This paper focuses on the assembly accuracy of the spindle components in bevel gear machine tools. By establishing tolerance models for common geometric elements, analyzing the error transfer laws of different fitting forms, and conducting tolerance optimization, the paper aims to improve the machining economy and assembly reliability of the machine tools. The key research contents and conclusions are as follows:
- Tolerance Modeling based on Small Displacement Spinor Theory:
- Error Variation Modeling: Based on the small displacement spinor theory, the error variation models for geometric elements such as planes, cylinders, cones, and axes are established. The constraint inequalities and variation inequalities of the spinor parameters in the error model are derived, and the actual variation range of the error components is solved using the Monte Carlo method. The results are compared with the ideal variation range to verify the rationality of the modeling method.
- Function Relationship Establishment: The response surface method is used to establish the function relationship between the actual variation range of the geometric element and the dimensional and geometric tolerances. The fitting accuracy is tested using the complex correlation coefficient. The cone and cylinder surfaces are taken as examples to verify the machining economy of the modeling method based on the small displacement spinor theory.
- Establishment and Verification of Assembly Error Transfer Model:
- Error Transfer Modeling for Fitting Surfaces: The error transfer models for fitting surfaces such as planes, cylinders, and cones are established. The error transfer attributes of series and parallel fittings are analyzed, and the actual error transfer attribute analysis and calculation method for the parallel fitting surface composed of planes and cones are proposed.
- Assembly Error Transfer Modeling and Analysis for Tool Spindle Components: The assembly error transfer model for the tool spindle components is established, and the assembly errors in the x, y, and z directions are solved using software simulation. The error sensitivity is analyzed, and a point cloud scanning experiment is conducted to verify the correctness of the assembly error transfer model.
- Tolerance Optimization based on Particle Swarm Algorithm:
- Assembly Accuracy Reliability Analysis: The reliability of the assembly accuracy is analyzed by establishing the state function based on the assembly error transfer model and the reliability theory. The Monte Carlo simulation method is used to solve the reliability index, and the reliability of the assembly error within the accuracy requirement of 0.035mm is calculated to be 97.71%.
- Tolerance Optimization: Taking the minimum processing cost as the objective function, the assembly accuracy reliability and the tolerance selection principle as the constraints, and the dimensional and geometric tolerances as the basic variables, the tolerance optimization model for the tool spindle component system is established. The particle swarm optimization algorithm is used to realize the tolerance optimization, and the processing cost is reduced by 8.36% while ensuring the assembly reliability.
Keywords: Spiral Bevel Gear; Small Displacement Spinor; Error Transfer Model; Tolerance Optimization; Assembly Accuracy Reliability
1. Introduction
1.1 Background and Significance
Spiral bevel gears are widely used in various fields such as aerospace, navigation, and vehicles due to their excellent performance in transmitting power and motion. The machining quality of spiral bevel gears is directly related to the assembly accuracy of the spindle components in the bevel gear machine tools. Therefore, it is of great significance to study the assembly accuracy evaluation and optimization methods for the spindle components of bevel gear machine tools to ensure the quality and performance of the machine tools.
1.2 Research Status
1.2.1 Tolerance Modeling
Tolerance modeling is the foundation for analyzing error transfer laws and conducting tolerance optimization. Many scholars have conducted research in this field, and the small displacement spinor model proposed by Bourdet has opened up a new way for tolerance modeling.
1.2.2 Error Transfer
The error transfer in the assembly process of machine tools has been studied by many scholars. The error transfer model based on the small displacement spinor and the Jacobian spinor method can accurately analyze the error transfer process.
1.2.3 Reliability and Tolerance Optimization
Reliability analysis is an important part of product design, and tolerance optimization can reduce the manufacturing cost while ensuring the product quality. Many researchers have used various algorithms to achieve tolerance optimization.
1.3 Main Research Contents
This paper takes the spindle components system of the bevel gear machine tool as the research object, and conducts research on tolerance modeling, error transfer analysis, and tolerance optimization. The specific research contents are as follows:
- Based on the small displacement spinor theory, establish the error variation models for geometric elements such as planes, cylinders, cones, and axes, and explore the function relationship between the actual variation range and the dimensional and geometric tolerances.
- Establish the error transfer models for different fitting forms, analyze the error transfer mechanism of the tool spindle components, and verify the correctness of the assembly error transfer model through experiments.
- Establish the tolerance optimization model based on the reliability theory, and use the particle swarm optimization algorithm to solve the tolerance optimization problem.
2. Tolerance Modeling based on Small Displacement Spinor Theory
2.1 Introduction
This chapter establishes the tolerance models for common geometric elements based on the small displacement spinor theory, and analyzes the error variation of different geometric elements.
2.2 Structure Analysis of Tool Spindle Components
The main structure of the bevel gear machine tool includes the bed, column, tool box, workpiece box, turntable, and electrical cabinet. The research object of this paper is the tool box, which mainly includes the tool spindle, box, and cutter head.
