In this study, I investigate an automatic monitoring method for the operating state of a pinion gear that drives a drum-type rotary filter screen under multi-source data fusion. The pinion gear is a critical transmission component in the rotary filter screen drive system, and its condition directly influences the reliability, maintainability, and operational safety of the entire filtration unit. A pinion gear does not operate in isolation. Its behavior is coupled with load variations, tooth contact conditions, lubrication states, alignment errors, and the dynamic response of the larger gear ring. Therefore, if I rely on only one measurement source, such as vibration or current, the resulting diagnosis may be incomplete or even misleading. To overcome this limitation, I fuse vibration signals and current signals into a unified state-space framework. The pinion gear state is treated as a latent variable, and the mapping from the latent state to the observed multi-source data is explicitly modeled. Then, I apply a Kalman filter to recursively predict and update the pinion gear state in real time. The experimental results show that the residuals of the pinion gear state detection are mainly concentrated in the interval \([-0.5, 0.5]\), which indicates that my method is reliable for automatic monitoring of the pinion gear operating state.
1. Introduction
The operating state of a pinion gear has multi-dimensional characteristics. These characteristics include geometric degradation, surface wear, pitting, scuffing, root crack initiation, misalignment, and load-induced elastic deformation. When I monitor only a single data source, the information that I obtain usually reflects only a partial projection of the actual pinion gear condition. For example, vibration signals are sensitive to impact events and tooth meshing variations, but they may be affected by structural resonance and sensor mounting conditions. Current signals are sensitive to load torque changes and drive motor behavior, but they may be influenced by supply voltage fluctuation, controller dynamics, and electromagnetic interference. Consequently, neither vibration alone nor current alone can provide a complete and robust description of the pinion gear state.
For this reason, I focus on multi-source data fusion for automatic monitoring of the pinion gear in a drum-type rotary filter screen drive. The term pinion gear appears repeatedly in this work because the pinion gear is the central object of condition assessment. My aim is not merely to detect whether the pinion gear is faulty, but to estimate its operating state continuously and automatically. I construct a state-space model in which the pinion gear state is a hidden variable. The observations are formed by vibration features and current features. The state equation describes how the pinion gear state evolves over time, while the observation equation describes how the hidden pinion gear state generates the measured signals. This modeling strategy allows me to combine heterogeneous information in a principled way.
The main contributions of my work can be summarized as follows. First, I design a multi-source data fusion architecture that combines vibration time-domain features, vibration frequency-domain features, and current load-related features for pinion gear monitoring. Second, I build a state-space model of the pinion gear, where the pinion gear condition is treated as a latent state and the observation mapping is explicitly defined. Third, I apply a Kalman filter to perform recursive prediction and update of the pinion gear state under real-time multi-source data. Fourth, I validate the proposed method on a drum-type rotary filter screen drive test rig and compare the residual distribution with other monitoring strategies. The results demonstrate that the pinion gear state residuals are concentrated in a narrow interval, indicating high reliability.
In the following sections, I present the system description, feature extraction, state-space modeling, Kalman filtering, experimental setup, parameter calculation, results, discussion, and conclusions. I use tables and equations throughout the paper to make the mathematical structure explicit and to summarize the key numerical settings. The use of multi-source data is central to my approach because it allows the pinion gear state to be observed from complementary viewpoints. In particular, vibration signals provide information about tooth contact and dynamic excitation, while current signals provide information about load and drive torque. When these two sources are fused, the pinion gear monitoring result becomes more stable and more accurate than a single-source result.
2. System Description and Measurement Sources
The drum-type rotary filter screen is a large rotating filtration device used in industrial water intake and cooling systems. The pinion gear transmits torque from the drive motor to the large gear ring mounted on the drum. As the pinion gear rotates, it meshes with the large gear ring and drives the drum at a low rotational speed. The pinion gear is therefore subjected to cyclic contact stress, sliding friction, bending stress, and possible misalignment. Because the drum rotates slowly and the load may vary with debris accumulation, the pinion gear state cannot be assumed to be constant. I need a dynamic monitoring method that can track the pinion gear condition over time.
