This article focuses on the research of tolerance modeling, assembly error transmission laws, and tolerance optimization for the spindle components of bevel gear machine tools. The geometric error model including dimensional and geometric tolerances is established based on the small displacement spinor theory. The error accumulation and transmission laws of common fitting forms are analyzed, and the assembly error transmission model is established. The assembly accuracy reliability evaluation index is established, and the tolerance optimization model is solved by using the particle swarm optimization algorithm. The research results can provide theoretical guidance for improving the tolerance processing economy of bevel gear machine tools.
1. Introduction
1.1 Research Background and Significance
Bevel gear machine tools are important equipment for processing bevel gears in the equipment manufacturing industry. The spindle components in the tool box directly affect the machining quality of bevel gears. In China, there is a gap in the precision and reliability of machine tools compared with foreign countries. Therefore, it is necessary to study the evaluation and optimization methods of assembly accuracy for the design of machine tools.
1.2 Research Status of Tolerance Modeling, Error Transmission, and Tolerance Optimization
- Tolerance Modeling: Many scholars have studied tolerance modeling methods, such as the small displacement spinor model, the drift tolerance band theory, and the T – MAP model.
- Error Transmission: The error transmission models of assembly bodies have been studied, including the models based on homogeneous coordinate transformation matrices, the low – order body array method, and the small displacement spinor theory.
- Tolerance Optimization: Tolerance optimization algorithms have been developed, such as the ant colony algorithm, the genetic algorithm, and the particle swarm optimization algorithm.
1.3 Main Research Contents
- Establish geometric error models for common geometric elements based on the small displacement spinor theory.
- Analyze the error accumulation and transmission laws of different fitting forms and establish assembly error transmission models.
- Establish assembly accuracy reliability evaluation indexes and solve tolerance optimization models.
2. Tolerance Modeling Based on the Small Displacement Spinor Theory
2.1 Introduction to the Small Displacement Spinor Theory
The small displacement spinor is a vector composed of six motion components of a rigid body with a small displacement. It is suitable for representing the deviation of an ideal shape feature.
2.2 Structure Analysis of the Tool Spindle Component
The main structure of a CNC bevel gear machine tool includes a bed, a column, a tool box, a workpiece box, a rotary table, and an electrical cabinet. The tool spindle component system mainly includes a tool spindle, a box body, and a tool disc, and the geometric elements include planes, cylinders, cones, and axes.
2.3 Tolerance Modeling Based on the Small Displacement Spinor
- Tolerance Principles: The independent principle is selected as the tolerance principle in this study.
- Error Variation Analysis of Geometric Elements: The error variation models of planes, cylinders, cones, and axes are established according to the specific conditions of geometric elements.
2.4 Solution of the Actual Variation Interval of Spinor Parameters Based on the Monte Carlo Method
The Monte Carlo simulation method is used to solve the actual variation range of spinor parameters by assuming that the error components follow a normal distribution.
2.5 Establishment of the Functional Relationship between Tolerance and the Actual Variation Range
The response surface method is used to establish the functional relationship between the actual variation interval bandwidth of spinor parameters and tolerance.
2.6 Example Analysis
Taking the cone and cylinder geometric elements as examples, the actual variation range of spinor parameters is solved and compared with the ideal variation range, and the function relationship between the actual variation interval bandwidth and tolerance is established.
3. Establishment and Verification of the Assembly Error Transmission Model
3.1 Introduction
The assembly process of machine tools causes the machining errors of parts to accumulate and form assembly errors. The error transmission forms of different fitting surfaces need to be analyzed to establish the assembly error transmission model.
3.2 Error Modeling of Fitting Surfaces
The fitting surfaces are mainly divided into plane, cylinder, and cone fitting surfaces. The error models of different fitting surfaces are established according to their error formation mechanisms.
3.3 Error Transmission Mechanism of Fitting Surfaces
- Relationship between Adjacent Fitting Surfaces: Fitting surfaces can be divided into series and parallel fitting surfaces according to the error transmission path.
- Error Transmission Attributes of Fitting Surfaces: The error transmission attributes of fitting surfaces can be divided into strong constraints, weak constraints, and no constraints.
- Analysis of the Actual Error Transmission Attributes of Fitting Surfaces: The actual error transmission attributes of parallel fitting surfaces are affected by the positioning order and assembly interference.
- Calculation of the Actual Error Transmission of Fitting Surfaces: The actual error transmission of the plane – cone parallel fitting in the tool spindle component is calculated as an example.
3.4 Error Transmission Modeling and Analysis of the Tool Spindle Component
- Error Transmission Model of the Tool Spindle Component: The assembly error transmission model of the tool spindle component system is established with the fitting surface as the error transmission node.
- Error Sensitivity Analysis: The error sensitivity analysis model is established to determine the influence of each error component on the overall assembly error.
3.5 Experimental Verification
A point cloud scanning experimental platform is built to verify the correctness of the assembly error transmission model and the assembly error calculation results.
4. Tolerance Optimization of the Error Model Based on the Particle Swarm Algorithm
4.1 Introduction
Based on the error transmission model of the tool spindle component system, a tolerance optimization model is established with the minimum processing cost as the target and the assembly accuracy reliability and tolerance selection principles as constraints.
