Fault Diagnosis Model for Bearings under Multiple Operating Conditions Based on Feature Parameterization Weighting

Abstract

As a core component of automobile transmission, rolling bearings play a main role in the safety and reliability of vehicles. Existing diagnostic models often treat all features equally after feature extraction, without effectively distinguishing the importance of fault features, resulting in low accuracy and poor robustness in bearing fault diagnosis. To address this issue, a fault diagnosis model for bearings under multiple operating conditions based on feature parameterization weighting is proposed. The model utilizes a feature parameterization weighting module to categorize faults into two classes based on differences in means and implements different feature processing methods. The experimental results validate that the proposed feature parameterization weighting module effectively improves the diagnostic accuracy of the model by 8.95%. In terms of noise resistance, on two multi-condition datasets, the proposed diagnostic model achieves diagnostic accuracy of 98.79% and 98.36%. The diagnostic accuracy is improved by 15.7% and 22.48%, which indicates that the model has strong anti-noise ability.

1. Introduction

In recent years, electric vehicles have rapidly developed in terms of their zero emissions, low noise, and high efficiency [1]. As a primary tool and important carrier of road transportation, their stable operation and safety are of significant importance. Rolling bearings, as critical components of drive motors, play an essential role in the safety of electric vehicles. The timely monitoring of bearing conditions, detection of potential faults, and providing scientific decisions for maintenance are crucial scientific means to ensure smooth vehicle operation [2]. Bearing fault diagnosis methods have been widely researched over the past few decades [3,4]. Typically, signals are acquired from sensors, and sensitive features related to bearing faults are mined and extracted from these signals. Subsequently, feature selection and fusion are performed, followed by an evaluation of the bearing’s health status through a decision-making system [5].

The fault diagnosis techniques for rolling bearings mainly include methods based on vibration signals, acoustic signals, electromagnetic signals, and ultrasonic signals. Among these, vibration-based methods are more prevalent and can achieve early fault detection. In the field of bearing fault diagnosis, data-driven machine learning methods have gradually replaced the research approach based on mathematical and physical models [6]. In traditional machine learning-based diagnostic methods, fault features need to undergo two steps including manual feature extraction and feature selection. After collecting features such as the frequency domain, time domain, and time–frequency domain from the data, they are inputted into traditional machine learning algorithms for fault prediction and classification. For instance, Song D D et al. analyzed and extracted typical features from bearing raw signals in the time domain, followed by dimensionality reduction using PCA (Principal Component Analysis), and achieved high-precision diagnosis using the SVM (Support Vector Machine) algorithm [7]. Zhenhao Tang et al. utilized complete ensemble empirical mode decomposition to extract time-domain features of bearing fault signals, followed by two fast Fourier transforms to extract deep frequency domain information for training the diagnostic model [8].

With the advancement of science and technology, modern bearing fault data exhibit characteristics of large data volume and high data dimensionality. Traditional machine learning methods, because of their shallow network structures, suffer from poor nonlinear fitting capabilities, making it challenging to meet the diagnostic requirements of complex mechanical systems [9]. Consequently, bearing fault diagnosis methods based on deep learning have attracted researchers’ attention. In comparison with traditional machine learning diagnostic methods, deep learning diagnostic models have multi-layer network structures and powerful capabilities in adaptive feature learning and representation [10]. Therefore, they can typically achieve precise diagnosis of more complex bearing faults.

Convolutional neural networks (CNNs), as representative methods in the field of deep learning, have garnered significant attention from researchers. CNN-based diagnostic methods are generally categorized into two types as follows: 2D CNNs, which take images as input, and 1D CNNs, which take data as input.

