A New Multisensor Data‐Level Fusion Method for Deep Learning–Based Fault Diagnosis of Rotating Machines: Considering Varying Sampling Frequencies and Different Sensor Mounting Directions in Vibration Signal Analysis

Abstract

Vibration signals are the most widely used condition monitoring data in deep learning–based fault diagnosis for rotating machines. However, relying solely on data from a single vibration sensor often limits the diagnostic accuracy of the diagnosis models. To overcome this challenge, researchers have explored multisensor data fusion techniques. Nevertheless, existing fusion approaches face challenges when dealing with variations in sampling frequencies and different sensor mounting orientations. In this paper, therefore, we propose a new data‐level fusion method, compensated synchronized resampling and weighted averaging fusion (CSR‐WAF), to enhance the accuracy of deep learning–based fault diagnosis in rotating machines. In this method, the CSR component first synchronizes the sampling frequencies of vibration data and compensates for sensor orientation. Subsequently, the WAF technique fuses the multisensor vibration data. The fused data are then processed using a one‐dimensional convolutional neural network (1DCNN) for fault diagnosis. Experiments conducted using motor bearing vibration signals sampled at 12 and 48 kHz show that the proposed CSR‐WAF‐1DCNN method achieves an accuracy of 99.87%. Furthermore, the proposed method is applied to gearbox fault diagnosis, accounting for different sensor mounting directions, and achieves an accuracy of 97.91%. These results confirm the reliable performance and practical applicability of CSR‐WAF‐1DCNN across diverse data acquisition scenarios.

1. Introduction

Rotating machines are essential components in many industrial sectors, including manufacturing, power generation, aviation, and transportation. They constitute a significant proportion—over 90%—of industrial machines [1]. Given that critical components within these machines often function under harsh and challenging circumstances for extended periods, rotating machines are susceptible to mechanical faults [2]. These faults can lead to unforeseen downtime, expensive repairs, and potential hazards [3]. Therefore, it is critical to continuously monitor the condition of rotating machines and accurately diagnose any faults to ensure the safety and optimal performance of industrial systems.

Over the years, industries and large institutions have increasingly adopted data-driven fault diagnosis methods to provide consistent support for proactive maintenance and minimize the economic and operational risks associated with unexpected failures [4]. In particular, intelligent fault diagnosis methods based on time-series vibration signals have gained significant popularity in this context due to several compelling advantages. First, vibration-based diagnosis offers a versatile and effective means of detecting various types of faults, whether they are distributed or localized [5]. This versatility enables the identification of diverse fault patterns within machinery. Second, the analysis of vibration signals offers crucial insights into the specific location of damage, allowing us to pinpoint the precise area where the fault is occurring. This capability facilitates targeted maintenance actions, ensuring that repairs are focused and efficient. Third, the ability to compare failure rates allows for a comprehensive evaluation of the overall health condition of the target machine. Analyzing vibration data over time allows for the assessment of degradation and failure rates, enabling proactive maintenance planning and resource allocation. As a result, vibration signals have become a prominent choice for deep learning–based fault diagnosis methods [6]. Janssens et al. [7] developed a feature learning model utilizing a convolutional neural network (CNN) for condition monitoring and fault finding in rotating equipment. Abdeljaber et al. [8] introduced an adaptive method based on a one-dimensional convolutional neural network (1DCNN) for structural damage detection. Similarly, Guo et al. [9] presented a hierarchical learning rate–adaptive deep CNN for bearing fault detection. Li et al. [10] also developed a fault diagnosis approach based on a 1DCNN, eliminating the need for preprocessing raw data. Zhu et al. [11] presented a CNN based on a capsule network for bearing fault diagnosis, which demonstrated strong generalizability. Yu et al. [12] introduced a hierarchical algorithm for bearing fault diagnosis based on stacked long short-term memory (LSTM), while Shi et al. [13] presented a fault diagnosis method employing sliding window stacked denoising autoencoders and LSTM models. This method effectively extracts timing relationships between data cycles using LSTM, thereby aiding in early fault detection. Similarly, Qiao et al. [14] combined CNN with LSTM for fault diagnosis under challenging conditions, such as strong noise and variable loads, resulting in improved fault diagnosis performance. Collectively, these studies show the significant advancements made in fault diagnosis through the utilization of vibration signals. However, relying solely on vibration data from a single sensor has limitations in accurately capturing the complete health condition of complex rotating systems under dynamic conditions, which can result in inaccurate fault diagnosis. These limitations arise due to factors such as the placement of the sensor, the direction of vibration, and the specific component being monitored. Moreover, signals from a single sensor are susceptible to external interference, which can further contribute to inaccurate diagnosis results.

Most recently, researchers have shifted their focus toward utilizing multiple vibration sensors, leading to the emergence of multisensor data fusion methods. The fusion of data from multiple sensors holds promise for providing extensive operational insights and improving the visibility of fault indications. According to the literature [15], these fusion methods can be categorized into data-level, feature-level, and decision-level fusion. Among these levels, data-level fusion methods are commonly employed to integrate raw homogeneous data obtained from vibration sensors. These fusion methods offer advantages such as preserving more information and requiring less expertise than other levels [16]. Consequently, various techniques have been employed for data-level fusion to enhance accuracy. For instance, the Kalman filtering has been utilized for diagnosing faults in wind turbine planetary gearboxes [17]. Principal component analysis has been employed for bearing fault detection and diagnosis [18], while independent component analysis has been applied for gearbox fault diagnosis [19]. Jing et al. [20] directly fused data from multiple sensors to construct a deep network for fault diagnosis in planetary gearboxes. Another study [21] presented a two-stage compressed sensing method to compress triaxial vibration signals for rotating equipment fault diagnosis. The studies referenced previously have collectively shown the effectiveness of fusing raw vibration data at the data level to improve the accuracy of fault diagnosis, overcoming the limitations of approaches relying on a single sensor.

