In the field of aerospace engineering, the demand for high-precision and lightweight components has led to the development of advanced gear systems, such as curvic couplings. These couplings are critical for connecting rotor sections in aircraft engines, offering superior repeatability, load capacity, and reliability. However, the introduction of annular grooves in these couplings presents unique challenges in gear grinding processes, particularly when using domestic CNC gear grinding machines. This article explores the optimization of grinding parameters for aviation curvic couplings with annular grooves, focusing on minimizing errors like grinding cracks and enhancing the efficiency of gear profile grinding. The goal is to achieve precise tooth profiles while avoiding defects that could compromise performance.
The gear grinding process for curvic couplings involves using a cup-shaped grinding wheel to machine the tooth surfaces. During gear profile grinding, the wheel rotates at high speeds and performs reciprocating feed motions to remove material. For couplings with annular grooves, the groove divides the working tooth surface, which complicates the grinding operation. If not properly managed, this can lead to grinding cracks due to excessive stress or improper wheel engagement. The relative position between the wheel and workpiece is crucial, as illustrated below:
In this setup, the convex part is ground using the inner edge of the wheel, while the concave part uses the outer edge. The gear grinding parameters must be carefully calculated to prevent issues like uneven material removal or grinding cracks. Key parameters include the grinding wheel radius, blade point width, and cutter profile radius, which depend on the helix angle. When the helix angle is zero, the calculations simplify, but for non-zero helix angles, more complex formulas are required. For instance, the grinding wheel radius \( r \) is determined based on the midpoint radius \( R_m \) and the span angle \( \delta \). The relationship is given by:
$$ \delta = 90^\circ \times \frac{N_x}{N} $$
where \( N_x \) is the number of half pitch intervals covered by the grinding wheel edge, and \( N \) is the number of teeth. For a non-zero helix angle \( \beta \), the wheel radius is calculated as:
$$ \varepsilon = \frac{\pi}{2} – (\delta – \beta) $$
$$ r = R_m \frac{\sin \delta}{\sin \varepsilon} $$
These equations ensure that the wheel engages properly with the tooth surface during gear profile grinding, reducing the risk of grinding cracks. Additionally, the blade point width must be optimized to avoid over-cutting or under-cutting in the annular groove regions. The maximum and minimum slot widths at different diameters (e.g., outer diameter, outer groove diameter, inner groove diameter, and inner diameter) are critical constraints. For the convex part, the slot widths are:
$$ W_{ov} = \frac{\pi D_o}{2N} – 2dr – 2h_b \tan \alpha_n + 2(H + F \tan \beta / 2) $$
$$ W_{ogv} = \frac{\pi D_{oc}}{2N} – 2dr – 2h_b \tan \alpha_n + 2(H – H_1 + (R_{oc} – R_m) \tan \beta) $$
$$ W_{igv} = \frac{\pi D_{ic}}{2N} – 2dr – 2h_b \tan \alpha_n + 2(H – H_2 – (R_m – R_{ic}) \tan \beta) $$
$$ W_{iv} = \frac{\pi D_i}{2N} – 2dr – 2h_b \tan \alpha_n + 2(H – F \tan \beta / 2) $$
Similarly, for the concave part:
$$ W_{oc} = \frac{\pi D_o}{2N} + 2dr – 2h_b \tan \alpha_n – 2(H + F \tan \beta / 2) $$
$$ W_{ogc} = \frac{\pi D_{oc}}{2N} + 2dr – 2h_b \tan \alpha_n – 2(H – H_1 + (R_{oc} – R_m) \tan \beta) $$
$$ W_{igc} = \frac{\pi D_{ic}}{2N} + 2dr – 2h_b \tan \alpha_n – 2(H – H_2 – (R_m – R_{ic}) \tan \beta) $$
$$ W_{ic} = \frac{\pi D_i}{2N} + 2dr – 2h_b \tan \alpha_n – 2(H – F \tan \beta / 2) $$
Here, \( dr \) is the increment in wheel radius due to equal-section correction, \( h_b \) is the dedendum height, \( \alpha_n \) is the pressure angle, and \( H \), \( H_1 \), and \( H_2 \) are arc heights. The blade point width must satisfy \( 0.5W_{\text{max}} < W_{cx} < W_{\text{min}} \) for the concave part and \( 0.5W_{\text{max}} < W_{vx} < W_{\text{min}} \) for the convex part to prevent grinding cracks. The pressure angle of the grinding wheel edge equals the gear pressure angle \( \alpha_n \), and the cutter profile radius is derived based on the blade point width.
To address the challenges in gear grinding for large-scale aviation curvic couplings, an optimization model is developed. The design variables are the span pitch number \( N_x \) and helix angle \( \beta \). The objective is to minimize the difference between the desired wheel radius \( R_{d1} \) and the theoretical radius \( R_d \), while constraints ensure that the ratio of maximum to minimum slot width is less than 2 and the root fillet radius meets design specifications. The optimization problem is formulated as:
$$ \min f = R_{d1} – R_d $$
$$ \text{subject to: } x_{i,\text{min}} \leq x_i \leq x_{i,\text{max}} \quad (i=1,2,\dots,n) $$
$$ \frac{W_{\text{max}}}{W_{\text{min}}} < 2 $$
$$ R_{ho} \geq R_1 $$
Using a genetic algorithm with a population size of 50, crossover rate of 0.9, and mutation rate of 0.1, the optimal values are found to be \( N_x = 27 \) and \( \beta = 0.70^\circ \) for a desired wheel radius of 150 mm. This optimization reduces the risk of grinding cracks by ensuring balanced material removal during gear profile grinding. The calculated machine adjustment parameters and wheel parameters are summarized in the following tables:
| Parameter | Convex Part | Concave Part |
|---|---|---|
| Radial Tool Position (mm) | 234.271 | 234.271 |
| Axial Position (mm) | 3.185 | 3.185 |
| Workpiece Installation Angle (°) | 90 | 90 |
| Angular Tool Position (°) | 180 | 0 |
| Wheel Parameter | Value |
|---|---|
| Wheel Radius (mm) | 154.397 (convex), 150.719 (concave) |
| Pressure Angle (°) | 30 |
| Blade Point Width (mm) | 4.064 (convex), 3.956 (concave) |
Experimental validation was conducted on a domestic H650GA CNC spiral bevel gear grinding machine. The workpiece geometry included an outer diameter of 390 mm, inner diameter of 330 mm, and 60 teeth. After gear grinding, the tooth profile errors were measured using a gear measurement center. The convex part exhibited a maximum profile error of 6.3 µm, while the concave part had 5.4 µm, both within acceptable limits for aerospace applications. Contact pattern tests using coloring agents showed uniform engagement, with no signs of grinding cracks or edge contact. This confirms the effectiveness of the optimized parameters in gear profile grinding.
In conclusion, the optimization of grinding parameters for aviation curvic couplings with annular grooves enhances the precision and reliability of gear grinding processes. By integrating mathematical models and genetic algorithms, the method minimizes errors such as grinding cracks and ensures efficient material removal. The experimental results demonstrate that the approach is viable for domestic CNC machines, contributing to advancements in aerospace component manufacturing. Future work could focus on real-time monitoring during gear grinding to further reduce defects and improve surface quality.