This article focuses on the assembly accuracy of the cutter spindle components in a bevel gear machine tool. It begins with an introduction to the research background and significance, highlighting the importance of accurate assembly for the quality of bevel gears and the overall performance of the machine tool. The study involves tolerance modeling, error transfer analysis, and tolerance optimization. Through a detailed exploration of geometric elements, error accumulation and transfer laws, and the application of reliability theory, a comprehensive approach to evaluating and optimizing assembly accuracy is presented. The research not only provides theoretical support for improving the manufacturing quality of bevel gear machine tools but also has practical implications for enhancing the efficiency and reliability of the manufacturing process.
1. Introduction
1.1 Research Background
Bevel gear machine tools play a crucial role in the manufacturing of bevel gears, which are widely used in aerospace, marine, and automotive industries. The cutter spindle components within these tools significantly impact the machining quality of bevel gears. In recent years, with the development of CNC technology, there has been a continuous pursuit of higher precision and reliability in machine tool manufacturing. However, compared to international standards, there is still a gap in the accuracy and reliability of domestic machine tools. This research is motivated by the need to address these issues and improve the assembly accuracy of the cutter spindle components.
1.2 Research Significance
The accurate assembly of cutter spindle components is essential for ensuring the quality of bevel gears and the overall performance of the machine tool. By studying tolerance modeling, error transfer, and tolerance optimization, this research aims to provide a theoretical basis for reducing manufacturing costs, improving assembly efficiency, and enhancing the reliability of the machine tool. It also has practical value in guiding the design and manufacturing process of bevel gear machine tools.
2. Tolerance Modeling Based on Small Displacement Torsor Theory
2.1 Cutter Spindle Component Structure Analysis
The CNC bevel gear machine tool consists of several main parts, including the bed, column, tool box, workpiece box, and rotary table. The cutter spindle components in the tool box are the focus of this study. These components contain various geometric elements such as planes, cylinders, cones, and axes.
2.2 Tolerance Modeling Principles
The small displacement torsor theory is used to model the tolerances of geometric elements. A torsor is a vector with six motion components representing the deviation of an ideal shape feature. The tolerance model includes both dimensional and geometric tolerances, and the torsor parameters are subject to constraint and variation inequalities.
2.3 Geometric Element Error Modeling
- Plane Error Modeling: For a plane geometric element, considering the influence of dimensional tolerance and perpendicularity tolerance, the error variation model is established. The small displacement torsor expression and the corresponding variation and constraint inequalities are derived.
- Cylinder Error Modeling: In the case of a cylinder, the error is controlled by the roundness of the surface and the parallelism of the generatrix to the axis. The torsor expression and the equations for calculating the error variation are determined.
- Cone Error Modeling: Based on the basic taper method, the error variation of the cone surface is analyzed. The torsor parameters and their variation inequalities are obtained.
- Axis Error Modeling: Taking the axis of a cylinder or cone as an example, the error variation model is established, considering the position and straightness tolerances.
2.4 Solving the Actual Variation Range of Torsor Parameters
The Monte Carlo method is employed to solve the actual variation range of torsor parameters. The distribution types of error components, such as normal distribution, are assumed, and random sampling experiments are conducted. The actual variation range is then compared with the ideal variation range determined by the variation inequalities.
2.5 Establishing the Function Relationship between Tolerance and Variation Range
Using the response surface method, a function relationship between the actual variation range of torsor parameters and dimensional and geometric tolerances is established. Polynomial functions are selected, and the fitting accuracy is tested using the complex correlation coefficient.
3. Assembly Error Transfer Model Establishment and Verification
3.1 Introduction to Error Transfer
During the assembly process of a machine tool, the machining errors of parts accumulate and transfer, resulting in assembly errors that affect the functionality of the machine tool. Understanding the error transfer mechanism is crucial for predicting and controlling assembly errors.
3.2 Modeling of Fitting Surface Errors
- Plane Fitting Surface: The error of a plane fitting surface is composed of the relative position changes between two ideal planes during assembly. The error transfer matrix is established, considering the error transfer from the ideal reference plane to the ideal assembly plane.
- Cylinder Fitting Surface: For a cylinder fitting surface, the error is related to the machining errors of the hole and shaft and the clearance between them. The error transfer process and the corresponding error expression are analyzed.
- Cone Fitting Surface: The cone fitting surface error is formed by the relative position changes of the two cone surfaces. Similar to the cylinder fitting surface, the error transfer mechanism is studied.
3.3 Error Transfer Mechanism of Fitting Surfaces
- Adjacent Fitting Surface Relationships: Fitting surfaces can be classified as series or parallel fitting surfaces based on the error transfer path. The error transfer properties of different fitting surface combinations are analyzed.
- Error Transfer Attributes: The six parameters of the small displacement torsor can be classified as strongly constrained, weakly constrained, or unconstrained. The error transfer attributes of different fitting surfaces are determined according to these classifications.
