I investigate the small-module helical gear rolling process because helical gears with a module smaller than 1 mm are increasingly important in precision instruments, automotive components, aerospace systems, and medical devices. The demand for high-quality small-module helical gears has grown rapidly with the development of new energy vehicles, medical equipment, and electronic products. Traditional manufacturing methods for these gears include gear shaping, hobbing, multi-axis machining, powder metallurgy, and stamping. However, cutting processes remove material, waste raw material, and interrupt the metal fiber flow, which reduces tooth strength. Powder metallurgy often produces lower-strength parts because of poor powder flowability. Stamping suffers from short die life, difficult demolding, and high manufacturing cost. Therefore, I study a plastic-forming route, namely helical gear rolling, to achieve high efficiency and high quality for small-module helical gears.
In the helical gear rolling process, rolling wheels and the workpiece perform generating motion. The teeth of the rolling wheel envelop the workpiece blank while pressing it, forcing the material to flow plastically and form the gear teeth. Compared with cutting, the process is shorter, avoids material removal, and preserves the continuous metal fiber flow. Compared with powder metallurgy, the rolled helical gear has higher strength. Compared with forging, the forming load is lower, demolding is simpler, and production efficiency is higher. Thus, helical gear rolling has strong potential for mass production of small-module helical gears. I focus on the axial rolling scheme with external meshing double rolling wheels and free indexing. This scheme avoids the large contact area and unstable radial feed of radial rolling, and it avoids the complex tooling of internal meshing. The main challenge is that the helical angle produces a significant axial force during bite-in. If the bite-in condition is not satisfied, the rolling wheel slips, indexing fails, and the workpiece may be damaged. Therefore, I build a mechanical model for the bite-in process, derive the bite conditions, design the rolling wheel, and conduct simulation and experimental research.
The workpiece material I use is a copper-zinc alloy, C27000, supplied by the cooperative company. Its material properties and chemical composition are summarized in Table 1. The target small-module helical gear has a normal module of 0.5 mm, a normal pressure angle of 20°, 11 teeth, a right-hand helix, a helix angle of 20°, a profile shift coefficient of 0, a pitch diameter of 5.853 mm, a root diameter of 4.603 mm, a tip radius of R0.15, and a root radius of R0.15. Table 2 lists the main dimensions. From the volume constancy principle, I determine the initial blank diameter. Assuming that the material only undergoes plastic deformation and no axial movement during rolling, the initial blank radius \(r_1\) is obtained from
$$V = \pi r_1^2 W,$$
where \(V\) is the volume of the formed helical gear and \(W\) is the tooth width. Using the measured end-face area of 26.1 mm\(^2\) from the three-dimensional model, I calculate the initial blank diameter as 5.7636 mm. This value is used in all subsequent simulations and experiments.
| Standard | Type | Relative elongation | Hardness |
|---|---|---|---|
| GB/T 2059-2000 | Copper-zinc alloy | 85–145 HV | — |
Table 1. Material parameters of C27000.
| Main parameter | Specification |
|---|---|
| Normal module | 0.5 mm |
| Normal pressure angle | 20° |
| Number of teeth | 11 |
| Helix direction | Right-hand |
| Helix angle | 20° |
| Profile shift coefficient | 0 |
| Pitch diameter | 5.853 mm |
| Root diameter | 4.603 mm |
| Tip radius | R0.15 |
| Root radius | R0.15 |
Table 2. Workpiece dimension parameters.
For the rolling scheme, I adopt external meshing with two rolling wheels, free indexing, and axial feeding. In free indexing, only the rolling wheel is driven by the motor and transmission mechanism. The workpiece is driven by the friction and pressure from the rolling wheel. This reduces the transmission accuracy requirement but demands a strict bite-in condition. To ensure correct indexing, the two rolling wheels must have the same phase. Before rolling, I adjust the circumferential position of one rolling wheel until the indentations from both wheels coincide. If the phases are inconsistent, the workpiece may suffer from乱齿 or even collapse. Therefore, the bite-in condition is the first critical problem. I analyze the forces on the workpiece when the rolling wheel first contacts the blank. The forces include the normal pressure from the tooth tip, the normal pressure from the tooth side, and the corresponding friction forces. The contact plane is taken as the normal plane of the workpiece helix. I decompose all forces into the axial direction and the tangential direction. The torque condition and the axial force condition are written as
$$M_1 + M_2 + M_{1f} + M_{2f} > 0,$$
$$F_{1z} + F_{2z} + F_{1fz} + F_{2fz} > 0,$$
where \(M_1\) and \(M_2\) are the torques generated by the tooth-side pressure and tooth-tip pressure, \(M_{1f}\) and \(M_{2f}\) are the torques from the corresponding friction forces, and \(F_{iz}\) and \(F_{ifz}\) are the axial components. The first inequality ensures that the workpiece rotates with the rolling wheel without slipping. The second inequality ensures that the workpiece moves axially in the feed direction. If either inequality is violated, the rolling wheel cannot bite in successfully.