4.2 Analysis of Assembly Accuracy Reliability
- Definition of Reliability: Reliability refers to the ability of a product to operate normally without failure under certain conditions and within a certain time.
- Solution Method of Reliability: The Monte Carlo simulation method is used to solve the assembly accuracy reliability.
4.3 Tolerance Optimization
- Tolerance – Cost Function: The tolerance – cost function is established according to the relationship between tolerance and manufacturing cost.
- Optimization Model of the Tool Spindle Component System: The optimization model of the tool spindle component system is established with the tolerance processing cost as the optimization target, various tolerances as optimization variables, and assembly accuracy reliability and tolerance value selection principles as constraints.
- Particle Swarm Optimization Algorithm: The particle swarm optimization algorithm is used to solve the tolerance optimization of the tool spindle component system.
4.4 Chapter Summary
The tolerance optimization model of the tool spindle component system is established and solved by using the particle swarm optimization algorithm, which reduces the processing cost without reducing the assembly reliability.
5. Summary and Outlook
5.1 Summary
- Geometric error models of common geometric elements are established based on the small displacement spinor theory.
- The error accumulation and transmission laws of different fitting forms are analyzed, and the assembly error transmission model is established.
- The assembly accuracy reliability evaluation index is established, and the tolerance optimization model is solved.
5.2 Outlook
- Establish a universal geometric element tolerance model.
- Consider the influence of human operation and assembly process on error transmission.
- Improve the error transmission model and tolerance optimization model to cover the entire machine tool.
- Design a more accurate and easier – to – operate experiment to verify the assembly error transmission model.
6. Appendix: Tables and Figures
Table 1: Geometric Elements and Their Error Variation Models
Table 2: Error Transmission Attributes of Common Fitting Surfaces
Table 3: Geometric Elements Tolerance and Simulation Values in the Tool Spindle Component System
| Fitting Surface | Geometric Element | Tolerance | Given Value Range |
|---|---|---|---|
| Cylinder Fit D1 | Inner Cylinder a | Hole Size Tolerance | [0.01,0.03] |
| Cylinder Degree | T = 0.022 | ||
| Axis Position Tolerance T3 = 0.005 | [0.003,0.012] | ||
| Axis Straightness T4 = 0.003 | [0.001,0.01] | ||
| Outer Cylinder b | Cylinder Degree Ts = 0.003 | [0.001,0.01] | |
| Axis Size Tolerance T6 = 0.015 | [0.01,0.03] | ||
| Cone Fit D2 and Plane Fit D3 | Size Tolerance | [0.002,0.02] | |
| Axis Straightness | T4 = 0.003 | [0.001,0.01] | |
| Size Tolerance | [0.002,0.02] | ||
| Tool Disc Inner Cone d | Size Tolerance | T8 = 0.004 | |
| Axis Position Tolerance T3 = 0.005 | [0.003,0.012] | ||
| Tool Disc Plane e | Size Tolerance T9 = 0.01 | [0.005,0.02] | |
| T10 = 0.003 | [0.001,0.01] | ||
| Main Shaft Plane f | T = 0.003 | [0.001,0.01] | |
| Size Tolerance Ti2 = 0.01 | [0.005,0.02] |
Figure 1: Machine Tool Structure Sketch
[Insert a sketch of the machine tool structure here, showing the main components such as the bed, column, tool box, workpiece box, rotary table, and electrical cabinet.]
Figure 2: Geometric Tolerances and Their Small Displacement Spinor Expressions
[Show diagrams of different geometric tolerances (e.g., straightness, plane-ness, face parallelism, coaxility) and their corresponding small displacement spinor expressions.]
Figure 3: Plane Error Variation Model
[Illustrate the plane error variation model with a diagram, showing the relationship between the plane, its normal, the size tolerance, and the perpendicularity tolerance.]
Figure 4: Cylinder Error Variation Model
[Present the cylinder error variation model with a diagram, indicating the cylinder, its axis, the diameter size tolerance, and the cylinder degree tolerance.]
Figure 5: Cone Error Variation Model
[Depict the cone error variation model with a diagram, including the cone, its axis, the size tolerance, and the cone angle.]
Figure 6: Axis Error Variation Model
[Show the axis error variation model with a diagram, highlighting the axis, its position tolerance, and the straightness tolerance.]
Figure 7: Series and Parallel Fitting Surfaces
[Draw diagrams of series and parallel fitting surfaces to illustrate the difference in error transmission paths.]
Figure 8: Assembly Error Transmission Model of the Tool Spindle Component System
[Present a detailed diagram of the assembly error transmission model of the tool spindle component system, showing the error transmission paths from different fitting surfaces to the tool disc.]
Figure 9: Point Cloud Scanning Experimental Platform
[Show a schematic of the point cloud scanning experimental platform, including the laser scanner, the tool spindle component, and the computer for data processing.]
7. Conclusion
This article has conducted in-depth research on the assembly accuracy evaluation and optimization of spindle components in bevel gear machine tools. The research results can provide theoretical support and practical guidance for improving the machining quality and efficiency of bevel gear machine tools, and have important theoretical and practical value for the development of the equipment manufacturing industry.