I consider two primary measurement sources. The first source is vibration, which is measured on the bearing housing or gearbox casing near the pinion gear. The vibration signal contains rich information about gear meshing, tooth impact, and structural response. The second source is current, which is measured from the drive motor. The current signal reflects the electromagnetic torque and hence the load variation of the pinion gear drive. By combining these two sources, I can observe the pinion gear state from both mechanical and electrical perspectives. This is especially important because a change in pinion gear condition may produce a clear vibration signature but only a weak current signature, or vice versa.
| Symbol | Meaning | Unit |
|---|---|---|
| \(x_t\) | Pinion gear state vector at time \(t\) | dimensionless |
| \(u_t\) | Input current vector at time \(t\) | A |
| \(y_t\) | Observation vector at time \(t\) | mixed |
| \(A\) | State transition matrix of the pinion gear | dimensionless |
| \(B\) | Input control matrix of the pinion gear | dimensionless |
| \(C\) | Output matrix of the pinion gear observation model | dimensionless |
| \(D\) | Direct transmission matrix | dimensionless |
| \(v_t\) | Process noise | dimensionless |
| \(w_t\) | Observation noise | mixed |
| \(Q\) | Process noise covariance matrix | dimensionless |
| \(R\) | Observation noise covariance matrix | mixed |
| \(K_t\) | Kalman gain matrix at time \(t\) | dimensionless |
The measurement sources are not synchronized by default, so I perform time alignment before fusion. The vibration signal is sampled at a high rate, while the current signal is sampled at a lower rate. I resample both signals to a common time base and then extract features within fixed windows. The window length is chosen to cover several pinion gear meshing cycles. This ensures that the extracted features represent the pinion gear state rather than transient noise. In my implementation, the window length is \(0.5\) s, and the overlap is \(50\%\). The feature extraction process is described in the next section.
3. Multi-Source Feature Extraction
Feature extraction is a critical step in pinion gear monitoring. If the features are not informative, the state-space model will not be able to distinguish different pinion gear conditions. I extract three groups of features: vibration time-domain features, vibration frequency-domain features, and current load features. The vibration time-domain features include root mean square, peak value, kurtosis, and crest factor. The vibration frequency-domain features include meshing frequency energy, sideband energy, and spectral entropy. The current features include mean current, current variance, and load torque proxy. These features are concatenated into the observation vector \(y_t\).
For a vibration signal segment \(s(n)\) with \(N\) samples, the root mean square is computed as:
$$s_{\mathrm{RMS}} = \sqrt{\frac{1}{N}\sum_{n=1}^{N} s(n)^2}$$
The peak value is:
$$s_{\mathrm{peak}} = \max_{1 \le n \le N} |s(n)|$$
The kurtosis is:
$$s_{\mathrm{kurt}} = \frac{\frac{1}{N}\sum_{n=1}^{N} (s(n)-\bar{s})^4}{\left(\frac{1}{N}\sum_{n=1}^{N} (s(n)-\bar{s})^2\right)^2}$$
where \(\bar{s}\) is the mean of the vibration segment. The crest factor is:
$$s_{\mathrm{crest}} = \frac{s_{\mathrm{peak}}}{s_{\mathrm{RMS}}}$$
For frequency-domain features, I apply a fast Fourier transform to the vibration segment. Let \(S(k)\) be the spectrum. The meshing frequency energy is:
$$E_{\mathrm{mesh}} = \sum_{k \in \mathcal{K}_{\mathrm{mesh}}} |S(k)|^2$$
where \(\mathcal{K}_{\mathrm{mesh}}\) is the frequency bin set around the pinion gear meshing frequency. The sideband energy is:
$$E_{\mathrm{side}} = \sum_{k \in \mathcal{K}_{\mathrm{side}}} |S(k)|^2$$
where \(\mathcal{K}_{\mathrm{side}}\) contains bins around the meshing frequency plus and minus the shaft rotation frequency. The spectral entropy is:
$$H_{\mathrm{spec}} = -\sum_{k=1}^{K} p_k \log p_k$$
with
$$p_k = \frac{|S(k)|^2}{\sum_{j=1}^{K} |S(j)|^2}$$
For the current signal \(i(n)\), I compute the mean current:
$$\bar{i} = \frac{1}{N}\sum_{n=1}^{N} i(n)$$
The current variance is:
$$\sigma_i^2 = \frac{1}{N}\sum_{n=1}^{N} (i(n)-\bar{i})^2$$
The load torque proxy is derived from the mean current and the motor torque constant \(k_t\):
$$T_{\mathrm{load}} \approx k_t \bar{i}$$
The observation vector is then formed as:
$$y_t = \begin{bmatrix}
s_{\mathrm{RMS}} & s_{\mathrm{peak}} & s_{\mathrm{kurt}} & s_{\mathrm{crest}} & E_{\mathrm{mesh}} & E_{\mathrm{side}} & H_{\mathrm{spec}} & \bar{i} & \sigma_i^2 & T_{\mathrm{load}}
\end{bmatrix}^T$$
In practice, I select a subset of these features to avoid over-dimensioning the observation model. The selected features are RMS, kurtosis, meshing frequency energy, sideband energy, mean current, and current variance. This subset provides a balance between sensitivity and computational cost. Table 2 summarizes the feature set and its role in pinion gear monitoring.
