This article delves into the significance of Delaunay triangulation in detecting wear in automotive spiral bevel gears. By addressing the challenges posed by traditional methods, it presents a comprehensive overview of how this innovative approach improves detection accuracy and reliability. Through in – depth analysis of the triangulation process, wear detection algorithms, and experimental verification, the article demonstrates the effectiveness of Delaunay triangulation in enhancing automotive safety and maintenance.
1. Introduction
Automobiles have become an essential part of modern life. However, as vehicles age, the wear of their components, especially automotive spiral bevel gears, becomes a significant concern. The spiral bevel gears in the rear axle of a car are prone to wear due to the coupled vertical vibration during driving. If left undetected, this wear can lead to a series of problems, such as reduced driving safety, increased fuel consumption, and even sudden mechanical failures.
Traditional gear wear detection methods, such as those based on wavelet – packet energy and modulation – signal bispectrum side – band estimation, face limitations. These methods often overlook the interference of noise, resulting in insufficient detection accuracy. For example, the method in [3] fails to account for noise, which can distort the detected signals and lead to inaccurate wear assessments. Another traditional approach, based on reverse engineering, has issues with the fitting of wear surfaces. It may include non – worn areas in the wear – surface fitting results, thus reducing the effectiveness of the detection. The method proposed in [4] is a case in point, where the inaccurate fitting affects the overall detection performance. The method based on improved Mask Scoring R – CNN also has its drawbacks. It neglects data denoising, which causes excessive noise in the sample data and decreases the detection accuracy.
To overcome these problems, the Delaunay triangulation – based wear detection method has emerged as a promising solution. This method can accurately process the discrete data on the gear surface, effectively describe the gear surface area, and precisely detect the wear area. It has the potential to revolutionize the field of automotive gear wear detection.
2. Gear Surface Non – characteristic Discrete Data Blocking with Delaunay Triangulation
2.1 NURBS Surface Construction of Automotive Spiral Bevel Gears
The first step in the Delaunay triangulation – based wear detection method is to construct the automotive spiral bevel gear model using the NURBS (Non – Uniform Rational B – Splines) surface. The NURBS surface is a powerful tool for representing complex geometric shapes. The formula for constructing the NURBS surface of a gear is as follows:
where is the NURBS surface function, and represent different directions, is the number of center points in direction , is the number of center points in direction , is the surface center point, and are the spline basis functions in directions and respectively, and is the weight factor.
This formula allows for a flexible and accurate representation of the gear’s surface geometry. By adjusting the parameters such as the number of center points, spline basis functions, and weight factors, we can fine – tune the shape of the NURBS surface to match the actual gear surface.
2.2 Principles and Procedures of Delaunay Triangulation
Delaunay triangulation is a key step in this process. In a Delaunay triangle network, the circum – circle of any triangle does not contain other points, and the triangles are formed by the nearest points. This property ensures the uniqueness and stability of the triangulation results, regardless of where the processing starts on the gear surface.
After measuring the discrete data on the gear surface through a scanning method, we apply the Delaunay triangulation principle. Let represent the scan line, with , and being two adjacent scan lines. () and () are the test points, where is the number of test points.
We first connect the starting and ending points of two adjacent scan lines, and then connect 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, as shown in Figure 1(a). Next, we detect the triangular and quadrilateral space regions. For the quadrilateral space regions, we use the minimum – interior – angle – maximum criterion to perform triangulation, and finally complete the Delaunay triangulation, as shown in Figure 1(b).
[Insert Figure 1: Adjacent scan line Delaunay triangulation here]
2.3 Data Blocking and Area Description
After the Delaunay triangulation, we perform data blocking. We select an arbitrary triangle on the NURBS tooth – surface model as the central triangle CenTri. Then, we calculate the values of all central triangles using the following formula:
where are the angles between two triangles, and is the central – triangle region value.
