Wear Detection of Automotive Spiral Bevel Gears Based on Delaunay Triangulation

his paper presents a novel approach for wear detection in automotive spiral bevel gears using Delaunay triangulation. The coupled vertical vibration of the rear axle spiral bevel gear in automobiles often results in wear that is difficult to accurately detect. To address this challenge, we propose a method that involves measuring discrete data on the gear surface using a scanning method, followed by Delaunay triangulation on adjacent scan lines to segment the non-characteristic discrete data. This segmented data is then analyzed using the Local Mean Decomposition (LMD) algorithm based on Hermite interpolation to calculate the amplitude of the Product Function (PF) components of the triangular mesh, thus detecting wear faults in the gear. Experimental results demonstrate that the proposed method yields wear depth and wear rate results that are consistent with measured values, and achieves a maximum detection accuracy of 98.7% for wear area detection. Therefore, the method is effective in accurately detecting gear wear.

Introduction

With the increasing popularity of automobiles, gear wear has become a crucial issue affecting the safety and performance of vehicles. In particular, the spiral bevel gears in the rear axle are prone to wear due to coupled vertical vibrations during vehicle operation. Timely and accurate detection of gear wear is essential to ensure safe driving conditions. However, conventional gear wear detection methods often suffer from limitations such as noise interference and insufficient detection accuracy.

This paper proposes a Delaunay triangulation-based method for detecting wear in automotive spiral bevel gears. By segmenting non-characteristic discrete data on the gear surface and analyzing these segments using Hermite interpolation, the method provides a comprehensive and accurate wear detection solution.

Literature Review

Previous studies on gear wear detection have explored various approaches, each with its strengths and limitations.

  • Wavelet Packet Energy and Bispectral Analysis: One method utilizes wavelet packet transform to decompose gear wear signals into multiple frequency bands and employs modulation signal bispectral analysis to extract tooth surface wear features [1]. However, this approach is susceptible to noise interference, resulting in reduced detection accuracy.
  • Reverse Engineering Technique: Another study proposed a reverse engineering-based approach that constructs a conversion matrix for the worn gear area and uses error compensation to fit a NURBS (Non-Uniform Rational B-Splines) surface for wear detection [2]. However, unworn areas may appear in the fitted wear surface, affecting detection accuracy.
  • Mask Scoring R-CNN: A more recent method applies an improved Mask Scoring R-CNN network, which combines residual networks and feature pyramid networks to extract wear features and detect wear areas through normalized local features [3]. However, noise in the sample data can significantly impact detection accuracy.

Proposed Method

To overcome the limitations of existing methods, we introduce a Delaunay triangulation-based approach for detecting wear in automotive spiral bevel gears. This method involves three main steps: data acquisition, Delaunay triangulation for data segmentation, and Hermite interpolation-based wear detection.

Step 1: Data Acquisition

The first step involves scanning the gear surface to obtain discrete data points. These points are then used to construct a NURBS surface model of the gear, as shown in Equation 1:

A(u,v)=i=0∑qj=0∑mNi,p​(u)Nj,q​(v)ωijPij

where Ni,p​(u) and Nj,q​(v) are the B-spline basis functions, ωij​ are the weights, and Pij​ are the control points.

Step 2: Delaunay Triangulation for Data Segmentation

Delaunay triangulation is applied to the discrete data points to segment the gear surface into non-characteristic discrete data blocks. This process ensures that the exterior circle of any triangle formed in the triangulation does not contain any other points, ensuring a consistent and accurate description of the gear surface.

adjacent scan lines are connected to form initial triangles, which are then refined according to the Delaunay criterion to optimize the triangulation. This results in a comprehensive segmentation of the gear surface.

Hermite Interpolation-Based Wear Detection

The segmented data blocks are analyzed using the Local Mean Decomposition (LMD) algorithm based on Hermite interpolation. This involves several steps:

  1. Noise Reduction: The raw discrete data is first denoised using a cascaded bistable stochastic resonance (CBSR) system.
  2. Envelope Extraction: Envelopes are generated around the extrema of the denoised data using Hermite interpolation.
  3. Calculation of PF Component Amplitudes: The product function (PF) component amplitudes are calculated for each triangular mesh using Equation 5:

f0​(t)=2π1​arccos(−2a22​a1​a3​​)a12​+a32​−4a22​(a1​a3​)2​​

where a1​, a2​, and a3​ are the coefficients of the Hermite polynomial approximating the local mean function.

  1. Wear Detection: The PF component amplitudes are compared to a threshold to identify wear regions.

Experimental Analysis

To validate the proposed method, experiments were conducted using spiral bevel gears from a 2015 Santana manual transmission. Twenty identical gears were selected for the study. The material properties of the gears are summarized in Table 1.

Table 1: Material Properties of Automotive Spiral Bevel Gears

PropertyValue
Material45Cr
Density (kg/m³)7,850
Young’s Modulus (GPa)206
Poisson’s Ratio0.3
The experimental process involved the following steps:
Data Collection: The gears were subjected to continuous operation for 200 hours, with data collected every 5 minutes.
Detection Methods: The proposed Delaunay triangulation method was compared with two existing methods: the reverse engineering technique [2] and the improved Mask Scoring R-CNN method [3].
Evaluation Metrics: Wear depth, wear rate, and wear area detection accuracy were used as evaluation metrics.
Results and Discussion
Wear Depth Detection:
As shown in Figure 4, the proposed method provides wear depth results that are consistent with actual measurements, outperforming the other two methods.

Wear Rate Detection:

The wear rate detected by the proposed method closely matches the actual wear rate, with a maximum error of less than 0.01 μm

Wear Area Detection Accuracy:

Table 2 summarizes the wear area detection accuracy results from ten experiments. The proposed method achieves a maximum accuracy of 98.7%, significantly higher than the other two methods.

Table 2: Wear Area Detection Accuracy

ExperimentProposed MethodMethod [2]Method [3]
196.9%77.9%83.9%
298.3%76.3%79.1%
1098.7%71.9%79.4%

Conclusion

This paper proposes a novel Delaunay triangulation-based method for detecting wear in automotive spiral bevel gears. By segmenting non-characteristic discrete data on the gear surface and analyzing these segments using Hermite interpolation, the method achieves high detection accuracy. Experimental results demonstrate that the proposed method outperforms existing techniques in terms of wear depth, wear rate, and wear area detection accuracy, achieving a maximum accuracy of 98.7%. This method provides a promising solution for ensuring safe and reliable gear operation in automobiles.

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