tooth thickness rolling tool. The geometric parameters of the rolling tool are summarized in Table 1.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Module m (mm) | 1.75 | Pressure angle α (°) | 20 |
| Workpiece tooth count z’₁ | 46 | Addendum coefficient h*ₐ | 1 |
| Rolling tool tooth count z’₂ | 128 | Tip clearance coefficient c* | 0.25 |
| Cutting-in section displacement coefficient X₁ | -2 to 0 | Cutting-in section width b₁ (mm) | 40 |
| Finishing section displacement coefficient X₂ | 0 | Finishing section width b₂ (mm) | 25 |
| Exit section displacement coefficient X₃ | -0.5 to 0 | Exit section width b₃ (mm) | 5 |
The finite element simulation of the axial rolling process for the gear shaft is conducted using DEFORM-3D software. The model includes the rolling tool, workpiece, core rod, axial baffle, and circumferential baffle. The workpiece is fixed, while the rolling tool rotates both around its own axis and the workpiece axis. The axial motion of the workpiece is equivalently represented by the reverse motion of the rolling tool. The workpiece material is AL-6061-T6-cold aluminum alloy at 20°C, discretized with 92,000 tetrahedral elements. Mesh refinement is applied to the outer 4 mm region to capture detailed deformation. The friction between the rolling tool and workpiece is modeled as shear friction with a coefficient of 0.15, while other contacts are frictionless. The rotational speed of the rolling tool is set to 13.75 rad/s for revolution and 4.94 rad/s for rotation, with an axial feed rate of 0.5 mm/s.
The rolling process is divided into four stages: tooth cutting, forming, finishing, and exit. During tooth cutting, the small end of the variable tooth thickness rolling tool gradually contacts the gear shaft workpiece, creating uniform grooves on the surface. In the forming stage, the displacement coefficient increases, leading to deeper penetration and plastic deformation that shapes the gear teeth. The finishing stage involves constant displacement coefficients, resulting in improved dimensional accuracy and surface quality. The exit stage allows for gradual separation, minimizing damage to the formed teeth. Stress and strain distributions are analyzed throughout these stages. Equivalent stress is concentrated at the contact regions during tooth cutting and increases during forming, becoming uniform in the finishing stage before decreasing upon exit. Similarly, equivalent strain is minimal initially but rises significantly during forming, particularly at the tooth tips and roots, and stabilizes during finishing.
Material flow behavior is investigated by monitoring points at different depths on the gear shaft workpiece. Points are categorized into surface layer (P1–P10), 1 mm depth (P11–P20), and 2 mm critical depth (P21–P30). The results indicate that material flow is more pronounced near the surface, with greater deformation in streamlines. The left side of each tooth profile experiences more significant flow changes than the right side due to frictional effects. A comparison between variable tooth thickness rolling tools and conventional rolling tools reveals superior performance in contact uniformity, material flow velocity, and stress conditions. The maximum material flow velocity for the variable tool is 11.71592 mm/s, which is 25.2% higher than that of the conventional tool. Additionally, the maximum forming force for the variable tool is 51.728 kN, compared to 67.896 kN for the conventional tool, representing a 31.26% reduction, which enhances process stability and gear shaft quality.
Experimental validation is performed on a custom-built axial rolling platform equipped with dual rolling tools, PLC-controlled servo drives, and an electric cylinder for axial feed. Workpieces with tooth counts z’₁ = 42, 46, and 50 are formed using the variable tooth thickness rolling tool. The formed gear shafts are measured using a 2D measuring instrument to evaluate tooth thickness, addendum circle radius, and dedendum circle radius. The theoretical tooth thickness is 2.749 mm for all workpieces. The average measured values and errors are summarized in Table 2. For z’₁ = 50, the average tooth thickness is 2.745 mm (error -0.004 mm), addendum circle radius is 45.575 mm (error +0.075 mm), and dedendum circle radius is 41.6038 mm (error +0.0413 mm). For z’₁ = 46, the averages are 2.759 mm (error +0.010 mm), 42.0339 mm (error +0.0339 mm), and 38.0241 mm (error -0.0384 mm). For z’₁ = 42, the averages are 2.754 mm (error +0.005 mm), 38.25 mm (error -0.250 mm), and 34.5319 mm (error -0.0306 mm). The maximum errors are +0.010 mm for tooth thickness, -0.250 mm for addendum circle radius, and +0.0413 mm for dedendum circle radius, indicating good agreement with simulation predictions and verifying the reliability of the process for producing high-quality gear shafts.
| Tooth Count z’₁ | Theoretical Tooth Thickness (mm) | Average Measured Tooth Thickness (mm) | Error (mm) | Theoretical Addendum Radius (mm) | Average Measured Addendum Radius (mm) | Error (mm) | Theoretical Dedendum Radius (mm) | Average Measured Dedendum Radius (mm) | Error (mm) |
|---|---|---|---|---|---|---|---|---|---|
| 50 | 2.749 | 2.745 | -0.004 | 45.5 | 45.575 | +0.075 | 41.5625 | 41.6038 | +0.0413 |
| 46 | 2.749 | 2.759 | +0.010 | 42.0 | 42.0339 | +0.0339 | 38.0625 | 38.0241 | -0.0384 |
| 42 | 2.749 | 2.754 | +0.005 | 38.5 | 38.25 | -0.250 | 34.5625 | 34.5319 | -0.0306 |
The tooth thickness tolerances are calculated based on mechanical design standards to assess the precision of the formed gear shafts. For z’₁ = 42, the 9th-grade tolerance is 95.872 μm and the 10th-grade is 127.534 μm; for z’₁ = 46 and 50, the 8th-grade tolerance is 86.799 μm, 9th-grade is 112.799 μm, and 10th-grade is 144.147 μm. The experimental results show that the gear shaft with z’₁ = 50 meets 10th-grade accuracy, z’₁ = 46 meets 9th-grade accuracy, and z’₁ = 42 meets 10th-grade accuracy, demonstrating the capability of the process to produce gear shafts with acceptable precision levels.
In conclusion, the axial rolling gradient forming process using variable tooth thickness rolling tools effectively addresses the limitations of conventional methods. The derivation of tooth profile equations and determination of the minimum displacement coefficient ensure non-interference and optimal tool design. Finite element simulations reveal that stress and strain distributions evolve gradually, with material flow being more intense near the surface and asymmetric between tooth profiles. Experimental results confirm that the formed gear shafts exhibit minimal deviations in tooth thickness, addendum circle radius, and dedendum circle radius, validating the process reliability. Future work could focus on optimizing process parameters and tool designs to further improve the consistency and accuracy of gear shaft manufacturing. The integration of simulation and experimental approaches provides a comprehensive framework for advancing rolling forming technologies in the production of high-performance gear shafts.