Spiral bevel gears are critical components in automotive rear axle systems, where coupled vertical vibrations often lead to uneven wear patterns. This paper presents a novel detection method combining Delaunay triangulation with Hermite interpolation-based Local Mean Decomposition (LMD) to address wear quantification challenges. The proposed approach achieves 98.7% detection accuracy through advanced surface modeling and signal processing techniques.
1. Surface Modeling and Data Segmentation
The NURBS surface representation for spiral bevel gears is formulated as:
$$
A(u,v) = \frac{\sum_{i=0}^m \sum_{j=0}^n N_{i,p}(u)N_{j,q}(v)w_{ij}P_{ij}}{\sum_{i=0}^m \sum_{j=0}^n N_{i,p}(u)N_{j,q}(v)w_{ij}}
$$
where $N_{i,p}$ and $N_{j,q}$ represent B-spline basis functions, $w_{ij}$ are weighting factors, and $P_{ij}$ denote control points.
Delaunay triangulation processes adjacent scan lines (αl, αl+1) through quadrilateral decomposition and minimum angle maximization. The triangular quality metric is defined as:
$$
Q_{\Delta} = \frac{4\sqrt{3}A}{a^2+b^2+c^2}
$$
where A is triangle area and a,b,c are side lengths. Triangles with QΔ > 0.8 are selected for surface reconstruction.
| Component | Material | Density (kg/m³) | Young’s Modulus (GPa) | Poisson’s Ratio |
|---|---|---|---|---|
| Gear | 45Cr | 7850 | 206 | 0.3 |
2. Wear Feature Extraction
The Hermite interpolation-enhanced LMD algorithm processes discrete surface data through:
- Noise reduction using Cascaded Bistable Stochastic Resonance (CBSR):
$$
\frac{dx}{dt} = ax – bx^3 + S(t) + \sqrt{D}\xi(t)
$$
where $S(t)$ is measured signal and $D$ noise intensity. - Local mean calculation:
$$
m(t) = \frac{e_{upper}(t) + e_{lower}(t)}{2}
$$ - Product Function (PF) decomposition:
$$
PF_i(t) = a_i(t)\cos\left(2\pi \int f_i(t)dt\right)
$$
| Method | Depth Error (%) | Area Accuracy (%) | Rate Error (μm/h) |
|---|---|---|---|
| Proposed | 1.3 | 98.7 | 0.008 |
| Reverse Engineering | 7.2 | 79.4 | 0.152 |
| Mask R-CNN | 5.6 | 84.3 | 0.113 |
3. Experimental Validation
Testing on 20 spiral bevel gears under 200-hour continuous operation demonstrated:
- Maximum wear depth correlation: R² = 0.987
- Wear rate estimation error: < 0.01 μm/h
- False positive rate: 1.2%
The wear quantification model calculates equivalent stress distribution:
$$
\sigma_{eq} = \sqrt{\frac{1}{2}\left[(\sigma_1-\sigma_2)^2 + (\sigma_2-\sigma_3)^2 + (\sigma_3-\sigma_1)^2\right]}
$$
4. Conclusion
This Delaunay triangulation-based methodology effectively addresses spiral bevel gear wear detection challenges through:
- Precise surface modeling with adaptive mesh density (15-25 nodes/mm²)
- Robust feature extraction using modified LMD (97.4% noise rejection)
- Comprehensive wear parameter estimation (depth, area, rate)
The technique shows particular effectiveness for spiral bevel gears in heavy-duty vehicles, achieving sub-micron resolution while maintaining computational efficiency (processing time < 15s per gear). Future work will integrate real-time monitoring capabilities for automotive predictive maintenance systems.