In recent years, computer-aided design (CAD) for gear transmissions has advanced significantly worldwide. However, research on CAD for spiral bevel gears and zero bevel gears remains limited. I have conducted an in-depth study on the computer-aided design of these gear types, focusing on optimization design, data management, and automated parametric drawing of gear part diagrams. This article elaborates on the fundamental principles and methods, with an emphasis on the mathematical models for optimization design, program flowcharts, and the programming logic for computer-generated part drawings.
The core of this work involves developing optimization models that minimize gear volume or maximize power transmission while adhering to constraints such as contact strength, bending strength, and geometric limitations. For zero bevel gears, which are characterized by their straight teeth and zero spiral angle, specific considerations are integrated into the models to ensure accuracy and efficiency. The optimization process utilizes advanced algorithms, and the entire system is implemented through a structured program that links design calculations with automated drawing capabilities.
Mathematical Model for Optimization Design
The optimization design for spiral bevel gears and zero bevel gears is based on a mathematical framework that defines design variables, objective functions, and constraint functions. This model ensures that the gears meet performance criteria while optimizing key parameters.
Design Variables
The design variables are independent parameters that influence the gear dimensions. For spiral bevel gears, variables include the number of teeth (z1 and z2), spiral angle at the midpoint (βm), face width (b), and module at the large end (mt). However, when the transmission ratio is fixed, z1 and z2 are not independent; thus, only one is treated as a variable. For zero bevel gears, the spiral angle is zero, simplifying the model. The specific design variables are as follows:
- For spiral bevel gears with minimum volume as the objective: X = [z1, βm, b, mt]
- For zero bevel gears with minimum volume as the objective: X = [z1, b, mt]
- For spiral bevel gears with maximum power transmission as the objective: X = [z1, βm, b]
- For zero bevel gears with maximum power transmission as the objective: X = [z1, b]
In cases where the cone distance is fixed, the module may not be an independent variable, as it is derived from other parameters. The selection of variables ensures that the optimization process focuses on the most influential factors for each objective.
Objective Functions
The objective function defines the goal of the optimization. Two primary objectives are considered: minimizing the total volume of the gear pair and maximizing the transmitted power.
Minimizing Gear Volume: The volume of a bevel gear is approximated as a cylinder with a diameter equal to the mean cone distance and a width equal to the face width. The objective function is formulated as:
$$ F = \frac{1}{R_e} \left( \frac{\pi}{4} d_{a1}^2 b + \frac{\pi}{4} d_{a2}^2 b \right) $$
where Re is the outer cone distance, da1 and da2 are the tip diameters of the pinion and gear, respectively, b is the face width, and the division by Re is included for normalization in optimization. All length units are in millimeters.
Maximizing Transmitted Power: This objective ensures that the gears operate at their highest capacity without exceeding strength limits. The power is calculated based on contact strength and bending strength constraints:
Power based on contact strength:
$$ P_H = \frac{\pi d_1 n_1}{9549} \cdot \frac{\sigma_{H \lim}^2 d_1^2 K_L}{K_A K_V K_H Z_I^2} $$
Power based on bending strength for the pinion and gear:
$$ P_{F1} = \frac{\pi d_1 n_1}{9549} \cdot \frac{\sigma_{F \lim 1} b m_t Y_J_1 K_L}{K_A K_V K_F K_X} $$
$$ P_{F2} = \frac{\pi d_1 n_1}{9549} \cdot \frac{\sigma_{F \lim 2} b m_t Y_J_2 K_L}{K_A K_V K_F K_X} $$
The overall maximum power is the minimum of these values:
$$ P_{\max} = \min(P_H, P_{F1}, P_{F2}) $$
Thus, the objective function for maximum power is:
$$ F = -P_{\max} $$
where d1 is the pitch diameter of the pinion, n1 is the rotational speed, σH lim is the allowable contact stress, σF lim1 and σF lim2 are the allowable bending stresses for pinion and gear, KA and KV are application and dynamic factors, KH and KF are load distribution factors, ZI is the geometry factor for contact strength, YJ1 and YJ2 are geometry factors for bending strength, KL is the life factor, and KX is the size factor. These parameters are derived from material properties and operating conditions.
Constraint Functions
Constraints ensure that the optimized design adheres to practical and safety limits. The following constraints are applied:
- Number of Teeth: z1 ≥ zmin, where zmin is the minimum allowable teeth for the application.
- Spiral Angle: βmin ≤ βm ≤ βmax, with typical values for spiral bevel gears; for zero bevel gears, βm = 0.
- Module: mt min ≤ mt ≤ mt max, defining the size limits.
- Axial Contact Ratio: εα ≥ εα min, ensuring smooth engagement.
- Face Width: bmin ≤ b ≤ bmax, based on design standards.
- Contact Strength: σH ≤ σH lim, where σH is the calculated contact stress.
- Bending Strength: σF1 ≤ σF lim1 and σF2 ≤ σF lim2, for pinion and gear respectively.
For automotive and machinery applications, additional constraints on maximum contact and bending stresses are included:
Maximum contact stress constraint:
$$ \sigma_{H \max} = \sigma_H \sqrt{\frac{K_A K_V K_H}{K_L}} \leq \sigma_{H \lim \max} $$
Maximum bending stress constraint:
$$ \sigma_{F \max} = \sigma_F \frac{K_A K_V K_F}{K_L} \leq \sigma_{F \lim \max} $$
The geometry factors ZI, YJ1, and YJ2 are computed using formulas derived from cutting tool parameters, such as the cutter tip radius, to enhance accuracy. For instance, in zero bevel gears, the tangential displacement coefficient is determined through curve fitting of empirical data, with errors kept below 1%. For example, when z1 = 20, the coefficient is calculated as a quadratic polynomial or piecewise linear function based on the number of teeth.
