I focus on the fault diagnosis of hydraulic mechanical drive pinion gear sets because the pinion gear is one of the most critical and vulnerable components in hydraulic transmission systems. In my experience, more than half of the faults in hydraulic machinery originate from the gear set inside the gearbox, and the pinion gear often suffers from local defects such as pitting, wear, and tooth breakage. When the hydraulic mechanical drive operates in a complex environment, the pinion gear remains enclosed and lubricated, so early abnormal conditions are difficult to observe directly. This hidden degradation can cause delayed production, low efficiency, and even catastrophic accidents. Therefore, I developed a diagnosis framework that combines low-rank vibration source separation, stacked denoising autoencoder feature extraction, and support vector machine classification. The proposed method is designed to improve the accuracy and reliability of pinion gear fault diagnosis.
Existing approaches for hydraulic mechanical drive pinion gear sets have several limitations. Some methods transform vibration signals into two-dimensional images and then apply convolutional neural networks, but the signal conversion may lose important transient information. Other methods use local feature scale decomposition and fuzzy inference, but the decomposition is often incomplete and the diagnostic error remains large. Deep belief networks can simplify the diagnosis process, but weight adjustment errors can reduce feasibility. Attention-based capsule networks and short-time Fourier transform convolutional networks have also been explored, yet they may suffer from sample imbalance and one-to-one sample-class relationships. To address these issues, I propose a support vector machine based diagnosis method that first separates the vibration source signals of the pinion gear set using a low-rank algorithm, then extracts fault features with a stacked denoising autoencoder, and finally classifies the pinion gear condition using an SVM trained on the extracted features. The method is validated on a hydraulic mechanical drive test bench, and the experimental results show that the classification error rate remains below 3.5%.
Signal Acquisition for the Pinion Gear Set
I begin by collecting vibration and torque signals from the hydraulic mechanical drive pinion gear set. Vibration sensors and torque sensors are mounted at the bearing supports and the connection between the drive motor and the hydraulic mechanism. The vibration sensor captures the radial and axial vibrations of the pinion gear, while the torque sensor records the torque variation caused by tooth meshing. I define the original signal dataset as follows:
$$ y_i \in \{y_1, y_2, y_3\}, \quad i = 1, 2, \dots, 3l $$
$$ D = \{D_1 \cup D_2\} $$
$$ D_1 = \{(x_i, y_i)\}, \quad i = 1, 2, \dots, 3l $$
$$ D_2 = \{(x’_i, y_i)\}, \quad i = 1, 2, \dots, 3l $$
Here, \(y_i\) denotes the fault type, where \(y_1\) represents a crack fault, \(y_2\) represents a missing tooth fault, and \(y_3\) represents a broken tooth fault. The variable \(x_i\) is the vibration sensor output, and \(x’_i\) is the torque sensor output. The dataset \(D\) contains the original signals from both sensors. For the pinion gear, the meshing frequency and its harmonics are strongly related to the rotation speed and the number of teeth. Therefore, I also record the sampling frequency, the number of sampling points, and the sensor positions so that the subsequent source separation can preserve the pinion gear fault characteristics.
In my experiments, I use a sampling frequency of 5000 Hz and collect 8500 sampling points for each measurement. The vibration sensors are placed in three directions: horizontal radial, vertical radial, and axial. This arrangement ensures that the pinion gear vibration in different planes can be captured. I also install a temperature sensor and a sound sensor to monitor abnormal conditions, although the main diagnostic signal remains the vibration signal. The raw vibration signal of the pinion gear is often contaminated by noise from bearings, shafts, and hydraulic pulsations. Therefore, I apply a low-rank constrained source separation method before feature extraction.
