Gear shafts are critical components in mechanical transmission systems across fields like wind power and rail transportation, enduring long-term alternating loads. Such operating conditions make them susceptible to tooth breakage faults due to material fatigue and stress concentration. These faults are often sudden, highly destructive, and can lead to equipment shutdowns or even safety incidents, resulting in significant economic losses. Therefore, real-time monitoring and early warning for gear shaft tooth breakage are of paramount importance for ensuring equipment safety and reducing maintenance costs.
Recent advancements in IoT and AI have spurred significant progress in automatic monitoring for gear shaft faults. Traditional methods, however, face considerable challenges under complex alternating load conditions. For instance, one common approach involves signal separation followed by the extraction of instantaneous angular speed (IAS) components using time-synchronous averaging for fault detection via entropy measures. The weak impact energy of incipient faults often leads to incomplete IAS extraction or insignificant entropy changes, compromising detection reliability. Another strategy combines oil debris image features with denoised vibration time-domain features, inputting them into Long Short-Term Memory (LSTM) networks optimized by algorithms like Particle Swarm Optimization (PSO). While promising, such models can struggle to adapt to the varying signal characteristics induced by different loading conditions. Techniques based on the Hilbert-Huang Transform (HHT) aim to extract fault features by analyzing Intrinsic Mode Functions (IMFs) and Hilbert spectra. However, they frequently fail to account for the full complexity of alternating loads, leading to inaccurate signal acquisition and poor feature localization. Similarly, methods employing Empirical Mode Decomposition (EMD) to select characteristic IMFs and subsequently using Hilbert Transform (HT) for feature vectors, fed into improved neural networks like dual Radial Basis Function (RBF), often provide an incomplete reflection of tooth breakage characteristics under varying loads, resulting in suboptimal detection accuracy.
The core limitations of existing methods lie in their inadequate anti-interference capability, poor adaptability to varying operational conditions, incomplete feature extraction, and inability to automatically mine complex relationships between features, leading to high false alarm rates and unreliable monitoring. To address these shortcomings, this paper proposes a novel automatic monitoring method for gear shaft tooth breakage under alternating loads.
The proposed methodology is implemented through several key stages. First, Fiber Bragg Grating (FBG) sensors are employed to acquire high-quality vibration signals from the gear shafts, effectively overcoming electromagnetic interference prevalent in industrial environments. Second, an adaptive signal reconstruction process is performed. A threshold-iteration loop is applied to the average spectrum of the acquired vibration signal to locate characteristic and estimated peak positions. Based on these peak intervals, a dictionary is constructed using Gaussian density functions, and the Matching Pursuit (MP) algorithm is utilized to reconstruct the signal, thereby effectively fitting the peak features crucial for fault characterization. Third, time-frequency domain analysis via Singular Spectrum Analysis (SSA) is conducted on the reconstructed signal to extract dominant feature parameters that comprehensively reflect the state of tooth breakage. Finally, the Gradient Boosting Decision Tree (GBDT) algorithm is applied for recursive feature classification. This ensemble learning method automatically mines the complex, non-linear relationships between the extracted features, enhancing the accuracy and reliability of the final fault diagnosis. The subsequent sections detail the implementation of this method and present experimental validation of its performance.
Implementation of the Automatic Monitoring Method Incorporating Gradient Boosting Decision Tree
Vibration Signal Acquisition for Gear Shafts Under Complex Alternating Load Conditions
Alternating loads, which vary periodically in magnitude and direction, induce complex vibrations in gear shafts, exhibiting different characteristics at different locations. Comprehensive monitoring requires high-precision measurement techniques capable of capturing this distributed response. Traditional piezoelectric accelerometers can be susceptible to electromagnetic interference (EMI) in industrial settings, leading to signal distortion. Fiber Bragg Grating (FBG) sensors offer a robust alternative. They provide high-resolution strain measurement, capturing minute strain variations caused by vibrations, and allow for quasi-distributed multipoint sensing, offering a reliable data foundation for vibration analysis. Crucially, FBG sensors use optical signal transmission, making them inherently immune to EMI, ensuring stable and accurate signal acquisition even in complex electromagnetic environments typical of heavy machinery.
