Under real-world operating conditions, vibration signals collected from bevel gearboxes are inevitably contaminated with noise and outliers. Simultaneously, acquiring sufficient fault samples for these critical components is often challenging, leading to class-imbalanced datasets. This paper presents a comprehensive methodology for the intelligent fault diagnosis of bevel gearboxes, addressing both robustness against signal contamination and efficacy in handling imbalanced data.
The core of our methodology is a novel Robustness Imbalanced Convex Hull-based Classification (RICHC) model. This model enhances the traditional convex hull framework by integrating two key strategies: a sample confidence weighting mechanism for robustness and an adaptive hull scaling strategy for imbalance correction.
1. Foundation: Convex Hull Classification
A convex hull provides a compact geometric representation of a dataset. For a set of samples $\Theta = \{\mathbf{z}_i\}_{i=1}^n$ where $\mathbf{z}_i \in \Re^d$, the convex hull $C(\Theta)$ is defined as the set of all convex combinations of the points:
$$
C(\Theta) = \left\{ \sum_{i=1}^{n} \beta_i \mathbf{z}_i \ \middle|\ \sum_{i=1}^{n} \beta_i = 1, \ 0 \leq \beta_i \leq 1 \right\}
$$
For a binary classification problem with positive class samples $\Theta^+ = \{\mathbf{z}_i^+\}_{i=1}^{n^+}$ and negative class samples $\Theta^- = \{\mathbf{z}_j^-\}_{j=1}^{n^-}$, the classification hyperplane $\mathbf{w} \cdot \mathbf{z} + b = 0$ is determined by finding the closest points between the two convex hulls $C(\Theta^+)$ and $C(\Theta^-)$.
2. The Proposed RICHC Model
The standard convex hull is sensitive to outliers, which can distort its boundary and degrade classifier performance. Furthermore, its decision boundary tends to be biased towards the minority class in imbalanced scenarios.
2.1 Robustness via Confidence-Weighted Convex Hulls
To mitigate the influence of noisy samples and outliers, we assign a confidence value $\tau_i$ to each sample based on its distance from the class centroid. Samples closer to the center of the distribution are deemed more reliable. The confidence function for a sample $\mathbf{z}_i$ is defined as:
$$
\tau(\mathbf{z}_i) = \frac{1}{1 + \exp(\gamma \|\mathbf{z}_i – \mathbf{cent}\|^2)} \in (0,1]
$$
where $\mathbf{cent} = \frac{1}{n}\sum_{i=1}^n \mathbf{z}_i$ is the centroid of the class and $\gamma$ is a confidence factor. The confidence-weighted convex hull for a class is then:
$$
C_{\tau}(\Theta) = \left\{ \sum_{i=1}^{n} \tau_i \beta_i \mathbf{z}_i \ \middle|\ \sum_{i=1}^{n} \beta_i = 1, \ 0 \leq \beta_i \leq 1 \right\}
$$
This formulation reduces the contribution of low-confidence (potentially noisy) points in shaping the hull boundary, leading to a more compact and accurate model of the true class distribution, which is crucial for reliable bevel gear fault diagnosis.
2.2 Handling Imbalance via Adaptive Scaling
To address class imbalance, we introduce an adaptive scaling constraint on the combination coefficients $\beta_i$. The upper bound for $\beta_i$ is modified from 1 to $(1 + \mu \rho)$ for the majority class and $(1 – \mu \rho)$ for the minority class, where $\mu$ is a scaling coefficient. The dynamic imbalance factor $\rho$ is defined as:
$$
\rho =
\begin{cases}
n^+ / n^- & \text{if } n^+ > n^- \quad \text{(Class + is majority)} \\
-n^- / n^+ & \text{if } n^+ < n^- \quad \text{(Class – is majority)}
\end{cases}
$$
where $n^+$ and $n^-$ are the number of positive and negative samples, respectively. This strategy dynamically shrinks the convex hull of the majority class and/or expands that of the minority class, effectively shifting the optimal separating hyperplane to a more unbiased position. This is vital for diagnosing rare faults in a bevel gearbox where normal condition samples vastly outnumber fault samples.
