In the development of wheeled vehicles, the straight bevel miter gear differential stands out due to its straightforward construction, ease of manufacturing, and simple maintenance. This system is central to distributing torque while allowing wheels to rotate at different speeds. Historically, the design process for such differentials, encompassing parameter calculation and detailed drafting, has been labor-intensive, time-consuming, and prone to human error. The traditional cycle involves extensive manual calculations and drafting, which significantly delays project timelines.
The integration of Computer-Aided Design (CAD) methodologies presents a transformative solution. By synthesizing computer technology with modern design principles, CAD enables a more efficient, precise, and automated approach. My work focuses on developing a specialized CAD system for the design of differentials employing straight bevel miter gears. This system aims to drastically shorten design cycles, enhance the accuracy and reliability of outputs, and facilitate an optimized, automated workflow. Building upon foundational research in single-pair bevel gear optimization, this project extends into a comprehensive system for the complete differential assembly, addressing both computational and graphical generation challenges.
System Architecture and Functional Overview
The developed CAD system is engineered to accommodate various open and closed differential configurations used across different types of wheeled vehicles. To cater to diverse user requirements, the system is built with a modular architecture offering the following core functionalities:
- Geometric and kinematic calculation of the differential assembly.
- Strength verification for critical components (gears, shafts).
- Optimization of the differential design based on specified objectives.
- Automated generation of detailed working drawings for major components.
The system interface employs natural language dialogues, supporting both Chinese and English input/output. The core computational engine is developed in FORTRAN and implemented on an IBM-PC/XT platform. Given the system’s scale—incorporating over four thousand statements and requiring support from optimization libraries, graphics packages, and font libraries—batch processing is utilized to manage memory constraints effectively.
The program structure is modular, with individual modules for specific tasks interacting through defined data pathways. A central main program controls the flow, calling upon subroutines for input, optimization algorithms, and various calculation modules (geometry, strength, etc.). The structure is designed for flexibility, allowing users to select the calculation type (KKK) and define the objective function for optimization (FFF). This modularity also permits future expansion, such as adding new objective function modules.
Mathematical Modeling for Optimal Design
The heart of the CAD system lies in its formulation of the differential design as an optimization problem. The choice of objective function is crucial and varies based on overarching vehicle design goals, such as matching system durability, minimizing space for ground clearance, or achieving specific torque-biasing characteristics.
Definition of Objective Functions
Two primary objective functions have been implemented, focusing on minimizing size and maximizing torque capacity.
1. Minimum Volume: For a given required torque capacity, the goal is to minimize the overall volume of the differential while ensuring the strength of the miter gears, side gears, and cross/planetary shafts. The objective function $F_1(\vec{X})$ representing the total volume can be expressed as:
$$ F_1(\vec{X}) = \frac{\pi}{4} \left[ B (D_{p1}^2 + D_{p2}^2) + Z \cdot L \cdot d_s^2 + 2 \cdot L_1 \cdot d_h^2 \right] $$
where $m$ is the module, $Z_p$ and $Z_s$ are the numbers of teeth on the planetary and side miter gears, $\delta_p$ and $\delta_s$ are their pitch angles, $B$ is the face width, $R$ is the cone distance, $D_{p1}, D_{p2}$ are the outer pitch diameters, $d_s$ is the planetary shaft diameter, $d_h$ is the half-shaft diameter, and $Z$ is the number of planetary gears.
When angle modification (profile shifting) is applied to the miter gears, maximizing the sum of shift coefficients $(x_1 + x_2)$ is also desirable to improve gear performance. A combined objective function using a weighted sum approach can be formulated:
$$ F_1′(\vec{X}) = W_1 \cdot F_1(\vec{X}) – W_2 \cdot (x_1 + x_2) $$
where $W_1$ and $W_2$ are weighting factors.