Under dynamic loading, the FBG sensor operates based on the shift of its center wavelength. The fundamental equation for the Bragg wavelength is:
$$ \lambda_B = 2n_{eff}\varLambda $$
where \( \lambda_B \) is the central Bragg wavelength, \( n_{eff} \) is the effective refractive index of the fiber core, and \( \varLambda \) is the grating period. An external strain applied to the fiber causes a coupled change in both \( n_{eff} \) and \( \varLambda \), resulting in a shift \( \Delta\lambda_B \) of the reflected wavelength. This shift is linearly related to the applied strain \( \epsilon \) by:
$$ \Delta\lambda_B = \lambda_B (1 – p_e)\epsilon $$
where \( p_e \) is the photo-elastic coefficient of the optical fiber. Based on this principle, structural vibrations excited by the rotating gear shaft are transmitted to the FBG sensor mounted on the housing surface. The induced strain causes a proportional wavelength shift. By demodulating this wavelength shift in real-time, the vibration information of the gear shaft under alternating load is directly obtained, completing the signal acquisition task.
Reconstruction of Gear Shaft Vibration Signals
Under alternating loads, gear shaft vibration signals often suffer from strong noise interference and low signal-to-noise ratios (SNR), while signal characteristics vary significantly across different operational states, making feature extraction challenging. An adaptive approach is needed. First, a threshold-iteration method is used to alternately locate the local maxima and minima in the gear shaft vibration signal sequence \( x(k) \). For minima detection, a point is identified as a local minimum if the drop from the preceding maximum exceeds a preset threshold \( \Delta \):
$$ x(p) – \min(x(p), …, x(k)) > \Delta $$
where \( x(p) \) is the previous extremum point. Similarly, a point is a local maximum if the rise from the preceding minimum exceeds the threshold:
$$ \max(x(p), …, x(k)) – x(k+1) > \Delta $$
The detected local maxima correspond to the center coordinates \( Z_i \) of the Raman/characteristic peaks. The width \( y_{Hi} \) of the i-th characteristic peak is estimated as:
$$ y_{Hi} = 2 \times (Z_i – A_i) $$
where \( A_i \) is the coordinate of the adjacent local minimum to the left of the i-th peak. This provides the position and width estimation for the characteristic peaks.
Subsequently, to effectively fit these peak features—which are critical for reflecting the state of the gear shafts—a signal reconstruction process is undertaken using a Gaussian dictionary and the Matching Pursuit algorithm. The steps are as follows:
- Dictionary Formation: Based on the estimated peak position intervals, a dictionary \( E \) is formed using Gaussian density functions \( g_{\gamma}(t) \), where \( \gamma \) parameterizes the center and width. The characteristic peak signal \( y_{Hi} \) can be represented as a sparse combination in this dictionary:
$$ y_{Hi} = E\tau_0 + \alpha_0 $$
where \( \tau_0 \) is the sparse representation vector and \( \alpha_0 \) is the initial residual. - Atom Selection: In the \( t \)-th iteration, the atom \( d_i \) in dictionary \( E \) that has the maximum inner product with the current residual \( \alpha_{t-1} \) is selected:
$$ \zeta_i^t = \max_{i \in q} | \langle d_i, \alpha_{t-1} \rangle | $$
The coefficient for this atom is updated in the sparse vector. - Residual Update: The sparse approximation is updated, and the new residual is calculated:
$$ \alpha_t = \alpha_{t-1} – \zeta_i^t \times d_i $$ - Stopping Criterion: The iteration continues until the energy of the residual \( \alpha_t \) falls below a predefined threshold \( \epsilon \).
- Signal Reconstruction: Upon termination, the reconstructed characteristic peak signal \( \hat{y}_{Hi} \) is obtained by summing the selected atoms:
$$ \hat{y}_{Hi} = \sum_t \sum_j \zeta_i^t S_i $$
where \( S_i \) contains the signal contribution from the selected atoms.
The effectiveness of this reconstruction is validated by comparing key metrics before and after processing, as summarized in Table 1.
| Test Metric | Before Reconstruction | After Reconstruction |
|---|---|---|
| SNR (dB) | 6.443 | 10.378 |
| PSNR (dB) | 7.862 | 11.345 |
| RMSE | 0.785 | 0.012 |
Analysis of Table 1 confirms the method’s efficacy. The significant increase in SNR and PSNR, coupled with a drastic reduction in RMSE, demonstrates successful noise suppression and enhanced signal fidelity, providing a cleaner signal for subsequent fault feature extraction critical for monitoring gear shafts.