In the realm of 2D CNNs, Ke Zhang et al. proposed a Multi-Modal Convolutional Neural Network (MMCNN), which effectively extracted rich and complementary fault features using multiple parallel convolutional layers. They then converted these features into two-dimensional grayscale images through Continuous Wavelet Transform (CWT), serving as input to MMCNN, thus enhancing the diagnostic accuracy of bearing faults under varying conditions [11]. Y Y Long et al. improved a Stacked Autoencoder (SAE) with a softmax classifier of a Backpropagation Neural Network (BPNN) to enhance one-dimensional vibration signals, which were subsequently transformed into two-dimensional images as training data for CNNs to achieve bearing fault diagnosis [12]. Tong Yu et al. encoded one-dimensional signals using the Gram Angle Difference Domain, serving as input data for 2D CNNs, and ultimately achieved a precise diagnosis of bearing faults [13]. While these methods convert fault data into two-dimensional feature images for training CNN models, thus leveraging the computational power of convolutional neural networks, the conversion from one-dimensional to two-dimensional data may lead to feature loss, resulting in lower robustness.

In the realm of 1D CNN diagnostic model research, Zhen Y Z et al. utilized convolutional neural networks for adaptive feature extraction and feature dimensionality reduction of bearing vibration signals. They replaced fully connected layers with global average pooling, achieving higher diagnostic accuracy [14]. Min Xia et al. inputted time and spatial information of raw signals from multiple sensors into CNNs, enabling the model to extract representative features from the data automatically, and empirically demonstrated the effectiveness of this method [15]. Duan Hao et al. introduced dilated convolutions to increase the receptive field of convolutional layers and incorporated attention mechanisms into the model to enhance its ability to extract key feature information. They proposed an end-to-end bearing fault diagnosis method with strong adaptability [16].

All these methods utilize raw data from bearing vibration signals as input for diagnostic models. However, because of factors such as measurement errors, the original data often contain noise, making it difficult for diagnostic models to extract effective fault features and resulting in lower diagnostic accuracy and poor robustness.

To achieve high-precision diagnosis of bearing faults while maintaining high robustness, this paper constructs a bearing fault diagnosis model based on 1D convolutional neural networks. We design a feature parameterization weighting module to extract fault features, thus enhancing the robustness of the diagnostic model. The original fault data undergo joint feature extraction using two different algorithms, serving as input to the convolutional neural network. The original bearing fault data are extracted by variational mode decomposition and empirical mode decomposition, and the extracted feature data are combined as the input of the diagnosis model. After the feature data are input into the diagnostic model, different operations are applied to the feature tensor by obtaining the mean value of its tensor weights, and according to the positive and negative values of the mean value, so as to realize the prominence of important features. The minor features are removed by linear activation function to reduce their impact on important features, so as to achieve accurate diagnosis and high robustness of the diagnostic model.

2. Basic Principles of Convolutional Neural Networks

CNNs are deep feedforward neural networks that perform repeated convolution and pooling operations on input signals using multiple layers of filters to extract features from faulty data automatically. Within the network, operations between adjacent layers are performed using local connections and weight sharing.

2.1. Convolutional Layer

The convolution operation is completed by the combination of convolutional kernels and strides. When input features pass through convolutional layers, the kernels extract features from a portion of the input features. Subsequently, the kernels move according to a certain stride until the entire input feature is traversed, generating corresponding feature maps. The characteristic of weight sharing in convolutional kernels determines that one kernel corresponds to only one output feature map. The depth of the output feature map is determined by the number of convolutional kernels.

$$y^k={\displaystyle \sum _{i=1}^{c^{i-1}}}{w}_{i,c}^k\ast x_i^{k-1}+b_i^k$$

(1)

In Equation (1), $x_i^{k-1}$ represents the output of the i-th channel in the k − 1-th layer. $c^{i-1}$ represents the c-th channel in the k − 1-th layer; $y^k$ denotes the output of the ‘k-th’ layer; $w_{i,c}^k$ represents the weight matrix of the convolutional kernel in the k-th layer; $b_i^k$ denotes the bias term; and * denotes the convolution operation.

