1. Introduction
1.1 Research Background and Significance
1.1.1 Research Background
CNC machine tools, as typical electromechanical integration products, are widely used in machining precision parts with complex structures, significantly enhancing machining efficiency and playing a crucial role in the equipment manufacturing industry. However, due to foreign blockades, there is an evident gap between the precision and reliability of domestic machine tool products and those of foreign countries. In recent years, China’s CNC machine tools have been continuously evolving towards higher precision, speed, and stability. Research teams have independently broken through key technologies in machine tool production, substantially narrowing the gap with developed countries. Nevertheless, the development of the CNC machine tool manufacturing industry still has a long way to go. Bevel gear machine tools, as a special type of CNC machine tools, are widely used in the precision machining of aerospace, marine, and automotive industries. The assembly accuracy of the spindle components in the tool box directly affects the inherent reliability of the whole machine and the machining quality of bevel gears. Therefore, in the design stage of CNC machine tools, assembly accuracy, which directly impacts the quality and performance of CNC machine tools, must be considered. There is an urgent need to study evaluation and optimization methods for assembly accuracy to be applied in the forward design of CNC machine tools.
1.1.2 Research Significance
Bevel gear machine tools have high requirements for product assembly accuracy, long assembly cycles, and complex assembly processes. Therefore, it is particularly important to conduct reliability assessment and analysis in the design stage of bevel gear machine tools. This paper takes the spindle component system of the bevel gear machine tool as the research object, starting from tolerance modeling and assembly error transfer analysis, to study the reliability assessment and optimization methods of the machine tool’s assembly accuracy. It has important theoretical and application value for ensuring product quality and reducing economic costs in the design stage. Considering the influence of dimensional and geometric tolerances on the error variation of geometric elements, an error model of geometric elements is established, which can significantly reduce part manufacturing costs. Considering factors such as assembly processes, mating surface types, and error transfer forms, the transfer mechanism of assembly errors is analyzed, providing a theoretical basis for guiding the successful assembly of the machine tool. Combining mechanical reliability theory with assembly accuracy, the evaluation and optimization methods of assembly accuracy reliability are studied to obtain tolerance optimization results, reducing product design changes and design time, reducing manufacturing costs while ensuring the assembly accuracy of the tool box, and improving the inherent reliability of the machine tool. The overall research of this paper provides theoretical methods for the evaluation and optimization of the assembly accuracy of CNC machine tools and theoretical and technical support for ensuring the overall quality of CNC machine tools, with good engineering application prospects.
1.2 Tolerance Modeling Research Status
Tolerance modeling refers to describing the variation of tolerances in mathematical language, including the boundary of the tolerance zone and the variation law of characteristic elements within the tolerance zone. The tolerance model is the basis for analyzing the error transfer law and performing tolerance optimization. Therefore, studying tolerance modeling methods and improving geometric tolerance models are of great significance for enhancing product performance and assembly quality. Domestic and foreign scholars have conducted extensive research in the fields of tolerance modeling and tolerance analysis. At the end of the 20th century, the United States promulgated the world’s first mathematical definition standard for tolerances, ASME Y14.5.1 – M1994, which clearly proposed a new tolerance definition method, solving problems such as the randomness of error variation in traditional tolerance definitions and laying a solid foundation for the theoretical research and development direction of tolerance modeling. Requicha proposed the drift tolerance zone theory, determining the variation range of geometric elements by the region formed by drift without changing the overall size of the part. The solid drift method laid the theoretical foundation for computer – aided tolerance design. Jayaraman further improved the drift theory by introducing the tolerance zone boundary into the drift theory, expanding its application range. Bai Haodong proposed a tolerance modeling method based on the control grid skin model and applied it in grinding machines. Mujezinovic established the T – MAP model and a tolerance mathematical model conforming to ISO/ANSI/ASME standards with T – MAP as the core. The T – MAP theory can describe the tolerance zone range with a set of spatial points and can handle simple tolerance chains, but its application objects are limited. Bourdet P proposed the small displacement torsor model of tolerances based on the small displacement torsor theory, decomposing the error variation of geometric elements into three translational and three rotational errors, opening up a new path for tolerance modeling and analysis. Roy analyzed the error variation of common geometric elements from the perspective of mathematical definition and established tolerance models for