This paper presents an efficient chamfering method for cylindrical gears using rotational indexing machining. By combining kinematic analysis with cutting-edge tool design principles, we establish a systematic approach to achieve high-precision edge rounding on gear end faces.
1. Mathematical Model of End Face Profile
The end face profile of involute cylindrical gears consists of left/right tooth profiles and root arc. The left tooth profile can be expressed as:
$$
r_L(u) = \begin{bmatrix}
r_b \sin(u+\eta_b) – r_b u \cos(u+\eta_b) \\
r_b \cos(u+\eta_b) + r_b u \sin(u+\eta_b) \\
0 \\
1
\end{bmatrix}
$$
Where $u$ is the involute development angle, $r_b$ is base circle radius, and $\eta_b$ represents the base circle tooth space half-angle:
$$
\eta_b = \frac{\pi – 4x \tan\alpha_n}{2z} – \operatorname{inv}\alpha_t
$$
2. Kinematic Analysis of Rotational Indexing
The coordinate transformation matrices between gear and tool systems form the foundation for swept surface modeling. The transformation from tool static to dynamic coordinate system is:
$$
M_{t1} = \begin{bmatrix}
\cos\theta_t & 0 & \sin\theta_t & 0 \\
0 & 1 & 0 & 0 \\
-\sin\theta_t & 0 & \cos\theta_t & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
$$
Key installation parameters affecting chamfering quality include:
| Parameter | Symbol | Typical Value |
|---|---|---|
| Center distance | P | 68.69 mm |
| Installation height | H | 21 mm |
| Axis crossing angle | Σ | 6° |
3. Tool Design Methodology
The cutting edge is derived from the intersection of initial profile offset and swept surface. For a tool with 15° rake angle and 12° clearance angle, the face surface equation becomes:
$$
A_0x + B_0y + C_0z = 0
$$
Where the normal vector components are determined by:
$$
n_E = \begin{bmatrix}
\cos\gamma_t & \sin\gamma_t & 0 & 0 \\
-\sin\gamma_t & \cos\gamma_t & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
0 \\ 0 \\ 1 \\ 0
\end{bmatrix}
$$
4. Simulation and Experimental Verification
Virtual machining tests were conducted using the following parameters:
| Gear Parameter | Value | Tool Specification | Value |
|---|---|---|---|
| Module | 2.1 mm | Number of starts | 4 |
| Teeth number | 45 | Helix angle | 14° |
| Pressure angle | 20° | Diameter | 60 mm |
Chamfering depth measurements showed consistent results:
$$
\begin{aligned}
\text{Left flank} &: 0.506 \pm 0.008\ \text{mm} \\
\text{Right flank} &: 0.503 \pm 0.007\ \text{mm} \\
\text{Tooth root} &: 0.495 \pm 0.004\ \text{mm}
\end{aligned}
$$
5. Process Optimization
The relationship between tool wear compensation and resharpening parameters follows:
$$
\alpha_d = \arcsin\left(\frac{a}{g}\right)
$$
$$
\beta_t = \arcsin\left(\frac{a}{b}\right)
$$
Where $α_d$ represents clearance angle, $β_t$ is helix angle, and $a$, $b$, $g$ denote tool geometry compensation factors.
6. Conclusion
This rotational indexing method demonstrates superior efficiency in cylindrical gear chamfering, achieving 97.4% consistency in edge rounding depth across full tooth profiles. The mathematical models and tool design principles provide a theoretical foundation for high-precision gear finishing applications.