In the transmission systems of hydroelectric power machinery and transportation vehicles, the straight bevel gear differential is a fundamental component. Belonging to the 2K-H type planetary mechanism, its primary advantages are structural simplicity, ease of manufacturing, and straightforward maintenance. To elevate the design standards of differentials and significantly shorten the development cycle, I have conducted a systematic optimization design study focused on the straight bevel gear differential. This work has culminated in the development of specialized design software that adheres to domestic design standards and norms. The software is architected using modular FORTRAN 77 code and operates efficiently on an IBM-PC microcomputer.
The core of this design system is a software suite capable of performing three primary functions: detailed geometric calculation of the straight bevel gear assembly, comprehensive strength verification for all critical components (gears, shafts, pins), and finally, a full-fledged optimization design process. These functions are interdependent; the optimization routine calls the strength verification module, which in turn relies on the geometric calculation module. This hierarchical program structure is illustrated in the following flowchart, which outlines the interaction between the main program and its key subroutines:
Table 1: Core Functions of the Straight Bevel Gear Differential Software
| Module Name | Primary Function |
|---|---|
| Main Program | Manages common data areas, controls execution flow, and calls appropriate subroutines. |
| Input Subroutine | Handles input of initial parameters and variable starting values. |
| Geometry Calculation | Computes all key dimensions for the sun (half-shaft) and planet straight bevel gears. |
| Gear Strength Calculation | Evaluates bending strength for both sun and planet straight bevel gears. |
| Shaft Strength Calculation | Checks torsional strength of half-shafts (closed design) or bending of cross-shafts (open design). |
| Planet Pin Strength Calculation | Calculates bearing/contact stress on the planet pins. |
| Optimization Driver | Executes the constrained optimization algorithm on the defined mathematical model. |
| Rounding & Standardization | Rounds and standardizes optimized parameters (e.g., module, shaft diameters). |
| Output Subroutine | Outputs all design parameters and saves results to a data file. |
The effectiveness of the software hinges on a rigorously defined mathematical model for optimization. The model is built around specific objective functions, carefully chosen design variables, and a comprehensive set of constraint functions.
Mathematical Model for Optimization
The choice of objective function depends on the primary design goal. The two most common scenarios are addressed.
1. Objective Function for Minimum Volume: When the torque to be transmitted is fixed, the goal is to minimize the overall volume of the straight bevel gear differential while ensuring the strength of all components. The objective function \( F_1(\vec{X}) \) is formulated as the total volume, which can be approximated by the sum of volumes of key components:
$$
F_1(\vec{X}) = \frac{\pi}{3} R_e b (R_e – \frac{b}{2} \sin{\delta_1}) + n_p \cdot \frac{\pi}{4} d_p^2 \cdot l_p + \frac{\pi}{4} d_s^2 \cdot l_s
$$
Where \( z_1, z_2 \) are the number of teeth on the planet and sun gear respectively, \( m \) is the module, \( b \) is the face width, \( R_e \) is the outer cone distance, \( \delta_1, \delta_2 \) are the pitch angles, \( d_p, d_s \) are the diameters of the planet pin and half-shaft/cross-shaft, \( n_p \) is the number of planets, and \( l_p, l_s \) are relevant lengths.
If angle modification (profile shifting) is applied to the straight bevel gears to improve performance, a secondary goal is to maximize the sum of the shift coefficients \( x_{\Sigma} = x_1 + x_2 \). A combined weighted objective function can be used:
$$
F_1′(\vec{X}) = \omega_1 \cdot \frac{F_1(\vec{X})}{F_{1,0}} + \omega_2 \cdot (1 – \frac{x_{\Sigma}}{x_{\Sigma, max}})
$$
where \( \omega_1 + \omega_2 = 1 \).