To solve the contact stresses, I assume a plane strain state in the normal plane of the helical gear. The material flow is assumed to occur only in that plane, and the strain along the helix direction is neglected. This assumption is supported by finite element simulation results. Under plane strain, the stress state at any point in the plastic zone can be represented by the Mohr circle. The stresses are
$$\sigma_x = \sigma_m – K \sin 2\theta,$$
$$\sigma_y = \sigma_m + K \sin 2\theta,$$
$$\tau_{xy} = K \cos 2\theta,$$
where \(\sigma_m\) is the hydrostatic stress, \(K\) is the maximum shear stress, and \(\theta\) is the angle of the maximum shear stress direction. For a rigid-perfectly plastic material, \(K\) equals the yield shear stress. The slip-line field method is used because it is mathematically rigorous for plane strain problems. The two families of slip lines, \(\alpha\) and \(\beta\), satisfy the Hencky equations:
$$\sigma_m – 2K\theta = \xi \quad \text{on } \alpha \text{ lines},$$
$$\sigma_m + 2K\theta = \eta \quad \text{on } \beta \text{ lines}.$$
Along a slip line, the change in hydrostatic stress is proportional to the change in the tangent angle. I construct the slip-line field for the tooth-tip contact region and the tooth-side contact region. The boundaries include a friction boundary, a free surface, and a uniform stress field. For a general friction boundary, the friction shear stress is
$$\tau_{xy} = mK,$$
where \(m\) is the shear friction factor. The angle of the slip line relative to the contact surface is determined from
$$\theta = \pm \frac{1}{2}\arccos m.$$
Using the Hencky equations and the boundary conditions, I derive the average normal stresses on the tooth-tip contact surface and the tooth-side contact surface. These are denoted as \(p_1\) and \(p_2\), respectively. Substituting them into the torque and axial force integrals gives the bite condition model. The integrals are evaluated along the contact length \(x\) from 0 to \(x_1\). The forces are
$$F_1 = \int_0^{x_1} f(x)\,dx, \quad F_2 = \int_0^{x_1} g(x)\,dx,$$
where \(f(x)\) and \(g(x)\) are the distributed contact pressures. The bite conditions become
$$\int_0^{x_1} f(x) d_3\,dx + \int_0^{x_1} g(x) d_4\,dx > 0,$$
$$\int_0^{x_1} f(x)(1-\mu_1)\,dx + \int_0^{x_1} g(x)(1-\mu_2)\,dx > 0,$$
where \(d_3\) and \(d_4\) are the effective moment arms, and \(\mu_1\) and \(\mu_2\) are the friction coefficients. I solve this model using MATLAB. The results show that the torque and axial force during bite-in depend on the number of teeth of the rolling wheel and the taper angle. The torque decreases as the taper angle increases, but the effect is small. The torque also decreases as the number of teeth increases. For the axial force, the same trend is observed. The axial force decreases with increasing taper angle and with increasing number of teeth. Based on the model, the number of teeth should be less than 119 to ensure that the axial force remains positive. When the number of teeth is 121 or larger, the torque condition may fail. When the number of teeth is 119 or 120, the axial force condition may fail at large taper angles. Therefore, I choose a rolling wheel with 113 teeth and a taper angle of 3.5°.