However, many existing data-level fusion methods often neglect the challenges related to varying sampling frequencies and sensor mounting orientations when analyzing vibration signals for fault diagnosis in rotating machines. In real-world situations, these machines operate with nonstationary signals and at varying speeds and exhibit different vibration characteristics. This variability presents significant challenges when combining data from multiple vibration sensors, as the signals may have different sampling frequencies and characteristic information. To address these challenges, several techniques have been developed. One such method is the synchroextracting frequency synchronous chirplet transform, which focuses on discerning time-varying instantaneous frequency information in signals, thereby aiding in accurate fault diagnosis [22]. In addition, a new averaging method has been proposed to extract periodic components without an external reference, enhancing the adaptability of time synchronous averaging for fault diagnosis [23]. Moreover, a resampling scheme has been devised to balance diagnostic sample sets and enhance feature learning richness for effective fault diagnosis in industrial machinery [24]. These methods collectively help to improve fault diagnosis accuracy in rotating machinery by addressing challenges related to varying speeds and nonstationary signals. Furthermore, the mounting orientations of vibration sensors can differ, leading to variations in coordinate systems and further complicating the fusion process. Solely on sensors with a single direction may not always be sufficiently sensitive to detect faults that exhibit position variability. Some studies have made efforts to consider sensor mounting directions in fusion methods. Chen et al. [25] investigated various fusion methods at the input layer of a CNN, considering both horizontal and vertical vibration data. Yang et al. [26] presented a CNN-based approach that fused multidirectional and multisource signals. Another study [27] focused on multiaxis vibration signal fusion using a deep residual network for fault diagnosis in rotating machinery and achieved good diagnostic accuracy. However, a limitation of these studies is the lack of direct measurement of fusion quality using appropriate metrics. Instead, the evaluation often relies solely on diagnostic performance metrics, neglecting the importance of assessing fusion quality independently to ensure reliable and robust fault diagnosis. Moreover, the impact of fusion quality on overall diagnostic accuracy has not been examined in existing studies. Understanding the relationship between fusion quality and diagnostic performance is crucial for developing reliable and effective fusion methods.

In this paper, we propose a new multisensor data-level fusion method, compensated synchronized resampling and weighted averaging fusion (CSR-WAF), in combination with a 1DCNN. The CSR synchronizes the sampling frequencies of vibration data and compensates for sensor orientation based on the provided orientation information. Subsequently, the weight averaging fusion (WAF) technique is employed to fuse the vibration data obtained from multiple sensors. The fused signals are then processed through the 1DCNN for fault diagnosis of rotating systems operating under dynamic conditions. The efficacy of the CSR-WAF-1DCNN is validated using motor bearing and gearbox vibration datasets, and the results prove its efficacy.

The main contributions of this work can be summarized as follows:

• 1.

  A new multisensor data-level fusion method, CSR-WAF, is proposed for fusing time-domain vibration signals from multiple sensors. This method addresses the research question of how to effectively fuse information from multiple sensors with varied sampling frequencies and different sensor mounting directions.

• 2.

  The quality of the fusion is evaluated using the cross-correlation metric. This evaluation contributes to the understanding of how to measure the quality of fused vibration data.

• 3.

  We compare the diagnostic accuracy of the CSR-WAF-1DCNN using both single-sensor data and fused data. This comparison facilitates an evaluation of how multisensor vibration data fusion enhances the accuracy of fault diagnosis in rotating machines.

The rest of the paper is organized as follows: Section 2 presents a comprehensive description of the proposed CSR-WAF-1DCNN method. The validation and experimental results of the proposed method are presented in Section 3. Section 4 concludes the paper with final remarks.

2. Details of the Proposed CSR-WAF-1DCNN Method

In this section, the proposed CSR-WAF-1DCNN method for improved fault diagnosis of rotating machines is presented. This method incorporates the CSR-WAF technique to synchronize, resample, and fuse vibration data acquired from multiple sensors, which inherently consist of time series. The fused vibration data are then input into the 1DCNN model for fault diagnosis. The specific details are described as follows.

2.1. Synchronized Resampling

In industrial applications, rotating machines often operate at varying speeds and exhibit different vibration characteristics. This variability poses challenges when combining data from multiple vibration sensors for fault diagnosis, as the signals may have different sampling frequencies and signal characteristics. To address this issue, synchronized sampling (SR) is employed in this study.

2.1.1. Rationale Behind SR

The adoption of SR is not arbitrary but is grounded in both signal processing theory and the requirements of data-driven fault diagnosis systems. In multisensor systems, vibration signals are often recorded at different sampling frequencies and with varying orientations of the sensors on the machine structure. These differences lead to temporal misalignment and spectral inconsistency, which can significantly degrade the performance of any subsequent data fusion or classification model. SR addresses these issues by aligning all sensor signals to a common sampling frequency and time base, thereby enabling coherent analysis and feature extraction [22].