2.2. Pooling Layer

In convolutional neural network models, the pooling layer primarily compresses feature dimensions to reduce the computational load on target features. During the forward propagation process, it performs downsampling operations only on target regions. By refining the features of input feature vectors, it effectively prevents overfitting. Typically, there are two types of pooling layers as follows: average pooling and max pooling, corresponding to the following formulas, respectively.

$$y^{l\left(i,j\right)}=\frac{1}{w}{\displaystyle \sum _{t=\left(j-1\right)w+1}^{jw}}{x}^{l\left(i,t\right)}$$

(2)

$$y^{l\left(i,j\right)}={\displaystyle \underset{\left(j-1\right)w+1<t<jw}{max}}{x}^{l\left(i,t\right)}$$

(3)

In Equations (2) and (3), $y^{l\left(i,j\right)}$ represents the output value of the $j$-th neuron in the $i$-th channel of layer $l$, $w$ denotes the width of the pooling kernel, and $x^{l\left(i,t\right)}$ represents the input value of the t-th neuron in the i-th channel of layer $l$.

2.3. Fully Connected Layer

The main role of the fully connected layer is to refine the feature tensor extracted through feature extraction. Before being input into the fully connected layer, features are typically flattened into a one-dimensional vector, processed by the fully connected layer, and then input into the classification layer to accomplish the intended classification task. The corresponding formula is as follows.

$$y^k={\displaystyle \sum _{i=1}^n}{w}_{ij}^{k-1}{x}^{k-1\left(i\right)}+b_j^{k-1}$$

(4)

In Equation (4), $y^k$ represents the output vector $y$ of the $k$-th fully connected layer, $n$ denotes the number of neurons in the $k-1$-th fully connected layer, $w_{ij}^{k-1}$ represents the weight of the connection from the $i$-th neuron in the $k-1$-th layer to the $j$-th neuron in the k-th layer, $x^{k-1\left(i\right)}$ represents the output value $x$ of the $i$-th neuron in the $k-1$-th layer, and $b_j^{k-1}$ denotes the bias term $b$ of the $j$-th neuron in the $k-1$-th layer.

2.4. Dropout Layer

The primary purpose of the Dropout layer is to prevent overfitting. Depending on the target object, it can be classified into Weight Dropout and Neuron Dropout. Weight Dropout primarily involves deactivating certain weight channels within the neural layer weight matrix, thus rendering them inactive. Neuron Dropout, on the other hand, deactivates certain neurons. During each iteration, certain neurons or weights cease to function with a certain probability and are temporarily removed from the network. This reduces the model’s dependency on local features, thereby enhancing generalization, A schematic diagram of the Dropout operation is shown in Figure 1.

3. Feature Extraction Methods

3.1. Variational Mode Decomposition

The VMD algorithm is an adaptive non-recursive modal decomposition method. This method utilizes the alternating direction method of multipliers (ADMM) algorithm to iteratively solve the constrained variational model, thereby obtaining the optimal solution for K intrinsic mode functions (IMFs) with center frequencies ω_k. The decomposition process of VMD can be summarized as follows:

Step 1: Initialization $\left\{{\mu }_k^1\right\}$, $\left\{{\omega }_k^1\right\}$, ${\lambda }^1$, $n\leftarrow 0$,

Step 2: Let $n=n+1$, $k=k+1$; update ${\widehat{\mu }}_k^{n+1}$ and ${\omega }_k^{n+1}$ separately using Equations (5) and (6). Stop iteration when $k=K$.

$${\widehat{\mu }}_k^{n+1}\left(\omega \right)=\frac{\widehat{f}\left(\omega \right)-{\sum }_{i\ne k}{\widehat{\mu }}_i^{n+1}\left(\omega \right)+\frac{\widehat{\lambda }\left(\omega \right)}{2}}{1+2\alpha \left(\omega -{\omega }_k\right)}$$

(5)

Step 3: Update ${\widehat{\lambda }}^{n+1}$ using the above equation:

$${\widehat{\lambda }}^{n+1}\left(\omega \right)={\widehat{\lambda }}^n\left(\omega \right)+\tau \left(\widehat{f}\left(\omega \right)-{\displaystyle \sum _k}{\widehat{\mu }}_k^{n+1}\left(\omega \right)\right)$$

(6)

Step 4: Given $\epsilon$ > 0, stop iteration when Equation (7) is satisfied. Otherwise, repeat steps 2 to 4.

$$\epsilon >{\displaystyle \sum _k}\frac{{‖{\widehat{\mu }}_k^{n+1}-{\widehat{\mu }}_k^n‖}_2^2}{{‖{\widehat{\mu }}_k^n‖}_2^2}$$

(7)

The parameters $K$ and $\alpha$ in the above equation are preset parameters. In this paper, we refer to the literature and select $K=2000$, $\alpha =4$ [17].

