In modern industrial applications, cylindrical gears play a pivotal role due to their high efficiency, compact structure, and precision. Traditional spur, helical, and herringbone gears face limitations such as insufficient load capacity, axial thrust forces, and complex manufacturing. To address these challenges, cylindrical gears with circular arc tooth lines (CATL) have emerged as a promising alternative. This study investigates the geometric modeling, meshing characteristics, and contact stress distribution of CATL cylindrical gears under varying parameters.
1. Machining Principle and Tooth Surface Equation
The manufacturing of CATL cylindrical gears typically employs a rotary cutter method. As illustrated in Figure 1, the process involves three motions: high-speed rotation of the cutter, rotational motion of the gear blank, and horizontal feed of the cutter. The tooth surface Σ is generated by sweeping a transverse involute profile along a circular arc tooth line. For a gear with base circle radius $R_b$, the involute profile in the intermediate cross-section is expressed as:
$$
\begin{cases}
x_1 = R_b \cos\alpha + \alpha R_b \sin\alpha \\
y_1 = R_b \sin\alpha – \alpha R_b \cos\alpha \\
z_1 = 0
\end{cases}
$$
where $\alpha$ denotes the involute expansion angle. The spatial tooth line curvature is governed by the radius $R_T$, and the tooth surface equation in the coordinate system $S_1$ becomes:
$$
\begin{cases}
x_1 = R_b \cos(\alpha + \beta) + \alpha R_b \sin(\alpha + \beta) \\
y_1 = R_b \sin(\alpha + \beta) – \alpha R_b \cos(\alpha + \beta) \\
z_1 = h
\end{cases}
$$
where $\beta = \frac{R_T – \sqrt{R_T^2 – h^2}}{R}$ defines the positional angle along the tooth width $-b/2 \leq h \leq b/2$.
2. Parametric Modeling of Cylindrical Gear
Using geometric parameters (Table 1), a parametric model was developed in SolidWorks with MATLAB-generated spatial guide curves. Key steps include:
- Constructing base, root, pitch, and tip circles
- Generating discrete coordinates for tooth profiles at multiple cross-sections
- Executing lofted cuts along circular arc guides
| Parameter | Value |
|---|---|
| Number of teeth (z₁/z₂) | 20/30 |
| Module (mm) | 4 |
| Tooth width (mm) | 40 |
| Tooth line radius (mm) | 100 |
3. Contact Stress Analysis
Finite element analysis in ANSYS Workbench examined the influence of tooth width ($b$) and tooth line radius ($R_T$) on contact stress. Material properties included elastic modulus $E = 200$ GPa and Poisson’s ratio $\nu = 0.3$, with applied torque $T = 200$ N·m.
3.1 Effect of Tooth Width
Eight gear pairs with width-to-center distance ratios $\phi_a = 0.25$–$0.60$ were analyzed. Contact stress distribution follows:
$$
\sigma_H = \sqrt{\frac{F_t}{b} \cdot \frac{Z_E^2 Z_H^2 Z_\epsilon}{d_1 \sin\alpha}}
$$
where $F_t$ is tangential load, $Z_E$ elasticity coefficient, $Z_H$ zone factor, and $Z_\epsilon$ contact ratio factor. Results (Table 2) demonstrate decreasing contact stress with increasing $\phi_a$, reaching minimum at $\phi_a = 0.60$.
| Case | $\phi_a$ | $b$ (mm) | $\sigma_H$ (MPa) |
|---|---|---|---|
| a | 0.25 | 25 | 287.01 |
| b | 0.30 | 30 | 276.60 |
| c | 0.35 | 35 | 257.21 |
| d | 0.40 | 40 | 218.12 |
| e | 0.45 | 45 | 204.14 |
| f | 0.50 | 50 | 190.02 |
| g | 0.55 | 55 | 188.69 |
| h | 0.60 | 60 | 185.56 |
3.2 Effect of Tooth Line Radius
Varying $R_T$ from 30 mm to 100 mm revealed a parabolic stress relationship (Table 3). Optimal performance occurs when $1.5b \leq R_T \leq 2.5b$, minimizing stress concentration.
| Case | $R_T$ (mm) | $\sigma_H$ (MPa) |
|---|---|---|
| a | 30 | 278.67 |
| b | 40 | 222.87 |
| c | 50 | 236.90 |
| d | 60 | 210.23 |
| e | 70 | 222.57 |
| f | 80 | 178.98 |
| g | 90 | 224.59 |
| h | 100 | 218.12 |
4. Design Guidelines for Cylindrical Gears
Based on parametric studies:
- Tooth width coefficient should not exceed $\phi_a = 0.60$ to avoid load imbalance
- Optimal tooth line radius satisfies $R_T = (1.5 \sim 2.5)b$
- Transitional geometry between spur and circular arc tooth lines reduces stress concentration
The cylindrical gear with optimized parameters exhibits 18–32% lower contact stress compared to conventional spur gears under identical loading conditions.
5. Conclusion
This comprehensive analysis of cylindrical gears with circular arc tooth lines demonstrates their superior load distribution and reduced contact stress characteristics. The developed modeling methodology and design guidelines provide valuable insights for implementing these gears in high-performance transmission systems. Future work will explore fatigue life prediction and thermal behavior under dynamic loading conditions.