2. Maximum Transmittable Torque: When the installation space is constrained, the objective shifts to maximizing the torque capacity of the differential within a given envelope. Since differential gears experience relatively few load cycles, pitting resistance is often secondary. Therefore, the limiting torque is governed by gear bending strength, planetary shaft bearing pressure, and half-shaft torsional strength. The objective function $F_2(\vec{X})$ is defined as the minimum of these three capacities:
$$ F_2(\vec{X}) = \min(T_{bend}, T_{bearing}, T_{torsion}) $$
where $T_{bend}$, $T_{bearing}$, and $T_{torsion}$ represent the maximum torque derived from gear bending, planetary shaft crushing, and half-shaft torsion limits, respectively.
Selection of Design Variables
The fundamental geometric parameters defining the differential include the number of teeth on the planetary gear ($Z_p$) and side gear ($Z_s$), the face width coefficient ($\phi_R = B/R$), the module ($m$), the planetary shaft diameter ($d_s$), and the half-shaft diameter ($d_h$). To manage problem dimensionality, $Z_p$ and $Z_s$ are often treated as input parameters cycled within a feasible range (e.g., $Z_p = 10-16$, $Z_s/Z_p = 1.4-2.0$). For gears with a $20^\circ$ pressure angle, addendum modification and tangential shift coefficients are typically obtained from standard tables. For a $22.5^\circ$ or $25^\circ$ pressure angle, angle modification coefficients ($x_1, x_2$) are introduced as design variables. The specific set of variables $\vec{X}$ depends on the objective:
| Objective | Pressure Angle | Key Given Parameter | Design Variable Vector $\vec{X}$ |
|---|---|---|---|
| Min Volume | $20^\circ$ | Torque Requirement | $[m, \phi_R, d_s, d_h]$ |
| Min Volume | $22.5^\circ/25^\circ$ | Torque Requirement | $[m, \phi_R, d_s, d_h, x_1, x_2]$ |
| Max Torque | $20^\circ$ | Cone Distance ($R$) | $[m, B, d_s, d_h]$ |
| Max Torque | $22.5^\circ/25^\circ$ | Cone Distance ($R$) | $[m, B, d_s, d_h, x_1, x_2]$ |
Formulation of Constraint Functions
To ensure a feasible, reliable, and manufacturable design, the optimization is bound by a comprehensive set of constraint functions $g_j(\vec{X}) \leq 0$:
- Gear Geometry and Assembly:
- Tooth number combinations for assembly: $(Z_{s,left} + Z_{s,right}) / Z_p =$ Integer.
- Face width limit: $0.25 \leq \phi_R \leq 0.3$.
- Module range: $m_{min} \leq m \leq m_{max}$.
- Component Strength:
- Gear bending safety: $S_{Fp} \geq [S_F]$, $S_{Fs} \geq [S_F]$.
- Planetary shaft bearing pressure: $\sigma_{bearing} \leq [\sigma]_{bearing}$.
- Half-shaft torsional stress: $\tau_{torsion} \leq [\tau]_{torsion}$.
- Gear Meshing Quality (for angle modification): Constraints are evaluated using the virtual spur gear equivalence of the straight bevel miter gear.
- Undercut prevention: $x \geq x_{min}$.
- Contact ratio: $\epsilon_{\alpha} \geq [\epsilon_{\alpha}]$.
- Tip tooth thickness: $s_a \geq [s_a]$.
- Absence of interference at the root fillet.
