This article focuses on the sensitivity analysis of machining deviation of spiral bevel gears. It begins with an introduction to the importance of spiral bevel gears and the impact of machine tool geometric errors on gear manufacturing accuracy. The two main sensitivity analysis methods, local sensitivity analysis and Sobol global sensitivity analysis, are described in detail. Through a case study of spiral bevel gear machining, the characteristics and application scenarios of the two methods are compared and analyzed. The results provide theoretical guidance for the selection of sensitivity analysis methods in spiral bevel gear machining, which is of great significance for improving gear manufacturing accuracy and optimizing machine tool parameters.
1. Introduction
Spiral bevel gears are widely used in various fields such as automobiles, aviation, and mining machinery due to their high contact ratio, strong load-carrying capacity, high transmission ratio, and high transmission efficiency. The manufacturing accuracy of spiral bevel gears is directly related to the errors of computer numerical control (CNC) machine tools. Machine tool errors include geometric errors, thermal errors, and servo control errors. Among them, geometric errors have the characteristics of repetition and stability and can be accurately compensated through the CNC system. Therefore, studying the characteristics of machine tool geometric errors and determining the key geometric error terms play an important role in tooth surface deviation compensation and the distribution and maintenance of machine tool motion accuracy.
Sensitivity analysis is a method to study the influence degree of input factor changes on output results in a system. The analysis results are represented by sensitivity coefficients. A larger sensitivity coefficient indicates a higher correlation between the input factor and the output result. In the field of machine tool geometric error sensitivity analysis, many scholars at home and abroad have conducted in-depth research. However, due to the complex tooth surface of spiral bevel gears and numerous factors affecting tooth surface machining accuracy, the reasonable selection of sensitivity analysis methods is crucial for the sensitivity analysis of tooth surface deviation.
2. Sensitivity Analysis Methods
2.1 Local Sensitivity Analysis Method
Local sensitivity analysis is a single-factor analysis method. It determines the sensitivity coefficient by making a small change to a single input factor each time while keeping all other factors unchanged, and observing the change in the output result with respect to the input parameter or the change in the output result due to the change in a single input factor. The concept of local sensitivity is clear, and the calculation is simple. It is mainly applicable to linear models and models with weak nonlinearity.
The analysis process is as follows:
- Input the change amount of the first parameter.
- Calculate the theoretical output result and the actual output result.
- Calculate the deviation value.
- Increment the parameter index.
- Output all deviation values.
2.2 Sobol Global Sensitivity Analysis
The global sensitivity analysis method not only considers the range and distribution law of each parameter but also substitutes all parameters into the analysis during the analysis and calculation process, considering the influence of the mutual coupling between input parameters on the result. However, the global sensitivity analysis method generally requires a certain scale of sampling to obtain the input parameters of the model and then performs sensitivity analysis. Especially for cases with more parameters, the calculation amount will be relatively large.
Taking the Sobol method as an example, it is a variance-based global sensitivity method. It calculates the sensitivity coefficient of input parameters mainly by decomposing the variance of the model terms. The specific calculation process involves complex mathematical formulas, but in general, it decomposes the function corresponding to the model into multiple terms considering the individual and combined effects of different input parameters, and then calculates the variance components and sensitivity coefficients based on these decompositions.
2.3 Monte Carlo Estimation
When using the Sobol method to solve the sensitivity coefficient, it involves solving multiple integrals. For complex models, solving multiple integrals is usually very difficult. Therefore, the Monte Carlo method is often used to approximately simulate the solution of multiple integrals. The general calculation method is to sample the input parameters twice independently to obtain two independent sampling matrices E and F. Based on these matrices, some matrices are constructed to calculate the first-order sensitivity coefficient and the global sensitivity coefficient approximately.
3. Spiral Bevel Gear Machine Tool Processing Model
3.1 Machine Tool Geometric Error Classification
In a CNC machine tool for processing spiral bevel gears, there are 30 geometric errors related to the five motion axes (A, B, X, Y, and Z) associated with the tooth surface generation process. Each axis has 6 geometric errors, including 3 linear errors and 3 angle errors. These geometric errors are numbered for easy analysis, as shown in Table 1.
| Coordinate Axis | Geometric Error | Order Number |
|---|---|---|
| δ2x, δ2x, δx | 1 – 3 | |
| δx, δ., δ | 4 – 6 | |
| δ.z, δ,2, δ | 7 – 9 | |
| δ, δ4, δ4 | 10 – 12 | |
| δ8, δ,8, δ | 13 – 15 | |
| Bax, Egx, Byx | 16 – 18 | |
| BaY, Egy, ETY | 19 – 21 | |
| XYZABXYZAB | BL, E, E22 | 22 – 24 |
| BaA, EB, Ey4 | 25 – 27 | |
| EaB, 30, E78 | 28 – 30 |
3.2 Tooth Surface Processing Process
The generation motion of the workpiece gear is determined by five axes (X, Y, Z, A, and B). The homogeneous transformation matrix from the A axis to the Y axis, combined with the tool equation, can obtain the ideal tooth surface equation rB. When considering machine tool geometric errors, the actual tooth surface equation is obtained by multiplying the geometric error matrices of each axis with the corresponding matrices in the ideal equation.
