In modern manufacturing, the production of internal gears is critical for various industrial applications, such as transmissions and precision machinery. Traditional methods like gear shaping have limitations in achieving high efficiency and accuracy. As an internal gear manufacturer, we constantly seek advanced techniques to enhance the quality of internal gears. Skiving, a high-precision and efficient gear machining process, offers a promising alternative. This article delves into the design and optimization of skiving tools for internal gears, focusing on mathematical modeling, error analysis, and profile modification to improve performance. The methodology is grounded in meshing theory and numerical simulations, ensuring robust tool design for internal gears production.
Internal gears are components with teeth on the inner surface of a cylinder, commonly used in planetary gear systems. The manufacturing of these gears demands high precision to ensure proper meshing and load distribution. Skiving, compared to processes like hobbing or shaping, allows for continuous cutting with higher speeds and better surface finishes. However, tool design complexities, such as the presence of rake and relief angles, introduce inherent errors in the tooth profile. This study addresses these challenges by developing a comprehensive model for skiving tool design and implementing a parabolic rack-based correction method. By optimizing the tool profile, we aim to minimize errors and enhance the machining of internal gears for manufacturers.
The fundamental principle of internal gear skiving involves the relative motion between the tool and the workpiece. The tool and gear rotate synchronously at a fixed shaft angle, while the tool feeds along the gear axis. The speed ratio is governed by the number of teeth: $$ \frac{\omega_1}{\omega_2} = \frac{n_2}{n_1} $$ where $\omega_1$ and $\omega_2$ are the angular velocities of the workpiece and tool, respectively, and $n_1$ and $n_2$ are their tooth counts. This generates a series of micro-cuts that form the gear profile. For internal gears, the tooth surface is derived from an involute curve. The parametric equations for the involute profile in the transverse plane are: $$ x_0 = -r_b \left[ \sin(\theta_{os} + \theta_s) – \theta_s \cos(\theta_{os} + \theta_s) \right] $$ $$ y_0 = r_b \left[ \cos(\theta_{os} + \theta_s) + \theta_s \sin(\theta_{os} + \theta_s) \right] $$ Here, $r_b$ is the base circle radius, $\theta_s$ is the angle parameter, and $\theta_{os} = \frac{\pi}{2n_2} – \text{inv} \alpha$, with $\alpha$ as the pressure angle. For helical internal gears, this is extended to a helicoidal surface: $$ x_1 = x_0 \cos \theta – y_0 \sin \theta $$ $$ y_1 = x_0 \sin \theta + y_0 \cos \theta $$ $$ z_1 = p \theta $$ where $p$ is the lead, and $\theta$ is the rotation angle. The normal vector components are: $$ n_x = p (x_0′ \sin \theta – y_0′ \cos \theta) $$ $$ n_y = -p (x_0′ \cos \theta + y_0′ \sin \theta) $$ $$ n_z = (n_x y_1 – n_y x_1) / p $$ These equations form the basis for modeling the gear tooth surface, essential for tool design in internal gear manufacturer processes.