| Feature Group | Feature | Symbol | Physical Meaning |
|---|---|---|---|
| Vibration time domain | Root mean square | \(s_{\mathrm{RMS}}\) | Overall vibration energy of the pinion gear |
| Vibration time domain | Kurtosis | \(s_{\mathrm{kurt}}\) | Impulsiveness caused by tooth impact |
| Vibration frequency domain | Meshing energy | \(E_{\mathrm{mesh}}\) | Energy at the pinion gear meshing frequency |
| Vibration frequency domain | Sideband energy | \(E_{\mathrm{side}}\) | Modulation due to load or fault |
| Current | Mean current | \(\bar{i}\) | Average load level of the drive motor |
| Current | Current variance | \(\sigma_i^2\) | Load fluctuation of the pinion gear drive |
The feature extraction process is performed in real time. For each time window, I compute the feature vector and feed it into the state-space model. The state-space model then estimates the hidden pinion gear state. Because the features come from different physical domains, I normalize them before fusion. The normalization is:
$$\tilde{y}_j = \frac{y_j – \mu_j}{\sigma_j}$$
where \(\mu_j\) and \(\sigma_j\) are the mean and standard deviation of the \(j\)-th feature over a training period. This step prevents features with large numerical values from dominating the observation model. After normalization, the observation vector is used in the Kalman filter update.
4. State-Space Model of the Pinion Gear
To represent the dynamic behavior of the pinion gear, I construct a state-space model that consists of a state equation and an observation equation. The pinion gear state is treated as a latent variable. The state vector includes the pinion gear health indicator and the load indicator. I define:
$$x_t = \begin{bmatrix} h_t \\ l_t \end{bmatrix}$$
where \(h_t\) is the pinion gear health state and \(l_t\) is the load state. The state equation is:
$$x_{t+1} = A x_t + B u_t + v_t$$
where \(u_t\) is the input current vector, \(v_t\) is process noise, and \(A\) and \(B\) are matrices that describe the state evolution and input influence. The observation equation is:
$$y_t = C x_t + D u_t + w_t$$
where \(y_t\) is the observation vector, \(w_t\) is observation noise, and \(C\) and \(D\) are matrices that map the state and input to the observations. This model allows me to fuse vibration and current information in a unified probabilistic framework. The pinion gear state is not directly measured, but it is inferred from the observations through the Kalman filter.
The state transition matrix \(A\) describes how the pinion gear state evolves from one time step to the next. In my model, I use a stable linear time-invariant form. The input control matrix \(B\) describes how the current input affects the pinion gear state. The output matrix \(C\) describes how the hidden pinion gear state influences the observed vibration and current features. The direct transmission matrix \(D\) describes the instantaneous effect of the input current on the observations. The process noise \(v_t\) accounts for unmodeled dynamics, and the observation noise \(w_t\) accounts for measurement uncertainty. Both noise terms are assumed to be zero-mean Gaussian.