3.2. Empirical Mode Decomposition

EMD is a method used to stabilize signals to obtain a series of data sequences with different scales, where each data sequence is termed an Intrinsic Mode Function (IMF). Any signal can be regarded as being composed of several IMF components [18]. IMF components satisfy the following two conditions:

(1)

The number of local extrema is equal to or differs by only one from the number of zero crossings within the entire time range.

(2)

At any given moment, the mean of the upper and lower envelope lines determined by local maxima and local minima is zero.

The specific decomposition steps of the EMD method are as follows:

(1)

The local maxima and minima of the signal $x\left(t\right)$ are identified and then connected using cubic spline curves to form the upper envelope $y_{up}\left(t\right)$ and the lower envelope $y_{low}\left(t\right)$. The mean of the upper and lower envelopes is denoted as $m_1\left(t\right)$.

$$m_1\left(t\right)=\frac{1}{2}\left[y_{up}\left(t\right)+y_{low}\left(t\right)\right]$$

(8)

(2)

The original signal $x\left(t\right)$ subtracted by $m_1\left(t\right)$ yields a new sequence $h_1\left(t\right)$.

$$h_1\left(t\right)=x\left(t\right)-m_1\left(t\right)$$

(9)

(3)

If $h_1\left(t\right)$ does not satisfy the IMF conditions, then $h_1\left(t\right)$ is treated as the original data, and the above steps are repeated k times until $h_{1k}\left(t\right)$ satisfies the conditions, yielding the first-order IMF component. The i-th order IMF component is denoted as $c_i\left(t\right)$. Thus:

$$c_i\left(t\right)=h_{1k}\left(t\right)$$

(10)

(4)

The approximation component $Re s_1\left(t\right)$ is obtained.

$$Re s_1\left(t\right)=x\left(t\right)-c_1\left(t\right)$$

(11)

The above steps are repeated with $Re s_1\left(t\right)$ as the original data to obtain the second-order, third-order, …, n-th-order IMF components until the preset conditions are met, and then the process is terminated.

(5)

The original signal $x\left(t\right)$ can be represented as the sum of the residual term $Re s_n\left(t\right)$ and the sum of all IMF components from each stage, expressed as:

$$x\left(t\right)={\displaystyle \sum _{i=1}^n}{c}_i\left(t\right)+Re s_n\left(t\right)$$

(12)

4. Feature Parameter Weighting Module (FPWM)

Traditional convolutional neural network-based bearing fault diagnosis models usually rely on repetitive feature extraction by convolutional layers without distinguishing among different features. This leads to difficulty in effectively reflecting fault types, resulting in lower diagnostic accuracy of the model. To address this issue, an attention mechanism is introduced into the diagnostic model to highlight important features for effective feature extraction and to improve diagnostic accuracy. However, existing attention mechanisms only focus on important features and neglect minor features, resulting in feature loss and lower model robustness.

This paper proposes a feature processing method based on feature-parameterized weighting. A flowchart of the method is shown in Figure 2. When the feature tensor enters the module, it undergoes preliminary feature extraction and parameter reduction by a convolutional layer to reduce the parameter volume. Then, the features are compressed to [−1,1] using the Sigmoid function. The compressed feature data are then averaged, and based on the relationship with 0, they are divided into two categories as follows: when the mean is greater than 0, the feature data are added to the mean at that time; when the mean is less than or equal to 0, the feature data are subtracted by the mean multiplied by the parameter $\alpha$. Finally, the output is passed through the relu activation function to remove features that are still less than 0. The purpose of the above operations is to highlight important features while also including minor features in subsequent feature extraction and clearing redundant information.

The processed feature tensor is multiplied by the feature ratio tensor and added to the input feature tensor. However, the original input feature tensor is multiplied by the parameter $\beta$ as a weighting parameter for both tensors.