To design the skiving tool, we establish coordinate systems for the machining process. Let $s_1$ and $s_2$ be fixed to the internal gear and tool, respectively, with $s_0$ and $s_p$ as reference frames. The tool is installed at a shaft angle $\gamma$, and the relative motion includes rotation and linear feed. The meshing condition requires that the relative velocity is perpendicular to the common normal at the contact point: $$ \mathbf{n} \cdot \mathbf{v}_{12} = 0 $$ where $\mathbf{v}_{12}$ is the relative velocity vector. In component form, for a point in coordinate system $s_p$: $$ v_{12x} = -\omega_1 y_p – \omega_2 z_p \sin \gamma + \omega_2 y_p \cos \gamma $$ $$ v_{12y} = \omega_1 x_p – \omega_2 (x_p – a) \cos \gamma $$ $$ v_{12z} = \omega_2 (x_p – a) \sin \gamma + v $$ Here, $a$ is the distance between origins, and $v$ is the feed velocity. The tool conjugate surface is derived by transforming the gear surface using coordinate transformations. The transformation matrices are: $$ \mathbf{M}_{20} = \begin{bmatrix} \cos \theta_2 & \sin \theta_2 & 0 & 0 \\ -\sin \theta_2 & \cos \theta_2 & 0 & 0 \\ 0 & 0 & 1 & h \\ 0 & 0 & 0 & 1 \end{bmatrix} $$ $$ \mathbf{M}_{p0} = \begin{bmatrix} 1 & 0 & 0 & a \\ 0 & \cos \gamma & \sin \gamma & 0 \\ 0 & -\sin \gamma & \cos \gamma & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$ $$ \mathbf{M}_{p1} = \begin{bmatrix} \cos \theta_1 & \sin \theta_1 & 0 & 0 \\ -\sin \theta_1 & \cos \theta_1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$ The conjugate surface point in tool coordinates is: $$ \begin{bmatrix} x_2 \\ y_2 \\ z_2 \\ 1 \end{bmatrix} = \mathbf{M}_{20}^{-1} \mathbf{M}_{p0}^{-1} \mathbf{M}_{p1} \begin{bmatrix} x_1 \\ y_1 \\ z_1 \\ 1 \end{bmatrix} $$ To represent the complex surfaces, we use Non-Uniform Rational B-Spline (NURBS) fitting. A NURBS surface is defined as: $$ \mathbf{P}(u, w) = \sum_{i=0}^m \sum_{j=0}^n \mathbf{p}_{ij} N_{i,3}(u) N_{j,3}(w) $$ where $N_{i,3}(u)$ and $N_{j,3}(w)$ are basis functions, and $\mathbf{p}_{ij}$ are control points. This allows accurate representation of the tool surface for internal gears machining.
The rake face of the skiving tool is inclined to balance cutting angles on both sides. In a local coordinate system $s$, the rake face normal is $\mathbf{n} = (0, 0, 1)$. After transformations to the tool motion system $s_1$, the normal becomes: $$ \mathbf{n}_1 = (\cos \beta \sin \alpha, -\sin \beta, \cos \beta \cos \alpha) $$ where $\alpha$ is the helix angle and $\beta$ is the rake angle. The transformation matrices are: $$ \mathbf{B}_1 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \beta & \sin \beta \\ 0 & -\sin \beta & \cos \beta \end{bmatrix} $$ $$ \mathbf{B}_2 = \begin{bmatrix} \cos \alpha & 0 & \sin \alpha \\ 0 & 1 & 0 \\ -\sin \alpha & 0 & \cos \alpha \end{bmatrix} $$ Thus, the rake face equation in tool coordinates is: $$ x_2 \sin \alpha \cos \beta + (y_2 – r_1) \sin \beta + z_2 \cos \beta \cos \alpha = 0 $$ where $r_1$ is the tool radius. This plane intersects the conjugate surface to define the cutting edge.
The cutting edge is obtained by solving the intersection between the NURBS-fitted conjugate surface and the rake face. Let the surface point be $\mathbf{P}(u, w) = (P_1(u,w), P_2(u,w), P_3(u,w))$. We define a function $f(x_2, y_2, z_2) = x_2 \sin \alpha \cos \beta + (y_2 – r_1) \sin \beta + z_2 \cos \beta \cos \alpha$. Using Newton’s method, we iterate over grid points to find roots. For an edge between parameters $(u_i, w_j)$ and $(u_{i+1}, w_{j+1})$, if $f$ changes sign, we apply: $$ w_{k+1} = w_k – \frac{w_k – w_{k-1}}{f(u_t, w_k) – f(u_t, w_{k-1})} f(u_t, w_k) $$ with tolerance $\xi = 0.0001$. The resulting $(u, w)$ values give the cutting edge points, which are fitted to a B-spline curve: $$ \mathbf{k}(u’) = \sum_{j=0}^n \mathbf{d}_j N_{j,3}(u’) $$ where $\mathbf{d}_j$ are control points. This data is imported into CAD software to build the 3D tool model, crucial for internal gear manufacturer applications.
