Internal gears are critical components in various mechanical systems, such as planetary gearboxes and compact drives, due to their high load-bearing capacity and efficient power transmission. However, the manufacturing of internal gears, especially double-circular-arc profiles, presents significant challenges for internal gear manufacturers, including difficulties in achieving high precision, poor surface finish, and low processing efficiency. Traditional methods like gear shaping and hobbing often fall short in meeting the demands of modern applications, leading to increased costs and extended production times. To address these issues, we propose the use of power skiving, an advanced machining technique that combines the benefits of gear hobbing and shaping, offering a viable solution for internal gears production. This study focuses on developing a comprehensive approach for power skiving double-circular-arc internal gears, encompassing mathematical modeling, simulation, and experimental validation to enhance the capabilities of internal gear manufacturers.
Power skiving involves the relative motion between a cutting tool and a workpiece with intersecting axes, enabling efficient material removal and high-precision gear profiling. For internal gears, the process requires careful control of parameters such as the shaft angle and center distance to ensure accurate tooth formation. The shaft angle Σ is defined as the difference between the helical angles of the tool and the workpiece, given by the equation: $$\Sigma = | \beta_1 – \beta_2 |$$ where β₁ is the skiving tool helical angle and β₂ is the internal gear helical angle. The center distance a between the tool and the internal gear is calculated as: $$a = r_2 – r_1$$ where r₂ is the pitch radius of the internal gear and r₁ is the pitch radius of the skiving tool. These parameters are crucial for internal gear manufacturers to optimize the machining process and achieve desired gear quality.
The kinematic relationship in power skiving dictates that the angular velocities of the tool and workpiece, along with the feed rate, must satisfy specific conditions to maintain synchronization. The angular velocity of the skiving tool ω₁ is related to the workpiece angular velocity ω₂ and the feed velocity v₁ by: $$\omega_1 = i_{21} \omega_2 – \frac{\sin \beta_2}{r_1 \cos \beta_1} v_1$$ where i₂₁ is the transmission ratio. This equation ensures that the tool and workpiece move in a coordinated manner, facilitating the generation of the double-circular-arc profile. For internal gears, this coordination is essential to avoid undercutting and ensure uniform tooth depth, which is a common concern for internal gear manufacturers.
To model the power skiving process mathematically, we establish coordinate systems that describe the relative positions and motions of the tool and workpiece. Let S (O-XYZ) be the fixed coordinate system of the skiving tool, with the Z-axis aligned with the tool’s rotation axis. Similarly, S_p (O_p-X_pY_pZ_p) is the fixed coordinate system of the workpiece, with the Z_p-axis along its rotation axis. The axes Z and Z_p intersect at an angle Σ, with a minimum distance a. The moving coordinate systems S₁ (O₁-x₁y₁z₁) for the tool and S₂ (O₂-x₂y₂z₂) for the workpiece account for their rotations and translations during machining. The transformation from the workpiece coordinate system S₂ to the tool coordinate system S₁ is expressed as: $$[x_1, y_1, z_1, 1]^T = M_{10} M_{0p} M_{p2} [x_2, y_2, z_2, 1]^T$$ where the transformation matrices are defined as: $$M_{10} = \begin{bmatrix} \cos \phi_1 & \sin \phi_1 & 0 & 0 \\ -\sin \phi_1 & \cos \phi_1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}, \quad M_{0p} = \begin{bmatrix} 1 & 0 & 0 & -a \\ 0 & \cos \Sigma & \sin \Sigma & 0 \\ 0 & -\sin \Sigma & \cos \Sigma & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}, \quad M_{p2} = \begin{bmatrix} \cos \phi_2 & \sin \phi_2 & 0 & 0 \\ -\sin \phi_2 & \cos \phi_2 & 0 & 0 \\ 0 & 0 & 1 & l \\ 0 & 0 & 0 & 1 \end{bmatrix}$$ Here, φ₁ and φ₂ represent the rotation angles of the tool and workpiece, respectively, and l is the axial feed distance. This mathematical framework allows internal gear manufacturers to simulate and optimize the gear generation process, ensuring that the double-circular-arc profile meets design specifications.
