Due to the coupled vertical vibration of the automotive rear axle spiral bevel gear, precise detection of gear wear remains challenging. Therefore, this paper proposes a method for automotive spiral bevel gear wear detection using Delaunay triangulation. By employing a scanning technique to measure discrete data on the gear surface, Delaunay triangulation is performed on adjacent scan lines to complete non-feature discrete data segmentation of the gear surface, thereby achieving a comprehensive description of the gear surface area. Based on the segmented data, the LMD (Local Mean Decomposition) algorithm combined with Hermite interpolation is used to traverse and calculate the amplitude of the PF (Product Function) component of the triangular mesh, enabling the detection of gear wear faults. Experimental results demonstrate that the proposed method’s wear depth and wear rate detection results are highly consistent with measured values, and it effectively detects the area of gear wear regions with a detection accuracy of up to 98.7%. Thus, the method proves capable of effectively detecting gear wear.
1. Introduction
With the continuous development of the automotive industry, cars have become indispensable in daily life. However, as vehicles age, gear wear, particularly of rear axle spiral bevel gears, becomes increasingly prominent. The coupled vertical vibration of the rear axle during vehicle operation leads to wear on the spiral bevel gears. If not detected promptly, this wear can significantly reduce the safety of automobile operation. Therefore, research on effective gear wear detection methods that enhance detection accuracy and ensure vehicle safety is crucial.
Several gear wear detection methods have been proposed in the literature. For instance, a method based on wavelet packet energy and modulation signal bispectrum sideband estimation has been proposed to decompose gear wear signals into multiple frequency bands and extract tooth surface wear characteristics through modulation signal bispectrum sideband estimation [1]. However, this method does not consider noise interference, resulting in insufficient gear wear detection accuracy. Another method, based on reverse engineering, constructs a conversion matrix for the gear wear area and performs error compensation, followed by NURBS surface fitting to complete gear wear detection [2]. Nevertheless, this method may produce unworn regions in the wear surface fitting results, impacting detection efficacy. Additionally, a method employing improved Mask Scoring R-CNN for gear wear detection has been introduced, which utilizes a residual network and feature pyramid network to extract gear wear features through the fusion of semantic information and detailed features [3]. However, this method does not address data denoising, leading to excessive noise in sample data and reduced gear wear detection accuracy.
To address the limitations of existing gear wear detection methods, this paper proposes a Delaunay triangulation-based method for detecting wear in automotive spiral bevel gears. This method utilizes Delaunay triangulation to process discrete data from adjacent scan lines on the gear surface, completing non-feature discrete data segmentation. Hermite interpolation is then employed to traverse and detect the segmented Delaunay triangular mesh, enabling gear wear detection.
2. Non-Feature Discrete Data Segmentation of Gear Surfaces
2.1 Construction of the NURBS Gear Surface Model
Non-Uniform Rational B-Splines (NURBS) are widely used in computer-aided design and manufacturing due to their ability to precisely represent complex shapes. In this study, NURBS surfaces are utilized to construct the automotive spiral bevel gear model. The NURBS surface can be expressed as follows:
A(u,v)=∑i=0n∑j=0mNi,k(u)Nj,l(v)wij∑i=0n∑j=0mNi,k(u)Nj,l(v)wijPij
where Ni,k(u) and Nj,l(v) are the B-spline basis functions in the u and v directions, respectively; Pij are the control points; wij are the weights; n and m are the numbers of control points in the u and v directions, respectively; and k and l are the degrees of the B-spline basis functions.
2.2 Delaunay Triangulation for Gear Surface Data Segmentation
Delaunay triangulation is a method for dividing a two-dimensional plane into triangles such that no point in the plane lies within the circumcircle of any triangle. This property ensures that the triangles are as equilateral as possible, providing an optimal triangulation for the given point set.
In this study, Delaunay triangulation is applied to process discrete data from adjacent scan lines on the gear surface. As shown in Figure 1, the process involves connecting the starting and ending points of two adjacent scan lines and then connecting each measurement point on one scan line to its nearest measurement point on the other scan line. This divides the space between the two scan lines into several regions, which are then detected as triangles or quadrilaterals. Based on the maximum-minimum angle criterion, quadrilateral regions are further subdivided into triangles, completing the Delaunay triangulation.
2.3 Non-Feature Discrete Data Segmentation
To achieve non-feature discrete data segmentation of the gear surface, Delaunay triangulation is employed. Starting from an arbitrary triangle on the NURBS gear surface model, the center triangle with the smallest region value is selected. The region value Y of a triangle is calculated as follows:
Y=max(∣∣θ1−θ2∣∣,∣∣θ2−θ3∣∣,∣∣θ1−θ3∣∣)−min(∣∣θ1−θ2∣∣,∣∣θ2−θ3∣∣,∣∣θ1−θ3∣∣)
where θ1, θ2, and θ3 are the angles between the two triangles.
The selected center triangle serves as the starting point for segmentation. The adjacent triangle with the smallest angle with the starting triangle is chosen as the next center triangle. This process continues iteratively until all triangles are traversed, completing the non-feature discrete data segmentation of the gear surface. Figure 2 depicts the non-feature discrete data segmentation of the gear model.