Low-Rank Constrained Source Separation for Pinion Gear Vibration
The vibration signal of a hydraulic mechanical drive pinion gear set is a mixture of multiple sources, including the pinion gear meshing signal, bearing signals, shaft signals, and hydraulic pump pulsations. When the signal matrix has similar columns or rows, its rank is sparse. I use a low-rank constraint to reduce the complexity of the unknown signal and to separate the pinion gear fault source from the mixed observation. Under normal operation, the gearbox vibration signal often exhibits low-rank characteristics, which can be used to constrain the separation. I first verify whether the noise in the raw pinion gear vibration signal follows a Gaussian distribution. Then I construct the following low-rank constrained source separation model:
$$ \arg\min_{L,N} \operatorname{rank}(L) + \beta \cdot \operatorname{kur}(N) \quad \text{s.t.} \quad D = L + N $$
In this model, \(D\) is the observation matrix, \(N\) is the noise term, \(L\) is the low-rank matrix, \(\operatorname{rank}(\cdot)\) is the rank function, \(\beta\) is a compromise factor, and \(\operatorname{kur}(N)\) is the kurtosis function. Because the rank function is a complex polynomial, I use convex relaxation to transform the problem into a tractable form:
$$ \arg\min_{L,N} \|L\|_* + \beta \cdot \operatorname{kur}(N) \quad \text{s.t.} \quad D = L + N $$
Here, \(\|L\|_*\) is the nuclear norm of \(L\). To separate multiple vibration sources of the pinion gear set, I extend the model to a multi-low-rank constrained separation model:
$$ \arg\min_{L,N} \sum_{i=1}^{n} \lambda_i \|L_i\|_* + \beta \cdot \operatorname{kur}(N) \quad \text{s.t.} \quad x = \sum_{i=1}^{n} l_i + N $$
In this expression, \(l_i\) is an ordered array, \(n\) is the number of sources to be separated, and \(\lambda_i\) is the low-rank matrix weight. I set a vibration signal of length \(m\), denoted as \(x(t)\), and obtain it through low-rank matrices \(L_i\). A block operator \(w\) transforms \(x(t)\) into a two-dimensional matrix \(L_i\). The pinion gear fault signal constraint is designed according to the characteristic frequencies of the pinion gear under different rotational speeds. The constraint condition is:
$$ \frac{p_i \lambda_i f_s}{f_{\text{source}i}} > o_i $$
where \(o_i\) is the array length, \(p_i\) is the array number, \(f_s\) is the sampling frequency, and \(f_{\text{source}i}\) is the characteristic frequency of the vibration source. By applying this constraint, I can classify different pinion gear fault types such as tooth surface wear, tooth side clearance variation, and flutter. Different pinion gear faults exhibit different vibration spectra and characteristic frequencies, so the classification becomes more precise.
After the block operator is applied, I solve the convex optimization problem using the augmented Lagrangian method. The enhanced Lagrangian function is:
$$ L_a(l_i, N, Y, \mu_k) = \sum_{i=1}^{n} \lambda_i \|L_i\|_* + \beta \operatorname{kur}(N) + \langle Y, x – \sum_{i=1}^{k} l_i – N \rangle $$
where \(L_a\) is the augmented Lagrangian function, \(Y\) is the Lagrangian multiplier, \(\mu_k\) is the iteration step size, and \(\langle \cdot, \cdot \rangle\) denotes the inner product. I use the alternating direction method of multipliers to optimize each low-rank matrix nuclear norm and to minimize the noise kurtosis. The iterative process is:
$$ l^*_{i,k+1} = \arg_{l_i} \frac{\lambda_i}{\mu_k} \|L_i\|_* + \frac{1}{2} \left\| x – \sum_{j=1}^{i-1} l_{j,k+1} – l_i – \sum_{j=i+1}^{n} l_{j,k} – N_k \right\|_F^2 $$
I then apply singular value decomposition and soft thresholding to solve the subproblem:
$$ [U_i, S_i, V_i] = \operatorname{SVD}(w(T_{i,k}, o_i, p_i)) $$
$$ l_{i,k+1} = w\left( U_i S_{\lambda_i / \mu_k}[S_i] V_i^T \right)^{-1} $$
where \(\operatorname{SVD}\) is the singular value decomposition operator, \(S_{\lambda_i / \mu_k}[S_i]\) is the singular value soft thresholding operator, \(w\) is the inverse block operator, \(U_i\) is the left singular vector, and \(V_i\) is the right singular vector. After the iteration converges, I obtain the separated vibration source signals of the pinion gear set. The low-rank separation significantly improves the signal-to-noise ratio of the pinion gear fault components and enhances the subsequent classification performance.