From a time-domain perspective, it is essential that all sensors capture fault-induced transients and events at the same time instances. Without synchronization, corresponding features may appear at different time points across sensors, making it difficult to correlate their behavior or detect patterns that are indicative of specific mechanical faults. By interpolating and resampling the sensor signals to a shared temporal framework, SR ensures accurate time alignment, which is a prerequisite for any meaningful fusion of multisensor data. Moreover, input consistency is critical in deep learning models [28]. Most neural network architectures, especially CNNs, expect inputs to be of fixed size and structure. Signals that differ in length, resolution, or alignment introduce noise into the training process and can reduce model accuracy [29]. SR ensures that all sensor data conform to a uniform format, thereby enhancing the learnability and generalization capability of the model.

Furthermore, SR improves the physical interpretability of vibration signals [2]. When sensor data are temporally and spectrally aligned, it becomes easier to interpret the physical phenomena underlying the signals, such as rotating imbalance, misalignment, or bearing defects. Finally, from an information-theoretic standpoint, synchronization maximizes the mutual information between sensor signals by reducing temporal uncertainty and redundancy. This leads to a more compact and informative representation of the dynamic behavior of the system, which is beneficial for both feature extraction and classification. Thus, the rationale for employing SR is a theoretically grounded and practically essential step in the proposed multisensor fusion framework. This approach enables temporal and spectral alignment, supports deep learning requirements, enhances physical interpretability, and improves the overall robustness of the fault diagnosis process.

2.1.2. Steps in Synchronized Resampling Process

The synchronized resampling process consists of two main steps: interpolation and resampling. During the interpolation process, missing data points in the original vibration signals are estimated via interpolation techniques such as linear interpolation, spline interpolation, or polynomial interpolation [30, 31]. The interpolation method relies on the specific needs and attributes of the data at hand. In this study, polynomial interpolation is chosen due to its flexibility in fitting curves to the data, its ability to produce smooth representations of missing data, and its high accuracy in estimating missing data points, particularly when the underlying relationships in the data exhibit polynomial characteristics.

For n vibration sensors, labeled as Sensor 1, Sensor 2, …, Sensor n, with corresponding sampling frequencies fs₁, fs₂, …, fs_n, and mounting directions, the interpolation process follows these steps:

• 1.

  Each sensor’s data is interpolated to a common time base, denoted as t_common, to ensure alignment and synchronization across sensors. The appropriate interpolation method is applied for each sensor, resulting in interpolated data: x_{1_interp}(t_common), x_{2_interp}(t_common), …, x_{n_interp}(t_common).

• 2.

  Polynomial interpolation is used to accurately fit a given set of data points. The Lagrange polynomial, which is a commonly used method, is employed. This polynomial function passes through the given data points exactly. Given n + 1 data points (x₀, y₀), (x₁, y₁), …, (x_n, y_n), where x₀ < x₁ < ⋯<x_n; the Lagrange polynomial is the n-th degree polynomial that passes through each of these points. It can be expressed as follows:

1 $\begin{array}{}y=\Sigma \left(y_1\ast \Pi \left(\frac{x-x_j}{x_i-x_j}\right)\right).\end{array}$

Here, y represents the interpolated value, x denotes the common time base, and (x_i, y_i) represents the original data points for a specific sensor. The summation is performed for i ranging from 0 to n, and the product is evaluated for j ranging from 0 to n, excluding i.

By evaluating this Lagrange polynomial at the common time base, t_common, the interpolated data for each sensor are obtained: $x_{1_{\text{interp}}}\left(t_{\text{common}}\right),$$x_{2_{\text{interp}}}\left(t_{\text{common}}\right)$, …, $x_{n_{\text{interp}}}\left(t_{\text{common}}\right)$. The interpolation step ensures that the data from different sensors are aligned to a common time base, allowing for accurate fusion and further analysis in the fault diagnosis process.

Once the data are interpolated to a common time base, they undergo resampling to ensure uniformly spaced data points and synchronization across all sensors [32]. Resampling involves selecting a target sampling frequency, fs_target and adjusting the data accordingly. For each sensor, the resampling factor, k, is calculated based on the ratio of the target sampling frequency (fs_target) to the original sampling frequency (fs₁, fs₂, …, fs_n). This factor aligns individual vibration signals to the common sampling frequency, which is crucial for the effective fusion and analysis of vibration data from rotating machinery. The resampling process involves adjusting the data to a common time base and sampling frequency, ensuring proper alignment, and facilitating comprehensive data analysis [33].

For example, for Sensor 1, the resampled data $x_{1_{\text{resampled}}}\left(t_{\text{common}}\right)$ are obtained by multiplying the interpolated data, x_{1_interp}(t_common), by the resampling factor, k₁. The resampling factor for Sensor 1 is calculated as follows: 2 $\begin{array}{}{k}_1=\frac{{fs}_{\text{target}}}{{fs}_1}.\end{array}$

Similarly, the resampled data for other sensors, such as Sensor 2 ($x_{2_{\text{resampled}}}\left(t_{\text{common}}\right)$) and Sensor 3 ($x_{3_{\text{resampled}}}\left(t_{\text{common}}\right)$), can be obtained using their respective resampling factors (k₂, k₃, …).

By applying resampling to each vibration signal using the appropriate resampling factors, the data from all sensors are aligned to a common time base and sampling frequency. This process ensures proper alignment and facilitates comprehensive data analysis, enabling accurate fault diagnosis and ensuring optimal operational safety.

2.2. Alignment of Sensor Mounting Directions

In the data fusion process, aligning the sensor mounting directions is crucial. This step becomes particularly important when the sensor data have been acquired in different directions. The orientation or mounting directions of sensors can vary due to various factors such as machine design, installation constraints, or specific measurement requirements. Aligning the sensor mounting directions is essential to ensure accurate fusion and analysis of the acquired data.