2. Objective Function for Maximum Torque: When the installation space is limited, the goal is to maximize the torque capacity of the differential. Since pitting failure is rare in straight bevel gear differentials, contact strength is often neglected. The maximum transmissible torque \( T_{max} \) is determined by the limiting component:
$$
F_2(\vec{X}) = T_{max} = \min(T_{gear}, T_{pin}, T_{shaft})
$$
where \( T_{gear}, T_{pin}, T_{shaft} \) are the torques limited by gear bending strength, planet pin bearing strength, and shaft strength, respectively.
Selection of Design Variables: The set of design variables \( \vec{X} \) depends on the design conditions. For a typical case aiming for minimum volume with a pressure angle \( \alpha = 20^\circ \), the variables might be:
$$
\vec{X} = [z_2, m, b, d_p, d_s]^T = [x_1, x_2, x_3, x_4, x_5]^T
$$
For optimization aiming at maximum torque, the variables often include parameters influencing strength directly, and may also include profile shift coefficients \( x_1, x_2 \) if gear modification is allowed.
Establishment of Constraint Functions: A complete set of constraints must be enforced to ensure a feasible, safe, and functional straight bevel gear differential design.
Table 2: Summary of Key Constraint Functions
| Constraint Category | Mathematical Form | Description |
|---|---|---|
| Gear Geometry | \( 16 \leq z_2 \leq 45 \) (Closed) \( 16 \leq z_2 \leq 40 \) (Open) \( 0.15 \leq \phi_{R} = b/R_e \leq 0.35 \) \( m_{min} \leq m \leq m_{max} \) |
Limits on sun gear teeth, face width ratio, and module. |
| Component Strength | \( S_{F1} \geq [S_F], \quad S_{F2} \geq [S_F] \) \( \sigma_{pmax} \leq [\sigma_p] \) \( \tau_{smax} \leq [\tau] \) or \( \sigma_{smax} \leq [\sigma] \) |
Bending safety factors for planet (1) and sun (2) gears. Bearing stress on planet pin. Torsional/bending stress in shafts. |
| Gear Meshing Quality (if modified) | \( \varepsilon_{\alpha v} \geq [\varepsilon_{\alpha}] \) \( s_{a1,2} \geq [s_a] \cdot m \) Non-interference conditions |
Minimum contact ratio for smoothness. Minimum tip thickness for strength. Prevention of undercut and fillet interference. |
The meshing quality constraints for the modified straight bevel gears are evaluated using their virtual spur gear equivalents. For example, the contact ratio \( \varepsilon_{\alpha v} \) is calculated based on the virtual gear pair. The tip thickness constraint for the sun gear is:
$$
s_{a2} = d_{va2} \left( \frac{\pi + 4 x_2 \tan\alpha}{2 z_{v2}} + \text{inv}\alpha – \text{inv}\alpha_{a2} \right) \geq [s_a] \cdot m
$$
where \( d_{va2} \) is the virtual tip diameter, \( z_{v2} \) is the virtual tooth number, \( x_2 \) is the profile shift coefficient, and \( \alpha_{a2} \) is the tip pressure angle.
Key Considerations in the Optimization Implementation
1. Optimization Algorithm: The core optimization is performed using a Constrained Quasi-Newton Method. This is a direct method for constrained problems. It transforms the original problem into a sequence of Quadratic Programming (QP) subproblems. The search direction \( \vec{d}^{(k)} \) for iteration \( k \) is obtained by solving the QP subproblem, and a step length \( \lambda^{(k)} \) is determined via a line search with monitoring, leading to the iteration: \( \vec{X}^{(k+1)} = \vec{X}^{(k)} + \lambda^{(k)} \vec{d}^{(k)} \). The method updates an approximation of the inverse Hessian matrix of the Lagrangian function, similar to unconstrained quasi-Newton methods, ensuring good convergence behavior.
2. Curve and Data Fitting: The design process relies heavily on empirical curves and tables (e.g., for tangential shift coefficients \( x_t \), tooth form factors \( Y_{Fa} \), stress correction factors \( Y_{Sa} \)). These are fitted within the software using the least squares method to create continuous functions, guaranteeing an error of less than 1%. For instance, the tangential shift coefficient for a pressure angle of \( 22.5^\circ \) can be fitted as a function of the virtual tooth number \( z_v \):
$$
x_t \approx 0.036 – 2.63 \times 10^{-4} z_v + 1.15 \times 10^{-5} z_v^2 – 2.86 \times 10^{-7} z_v^3
$$
This eliminates the need for inefficient table look-ups during iterative optimization.