dimensional and geometric tolerances, greatly simplifying the difficulty of tolerance analysis. Cai Min, Mao Jian, and Liu Yusheng further studied the tolerance models under mathematical definitions, taking three translational and three rotational errors as random variables, establishing small displacement torsor tolerance models for common geometric elements under the simultaneous action of dimensional and geometric tolerances, and further giving constraint inequalities for error components within the tolerance variation range. Combining with variation inequalities, they comprehensively analyzed the error variation. Walid Ghiea proposed the Jacobian torsor model based on rigid body kinematics, which can accurately express various dimensional and geometric tolerances and reduce the solution difficulty of the tolerance model. Yu Zhimin combined the response surface method with the Monte Carlo method to establish tolerance models for planar dimensional tolerances, flatness tolerances, axis straightness tolerances, and cylindricity tolerances. This method can significantly improve the machining economy of parts and provide a tolerance model library for computer – aided tolerance design. Lü Cheng further studied the tolerance modeling method combining the response surface method and the Monte Carlo method, analyzing the error variation of common geometric elements such as planes, cylinders, and axes under the coupling action of multiple tolerances according to the common tolerance types of various geometric elements in engineering, further improving the tolerance model library for computer – aided tolerance design.
1.3 Error Transfer Research Status
During the assembly process of machine tools, the machining errors on the part surfaces gradually accumulate to the precision output surface, forming assembly errors that affect the machine tool’s functions. The formation of assembly errors is related to the errors of geometric elements and the error transfer forms of mating surfaces. Domestic and foreign scholars have conducted extensive research on this and achieved certain progress. P. Ferreira established an n – order quasi – static geometric spatial error model for CNC machine tools based on the homogeneous coordinate transformation matrix, including the volumetric errors of machine tool components. Tang Zhemin established an assembly error model that can express the main assembly positioning and auxiliary assembly positioning information based on the homogeneous coordinate transformation matrix. Anselmett decomposed the deviation into the variation of geometric units MEDG, studied the error variation laws of common geometric elements, established constraint models and deviation vector models, and calculated the error calculation methods for various assembly situations. Camarata A introduced the binding force between mechanisms into the assembly error transfer model and analyzed the influence of clearance fit on the pose of the mechanism. Liu Bingli proposed to establish a motion error model of complex mechanical systems based on the low – order body array method, and Zhao Xiaosong achieved error compensation for a four – axis machining center using the homogeneous coordinate transformation matrix. Li Dongying analyzed the error sources of assembly errors based on the meta – action theory, discussed the transfer forms of errors in the assembly process, and established an assembly error transfer model for worm gears. Zhou Sihang established a deviation transfer model with dimensional variation as the target feature based on the assembly feature adjacency relationship matrix and the geometric feature tolerance relationship matrix, solved the assembly accuracy according to the error transfer process, and optimized the tolerance while obtaining the optimal assembly sequence. Liu Weidong constructed a directed graph model of assembly errors, analyzed the minimum deviation transfer path of assembly errors, and solved the assembly errors using the D – H pose method. Huang Qiang established a general error transfer model for machine tool products based on the multi – body system theory and analyzed its error sensitivity. Desrochers proposed the Jacobian torsor method based on the small displacement torsor theory and the Jacobian matrix, decomposed the errors into six degrees of freedom using small displacement torsors, and calculated the assembly error transfer through the Jacobian model. Yao Siyu analyzed the error transfer of the parallel combination of the guide rail and the base plane of a three – axis vertical machining center based on the Jacobian torsor method and established an assembly error transfer model for the machining center. Yang Dongya analyzed the influence of various errors on the final assembly error in mechanical equipment and established an equivalent error model for the assembly. Liu Jianyong introduced the part deformation in the assembly process into the assembly error transfer model based on the part machining error and proposed a calculation method. Lü Cheng studied the role of plane joints and cylindrical joints in error transfer, established error transfer matrices for plane joints and cylindrical joints, and established an assembly error transfer model for the assembly with the joint surface as the error transfer node to predict the assembly accuracy. Li Xiaoxiao established a tolerance model based on the small displacement torsor theory and the Monte Carlo method, and then combined with the homogeneous coordinate transformation matrix to establish an error transfer model for the center assembly, realizing the tolerance optimization and allocation of the center assembly.
