Spiral bevel gears have high coincidence, strong load-bearing capacity, high transmission ratio and high transmission efficiency, and are widely used in automotive, aerospace and mining industries The manufacturing accuracy of spiral bevel gears is directly related to the errors of CNC machine tools. CNC machine tool errors include geometric errors, thermal The geometric error of the machine tool has the characteristics of repetition and stability, and can be accurately compensated through the numerical control system Therefore, studying the characteristics of geometric errors of machine tools and determining key geometric error terms are important for tooth surface deviation compensation and the allocation and maintenance of machine tool motion accuracy Sensitivity analysis is an analytical method that studies the impact of changes in input factors in a system on the output results. The analysis results are expressed through sensitivity coefficients, and a large sensitivity coefficient indicates a high correlation between the input factor and the output result. Sensitivity analysis is an important method for studying the relationship between machine tool geometric errors and machining accuracy. Scholars at home and abroad have conducted in-depth research on sensitivity analysis of machine tool geometric errors. The sensitivity coefficients of 37 geometric error terms of the machine tool were calculated by matrix differential method; the sensitivity analysis of 18 geometric errors of the vertical machining center was conducted by using the Sobol method; and the influence of five-axis CNC grinding machine was analyzed by using Morris method
The key geometric error terms of accuracy; using the extended Fourier amplitude test method to identify strong coupling geometric error terms and key geometric error terms. At present, although there are many sensitivity analysis methods, due to the complexity of the tooth surface of spiral bevel gears and the numerous factors affecting the machining accuracy of the tooth surface, it is crucial to reasonably select a sensitivity analysis method for the sensitivity analysis of tooth surface deviation. Firstly, the calculation principles and characteristics of local sensitivity and global sensitivity methods are compared; then, combined with the CNC machining principle of spiral bevel gears and sensitivity analysis methods, the sensitivity relationship between machine tool geometric errors and tooth surface deviation of spiral bevel gears is studied; finally, through case analysis of the characteristics of two sensitivity analysis methods, theoretical guidance is provided for the selection of sensitivity analysis methods for tooth surface deviation of spiral bevel gears.
Sensitivity analysis
There are two main types of sensitivity analysis methods: local sensitivity analysis methods represented by differential methods, difference methods, and perturbation methods, and regression analysis methods The global sensitivity analysis method, represented by the method of mean, variance, and screening. Through sensitivity analysis, key input factors and non-key Input factors. In practical analysis and calculation, non-critical input factors can be ignored first, which can significantly reduce the complexity and analytical difficulty of the system, as well as greatly reduce the computational load and processing difficulty of related data.
Local sensitivity analysis is a method of single-factor analysis, in which small changes are made to a single input factor at a time, while all other factors remain unchanged, and the sensitivity coefficient is determined by the output’s differential with respect to the input parameter or the change in output with respect to a single input factor change.
The global sensitivity analysis method not only considers the range and distribution of each parameter, but also incorporates all parameters into the analysis during the analysis and calculation process, considering the impact of mutual coupling between input parameters on the results. However, the global sensitivity analysis method generally requires a certain The input parameters of the model are obtained through sampling of a certain scale, and then sensitivity analysis is conducted. In particular, for cases with many parameters, the computational load can be significant.
Spiral bevel gear machine tool machining model
According to the set numerical control program, the A, B, X, Y, and Z axes can be linked to process the Gleason-made spiral bevel gear. The C axis drives the cutter head to rotate without affecting the tooth surface development process. For the five motion axes related to the tooth surface development process, each axis has geometric errors due to factors such as manufacturing and assembly, which directly lead to tooth surface deviations. Each axis has six geometric errors, including three linear errors and three angular errors. For the entire machine tool, 30 geometric errors need to be considered.
To conduct sensitivity analysis using the local sensitivity analysis method, it is necessary to ensure that the variations in the same type of input parameters are the same, so that the output results are Comparability. The selection criteria are independent of the actual range and distribution of input parameters, so the actual input parameter range changes It will not affect the results of local sensitivity analysis. In the previous article, the selection of the input parameter range for the global sensitivity analysis method was based on the characteristics of the local sensitivity analysis method, ensuring that the range and distribution of the same type of parameters are the same. However, in real cases, the value ranges of input parameters are generally different and the variation rules are often different. Therefore, considering the actual possible situations, taking the geometric error with number 4 as an example, let the value range of this geometric error change from 0-10μm to 0-20μm while keeping other conditions unchanged. For the global sensitivity analysis method, after changing the value range of the input parameter, the results of sensitivity analysis change significantly. Among them, the geometric error item with number 4, δxY, has a significant change in global sensitivity coefficient, increasing from 0.023 to 0.154. The global sensitivity coefficients of other items only have minor changes, and there is no significant change in the relevant proportion.
Result analysis
By comparing the calculation results of the two sensitivity analysis methods before and after the change in the value range, it can be found that for the machining of spiral bevel gear tooth surfaces The deviation calculation model and two sensitivity analysis methods have the following characteristics:
(1) In the ideal case where the input parameter value range and distribution rule are exactly the same, the linear errors obtained by the two sensitivity analysis methods are The sensitivity coefficient distribution of the tooth surface deviation of spiral bevel gears is the same. The sensitivity coefficients of the linear error terms for the five axes all exhibit a y-direction (Geometric errors with numbers 2, 5, 8, 11, and 14) > x direction (geometric errors with numbers 1, 4, 7, 10, and 13) > z direction (geometric errors with numbers 3, 6, 9, 12, and 15).
(2) In the ideal case where the input parameter value range and distribution rule are exactly the same, the angles of each axis obtained by the two sensitivity analysis methods are The distribution law of the sensitivity coefficient of error to the tooth surface deviation of spiral bevel gears is relatively small due to the use of probability calculation formulas and sampling sample limitations However, the overall trend is the same. The key angle errors of each axis are the same (geometric errors of S/N 17, 18, 19, 20, 23, 26, and 29).
(3) When the input parameter range changes, global sensitivity analysis can be used to discover that the input of non-critical geometric errors Parameters can be changed to key geometric errors. However, for local sensitivity analysis methods, the analysis results are independent of the distribution range of the input parameters. When the distribution range of the input parameters varies significantly, the accuracy of the analysis results of this method will significantly decrease, and it may even no longer be applicable to the sensitivity analysis of the model.
Conclusion
By applying two sensitivity analysis methods to the sensitivity analysis of the spiral bevel gear tooth surface machining deviation model, it can be found that:
(1) In the ideal case where the input parameter value range and distribution rule are exactly the same, the computational load of the local sensitivity analysis method is relatively small, The preliminary analysis results can be obtained quickly, and the distribution pattern of the analysis results is the same as that of the global sensitivity analysis results, which has certain reference value.
(2) The Sobol global sensitivity analysis method considers the impact of the range and distribution of input parameters on the analysis results, making it more scientific, more reliable, and more widely applicable. In addition, most global sensitivity analysis methods can be used to study the quantitative relationship between input parameters and output results, as well as the impact of interactions between parameters on output results. However, compared to local sensitivity analysis methods, the computational load increases significantly, requiring more time.
(3) For linear or nonlinear problems with unclear distribution range of input parameters, limited available time, and similar distribution conditions For models, local sensitivity analysis can be considered; for nonlinear models with complex distribution ranges of input parameters and high accuracy requirements, global sensitivity analysis methods can be considered.