Abstract:
Spiral bevel gear milling machines are crucial equipment in spiral bevel gear manufacturing. The precision of these machines directly influences the machining precision of the gears, further impacting gear transmission performances such as accuracy, stability, and load uniformity. Due to the complexity of the machine’s structure and working principle, establishing a mapping relationship between machine precision and gear machining precision is challenging. Traditional precision design for spiral bevel gear milling machines is still experience-based, which may increase manufacturing costs and difficulty. To address this issue, this thesis proposes and studies a gear transmission performance-oriented design methodology for the precision of spiral bevel gear milling machines.
1. Introduction
1.1 Introduction
Spiral bevel gears are fundamental transmission components for transmitting motion between non-parallel shafts. They are widely used in high-speed and heavy-load applications due to their smooth transmission, high load-carrying capacity, low noise, and long service life. In recent years, straight bevel gear pairs in aviation, marine, automotive, and various precision machine tools have gradually been replaced by spiral bevel gear pairs.
1.2 Overview of Spiral Bevel Gear Design Theory
Spiral bevel gears are classified into circular arc tooth bevel gears, prolonged hypocycloid tooth bevel gears, and quasi-involute tooth bevel gears based on the curve type of their tooth surface pitch line. Major manufacturers such as Gleason adopt the circular arc tooth system, Oerlikon uses the prolonged hypocycloid tooth system, and Klingelnberg employs the quasi-involute tooth system. In China, the majority of spiral bevel gear cutting machines used in industries represented by the automotive industry are Gleason-type, thus this thesis focuses on Gleason spiral bevel gear manufacturing technology.
1.3 Overview of Spiral Bevel Gear Machine Tool Development
Table 1: Development Status of Spiral Bevel Gear Machine Tools
| Region | Development Focus | Key Characteristics |
|---|---|---|
| International | Advanced numerical control technology | High precision, efficiency, and automation |
| China | CNC retrofitting of traditional mechanical swing-table machines | Improving machine precision and manufacturing capabilities |
1.4 Overview of Machine Tool Precision Design Research
Machine tool precision design is a critical aspect of machine tool development, divided into precision analysis and precision synthesis. Precision analysis predicts machine tool processing accuracy based on known component precisions, while precision synthesis optimizes component precisions before machine tool manufacturing to achieve optimal balance between accuracy and cost.
1.4.1 Machine Tool Precision Analysis
Research on machine tool precision analysis has a long history, evolving various precision analysis methods. Early methods included using triangular relationships and vector methods. Subsequent research, based on rigid body motion and small angle error assumptions, established analytical second-order models for geometric errors of three-axis machines. More recently, the variational method has been used to derive generalized accuracy models for arbitrarily configured machine tools, although these models often overestimate workpiece accuracy due to neglecting processing techniques.
1.4.2 Machine Tool Precision Synthesis
Traditional precision synthesis methods determine tolerance types and grades for each component based on designers’ experience, standards, and manuals. However, with the increasing number of design variables in modern machine tool precision synthesis, traditional methods are inadequate. Recent research includes precision synthesis methods based on sensitivity analysis, Monte Carlo simulation, neural networks, and genetic algorithms.
1.4.3 Current Issues in Precision Design of Spiral Bevel Gear Milling Machines
Despite extensive research on machine tool precision design, specific issues remain for spiral bevel gear milling machines:
- Precision design focuses on individual tooth surface errors, such as tooth profile deviation or pitch deviation, failing to comprehensively reflect the impact of tooth surface machining errors on gear transmission performance.
- Geometric error models do not comprehensively consider geometric errors of all machine components, leading to difficulties in supporting subsequent precision synthesis.
- The complex processing principle of spiral bevel gears makes the quantitative relationship between machine geometric errors and tooth surface errors time-consuming to solve, rendering traditional precision synthesis methods unsuitable.
1.5 Main Research Content
Addressing the current research status and deficiencies, this thesis proposes a precision design methodology for spiral bevel gear milling machines oriented towards gear transmission performance. This involves determining basic gear parameters and processing methods, obtaining machine adjustment parameters through tooth contact analysis to meet transmission performance requirements, establishing component transformation matrices incorporating geometric errors using multibody system kinematics, and then deriving error tooth surface equations. Tooth profile deviations and pitch deviations are used to characterize the impact of geometric errors of various machine components on gear pair transmission performance. Approximate models for tooth profile deviations and pitch deviations under the combined action of geometric errors are established. Finally, based on tooth surface machining accuracy requirements, Monte Carlo methods and optimization algorithms are applied to optimize various geometric accuracies of the machine, aiming to reasonably control manufacturing costs while ensuring gear transmission performance.