To achieve alignment, the CSR-WAF method employs coordinate transformations. Specifically, coordinate conversion techniques are utilized as part of the compensation process. These transformations consider the known or estimated mounting directions of the sensors and are aimed at aligning the measurements with a common reference frame. Consider the scenario where n vibration sensors are mounted in different directions, represented by the angles θ₁, θ₂, …, θ_n. The mathematical steps involved in aligning the sensor data using rotation matrices can be summarized as follows:

• 1.

  A reference coordinate system is established as a standard frame of reference. This frame serves as a basis for aligning the sensor measurements. The reference frame can be defined using Cartesian coordinates (x, y, and z).

• 2.

  The mounting direction of each sensor is determined based on its physical orientation concerning the reference frame. This information can be provided by the sensor manufacturer or measured during the installation process. The mounting direction of the i-th sensor as θ_i.

• 3.

  Coordinate conversion techniques, such as rotation matrices, are applied to adjust the acquired sensor data based on the determined mounting directions. A rotation matrix R(θ_i) is used to rotate the sensor data from the i-th sensor to align it with the reference frame. The rotation matrix R(θ_i) is a mathematical representation of the transformation needed to adjust the sensor data. The specific form of the rotation matrix depends on the coordinate system being used.

• 4.

  The acquired sensor data are adjusted using the results of the coordinate conversion. Each measurement from the i-th sensor, denoted as D_i, is multiplied by the corresponding rotation matrix R(θ_i) to bring the measurements to a consistent reference frame. The adjusted sensor data as D_{i_adjusted} can be calculated as follows:

3 $\begin{array}{}{D}_{i\_\text{adjusted}}=R\left({\theta }_i\right)\ast D_i.\end{array}$

By applying this equation to each sensor, the acquired sensor data can be adjusted using the corresponding rotation matrix, aligning them with the reference frame. This alignment enables accurate fusion and analysis of the sensor data.

2.3. Weighted Averaging

Following the alignment of the sensor mounting directions, the next step is to perform weighted averaging. This technique combines the information from multiple sensors, taking into account factors such as reliability, sensor quality, or relevance to the fault diagnosis task [33, 34]. The weight of each sensor reflects its importance in the fused output, and the weights can be adjusted based on specific requirements or sensor characteristics.

For n vibration sensors, the weights assigned to each sensor are denoted as w₁, w₂, …, w_n. These weights reflect the importance of each sensor’s contribution to the fused output and can be adjusted based on specific requirements or sensor characteristics. The weighted averaging process involves multiplying each sensor’s resampled data by its corresponding weight and summing them together. Mathematically, the fused output signal, denoted as x_fused(t_common), can be calculated as follows: 4 $\begin{array}{}{x}_{\text{fused}}\left(t_{\text{common}}\right)=w_1\ast x_{1\_\text{resampled}}\left(t_{\text{common}}\right)+w_2\ast x_{2\_\text{resampled}}\left(t_{\text{common}}\right)+\cdots +w_n\ast x_{n\_\text{resampled}}\left(t_{\text{common}}\right).\end{array}$

In this equation, x_fused(t_common) represents the fused output signal at the common time base t_common. $x_{i_{\text{resampled}}}\left(t_{\text{common}}\right)$ corresponds to the resampled data from the i-th sensor. w_i denotes the weight assigned to the i-th sensor’s resampled data.

This equation illustrates how the weighted averaging method combines the information from each sensor, considering their contributions based on the assigned weights. By adjusting the weights based on specific requirements or sensor characteristics, the fused output can be tailored to emphasize information from sensors that are more reliable, accurate, or relevant to the fault diagnosis task.

2.4. 1DCNN

The fused time-domain vibration data are sent to the 1DCNN model for fault diagnosis. The network architecture comprises 1D convolutional layers, 1D max-pooling layers, a flatten layer, fully connected layers, and an output layer, as shown in Figure 1. The first convolutional layer applies a set of filters to the input data and uses the ReLU activation function. This is followed by the first max-pooling layer, which reduces the spatial dimensions of the feature maps. The second convolutional layer also applies a set of filters to the output of the previous layer and uses the ReLU activation function. This is followed by the second max-pooling layer, further reducing the spatial dimensions of the feature maps. The flatten layer then converts the output of the second max-pooling layer into a 1D feature vector. This 1D feature vector is then passed through two fully connected layers, both of which use the ReLU activation function. Finally, the softmax classifier is used as the output layer to produce the final classification predictions. Detailed descriptions of each layer can be found in our previous work [35].

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The network parameters of the constructed 1DCNN model are determined through a grid search with a 10-fold cross-validation strategy. This approach is commonly used to ensure the selected parameters generalize well to unseen data. In this comprehensive approach, the available input data is first divided into 10 equal-sized subsets or folds, as shown in Figure 2. The model is then trained and evaluated 10 times, each time using 9 folds for training and the remaining fold for validation. This allowed the model to be tested on different data subsets, which can mitigate the risk of overfitting. The grid search component systematically explores a predefined range of parameter values and evaluates the performance of the model for each combination. By combining the grid search with the 10-fold cross-validation, the most optimal parameters are identified that can maximize the robustness and reliability of the model on new, unseen data. This optimization approach ensures that the network parameters presented in Table 1 are well-suited for the classification task at hand.

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Table 1 Network parameters of the 1DCNN model.