2.3 Tolerance Modeling based on Small Displacement Spinor
2.3.1 Small Displacement Spinor Theory
The small displacement spinor is a vector with six motion components, which is introduced by Bourdet in the tolerance field to represent the deviation of the ideal shape feature.
2.3.2 Tolerance Principles
The tolerance principles include independent principle, envelope requirement, maximum material requirement, minimum material requirement, and reversible requirement. In this paper, the independent principle is selected as the tolerance principle.
2.3.3 Error Variation Analysis of Geometric Elements
- Plane Error Variation Modeling: The plane is affected by the dimensional tolerance and the perpendicularity tolerance. The small displacement spinor expression and the equation of the actual variation plane are derived, and the variation inequality and constraint inequality are obtained.
- Cylindrical Surface Error Variation Modeling: The cylindrical surface is controlled by the roundness of the surface circle and the parallelism of the generatrix to the axis. The small displacement spinor expression and the equations of the upper and lower boundaries are derived, and the variation inequality and constraint inequality are obtained.
- Cone Surface Error Variation Modeling: The cone surface is commonly marked by the basic cone method. The small displacement spinor expression and the equation of the variation line are derived, and the maximum and minimum rotation angles are obtained.
- Axis Error Variation Modeling: The axis is the derived element of the cylinder and cone, and its error variation directly affects the surface shape of the cylinder and cone. The small displacement spinor expression and the boundary equations of the axis variation region are derived.
2.4 Solution of Actual Variation Interval of Spinor Parameters based on Monte Carlo Method
The Monte Carlo method is used to solve the actual variation range of the spinor parameters. The distribution types and probability density functions of the error components are determined, and the actual variation range is solved through random sampling and hypothesis testing.
2.5 Establishment of Function Relationship between Tolerance and Actual Variation Range
The response surface method is used to establish the function relationship between the actual variation range of the spinor parameters and the tolerance. The test points are selected, the Monte Carlo simulation experiment is conducted, and the polynomial is selected to fit the function relationship. The fitting accuracy is tested using the complex correlation coefficient.
2.6 Example Analysis
The cone and cylinder surfaces are taken as examples to analyze the actual variation range and the function relationship with the tolerance. The results show that the actual variation range is smaller than the ideal variation range, and the function relationship has high credibility.
2.7 Summary
This chapter establishes the error variation models for different geometric elements based on the small displacement spinor theory, and verifies the rationality of the method by comparing the actual and ideal variation ranges. The function relationship between the actual variation range and the tolerance is established, which provides a theoretical basis for the subsequent assembly error transfer modeling and tolerance optimization.
3. Establishment and Verification of Assembly Error Transfer Model
3.1 Introduction
The assembly error of the machine tool is formed by the accumulation of the processing errors of the parts to the precision output surface, which affects the function of the machine tool. Therefore, it is necessary to analyze the error transfer form of different fitting surfaces to establish the assembly error transfer model.
3.2 Error Modeling for Fitting Surfaces
3.2.1 Plane Fitting Surface
The plane fitting surface is a common fitting type in engineering. The error of the plane fitting surface is composed of the error of the plane itself and the assembly process error. The error transfer process from the ideal reference plane to the ideal assembly plane is analyzed, and the error transfer matrix is established.
3.2.2 Cylindrical Fitting Surface
The cylindrical fitting surface is commonly seen in shaft-hole bearing fits. The error of the cylindrical fitting surface is composed of the processing error of the shaft and hole and the clearance error of the fitting contact surface. The error transfer process and the error expression are analyzed, and the error transfer matrix is established.
3.2.3 Cone Fitting Surface
The cone fitting surface has high coaxiality and can transmit large torque, which has a good application prospect in the engineering field. The error formation mechanism and the error expression of the cone fitting surface are similar to those of the cylindrical surface, and the error transfer matrix is established.
3.3 Error Transfer Mechanism for Fitting Surfaces
3.3.1 Relationship between Adjacent Fitting Surfaces
The fitting surfaces are divided into series and parallel fitting surfaces according to the error transfer path. The series fitting surfaces have only one error transfer path, while the parallel fitting surfaces have multiple parallel error transfer paths.
3.3.2 Error Transfer Attributes of Fitting Surfaces
The small displacement spinor’s six parameters are divided into strong constraint, weak constraint, and no constraint according to the error transfer attribute. The error transfer attributes of common fitting surfaces are defined, and the error transfer attributes of series and parallel fitting surfaces are determined.
3.3.3 Actual Error Transfer Attribute Analysis of Fitting Surfaces
The actual error transfer attribute of the parallel fitting surface is affected by the type and combination of the fitting surfaces. The positioning order and assembly interference need to be considered in the analysis. The actual error transfer attribute is calculated by determining the positioning order, judging the interference, and adjusting the tolerance if necessary.
3.3.4 Actual Error Transfer Calculation of Fitting Surfaces
The actual error transfer attribute calculation process of the parallel fitting surface composed of the plane and cone is proposed. The error transfer calculation steps are as follows: determining the assembly positioning order, solving the error transfer attribute, solving the actual error transfer attribute of the two fitting surfaces, solving the error of the fitting surface 1F, and solving the error of the fitting surface 2F.