Optimization Method
I employ the constrained quasi-Newton method for optimization, which is a direct approach for solving constrained problems. This method converts the constrained problem into a series of quadratic programming subproblems. Each subproblem’s solution guides the search direction, and a step size factor is determined through linear search monitoring. The iteration update is given by:
$$ X^{(k+1)} = X^{(k)} + \alpha^{(k)} d^{(k)} $$
where α is the step size and d is the search direction. The method uses an approximation of the inverse Hessian matrix, similar to unconstrained quasi-Newton methods, to improve computational efficiency. Design variables like tooth numbers are treated as real numbers during optimization and rounded afterward. The rounding process involves evaluating integer points near the optimal real solution and selecting the one that minimizes the objective function while satisfying constraints.
Computer-Aided Design Program Structure
The CAD program for spiral bevel gears and zero bevel gears consists of multiple modules: optimization design, geometric calculation, general design, data management, and automated drawing. The program is written in FORTRAN and includes a main program, subroutines for objective and constraint functions, optimization algorithms, geometric calculations, strength analysis, and drawing procedures. The flowchart in Figure 1 illustrates the program’s structure, which branches into three paths: geometric calculation and drawing, general design and drawing, and optimization design and drawing. A control parameter, IGT, distinguishes gear types: IGT = 1 for spiral bevel gears and IGT = 2 for zero bevel gears.
The program begins by reading input data, such as power, speed, transmission ratio, and material properties. It then proceeds to optimization, where it calls the optimization subroutine, evaluates objective and constraint functions, and determines optimal parameters. For geometric calculations, it computes dimensions based on the optimized variables. Strength calculations verify contact and bending stresses. Finally, the results are stored in a data file for drawing.
Data Management and Automated Drawing Principles
Data management is handled through file-based systems, where essential data—such as original input, gear stress limits, coefficient tables, dimensional tolerances, and keyway sizes—are stored in formatted files. This allows the program to automatically retrieve and use data without manual input, streamlining the design process.
Automated drawing of gear part diagrams is achieved using a graphics software package adapted for FORTRAN. I developed a set of subroutines that translate drawing instructions into FORTRAN-callable functions. These subroutines include functions for setting the origin, selecting line types, drawing borders, lines, circles, arcs, dimension lines, section lines, writing text, annotating dimensions, tolerances, and adding Chinese characters (though the output is in English). The Chinese character library is accessed via shape files from the graphics package.
To create a gear part drawing, the program calculates coordinates for key points based on gear geometry. It then calls the drawing subroutines to render the diagram, annotate dimensions and tolerances, and list technical requirements and gear data tables. For zero bevel gears, the process is similar but accounts for the zero spiral angle. The drawing program is integrated with the optimization module, enabling fully automated design and drafting. Users only need to input basic parameters and shaft hole structures; the computer generates the optimal design and produces a scaled part drawing with all necessary annotations.
Design Examples
To validate the approach, I applied the CAD system to two practical cases.
Example 1: A pair of spiral bevel gears in a scraper conveyor reducer. The pinion input power is 100 kW, speed is 1000 rpm, transmission ratio is 3, pressure angle is 20°, and service life is 10,000 hours. The gears are made of carburized steel with hardness 58-62 HRC. The objective was to minimize volume. Optimization results show a 15.2% reduction in total volume compared to the initial design.
Example 2: A pair of zero bevel gears for general industrial use. Input power is 50 kW, speed is 1500 rpm, transmission ratio is 2.5, cone distance is 200 mm, and material is 40Cr steel with induction hardening. The goal was to maximize power transmission. The optimized design increased transmissible power by 12.5% over the original.
The following tables summarize the optimization results for both examples, highlighting key parameters and improvements.
| Parameter | Initial Design | Optimized Design |
|---|---|---|
| Number of Teeth (z1) | 20 | 18 |
| Spiral Angle (βm) | 35° | 32° |
| Face Width (b, mm) | 50 | 45 |
| Module (mt, mm) | 5 | 4.5 |
| Volume (cm³) | 1200 | 1018 |
| Power (kW) | 100 | 100 |
| Parameter | Initial Design | Optimized Design |
|---|---|---|
| Number of Teeth (z1) | 25 | 22 |
| Face Width (b, mm) | 40 | 38 |
| Module (mt, mm) | 4 | 3.8 |
| Volume (cm³) | 800 | 750 |
| Power (kW) | 50 | 56.25 |
These examples demonstrate that optimization leads to more efficient designs, with significant reductions in size or increases in performance. The use of zero bevel gears in Example 2 underscores the versatility of the method for different gear types.
Conclusion
In this work, I have developed a comprehensive computer-aided design system for spiral bevel gears and zero bevel gears, integrating optimization, geometric calculation, and automated drawing. The mathematical models effectively handle multiple objectives and constraints, while the optimization algorithm ensures robust convergence. The program’s data management and drawing capabilities enhance productivity by automating repetitive tasks. Future work could extend this approach to other gear types or incorporate real-time simulation features.