A Rolling Bearing Fault Diagnosis Method Based on Switchable Normalization and a Deep Convolutional Neural Network

Abstract

Aiming to address the problems of a low fault detection rate and poor diagnosis performance under different loads and noise environments, a rolling bearing fault diagnosis method based on switchable normalization and a deep convolutional neural network (SNDCNN) is proposed. The method effectively extracted the fault features from the raw vibration signal and suppressed high-frequency noise by increasing the convolution kernel width of the first layer and stacking multiple layers’ convolution kernels. To avoid losing the intensity information of the features, the K-max pooling operation was adopted at the pooling layer. To solve the overfitting problem and improve the generalization ability, a switchable normalization approach was used after each convolutional layer. The proposed SNDCNN was evaluated with two sets of rolling bearing datasets and obtained a higher fault detection rate than SVM and BP, reaching a fault detection rate of over 90% under different loads and demonstrating a better anti-noise performance.

1. Introduction

Rolling bearings are one of the main components of mechanical equipment. The online monitoring of the real-time operation of bearings and the accurate identification of fault types and fault degrees can ensure the reliability of mechanical equipment operation [1,2]. One of the most widely used methods to detect rolling bearing faults by vibration signals is to extract the characteristics of the vibration signals through techniques such as time-domain analysis [3], wavelet analysis, wavelet packet analysis [4,5,6], empirical wavelet transform [7,8], empirical mode decomposition [9], ensemble empirical mode decomposition [5,10], sparse decomposition [11], independent component analysis [12], and spectral kurtosis [13,14]. However, the characteristics of bearing faults under different operating conditions and noise environments are complex, and effectively mining information from bearing vibration data has always been an important problem in rolling bearing fault diagnosis [15,16,17].

Fault diagnosis is essentially pattern recognition. The currently commonly used approaches of support vector machines (SVMs) [18,19,20], random forests [21], AdaBoost [22], backpropagation neural networks [23], and radial basis neural networks [24] are shallow learning models with a limited fault feature expression ability, and the accuracy of the diagnosis is subject to the feature extraction results. In recent years, researchers have turned their attention to various deep learning models, such as convolutional neural networks (CNNs) [25], recurrent neural networks [26], autoencoders [27,28,29], and transfer learning [30]. Among these, CNNs can convert the original features of the input layer by layer into features that are easy to recognize and realize the deep characterization of rolling bearing faults, representing a hotspot of current research [31,32]. Several studies have applied CNNs as a classifier, using wavelet packets [33,34,35], continued wavelet transform [36,37], dual-tree complex wavelet transform [38,39], fast Fourier transform (FFT) [40], discrete wavelet transform (DWT) [40], and other signal processing methods [41] to pre-process one-dimensional vibration signals, which are converted into the time domain or the time-frequency domain and then input into the CNN for fault diagnosis [42]. This processing method fails to fully exploit CNNs’ deep learning ability. Therefore, dislocated time series (DTS) [43]; gray-scale images [44,45], infrared thermal images, and other image data [46,47]; one-dimensional CNNs [48,49]; and other methods have been proposed to reshape the input data dimension and get rid of the dependence on expert knowledge for vibration signal feature extraction. However, in most current studies, the pooling operation of CNNs mostly adopts the max-pooling method. Since the rolling bearing is a periodic time-series signal, this processing method causes the intensity information of the feature to be lost. In response to this shortcoming, the deep convolutional neural network (DCNN) model presented in this paper used the K-max pooling method, which could retain the Top-K score of all feature values, thereby avoiding the problem of the loss of intensity information in the signal.

Furthermore, although feature extraction can be effectively performed through the superposition of multiple convolutional layers and pooling layers, it is easy to cause overfitting. Therefore, in this study we adopted the switchable normalization (SN) method [50], which could choose the appropriate normalization method for each normalization layer of the DCNN. Experiments showed that the average recognition rate of the proposed SNDCNN model was 99.68%, and it achieved good diagnostic results under different loads and noise environments.

The remainder of this paper is structured as follows. The next section presents an interpretation of the relevant literature, including works on CNN and K-max pooling. In Section 3, we propose a novel model combining SN and DCNN. Then, in Section 4, we describe in detail the experimental procedures and diagnosis result analysis based on two rolling bearing datasets. Finally, the conclusions are presented in Section 5.

2. Convolutional Neural Network

The main CNN hierarchy is shown in Figure 1.

2.1. Convolution Layer

In the convolution layer, the convolution calculation is performed on the input through the convolution kernel, and each small block in the neural network is analyzed more deeply to obtain higher abstraction features.

$$X_j^k=f\left({\sum }_{i\in M_j}{X}_i^{k-1}\ast W_{ij}^k+b_j^k\right)$$

(1)

where X_j^k is the jth element of the kth layer; M_j is the input feature vector; X_i^k−1 is the element; W_ij^k is the weight matrix of the convolution kernel; b_j^k is the bias term; and ƒ(·) is the activation function, which is a non-linear function that converts the output of each neuron from linear to non-linear. In this model, the Relu function is chosen as the activation function.

$$f\left(x\right)=\max \left(x,0\right)$$

(2)

2.2. Pooling Layer

After the feature extraction of the input data through the convolution layer, the resulting data dimension is still relatively large. The pooling layer does not change the depth of the data matrix of the previous layer. By reducing the size of the data matrix, the goal of reducing the parameters in the neural network is achieved. The mathematical expression of the pooling layer is shown in Equation (3):

$$X=f\left(\alpha \ast down\left(x\right)+b\right)$$

(3)

where x is the output, ƒ(·) is the activation function, α is the multiplicative bias, down( ) is the downsampling function, and b is the bias term.