I also perform finite element simulations using Deform-3D to verify the bite condition model and to study the effects of rolling wheel speed, taper angle, and number of teeth. The simulation model consists of two rolling wheels, a workpiece blank, and a fixed mandrel. The workpiece is set as a plastic body with material C26000, which is similar to C27000. The rolling wheels are set as rigid bodies. The shear friction coefficient between the workpiece and the rolling wheels is 0.2. The mesh number is about 157,353. The rolling wheel rotates at a specified angular velocity, and the workpiece is driven by the rolling wheel. The axial feed speed is set according to the transmission ratio. Table 3 lists the simulation parameters for different numbers of teeth. Table 4 lists the simulation parameters for different taper angles.
| Number of teeth | Transmission ratio | Rolling wheel speed (rad/s) | Workpiece speed (rad/s) | Feed speed (mm/s) |
|---|---|---|---|---|
| 79 | 11/79 | 0.5 | 3.5909 | 1.2832 |
| 87 | 11/87 | 0.5 | 3.9545 | 1.4132 |
| 99 | 11/99 | 0.5 | 4.5000 | 1.6080 |
| 111 | 11/111 | 0.5 | 5.0454 | 1.8030 |
Table 3. Simulation parameters for different numbers of teeth.
| Taper angle | Rolling wheel speed (rad/s) | Workpiece speed (rad/s) | Feed speed (mm/s) |
|---|---|---|---|
| 3.5° | 0.5 | 4.5000 | 1.6080 |
| 4.5° | 0.5 | 4.5000 | 1.6080 |
| 5.5° | 0.5 | 4.5000 | 1.6080 |
Table 4. Simulation parameters for different taper angles.
The simulation results show that the workpiece material flows along the normal direction of the contact surface, which confirms the plane strain assumption. The stress distribution on the workpiece during bite-in is almost the same for single-wheel and double-wheel rolling, which means that the effect of the opposite wheel can be neglected in the normal plane. The torque on the workpiece during bite-in decreases as the number of teeth increases. For example, when the number of teeth is 79, the maximum torque is 90.9 N·mm; for 87 teeth, it is 75.4 N·mm; for 99 teeth, it is 61.9 N·mm; and for 111 teeth, it is 54.8 N·mm. These results agree with the bite condition model. The axial force during the entire rolling process also decreases slightly as the number of teeth increases, but the maximum axial force changes little. For the taper angle, the simulation results show that the torque during bite-in decreases as the taper angle increases. When the taper angle is 3.5°, the maximum torque is 62.7 N·mm; for 4.5°, it is 61.4 N·mm; and for 5.5°, it is 58.8 N·mm. The axial force during bite-in also decreases with increasing taper angle. However, the maximum axial force during the entire rolling process increases slightly with increasing taper angle. The axial displacement of the workpiece material shows that a larger taper angle increases the effective length of the formed helical gear, but the effect is not dominant. Based on these results, I select a small taper angle to ensure stable bite-in and sufficient finishing length.
The rolling wheel speed is an important process parameter. I simulate different rolling wheel speeds: 0.3, 0.5, 0.7, and 0.9 rad/s. The results show that as the rolling wheel speed increases, the axial force during bite-in and the maximum axial force during rolling both increase. For example, at 0.3 rad/s, the axial force at maximum penetration is 8.7 N; at 0.5 rad/s, it is 15.8 N; at 0.7 rad/s, it is 19.6 N; and at 0.9 rad/s, it is 22 N. The maximum axial force also increases from 659.6 N to 719.0 N. This is because a higher rolling wheel speed leads to a higher axial feed speed, which increases the penetration depth per unit time and intensifies material flow. Therefore, the forming load increases. Although a higher speed can improve productivity, it may reduce forming accuracy and accelerate tool wear. I therefore need to determine the maximum speed that still satisfies the precision requirement.