We choose the central triangle with the smallest region value as the starting triangle, as shown in Figure 2(a). Then, we select the adjacent triangle with the smallest angle to the starting triangle as the new central triangle. If it meets a certain condition (such as the angle relationship), we calculate the angles between the surrounding triangles and the starting triangle and select the one with the smallest angle as the central triangle. Otherwise, we re – select the starting triangle. By repeating this process, we can traverse all the triangles and complete the non – characteristic discrete data blocking of the gear, as shown in Figure 2(b).
[Insert Figure 2: Non – characteristic discrete data blocking of gear model here]
Through the triangular – region interpolation algorithm, we can calculate the tooth – surface data and achieve a comprehensive description of the gear – surface area. This data – blocking and area – description process is crucial for accurately detecting gear wear in the subsequent steps.
3. Gear Wear Detection Using Hermite Interpolation Algorithm
3.1 Detection Steps
Based on the results of the non – characteristic discrete data blocking of the gear surface, we use the local – mean – decomposition (LMD) algorithm in the Hermite interpolation algorithm to detect tooth – surface wear. The specific steps are as follows:
- Set the original discrete data as and transform the original strong – noise discrete data into small – parameter data . This step is necessary to pre – process the data and make it more suitable for subsequent operations. For example, in a noisy measurement environment, the original data may contain a large amount of interference information. Transforming it into small – parameter data can reduce the impact of noise on the detection results.
- Denoise the small – parameter data obtained in step 1 in the cascaded bistable stochastic resonance (CBSR) system to get the denoised discrete data . The CBSR system is an effective tool for removing noise from signals. It can enhance the weak signals related to gear wear while suppressing the noise, improving the signal – to – noise ratio of the data.
- Perform extension processing on the extreme points of the denoised discrete data to obtain a new sequence . This extension processing can help to better capture the characteristics of the data and improve the accuracy of the subsequent analysis.
- Select the maximum or minimum point from one end of the new sequence and use the Hermite interpolation algorithm to generate two envelope lines. The Hermite interpolation algorithm can provide a smooth interpolation between data points, enabling us to accurately represent the upper and lower envelopes of the data.
- Calculate the local – mean function using the formula , where is the local – mean function, is the upper envelope line, and is the lower envelope line. The local – mean function can reflect the local trend of the data and is an important intermediate parameter in the LMD analysis.
- Calculate the envelope – estimation function with the formula , where is the envelope – estimation function. This function can further describe the characteristics of the data envelope and help in identifying the wear – related features.
- Use the LMD analysis method to separate the envelope data from the original tooth – surface data and calculate the amplitude of the PF (product function) component. The formula for calculating the PF – component amplitude is , where is the component amplitude. The value range of is . If the calculation result of this formula is not within the value range, it indicates that wear has occurred in that area.
We repeat these steps for all the triangular meshes of the gear until all the meshes are traversed and detected.
3.2 Significance of Each Step
Each step in the above – mentioned detection process plays a crucial role. The data pre – processing steps, such as transforming the original data into small – parameter data and denoising, can improve the quality of the data. The generation of envelope lines and the calculation of local – mean and envelope – estimation functions help to extract the features related to gear wear. The calculation of the PF – component amplitude is the key to determining whether wear has occurred. By carefully performing these steps, we can accurately detect the wear area of the gear.
4. Experimental Analysis
4.1 Experimental Setup
To verify the effectiveness of the Delaunay – triangulation – based automotive spiral bevel gear wear detection method, we conducted a comparative test experiment. We selected the spiral bevel gears of the Santana 2015 manual – transmission car as the research objects, with 20 identical gears used in the experiment. These gears have characteristics such as smooth operation, large transmission ratio, high power – transmission capacity, and compact structure, which meet the requirements of this experiment.
The material properties of the automotive spiral bevel gears are shown in Table 1. The experimental equipment used is shown in Figure 3.
[Insert Table 1: Material properties of automotive spiral bevel gears here]
[Insert Figure 3: Experimental equipment here]
4.2 Experimental Process
We planned the experimental process to obtain more accurate results:
- Keep the gear – wear experimental equipment running continuously for 200 hours. During this period, we collected gear – wear data at 5 – minute intervals. In one hour, 12 sets of wear data can be collected. This long – term and high – frequency data collection can comprehensively reflect the wear process of the gears.