2.3 Final Model Formulation
Integrating both mechanisms, the objective of the RICHC model for finding the closest points between the two modified convex hulls is formulated as the following quadratic programming problem:
$$
\min_{\boldsymbol{\beta}} \left\| \sum_{i=1}^{n^+} \tau_i^+ \beta_i^+ \mathbf{z}_i^+ – \sum_{j=1}^{n^-} \tau_j^- \beta_j^- \mathbf{z}_j^- \right\|^2
$$
subject to:
$$
\begin{aligned}
&\sum_{i=1}^{n^+} \beta_i^+ = 1, \quad 0 \leq \beta_i^+ \leq 1 + \mu \rho, \\
&\sum_{j=1}^{n^-} \beta_j^- = 1, \quad 0 \leq \beta_j^- \leq 1 – \mu \rho.
\end{aligned}
$$
Once solved, the optimal coefficients $\boldsymbol{\beta}^*$ define the support vectors. The normal vector $\mathbf{w}^*$ and bias $b^*$ of the separating hyperplane are derived from these vectors. For final decision-making, the distance output is mapped to a posterior probability using a sigmoid function, making the classifier more probabilistic and robust than a hard-decision rule.
For nonlinear problems inherent in complex vibration signals from a bevel gearbox, a kernel trick (e.g., Gaussian kernel $K(\mathbf{z}_i, \mathbf{z}_j) = \exp(-\|\mathbf{z}_i – \mathbf{z}_j\|^2 / (2\varepsilon^2))$) is employed to map data into a higher-dimensional feature space.
3. Fault Diagnosis Framework for Bevel Gearboxes
The application of the RICHC model for bevel gearbox fault diagnosis follows a systematic pipeline:
- Signal Acquisition & State Definition: Vibration signals are collected from the bevel gearbox under various health states (e.g., normal, gear chipped tooth, gear crack, bearing inner/outer race fault).
- Feature Extraction: To characterize the state of the bevel gearbox, multiple time-domain and frequency-domain statistical features are extracted from each signal segment. A standard set includes 22 features such as mean, variance, kurtosis, skewness, and spectral moments.
- Feature Selection: The Fisher Score (FS) is used to rank features by their discriminative power. Features with high inter-class variance and low intra-class variance are selected to reduce dimensionality and improve model efficiency. The Fisher Score for the $m$-th feature is calculated as:
$$
F(m) = \frac{\sum_{k=1}^{c} n_k (v_k^m – v^m)^2}{\sum_{k=1}^{c} n_k (\sigma_k^m)^2}
$$
where $c$ is the number of classes, $n_k$ is the number of samples in class $k$, $v_k^m$ and $\sigma_k^m$ are the mean and standard deviation of the $m$-th feature in class $k$, and $v^m$ is the overall mean of the $m$-th feature. - Model Training & Testing: The selected features from the training data are used to build the RICHC model with optimal hyperparameters ($\gamma, \mu, \varepsilon$) determined via cross-validation. The trained model is then evaluated on the independent test set to diagnose the bevel gearbox state.
4. Experimental Validation and Analysis
The proposed method was validated using a bevel gearbox dataset with 7 health states. The optimal sensitive features selected by the Fisher Score are listed below:
| Index | Feature Description | Index | Feature Description |
|---|---|---|---|
| 1 | Mean | 14 | Spectral Variance |
| 2 | Square Mean Root | 15 | Spectral Skewness |
| 3 | Root Mean Square | 19 | Frequency Center |
| 4 | Absolute Mean | 20 | RMS Frequency |
| 7 | Kurtosis Index | 22 | Spectral Kurtosis |
The RICHC model was compared against several state-of-the-art methods, including SVM, ELM, MMCCH, EDHD, and MECHA. The optimal parameters for all models were tuned via cross-validation.
| Model | Optimal Parameters |
|---|---|
| SVM | Kernel parameter=0.5, Penalty coefficient=1 |
| ELM | Activation=’tribas’, Hidden neurons=1000 |
| MMCCH | Flexibility=0.85, Shrinkage=0.90, $\varepsilon$=1 |
| EDHD | Extension=0.8, Displacement=0.95, $\varepsilon$=1 |
| MECHA | $\varepsilon$=0.95, Vertex proportion=0.85 |
| RICHC | $\gamma$=0.2, $\mu$=0.0625, $\varepsilon$=1 |
4.1 Overall Diagnostic Performance
Over 10 independent trials, RICHC achieved the highest and most stable average diagnostic accuracy.