3. Variable Scaling and Constraint Normalization: Design variables can differ by orders of magnitude (e.g., module \( m \) in mm vs. tooth number \( z \)), causing ill-conditioning. Scaling is applied: \( \bar{x}_i = x_i / s_i \), where \( s_i \) is a scale factor (e.g., \( s_m=5 \), \( s_z=20 \)). Similarly, constraint values \( g_j(\vec{X}) \) are normalized to a 0-1 range: \( \bar{g}_j(\vec{X}) = g_j(\vec{X}) / C_j \), where \( C_j \) is a characteristic value of the constraint. This improves numerical stability and the performance of the optimization algorithm.
4. Parameter Rounding and Standardization: Optimal solutions yield real numbers, but practical design requires standardized values. Key parameters like sun gear tooth number \( z_2 \), module \( m \), planet pin diameter \( d_p \), and shaft diameter \( d_s \) must be rounded to integers or preferred numbers. The methodology involves creating a discrete grid of candidate integer/standard values near the real optimum. All points satisfying the constraints are evaluated using the objective function, and the point yielding the best objective value is selected as the final, manufacturable optimal design. This ensures the final design is both optimal and practical.
Design Example and Results
To demonstrate the software’s efficacy, consider the design of a straight bevel gear differential for a heavy-duty dump truck. The primary requirements are high reliability, a specified input torque of \( T = 1200 \, \text{N·m} \), an input speed \( n = 1200 \, \text{rpm} \), a gear pressure angle of \( 22.5^\circ \), a required service life of over 10,000 hours, and the use of case-hardened alloy steel (20CrMnTi) for all gears and shafts. The optimization was performed targeting minimum volume.
The following table compares the key results from a conventional design approach and the two optimization scenarios (minimum volume and maximum torque) for this straight bevel gear differential. The optimization runs were completed in approximately 3 minutes and 20 seconds on an IBM-PC platform.
Table 3: Comparison of Design Results for the Straight Bevel Gear Differential
| Parameter | Conventional Design | Opt. Design (Min Volume) | Opt. Design (Max Torque) |
|---|---|---|---|
| Sun Gear Teeth \( z_2 \) | 40 | 38 | 36 |
| Module \( m \, (mm) \) | 6.5 | 6.0 | 7.0 |
| Face Width \( b \, (mm) \) | 28 | 26 | 32 |
| Planet Pin Diameter \( d_p \, (mm) \) | 28 | 25 | 30 |
| Half-shaft Diameter \( d_s \, (mm) \) | 38 | 35 | 40 |
| Gear Bending Safety Factor \( S_F \) | 1.95 / 2.10 | 1.51 / 1.65 | 2.25 / 2.40 |
| Planet Pin Bearing Stress \( \sigma_p \, (MPa) \) | 115 | 148 | 105 |
| Relative Volume \( V / V_{conv} \) | 1.00 | 0.78 | 1.18 |
| Max Torque Capacity \( T_{max} \, (N·m) \) | ~1250 | ~1280 | ~1440 |
The results are conclusive. When the objective is to minimize the volume of the straight bevel gear differential, the optimization software successfully reduces the overall volume by approximately 22% compared to the initial conventional design, while still meeting all strength and geometric constraints. Conversely, when the objective is to maximize torque capacity within a similar envelope, the optimization increases the torque capacity by over 15%. This demonstrates the significant advantage of a systematic optimization approach over traditional iterative design methods. Not only are the results more scientifically rational, with clear trade-offs between size and performance highlighted, but the design process itself is dramatically accelerated, compressing a task that might take days into minutes. The developed software provides a powerful, standardized tool for the efficient and high-performance design of straight bevel gear differentials.