Spiroid face gear drives constitute a novel spatial crossed-axis transmission system comprising an involute spiroid pinion and a spiroid face gear. These systems offer advantages including high contact ratios (often exceeding 2.0), compact structure, high transmission ratios (ranging 10:1–400:1), low noise, and robust load-bearing capacity. Spiroid face gears are typically manufactured via die-casting for applications like fishing reels. However, mold precision limitations and thermal distortions result in poor surface quality (Ra ≈ 0.8 μm), accelerating wear and degrading transmission performance. This research introduces precision gear shaving as a post-casting solution.
Mathematical Foundation of Spiroid Face Gear Tooth Surfaces
The tooth surface of the spiroid face gear (Σ₂) is generated through the envelope of an imaginary pinion cutter (Σ_c). The pinion cutter’s involute helicoid surface equations in coordinate system $S_c$ are:
$$ \mathbf{r}_c^{(k)} = \begin{bmatrix}
r_b \cos(\theta_c \pm \mu_c \sin\lambda_b) \\
r_b \sin(\theta_c \mp \mu_c \cos\lambda_b) \\
p_c \mu_c \\
1
\end{bmatrix} $$
$$ \mathbf{n}_c^{(k)} = \begin{bmatrix}
\sin\lambda_b \sin\theta_c \\
-\sin\lambda_b \cos\theta_c \\
\mp \cos\lambda_b
\end{bmatrix} \quad (k=I, II) $$
where $r_b$ is base radius, $\lambda_b$ is base helix angle, $p_c$ is helix parameter, $\mu_c$ and $\theta_c$ are surface parameters. Coordinate systems for generation (Fig. 1) define the spatial relationship between cutter and gear. The meshing equation is derived from the orthogonality condition between the relative velocity $\mathbf{v}_c^{(c2)}$ and surface normal $\mathbf{n}_c$:
$$ f_c^{(k)} = \mathbf{n}_c^{(k)} \cdot \mathbf{v}_c^{(c2)} = 0 $$
$$ \mathbf{v}_c^{(c2)} = \mathbf{v}_c^{(c)} – \mathbf{v}_c^{(2)} = (\boldsymbol{\omega}_c – \boldsymbol{\omega}_2) \times \mathbf{r}_c – \boldsymbol{\omega}_2 \times \mathbf{R} $$
The gear tooth surface Σ₂ is then expressed parametrically:
$$ \mathbf{r}_2 = \mathbf{M}_{2c}(\phi_c) \mathbf{r}_c, \quad f_c^{(k)}(\mu_c, \theta_c, \phi_c) = 0 $$
Tooth geometry is constrained by undercutting and pointing avoidance. The minimum inner radius $L_1$ preventing undercutting solves:
$$ F(\mu_c, \theta_c, \phi_c) = \begin{vmatrix}
\frac{\partial f}{\partial \mu_c} & \frac{\partial f}{\partial \theta_c} & \frac{\partial f}{\partial \phi_c} \\
\frac{\partial x_c}{\partial \mu_c} & \frac{\partial x_c}{\partial \theta_c} & \frac{\partial x_c}{\partial \phi_c} \\
\frac{\partial y_c}{\partial \mu_c} & \frac{\partial y_c}{\partial \theta_c} & \frac{\partial y_c}{\partial \phi_c}
\end{vmatrix} = 0 \quad \text{at} \quad \theta_c^* = \sqrt{\frac{r_{ac}^2 – r_{bc}^2}{r_{bc}}} $$
The maximum outer radius $L_2$ preventing pointing is determined by solving:
$$ \mathbf{r}_2^{(I)} = \mathbf{r}_2^{(II)}, \quad z_2 = -r_{fc} $$
| Design Parameter | Symbol | Value |
|---|---|---|
| Pinion Teeth | $N_c$ | 7 |
| Normal Module | $m_n$ | 0.65 mm |
| Min Inner Radius | $L_1$ | 11.7 mm |
| Max Outer Radius | $L_2$ | 14.5 mm |
Gear Shaving Principle and Kinematic Model
Gear shaving utilizes a modified involute spiroid pinion as a shaving cutter. The cutter features axial slots creating cutting edges at the intersection with the tooth flank. The kinematic model (Fig. 2) involves three motions:
- Cutter rotation ($\omega_s$) about axis $z_s$
- Workpiece rotation ($\omega_g^*$) driven by the cutter
- Radial feed ($v_{s0}$) along the gear’s radius
The corrected workpiece angular velocity accounts for radial feed:
$$ \omega_g^* = m_{gs} \omega_s – \frac{N_s}{N_g p_s} v_{s0} \quad \text{where} \quad m_{gs} = \frac{N_g}{N_s} $$
The relative velocity $\mathbf{v}_{sg}$ in coordinate system $S_s$ is:
$$ \mathbf{v}_{sg} = \begin{bmatrix}
\omega_s y_s – \omega_g^* \cos\phi_s (z_s + l_0 + \Delta l) – \omega_g^* \sin\phi_s E \\
-\omega_s x_s + \omega_g^* \sin\phi_s (z_s + l_0 + \Delta l) – \omega_g^* \cos\phi_s E \\
v_{s0} + \omega_g^* (\cos\phi_s x_s – \sin\phi_s y_s)
\end{bmatrix} $$
The meshing equation for gear shaving becomes:
$$ f_{sg}^{(k)} = \mathbf{n}_s^{(k)} \cdot \mathbf{v}_{sg} = 0 \quad (k=I, II) $$
The generated gear surface during gear shaving is:
$$ \mathbf{r}_g^{(k)} = \mathbf{M}_{gs}(\phi_s) \mathbf{r}_s^{(k)}, \quad f_{sg}^{(k)}(\mu_s, \theta_s, \phi_s) = 0 $$
Shaving Cutter Design and Manufacturing
The cutter material is W6Mo5Cr4V2 high-speed steel (HRC 63-66). Key geometric parameters calculated via helical gear formulas are:
| Parameter | Symbol | Value |
|---|---|---|
| Normal Module | $m_n$ | 0.65 mm |
| Teeth Number | $N_s$ | 7 |
| Pressure Angle | $\alpha_n$ | 20° |
| Helix Angle | $\beta$ | 52° (LH) |
| Tip Diameter | $d_{as}$ | 6.25 mm |
| Root Diameter | $d_{fs}$ | 4.46 mm |
| Slot Width | $B_s$ | 1.0 mm |
| Slot Depth | $H_s$ | 2.0 mm |
Axial rectangular slots were machined (Fig. 3). Manufacturing tolerances adhered to AGMA 12 standards:
$$ \Delta f_h \leq 0.004 \text{mm}, \quad \Delta F_{\beta} = \pm 0.006 \text{mm}, \quad \Delta f_{pb} = \pm 0.003 \text{mm} $$
Numerical Control Simulation of Gear Shaving
A 5-axis CNC machine model (X, Y, Z, A, C) was built in VERICUT. The simulation workflow comprised:
- Machine modeling: Defined kinematics chain $$ \text{Base} \rightarrow \text{Y} \rightarrow \text{X} \rightarrow \text{A} \rightarrow \text{Fixture} \rightarrow \text{Workpiece} $$
$$ \text{Base} \rightarrow \text{Z} \rightarrow \text{C} \rightarrow \text{Spindle} \rightarrow \text{Cutter} $$ - Tool/Workpiece Import: STEP models of cutter and blank
- G-code generation for synchronized motion:
N6 R1=0 R2=0 R3=44.2872535 R4=0
N7 A=R1 C=R2 Z=R3
N8 MARK: R1=11.07653*R4 R2=57.29578*R4 R3=R3-0.0001 R4=R4+0.006
A=R1 C=R2 Z=R3
N9 IF R3>=38.287 GOTOB MARK
Simulation results (Fig. 4) confirmed the theoretical tooth form. Deviation analysis showed:
$$ \Delta_{\text{max}}^{\text{working flank}} = 9.8 \mu\text{m}, \quad \Delta_{\text{root}} \approx 25 \mu\text{m} \text{(non-contact)} $$
Experimental Validation and Surface Analysis
Gear shaving experiments used a CNC machining center with parameters:
| Parameter | Value |
|---|---|
| Spindle Speed | 2000 rpm |
| Radial Feed ($v_{s0}$) | 50 mm/min |
| Axis Angle ($\gamma_m$) | 90° |
| Offset ($E$) | 7 mm |
Surface roughness was measured using InfiniteFocus G5 optical profilometer:
| Surface | Ra Before Shaving (μm) | Ra After Shaving (μm) | Reduction |
|---|---|---|---|
| Concave Flank | 0.8365 | 0.6311 | 24.6% |
| Convex Flank | 0.8527 | 0.6419 | 24.7% |
The significant Ra reduction validated gear shaving effectiveness for die-cast spiroid gears.
Conclusions and Future Work
This research established a complete methodology for gear shaving of spiroid face gears:
- Mathematical models for tooth generation and gear shaving kinematics were derived
- Axial-slot shaving cutters were designed/manufactured
- VERICUT simulations verified CNC toolpaths (deviation < 9.8μm)
- Experiments demonstrated 24.6% surface roughness reduction
Future work will focus on:
- Cutter flank modification for localized bearing contact
- Dynamics analysis under gear shaving-modified surface topography
- Groove optimization on concave flanks for vibration suppression
- Extended durability testing of shaved gear pairs
The proposed gear shaving technique enables high-quality finishing of die-cast spiroid gears using standard CNC equipment, enhancing performance in precision transmission systems.