3. Gear Wear Detection Using Hermite Interpolation Algorithm
To accurately detect gear wear regions, the Hermite interpolation algorithm, specifically the Local Mean Decomposition (LMD) method, is employed based on the non-feature discrete data segmentation results. The detection steps are outlined below:
3.1 Data Preprocessing
- Noise Reduction: The original noisy discrete data c(y) is transformed into small parameter data and then denoised using the Cascaded Bistable Stochastic Resonance (CBSR) system to obtain denoised discrete data cn(y).
- Extrema Point Extension: The extrema points ml of the denoised data are extended to generate a new sequence cl.
3.2 Envelope Generation
- Selection of Maximum/Minimum Points: Starting from one end of the new sequence, the maximum or minimum points are selected to generate two envelope lines using the Hermite interpolation algorithm.
- Calculation of Local Mean and Envelope Estimation Functions:
- Local Mean Function:q(y)=2Rup(y)+Rdown(y)
- Envelope Estimation Function:s(y)=2Rup(y)−Rdown(y)
3.3 PF Component Amplitude Calculation
The LMD method is used to separate the envelope data from the original gear surface data, and the PF component amplitude is calculated as follows:
f0(y)=2π1arccos∫y2dy∫f2(y)dy∫yf(y)dy
where f0(y) is the PF component amplitude, with a value range of [0, 1]. If the calculation result falls outside this range, it indicates the presence of wear in that region.
3.4 Traversal Detection
The above steps are repeated until all triangular meshes on the gear surface have been traversed and detected.
4. Experimental Analysis
4.1 Experimental Setup
To validate the effectiveness of the proposed Delaunay triangulation-based gear wear detection method, comparative tests were conducted. A spiral bevel gear from a 2015 Santana manual transmission car was selected for the study, with 20 identical gears used in the experiments. The material properties of the spiral bevel gear are shown in Table 1.
Table 1: Material Properties of Automotive Spiral Bevel Gears
Component | Material | Density (kg/m³) | Young’s Modulus (GPa) | Poisson’s Ratio |
---|---|---|---|---|
Gear | 45Cr | 7850 | 206 | 0.3 |
4.2 Experimental Procedures
- Data Collection: The gear wear experimental equipment was operated continuously for 200 hours, with wear data collected every 5 minutes. This resulted in 12 sets of wear data per hour.
- Test Indicators: Gear wear depth, wear rate, and wear area detection accuracy were set as the test indicators.
- Comparison Methods: The proposed method was compared with two existing methods: the reverse engineering-based method proposed in Reference [4] and the improved Mask Scoring R-CNN-based method proposed in Reference [5].
- Experimental Validation: The three methods were tested according to the set indicators.
4.3 Results and Analysis
4.3.1 Gear Wear Depth
Comparing the wear depth detection results of the three methods with the actual measurements, it is evident that the proposed method’s results are highly consistent with the actual values. In contrast, the methods from References [4] and [5] exhibit significant deviations. This demonstrates that the proposed method can effectively analyze gear wear depth.
4.3.2 Gear Wear Rate
The proposed method’s wear rate results remain consistent with the actual wear rates throughout the study, with a maximum error of no more than 0.01 µm. In contrast, the methods from References [4] and [5] show larger deviations. This indicates that the proposed method can achieve more accurate gear wear rate results, enhancing gear wear detection effectiveness.
4.3.3 Gear Wear Area Detection Accuracy
Experiment Number | Proposed Method (%) | Method from Reference [4] (%) | Method from Reference [5] (%) |
---|---|---|---|
1 | 96.9 | 77.9 | 83.9 |
2 | 98.3 | 76.3 | 79.1 |
3 | 97.1 | 70.2 | 80.7 |
4 | 96.2 | 73.1 | 82.2 |
5 | 97.3 | 79.4 | 81.4 |
6 | 98.3 | 72.2 | 84.3 |
7 | 96.8 | 74.1 | 80.9 |
8 | 98.1 | 74.7 | 82.4 |
9 | 98.7 | 73.6 | 80.6 |
10 | 96.8 | 71.9 | 79.4 |
As shown in Table 2, the proposed method achieves the highest wear area detection accuracy of up to 98.7%, significantly outperforming the methods from References [4] and [5], which reach maximum accuracies of 79.4% and 84.3%, respectively. This demonstrates that the proposed method can accurately detect gear wear areas.
5. Conclusion
Effective detection of automotive spiral bevel gear wear is crucial for ensuring vehicle driving safety. This paper proposes a Delaunay triangulation-based method for detecting wear in automotive spiral bevel gears. By performing Delaunay triangulation on adjacent scan lines on the gear surface, non-feature discrete data segmentation is achieved. Based on the segmented data, Hermite interpolation is employed to calculate PF component amplitudes, and all triangular meshes are traversed to detect wear regions.
Experimental results show that the proposed method’s wear depth and wear rate detection results are highly consistent with actual values, and it can accurately detect gear wear areas with a maximum accuracy of 98.7%. This method effectively addresses the limitations of traditional methods and introduces new ideas into automotive spiral bevel gear wear detection technology.