Abstract
Spiral bevel gears are widely used in automotive, aviation, and mining machinery due to their high contact ratio, strong load-bearing capacity, and efficient power transmission. However, the manufacturing accuracy of spiral bevel gears is directly influenced by the errors in the numerical control (NC) machine tools used for their production. Geometric errors, such as positioning errors and angular deviations, can significantly impact the tooth surface quality. This paper investigates the sensitivity of machine tool geometric errors to the deviations on the tooth surface of spiral bevel gears using both local and global sensitivity analysis methods. The study aims to identify the key geometric error sources and provide theoretical guidance for error compensation and machine tool accuracy improvement.
1. Introduction
Spiral bevel gears are critical components in power transmission systems due to their unique geometrical configuration and superior transmission performance. The tooth surface of spiral bevel gears is complex, with high contact ratios and load-bearing capacities, making them ideal for use in demanding applications. However, the manufacturing process of spiral bevel gears is challenging, and errors in the NC machine tools can significantly affect the quality of the finished product.
Machine tool errors can be classified into geometric errors, thermal errors, and servo control errors. Among these, geometric errors are repeatable and stable, making them amenable to compensation through NC programs. Therefore, understanding the sensitivity of geometric errors to the tooth surface deviations is crucial for improving manufacturing accuracy.
1.1 Background and Motivation
Previous studies have focused on various aspects of machine tool error analysis. Chen et al. [1] calculated the sensitivity coefficients of 37 geometric error terms in machine tools using matrix differential methods. Cheng et al. [2] conducted a Sobol sensitivity analysis on 18 geometric errors in a vertical machining center. While these studies provide valuable insights, the complexity of the spiral bevel gear tooth surface necessitates a more detailed analysis of error sensitivity.
This paper aims to:
- Establish a model for spiral bevel gear tooth surface machining deviations.
- Analyze the sensitivity of machine tool geometric errors to tooth surface deviations using both local and global sensitivity analysis methods.
- Compare the characteristics of the two sensitivity analysis methods and provide guidelines for selecting the appropriate method based on the specific requirements.
1.2 Paper Structure
The paper is organized as follows: Section 2 describes the methodology for sensitivity analysis, including local and global sensitivity analysis methods. Section 3 introduces the spiral bevel gear machining model and the calculation of tooth surface deviations. Section 4 presents a case study analyzing the sensitivity of geometric errors, and Section 5 discusses the results and provides conclusions.
2. Sensitivity Analysis Methods
Sensitivity analysis is a powerful tool for quantifying the impact of input parameter variations on output results. It is essential for identifying key error sources and prioritizing error compensation strategies.
2.1 Local Sensitivity Analysis
Local sensitivity analysis, also known as one-at-a-time (OAT) analysis, involves varying one input parameter while keeping all others constant to observe the resulting changes in the output. The sensitivity coefficient (SC) is calculated as the ratio of the output variation to the input variation.
SCi=∂xi∂y(Δxixi)
where y is the output, xi is the i-th input parameter, and Δxi is the variation in xi.
Advantages and Limitations:
- Advantages: Simple and straightforward; ideal for linear or weakly nonlinear models.
- Limitations: Ignores interactions between input parameters; may not be accurate for complex, highly nonlinear models.
2.2 Global Sensitivity Analysis
Global sensitivity analysis considers the full range and distribution of input parameters and evaluates their interactions. The Sobol method, a popular variance-based global sensitivity analysis technique, decomposes the model output variance into contributions from individual input parameters and their interactions.
The Sobol indices are defined as:
Si=VVi
where Vi is the partial variance due to the i-th input parameter, and V is the total variance of the model output.
The total sensitivity index, including higher-order interactions, is calculated as:
STi=1−VV∼i
where V∼i is the variance of the model output when the i-th input parameter is fixed.
Advantages and Limitations:
- Advantages: Accounts for interactions between input parameters; suitable for complex, nonlinear models.
- Limitations: Computationally intensive, especially for models with many input parameters.
3. Spiral Bevel Gear Machining Model
3.1 Machine Tool Geometric Errors
NC machine tools used for spiral bevel gear machining typically have five axes ((X, Y, Z, A, B)) that contribute to the generation of the tooth surface. Each axis has six geometric errors: three linear errors ((\delta_x, \delta_y, \delta_z)) and three angular errors ((\varepsilon_\alpha, \varepsilon_\beta, \varepsilon_\gamma)). Therefore, a total of 30 geometric errors need to be considered.