For the rolling wheel design, I first derive the tooth profile equations. The helical gear tooth surface is generated by an involute curve that rotates along the helix. The right-side tooth surface in the workpiece coordinate system is
$$x_1 = r_b \sin\theta_K – r_b\theta_K \cos\theta_K,$$
$$y_1 = r_b \sin(\theta_K + \varphi) – r_b\theta_K \cos(\theta_K + \varphi),$$
$$z_1 = \sqrt{x_1^2 + y_1^2} \tan\beta,$$
where \(r_b\) is the base radius, \(\theta_K\) is the involute parameter, \(\varphi\) is the initial angle, and \(\beta\) is the helix angle. The left-side tooth surface is obtained similarly. Using the meshing principle, the rolling wheel tooth surface is derived by coordinate transformation and the meshing equation. The meshing equation is
$$\mathbf{v}_0 \cdot \mathbf{n}_{14} = 0,$$
where \(\mathbf{v}_0\) is the relative velocity and \(\mathbf{n}_{14}\) is the normal vector of the workpiece tooth surface. By transforming the workpiece tooth surface to the rolling wheel coordinate system, I obtain the normal tooth profile of the rolling wheel. The rolling wheel material is selected as DC53 cold-work die steel. After heat treatment, its hardness can reach HRC 62–63 while maintaining good toughness. Table 5 lists the material properties of DC53.
| Parameter | Value |
|---|---|
| Elastic modulus | 212.64 GPa |
| Yield strength | 1442 MPa |
Table 5. Material properties of DC53.
The number of teeth of the rolling wheel should be mutually prime with the number of teeth of the workpiece to avoid periodic errors. It should also be large enough to reduce wear and improve tooth profile accuracy, but not so large that the bite condition fails. Based on the bite condition model and simulation results, I choose 113 teeth. The tooth width is 20 mm, and the taper angle is 3.5°. I check the bending fatigue strength of the rolling wheel tooth root. The bending fatigue strength condition for a helical gear is
$$\sigma_F = \frac{2 K_F T_1 Y_{Fa} Y_{Sa} Y_\epsilon Y_\beta}{m_n^3 z_1^2 \phi_{Da}} \leq [\sigma_F],$$
where \(K_F\) is the load factor, \(T_1\) is the torque, \(Y_{Fa}\) is the tooth form factor, \(Y_{Sa}\) is the stress correction factor, \(Y_\epsilon\) is the contact ratio factor, \(Y_\beta\) is the helix angle factor, \(m_n\) is the normal module, \(z_1\) is the number of teeth, and \(\phi_{Da}\) is the face width factor. The values are listed in Table 6. The load factor is calculated from
$$K_F = K_A K_V K_{Fa} K_{F\beta},$$
where the values are given in Table 7. The calculated bending stress is 92.799 MPa, which is much lower than the allowable stress of 350 MPa. Therefore, the rolling wheel tooth root strength is sufficient.
| Parameter | Value |
|---|---|
| \(Y_{Fa}\) | 2.28 |
| \(Y_{Sa}\) | 1.73 |
| \(Y_\epsilon\) | 0.68 |
| \(Y_\beta\) | 0.90 |
| \(\phi_{Da}\) | 0.72 |
| \(m_n\) | 0.5 mm |
| \(z_1\) | 99 |
| \([\sigma_F]\) | 350 MPa |
Table 6. Parameters for bending fatigue strength calculation.
| Parameter | Value |
|---|---|
| \(K_A\) | 1.5 |
| \(K_V\) | 1.18 |
| \(K_{Fa}\) | 1.2 |
| \(K_{F\beta}\) | 1.13 |
Table 7. Load factor parameters.
For the experimental research, I use a horizontal thread rolling CNC machine. The main technical parameters are listed in Table 8. The machine uses free indexing. I adjust the phase of the two rolling wheels before rolling. The workpiece is placed in the entry section of the rolling wheels. I move the rolling wheels radially until a clear indentation appears on the workpiece surface. Then I rotate the spindle to make the workpiece rotate more than half a turn. If the indentations from the two wheels coincide, the phases are consistent. Otherwise, I repeat the adjustment. After phase adjustment, I conduct rolling experiments at different rolling wheel speeds: 20, 25, 30, and 35 rpm. The workpiece blank length is chosen as 12 mm to reduce the end-face material loss. The simulation results show that the effective length ratio increases from 65.227% for an 8 mm blank to 93.665% for a 12 mm blank. Therefore, a longer blank improves the effective length of the formed helical gear.
| Parameter | Specification |
|---|---|
| Workpiece size range | 3–80 mm |
| Maximum rolling wheel diameter | 235 mm |
| Spindle speed | 0–150 rpm |
| Spindle deflection angle | ±5° |
| Phase adjustment accuracy | ±5′ |
| Maximum rolling force | 235 kN |
Table 8. Main technical parameters of the experimental platform.