The dimensions of the matrices are determined by the number of states and observations. In my implementation, the state dimension is \(n_x = 2\), the input dimension is \(n_u = 2\), and the observation dimension is \(n_y = 6\). Therefore, \(A\) is \(2 \times 2\), \(B\) is \(2 \times 2\), \(C\) is \(6 \times 2\), and \(D\) is \(6 \times 2\). The process noise covariance \(Q\) is \(2 \times 2\), and the observation noise covariance \(R\) is \(6 \times 6\). These dimensions are summarized in Table 3.
| Matrix | Dimension | Description |
|---|---|---|
| \(x_t\) | \(2 \times 1\) | Pinion gear state vector |
| \(u_t\) | \(2 \times 1\) | Input current vector |
| \(y_t\) | \(6 \times 1\) | Observation vector |
| \(A\) | \(2 \times 2\) | State transition matrix |
| \(B\) | \(2 \times 2\) | Input control matrix |
| \(C\) | \(6 \times 2\) | Output matrix |
| \(D\) | \(6 \times 2\) | Direct transmission matrix |
| \(Q\) | \(2 \times 2\) | Process noise covariance |
| \(R\) | \(6 \times 6\) | Observation noise covariance |
The state-space model is linear, which allows me to apply the Kalman filter directly. However, the pinion gear system may exhibit nonlinear behavior under severe fault conditions. In future work, I plan to extend the model to nonlinear state-space forms, such as the extended Kalman filter or unscented Kalman filter. For the current study, the linear model is sufficient because the pinion gear operates mostly in a stable regime during the tests. The model captures the essential coupling between the pinion gear state, load variation, and the measured vibration and current features.
5. Kalman Filtering for Pinion Gear State Estimation
After constructing the state-space model, I use the Kalman filter to estimate the pinion gear state recursively. The Kalman filter is optimal for linear Gaussian systems. It alternates between a prediction step and an update step. In the prediction step, I use the state equation to predict the next pinion gear state and its covariance. In the update step, I use the observation equation and the actual measurements to correct the prediction. This recursive process provides a real-time estimate of the pinion gear state.
The initial state estimate is set as:
$$\hat{x}_0 = E[x_0]$$
The initial covariance is:
$$P_0 = E[(x_0 – \hat{x}_0)(x_0 – \hat{x}_0)^T]$$
For each time step \(t\), the prediction equations are:
$$\hat{x}_{t|t-1} = A \hat{x}_{t-1|t-1} + B u_t$$
$$P_{t|t-1} = A P_{t-1|t-1} A^T + Q$$
The Kalman gain is computed as:
$$K_t = P_{t|t-1} C^T (C P_{t|t-1} C^T + R)^{-1}$$
The update equations are:
$$\hat{x}_{t|t} = \hat{x}_{t|t-1} + K_t (y_t – C \hat{x}_{t|t-1} – D u_t)$$
$$P_{t|t} = (I – K_t C) P_{t|t-1}$$
where \(I\) is the identity matrix. The term \(y_t – C \hat{x}_{t|t-1} – D u_t\) is the innovation, which represents the difference between the actual observation and the predicted observation. The Kalman gain determines how much the innovation should influence the state estimate. If the observation noise is small, the gain is large, and the filter trusts the measurement more. If the process noise is small, the gain is small, and the filter trusts the model more.
The recursive nature of the Kalman filter makes it suitable for automatic monitoring. Once the model matrices are identified, the filter can run continuously without manual intervention. The output is the estimated pinion gear state \(\hat{x}_{t|t}\), which can be used for condition assessment, trend analysis, and alarm generation. I summarize the Kalman filter steps in Table 4.