| Model | Average Accuracy (%) | Standard Deviation (%) | Avg. Training Time (s) |
|---|---|---|---|
| SVM | 96.70 | 1.01 | 0.7105 |
| ELM | 97.10 | 0.49 | 0.1563 |
| MMCCH | 97.85 | 0.72 | 0.9265 |
| EDHD | 97.74 | 0.55 | 2.2857 |
| MECHA | 97.47 | 0.81 | 3.1740 |
| RICHC | 99.20 | 0.41 | 0.8816 |
4.2 Robustness Against Noise and Outliers
To test robustness, Gaussian white noise with different Signal-to-Noise Ratios (SNR) was added to the vibration signals. RICHC maintained superior performance, especially under strong noise.
| Model | Accuracy at Different Noise Levels (dB, %) | ||||
|---|---|---|---|---|---|
| 0 dB | 2 dB | 4 dB | 6 dB | 8 dB | |
| SVM | 86.14 | 91.15 | 91.35 | 92.56 | 94.69 |
| ELM | 87.53 | 88.03 | 89.24 | 92.21 | 93.60 |
| MMCCH | 88.96 | 91.14 | 90.95 | 93.11 | 95.00 |
| EDHD | 86.44 | 89.63 | 90.23 | 92.33 | 92.54 |
| MECHA | 89.11 | 90.25 | 91.75 | 92.47 | 95.11 |
| RICHC | 92.53 | 94.65 | 94.90 | 96.90 | 97.80 |
Furthermore, when artificial outliers were introduced into the training set, RICHC demonstrated the strongest resistance to performance degradation, proving the effectiveness of the confidence weighting mechanism for maintaining reliable bevel gear fault diagnosis.
4.3 Performance on Imbalanced Data
We constructed datasets with varying imbalance ratios (k = majority/minority samples) to simulate realistic scenarios where fault samples for the bevel gearbox are scarce. Performance was measured using Accuracy, Geometric Mean (G-mean), and F-value. RICHC consistently outperformed other models across all imbalance levels.
| Imbalance Set (k) | SVM (%) | ELM (%) | MMCCH (%) | EDHD (%) | MECHA (%) | RICHC (%) |
|---|---|---|---|---|---|---|
| A (k=10) | 92.20 | 91.50 | 92.70 | 93.70 | 92.50 | 96.80 |
| B (k=5) | 91.76 | 92.92 | 92.23 | 93.76 | 92.59 | 96.79 |
| C (k=3.33) | 91.77 | 93.70 | 94.53 | 96.41 | 93.07 | 98.90 |
| D (k=2.5) | 98.00 | 97.60 | 98.20 | 98.01 | 88.35 | 99.27 |
| E (k=2.0) | 88.90 | 98.28 | 99.01 | 98.59 | 92.98 | 99.30 |
The adaptive scaling strategy successfully mitigated the bias toward the majority class, enabling RICHC to maintain high sensitivity to minority-class faults in the bevel gearbox.
5. Conclusion
This work presents a robust and effective framework for the intelligent fault diagnosis of bevel gearboxes. The proposed RICHC model innovatively integrates a confidence-based weighting scheme and an adaptive scaling strategy into the convex hull classification framework. The confidence function effectively suppresses the influence of noise and outliers, which are prevalent in real-world vibration signals from bevel gear drives. Concurrently, the dynamic scaling mechanism adeptly handles the class imbalance problem commonly encountered in fault diagnosis, where normal operation data is abundant but fault samples are limited.
Comprehensive experimental results on a bevel gearbox dataset demonstrate that RICHC achieves superior diagnostic accuracy, stability, and robustness compared to several state-of-the-art methods. It exhibits remarkable resistance to signal contamination and maintains excellent performance even under high imbalance ratios. This makes the RICHC-based methodology a highly promising and practical tool for ensuring the reliability and safety of mechanical systems relying on critical bevel gear components.