Automatic Monitoring of Gear Shaft Tooth Breakage Anomaly via Time-Frequency Analysis and GBDT
Following signal reconstruction, Singular Spectrum Analysis (SSA) is employed for time-frequency feature analysis to quantitatively characterize the parameters associated with tooth breakage in gear shafts under alternating stress. The one-dimensional reconstructed signal \( \hat{y}_{Hi} \) (time series \( x_n \)) is embedded into a multi-dimensional trajectory matrix \( \mathbf{X} \). After Singular Value Decomposition (SVD) of \( \mathbf{X} \), the trajectory matrix can be expressed as a sum of elementary matrices:
$$ \mathbf{X} = \mathbf{X}_1 + \mathbf{X}_2 + … + \mathbf{X}_L = \sum_{j=1}^{L} \sqrt{\lambda_j} U_j V_j^T $$
where \( \lambda_j \) are the eigenvalues, and \( U_j \), \( V_j \) are the corresponding left and right singular vectors. Grouping these elementary matrices \( \mathbf{X}_i \) into disjoint subsets \( \{R_1, R_2, …, R_n\} \) and summing within each group yields grouped matrices \( \mathbf{X}_{R_j} \). The matrix \( \mathbf{X} \) is thus decomposed as:
$$ \mathbf{X} = \mathbf{X}_{R_1} + \mathbf{X}_{R_2} + … + \mathbf{X}_{R_n} $$
Each grouped matrix \( \mathbf{X}_{R_j} \) is then transformed back into a time series \( g_k \) of length \( n \) via diagonal averaging:
$$ g_k = \frac{1}{n} \sum_{j} \mathbf{X}_{R_j}(k) $$
The final extracted dominant feature component \( \tilde{x}_n \), representing the gear shaft’s condition, is obtained by selectively reconstructing from these grouped components:
$$ \tilde{x}_n = \sum_{n} x_n \cdot g_k $$
While SSA extracts relevant feature parameters, the relationships between them can be complex and non-linear. Traditional analytical methods struggle to automatically mine these relationships, limiting monitoring accuracy. To overcome this, the proposed method employs a Gradient Boosting Decision Tree (GBDT) algorithm for recursive classification. GBDT is an ensemble learning technique that builds a strong predictive model by sequentially combining multiple weak learners (decision trees), each correcting the errors of its predecessor.
Given a training dataset \( \{(x_1, y_1), (x_2, y_2), …, (x_N, y_N)\} \) where \( x_i \) is the feature vector (containing extracted parameters like \( \tilde{x}_n \)) and \( y_i \) is the label (e.g., normal or faulty), GBDT operates as follows:
- Initialization: Start with an initial constant model, typically the mean of the labels:
$$ F_0(x) = \arg\min_{\gamma} \sum_{i=1}^{N} L(y_i, \gamma) $$
where \( L(y_i, \gamma) \) is a differentiable loss function (e.g., mean squared error for regression, deviance for classification). - Iterative Boosting (for m=1 to M):
- Compute the negative gradient (pseudo-residuals) for each instance:
$$ r_{im} = -\left[\frac{\partial L(y_i, F(x_i))}{\partial F(x_i)}\right]_{F(x)=F_{m-1}(x)} $$ - Fit a decision tree \( h_m(x) \) to the pseudo-residuals \( r_{im} \), creating terminal regions \( R_{jm} \).
- For each terminal region \( R_{jm} \), calculate the output value \( \gamma_{jm} \) that minimizes the loss:
$$ \gamma_{jm} = \arg\min_{\gamma} \sum_{x_i \in R_{jm}} L(y_i, F_{m-1}(x_i) + \gamma) $$ - Update the model:
$$ F_m(x) = F_{m-1}(x) + \nu \cdot \sum_{j=1}^{J} \gamma_{jm} I(x \in R_{jm}) $$
where \( \nu \) is the learning rate (shrinkage parameter) controlling the contribution of each tree, and \( I(\cdot) \) is the indicator function.
- Compute the negative gradient (pseudo-residuals) for each instance:
- Final Model: After \( M \) iterations, the final GBDT model for prediction is:
$$ F_M(x) = F_0(x) + \nu \sum_{m=1}^{M} \sum_{j=1}^{J} \gamma_{jm} I(x \in R_{jm}) $$
For the task of monitoring gear shafts, the extracted feature vector (including \( \tilde{x}_n \) and other time-frequency statistics) serves as the input \( x \). The model \( F_M(x) \) outputs a decision score. A threshold \( \Theta \) is applied to this score to obtain the final monitoring verdict \( P_{x,y} \):
$$ P_{x,y} = \begin{cases}
\text{Anomaly (Tooth Breakage)}, & \text{if } F_M(x) \geq \Theta \\
\text{Normal}, & \text{otherwise}
\end{cases} $$
This integrated approach, combining robust signal acquisition, adaptive reconstruction, comprehensive feature extraction via SSA, and powerful non-linear pattern recognition via GBDT, achieves automatic and accurate monitoring of tooth breakage anomalies in gear shafts under challenging alternating load conditions.
