Gear transmission serves as a critical mechanism for transferring motion and power between arbitrary axes, offering advantages including compact structure, smooth operation, high efficiency, strong load capacity, and extended service life. Achieving optimal transmission performance requires meticulous attention to gear tooth surface design, material selection, and surface treatments. As a pivotal finishing process in gear manufacturing, gear shaving corrects errors such as radial runout, pitch deviation, tooth profile inaccuracies, and tooth alignment deviations. This enhances operational stability and contact strength. However, a persistent challenge arises when using standard involute helical shaving cutters: the emergence of concave deformations near the pitch circle on the shaved gear tooth flank, known as the middle-concave error. This defect induces elevated noise and reduced lifespan in gear systems. This study addresses this problem through theoretical deformation analysis and tool modification curve fitting.
1. Deformation Analysis in Gear Shaving
1.1 Contact Deformation During Gear Shaving
Gear shaving emulates the meshing of crossed-helical gears without backlash. Contact deformation (\(\delta_e\)) between the shaving cutter and workpiece gear, governed by Hertzian contact theory, is expressed as:
$$ \delta_e = \frac{3P}{2a} (\kappa_1 + \kappa_2) K(e) $$
where \(P\) represents the normal load, \(a\) denotes the contact ellipse semi-major axis, \(\kappa_1\) and \(\kappa_2\) are material-dependent constants, and \(K(e)\) is the complete elliptic integral of the first kind. For scenarios where contact overlap is less than 2 (typical in middle-concave occurrences), contact points range between 2 and 4. Force equilibrium equations are formulated for varying contact point counts:
Two contact points:
$$ \begin{cases} F_1 \cos \alpha_y = F_2 \cos \alpha_y \\ F_1 \cos \alpha_x + F_2 \cos \alpha_x = F_r \end{cases} $$
Three contact points:
$$ \begin{cases} F_1 \cos \alpha_x + F_2 \cos \alpha_x + F_3 \cos \alpha_x = F_r \\ F_1 \cos \alpha_y + F_2 \cos \alpha_y = F_3 \cos \alpha_y \\ F_1 \cos \alpha_y L_1 + F_2 \cos \alpha_y L_2 = F_3 \cos \alpha_y L_3 \end{cases} $$
Four contact points:
$$ \begin{cases} F_1 \cos \alpha_x + F_2 \cos \alpha_x + F_3 \cos \alpha_x + F_4 \cos \alpha_x = F_r \\ F_1 \cos \alpha_y + F_2 \cos \alpha_y = F_3 \cos \alpha_y + F_4 \cos \alpha_y \\ F_1 \cos \alpha_y L_1 + F_2 \cos \alpha_y L_2 = F_3 \cos \alpha_y L_3 + F_4 \cos \alpha_y L_4 \\ \delta_{e1} + \delta_{e2} = \delta_{e3} + \delta_{e4} \end{cases} $$
Here, \(F_i\) designates load magnitude at contact point \(i\), \(\alpha_x\) and \(\alpha_y\) define force direction angles relative to coordinate axes, and \(L_i\) signifies the perpendicular distance from contact point \(i\) to the y-axis. Solving these systems enables precise contact deformation (\(\delta_e\)) computation per meshing position.