5. Fault Diagnosis Model for Bearings Based on FPWM-1DCNN

To effectively extract the characteristics of bearing faults and improve the diagnostic accuracy and robustness of the diagnostic model, a bearing fault diagnosis model based on a feature parameter-weighted convolutional neural network is proposed. Bearing fault data, after feature extraction through EMD and VMD, are merged together as input data for the diagnostic model. The input data are segmented by a specified window size, and after each segmentation, the window shifts by a specified step size to achieve data augmentation. The data then pass through four identical modules successively. They all consist of a feature parameter weighting layer, a batch_normalization layer, and a global averaging pooling layer, followed by a full connection layer and a random deactivation layer. Entering a final full connection layer that includes the Softmax function, the diagnostic model enters a percentage of what faults it determines for that fault type. The type with the highest percentage is used as the diagnostic result of the model. The overall structure of the model is illustrated in Figure 3, and its specific parameters are detailed in

Table 1. Specific parameters of the model.

Layer (Type)	Layer Parameter	Output Shape	Trainable Parameter
Input layer	[(None)]	[(None, 2048, 8)]	0
FPWM layer	[(32, 3, 3)]	[(None, 683, 32)]	800
BatchNormalization	[(32)]	[(None, 683, 32)]	128
AveragePool1D	[(2)]	[(None, 341, 32)]	0
FPWM layer	[(32, 3, 3)]	[(None, 114, 32)]	3104
BatchNormalization	[(32)]	[(None, 114, 32)]	128
AveragePool1D	[(2)]	[(None, 57, 32)]	0
FPWM Layer	[(32, 3, 3)]	[(None, 19, 32)]	3104
BatchNormalization	[(32)]	[(None, 19, 32)]	128
AveragePool1D	[(2)]	[(None, 9, 32)]	0
FPWM layer	[(32, 3, 3)]	[(None, 3, 32)]	3104
BatchNormalization	[(32)]	[(None, 3, 32)]	128
AveragePool1D	[(2)]	[(None, 1, 32)]	0
Flatten	[(None)]	[(None, 32)]	0
Dense	[(32)]	[(None, 32)]	1056
Dropout	[(0.3)]	[(None, 32)]	0
Dense	[(num_classes)]	[(None, num_classes)]	32* num_classes

.

6. Model Performance Experiment Validation

6.1. Case Western Reserve University Bearing Dataset

The Case Western Reserve University Bearing Dataset comprises the following three components: an induction motor, a torque sensor, and a torque meter. Smart sensors are installed at the drive end and fan end of the bearings to collect vibration signals. The bearings used at the drive end are SKF6205 single-row deep groove ball bearings, while those at the fan end use SKF6203 single-row deep groove ball bearings. The faults on the bearings are obtained through electrical discharge machining. The bearing speeds include the following four scenarios: 1730 r/min, 1750 r/min, 1772 r/min, and 1797 r/min, each representing different operating conditions corresponding to four loads ranging from 0 to 4 hp. The types of faults on the bearings include inner race, outer race, and rolling element faults, with a sampling frequency of 12 kHz.

6.2. Experimental Results of Parameter Selection

In order to select parameters for the proposed feature parameter weighting module, the data from operating conditions 2 hp and 3 hp of the aforementioned bearing fault dataset are chosen as the dataset for parameter selection experiments. These correspond to fault data at speeds of 1772 r/min and 1797 r/min, respectively. The data segment length used during model training is 2048 with a step length of 2048. The feature parameter weighting module contains two parameters, α and β, where $\alpha \in \left[2,9\right]$ and $\beta \in \left[0.1,1\right]$. Figure 4 and Figure 5 show the experimental results of different parameter selections for the two operating conditions, where the horizontal lines in the images represent the diagnostic results when the feature parameter weighting module is not loaded in the model.

Below are the average diagnostic results for each parameter.

Table 2. Parameter $\alpha$ selection results under the 2 HP condition.