For illustration, consider a case study with parameters typical for internal gears production. The tool and gear specifications are summarized in the table below:
| Parameter | Skiving Tool | Internal Gear |
|---|---|---|
| Module (mm) | 3 | 3 |
| Number of Teeth | 31 | 71 |
| Face Width (mm) | 15 | 15 |
| Pressure Angle (°) | 20 | 20 |
| Helix Angle (°) | 5 | 5 |
| Tip Relief Angle (°) | 4 | — |
| Rake Angle (°) | 5 | — |
Using these parameters, the cutting edge points are computed, as shown in the following sample data:
| Point | x (mm) | y (mm) | z (mm) |
|---|---|---|---|
| 1 | 21.2398 | -1.9928 | 9.0391 |
| 2 | 21.2877 | -1.9926 | 9.0357 |
| 3 | 21.3451 | -1.9915 | 9.0317 |
| 4 | 21.4123 | -1.9892 | 9.0274 |
| 5 | 21.4893 | -1.9854 | 9.0227 |
| 6 | 21.5762 | -1.9798 | 9.0177 |
| 7 | 21.6732 | -1.9721 | 9.0125 |
| 8 | 21.7803 | -1.9618 | 9.0072 |
| 9 | 21.8979 | -1.9486 | 9.0018 |
| … | … | … | … |
| 39 | 21.3495 | -0.2425 | 9.4918 |
| 40 | 21.3019 | -0.2385 | 9.4963 |
After fitting and CAD modeling, the tool’s 3D structure is assembled, enabling precise machining of internal gears. However, inherent errors arise due to the projection of the cutting edge onto the transverse plane. The rake and relief angles cause deviations from the ideal involute profile. Specifically, the error increases from tooth tip to root and is asymmetric between left and right flanks. For instance, with a rake angle of 5°, the right flank error is larger than the left. This principle error $\Delta$ can be expressed as a function of the tool parameters: $$ \Delta = f(\alpha, \beta, r_d, r_e, r_f) $$ where $r_d$, $r_e$, and $r_f$ are tip, pitch, and root radii. Analysis shows that error magnitude grows with increasing rake angle, impacting the quality of internal gears produced.
To correct these errors, a parabolic rack profile is applied to the tool tooth flanks. The rack coordinate system $s_a$ has a profile defined by: $$ \mathbf{r}_a = [b u^2, u, l, 0]^T $$ where $b$ is the modification coefficient, $u$ is the profile parameter, and $l$ is the length along the tooth. Transforming to the rack coordinate system $s_c$: $$ \mathbf{r}_c(u, l) = \mathbf{M}_{ca} \mathbf{r}_a(u, l) $$ The normal vector is: $$ \mathbf{n}_c(u, l) = \frac{\partial \mathbf{r}_c}{\partial u} \times \frac{\partial \mathbf{r}_c}{\partial l} $$ In the tool generation process, the rack and tool mesh at a point where the common normal passes through the instantaneous center of rotation. The tool rotation angle $\phi$ is: $$ \phi = \left[ (l \tan \alpha_0 + y_c) \frac{n_{cx}}{n_{cy}} – x_c \right] / r_a $$ where $\alpha_0$ is the relief angle, and $x_c$, $y_c$ are components of $\mathbf{r}_c$. The modified tool surface is: $$ \mathbf{r}_s(u, l) = \mathbf{M}_{sc} \mathbf{r}_c(u, l) $$ $$ \mathbf{n}_s(u, l) = \mathbf{M}_{sc} \mathbf{n}_c(u, l) $$ This modification reduces profile errors significantly. The effect of different modification coefficients $b$ on error reduction is analyzed below. For the right flank, as $b$ increases, error decreases to a minimum at $b = 0.0010$, then rises. Similarly, for the left flank, the optimal $b$ is 0.0002. The table summarizes error trends:
| Flank | Optimal b | Max Error Reduction (μm) |
|---|---|---|
| Right | 0.0010 | 20 |
| Left | 0.0002 | 4 |
The parabolic modification effectively compensates for the inherent errors, making it a valuable technique for internal gear manufacturer seeking high precision. In practice, the choice of $b$ depends on specific tool and gear parameters, and further optimization can be done using iterative simulations. This approach enhances the skiving process for internal gears, ensuring better performance in applications like automotive and aerospace industries.
In conclusion, the design and optimization of skiving tools for internal gears involve detailed mathematical modeling and error correction. By leveraging meshing theory, NURBS surfaces, and parabolic rack modification, we can achieve significant improvements in tooth profile accuracy. This methodology supports internal gear manufacturer in producing high-quality gears efficiently. Future work could explore real-time adaptive control and multi-axis machining to further advance the skiving technology for internal gears.