1.2 Bending Deformation During Gear Shaving
Given helical gear bending complexity, the workpiece gear is approximated as a spur gear. For a single force application (Figure 1), the tooth is modeled as a cantilever beam with variable cross-section. The bending deformation \(\delta_w\) at load point \(A\) under force \(F_A\) along the y-direction is derived via Castigliano’s theorem:
$$ \delta_w = \sum_{i=1}^{n} \int \frac{M(x)}{E_e I_i} \frac{\partial M(x)}{\partial F} dx $$
For dual contact points, two configurations exist (Figure 2). The bending deformation at point \(A\) for Figure 2a becomes:
$$ \delta_w = \sum_{i=1}^{n_1} \delta_{F_A A_{wi}} – \sum_{i=1}^{n_2} \delta_{F_B B_{wi}} – (x_A – x_B) \sum_{i=1}^{n_2} \theta_{F_B B_i} $$
Conversely, for Figure 2b, it simplifies to:
$$ \delta_w = \sum_{i=1}^{n} ( \delta_{F_A A_{wi}} – \delta_{F_B A_{wi}} ) $$
The total deformation (\(\delta\)) during gear shaving combines contact and bending contributions:
$$ \delta = \delta_e \pm \delta_w $$
The sign depends on force application: negative (\(-\)) for unilateral loading and positive (\(+\)) for bilateral loading. As contact points dynamically shift during gear shaving, deformations require computational processing for practical implementation.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Normal Module (mm) | 3 | Tip Diameter (mm) | 231.7 |
| Pressure Angle (°) | 20 | Addendum Coefficient | -0.17 |
| Number of Teeth | 73 | Face Width (mm) | 20 |
| Helix Angle (°) | 15 | Radial Force (N) | 750 |
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Normal Module (mm) | 3 | Tip Diameter (mm) | 45.231 |
| Pressure Angle (°) | 20 | Addendum Coefficient | 0.3094 |
| Number of Teeth | 11 | Face Width (mm) | 15 |
| Helix Angle (°) | 28 | Radial Force (N) | 750 |
2. Modification Methodology for Middle-Concave Mitigation
Common strategies against middle-concave errors include cutter modification, balanced shaving, negative shift shaving, altered pressure angle shaving, and process adjustments. Cutter modification proves particularly effective. It intentionally introduces a concave profile near the shaving cutter’s pitch line, counteracting the anticipated concave deformation on the workpiece gear. Accurately defining this modification’s position and magnitude is paramount. This study employs numerical analysis and least-squares fitting to derive the modification curve based on calculated deformations.
The normal equations for a quadratic polynomial fit (\(y = a_0 + a_1 x + a_2 x^2\)) are:
$$
\begin{bmatrix}
n & \sum \theta_i & \sum \theta_i^2 \\
\sum \theta_i & \sum \theta_i^2 & \sum \theta_i^3 \\
\sum \theta_i^2 & \sum \theta_i^3 & \sum \theta_i^4
\end{bmatrix}
\begin{bmatrix}
a_0 \\
a_1 \\
a_2
\end{bmatrix}
=
\begin{bmatrix}
\sum \delta_i \\
\sum \delta_i \theta_i \\
\sum \delta_i \theta_i^2
\end{bmatrix}
$$
where \(\theta_i\) is the involute roll angle and \(\delta_i\) is the corresponding total deformation. Using Mathematica 6.0 and the parameters from Tables 1 & 2, contact deformation (\(\delta_e\)), bending deformation (\(\delta_w\)), and total deformation (\(\delta\)) trends versus roll angle were computed and visualized (Figures 3, 4, 5). The quadratic curve fitted to the total deformation data using least squares is shown alongside the actual deformation curve in Figure 6.
The fitted curve closely tracks the actual deformation, exhibiting the characteristic “middle-concave” profile with minimal error. This smooth curve provides a foundational profile for precise shaving cutter modification. Further refinement yields the ideal modification curve, directly applicable to gear shaving cutter design for enhanced accuracy and reduced middle-concave error.
3. Conclusion
(1) This study integrates theoretical modeling with computational analysis to investigate the middle-concave phenomenon in gear shaving. Contact and bending deformation mechanisms were rigorously analyzed. Force distributions were solved for 2, 3, and 4 contact point scenarios under low overlap conditions, enabling precise contact deformation calculation. Cantilever beam models with segmented superposition techniques quantified bending deformations under unilateral and bilateral loading, culminating in total deformation prediction.
(2) A cutter modification strategy was developed to counteract the middle-concave error. Precise deformation calculations, utilizing specified numerical examples processed via Mathematica, facilitated modification curve fitting. Results confirm that the least-squares fitted quadratic curve accurately replicates the characteristic “middle-concave” deformation trend with minimal deviation. This methodology provides a critical foundation for designing optimized shaving cutter profiles, significantly enhancing gear shaving precision and product quality.