This paper presents a systematic approach for designing universal skiving tools applicable to involute cylindrical gears with arbitrary rake geometries. By leveraging line-contact meshing theory between staggered-axis involute helicoids, the methodology eliminates theoretical edge errors while maintaining regrinding stability. Key geometrical relationships and parametric derivations are demonstrated through analytical modeling and numerical verification.
1. Fundamentals of Skiving Process
The skiving mechanism employs crossed-axis kinematics where the tool (angular velocity ωt) and workpiece (ωp) rotate synchronously with fixed shaft angle Σ. The process achieves material removal through controlled relative motion:
$$ \mathbf{v}_{rel} = \mathbf{v}_t – \mathbf{v}_p = (\omega_t \times \mathbf{r}_t) – (\omega_p \times \mathbf{r}_p) $$
2. Line Contact Conditions for Staggered-Axis Involute Helicoids
For conjugate meshing between two involute helicoids, three essential conditions must be satisfied:
| Parameter | Expression |
|---|---|
| Shaft angle | Σ = βb1 ± βb2 |
| Center distance | a = rb1 ± rb2 |
| Velocity ratio | ω1/ω2 = (rb2cosβb2)/(rb1cosβb1) |
The parametric equation of conjugate helicoids is expressed as:
$$ \begin{cases}
x = r_b\cos(\theta \pm \mu_b) + t\cos\lambda_b\sin(\theta \pm \mu_b) \\
y = r_b\sin(\theta \pm \mu_b) – t\cos\lambda_b\cos(\theta \pm \mu_b) \\
z = p\theta \mp t\sin\lambda_b
\end{cases} $$
3. Cutting Edge Formulation
The cutting edge is derived from the intersection between rake face and involute helicoid. For a planar rake face with normal vector nr:
$$ \mathbf{n}_r \cdot (\mathbf{r} – \mathbf{r}_0) = 0 $$
Substituting helicoid coordinates yields the edge equation:
$$ \begin{cases}
T = \frac{p\theta\cos\beta_b + r_b[\sin(\theta-\mu_b)\sin\beta_b – \cos(\theta-\mu_b)\tan\gamma]}{ \sin\lambda_b\cos\beta_b + \cos\lambda_b[\cos(\theta-\mu_b)\sin\beta_b + \sin(\theta-\mu_b)\tan\gamma]} \\
x_e = r_b\cos(\theta-\mu_b) + T\cos\lambda_b\sin(\theta-\mu_b) \\
y_e = r_b\sin(\theta-\mu_b) – T\cos\lambda_b\cos(\theta-\mu_b) \\
z_e = p\theta – T\sin\lambda_b
\end{cases} $$
4. Flank Surface Modeling
The flank surface maintains constant edge accuracy after regrinding through helical motion:
| Parameter | Definition |
|---|---|
| Helix parameter | pc = rb\tan(\pi/2 – \beta_b) |
| Relief angle control | Δμ = pc\theta_c[\tan(\beta_b+\alpha_c) – \tan\beta_b]/rb |
The flank surface equation becomes:
$$ \mathbf{r}_f(\theta,\theta_c) = \begin{bmatrix}
\cos\theta_c & -\sin\theta_c & 0 & 0 \\
\sin\theta_c & \cos\theta_c & 0 & 0 \\
0 & 0 & 1 & p_c\theta_c \\
0 & 0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
x_e(\theta,\Delta\mu) \\
y_e(\theta,\Delta\mu) \\
z_e(\theta,\Delta\mu) \\
1
\end{bmatrix} $$
5. Process Parameter Calculation
Critical skiving parameters for cylindrical gear manufacturing:
| Parameter | External Gear | Internal Gear |
|---|---|---|
| Shaft angle (Σ) | βbt + βbp | |βbt – βbp| |
| Center distance (a) | rbp + rbt | rbp – rbt |
| Offset (ρ) | ρ = rfp\cos\theta_p + rat\cosΣ\cos\theta_t | ρ = rfp\cos\theta_p – rat\cosΣ\cos\theta_t |
| Speed relationship | ωt = (rbp\cos\beta_{bp}/rbt\cos\beta_{bt})ω_p – (\sin\beta_{bp}/rbt\cos\beta_{bt})f | |
6. Verification and Application
Numerical verification using VERICUT demonstrates the universal tool’s capability for various cylindrical gears:
| Workpiece | Type | Teeth | Helix Angle | Max Error (mm) |
|---|---|---|---|---|
| Gear A | External spur | 125 | 0° | 0.008 |
| Gear B | Internal spur | 125 | 0° | 0.009 |
| Gear C | External helical | 70 | 18.7° | 0.012 |
| Gear D | Internal helical | 100 | 14.1° | 0.011 |
The developed methodology enables error-free generation of cylindrical gear tooth profiles across different helix angles and tooth forms. This universal approach significantly reduces tool inventory requirements while maintaining machining precision.