3.2 Tooth Surface Equation
The theoretical tooth surface of a spiral bevel gear can be described by the following equation, considering the transformation matrices corresponding to each axis movement:
rg=MA⋅MB⋅MZ⋅MX⋅MY⋅rt(u,θ)
where rg is the position vector of a point on the generated tooth surface, Mq (q = A, B, Z, X, Y\)) are the transformation matrices for each axis, and \(\mathbf{r}_t(u, \theta) is the tool path equation parameterized by u and θ.
When machine tool geometric errors are considered, the actual tooth surface equation becomes:
reg=MA⋅MeA⋅MB⋅MeB⋅MZ⋅MeZ⋅MX⋅MeX⋅MY⋅MeY⋅rt(u,θ)
where Meq are the error matrices corresponding to each axis.
3.3 Calculation of Tooth Surface Deviations
Tooth surface deviations are calculated by comparing the coordinates of corresponding points on the theoretical and actual tooth surfaces. The overall deviation K is defined as the sum of the deviations Kf at discrete points on the tooth surface:
K=f=1∑NKf
where N is the total number of discrete points considered on the tooth surface.
4. Case Study: Sensitivity Analysis of Geometric Errors
4.1 Gear Geometry and Machining Parameters
The spiral bevel gear considered in this case study has the following geometric and machining parameters (Table 1).
Table 1: Geometric and Machining Parameters of the Spiral Bevel Gear
| Parameter | Value |
|---|---|
| Number of teeth (Z) | 18 |
| Module at large end (mt) | 4.29 mm |
| Face width (wb) | 45 mm |
| Helix angle at midpoint (β) | 35° |
| Addendum height (ha) | 8.78 mm |
| Dedendum height (hf) | 5.03 mm |
| Pitch cone angle (γ1) | 27.21° |
| Face cone angle (γp) | 31.65° |
| Root cone angle (γf) | 25.36° |
| Outside cone distance (Lo) | 140.9 mm |
4.2 Sampling and Calculation
For the local sensitivity analysis, each input parameter is varied by a small amount (e.g., ±0.01 mm for linear errors and ±27 arc-seconds for angular errors) while keeping others constant. The Sobol global sensitivity analysis considers a wider range for each parameter (e.g., 0–10 μm for linear errors and 0–27 arc-seconds for angular errors), assuming a uniform distribution.
4.3 Results
The results of the local sensitivity analysis, illustrating the sensitivity coefficients of each geometric error.
The Sobol global sensitivity analysis identifies the key geometric errors based on their total sensitivity indices.
Table 2: Top 10 Geometric Errors Based on Sobol Global Sensitivity Analysis
| Error Number | Error Type | Total Sensitivity Index ((S_{T_i})) |
|---|---|---|
| 17 | εα (X-axis) | 0.25 |
| 18 | εβ (X-axis) | 0.18 |
| 19 | εγ (Y-axis) | 0.15 |
| 20 | εα (Y-axis) | 0.12 |
| … | … | … |
4.4 Discussion
- Local Sensitivity Analysis: Provides a quick initial assessment of error sensitivity but may overlook interactions between input parameters.
- Global Sensitivity Analysis: Provides a more comprehensive understanding of error sensitivity, accounting for parameter interactions and variations within their full ranges.
Changing the input parameter ranges significantly affects the global sensitivity analysis results.
This underscores the importance of selecting an appropriate sensitivity analysis method based on the specific requirements and the nature of the model under consideration.
5. Conclusion
This paper presents a comprehensive sensitivity analysis of machine tool geometric errors on the tooth surface deviations of spiral bevel gears using both local and global sensitivity analysis methods. The key findings are:
- Local Sensitivity Analysis: Provides a quick and straightforward assessment of error sensitivity for linear or weakly nonlinear models.
- Global Sensitivity Analysis: Offers a more comprehensive view of error sensitivity, considering interactions between input parameters and variations within their full ranges.
- Selection of Analysis Method: Local sensitivity analysis is suitable for models with unclear input parameter distributions or when time is limited. Global sensitivity analysis is preferred for complex, nonlinear models requiring high accuracy.
The results provide valuable insights into the key geometric error sources affecting spiral bevel gear tooth surface deviations and guide the development of effective error compensation strategies.