| Step | Equation | Purpose |
|---|---|---|
| Initialization | \(\hat{x}_0 = E[x_0]\), \(P_0 = E[(x_0-\hat{x}_0)(x_0-\hat{x}_0)^T]\) | Set initial pinion gear state and uncertainty |
| Prediction | \(\hat{x}_{t|t-1} = A \hat{x}_{t-1|t-1} + B u_t\) | Predict next pinion gear state |
| Prediction | \(P_{t|t-1} = A P_{t-1|t-1} A^T + Q\) | Predict state covariance |
| Gain | \(K_t = P_{t|t-1} C^T (C P_{t|t-1} C^T + R)^{-1}\) | Compute Kalman gain |
| Update | \(\hat{x}_{t|t} = \hat{x}_{t|t-1} + K_t (y_t – C \hat{x}_{t|t-1} – D u_t)\) | Update pinion gear state estimate |
| Update | \(P_{t|t} = (I – K_t C) P_{t|t-1}\) | Update state covariance |
In my implementation, the Kalman filter is initialized with \(\hat{x}_0 = [1, 0]^T\), where the first element represents a healthy pinion gear state and the second element represents a zero load offset. The initial covariance \(P_0\) is set to \(0.1 I\). The process noise covariance \(Q\) and observation noise covariance \(R\) are tuned using a training data set. The filter is then applied to the test data. The output is the estimated pinion gear state over time. Because the filter fuses vibration and current features, the estimated pinion gear state is more stable than estimates obtained from either source alone.
6. Multi-Source Fusion and Noise Modeling
The quality of the pinion gear state estimate depends on the noise models. I assume that the process noise and observation noise are independent and Gaussian. The process noise covariance \(Q\) describes the uncertainty in the pinion gear state evolution. The observation noise covariance \(R\) describes the uncertainty in the vibration and current features. If \(Q\) is too small, the filter may become overconfident in the model and ignore actual changes in the pinion gear state. If \(Q\) is too large, the filter may become too sensitive to noise. Similarly, if \(R\) is too small, the filter may overfit the measurements. If \(R\) is too large, the filter may ignore useful information.
I estimate \(Q\) and \(R\) from the training data using the innovation sequence. The innovation covariance is:
$$S_t = C P_{t|t-1} C^T + R$$
The normalized innovation squared is:
$$\epsilon_t = \nu_t^T S_t^{-1} \nu_t$$
where \(\nu_t = y_t – C \hat{x}_{t|t-1} – D u_t\). If the model is correct, \(\epsilon_t\) should follow a chi-square distribution with \(n_y\) degrees of freedom. I use this property to check the consistency of the noise models. If the average normalized innovation squared is significantly larger than \(n_y\), I increase \(Q\) or \(R\). If it is significantly smaller, I decrease \(Q\) or \(R\). This tuning process ensures that the filter remains consistent.
The multi-source fusion is achieved through the observation matrix \(C\). The rows of \(C\) correspond to the vibration and current features. The columns of \(C\) correspond to the pinion gear health state and load state. By assigning appropriate weights in \(C\), I can control how much each feature contributes to the pinion gear state estimate. For example, the meshing frequency energy and sideband energy are strongly related to the pinion gear health state, so they receive larger weights in the first column of \(C\). The mean current and current variance are strongly related to the load state, so they receive larger weights in the second column of \(C\). The vibration RMS and kurtosis are influenced by both health and load, so they receive moderate weights in both columns.
| Observation Feature | Weight on \(h_t\) | Weight on \(l_t\) | Rationale |
|---|---|---|---|
| Vibration RMS | 0.6 | 0.3 | Reflects overall vibration and load |
| Vibration kurtosis | 0.8 | 0.1 | Sensitive to tooth impact and pitting |
| Meshing energy | 1.0 | 0.0 | Directly related to pinion gear meshing |
| Sideband energy | 0.9 | 0.2 | Indicates modulation and fault growth |
| Mean current | 0.1 | 1.0 | Reflects average load torque |
| Current variance | 0.2 | 0.8 | Reflects load fluctuation |
The fusion process is not simply averaging the features. It is a model-based combination that accounts for uncertainty. The Kalman filter computes the optimal weights dynamically through the gain matrix \(K_t\). When a feature is noisy, its contribution is reduced. When a feature is informative, its contribution is increased. This adaptive weighting is a major advantage of the state-space approach. It allows the pinion gear monitoring system to remain robust even when one sensor is temporarily degraded or disturbed.
7. Experimental Setup
I built a test platform to evaluate the proposed pinion gear monitoring method. The test platform represents a drum-type rotary filter screen drive. The drum diameter is \(11,000\) mm, and the drum width is \(3,000\) mm. The drive motor has a rated power of \(18.5\) kW. For low-speed operation at \(5.0\) m/min, the margin coefficient is \(1.5\). For medium and high-speed operation between \(10.0\) and \(20.0\) m/min, the margin coefficient is \(1.3\). These parameters define the operating envelope of the pinion gear drive.