Experimental Validation
Experimental Platform and Parameter Settings
To validate the effectiveness of the proposed automatic monitoring method for gear shaft tooth breakage under alternating loads, a comprehensive experimental analysis was conducted. The test platform, as described, primarily consists of a drive motor, a gearbox, a load simulator, sensors, and a speed controller. The load simulator applies programmable alternating loads to simulate real-world high-cycle fatigue conditions on the gear shafts. Vibration signals during operation are acquired by sensors (including the FBG sensor) installed on the gearbox bearing housings.
The key parameters for the experiment are set as follows:
| Component | Parameter | Value/Specification |
|---|---|---|
| FBG Sensor | Center Wavelength | 1550 nm |
| Wavelength Sensitivity | 1.0 pm/με | |
| Interrogator Range & Resolution | 80 nm, < 2 pm | |
| Sampling Rate | 10 kHz | |
| Source Power | 5 mW | |
| Gear Shaft | Rotational Speed | 300 r/min |
| Load Torque | 1000 N·m | |
| Number of Teeth | 28 | |
| Algorithm | Iteration Count (MP) | 200 |
| Scale Parameter | 0.008 | |
| Number of Trees (GBDT) | 500 | |
| Max Tree Depth (GBDT) | 7 |
Experimental Procedure
- Baseline Data Collection: The system was operated under normal (healthy) conditions with alternating loads ranging from 5 kN to 25 kN, each for 1 hour, to collect baseline vibration signals.
- Fault Seeding: An early-stage tooth crack defect was simulated by introducing a 0.2 mm wide notch via wire cutting at the tooth root of a test gear shaft.
- Faulty Data Collection: For each load level, the system with the seeded fault was run for 24 hours, with 10 minutes of vibration data collected every hour (resulting in 1440 data samples per load condition).
- Comparative Analysis: The proposed method was compared against four established methods from the literature: the Piecewise Mean Difference (PMD) method, the PSO-LSTM method, the Self-Mixing Interference (SMI) based method, and the Radial Basis Function Neural Network (RBF-NN) method. Performance was evaluated using metrics such as monitoring outcome visualization, false alarm rate, F1-score, and accuracy across different operational scenarios.
Experimental Analysis and Results
The monitoring results under varying alternating loads are first visualized. The proposed method successfully tracks the changes in gear shaft vibration frequency and accurately triggers an anomaly alert when the signal amplitude increases significantly due to the simulated tooth breakage. In contrast, the outputs from the four comparison methods show poorer alignment with the actual fault transients. This qualitative observation suggests the proposed method yields more accurate monitoring results for gear shaft anomalies.
A quantitative analysis of the false alarm rate (FAR) under increasing alternating load is presented in Figure 1 and the corresponding data. As the load increases, the false alarm rate for all five methods rises due to the increased complexity of gear shaft operation and signal interference. However, the proposed method maintains the lowest and most stable false alarm rate across the load spectrum. Crucially, in the high-load region (above 15 kN), while the FAR of other methods increases sharply (e.g., PMD: 1.9% to 4.0%, PSO-LSTM: 2.0% to 4.5%, SMI: 2.2% to 5.3%, RBF-NN: 2.9% to 5.8%), the proposed method shows only a gradual linear increase from 1.1% to 2.9%, demonstrating superior load adaptability and reliability for monitoring industrial gear shafts.