The formulas for virtual gear parameters and constraints, such as the contact ratio for modified bevel gears, are computed as follows, where $z_v$ is the virtual tooth number ($z_v = z / \cos \delta$):
$$ \epsilon_{\alpha} = \frac{1}{2\pi} [ z_{v1}(\tan\alpha_{a1} – \tan\alpha’) + z_{v2}(\tan\alpha_{a2} – \tan\alpha’) ] $$
with the operating pressure angle $\alpha’$ derived from:
$$ \text{inv}\,\alpha’ = \text{inv}\,\alpha + 2 \frac{x_1 + x_2}{z_{v1} + z_{v2}} \tan\alpha $$
Implementation Details in the Optimization Engine
Optimization Algorithm
Given the computational expense of evaluating the objective and constraint functions—which involve complex geometry and strength calculations—an efficient constrained optimization method is essential. The system employs a Constrained Variable Metric Method, specifically a variant of the Sequential Quadratic Programming (SQP) method from our optimization library. This CFSQP-type algorithm is well-suited for nonlinear problems with multiple constraints, as it approximates the Hessian matrix to achieve superlinear convergence near the optimum, reducing the total number of function evaluations required.
Data Handling and Pre-processing
Curve and Table Fitting: The design process relies on numerous empirical curves and tables (e.g., for tangential modification coefficients, stress correction factors, fatigue limits). To integrate these seamlessly into the automated system, all such data are approximated using least-squares polynomial fitting. The accuracy of these fits is maintained within a 1% error margin. For example, the tangential shift coefficient $x_t$ for a $20^\circ$ pressure angle and pinion tooth count $Z_p=11$ is represented as:
$$ x_t = 0.275 – 0.0125 \times Z_p + 0.0005 \times Z_p^2 $$
Scaling of Variables and Normalization of Constraints: Design variables often differ by orders of magnitude (e.g., $\phi_R \approx 0.3$ vs. $d_h \approx 40\,mm$), causing ill-conditioning in the optimization search. To improve numerical stability and convergence, variables are scaled to a similar order of magnitude. Similarly, constraint functions are normalized to a range around [-1, 1]. For a constraint $g(\vec{X}) = c – c_{max} \leq 0$, the normalized form is:
$$ \hat{g}(\vec{X}) = \frac{c}{c_{max}} – 1 \leq 0 $$
This prevents any single constraint from dominating the search process due to its raw numerical value.
Post-Processing: Parameter Rounding and Standardization
Optimal solutions yield continuous values, but manufacturing requires standardized or integer values for parameters like tooth numbers ($Z_s$), module ($m$), and shaft diameters ($d_s, d_h$). A dedicated post-processing module handles this. It creates a discrete grid of integer/standardized values around the continuous optimum. All points on this grid satisfying the constraints are evaluated using the objective function. The point yielding the best objective value is selected as the practical optimum. This rounding process must respect additional constraints:
- Revised Assembly Condition: After rounding $Z_s$, the condition $(Z_{s,left} + Z_{s,right})/Z_p = \text{Integer}$ must be re-checked.
- Cone Distance Tolerance: For the “Max Torque” objective, the cone distance $R$ is fixed. Rounding $m$ and $Z$ will alter the theoretical $R’$. The change must be within a permissible relative error $\Delta$:
$$ \frac{|R’ – R|}{R} \leq \Delta $$
Parametric Automated Drafting Technology
The automated drawing generation is a cornerstone of this CAD system. It utilizes the GRAFTRA-86 graphics package as the foundational software. Since GRAFTRA-86 is inherently command-driven, a bridge was built to enable parametric drawing from within the FORTRAN environment. This was achieved by developing an interface library that encapsulates GRAFTRA-86’s plotting commands into callable FORTRAN subroutines. This library includes routines for file management, basic drawing primitives (line, circle, arc), and specialized functions for dimensioning, tolerancing, and creating title blocks.
The drawing program for a specific component, such as a planetary miter gear, is created by combining these interface subroutines with a parameterized geometric model of the part. The execution flow is:
- Open the data file containing the finalized design parameters from the optimization module.
- Establish a coordinate system and compute the coordinates of all geometric features (centers, endpoints, circle radii) as variables based on input parameters (e.g., $m$, $Z$, $B$, $d_s$).
- Sequentially call drawing primitives to construct the geometry (front view, section view).
- Call dimensioning subroutines to annotate the drawing with sizes and tolerances.
- Call text subroutines to fill in the title block, technical specifications table, and notes.