Layer type	Filters	Kernel size	Stride	Pooling size	Neurons	Activation function
Conv1D_1	32	3	1	—	—	ReLU
MaxPooling1D_1	—	—	—	2	—	—
Conv1D_2	64	3	1	—	—	ReLU
MaxPooling1D_2	—	—	—	2	—	—
Flatten	—	—	—	—	—	—
Fully connected_1	—	—	—	—	128	ReLU
Fully connected_2	—	—	—	—	64	ReLU
Output layer	—	—	—	—	Na	Softmax

In the training process of the 1DCNN model, the cross-entropy loss function is utilized. This loss function quantifies the disparity between the predicted output of the network (y_pred) and the true category label of the sample [37]. The cross-entropy loss function is mathematically expressed as follows: 5 $\begin{array}{}\text{Loss}=-\frac{1}{S}\sum _{S=1}^S\left[{y_{\text{true}}}^{\left(S\right)}\log {y_{\text{pred}}}^{\left(S\right)}+\left(1-{y_{\text{true}}}^{\left(S\right)}\right)\log \left(1-{y_{\text{pred}}}^{\left(S\right)}\right)\right]+L_2.\end{array}$

In this equation, S is the number of samples, and L₂ is the regularization term used to mitigate the issue of overfitting by penalizing large weights in the network [38]. The regularization term is calculated as follows: 6 $\begin{array}{}{L}_2=\frac{\lambda }{2S}\sum _{l=1}^L{‖w^l‖}^2,{‖w^l‖}^2=\sum _{j=1}^{N_l}\sum _{k=1}^{N_{l+1}}{\left({w^l}_{k.j}\right)}^2,\end{array}$ where ${w^l}_{k.j}$ is the weight between the k-th neuron in the l-th layer and the j-th neuron in the next layer. λ is the regularization factor.

During the training process, the Adam optimizer is employed to optimize the weights and biases of the 1DCNN model. The Adam optimizer updates these parameters by computing the first and second moment estimates, represented as s_t and r_t, respectively. The update process at time step t can be expressed as follows: 7 $\begin{array}{}{s}_t={\rho }_1{s}_{t-1}+\left(1-{\rho }_1\right)g_t,\end{array}$ 8 $\begin{array}{}{r}_t={\rho }_2{r}_{t-1}+\left(1-{\rho }_2\right)g_t,\end{array}$ 9 $\begin{array}{}\stackrel{\wedge }{s_t}=\frac{s_t}{1-{{\rho }_2}^t},\end{array}$ 10 $\begin{array}{}\stackrel{\wedge }{r_t}=\frac{r_t}{1-{{\rho }_2}^t},\end{array}$ 11 $\begin{array}{}{w}_t=w_{t-1}-\eta \frac{\stackrel{\wedge }{s_t}}{\sqrt{\stackrel{\wedge }{r_t}-\delta }},\end{array}$ where g_t is the gradient, ρ₁ and ρ₂ are the exponential decay rates, δ is a small constant, and η is the learning rate.

The parameter optimization process involves adjusting the learning rate, batch size, and other hyperparameters to enhance model performance. In this study, the learning rate is configured to 0.001, and the batch size is set to 64. The batch size determines the number of samples processed in each iteration. The number of epochs is specified as 100, indicating the number of complete passes through the training dataset during the training process. These parameters are consistent with those used in our previously published work [39].

2.5. Fault Diagnosis Procedures

The CSR-WAF-1DCNN approach presented in this study is designed to improve the fault diagnosis accuracy of rotating machines. The method utilizes multisensor time-series vibration signals as input data and incorporates data-level fusion through the CSR-WAF method. The fused data are subsequently input into a 1DCNN model to facilitate accurate fault diagnosis. The detailed procedures for fault diagnosis using this method are as follows:

Step 1: Acquire time-series vibration signals from multiple sensors installed on the rotating machines, considering varying sampling frequencies and sensor mounting directions for different fault types.

Step 2: The acquired vibration data are segmented into samples via time window sliding segmentation.

Step 3: Perform data-level fusion using the CSR-WAF method to combine the vibration signals from different sensors. This fusion process yields a fused data representation that encapsulates the combined information from all the sensors.

Step 4: Evaluate the quality of the fused data using the cross-correlation metric. If the quality is deemed sufficient, move on to the next step. If not, go back to Step 3 and repeat the fusion process.

Step 5: Split the high-quality fused dataset into training, validation, and testing sets. This partition is essential for training the 1DCNN model and assessing its performance.

Step 6: Initialize the network parameters and select the optimal batch size and learning rate.

Step 7: Train the 1DCNN model using the training set, optimizing the model parameters with the cross-entropy loss function and the Adam optimizer. This training process enables the model to learn and extract pertinent features from the fused vibration data.

Step 8: Evaluate the diagnostic performance of the trained 1DCNN model on the testing set using metrics such as classification accuracy and confusion matrix. t-SNE visualization is also employed to analyze the feature learning ability of the model. If the performance is satisfactory, proceed to the next step. Otherwise, return to Step 6 and further optimize the model.

Step 9: Compare the accuracy rates obtained from the single-sensor data with those from the fused data to evaluate the improvement achieved through fusion. A significant improvement in accuracy is observed with the fused data, which indicates successful enhancement of fault diagnosis accuracy using the CSR-WAF fusion method. If no notable difference or improvement is observed, further investigation is required to understand the underlying reasons.

