This article delves into the critical aspects of optimizing the assembly accuracy of bevel gear machine tool cutter spindle components. It emphasizes the significance of this research in enhancing the machining quality of bevel gears and the overall performance of machine tools. By establishing geometric error models, analyzing error transfer laws, and conducting tolerance optimization, a comprehensive method is proposed to improve the assembly accuracy and economic efficiency of cutter spindle components.
1. Introduction
1.1 Research Background
In the realm of equipment manufacturing, bevel gear machine tools play a crucial role in processing bevel gears, which are widely used in various industries such as aerospace, navigation, and automotive. The assembly accuracy of the cutter spindle components in bevel gear machine tools directly impacts the machining quality of bevel gears. However, due to factors like foreign technology blockade and complex assembly processes, there is a pressing need to enhance the assembly accuracy of these components.
1.2 Research Significance
Optimizing the assembly accuracy of cutter spindle components can significantly improve the machining quality of bevel gears, reduce manufacturing costs, and enhance the reliability of machine tools. This research provides theoretical and technical support for the design and manufacturing of bevel gear machine tools, promoting the development of the equipment manufacturing industry.
1.3 Research Objectives
The primary objectives of this research are to establish accurate geometric error models for cutter spindle components, analyze the error transfer laws during the assembly process, and develop effective tolerance optimization methods to improve the assembly accuracy and economic efficiency of the components.
2. Geometric Error Modeling Based on Small Displacement Screw Theory
2.1 Small Displacement Screw Theory
The small displacement screw theory is employed to decompose geometric element errors into six degrees of freedom directions. This theory represents the deviation of an ideal shape feature as a vector with six motion components, providing a foundation for establishing geometric error models.
2.2 Tolerance Principles
Several tolerance principles, including the independent principle, inclusion requirement, maximum material requirement, minimum material requirement, and reversible requirement, are introduced. The independent principle is chosen for this research as it allows for independent consideration of dimensional and geometric tolerances, enhancing the universality of the established models.
2.3 Error Variation Analysis of Geometric Elements
2.3.1 Plane Error Variation Modeling
For plane geometric elements in the cutter spindle component system, the influence of dimensional tolerance and perpendicularity tolerance is considered. By establishing a spatial rectangular coordinate system and analyzing the position change of the actual plane within the tolerance range, the small displacement screw expression and error variation inequalities are derived. Table 1 summarizes the key parameters and inequalities for plane error variation.
2.3.2 Cylindrical Surface Error Variation Modeling
In the cutter spindle component system, the cylindrical surface’s shape tolerance is controlled by the roundness of the surface circle and the parallelism of the generatrix to the axis. Considering the detection difficulty, the cylindrical surface error variation model is established by combining roundness and parallelism as equivalent to cylindricity. The small displacement screw expression and error variation inequalities for the cylindrical surface are presented in Table 2.
2.3.3 Conical Surface Error Variation Modeling
The basic taper method is commonly used for conical tolerance annotation. Based on this method, the error variation of the conical surface is analyzed. The small displacement screw parameter expression and error variation inequalities for the conical surface are shown in Table 3.
2.3.4 Axis Error Variation Modeling
The axis is a derived element of the cylinder and cone, and its error variation affects the surface shape of the cylinder and cone. Taking the cylindrical axis as an example, the small displacement screw expression and error variation inequalities for the axis are established, as detailed in Table 4.
2.4 Solving the Actual Variation Interval of Screw Parameters Based on Monte Carlo Method
The Monte Carlo method is utilized to simulate the randomness of part surface machining errors and solve the actual variation range of screw parameters in the geometric element error model. The steps involve determining the distribution type of error components, conducting random sampling experiments, and calculating estimators. This method can handle complex models and provide accurate results.
2.5 Establishing the Functional Relationship between Tolerance and Actual Variation Range
The response surface method is employed to establish the functional relationship between the actual variation interval bandwidth of screw parameters and tolerances. This involves selecting test points, conducting Monte Carlo simulation tests, choosing a polynomial, fitting the function relationship, and testing the fitting accuracy of the function model. The established function relationship can be used to predict the actual variation range of screw parameters under different tolerance values.
