This study focuses on the development and stress analysis of cylindrical gears featuring circular arc tooth profiles and tooth lines. By combining the advantages of multi-point meshing and convex-concave contact characteristics, this gear design demonstrates superior load-bearing capacity compared to traditional involute gears. The mathematical framework, finite element modeling, and stress distribution patterns are systematically explored to establish a theoretical foundation for practical applications.
1. Geometric Design Principles
The fundamental geometry of cylindrical gears with circular arc profiles can be derived using coordinate transformation theory. The tooth surface equation is established through three critical coordinate systems:
$$
\begin{cases}
x_n^2 = \rho_2 \sin\alpha_2 + E_2 \\
y_n^2 = \rho_2 \cos\alpha_2 + F_2 \\
z_n^2 = 0 \quad (\alpha_2 \in [\alpha’, \alpha”])
\end{cases}
$$
Where \((\rho_2, E_2, F_2)\) define the arc radius and center coordinates. The tooth surface generation process involves spatial transformations:
$$
\mathbf{r}_t = \begin{bmatrix}
\rho_2 \sin\alpha_2 + E_2 \\
(\rho_2 \cos\alpha_2 + F_2 + R_r)\cos\theta_2 \\
(\rho_2 \cos\alpha_2 + F_2 + R_r)\sin\theta_2 \\
1
\end{bmatrix}
$$
2. Mathematical Modeling of Tooth Surface
The meshing equation governing cylindrical gear generation is derived as:
$$
(\rho_2 \cos\alpha_2 + F_2 + R_r)\cos(\delta_2 + \theta_2)\frac{R_r}{r}\sin\alpha_2 – (\rho_2 \sin\alpha_2 + E_2)\frac{R_r}{r}\cos\alpha_2\cos(\delta_2 + \theta_2) – u_2 \sin\delta_2\frac{R_r}{r}\sin\alpha_2 + \cos\delta_2\cos\alpha_2\cos(\delta_2 + \theta_2) + \sin\delta_2\cos\alpha_2\sin(\delta_2 + \theta_2) = 0
$$
3. Multi-Arc Profile Optimization
Comparative analysis of double-arc and quadruple-arc profiles reveals significant performance differences:
| Parameter | Double-Arc | Quadruple-Arc |
|---|---|---|
| Contact Points | 4 | 8 |
| Max Stress (MPa) | 422 | 337 |
| Stress Reduction | – | 20% |
The stress reduction mechanism can be expressed as:
$$
\sigma_{quad} = 0.8\sigma_{double} + \Delta\varepsilon_{contact}
$$
4. Finite Element Analysis
Using ABAQUS with material parameters \(E = 206\ \text{GPa}\) and \(\mu = 0.3\), the contact stress distribution confirms:
- Stress concentration occurs at pitch circle regions
- Quadruple-arc design demonstrates 32% larger contact area
- Equivalent stress follows parabolic distribution:
$$
\sigma_{eq} = \sqrt{\frac{(\sigma_1 – \sigma_2)^2 + (\sigma_2 – \sigma_3)^2 + (\sigma_3 – \sigma_1)^2}{2}}
$$
5. Manufacturing Considerations
Key parameters for cylindrical gear production:
$$
\begin{array}{|c|c|c|}
\hline
\text{Parameter} & \text{Symbol} & \text{Value} \\
\hline
\text{Module} & m & 8\ \text{mm} \\
\text{Teeth Number} & z & 20/30 \\
\text{Face Width} & b & 120\ \text{mm} \\
\text{Cutter Radius} & R_r & 6” \\
\hline
\end{array}
$$
6. Dynamic Performance Characteristics
The improved cylindrical gear design exhibits:
- Axial force cancellation without requiring relief grooves
- Enhanced bending resistance through arching effect (\(K_b = 1.2-1.5\times\))
- Reduced sliding friction coefficient (\(\mu_{dynamic} \leq 0.06\))
7. Conclusion
This systematic investigation validates that cylindrical gears with circular arc profiles and tooth lines achieve superior mechanical performance through optimized contact patterns. The quadruple-arc configuration demonstrates particular advantages in stress reduction and load distribution, providing valuable insights for heavy-duty gear applications.