The pinion gear is made of MC nylon, and the large gear ring is made of cast iron. This material combination creates a difference in elastic modulus, which helps reduce tooth surface contact stress and improves resistance to scuffing. The pinion gear parameters are listed in Table 6. The large gear parameters are also listed. The pinion gear has 21 teeth, a module of \(2.0\) mm, a modification coefficient of \(0.20\), a contact ratio of \(1.35\), and a relative sliding ratio of \(1.56\). The large gear has 82 teeth, a module of \(2.0\) mm, a modification coefficient of \(0.1616\), a contact ratio of \(1.35\), and a relative sliding ratio of \(1.56\).
| Parameter | Pinion Gear | Large Gear |
|---|---|---|
| Module | \(2.0\) mm | \(2.0\) mm |
| Modification coefficient | \(0.20\) | \(0.1616\) |
| Contact ratio | \(1.35\) | \(1.35\) |
| Relative sliding ratio | \(1.56\) | \(1.56\) |
| Number of teeth | \(21\) | \(82\) |
Vibration is measured using an accelerometer mounted on the bearing housing near the pinion gear. The sampling frequency is \(20\) kHz. Current is measured using a Hall-effect current sensor on the motor supply line. The sampling frequency is \(2\) kHz. Both signals are collected simultaneously for \(80\) s. The data are divided into \(0.5\) s windows with \(50\%\) overlap. For each window, I extract the features described in Section 3. The resulting feature matrix is used to identify the state-space model and to run the Kalman filter.
The test conditions include normal operation, slight load variation, and induced pinion gear wear. The induced wear is created by running the pinion gear under accelerated conditions for a period of time. I do not artificially damage the pinion gear severely, because the goal is to evaluate early-state monitoring rather than catastrophic failure detection. The pinion gear state is labeled by an expert using offline inspection data. These labels are used only for validation, not for training the Kalman filter. The filter is trained on normal data and then tested on all conditions.
8. Parameter Calculation and Matrix Instantiation
Using the collected vibration and current data, I identify the state-space matrices. The state transition matrix is identified as:
$$A = \begin{bmatrix} 0.9 & 0.1 \\ 0 & 0.8 \end{bmatrix}$$
The input control matrix is:
$$B = \begin{bmatrix} 0.01 & 0.02 \\ 0 & 0.05 \end{bmatrix}$$
The output matrix is:
$$C = \begin{bmatrix} 1 & 0 \\ 0.5 & 0.2 \\ 0 & 1 \end{bmatrix}$$
In the actual implementation, the observation dimension is 6, so \(C\) is expanded to include the selected features. The expanded output matrix is:
$$C = \begin{bmatrix}
1 & 0 \\
0.5 & 0.2 \\
0 & 1 \\
0.8 & 0.1 \\
0.9 & 0.2 \\
0.1 & 1.0
\end{bmatrix}$$
The direct transmission matrix is:
$$D = \begin{bmatrix}
0 & 0 \\
0 & 0 \\
1 & 0 \\
0 & 0 \\
0 & 0 \\
0 & 0
\end{bmatrix}$$
The process noise covariance is:
$$Q = \begin{bmatrix} 0.01 & 0 \\ 0 & 0.02 \end{bmatrix}$$
The observation noise covariance is:
$$R = \begin{bmatrix}
0.1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0.05 & 0 & 0 & 0 & 0 \\
0 & 0 & 0.2 & 0 & 0 & 0 \\
0 & 0 & 0 & 0.1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0.15 & 0 \\
0 & 0 & 0 & 0 & 0 & 0.25
\end{bmatrix}$$
These matrices are used in the Kalman filter. The process noise covariance \(Q\) is relatively small, which means the pinion gear state is assumed to change slowly. The observation noise covariance \(R\) is larger for the current variance and sideband energy, because these features are more affected by external disturbances. The observation noise covariance is smaller for the meshing energy and mean current, because these features are more stable. This careful tuning improves the accuracy of the pinion gear state estimate.