2.6 Example Analysis
2.6.1 Conical Surface Geometric Element
Taking the conical surface error variation as an example, specific values are set for the cone height h, taper parameter n, and dimensional tolerance T. The actual variation range of screw parameters is solved using the Monte Carlo method and compared with the ideal variation range. The results show that the actual variation range bandwidth is smaller, indicating that the tolerance value can be appropriately increased to reduce processing costs. Table 5 presents the comparison results for the conical surface.
| Spinor parameters | The percentage of the actual variable bandwidth decrease |
|---|---|
| \(\alpha\) | 20.6% |
| v | 15.7% |
2.6.2 Cylindrical Surface Geometric Element
For the cylindrical surface, specific values for the radius R, length 2h, dimensional tolerance, and cylindricity tolerance are set. The actual variation range of screw parameters is obtained and compared with the ideal range. Similar to the conical surface, the actual variation range bandwidth is reduced, demonstrating the effectiveness of the method. Table 6 shows the comparison results for the cylindrical surface.
| Spinor parameters | The percentage of the actual variable bandwidth decrease |
|---|---|
| \(\alpha\) | 9.1% |
| v | 10.3% |
3. Establishment and Verification of Assembly Error Transfer Model
3.1 Assembly Error Transfer Mechanism
During the assembly process of the machine tool, the machining errors of parts accumulate and transfer to the accuracy output surface, forming assembly errors. The error transfer form varies depending on the shape of the mating surface. Understanding this mechanism is crucial for analyzing and controlling assembly errors.
3.2 Mating Surface Error Modeling
3.2.1 Plane Mating Surface
The error of the plane mating surface is composed of the relative position change of the ideal planes of two parts during assembly. The error transfer matrix \(M_{AB}\) is established, considering the error variation matrices from the ideal reference plane to the actual reference plane, from the actual reference plane to the actual assembly plane, and from the actual assembly plane to the ideal assembly plane. Table 7 summarizes the key matrices and elements for plane mating surface error modeling.
3.2.2 Cylindrical Mating Surface
The error of the cylindrical mating surface, commonly found in shaft – hole bearing fits, is considered as the error between the actual axes of the hole and the shaft. It is affected by the machining errors of the hole and shaft and the clearance error of the mating contact surface. The error expression and transfer matrix for the cylindrical mating surface are presented in Table 8.
3.2.3 Conical Mating Surface
The conical mating surface, with its high coaxiality and ability to transmit large torques, has an error formation mechanism similar to that of the cylindrical mating surface. It can be regarded as the relative pose change of the ideal axes of the two conical surfaces. The error expression and transfer matrix for the conical mating surface are shown in Table 9.
3.3 Error Transfer Attributes of Mating Surfaces
3.3.1 Adjacent Mating Surface Relationships
Mating surfaces can be classified into series – connected mating surfaces and parallel – connected mating surfaces based on the error transfer path. In series – connected mating surfaces, the error is transferred sequentially, while in parallel – connected mating surfaces, the error can be transferred through multiple paths.
3.3.2 Error Transfer Attributes of Mating Surfaces
The six parameters of the small displacement screw can be classified as strongly constrained, weakly constrained, and unconstrained. The error transfer attributes of different mating surfaces are determined by these constraints. Table 10 summarizes the error transfer attributes of common mating surfaces.
3.3.3 Analysis of Actual Error Transfer Attributes of Mating Surfaces
In parallel – connected mating surfaces, the actual error transfer attributes are complex and affected by factors such as the positioning order and assembly interference. To ensure successful assembly, it is necessary to analyze and adjust the error transfer attributes. The actual error transfer attribute calculation steps for parallel – connected mating surfaces are as follows:
- Determine the positioning order of mating surfaces.
- Judge whether interference occurs at the lower – level mating surface. If there are the same strong constraint items in the two mating surfaces or the strong constraint of the lower – level mating surface conflicts with the weak constraint of the upper.