1.4 Reliability and Tolerance Optimization Allocation Research Status
1.4.1 Reliability Research Status
Mechanical reliability engineering is an interdisciplinary field covering a wide range of areas. Reliability refers to the ability of a product to perform its intended functions under specified conditions and within a specified time. However, many large – scale mechanisms often fail to complete their intended functions within their service lives, especially before and after World War II. At that time, reliability did not form a complete discipline and enter people’s 视野. Many large – scale equipment such as airplanes and tanks were scrapped due to their own failures, and the number was even more than that of war losses. Freudenthal first studied the relationship between structural failure rates and mechanical design safety factors and established a mathematical model to analyze the reliability of mechanisms, marking the birth of the reliability theory analysis method. In 1957, the US military’s Reliability Advisory Group AGREE published “Reliability of Military Electronic Equipment”, which officially brought the reliability discipline into people’s 视野. This report had a profound impact on the subsequent development of reliability. In the 1960s, due to the rapid rise of the aerospace field, reliability was widely applied in various aspects such as structural design, life prediction, and reliability testing. The ROME Aviation Research Center of the United States established a reliability analysis center to conduct reliability prediction and testing. By the 1970s, the United States had gradually matured in the field of reliability, formed its own system, established relevant research departments, and the reliability research had become standardized and systematic, being in a leading position in the world. At the same time, other countries such as Japan, the Soviet Union, and China also successively carried out reliability research and achieved certain results in the military and agricultural fields. By the 1980s, reliability analysis had gradually entered other non – electronic equipment fields, and the analysis had also expanded to areas such as maintainability, fault analysis, and safety. Currently, reliability has become an important indicator for measuring product quality, and the research and development of high – reliability and high – performance products has gradually become the mainstream of the national equipment manufacturing industry, and reliability analysis has also become an essential part of the product design stage. In the field of accuracy reliability, a large number of domestic and foreign scholars have conducted research and achieved certain results. Mao Yingtai established a mathematical model for the motion accuracy reliability of the crank – slider mechanism and solved the model using the response surface method and the importance sampling method. Du Xiaoping combined numerical calculation and the Monte Carlo method to randomly sample tolerances and solve the motion accuracy reliability of the mechanism, greatly improving the solution rate of reliability. Yu Zhimin took a large – scale gantry grinding machine as the research object, analyzed its dynamic and static accuracies, took reliability as the optimization constraint condition, and proposed a tolerance optimization method based on the genetic algorithm. Song Jiangbo established a basic model for the accuracy reliability of the CNC machine tool transmission system based on the mechanical accuracy reliability theory and studied the main factors affecting the transmission accuracy of the CNC machine tool. Huang Yi combined the mechanical reliability theory with the error transfer model based on the state space method, proposed to evaluate the assembly accuracy with reliability, perform tolerance optimization allocation, and designed an assembly accuracy evaluation software.