I measure the tooth profile error and tooth trace error of the rolled helical gears. The results for different rolling wheel speeds are summarized in Tables 9–12. For 20 rpm, the maximum left-side profile error is 5.3 µm, which satisfies grade 6; the average left-side profile error is 3.5 µm, which satisfies grade 5. The maximum right-side profile error is 5.7 µm, which satisfies grade 8; the average right-side profile error is 4.5 µm, which satisfies grade 7. For 25 rpm, the left-side maximum profile error is 10.2 µm (grade 8), and the right-side maximum is 10.5 µm (grade 8). For 30 rpm, the left-side maximum profile error is 14.5 µm (grade 10), and the right-side maximum is 11.5 µm (grade 8). For 35 rpm, the left-side maximum profile error is 23.0 µm (grade 11), and the right-side maximum is 17.9 µm (grade 9). The tooth trace errors show a similar trend. At 20 rpm, the maximum left-side trace error is 14.8 µm (grade 8), and the average is 5.8 µm (grade 5). At 25 rpm, the maximum left-side trace error is 12.6 µm (grade 8), and the average is 10.0 µm (grade 7). At 30 rpm, the maximum left-side trace error is 20.6 µm (grade 9), and the average is 12.8 µm (grade 8). At 35 rpm, the maximum left-side trace error is 26.6 µm (grade 10), and the average is 12.2 µm (grade 8).
| Item | Profile error (µm) | DIN grade |
|---|---|---|
| Left 001 | 3.4 | 5 |
| Left 004 | 3.7 | 5 |
| Left 007 | 2.9 | 5 |
| Left 009 | 5.3 | 6 |
| Left average | 3.5 | 5 |
| Right 001 | 5.3 | 7 |
| Right 004 | 4.6 | 7 |
| Right 007 | 3.0 | 5 |
| Right 009 | 5.7 | 8 |
| Right average | 4.5 | 7 |
Table 9. Tooth profile error results at 20 rpm.
| Item | Profile error (µm) | DIN grade |
|---|---|---|
| Left 001 | 4.8 | 7 |
| Left 004 | 10.2 | 8 |
| Left 007 | 7.5 | 7 |
| Left 009 | 3.2 | 6 |
| Left average | 5.4 | 6 |
| Right 001 | 4.8 | 7 |
| Right 004 | 7.4 | 7 |
| Right 007 | 10.5 | 8 |
| Right 009 | 4.8 | 6 |
| Right average | 6.5 | 7 |
Table 10. Tooth profile error results at 25 rpm.
| Item | Profile error (µm) | DIN grade |
|---|---|---|
| Left 001 | 9.4 | 8 |
| Left 004 | 12.4 | 9 |
| Left 007 | 14.5 | 10 |
| Left 009 | 11.2 | 9 |
| Left average | 11.8 | 9 |
| Right 001 | 6.9 | 7 |
| Right 004 | 7.4 | 8 |
| Right 007 | 11.2 | 8 |
| Right 009 | 11.5 | 8 |
| Right average | 9.0 | 8 |
Table 11. Tooth profile error results at 30 rpm.
| Item | Profile error (µm) | DIN grade |
|---|---|---|
| Left 001 | 12.3 | 9 |
| Left 004 | 8.5 | 8 |
| Left 007 | 18.1 | 10 |
| Left 009 | 6.5 | 5 |
| Left average | 10.0 | 7 |
| Right 001 | 6.7 | 6 |
| Right 004 | 7.4 | 6 |
| Right 007 | 7.5 | 6 |
| Right 009 | 3.1 | 3 |
| Right average | 6.2 | 5 |
Table 12. Tooth profile error results at 35 rpm.
