Hardened gears are essential for automotive transmissions due to their load capacity, wear resistance, and compact size. Gear honing stands out among hard-surface processing methods for its ability to improve profile accuracy and reduce noise while inducing beneficial compressive residual stresses. Unlike internal gear honing, external variants face challenges like concave tooth profiles caused by variable honing forces in single/double mesh zones. Our solution leverages tooth-skipped meshing—where a modified worm engages alternate gear teeth—to eliminate these defects and enhance gear honing precision.
Tooth-skipped meshing divides into three phases: gear addendum meshing, standard involute helicoid meshing, and worm addendum meshing. We model these using spatial geometry and meshing theory. The involute helicoid surface is defined parametrically by axial position \(z\) and pressure angle \(\theta\):
$$ \mathbf{r} = \begin{bmatrix} r_b \cos(\tau + \mu + \theta) + \theta r_b \sin(\tau + \mu + \theta) \\ r_b \sin(\tau + \mu + \theta) – \theta r_b \cos(\tau + \mu + \theta) \\ z \end{bmatrix} $$
where \(\mu = z \tan\beta_b / r_b\), \(r_b\) is the base radius, and \(\beta_b\) is the base helix angle. The surface normal vector is:
$$ \mathbf{n} = \left[ -\sin\alpha_t, \cos\alpha_t, -\tan\beta_b \right]^T $$
For standard meshing, contact points satisfy the equation:
$$ \phi_1 r_{b1} \cos\beta_{b1} + \phi_2 r_{b2} \cos\beta_{b2} = r_{b1} \cos\beta_{b1} (\alpha_{t1} – \tan\alpha_{t1}) + r_{b2} \cos\beta_{b2} (\alpha_{t2} – \tan\alpha_{t2}) $$
Addendum meshing treats the gear tip as a helix:
$$ \mathbf{r}_2 = \begin{bmatrix} r_{a2} \cos(\tau_2 + z_2 \tan\beta_{b2}/r_{b2} + \theta_{a2} – \alpha_{a2}) \\ r_{a2} \sin(\tau_2 + z_2 \tan\beta_{b2}/r_{b2} + \theta_{a2} – \alpha_{a2}) \\ z_2 \end{bmatrix} $$
Contact occurs where this helix tangentially meets the opposing helicoid. Transition points between phases are calculated using geometric constraints. For example, gear addendum-to-standard meshing starts when:
$$ \phi_{23} = \frac{\alpha_{t2} \cos^2\beta_{b2} – \theta_{a2} + \tan\alpha_{t2} \sin^2\beta_{b2}}{\cos^2\beta_{b2}} $$
Simulations for a 3-start worm and 28-tooth gear reveal distinct contact patterns. Standard meshing dominates the tooth flank, while addendum phases concentrate near tips/roots. Relative velocity analysis during gear honing shows:
| Meshing Phase | Velocity Direction | Magnitude Trend |
|---|---|---|
| Standard | Axial-dominant | Decreases from root to tip |
| Worm Addendum | Constant | Near-constant |
| Gear Addendum | Root-to-tip | Increases away from standard zone |
Relative velocity vectors cross at the pitch line, creating micro-crosshatch patterns that improve lubrication in gear honing applications. Integrated error curves from our model align perfectly with solid-intersection validation, confirming accuracy. This framework enables optimized worm design for gear honing—exploiting addendum phases for tip rounding and root relief while eliminating mid-profile concavity. Future work will refine velocity vector angles to enhance surface texture.