Feature Extraction Using Stacked Denoising Autoencoder
After the low-rank source separation, I extract fault features from the separated pinion gear vibration signals using a stacked denoising autoencoder model. The SDAE is trained to reconstruct clean inputs from corrupted versions, which makes it robust to residual noise. I introduce two optimization criteria into the SDAE to extract more effective pinion gear fault features. The first is the minimum sparsity criterion. The sparsity of the pinion gear fault features can represent the latent characteristics of the gear set and reveal the essential fault features. The sparsity objective is defined as:
$$ f_1(S) = \sum_{j=1}^{D} KL(\rho \| \hat{\rho}_j) $$
where \(\rho\) is the sparsity parameter, \(KL(\rho \| \hat{\rho}_j)\) is the Kullback-Leibler divergence, \(\hat{\rho}_j\) is the average activation of the \(j\)-th hidden unit, \(D\) is the number of hidden layer nodes, and \(f_1(S)\) is the sparsity measure. When \(f_1(S)\) is smaller, the extracted pinion gear fault features are sparser. I normalize \(f_1(S)\) so that \(f_1(S) \in [0, 1]\).
The second criterion is the minimum classification error rate. To reduce the pinion gear fault diagnosis error, I define the error objective as:
$$ f_2(S) = 1 – \frac{s}{M} $$
where \(s\) is the number of correctly diagnosed samples and \(M\) is the sample size. Based on these two objectives, I construct the SDAE fault feature extraction model with the following objective function:
$$ \min F(S) = \{f_1(S), f_2(S)\} $$
The feature extraction process is summarized as follows:
| Step | Operation |
|---|---|
| 1 | Input the separated pinion gear vibration spectrum \(X\) and labels \(y\). |
| 2 | Set the SDAE parameters and determine the population neighbors. |
| 3 | Calculate the objective function values of the decision vector and find the minimum objective value. |
| 4 | Select two neighbors from the population, perform crossover, and update the population and minimum objective value. |
| 5 | If the maximum number of iterations is reached, stop; otherwise return to step 4. |
| 6 | Obtain the best SDAE model and extract the diverse pinion gear fault features. |
The extracted features include time-domain statistics, frequency-domain amplitudes at the pinion gear meshing frequency and its harmonics, and envelope spectrum features. These features form a compact representation of the pinion gear condition and are used as inputs to the SVM classifier.
SVM-Based Fault Diagnosis for the Pinion Gear Set
I model the pinion gear fault degradation as a low-rank plus sparse problem. Suppose the clean low-rank matrix \(L \in \mathbb{R}^{m \times n}\) is corrupted by a sparse error matrix \(S \in \mathbb{R}^{m \times n}\). The degradation model is:
$$ D = L + S $$
where \(D \in \mathbb{R}^{m \times n}\) is the observed data matrix, \(L\) and \(S\) are unknown. The goal is to recover the low-rank matrix \(L\). I formulate the problem as finding the lowest-rank matrix \(L\) that can generate the observed data while constraining the error matrix \(S\) to be sparse, i.e., \(\|S\|_0 \leq k\). To find the best low-rank matrix, I minimize the mean squared error between the observation matrix and the low-rank matrix:
$$ \min_{L} \|L – D\|_F^2 \quad \text{s.t.} \quad \operatorname{rank}(L) = r $$
where \(\|\cdot\|_F\) is the Frobenius norm, \(\operatorname{rank}(L)\) is the rank of \(L\), and \(r\) is the expected rank. The expected rank \(r\) is then fed into the SVM training process. I set the SVM sample set as \((x_i, y_i)\), where \(x_i\) is the class attribute and \(y_i\) is the class label. The SVM maps the low-dimensional space to a high-dimensional space and finds the optimal hyperplane:
$$ f(x) = \sum_{i=1}^{n} y_i a_i K(x, x_i) + b $$
where \(a_i\) is the Lagrange multiplier, \(K(x, x_i)\) is the kernel function, and \(b\) is the bias. I use the Gaussian radial basis function kernel:
$$ K(x, x_i) = \exp\left( -\frac{\|x – x_i\|^2}{2\sigma^2} \right) $$
where \(\sigma\) is the variance and \(g = 1 / (2\sigma^2)\) is the kernel parameter. With this kernel, the SVM can separate non-linear pinion gear fault features effectively. The diagnosis workflow is as follows:
| Step | Operation |
|---|---|
| 1 | Collect vibration and torque signals from the hydraulic mechanical drive pinion gear set under multiple fault conditions. |
| 2 | Apply low-rank constrained source separation to obtain the separated pinion gear vibration sources. |
| 3 | Extract pinion gear fault features using the SDAE model and divide them into training and testing samples. |
| 4 | Train the SVM with the training samples and optimize the kernel parameters. |
| 5 | Classify the testing samples and output the pinion gear fault diagnosis result. |
Experimental Setup and Data Description
I validate the proposed method on a hydraulic mechanical drive pinion gear set test bench. The test bench consists of an electric motor, a torque sensor, a reducer, and an actuator. The torque sensor is installed at the bearing support of the pinion gear set. The reducer is mounted between the drive motor and the hydraulic mechanism. The actuator is connected to the key transmission components. A data acquisition device records the sensor signals. A temperature sensor is installed on the gearbox and a sound sensor is placed around the mechanical components. The experimental parameters are listed below.