For the tooth trace error, the results at 20 rpm show that the maximum left-side trace error is 14.8 µm (grade 8), and the average is 5.8 µm (grade 5). The right-side maximum is 12.7 µm (grade 7), and the average is 5.3 µm (grade 5). At 25 rpm, the left-side maximum is 12.6 µm (grade 8), and the average is 10.0 µm (grade 7). The right-side maximum is 7.5 µm (grade 6), and the average is 6.2 µm (grade 5). At 30 rpm, the left-side maximum is 20.6 µm (grade 9), and the average is 12.8 µm (grade 8). The right-side maximum is 18.9 µm (grade 9), and the average is 10.3 µm (grade 7). At 35 rpm, the left-side maximum is 26.6 µm (grade 10), and the average is 12.2 µm (grade 8). The right-side maximum is 20.5 µm (grade 9), and the average is 11.9 µm (grade 8).
| Item | Trace error (µm) | DIN grade |
|---|---|---|
| Left 001 | 14.8 | 8 |
| Left 004 | 7.3 | 6 |
| Left 007 | 4.2 | 4 |
| Left 009 | 5.7 | 5 |
| Left average | 5.8 | 5 |
| Right 001 | 12.0 | 7 |
| Right 004 | 12.7 | 8 |
| Right 007 | 3.0 | 3 |
| Right 009 | 2.8 | 3 |
| Right average | 5.3 | 5 |
Table 13. Tooth trace error results at 20 rpm.
| Item | Trace error (µm) | DIN grade |
|---|---|---|
| Left 001 | 9.8 | 7 |
| Left 004 | 12.6 | 8 |
| Left 007 | 10.9 | 7 |
| Left 009 | 6.5 | 5 |
| Left average | 10.0 | 7 |
| Right 001 | 6.7 | 6 |
| Right 004 | 7.4 | 6 |
| Right 007 | 7.5 | 6 |
| Right 009 | 3.1 | 3 |
| Right average | 6.2 | 5 |
Table 14. Tooth trace error results at 25 rpm.
| Item | Trace error (µm) | DIN grade |
|---|---|---|
| Left 001 | 9.5 | 7 |
| Left 004 | 1.8 | 1 |
| Left 007 | 19.4 | 9 |
| Left 009 | 20.6 | 9 |
| Left average | 12.8 | 8 |
| Right 001 | 13.1 | 8 |
| Right 004 | 3.2 | 4 |
| Right 007 | 12.1 | 8 |
| Right 009 | 18.9 | 9 |
| Right average | 10.3 | 7 |
Table 15. Tooth trace error results at 30 rpm.
| Item | Trace error (µm) | DIN grade |
|---|---|---|
| Left 001 | 12.6 | 8 |
| Left 004 | 5.2 | 5 |
| Left 007 | 14.4 | 8 |
| Left 009 | 26.6 | 10 |
| Left average | 12.2 | 8 |
| Right 001 | 7.0 | 6 |
| Right 004 | 4.4 | 4 |
| Right 007 | 19.2 | 9 |
| Right 009 | 20.5 | 9 |
| Right average | 11.9 | 8 |
Table 16. Tooth trace error results at 35 rpm.
The experimental results show that when the rolling wheel speed is 20 or 25 rpm, the formed helical gears meet the grade 8 accuracy requirement, and after subsequent grinding they can reach grade 7. When the speed is 30 or 35 rpm, the tooth profile and tooth trace errors increase significantly, and the accuracy degrades to grade 9 or worse. Moreover, at 30 rpm, I observe visible wear on the rolling wheel teeth after processing 59 workpieces of 2 m length. The wear reduces the forming accuracy further. Therefore, considering productivity, tool life, and accuracy, I select 20 rpm as the rolling wheel speed for the small-module helical gear rolling process. At this speed, the bite condition is satisfied, the forming load is moderate, and the rolled helical gears meet the required accuracy.
In summary, I have studied the small-module helical gear rolling process through theoretical analysis, finite element simulation, and experiments. I established a bite condition model based on the forces during bite-in, used the slip-line field method to calculate contact stresses, and solved the model with MATLAB. The results show that the number of teeth and the taper angle of the rolling wheel significantly affect the bite condition. I designed a rolling wheel with 113 teeth, a taper angle of 3.5°, a tooth width of 20 mm, and DC53 material. The tooth root bending fatigue strength is sufficient. The experimental results confirm that the designed rolling wheel and process parameters can produce small-module helical gears with grade 8 accuracy at a rolling wheel speed of 20 rpm. This research provides a theoretical basis and practical guidance for the industrial production of small-module helical gears by rolling. The helical gear rolling process is efficient, material-saving, and capable of producing strong gears with continuous fiber flow. Future work can further optimize the taper angle and number of teeth over a wider range, improve the slip-line field model for transient contact, and extend the process to other small-module helical gear sizes.