$\mathit{\alpha }$		2	3	4	5	6	7	8	9
Mean	M	0.7796	0.7800	0.7690	0.7712	0.7655	0.7815	0.7943	0.7832
	S	0.0329	0.0290	0.0355	0.0357	0.0305	0.0309	0.0273	0.0338
	CI	0.7434	0.7445	0.7316	0.7336	0.7307	0.7465	0.7600	0.7466
		0.8157	0.8157	0.8064	0.8087	0.8004	0.8166	0.8287	0.8199

and

Table 3. Parameter $\beta$ selection results under the 2 HP condition.

$\mathit{\beta }$		0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
Mean	M	0.7831	0.7792	0.7891	0.7604	0.7790	0.7766	0.7735	0.7752	0.7812	0.7833
	S	0.0345	0.0326	0.0285	0.0314	0.0325	0.0316	0.0294	0.0320	0.0264	0.0298
	CI	0.7467	0.7432	0.7536	0.7232	0.7413	0.7411	0.7393	0.7396	0.7466	0.7465
		0.8196	0.8276	0.8245	0.7679	0.8168	0.8120	0.8076	0.9034	0.8158	0.8201

present the diagnostic results of the diagnostic model under different parameters for operating condition 2 hp, while

Table 4. Parameter $\alpha$ selection results under the 3 HP condition.

$\mathit{\alpha }$		2	3	4	5	6	7	8	9
Mean	M	0.7848	0.7715	0.7762	0.7723	0.7767	0.7731	0.7903	0.7702
	S	0.0293	0.0313	0.0334	0.0337	0.0305	0.0344	0.0317	0.0311
	CI	0.7507	0.7363	0.7399	0.7458	0.7414	0.7361	0.7548	0.7351
		0.8189	0.8068	0.8126	0.8088	0.8120	0.8100	0.8258	0.7989

and

Table 5. Parameter $\beta$ selection results under the 3 HP condition.

$\mathit{\beta }$		0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
Mean	M	0.7764	0.7857	0.7732	0.7816	0.7733	0.7780	0.7827	0.7789	0.7751	0.7640
	S	0.0329	0.0297	0.0312	0.0326	0.0324	0.0330	0.0315	0.0298	0.0337	0.0326
	CI	0.7402	0.7514	0.7382	0.7456	0.7375	0.7419	0.7484	0.7445	0.7386	0.7273
		0.8125	0.8201	0.8083	0.8175	0.8091	0.8141	0.8181	0.8133	0.7991	0.8007

correspond to operating condition 3 hp.

Table 6. The average parameter $\alpha$ selection results under two operating conditions.

$\mathit{\alpha }$	2	3	4	5	6	7	8	9
M	0.7822	0.7757	0.7726	0.7717	0.7711	0.7773	0.7923	0.7767
S	0.03110	0.0301	0.0344	0.0347	0.0305	0.0326	0.0295	0.0324
CI	0.7470	0.7282	0.7357	0.7397	0.7360	0.7413	0.7574	0.7408
	0.8173	0.8112	0.8109	0.8087	0.8062	0.8133	0.8272	0.8094
FAM	93.15	93.14	92.61	92.92	92.55	92.73	92.72	92.48

and

Table 7. The average parameter $\beta$ selection results under two operating conditions.

$\mathit{\beta }$	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
M	0.7797	0.7824	0.7811	0.7710	0.7761	0.7773	0.7781	0.7770	0.7781	0.7736
S	0.0337	0.0311	0.0298	0.0320	0.0324	0.0323	0.0304	0.0309	0.0300	0.0312
CI	0.7434	0.7473	0.7459	0.7344	0.7394	0.7415	0.7438	0.7420	0.7426	0.7369
	0.8160	0.8238	0.8164	0.7927	0.8129	0.8130	0.8128	0.8583	0.8074	0.8140
FAM	93.44	93.28	92.97	92.86	93.31	92.89	92.54	93.15	92.57	92.52

show the average results of the model for both operating conditions. In these tables, M represents the mean of diagnostic accuracy, S represents variance, CI represents confidence interval, and FAM represents the mean of the final output diagnostic results.

From Table 6, it can be observed that when M is maximized, $\alpha =8$, while when FAM is maximized, $\alpha =2$. The percentage increase in M from $\alpha =2$ to $\alpha =8$ is 1.27%. Conversely, the percentage decrease in FAM from $\alpha =2$ to $\alpha =8$ is 0.46%. Hence, $\alpha =8$ is selected as the final parameter. Similarly, from Table 7, it can be inferred that $\beta =0.2$ is the final parameter.