| Matrix | Values | Interpretation |
|---|---|---|
| \(A\) | \(\begin{bmatrix} 0.9 & 0.1 \\ 0 & 0.8 \end{bmatrix}\) | Pinion gear state evolves smoothly |
| \(B\) | \(\begin{bmatrix} 0.01 & 0.02 \\ 0 & 0.05 \end{bmatrix}\) | Current input has mild effect on state |
| \(C\) | \(\begin{bmatrix} 1 & 0 \\ 0.5 & 0.2 \\ 0 & 1 \\ 0.8 & 0.1 \\ 0.9 & 0.2 \\ 0.1 & 1.0 \end{bmatrix}\) | Features map to health and load states |
| \(D\) | \(\begin{bmatrix} 0 & 0 \\ 0 & 0 \\ 1 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{bmatrix}\) | Input directly affects mean current |
| \(Q\) | \(\begin{bmatrix} 0.01 & 0 \\ 0 & 0.02 \end{bmatrix}\) | Low process uncertainty |
| \(R\) | \(\mathrm{diag}(0.1, 0.05, 0.2, 0.1, 0.15, 0.25)\) | Feature-dependent observation uncertainty |
The initial state estimate is set to \(\hat{x}_0 = [1, 0]^T\), and the initial covariance is \(P_0 = 0.1 I\). The Kalman filter is run for the entire \(80\) s data set. The estimated pinion gear health state is compared with the expert label. The residual is defined as:
$$r_t = h_t – \hat{h}_{t|t}$$
where \(h_t\) is the true pinion gear health state and \(\hat{h}_{t|t}\) is the estimated state. A small residual indicates accurate monitoring. I also compute the root mean square error and mean absolute error:
$$\mathrm{RMSE} = \sqrt{\frac{1}{T}\sum_{t=1}^{T} r_t^2}$$
$$\mathrm{MAE} = \frac{1}{T}\sum_{t=1}^{T} |r_t|$$
These metrics are used to compare my method with other approaches. The results are presented in the next section.
9. Results and Residual Analysis
I evaluate the residual distribution of the pinion gear state detection. The residuals are computed for the entire test data set. The histogram of the residuals shows that most values are concentrated in the interval \([-0.5, 0.5]\). This indicates that the estimated pinion gear state is close to the true state. The probability density is highest near zero, and it decays rapidly as the residual magnitude increases. The residual distribution is approximately symmetric, which suggests that the filter is not biased in either direction.
For comparison, I also evaluate two reference methods. The first reference method is based on an improved ensemble KNN regression algorithm for wind turbine gearbox condition monitoring. The second reference method is based on a DSP-LightGBM algorithm for high-speed shaft gear health monitoring. Both reference methods are applied to the same data set. Their residual distributions are wider than my method. This comparison shows that the multi-source fusion and Kalman filtering approach provides a more accurate pinion gear state estimate.
| Method | Residual Interval | RMSE | MAE | Concentration |
|---|---|---|---|---|
| Improved ensemble KNN regression | \([-1.8, 1.8]\) | 0.82 | 0.61 | Moderate |
| DSP-LightGBM | \([-1.6, 1.6]\) | 0.74 | 0.55 | Moderate |
| My multi-source fusion method | \([-0.5, 0.5]\) | 0.18 | 0.13 | High |
The residual concentration in \([-0.5, 0.5]\) is a strong indicator of reliability. In my method, the vibration and current signals are fused, so the pinion gear state estimate is not dominated by a single source. The Kalman filter also suppresses noise and handles missing data gracefully. When one feature is temporarily corrupted, the filter still produces a reasonable estimate because the other features compensate. This robustness is important for industrial applications, where sensors may be exposed to dust, moisture, and electromagnetic interference.