- Set the gear – wear depth, gear – wear rate, and wear – area – detection accuracy as the test indicators for this experiment. These indicators can effectively evaluate the performance of the wear – detection method.
- Select the detection method based on reverse engineering proposed in [4] and the detection method based on improved Mask Scoring R – CNN proposed in [5] as the comparison methods for this experiment. By comparing with these two methods, we can clearly show the advantages of the Delaunay – triangulation – based method.
- Conduct experimental verification using the three different methods according to the set experimental indicators.
4.3 Experimental Results and Analysis
4.3.1 Gear Wear Depth
Gear wear mainly occurs between the driving gear and the driven gear. The wear depth of the gear changes with the increase of running time. The wear – depth results of the driving gear and the driven gear are shown in Figure 4. In Figure 4, a wear depth of 0 represents the initial surface state, and the areas with higher gear – wear degrees are the tooth tip and the tooth root.
[Insert Figure 4: Gear wear depth results here]
By comparing the wear – depth detection results of the three methods with the actual results, we found that the wear – depth results of the proposed method are basically consistent with the actual measurements. In contrast, the wear – depth values obtained by the methods in [4] and [5] differ significantly from the actual measurement results. This indicates that the proposed method can effectively analyze the wear depth of the gear.
4.3.2 Gear Wear Rate
The gear – wear rate, which refers to the amount of gear – material wear per unit time, can effectively reflect the wear situation of the gear. We also compared the wear – rate detection results of the three methods with the actual wear – rate calculation results. The wear – rate results are shown in Figure 5.
[Insert Figure 5: Gear wear rate results here]
From the comparison results of the gear – wear rate in Figure 5, we can see that during the study of 20 gears, the wear rate obtained by the proposed method is always consistent with the actual wear rate, with a maximum error of no more than 0.01μm. However, the wear rates obtained by the methods in [4] and [5] differ greatly from the actual wear rate. This shows that the proposed method can obtain more accurate gear – wear – rate results and improve the detection effect of gear wear.
4.3.3 Gear Wear Area Detection Accuracy
To further verify the gear – wear – detection performance of the proposed method, we used the gear – wear – area – detection accuracy as an indicator and compared the proposed method with the methods in [4] and [5]. In each experiment, we detected the wear – area of 20 gears. A total of 10 experiments were conducted, and the average detection results of the wear – area in each experiment are shown in Table 2.
[Insert Table 2: Gear wear area detection accuracy here]
From the comparison results of the gear – wear – area – detection accuracy in Table 2, we can see that compared with the methods in [4] and [5], the wear – area – detection accuracy of the proposed method can reach up to 98.7%, while the maximum detection – accuracy values of the two comparison methods are only 79.4% and 84.3% respectively. This indicates that the proposed method can accurately detect the area of the gear – wear region.
5. Conclusion
The detection of automotive spiral bevel gear wear is of great significance for ensuring driving safety and the stable operation of vehicles. The Delaunay – triangulation – based automotive spiral bevel gear wear detection method proposed in this article has several advantages. By performing Delaunay triangulation on the adjacent scan lines of the tooth surface, we can complete the non – characteristic discrete data blocking of the tooth surface. Based on the blocking results, we use the Hermite interpolation algorithm to calculate the PF – component amplitude and traverse all the triangular meshes to complete the wear – area detection.
Experimental data shows that the wear – depth and wear – rate results of the proposed method are basically consistent with the actual values, and it can accurately detect the wear – area with a maximum detection accuracy of 98.7%. This method effectively solves the problems existing in traditional methods and provides a new and reliable solution for automotive spiral bevel gear wear – detection technology. In the future, with the continuous development of technology, this method can be further optimized and integrated with other advanced techniques to achieve more accurate and intelligent gear – wear detection.