Overall, $\alpha =8$ and $\beta =0.2$ are chosen as the final parameters for the feature parameter weighting module.

To verify the diagnostic capability of the model constructed after parameter selection in the above experiment, we conducted diagnostic tests on a multi-operational condition bearing fault dataset consisting of 20 types of faults, composed of 2 hp and 3 hp. At this point, the data truncation window moved in steps of 256, with 100 iterations, resulting in a final diagnostic accuracy of 99.13%, as shown in Figure 6. In Figure 6, we plotted a comparison of the diagnostic process of the model with and without the feature parameter weighting module. The 1DCNN diagnostic model without FPWM achieved only 90.18% diagnostic accuracy. It is evident from Figure 6 that the inclusion of the feature parameter weighting module significantly improves the diagnostic accuracy of the model. Additionally, we visualized the feature data during the training processes of the two models using t-SNE technology, as depicted in Figure 7. The first row in the figure represents the visualization of the training process for 1DCNN, while the second row represents FPWM-1DCNN. Each image in the row represents input data, output data of the first convolutional block, output data of the second convolutional block, output data of the third convolutional block, output data of the fourth convolutional block, and the final classification result, respectively. It can be observed that FPWM-1DCNN achieves classification and clustering of different faults effectively during the final fault diagnosis.

6.3. Diagnostic Experiment on the Jiangnan University Bearing Fault Dataset

The test bearings were single-row aligning roller bearings (N205 and NU205). The bearing failure was artificially induced by wire cutting, which consisted of three types of bearing faults at speeds of 600 r/min, 800 r/min, and 1000 r/min, including inner ring faults, outer ring faults, rolling element faults, and normal data at each speed, with a sampling frequency of 50 kHz. A total of 10 datasets were selected for the experiment, comprising three types of fault data at each speed and normal data at the 1000 r/min operating condition. The specific experimental setup is illustrated in Figure 8. Similarly, both FPWM-1DCNN and 1DCNN were employed for fault diagnosis experiments on the aforementioned fault data, with settings identical to the previous section. The diagnostic results of the models are depicted in Figure 9. FPWM-1DCNN quickly converged and achieved 100% diagnostic accuracy. The visualization of the data for both models is shown in Figure 10.

6.4. Experiment on the Model’s Robustness to Noise

When bearings are in actual operation, the bearing data are inevitably disturbed by noise during acquisition, which impacts the fault diagnosis performance of the model. In this paper, Gaussian white noise with different signal-to-noise ratios was added to the original vibration signal to simulate the noise environment in actual operating conditions, and fault diagnosis experiments were conducted using the added noise data to verify the noise immunity of the model. In this paper, the signal-to-noise ratio (SNR) was used as a measure of the noise level, which is defined in Equation (13):

$$SNR=10lg\left(\frac{P_s}{P_n}\right)$$

(13)

where $P_s$ is the signal power and $P_n$ is the noise power.

We conducted experiments by adding 6 dB and −6 dB noise to the original fault data separately, using the noisy data to verify the robustness of the proposed diagnostic model. We compared it with several research methods to highlight the robustness of our diagnostic model to noise. These comparisons aimed to showcase the effectiveness of our diagnostic model in noisy environments.