- Actual Error Transfer Property Analysis: For parallel fitting surfaces, the actual error transfer properties are affected by the positioning order and the possibility of interference. The methods for analyzing and calculating the actual error transfer properties are presented.
3.4 Modeling and Analysis of Cutter Spindle Component Error Transfer
- Error Transfer Model Establishment: Based on the fitting surface error transfer model, an error transfer model for the cutter spindle component system is established. The model takes into account the error transfer from the housing to the spindle and finally to the cutter head in the x, y, and z directions.
- Error Sensitivity Analysis: The sensitivity of each error component to the overall assembly error is analyzed. The error component with the greatest influence on the assembly error is identified, which provides a basis for error control and optimization.
3.5 Experimental Verification
A point cloud scanning experimental platform is built to verify the accuracy of the assembly error transfer model and the error calculation results. The RANSAC algorithm is used to fit the space circle in the point cloud data, and the assembly errors calculated from the experimental data are compared with the simulated values.
4. Tolerance Optimization Based on Particle Swarm Algorithm
4.1 Introduction to Tolerance Optimization
Tolerance optimization aims to minimize the manufacturing cost while ensuring the assembly accuracy and reliability of the product. In this study, a tolerance optimization model for the cutter spindle component system is established, taking into account the tolerance-cost function and the constraints of assembly accuracy reliability and tolerance values.
4.2 Assembly Accuracy Reliability Analysis
- Reliability Definition: The reliability of a mechanical product refers to its ability to operate normally within a certain period under certain conditions. In this study, the assembly accuracy reliability is analyzed by considering the basic variables (tolerances), response quantities (assembly errors), and state functions.
- Reliability Solving Methods: The reliable degree is calculated using methods such as the Monte Carlo simulation method. The state function is established based on the assembly error transfer model, and the reliable degree is determined by integrating the state function within the reliable domain.
4.3 Tolerance Optimization Model Establishment
- Tolerance-Cost Function: Different types of tolerance-cost functions for various geometric elements are introduced. These functions describe the relationship between tolerance and manufacturing cost.
- Optimization Model of Cutter Spindle Component System: The optimization model of the cutter spindle component system is established with the minimum manufacturing cost as the objective function and the assembly accuracy reliability and tolerance value principles as constraints. The model includes dimensional tolerances, geometric tolerances, and position tolerances as optimization variables.
4.4 Particle Swarm Optimization Algorithm Application
The particle swarm optimization algorithm is used to solve the tolerance optimization model. The algorithm searches for the optimal solution by updating the positions and velocities of particles in the solution space. The performance of the algorithm is improved by adjusting the inertia factor and other parameters.
5. Summary and Outlook
5.1 Research Summary
- Tolerance Modeling: The error variation models of geometric elements are established based on the small displacement torsor theory. The actual variation ranges of error components are obtained, and the function relationships between tolerances and variation ranges are established.
- Error Transfer Modeling: The error transfer models of fitting surfaces and the cutter spindle component system are established. The error transfer mechanisms and actual error transfer properties are analyzed, and the error sensitivity is studied.
- Tolerance Optimization: The assembly accuracy reliability is evaluated, and a tolerance optimization model is established. The particle swarm optimization algorithm is used to solve the optimization model, achieving a reduction in manufacturing cost while maintaining assembly accuracy reliability.
5.2 Research Outlook
- Universal Tolerance Modeling: Future research could focus on establishing more universal geometric element tolerance models to cover a wider range of applications.
- Consideration of More Factors in Error Transfer: The influence of factors such as human operation and part deformation during assembly on error transfer could be further studied and incorporated into the error transfer model.
- Expansion of Error Transfer and Optimization Models: The error transfer and optimization models could be extended to cover the entire machine tool, including the bed and column, to improve the accuracy of error calculation.
- Improvement of Experimental Verification Methods: More accurate and user-friendly experimental verification methods could be developed to further validate the research results.
In conclusion, this research provides a comprehensive approach to evaluating and optimizing the assembly accuracy of bevel gear machine tool cutter spindle components. The findings have important theoretical and practical implications for the manufacturing industry, contributing to the improvement of machine tool quality and manufacturing efficiency.
6. Geometric Element Tolerance Modeling in Detail
6.1 Plane Geometric Element Tolerance Modeling
The tolerance of a plane geometric element is affected by both dimensional tolerance and geometric tolerance. In this study, the perpendicularity tolerance of the plane is considered. The small displacement torsor expression for the plane is . The error variation model is established based on the relationship between the actual variation plane and the dimensional and perpendicularity tolerance domains.
The variation inequalities and constraint inequalities for the plane are derived as follows:
These inequalities help to determine the actual variation range of the plane geometric element and provide a basis for tolerance analysis and optimization.
6.2 Cylinder Geometric Element Tolerance Modeling
For a cylinder geometric element, the error is controlled by the roundness of the surface and the parallelism of the generatrix to the axis. The small displacement torsor expression for the cylinder is . Considering the diameter dimension tolerance and the cylinder degree tolerance, the error variation model is established.
