Traditional manufacturing methods for double-circular-arc internal gears face significant challenges in precision and efficiency, often leading to premature gear failure due to poor tooth profile accuracy. This research establishes a mathematical foundation for power skiving using spatial coordinate transformations. The workpiece and tool coordinate systems are interrelated through transformation matrices:
$$
\begin{bmatrix} x_1 \\ y_1 \\ z_1 \\ 1 \end{bmatrix} = \mathbf{M_{10}} \mathbf{M_{0p}} \mathbf{M_{p2}} \begin{bmatrix} x_2 \\ y_2 \\ z_2 \\ 1 \end{bmatrix}
$$
Where $\mathbf{M_{10}}$, $\mathbf{M_{0p}}$, and $\mathbf{M_{p2}}$ represent rotation and translation matrices. The axis angle $\Sigma$ and relative velocities are governed by:
$$
\Sigma = | \beta_1 – \beta_2 |, \quad \omega_1 = i_{21}\omega_2 – \frac{\sin \beta_2}{r_1 \cos \beta_1} v_1
$$
Vericut simulations validate the tooth formation mechanism, demonstrating precise conformity with GB/T 12759-1991 standards. The progressive material removal during skiving is visualized below:
Abaqus explicit dynamics simulations model cutting forces during engagement. The orthogonal array L9(3⁴) identifies critical parameter interactions influencing gear failure risks:
| Run | A: Speed (r/min) | B: Feed (mm/s) | C: Depth (mm) | Resultant Force (kN) |
|---|---|---|---|---|
| 1 | 500 | 80 | 0.08 | 24.77 |
| 2 | 500 | 100 | 0.10 | 31.39 |
| 3 | 500 | 120 | 0.12 | 34.55 |
| 4 | 600 | 80 | 0.10 | 24.13 |
| 5 | 600 | 100 | 0.12 | 27.98 |
| 6 | 600 | 120 | 0.08 | 30.96 |
| 7 | 700 | 80 | 0.12 | 25.06 |
| 8 | 700 | 100 | 0.08 | 27.01 |
| 9 | 700 | 120 | 0.10 | 30.82 |
ANOVA reveals that feed rate (Factor B) dominates cutting forces ($F=33.814$, $p=0.029$), directly impacting tool stress and potential gear failure:
| Factor | Sum of Squares | F-value | Significance |
|---|---|---|---|
| B (Feed) | 83.763 | 33.814 | p=0.029 |
| A (Speed) | 13.285 | 5.363 | p=0.157 |
| C (Depth) | 4.204 | 1.697 | p=0.371 |
Experimental trials on a YK2260MC skiving machine with S390 carbide tools confirm the model’s accuracy. A 13-pass strategy (10 roughing, 3 finishing) achieves surface integrity critical for preventing gear failure. Precision measurements show:
| Parameter | Standard (GB 8H) | Measured | Accuracy Class |
|---|---|---|---|
| Helix deviation $F_β$ (μm) | ≤27 | 20.7 (L), 23.5 (R) | Grade 7 |
| Pitch deviation $f_p$ (μm) | ≤22 | 18 (L), 15.1 (R) | Grade 7 |
| Cumulative pitch $F_p$ (μm) | ≤94 | 38.3 | Grade 5 |
| Radial runout $F_r$ (μm) | ≤75 | 48.7 | Grade 7 |
Power skiving reduces processing time by 60% compared to hobbing while achieving Grade 5 pitch accuracy. The linear relationship between feed rate and cutting forces ($R²=0.98$) provides actionable optimization guidelines:
$$
F_n = 0.214B + 0.018A^{-1} + 12.5C + 8.37
$$
This research demonstrates that controlled skiving parameters significantly reduce tooth root stresses and micro-geometric deviations that initiate gear failure. The technology enables reliable planetary double-circular-arc transmissions where gear failure prevention is critical for high-torque applications.