4. Sensitivity Analysis Example
4.1 Sampling Calculation
Taking a gear with specific parameters (such as number of teeth, module, tooth width, etc.) as an example, the theoretical tooth surface of the spiral bevel gear can be obtained according to the relevant equation. Considering the influence of machine tool geometric errors in actual processing, the actual tooth surface can also be calculated. A discrete 15×9 dot matrix is used to represent the tooth surface. By comparing the theoretical and actual tooth surface dot matrices, the deviation values of corresponding tooth surface points can be calculated, and the tooth surface deviation K is obtained as the output result of the model.
For local sensitivity analysis, given the change amounts of linear and angle parameters, the sensitivity analysis result is obtained. For Sobol global sensitivity analysis, by randomly sampling the machine tool geometric error parameters within a certain range and calculating the corresponding theoretical and actual tooth surfaces, the global sensitivity coefficients of 30 machine tool geometric errors are obtained and sorted in descending order of magnitude.
4.2 Input Parameter Situation Discussion
In local sensitivity analysis, the change amounts of the same type of input parameters are required to be the same, and the analysis result is not affected by the actual range and distribution law of the input parameters. In global sensitivity analysis, when the range of an input parameter changes (for example, from 0 – 10 μm to 0 – 20 μm for a certain geometric error), the sensitivity analysis result changes significantly.
4.3 Result Analysis
- In the ideal case where the input parameter range and distribution law are completely the same, the distribution laws of the sensitivity coefficients of linear errors to the tooth surface deviation of spiral bevel gears obtained by the two sensitivity analysis methods are the same. The sensitivity coefficients of linear errors of each axis show a trend of y direction > x direction > z direction.
- In the ideal case where the input parameter range and distribution law are completely the same, the distribution laws of the sensitivity coefficients of angle errors to the tooth surface deviation of spiral bevel gears obtained by the two sensitivity analysis methods have a small difference due to different probability calculation formulas and sampling sample limitations, but the overall trend is the same. The key angle errors of each axis are the same.
- When the input parameter range changes, through global sensitivity analysis, an originally non-key geometric error input parameter may become a key geometric error. However, for the local sensitivity analysis method, its analysis result is independent of the distribution range of the input parameters. When the distribution range difference of the input parameters is large, the accuracy of the analysis result will decrease significantly, and it may even no longer be applicable to the sensitivity analysis of the model.
5. Comparison and Selection of Sensitivity Analysis Methods
5.1 Comparison of the Two Methods
- In the ideal case where the input parameter range and distribution law are the same, the distribution laws of the sensitivity coefficients obtained by the two methods are similar for both linear errors and angle errors.
- The local sensitivity analysis method has a smaller calculation amount and can obtain preliminary analysis results faster. The Sobol global sensitivity analysis method considers the influence of input parameter range and distribution law on the analysis result, is more scientific, and the obtained conclusion is more reliable and has a wider application range.
- When the input parameter range changes, the local sensitivity analysis method may not be applicable, while the global sensitivity analysis method can reflect the change of key geometric errors.
5.2 Selection Principles of Sensitivity Analysis Methods
For models where the distribution range of input parameters is not clear or the linear or nonlinear distribution is similar, local sensitivity analysis can be considered. For nonlinear models with complex distribution of input parameters and high accuracy requirements, the global sensitivity analysis method is suitable.
6. Conclusion
- In the ideal case where the input parameter range and distribution law are the same, the local sensitivity analysis method has a smaller calculation amount and can provide a reference value. The Sobol global sensitivity analysis method is more scientific and reliable.
- The selection of sensitivity analysis methods should be based on the characteristics of the model and the input parameters. For different models, choosing the appropriate method can improve the accuracy of the analysis result and provide theoretical support for improving gear manufacturing accuracy and optimizing machine tool parameters.
In conclusion, this study provides theoretical guidance for the selection of sensitivity analysis methods in spiral bevel gear machining, which is of great significance for the development of the gear manufacturing industry.