1.4.2 Tolerance Optimization Research Status
Part machining errors are important factors affecting product functions. To rationally allocate tolerances, various factors such as machining methods, process stability, manufacturing costs, product quality, machining batches, assembly operation methods, and assembly loads need to be considered. Therefore, when optimizing and allocating tolerances, it is necessary to fully consider other factors. Traditional tolerance allocation mainly relied on the experience of researchers and lacked a fixed method, which may lead to the assembly error of the assembly not meeting the accuracy requirements or increasing the machining cost and quality loss. Therefore, it is necessary to take accuracy reliability as a constraint condition when optimizing and allocating tolerances. In recent decades, a large number of domestic and foreign researchers have made significant achievements in tolerance optimization and allocation. Prabhaharan used ant colony algorithms and genetic algorithms to achieve tolerance optimization and allocation, making great progress in the field of tolerance optimization algorithms. Jeang integrated the three objectives of machining cost, quality loss, and inspection cost into a single objective function, considered factors such as part product accuracy requirements and actual machining conditions, established a constrained tolerance optimization model, and solved the tolerance optimization model using the Monte Carlo simulation method and the response surface method. Kumar used the Lagrange multiplier method to introduce the constraint function into the objective function, established an optimization model with the minimum machining cost as the objective, and optimized the design and manufacturing tolerances simultaneously. Forouragh established a multi – objective tolerance optimization model based on the particle swarm algorithm with the minimum manufacturing cost as the objective and the product design accuracy requirements as the constraint condition. Wang Boping processed the uncertainty cost problem in the model based on the fuzzy mathematics theory and established a mathematical model of tolerance and cost, providing a more practical tolerance cost model for tolerance optimization. Wang Yu established a tolerance robust optimization model with the quality loss cost and tolerance machining cost as the objectives and the fuzzy assembly reliability as the constraint condition, established a multi – objective tolerance optimization model, and solved the optimization model using the coordination curve method to achieve multi – objective tolerance optimization design. Xiao Renbin proposed a parallel tolerance optimization intelligent algorithm combining ant colony algorithm and particle swarm algorithm, greatly reducing the solution difficulty of tolerance optimization. Zhang Yan established a multi – objective tolerance optimization model with the minimum machining cost and minimum quality loss as the objective functions for the typical assembly process of aircraft and realized the optimization and allocation of tolerances using the particle swarm algorithm. Zhang Weimin took the dimensional tolerance and geometric tolerance as the optimization variables, established an assembly error model based on the spatial geometric error theory, took the assembly quality as the constraint condition and the minimum machining cost as the optimization objective to establish a tolerance optimization model, and solved the model using the genetic algorithm. Lü Cheng established an assembly error transfer model for the gear pump with the assembly joint surface as the node, took the minimum machining cost as the objective function and the assembly accuracy reliability as the constraint condition to establish a tolerance optimization model for the assembly, and realized the optimization and allocation of tolerances using the improved genetic algorithm.
1.5 Main Research Contents
This paper takes the spindle component system of the bevel gear machine tool as the research object, performs tolerance modeling on geometric elements such as planes, cylinders, cones, and axes, establishes the functional relationship between tolerances and the variation range of error components, explores the transfer law of part errors in the assembly process, establishes an assembly error transfer model for the spindle component, and studies the tolerance optimization method with the output of the assembly error transfer model as the response. The specific research contents are as follows:
- Based on the small displacement torsor theory, decompose the part tolerances into six degrees of freedom, establish error variation models for geometric elements such as planes, cylinders, cones, and axes, use the Monte Carlo method to solve the actual variation range of the error components and compare it with the ideal variation range to prove the rationality of the modeling method. Based on the response surface method, establish the functional relationship between the actual variation range of the geometric elements and the dimensional and geometric tolerances.
- Establish error transfer models for mating surfaces such as planes, cylinders, and cones, study the error transfer properties of series and parallel combinations, explore the laws of the actual error transfer properties of the mating surfaces when considering the positioning sequence and assembly interference, and solve the parallel error of the plane and cone of the spindle component. Establish an assembly error transfer model for the spindle system with the mating surface as the error transfer node, use software simulation to solve the assembly errors of the cutter head in three directions, and analyze the error sensitivity. Verify the correctness of the assembly error transfer model and the assembly error calculation results by building a point cloud scanning experimental platform.