To further evaluate performance under sustained high stress, a 12-hour endurance test was conducted under a constant 25 kN alternating load. The average F1-score, which balances precision and recall, was recorded at intervals. The results are summarized in Table 3.
| Running Time (hours) | Average F1-Score | ||||
|---|---|---|---|---|---|
| Proposed Method | PMD Method | PSO-LSTM Method | SMI Method | RBF-NN Method | |
| 2 | 0.96 | 0.85 | 0.84 | 0.82 | 0.81 |
| 4 | 0.94 | 0.83 | 0.82 | 0.79 | 0.80 |
| 6 | 0.92 | 0.80 | 0.78 | 0.76 | 0.77 |
| 8 | 0.91 | 0.78 | 0.74 | 0.72 | 0.73 |
| 10 | 0.90 | 0.75 | 0.70 | 0.68 | 0.69 |
| 12 | 0.89 | 0.72 | 0.67 | 0.64 | 0.66 |
Table 3 clearly shows that while all methods experience performance degradation over extended operation, the proposed method maintains the highest F1-score throughout (0.89 to 0.96), with the smallest relative decline (7.3%). This indicates strong robustness and an ability to effectively resist performance decay from prolonged operation under high alternating loads, a critical requirement for continuous monitoring of gear shafts.
Finally, to verify practical applicability, the test platform was used to simulate various real-world operational scenarios for gear shafts: light-medium-heavy load cycles (5-25 kN), random variable loads, and step-variable loads. Each scenario was tested for 24 hours. The monitoring accuracy for each method was recorded over five independent runs per scenario. The results are consolidated in Table 4.
| Test Scenario | Run # | Accuracy | ||||
|---|---|---|---|---|---|---|
| Proposed Method | PMD Method | PSO-LSTM Method | SMI Method | RBF-NN Method | ||
| Load Cycle | 1 | 0.94 | 0.92 | 0.88 | 0.85 | 0.84 |
| 2 | 0.94 | 0.91 | 0.87 | 0.83 | 0.82 | |
| 3 | 0.95 | 0.90 | 0.85 | 0.80 | 0.79 | |
| 4 | 0.96 | 0.92 | 0.88 | 0.84 | 0.83 | |
| 5 | 0.94 | 0.90 | 0.85 | 0.80 | 0.79 | |
| Random Load | 1 | 0.92 | 0.85 | 0.81 | 0.78 | 0.76 |
| 2 | 0.90 | 0.84 | 0.80 | 0.77 | 0.75 | |
| 3 | 0.91 | 0.81 | 0.77 | 0.74 | 0.72 | |
| 4 | 0.91 | 0.82 | 0.78 | 0.75 | 0.73 | |
| 5 | 0.92 | 0.81 | 0.77 | 0.74 | 0.72 | |
| Step Load | 1 | 0.91 | 0.86 | 0.84 | 0.81 | 0.80 |
| 2 | 0.92 | 0.87 | 0.85 | 0.82 | 0.81 | |
| 3 | 0.92 | 0.85 | 0.83 | 0.80 | 0.79 | |
| 4 | 0.91 | 0.83 | 0.81 | 0.78 | 0.77 | |
| 5 | 0.90 | 0.82 | 0.80 | 0.77 | 0.76 | |
The results in Table 4 demonstrate the consistent superiority of the proposed method. Across all three complex, realistic scenarios, it achieves the highest accuracy ratings (consistently 0.90 or above) with the least variability between runs. This performance highlights the method’s excellent adaptability and robustness to diverse, changing operational conditions, making it a reliable solution for the practical monitoring of gear shafts in the field.
Conclusion
To address the limitations of traditional and existing automated methods in monitoring gear shaft tooth breakage—such as susceptibility to environmental interference, poor feature fitting, and high false alarm rates—this paper proposed a novel automatic monitoring method designed for operation under alternating loads. The method integrates several key innovations: using FBG sensors for robust signal acquisition, applying adaptive signal reconstruction via a Gaussian dictionary and Matching Pursuit to accurately fit characteristic peak features, employing Singular Spectrum Analysis for comprehensive time-frequency feature extraction, and leveraging the Gradient Boosting Decision Tree algorithm to automatically mine complex feature relationships for precise classification.
Experimental studies validated the method’s effectiveness. It significantly improves the quality of reconstructed gear shaft vibration signals (e.g., raising SNR from 6.44 dB to 10.38 dB). It maintains a low and stable false alarm rate even under high loads (increasing only to 2.9% at 25 kN). It demonstrates robust long-term performance, retaining a high F1-score (≥0.89) over a 12-hour high-stress test. Most importantly, it achieves consistently high accuracy (≥0.90) across various simulated real-world operational scenarios, including load cycles, random loads, and step loads. These results confirm that the proposed method substantially enhances the accuracy and reliability of automatic tooth breakage anomaly monitoring for gear shafts, which is of great significance for ensuring their safe and continuous operation in demanding industrial applications.