| Parameter | Value |
|---|---|
| Sampling frequency | 5000 Hz |
| Sampling points | 8500 |
| Vibration directions | Horizontal radial, vertical radial, axial |
| Number of fault types | 5 |
| Feature vectors per fault type | 40 |
| Total feature vectors | 200 |
| Training samples per fault type | 20 |
| Testing samples per fault type | 20 |
| Number of repeated experiments | 5 |
The gear fault categories used in my experiments are described in the following table. I use these categories to represent typical pinion gear and gear set faults in hydraulic mechanical drives.
| Category label | Fault type |
|---|---|
| 1 | Input stage pinion gear broken |
| 2 | Input stage pinion gear tooth surface scuffing and spalling |
| 3 | Intermediate stage pinion gear tooth surface scuffing and spalling |
| 4 | Output stage pinion gear tooth surface scuffing and spalling with broken tooth |
| 5 | Output stage pinion gear missing tooth |
| 6 | Input stage pinion gear root crack |
| 7 | Intermediate stage pinion gear root crack |
| 8 | Output stage pinion gear root crack |
| 9 | Input stage pinion gear broken plus intermediate stage pinion gear tooth surface scuffing and spalling |
| 10 | Healthy |
For each fault type, I extract 40 feature vectors. Among them, 20 are used as training samples and 20 are used as testing samples. I repeat the training and testing process five times and calculate the average accuracy. The classification error rate is used as the evaluation metric:
$$ A_i = 1 – \frac{N_{pi}}{N_{Ai}} \times 100\% $$
where \(A_i\) is the classification error rate of the \(i\)-th fault type, \(N_{pi}\) is the number of correctly diagnosed samples of the \(i\)-th fault type, and \(N_{Ai}\) is the total number of testing samples of the \(i\)-th fault type.
Classification Error Rate Analysis
I compare the proposed method with two reference methods: an improved convolutional neural network method and a local feature scale decomposition method. The classification error rates for different numbers of feature data are shown in the following table. As the number of feature data increases, the error rates of all methods increase slightly, but the proposed method maintains the lowest error rate.
| Feature data | Proposed method (%) | Method 2 (%) | Method 3 (%) |
|---|---|---|---|
| 100 | 1.2 | 2.3 | 3.2 |
| 200 | 1.3 | 2.6 | 3.6 |
| 300 | 1.5 | 2.7 | 3.9 |
| 400 | 1.7 | 2.9 | 4.1 |
| 500 | 2.1 | 3.3 | 4.3 |
| 600 | 2.2 | 3.5 | 4.6 |
| 700 | 2.6 | 3.8 | 4.7 |
| 800 | 2.9 | 4.0 | 5.2 |
| 900 | 3.1 | 4.3 | 5.6 |
| 1000 | 3.5 | 4.5 | 5.8 |
The proposed method achieves a classification error rate below 3.5% even when 1000 feature data are used. The improvement is mainly attributed to the low-rank constrained source separation, which removes the interference from other vibration sources and enhances the pinion gear fault signature. The SDAE feature extraction also selects discriminative features that are sensitive to pinion gear cracks, missing teeth, and broken teeth. In contrast, the reference methods suffer from signal conversion loss and incomplete decomposition, leading to higher error rates.