Step 10: Apply the trained 1DCNN model to new, unseen vibration signals from rotating machines for real-time or near-real-time fault diagnosis. The model predicts fault conditions or types based on the learned patterns and features, allowing timely detection and diagnosis of faults in rotating machines.

2.6. Evaluation Metrics

2.6.1. Fusion Quality

To assess the quality of fusion in this study, a cross-correlation metric is utilized. Cross-correlation evaluates the degree of similarity between two signals by analyzing their temporal alignment and relationship. It can provide insights into the alignment and consistency of signals in the time form. The cross-correlation between the fused vibration data (F) and the individual sensor data (X and Y) can be calculated as follows: 12 $\begin{array}{}\text{Cross}-\text{correlation}\left(F,X\right)=\frac{1}{M}\sum F_i{X}_i,\end{array}$ 13 $\begin{array}{}\text{Cross}-\text{correlation}\left(F,Y\right)=\frac{1}{M}\sum F_i{Y}_i,\end{array}$ where F_i, X_i, and Y_i represent the individual samples of the fused vibration data, the individual sensor data X, and the individual sensor data Y, respectively, at each time index i. M is the number of samples.

A high cross-correlation value (i.e., close to 1) signifies strong similarity and alignment between the fused data and the individual sensor data. On the other hand, a low cross-correlation value (i.e., close to 0) suggests a weak or no relationship between the signals.

2.6.2. Diagnosis Performance

In this study, diagnostic performance metrics, such as the classification accuracy, confusion matrix, and t-SNE, were employed to assess the efficacy of the model. The classification accuracy assesses the overall accuracy of the predicted data compared to the actual data. This measure is calculated as follows: 14 $\begin{array}{}\text{Accuracy}\left(\%\right)=\frac{\left|\text{TP}\right|+\left|\text{TN}\right|}{\left|\text{TP}\right|+\left|\text{FP}\right|+\left|\text{FN}\right|+\left|\text{TN}\right|},\end{array}$ where |TP| is the true positive, |TN| is the true negative, |FP| is the false positive, and |FN| is the false negative.

The confusion matrix is a powerful visualization tool that provides a detailed breakdown of a model’s predictive performance. It takes the form of a square matrix, where the dimensions correspond to the number of target classes in the classification task. This matrix categorizes the predictions of the model into four distinct groups: true positives, true negatives, false positives, and false negatives. This comprehensive representation allows for a deep understanding of the strengths, weaknesses, and overall classification accuracy of the model across the different target classes. Figure 3 shows a typical binary confusion matrix.

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Furthermore, t-SNE can be employed as a nonlinear dimensionality reduction algorithm to visualize the extracted features. By mapping high-dimensional feature vectors into a lower dimensional space, t-SNE facilitates the identification of clustering or grouping patterns among the features. This visualization technique aids in understanding the ability of the model to learn meaningful representations of the data.

3. Experimental Validation and Analysis

In this section, the efficacy of the presented CSR-WAF-1DCNN method is validated through two case studies: motor bearing and gearbox fault diagnosis. The datasets utilized in each case study are described, followed by a comprehensive analysis of the results and insightful discussion.

3.1. Case I: Motor Bearing Fault Diagnosis

3.1.1. Dataset Description

For the first case study dataset, motor bearing vibration data from the Case Western Reserve University (CWRU) test rig [41] are utilized. This publicly available dataset includes recordings from four different motor loads: 0, 1, 2, and 3 horsepower (hp), corresponding to rotational speeds of 1730, 1750, 1772, and 1797 rpm, respectively. The motor housing was equipped with two accelerometers, one at the drive end and the other at the fan end, as shown in Figure 4. The focus of this recent study was on drive end bearing vibration data, which were collected at two distinct sampling frequencies: 12 and 48 kHz. Each loading condition comprised data collected under four health conditions: normal, outer race fault, inner race fault, and ball fault. Furthermore, each fault type had three different damage diameters: 0.007, 0.014, and 0.021 in. Consequently, each loading condition encompassed nine fault states, along with one health state, resulting in a total of 10 states within each dataset.

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To account for variable working conditions in practical scenarios, a combined load dataset is considered in this study. The dataset includes 10 different health conditions, labeled from 0 to 9, as shown in Table 2. The data are segmented into multiple samples using a sliding time window approach. Each data sample consists of 2048 data points, with a stride of 64 employed during the segmentation process. For this study, a total of 6600 data samples are randomly selected as the training dataset, 70 samples are allocated for the validation set, and 250 samples are chosen as the testing dataset. The chosen number of samples ensures consistency with other studies in the field [42, 43].

Table 2 Detailed description of the CWRU motor bearing dataset.

Sampling frequency	Load	Health condition	Damage diameter (inch)	Training/validation/testing samples	Label
12 and 48 kHz	0~3 hp	Normal	0	6600/70/250	0
Ball fault	0.007	1
Ball fault	0.014	2
Ball fault	0.021	3
Inner race fault	0.007	4
Inner race fault	0.014	5
Inner race fault	0.021	6
Outer race fault	0.007	7
Outer race fault	0.014	8
Outer race fault	0.021	9

3.1.2. Result Analysis

In this study, the acquired vibration signals at a sampling frequency of 48 kHz were resampled to a lower frequency of 12 kHz using the synchronized resampling technique. This resampling process is aimed at aligning the data from different sensors to a common time base, facilitating effective fusion and analysis. The WAF method was then employed to fuse the vibration data, followed by the evaluation of the quality of the fused data using the cross-correlation metric.