The equations for calculating the upper and lower boundaries of the cylinder degree tolerance domain are . By analyzing the maximum inclination situation of the tolerance region, the variation inequality and constraint inequality for are obtained as follows:
The error variation of the cylinder surface can be reflected by the error variation of the generatrix. The error components of the generatrix are calculated using a one-dimensional linear regression method.
6.3 Cone Geometric Element Tolerance Modeling
Based on the basic taper method, the cone geometric element error is analyzed. The small displacement torsor expression for the cone is , and considering the symmetry of the error variation range, it can be simplified to . The equations for calculating the boundaries of the dimension tolerance domain are .
The maximum and minimum rotation angles of the cone generatrix are calculated using geometric relationships. The variation inequality for the cone generatrix is . The constraint inequality for the cone generatrix is .
6.4 Axis Geometric Element Tolerance Modeling
Taking the axis of a cylinder or cone as an example, the error variation model is established. The small displacement torsor expression for the axis is . Considering the position and straightness tolerances, the boundary equations of the axis variation region are .
The variation inequality and constraint inequality for are obtained as follows:
The error variation of the axis can be simulated and calculated using a one-dimensional linear regression method similar to that of the cylinder generatrix.
7. Fitting Surface Error Transfer in Detail
7.1 Plane Fitting Surface Error Transfer
The error transfer of a plane fitting surface occurs during the assembly process when two ideal planes have relative position changes. The error transfer matrix from the ideal reference plane to the ideal assembly plane is .
The small displacement torsor expressions for and are and respectively. In the case of perfect coincidence of the two assembly planes, is a 4th-order identity matrix.
After ignoring the high-order small terms, the error transfer matrix can be simplified to . The relationship between the error components of the actual assembly plane to the ideal assembly plane and those of the ideal plane to the actual plane is established using a coordinate transformation matrix around the z-axis.
7.2 Cylinder Fitting Surface Error Transfer
The error of a cylinder fitting surface is related to the machining errors of the hole and shaft and the clearance between them. The error transfer process from the shaft ideal axis to the shaft actual axis , then to the hole actual axis , and finally back to the hole ideal axis is considered.
The error expression for the cylinder fitting surface is . The error components and are related to the machining error of the shaft, and are related to the clearance between the hole and shaft, and and are related to the machining error of the hole.
The error transfer matrix for the cylinder fitting surface is . The values of the torsor parameters and can be calculated according to the error transfer process.
7.3 Cone Fitting Surface Error Transfer
The cone fitting surface error is formed by the relative position changes of the two cone surfaces. Similar to the cylinder fitting surface, the error transfer process from the cone shaft ideal axis to the cone shaft actual axis , then to the cone hole actual axis , and finally back to the cone hole ideal axis is considered.
The error expression for the cone fitting surface is \left\{\begin{array}{l} \alpha_{34}=\alpha_{34′}+\alpha_{3′ 4′}+\alpha_{4′ 4} \\ \mu_{34}=\mu_{34′}+\mu_{3′ 4′}+\mu_{4′ 4} \end ^{\circ} The error components and are related to the machining error of the inner cone, and are related to the clearance between the inner and outer cones, and and are related to the machining error of the outer cone.
The error transfer matrix for the cone fitting surface is . The values of the torsor parameters and can be calculated according to the error transfer process.
8. Error Transfer Mechanism of Fitting Surfaces in Detail
8.1 Adjacent Fitting Surface Relationships and Error Transfer Attributes
Fitting surfaces can be classified as series or parallel fitting surfaces based on the error transfer path. The error transfer properties of different fitting surface combinations are analyzed according to the classification of the six parameters of the small displacement torsor as strongly constrained, weakly constrained, or unconstrained.
The error transfer attributes of series and parallel fitting surfaces are determined by the combination of the error attributes of the associated fitting surfaces. For series fitting surfaces, the strong constraint set includes all the strong constraints of the fitting surfaces, and the weak constraint set includes the weak constraints excluding the intersection of weak and unconstrained sets. For parallel fitting surfaces, the strong constraint set includes all the strong constraints of the fitting surfaces, and the weak constraint set includes the weak constraints excluding the intersection of strong and weak constraints.
8.2 Actual Error Transfer Property Analysis of Parallel Fitting Surfaces
For parallel fitting surfaces, the actual error transfer properties are affected by the positioning order and the possibility of interference. When the intersection of the transfer attributes of each fitting surface is an empty set, the actual error transfer properties do not affect each other. When the intersection is not an empty set, the torsor parameters of some fitting surfaces may not be transferred.
The actual error transfer property calculation of parallel fitting surfaces involves determining the positioning order of the fitting surfaces, judging whether interference occurs at the lower-level fitting surface, and adjusting the tolerance values or positions accordingly. The methods for analyzing and calculating the actual error transfer properties are presented with examples of plane, cylinder, and cone parallel fitting surfaces.
9. Cutter Spindle Component Error Transfer Modeling and Analysis in Detail
9.1 Error Transfer Model Establishment
Based on the fitting surface error transfer model, an error transfer model…