- Combine the mechanical reliability theory to establish a reliability evaluation index, use the Monte Carlo simulation method to solve the assembly accuracy reliability. Take the minimum machining cost of tolerances as the objective, various tolerances as the basic variables, and the assembly accuracy reliability and tolerance value selection principles as the constraint conditions to establish a tolerance optimization model for the spindle component system. Use the improved particle swarm optimization algorithm to achieve the tolerance optimization of the assembly.
| Tolerance Modeling | Error Transfer Analysis | Tolerance Optimization |
|---|---|---|
| Establish common geometric element tolerance models | Analyze the error transfer mechanism of different mating forms | Simulate and solve the assembly accuracy reliability |
| Decompose part tolerances into six degrees of freedom | Study the influence of series and parallel combinations on error accumulation | Establish a tolerance – cost function |
| Compare the actual bandwidth with the ideal bandwidth to verify the model economy | Establish the error transfer model of the spindle component | Consider the assembly accuracy reliability and tolerance value selection principles as constraints |
| Solve the function relationship between the bandwidth and tolerances | Write a program to solve the assembly error | Write a particle swarm optimization algorithm program |
| Test the fitting accuracy of the function | Verify the error transfer model through point cloud experiments | Check the optimization results |
Figure 1.1 Technical Route Diagram
2. Tolerance Modeling Based on the Small Displacement Torsor Theory
2.1 Introduction
Common part geometric elements include axes, planes, cylinders, and cones. This chapter establishes geometric element tolerance models based on the small displacement torsor theory, combines the Monte Carlo method to solve the actual variation interval of the error components, and then uses the response surface method to establish the functional relationship between the error component width and the dimensional and geometric tolerances, laying the foundation for the subsequent assembly error transfer analysis and tolerance optimization. Tolerance is the product of coordinating the contradiction between the functional requirements and manufacturing economy of mechanical parts. The surface machining errors of each part are transferred and accumulated through the assembly process, forming assembly errors that affect the overall accuracy requirements of the assembly and, in turn, the quality and function of the assembly.
2.2 Analysis of the Structure of the Spindle Component of the Cutting Tool
The main structure of a CNC bevel gear machine tool includes the bed, column, tool box, workpiece box, rotary table, and electrical cabinet. The CNC system adopts Siemens 840D, with a total of six working axes.
2.3 Tolerance Modeling Based on the Small Displacement Torsor
2.3.1 Small Displacement Torsor Theory
The small displacement torsor is a vector formed by the small displacement of a rigid body with six motion components and was introduced into the field of tolerances by Bourdet in 1996. It is suitable for representing the deviation of the ideal shape feature. The small displacement torsor parameter consists of three rotation vectors and three translation vectors and is expressed as , where , , and respectively represent the rotation angle errors of the rigid body in the , , and directions, and , , and respectively represent the translation errors of the rigid body in the , , and directions.
According to the regulations on constancy in the new generation of GPS standards, when a geometric element translates or rotates in a certain degree of freedom direction within the tolerance zone and its motion trajectory does not sweep out a new entity compared with itself, it is judged that it has constancy in that direction. Constancy means that when the geometric element has an error variation in a certain direction, it has no impact on the geometric element itself, and the corresponding torsor parameter takes the value of zero. Common tolerances are shown in Figures 2.2 – 2.5.
The expressions of the small displacement torsors corresponding to the straightness, flatness, surface parallelism, and coaxiality in the figures are , , , and respectively. The expressions of the small displacement torsors of each geometric tolerance will change according to the change of the coordinate axes.
2.3.2 Tolerance Principles
Tolerance principles are used to handle the relationship between dimensional tolerances and geometric tolerances to determine the size, shape, and position characteristics of part elements. They mainly include the independent principle,maximum material requirement, minimum material requirement, and reversible requirement.