The WAF method assigned weights to the drive-end and fan-end sensors based on their physical proximity to the fault source and empirical diagnostic relevance. The drive-end accelerometer, mounted closer to the bearing, captures stronger fault-induced vibrations due to reduced signal attenuation. Empirical analysis of the CWRU dataset confirmed that the drive-end sensor contributed 70% of the discriminative features for bearing fault detection. In contrast, the fan-end sensor, farther from the fault location, contributed 30% of complementary information. Thus, fixed weights of w_drive-end = 070 and w_fan-end = 0.30 were applied to prioritize the drive-end signal while retaining auxiliary data from the fan end.

After resampling the 48 kHz data to 12 kHz for synchronization, the fused signal achieved cross-correlation values of 0.97 with the drive-end sensor and 0.92 with the fan-end sensor (see Table 3), confirming effective integration of both signals. The fused data improved diagnostic accuracy to 99.87%, surpassing single-sensor results (drive end: 96.75%, fan end: 93.20%). This 3.12% accuracy gain demonstrates the ability of the WAF method to amplify critical fault signatures through sensor geometry-aware weighting.

Table 3 Case I: Motor bearing diagnostic results.

Sensor position	Diagnosis accuracy	Assigned weight	Cross-correlation
Drive end (12 kHz)	96.75%	0.70	0.97
Fan end (48 kHz)	93.20%	0.30	0.92
Fused data	99.87%	—	—

Figure 5 presents the plots of training and validation accuracies as well as losses during the training process of the CSR-WAF-1DCNN method on the fused vibration data. The training accuracy plot demonstrates a gradual improvement in accuracy as the model learns from the training data over successive epochs. Similarly, the validation accuracy plot depicts the performance of the model on a separate validation dataset. This plot serves as a measure of the generalization ability of the model. The loss plots for both training and validation reflect the optimization process, with the loss decreasing as the model better aligns the predicted fault labels with the actual labels. Remarkably, from the observed stability in the training process after approximately 30 epochs on the fused data, it can be inferred that the model has effectively learned the necessary patterns and features for accurate fault classification. This finding supports the effectiveness of the CSR-WAF-1DCNN method in training on fused vibration data and achieving stable performance.

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Figure 6 shows the confusion matrix resulting from the first trial of the proposed CSR-WAF-1DCNN method on the fused vibration data. This matrix provides a comprehensive overview of the model’s classification performance by comparing true labels (horizontal coordinates) with predicted labels (vertical coordinates). The results reveal that the CSR-WAF-1DCNN identified 100% of seven health states, indicating its ability to accurately classify the majority of the samples. However, misclassifications occurred in Classes 1, 2, and 3, which correspond to ball faults with damage diameters of 0.007 ^″, 0.014 ^″, and 0.021 ^″, respectively. The potential reasons for these misclassifications include the overlapping vibration patterns associated with increasing damage severity; larger ball faults can exhibit nonlinear behaviors that may confuse the model with smaller faults. Despite these misclassifications, the average testing accuracy remains at an acceptable level, indicating that the proposed method maintains a consistently high level of accuracy overall. These results provide strong evidence for the effectiveness and robustness of the CSR-WAF-1DCNN method in handling complex fault conditions within the rolling bearing dataset.

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The t-SNE visualizations in Figure 7 provide insights into the feature extraction capabilities of the CSR-WAF-1DCNN model on fused vibration data. Starting from the scatter plot after the first Conv1D layer (Figure 7a), it becomes evident that the convolutional layers can extract discriminative features that aid in distinguishing between different fault categories compared to the raw time-domain signal. This improvement in feature representation continues as the layers progress. Figure 7b, which depicts the scatter plot after the second Conv1D layer, demonstrates further enhancement in feature separability. The extracted features enable better differentiation between fault categories, indicating the effectiveness of the convolutional layers in capturing relevant information. With respect to the first dense layer (Figure 7c), there is a noticeable improvement in feature discrimination. The scatter plot shows clearer clusters for each fault category, indicating that the dense layer has learned to transform the extracted features into a more discriminative representation. The scatter plot after the last dense layer (Figure 7d) reveals that the features have been projected into a lower-dimensional space with enhanced separability. This suggests that the dense layers have successfully learned high-level representations that enable effective fault category separation. Finally, the scatter plot at the output layer (Figure 7e) demonstrates a complete absence of overlap between fault categories. This indicates strong generalization capability and minimal misclassification, further validating the superior performance of the CSR-WAF-1DCNN method. The progressive improvement in feature separability throughout the layers demonstrates the capacity of the CSR-WAF-1DCNN to distinguish between different fault categories. These findings strongly support the superior performance of the method and validate its feature extraction capabilities and generalizability for fault diagnosis tasks.

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3.2. Case II: Gearbox Fault Diagnosis

To further evaluate the effectiveness of the presented CSR-WAF-1DCNN method, experiments were conducted using the MCC5-THU gearbox dataset [44]. The schematic of the test rig used in the experiment is shown in Figure 8. The collected data pose the challenge of different sensor mounting directions on fault diagnosis accuracy, where a triaxis accelerometer is employed to capture the three-axis vibration signals of the gearbox intermediate shaft. The data were acquired at a sampling rate of 12.8 kHz.

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The MCC5-THU gearbox dataset employed in this study encompasses various fault types, fault degrees, and working conditions. Detailed information about these aspects is presented in Table 4. Note that the dataset includes both single faults and compound faults, adding to its complexity and relevance for fault diagnosis research.

Table 4 Detailed description of the MCC5-THU gearbox dataset.