The key to parameterization is that all coordinate calculations are expressed as functions of the primary design variables, not fixed numbers. Changing the input data file automatically generates a new, correctly proportioned drawing. The modular structure of the drawing code allows easy reordering of views or notes for different drafting standards.
System Data Flow and Management Strategy
Robust data management is critical for a complex CAD system. The data flow begins with user input—either via interactive dialogue or from a pre-defined specification file—which provides the initial design requirements and constraints. The parameter calculation/optimization module processes this input, performs the optimal design search, and outputs the final component dimensions and properties to a structured results file. This file acts as the sole input for the automated drafting module.
To manage the vast amounts of data involved—material properties, geometric interdependencies, strength formulas, standard tolerance tables—a hybrid approach is used:
- File-Based Management: Large, stable datasets (standard gear tooth forms, material grade properties, default tolerance tables) are stored in dedicated disk files. This separates data from program logic, allowing easy updates without recompiling the core software.
- Program-Embedded Management: Volatile or frequently accessed logic, such as the sequence of calculation steps or simple interpolation formulas, is implemented directly within the FORTRAN code using DATA statements and conditional blocks, ensuring fast runtime access.
Design Example and System Evaluation
To validate the system, a case study was performed to redesign the differential for a medium-duty truck. The primary goal was enhanced reliability. Key input parameters were:
| Parameter | Value |
|---|---|
| Input Torque to Final Drive | 1500 Nm |
| Pinion Speed | 2500 rpm |
| Final Drive Ratio | 5.0 |
| Required Gear Life | 5000 hours |
| Gear Material | 20CrMnTi, Case-Hardened |
| Gear Accuracy Grade | 7-6-6 (AGMA) |
| Number of Planetary Gears | 4 |
The system was executed on an IBM-PC/XT. The optimization calculation converged in approximately 2 minutes and 30 seconds. The subsequent generation of a complete working drawing for the planetary miter gear took about 1 minute and 40 seconds. A comparison between a baseline design and the optimized results is summarized below:
| Design Parameter / Result | Baseline Design | Optimized for Min Volume | Optimized for Max Torque |
|---|---|---|---|
| Planetary Gear Teeth ($Z_p$) | 11 | 10 | 12 |
| Side Gear Teeth ($Z_s$) | 18 | 20 | 18 |
| Module, $m$ (mm) | 5.5 | 5.0 | 5.75 |
| Face Width, $B$ (mm) | 32 | 28 | 35 |
| Planetary Shaft Diameter, $d_s$ (mm) | 18 | 16 | 20 |
| Calculated Volume (cm³) | ~1250 | ~1020 (-18.4%) | ~1380 |
| Max Transmittable Torque (Nm) | ~1650 | ~1580 | ~1850 (+12.1%) |
| Bending Safety Factor (Planetary Gear) | 1.8 | 2.1 | 2.0 |
The results clearly demonstrate the system’s efficacy. The “Min Volume” optimization achieved an 18.4% reduction in envelope size while maintaining required torque capacity and improving safety margins. Conversely, the “Max Torque” optimization boosted torque capacity by over 12% within a slightly larger but still feasible space. The automated drawings produced were fully compliant with industrial standards, containing all necessary views, dimensions, tolerances, and technical notes.
Conclusion: The developed CAD system for straight bevel miter gear differentials proves to be a powerful and flexible engineering tool. It successfully integrates optimal design theory with automated drafting, creating a streamlined workflow from specification to manufacturing drawing. The system’s modular architecture, robust numerical methods, and parametric approach allow it to adapt to various design requirements, consistently yielding superior designs compared to conventional iterative methods. The significant reduction in design time—from days to minutes—along with improved design quality, underscores its substantial practical value for the automotive and machinery industries. Future work will focus on expanding the component library, integrating with 3D solid modeling, and enhancing the user interface to transition towards a comprehensive commercial CAD product.