In the simulation phase, we utilized Vericut software to model the tooth formation process for double-circular-arc internal gears. The machine tool setup included linear axes (X, Y, Z) for tool positioning and rotational axes (U, V, W) for tool and workpiece rotations. The skiving tool was a straight-tooth cutter with 24 teeth, while the internal gear had 76 teeth and a helical angle of 20°. The shaft angle Σ was set to 20° based on the helical angles, and the center distance was adjusted accordingly. The workpiece was moved axially over a distance of 35 mm, with a tool rotational speed of 50 rpm, corresponding workpiece speed of 15.1315 rpm, and an axial feed rate of 0.01 mm/min. The simulation demonstrated the progressive formation of the double-circular-arc tooth profile, confirming that the generated gear teeth adhered to the standard geometric requirements. This validation is crucial for internal gear manufacturers to predict machining outcomes and reduce trial-and-error costs.
For cutting force analysis, we employed Abaqus explicit dynamics module to simulate the power skiving process. A simplified model with a 5-tooth skiving tool and a 5-tooth internal gear segment with 1 mm of stock allowance was used to reduce computational complexity. The tool was modeled as a rigid body, while the workpiece was defined as a deformable body with material properties accounting for plasticity. Meshing was refined near the cutting edges, with element sizes of 0.5 mm for the tool and 0.1 mm for the workpiece to capture accurate stress distributions. The simulation parameters included a tool rotational speed of 600 rpm, workpiece rotational speed corresponding to the transmission ratio, and an axial feed rate of 100 mm/s. The friction between the tool and workpiece was defined using a shear friction model with a coefficient of 0.23, and thermal properties were set to simulate heat generation during cutting. The simulation time was set to 0.2 seconds with 100 increments to analyze transient cutting forces. The results showed varying stress patterns as different tool teeth engaged with the workpiece, with peak forces occurring when both left and right cutting edges were fully engaged. The cutting forces were decomposed into three components: main cutting force F_z, radial force F_y, and axial force F_x, with the resultant force F_n calculated as: $$F_n = \sqrt{F_x^2 + F_y^2 + F_z^2}$$ This analysis helps internal gear manufacturers understand force distributions and optimize tool design to minimize wear and improve surface quality for internal gears.
To investigate the influence of machining parameters on cutting forces, we conducted an orthogonal experiment using an L9(3^4) array, with three factors at three levels each. The factors included skiving tool rotational speed (A), workpiece axial feed rate (B), and depth of cut (C), while the response variable was the maximum resultant cutting force. The experimental design and results are summarized in the table below:
| Run | A: Speed (rpm) | B: Feed Rate (mm/s) | C: Depth of Cut (mm) | D: Empty | Resultant Force (kN) |
|---|---|---|---|---|---|
| 1 | 500 | 80 | 0.08 | 1 | 24.77 |
| 2 | 500 | 100 | 0.10 | 2 | 31.39 |
| 3 | 500 | 120 | 0.12 | 3 | 34.55 |
| 4 | 600 | 80 | 0.10 | 3 | 24.13 |
| 5 | 600 | 100 | 0.12 | 2 | 27.98 |
| 6 | 600 | 120 | 0.08 | 1 | 30.96 |
| 7 | 700 | 80 | 0.12 | 2 | 25.06 |
| 8 | 700 | 100 | 0.08 | 3 | 27.01 |
| 9 | 700 | 120 | 0.10 | 1 | 30.82 |
Analysis of variance (ANOVA) was performed to determine the significance of each factor on the cutting force. The results indicated that factor B (axial feed rate) had the most substantial effect, with an F-value of 33.814 and a significance probability P=0.029, which is below the 0.05 threshold. Factors A (tool speed) and C (depth of cut) were less significant, with P-values of 0.157 and 0.371, respectively. The order of influence was B > A > C, meaning that axial feed rate is the primary parameter affecting cutting forces in power skiving of internal gears. This insight allows internal gear manufacturers to prioritize feed rate control to minimize forces and enhance tool life. Additionally, trend analysis revealed that cutting forces increase linearly with axial feed rate and depth of cut, but decrease with higher tool speeds up to a certain point, beyond which the effect diminishes. For instance, at a feed rate of 100 mm/s and depth of cut of 0.1 mm, increasing the tool speed from 500 rpm to 700 rpm reduced the resultant force from 31.39 kN to 27.01 kN, highlighting the importance of speed optimization for internal gears production.