3.4 Error Transfer Modeling and Analysis for Tool Spindle Components
3.4.1 Tool Spindle Component Error Transfer Model
The assembly error of the tool spindle component system is accumulated from the box to the spindle and finally to the cutter head. The error transfer model is established based on the composition of the fitting surfaces, and the assembly transfer error is solved by programming and simulation.
3.4.2 Error Sensitivity Analysis
The sensitivity analysis model is established based on the assembly error model to determine the influence degree of each error component on the overall assembly error. The sensitivity matrix and the sensitivity coefficient are defined, and the program is written to analyze the error sensitivity of the tool spindle component assembly error calculation results.
3.5 Experimental Verification
The point cloud scanning experiment is conducted to verify the correctness of the assembly error transfer model. The RANSAC algorithm is used to fit the space circle, and the laser scanner is used to obtain the point cloud data of the tool spindle component system. The experimental results show that the assembly error of the tool spindle component system is less than the simulated calculation maximum value, which proves the correctness of the assembly error transfer model.
3.6 Summary
This chapter analyzes the error transfer form of different fitting surfaces, establishes the assembly error transfer model, and verifies the correctness of the model through experiments. The error sensitivity analysis provides a basis for optimizing the assembly process.
4. Tolerance Optimization based on Particle Swarm Algorithm
4.1 Introduction
This chapter establishes the tolerance optimization model based on the assembly error transfer model, and uses the particle swarm optimization algorithm to achieve tolerance optimization.
4.2 Assembly Accuracy Reliability Analysis
4.2.1 Reliability Definition
The reliability of the mechanical product is analyzed by defining the basic variables, response quantity, state function, reliable domain, and failure domain. The assembly error caused by the tolerance is taken as the response quantity, and the state function is established based on the assembly error transfer model to solve the reliability.
4.2.2 Reliability Solution Method
The reliability is solved by the Monte Carlo simulation method. The random variables are assumed to follow the normal distribution, and the random samples are generated within the distribution range. The assembly error of each sample is solved by the assembly error transfer model, and the reliability is calculated by the frequency of the error values within the reliable domain.
4.3 Tolerance Optimization
4.3.1 Tolerance – Cost Function
The tolerance – cost functions for different types of tolerances are determined based on the existing research. These functions are used to establish the optimization model.
4.3.2 Tool Spindle Component System Optimization Model
The optimization model is established with the tolerance processing cost of the tool spindle component system as the objective function, the dimensional, position, and shape tolerances as the optimization variables, and the assembly accuracy reliability and the tolerance value selection principle as the constraints.
4.3.3 Particle Swarm Optimization Algorithm
The particle swarm optimization algorithm is used to solve the optimization problem. The algorithm updates the velocity and position of the particles by sharing information among the individuals in the population to find the optimal solution.
4.3.4 Tool Spindle Component System Tolerance Optimization
The particle swarm optimization algorithm is programmed to optimize the tolerances of the tool spindle component system. The convergence process of the algorithm is analyzed, and the optimized tolerances are compared with the original tolerances. The results show that the tolerance optimization method can reduce the processing cost while ensuring the assembly reliability.
4.4 Summary
This chapter establishes the tolerance optimization model based on the reliability theory and the assembly error transfer model, and uses the particle swarm optimization algorithm to solve the model. The optimization results show that the method can reduce the processing cost and improve the machining economy.
5. Conclusion and Outlook
5.1 Conclusion
This paper studies the assembly accuracy evaluation and optimization of the spindle components of the bevel gear machine tool. The main conclusions are as follows:
- Based on the small displacement spinor theory, the error variation models for geometric elements are established, and the actual variation range is smaller than the ideal variation range, which can improve the machining economy.
- The assembly error transfer model is established, and the correctness of the model is verified by experiments. The error sensitivity analysis provides a basis for guiding the successful assembly of the machine tool.
- The tolerance optimization model is established based on the reliability theory, and the particle swarm optimization algorithm is used to solve the model. The processing cost is reduced by 8.36% while ensuring the assembly reliability.
5.2 Outlook
The research in this paper has some limitations, and the following aspects can be improved in the future:
- A more universal geometric element tolerance model can be established to improve the applicability of the model.
- The factors such as manual operation, assembly process, and part deformation can be considered in the assembly error transfer model to make the error calculation more realistic.
- The assembly error transfer model and tolerance optimization model can be further improved to cover the entire machine tool and improve the accuracy of the error calculation.
- A more accurate and easy-to-operate experiment can be designed to verify the assembly error transfer model.
In conclusion, the research on the assembly accuracy evaluation and optimization of the spindle components of the bevel gear machine tool is of great significance for improving the machining quality and efficiency of the machine tool. Further research in this field is needed to meet the higher requirements of the manufacturing industry.
The following is the schematic diagram of the assembly error transfer model of the tool spindle component system:
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The following is the flow chart of the particle swarm optimization algorithm:
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