I also analyze the time evolution of the residual. During normal operation, the residual remains close to zero. When the pinion gear experiences slight wear, the residual increases slightly but remains within the \([-0.5, 0.5]\) interval. The filter responds quickly to changes in the pinion gear state, but it does not overreact to transient disturbances. This balance between sensitivity and stability is achieved by the optimal gain matrix \(K_t\). The gain adapts to the uncertainty in the state and observations, so the filter remains consistent across different operating conditions.
| Time Segment | Pinion Gear Condition | Mean Residual | Standard Deviation | Maximum Absolute Residual |
|---|---|---|---|---|
| 0–20 s | Normal | 0.02 | 0.08 | 0.21 |
| 20–40 s | Slight load variation | 0.05 | 0.11 | 0.31 |
| 40–60 s | Early wear | 0.08 | 0.14 | 0.42 |
| 60–80 s | Recovery | 0.03 | 0.09 | 0.24 |
The maximum absolute residual never exceeds \(0.5\) in the test data. This means that the estimated pinion gear state is always within half a unit of the true state. In practical terms, this level of accuracy is sufficient for early warning and condition-based maintenance. If the residual exceeds a predefined threshold, the monitoring system can issue an alarm. The threshold can be set based on the acceptable risk level. For example, a threshold of \(0.5\) would trigger an alarm only when the pinion gear state deviates significantly.
10. Discussion
The proposed method has several advantages. First, it uses multi-source data, which reduces the risk of false alarms and missed detections. Second, it uses a state-space model, which provides a clear probabilistic description of the pinion gear state and observations. Third, it uses the Kalman filter, which is computationally efficient and suitable for real-time implementation. Fourth, it provides a residual signal that can be used for condition monitoring and fault diagnosis. The residual is easy to interpret, and it can be tracked over time to detect gradual degradation of the pinion gear.
The method also has limitations. The state-space model is linear and time-invariant. If the pinion gear system exhibits strong nonlinearity or time-varying dynamics, the linear model may not be sufficient. In future work, I plan to use nonlinear filtering methods, such as the unscented Kalman filter, to handle more complex pinion gear behavior. I also plan to include additional data sources, such as temperature, acoustic emission, and oil debris, to further improve the monitoring accuracy. These additional sources could provide complementary information about lubrication, friction, and wear particles.
Another limitation is the need for training data to identify the model matrices. In my study, I used a training period to estimate \(A\), \(B\), \(C\), \(D\), \(Q\), and \(R\). In practical applications, the training data may not be available for every pinion gear installation. To address this, I plan to develop adaptive identification methods that can update the model parameters online. This would allow the monitoring system to adapt to different pinion gear units and operating conditions without extensive offline training.
The computational cost of the method is modest. The Kalman filter requires matrix multiplications and inversions. For a state dimension of 2 and an observation dimension of 6, the computational load is very low. The filter can run on an embedded controller or a programmable logic controller. This makes the method suitable for industrial deployment. The feature extraction is also lightweight, especially when only a subset of features is used. The overall system can operate in real time with a sampling interval of \(0.5\) s.
From an application perspective, the pinion gear monitoring method can be integrated into the control system of the drum-type rotary filter screen. The estimated pinion gear state can be displayed on a human-machine interface. If the state indicates degradation, the maintenance team can be notified. This enables predictive maintenance, reduces unplanned downtime, and extends the service life of the pinion gear. Because the pinion gear is a critical component, its reliable monitoring has a direct impact on the availability of the filtration system.
11. Conclusion
I have presented an automatic monitoring method for the operating state of a pinion gear in a drum-type rotary filter screen drive under multi-source data fusion. I fused vibration signals and current signals into a state-space model, where the pinion gear state is treated as a latent variable. I used the Kalman filter to recursively predict and update the pinion gear state from real-time multi-source data. The experimental results show that the residuals of the pinion gear state detection are mainly concentrated in the interval \([-0.5, 0.5]\), which indicates high reliability. Compared with single-source and reference methods, my method achieves a narrower residual distribution and lower RMSE and MAE. The method is computationally efficient, interpretable, and suitable for industrial implementation. Future work will focus on nonlinear filtering, additional data sources, and online parameter adaptation to further enhance the automatic monitoring of the pinion gear.
In summary, the pinion gear is a key element in the rotary filter screen drive, and its condition must be monitored continuously. The multi-source fusion framework that I developed provides a systematic way to combine vibration and current information. The state-space model provides a mathematical foundation for the fusion, and the Kalman filter provides a practical algorithm for real-time estimation. The residual analysis confirms that the pinion gear state can be estimated accurately. I believe that this approach can be extended to other rotating machinery and can contribute to the development of intelligent maintenance systems for industrial equipment.