In the experimental phase, we conducted power skiving trials on a YK2260MC CNC skiving machine to manufacture double-circular-arc internal gears. The skiving tool was made of S390 hardened steel with a coating, achieving a hardness of 66 HRC, which is typical for tools used by internal gear manufacturers to withstand high cutting loads. The workpiece was a gear blank with specifications including 76 teeth and a helical angle of 20°. The machine setup involved adjusting the tool and workpiece positions to achieve the required shaft angle of 20° and center distance. Key machining parameters are listed in the table below:
| Parameter | Value |
|---|---|
| Tool Holder Length (mm) | 90 |
| Fixture Length (mm) | 120 |
| Initial Machine B-Axis Angle (°) | 20 |
| Helical Feed Rate (mm/rev) | 0.07 |
| Tool Rotational Speed (rpm) | 600 |
| Total Cutting Stroke (mm) | 40 |
| Helical Air Stroke (mm) | 5 |
The machining process employed a multi-pass strategy to ensure surface finish and accuracy. Roughing involved 10 passes: the first 5 passes with a depth of cut of 1 mm each, and the next 5 passes with 0.5 mm each. Finishing consisted of 3 passes: the first with 0.3 mm depth of cut, and the last two with 0.1 mm each. This approach reduced cutting forces and minimized thermal deformation, resulting in a total machining time of approximately 2 hours, which is significantly faster than traditional methods like gear shaping. The successful production of the internal gears demonstrates the efficiency of power skiving, providing a competitive edge for internal gear manufacturers.
Gear accuracy was evaluated using a 650GMA gear measurement center, with reference to relevant standards for double-circular-arc gears. Due to limitations in dedicated measurement modules for double-circular-arc profiles, we used an approximate involute method to assess key parameters such as helix deviation, pitch deviation, and radial runout. The workpiece was aligned based on the pitch circle, with 16 measurement points selected for consistency. The results, compared to tolerance grades, are summarized in the table below:
| Inspection Item | Unit | Tolerance (Grade 8H) | Measured Value (Left Flank) | Measured Value (Right Flank) | Judgment |
|---|---|---|---|---|---|
| Helix Deviation F_β | μm | ≤27 | 20.7 | 23.5 | Conform |
| Pitch Deviation f_p | μm | ≤22 | 18.0 | 15.1 | Conform |
| Cumulative Pitch Deviation F_p | μm | ≤94 | 38.3 | 34.6 | Conform |
| Radial Runout F_r | μm | ≤75 | 48.7 | – | Conform |
The measurements indicate that the internal gears achieved a Grade 7 overall accuracy, with cumulative pitch deviation reaching Grade 5 (40 μm) and radial runout within Grade 7 (50 μm). This level of precision meets the design requirements and validates the power skiving process for high-quality internal gears production. For internal gear manufacturers, this translates to reduced post-processing needs and improved product reliability.
In conclusion, our study establishes a robust framework for power skiving double-circular-arc internal gears, addressing key challenges faced by internal gear manufacturers. The mathematical model and simulations provide accurate predictions of tooth profiles and cutting forces, while the experimental results confirm the process efficiency and gear quality. The orthogonal experiment highlights axial feed rate as the most critical parameter for cutting force control, enabling optimization of machining conditions. By adopting power skiving, internal gear manufacturers can enhance production rates, achieve higher precision, and reduce costs, paving the way for advanced applications in planetary drives and other compact transmission systems. Future work could focus on tool wear analysis and adaptive control strategies to further improve the manufacturing of internal gears.