The main research content includes:
- Establishing and validating mathematical models for conjugate tooth surfaces and transition surfaces for generating method large wheels and modified method small wheels.
- Performing tooth contact analysis to determine tooth surface contact regions and optimizing machine adjustment parameters.
- Establishing geometric error models for spiral bevel gear milling machines and deriving error tooth surface equations.
- Analyzing the influence of machine geometric errors on tooth profile deviations and pitch deviations.
- Summarizing the technical process for precision design of spiral bevel gear milling machines oriented towards gear transmission performance and developing a software system to assist designers in quickly completing precision design.
2. Establishment of Mathematical Models for Spiral Bevel Gear Tooth Surfaces
2.1 Spiral Bevel Gear Cutting Methods
Spiral bevel gears are conventionally classified as spiral bevel gears (intersecting axes) and hypoid gears (offset axes). This thesis focuses on hypoid gears, with spiral bevel gears being a special case. Gear cutting methods are generally divided into direct generative and indirect generative methods.
2.2 Mathematical Model for Conjugate Tooth Surfaces
Based on the principle of local conjugation, mathematical models for conjugate tooth surfaces and transition surfaces are established for both the generating method and modified method.
Table 2: Key Parameters for Large Wheel Processing
| Parameter | Description |
|---|---|
| 2S, 2q | Blade position parameters |
| 2X | Axial wheel position |
| B2X | Bed position |
| M2δ | Wheel blank mounting angle |
| 02E | Vertical wheel position |
| 02i | Ratio of large wheel to generating wheel angular velocities (rolling ratio) |
2.3 Verification of Tooth Surface Models
The correctness of the tooth surface models is verified through comparison with actual tooth surfaces, ensuring the accuracy of subsequent analysis and design.
3. Tooth Contact Analysis
Tooth contact analysis is conducted to determine tooth surface contact regions and optimize machine adjustment parameters to meet transmission performance requirements. This involves calculating tooth surface contact trajectories and contact areas, and modifying contact areas as needed.
4. Geometric Error Modeling and Machining Error Modeling for Gear Milling Machines
4.1 Geometric Error Modeling for Gear Milling Machines
The topological structure of the gear milling machine is described, and geometric error elements are analyzed. Component transformation matrices are established to comprehensively incorporate geometric errors of the machine with multibody kinematics.
4.2 Derivation of Error Tooth Surface Equations
Based on the component transformation matrices, error tooth surface equations are derived to quantitatively analyze the impact of machine geometric errors on tooth surface accuracy.
4.3 Influence of Machine Geometric Errors on Tooth Profile and Pitch Deviations
The influence of machine position errors and motion errors on tooth profile deviations and pitch deviations is analyzed. This analysis provides a basis for subsequent precision synthesis.
5. Precision Design of Spiral Bevel Gear Milling Machines Oriented Towards Gear Transmission Performance
5.1 Precision Analysis of Spiral Bevel Gear Milling Machines
Machine geometric accuracy and tolerances are analyzed, and tooth surface machining accuracy requirements are determined. Monte Carlo analysis is applied to assess tooth surface machining accuracy.
5.2 Precision Synthesis of Spiral Bevel Gear Milling Machines
Based on the precision analysis results, precision synthesis is conducted to optimize various geometric accuracies of the machine.
5.3 Precision Design Process
The technical process for precision design is summarized, integrating various research findings.
5.4 Development of Precision Design Software
A software system is developed for precision design, utilizing CAD and database technology to assist designers in efficiently completing precision design tasks.
6. Conclusion
This thesis proposes a precision design methodology for spiral bevel gear milling machines oriented towards gear transmission performance. By comprehensively considering machine geometric errors and their impact on tooth surface accuracy, and optimizing machine precision based on tooth surface machining accuracy requirements, this methodology aims to reasonably control manufacturing costs while ensuring gear transmission performance. The research findings provide theoretical support and practical guidance for the precision design of spiral bevel gear milling machines.