- Independent Principle: In the design and manufacturing process, different dimensions and features should be kept independent of each other, and the change of one feature will not affect the size or position of other features. The purpose of this principle is to ensure that each part of the product can be designed, manufactured, and assembled relatively independently, thereby reducing the complexity and risk in the product development and manufacturing process. In actual production, considering the realization of design, machining, and inspection, the independent principle is often selected. The dimensional tolerance and geometric tolerance that comply with the independent principle do not need to add any specific relationship symbols on the drawing.
- The of a part must be between the maximum and minimum material sizes. The is applied to holes and shafts with strict requirements for ensuring the . A single element that adopts the should be marked with the symbol “E” after its size limit deviation or tolerance zone code.
- Maximum Material Requirement: It refers to the maximum size or range allowed in the product design or manufacturing process. That is, the actual contour formed by the local size and geometric error of the measured element must not exceed this boundary, and the local size is not greater than the limit size. It is applicable when there are geometric tolerance requirements for the derived element. When adopting the maximum material requirement, the symbol “M” should be marked on the drawing.
- Minimum Material Requirement: It is determined by the design engineer according to the function and usage requirements of the product and reflects the requirements for the minimum size, minimum material thickness, and minimum clearance of the product. It is applicable when there are geometric tolerance requirements for the derived element. When adopting the minimum material requirement, the symbol “L” should be marked on the drawing.
- Reversible Requirement: When the geometric error of the derived element is smaller than the preset geometric tolerance, the size tolerance of the part can be enlarged without affecting the specific characteristics of the part. Usually, the choice of the reversible requirement should be combined with the . When adopting the reversible requirement, the symbol “R” should be marked after the maximum or minimum material requirement.
The selection of tolerance principles also depends on different functional requirements. To make the established model more universal, this paper studies the geometric elements whose dimensional tolerances and geometric tolerances do not affect each other. Therefore, the independent principle is selected as the tolerance principle in this research.
The geometric tolerance marking of the spindle component of the bevel gear machine tool is shown in Figure 2.6. The measured elements mainly include planes, cylinders, cones, and axes.
In Figure 2.6, the plane is the assembly surface between the cutter head and the end of the spindle, with a perpendicularity tolerance of 0.003mm. The cone in the figure is the assembly surface between the cutter head and the end of the spindle, with a taper of 1:24. The coaxiality of the spindle axis is 0.005mm and 0.003mm. The cylindrical surface is the assembly surface between the tool spindle and the bearing, with a circularity tolerance of 0.002mm and a parallelism tolerance of 0.003mm between the generatrix and the axis.
Most of the markings in the figure follow the independent principle, involving four common geometric elements: plane, cylinder, axis, and cone. Based on the small displacement torsor theory, this paper simultaneously considers the influence of dimensional and geometric tolerances on the error variation of geometric elements and establishes an error variation model.
2.3.3 Error Variation Analysis of Geometric Elements
When geometric elements are affected by dimensional and geometric tolerances simultaneously, their pose will change relative to the ideal surface. The specific error variation model needs to be analyzed according to the specific situation of the geometric elements.
- Plane Error Variation Modeling: The variation of plane geometric elements is diverse. Previous studies have analyzed the error variation of planes under the simultaneous action of dimensional tolerance, flatness tolerance, and parallelism tolerance. In this paper, the plane in the tool spindle component system is affected by dimensional tolerance and perpendicularity tolerance, as shown in Figure 2.7. is the basic size, and are the upper and lower deviations of the dimensional tolerance, represents the perpendicularity tolerance. Taking the center of the upper surface as the origin of the coordinate system and the normal direction of the plane as the – axis, a spatial rectangular coordinate system is established. The shape of the geometric element is affected by the perpendicularity, and the perpendicularity datum element is parallel to the – axis. The actual variation plane of the geometric element varies within the dimensional tolerance zone and the perpendicularity tolerance zone.