Fault Diagnosis Effect Test
I also test the fault diagnosis performance under three conditions: normal, missing tooth, and crack. For each condition, I collect 50 samples. I use 30 samples for training and 20 samples for testing. The SVM labels are set as \(y=1\) for normal, \(y=2\) for missing tooth, and \(y=3\) for crack. The diagnosis results are compared with the actual classification results.
| Test sample | Actual classification | Proposed method | Method 2 | Method 3 |
|---|---|---|---|---|
| 10 | 10 | 10 | 7 | 8 |
| 20 | 20 | 20 | 18 | 16 |
| 30 | 30 | 30 | 28 | 25 |
| 40 | 39 | 40 | 33 | 36 |
| 50 | 50 | 50 | 40 | 42 |
| 60 | 60 | 60 | 51 | 54 |
When the number of test samples is 10, the proposed method matches the actual classification exactly. Method 2 has three deviated samples and Method 3 has two deviated samples. When the number of test samples is 40, the proposed method has only one deviated sample, while Method 2 has seven and Method 3 has four. These results confirm that the proposed method provides more accurate pinion gear fault diagnosis and higher feasibility.
Fault Diagnosis Efficiency Analysis
Diagnosis time is an important indicator for practical deployment. I evaluate the fault diagnosis duration for 100 iterations using the three methods. The results are shown in the following table.
| Iteration | Proposed method (s) | Method 2 (s) | Method 3 (s) |
|---|---|---|---|
| 10 | 9 | 19 | 35 |
| 20 | 8 | 20 | 36 |
| 30 | 10 | 21 | 35 |
| 40 | 9 | 22 | 34 |
| 50 | 8 | 23 | 33 |
| 60 | 6 | 25 | 32 |
| 70 | 9 | 24 | 33 |
| 80 | 10 | 22 | 32 |
| 90 | 8 | 23 | 35 |
| 100 | 7 | 24 | 34 |
The proposed method requires at most 10 seconds, while Method 2 reaches 25 seconds and Method 3 reaches 36 seconds. The lower computation time of the proposed method comes from the efficient low-rank separation and the compact SDAE feature representation. The SVM training is also fast because the feature dimension is reduced and the kernel parameters are optimized with a small number of support vectors.
Diagnosis Under Different Load Conditions
To verify the robustness of the proposed method under diverse working conditions, I test four load types: constant load, sinusoidal load, slope load, and pulse load. The diagnostic errors of the three methods are compared.
| Load type | Proposed method errors | Method 2 errors | Method 3 errors |
|---|---|---|---|
| Constant load | 0 | 2 | 2 |
| Sinusoidal load | 1 | 3 | 2 |
| Slope load | 1 | 2 | 2 |
| Pulse load | 0 | 2 | 1 |
The proposed method has only two errors across all load conditions, while Method 2 has nine errors and Method 3 has seven errors. This demonstrates that the proposed method maintains high diagnostic accuracy under varying loads, which is important for hydraulic mechanical drive pinion gear sets operating in real industrial environments.
Discussion of Pinion Gear Fault Signatures
The vibration signature of a pinion gear depends on its geometry, rotational speed, and meshing conditions. For a pinion gear with \(N_p\) teeth rotating at \(f_r\) revolutions per second, the meshing frequency is \(f_m = N_p f_r\). When a crack or a broken tooth occurs on the pinion gear, the vibration signal contains impulses at the rotation frequency and its harmonics. The envelope spectrum around the meshing frequency shows sidebands separated by \(f_r\). In my method, the low-rank separation preserves these sidebands by removing the low-rank background components. The SDAE then extracts the envelope features and the SVM classifies the pinion gear condition. I have observed that the proposed method is particularly effective for early root cracks and tooth surface spalling, because these faults produce weak impulse components that are often buried in noise. The low-rank separation enhances the impulsiveness, and the kurtosis term in the separation model further highlights the transient pinion gear fault features.
The choice of the compromise factor \(\beta\) and the low-rank weights \(\lambda_i\) affects the separation performance. In my experiments, I set \(\beta = 0.1\) and \(\lambda_i = 1\) for all sources. I also select the number of sources \(n\) according to the number of dominant vibration components in the spectrum. For the hydraulic mechanical drive pinion gear set, I typically set \(n = 4\) to separate the pinion gear meshing signal, the bearing signal, the shaft signal, and the hydraulic pulsation. The constraint condition in the separation model uses the characteristic frequency of each source, which I compute from the gear geometry and the rotational speed. This design ensures that the pinion gear fault source is not over-smoothed or merged with other sources.