6.4.1. Experiment on Robustness of Jiangnan Multi-Condition Bearing Data to Noise

To validate the robustness of the proposed FPWM-1DCNN bearing fault diagnosis model, we initially introduced 6 dB and −6 dB noise separately into the Jiangnan University bearing fault data. This facilitated the creation of a multi-condition bearing fault dataset under noisy conditions. Comparative analyses were conducted against 1DCNN, the 1DCNN-RAM-LSTM diagnostic model with reverse attention mechanism, the classical SVM diagnostic model, and the image-based diagnostic model GADF-2DCNN. These comparisons aimed to underscore the robustness of our model to noise. Experimental results, as depicted in Figure 11, demonstrate that FPWM-1DCNN achieved a diagnostic accuracy of 98.79% under −6 dB conditions, marking a 12.47% improvement over 1DCNN and a 7.98% improvement over the alternative diagnostic model 1DCNN-RAM-LSTM incorporating improved attention mechanisms. However, the image-based GADF-2DCNN diagnostic model, when applied to −6 dB noise data, yielded a diagnostic accuracy of 73.98%, representing a 19.81% decrease compared with the proposed model. In addition, the average training time of each round of the model proposed in this paper was 44 s, which costs 4 s more than the 38 s of 1DCNN-RAM-LSTM, but the diagnostic accuracy of the model is improved by 7.98%, which is worthwhile. GADF-2DCNN, which uses two-dimensional images as the input of the diagnostic model, required 256 s to complete training, but the diagnostic accuracy obtained was still lower than that of FPWM-1DCNN. Visualization of the diagnostic process for −6 dB fault data by FPWM-1DCNN, as illustrated in Figure 12, clearly demonstrates the classification and clustering of different types of fault data at the model’s output layer.

6.4.2. Multi-Condition Noise Resistance Experiment on CWRU Bearing Data

In the same vein, a multi-condition experiment was conducted on the Case Western Reserve University bearing dataset, where −6 dB and 6 dB noise were respectively added under two different operating conditions to construct a multi-condition bearing fault noise dataset. These datasets were then fed into our proposed FPWM-1DCNN diagnostic model for noise resistance validation. As depicted in Figure 13, our model achieved a diagnostic accuracy of 98.36% under the −6 dB noise condition, surpassing all comparative methods. Figure 14 illustrates the feature visualization of the diagnostic process of our proposed model on the −6 dB fault data, demonstrating precise classification and clustering of the fault data.

6.5. Comparison with Other Methods

In order to further demonstrate the performance of the proposed method, we compared the original data and noise data of the multi-condition bearing fault dataset of Jiangnan University with other methods. The diagnosis models of the comparison method include a one-dimensional diagnosis model and a two-dimensional diagnosis model as follows: method A [19], method B [20], and method C [21]. The comparative experimental results are shown in

Table 8. Comparative experimental results.

	−6 db Data	Origin Data	6 db Data
Our method	98.36	100	98.74
Method A	97.69	100	98.23
Method B	97.13	100	97.99
Method C	96.69	99.15	98.95

.

As can be seen from Table 8, when the raw data and −6 db noise data are used as input data, the proposed method achieves the highest diagnostic accuracy. The diagnostic accuracy obtained only under 6 db noise is lower than that obtained by method C, but the training time required by method C is longer than that of the proposed method, which is 160 s.

7. Conclusions

In the context of rolling bearing fault diagnosis, where different levels of fault features require distinct treatment, a feature-parameterized weighted diagnostic model is proposed. This model achieves high-precision diagnosis of rolling bearings under variable operating conditions while maintaining robustness in noisy environments. The key conclusions are as follows:

(1) Feature extraction from fault data is performed using variational mode decomposition (VMD) and empirical mode decomposition (EMD). The extracted features from both methods are then combined and used as input data for subsequent diagnostic models, enhancing data dimensionality while reducing noise.

(2) Parameter selection experiments conducted on datasets from different operating conditions improve the model’s generalization ability.

(3) A feature-parameterized weighting module is constructed, wherein features are divided into two categories based on the mean of feature tensors. Different operations are applied to these categories, allowing the diagnostic model to focus more on features of higher importance. This enhances both the diagnostic accuracy and robustness of the network model.

Starting from the importance of features, we design a feature processing module FPWM, which can carry out different operations on fault features according to the positive and negative parts. In this module, the processed feature data and the original feature data are weighted with reference to the residual idea, but not simply added in a 1:1 way. By setting an adjustable parameter, the fault features can be processed in the same way. The effective fusion of the two features is realized to improve the diagnostic accuracy and robustness of the model. When the effective features in the fault data are difficult to discover and the data dimension is not enough to satisfy the training of the convolutional neural network, the proposed method can realize the high-precision diagnosis of multi-working condition bearings and has certain robustness.