Fault type	Health condition	Speed (rpm)	Torque (Nm)	Label
Single fault	Healthy	1000, 2000, and 3000	10 and 20	0
Gear pitting	1
Gear wear	2
Miss teeth	3
Teeth crack	4
Compound fault	Teeth break and inner bearing fault	1000, 2000, and 3000	10 and 20	5
Teeth break and outer bearing fault	6

For the triaxis accelerometer, weights were assigned based on axis-specific sensitivity to fault types, derived from mechanical principles. Gear faults (e.g., pitting and cracks) generate dominant radial vibrations (x-axis), while bearing faults involve axial components (z-axis). Empirical analysis of the dataset revealed that the x-axis contributed 55% of discriminative features (e.g., gear mesh frequency harmonics), compared to 30% (y-axis) and 15% (z-axis). Fixed weights of w_x = 0.55, w_y = 0.30, and w_z = 0.15 were applied to emphasize the x-axis while retaining multidirectional information. These weights ensure that dominant gear fault signatures are emphasized, while auxiliary bearing-related vibrations (z-axis) and transverse components (y-axis) are incorporated at reduced contributions.

The fused gearbox data achieved a diagnostic accuracy of 97.91%, outperforming single-axis results (x-axis: 95.2%, y-axis: 88.3%, and z-axis: 82.1%). Cross-correlation values between the fused signal and individual axes were 0.96 (x), 0.85 (y), and 0.80 (z) (Table 5), validating the relevance of axis-specific weighting.

Table 5 Case II: Gearbox diagnostic results.

Axis	Diagnosis accuracy	Assigned weight	Cross-correlation
x	95.2%	0.55	0.96
y	88.3%	0.30	0.85
z	82.1%	0.15	0.80
Fused data	97.91%	—	—

To further analyze the diagnosis results, the training process of the CSR-WAF-1DCNN model was first evaluated, with 10 repetitions to account for randomness. Figure 9 shows the training accuracy and loss curves from a single experiment. The curves show that the proposed CSR-WAF-1DCNN model exhibited a consistent increase in training accuracy throughout the epochs, ultimately reaching 100%. Simultaneously, the training loss decreased rapidly in the initial epochs and then stabilized at a low value close to 0 after approximately 30 epochs. These observations indicate the effectiveness of the model in learning and extracting relevant information from the triaxial accelerometer data, considering sensor fusion for improved accuracy.

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Subsequently, the trained CSR-WAF-1DCNN model was evaluated on the testing samples, resulting in a diagnostic accuracy of 97.91%. Although a slight decrease in performance compared to the training accuracy was observed, this is expected as the model encounters unseen data during testing. However, it is important to note that the achieved accuracy demonstrates the effectiveness of the model in gearbox fault diagnosis using the MCC5-THU dataset. This result showcases the ability of the model, which incorporates various sensor mounting directions and sensor fusion, to accurately identify faults in rotating machines, even under diverse working conditions. Thus, the presented CSR-WAF-1DCNN method holds significant potential as a valuable tool for implementing preventive maintenance strategies in rotating machinery.

4. Conclusions

In this study, a new data-level fusion method, called CSR-WAF, is proposed to enhance the fault diagnosis accuracy of deep learning–based methods for rotating machines, considering discrepancies in sampling frequencies and sensor mounting directions. The CSR-WAF effectively synchronized varying sampling frequencies of vibration data and compensated for different sensor mounting directions. The fused vibration data were input into a 1DCNN for fault diagnosis. Based on the two experimental cases presented, the following main conclusions are drawn:

• 1.

  The CSR-WAF fusion method successfully fused the raw time-based vibration signals, as demonstrated by a cross-correlation metric approximation of 1. This highlights the effectiveness of the proposed data-level fusion method.

• 2.

  The CSR-WAF-1DCNN method achieved an impressive accuracy of 99.87% when evaluating motor bearing vibration signals sampled at different frequencies (i.e., 12 and 48 kHz). This improvement of 3% over the single-sensor approach showcases the superiority of multisensor data-based fault diagnosis in terms of accuracy.

• 3.

  The CSR-WAF-1DCNN method was also applied to gearbox fault diagnosis, considering various sensor mounting directions, resulting in an accuracy of 97.91%. This confirms the diagnostic performance of the CSR-WAF-1DCNN across diverse data acquisition settings.

In general, this study successfully addresses the limitations identified in previous studies by emphasizing the independent assessment of fusion quality and investigating its impact on diagnostic accuracy. The findings provide valuable insights and serve as a valuable resource for researchers, practitioners, and industrial professionals in the field of intelligent fault diagnosis in rotating systems. These insights hold practical implications for industries and large institutions that are consistently concerned with capacity adjustments and cost minimization to meet demand without delays or excessive resource utilization.

However, the focus of the present study was specifically on integrating homogeneous sensor data from multiple vibration sensors to enhance fault diagnosis in rotating machines. Future research endeavors will explore the integration of diverse signal types, such as vibration, temperature, and sound, to further improve the accuracy and reliability of deep learning–based intelligent fault diagnosis for rotating machines. By incorporating these diverse signal types, a comprehensive and robust approach to fault diagnosis can be developed. This approach holds great potential for advancing reliability and safety in the field of rotating machinery.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest

The authors declare no conflicts of interest.

Funding

No funding was received for this manuscript.

Acknowledgments

The authors gratefully acknowledge the Artificial Intelligence and Robotics Center of Excellence at Addis Ababa Science and Technology University, Ethiopia, for providing the space and equipment for this work.

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