Feature Sensitivity and SVM Parameter Selection
The SVM performance depends on the kernel parameter \(g\) and the regularization parameter \(C\). I use a grid search with five-fold cross-validation to select \(g\) and \(C\). The search range is \(g \in [2^{-10}, 2^{10}]\) and \(C \in [2^{-10}, 2^{10}]\). The optimal parameters are obtained by minimizing the validation error. In my experiments, the best values are \(g = 0.125\) and \(C = 8\). The Gaussian radial basis function kernel provides a smooth decision boundary and is suitable for the non-linear pinion gear fault features. I also compare the SVM with a linear kernel and a polynomial kernel. The Gaussian kernel achieves the lowest classification error rate, especially for the mixed fault category that includes both a broken pinion gear and tooth surface scuffing. The feature sensitivity analysis shows that the envelope spectrum entropy and the kurtosis of the separated pinion gear signal are the most discriminative features. The time-domain root mean square and the frequency-domain amplitude at the meshing frequency also contribute significantly to the classification.
Comparison with Existing Methods
I compare the proposed method with several existing approaches. The improved convolutional neural network method converts the vibration signal into a two-dimensional image and extracts features with convolutional layers. Although it reduces diagnosis time in some cases, the signal conversion can lose the phase information of the pinion gear fault. The local feature scale decomposition method decomposes the signal into intrinsic scale components and uses symbol entropy and an adaptive neuro-fuzzy inference system. However, the decomposition may not be complete, and the fuzzy inference can introduce errors when the pinion gear fault is weak. The deep belief network method uses a hierarchical representation and transfer learning, but the weight adjustment may not converge to the global optimum. The attention-based capsule network and the short-time Fourier transform convolutional network also have limitations in sample balance and sample-class mapping. In contrast, the proposed method combines the physical interpretability of low-rank source separation with the robust feature learning of SDAE and the strong generalization of SVM. The experimental results confirm that the proposed method achieves lower classification error, higher diagnosis accuracy, and shorter diagnosis time.
Practical Implications for Hydraulic Mechanical Drive Pinion Gear Sets
In industrial applications, the pinion gear set often operates under variable load and speed. The proposed method can be implemented as an online monitoring module. The vibration sensors collect the pinion gear signals, and the low-rank separation is performed in a sliding window. The SDAE features are extracted periodically, and the SVM classifier outputs the pinion gear condition. When a fault is detected, the maintenance team can schedule inspection or replacement before the pinion gear fails completely. The method is also suitable for other gear types in hydraulic mechanical drives, such as helical gears and bevel gears, because the low-rank separation and SDAE feature extraction are not limited to a specific gear geometry. The key requirement is to adjust the characteristic frequency constraint according to the gear tooth number and the rotational speed.
Conclusion
I have presented a support vector machine based fault diagnosis method for hydraulic mechanical drive pinion gear sets. The method first collects vibration and torque signals from the pinion gear set. Then it applies a low-rank constrained source separation model to separate the pinion gear vibration source from the mixed observation. The separation model uses rank minimization and kurtosis minimization, and it is solved with the alternating direction method of multipliers and singular value decomposition. After separation, a stacked denoising autoencoder extracts fault features with minimum sparsity and minimum classification error objectives. Finally, a support vector machine with a Gaussian radial basis function kernel classifies the pinion gear condition. I validated the method on a hydraulic mechanical drive test bench with multiple pinion gear fault types, including broken teeth, missing teeth, root cracks, and tooth surface scuffing and spalling. The experimental results show that the classification error rate does not exceed 3.5%, the diagnosis time is shorter than that of the reference methods, and the diagnosis accuracy is higher under constant, sinusoidal, slope, and pulse loads. The proposed method provides a reliable and efficient solution for pinion gear fault diagnosis in hydraulic mechanical drives. In future work, I plan to extend the method to online monitoring with edge computing and to test it on a wider range of pinion gear geometries and operating conditions.