Commonly used pooling layer processing methods include general pooling, overlapping pooling, average pooling, and max pooling. In this study, K-max pooling was selected, which is an improved maximum pooling method. A comparison chart for max pooling and K-max pooling when K = 2 is shown in Figure 2. The max pooling method has an obvious disadvantage in that sometimes certain strong features appear multiple times. The more occurrences, the stronger the feature. Because maximum pooling only retains a maximum value, which means that the same feature has incomplete intensity information, K-max pooling can retain all feature values scored in Top-K and the original order of these feature values, thus providing more complete feature information for subsequent use.

2.3. Fully Connected Layer

The fully connected layer acts as a classifier in the CNN. It accepts the input of the previous neural layer and outputs an N-dimensional vector, where N is the total number of all possible categories. After the processing of multiple convolutional layers and pooling layers, the final classification results are generally provided by one or two fully connected layers. This study used the Softmax layer for classification.

3. Rolling Bearing Fault Diagnosis Method Based on SNDCNN

Although traditional CNNs have a high recognition rate for the fault diagnosis of rolling bearings, the recognition rate still needs to be improved, as it cannot meet the challenge of bearing fault diagnosis under different loads and noise environments. In response to these problems, a new rolling bearing fault diagnosis method based on switchable normalization and a deep convolutional network (SNDCNN) was proposed, as shown in Figure 3.

3.1. Structural Parameters of the SNDCNN

In this study, by increasing the convolution kernel width of the DCNN’s first layer and setting the convolution kernel size of the first layer to 80 × 1, a larger receptive field could be obtained, which could effectively extract one-dimensional vibration signal characteristics and suppress high-frequency noise. The subsequent neural layers adopted a multi-layer convolution kernel stacking to achieve multi-layer nonlinear mapping. However, increasing the width of the convolution kernel and superimposing the convolution layer increases the complexity of the DCNN model and the training parameters, which may cause overfitting problems. To suppress overfitting and increase the training speed, the switchable normalization method was adopted, and the SN process was performed between each convolutional layer and the active layer of the DCNN. The structural parameters of the SNDCNN are shown in

Table 1. Structural parameters of SNDCNN.

No.	Layer Type	Size	Stride	Kernel Number	Padding (Yes or No)
1	C1	80 × 1	8*1	8	Y
2	P1	4 × 1	2*1	16	N
3	C2	5 × 1	1*1	32	Y
4	P2	2 × 1	2*1	32	N
5	C3	3 × 1	1*1	64	Y
6	P3	2 × 1	2*1	64	N
7	C4	3 × 1	1*1	64	Y
8	P4	2 × 1	2*1	64	N
9	C5	3 × 1	1*1	64	Y
10	P5	2 × 1	2*1	64	N
11	C6	3 × 1	1*1	64	Y
12	P6	2 × 1	2*1	64	N
13	F1	100	100	1	N
14	Softmax	10	10	1	N

.

3.2. Switchable Normalization

Normalization technology plays an important role in the deep learning model, as it can effectively suppress overfitting and improve the generalization ability of the DCNN. Commonly used normalization methods include batch normalization (BN), instance normalization (IN), layer normalization (LN), and group normalization (GN). However, each normalization method has its limitations in terms of the scenarios to which it can be applied. Although normalization can improve DCNN performance, solving different problems often requires different normalization methods. A DCNN often contains several or even dozens of normalization layers. Selecting the appropriate normalization operation for each normalization layer using a manual takes a lot of time. Thus, the same normalization method is used in traditional deep neural networks.

In this study, the switchable normalization method was proposed to solve the above problems. SN contains three normalization methods: IN, LN, and BN. Using differential learning, the appropriate normalization method for different normalization layers in the deep neural network can be automatically selected. Suppose that the input data of the convolutional layer of a DCNN can be expressed in four dimensions, with each dimension representing the number of samples N, the number of channels C, the height of the channel H, and the width of channel W. Each pixel that is normalized based on the SN method is represented as h_ncij:

$${\widehat{h}}_{ncij}=\gamma \frac{h_{ncik}-{\sum }_{k\in \Omega }{\omega }_k{\mu }_k}{\sqrt{{\sum }_{k\in \Omega }{\omega }_k^{\prime }{\sigma }_k{}^2+\theta }}+\beta$$

(4)

where ${\widehat{h}}_{ncij}$ is the normalized pixel value, and n, c, i, and j are the subscripts of the four dimensions. The above definition in Equation (5) is similar to those of BN, IN, and LN. They all learn the scaling factor γ and the offset factor β. The main difference is the statistical information included in SN (mean μ and variance σ²), making it unlike IN, which is only calculated in one channel, and LN, which is only calculated in one layer. In SN, a set is defined as Ω = IN, LN, BN, and the set weighted average is chosen by a suitable approach. ω_k and ω_k′ are the weight coefficients for the corresponding statistics. The weighting coefficient ω_k of the mean is calculated as follows:

$${\omega }_k=\frac{e^{{\lambda }_k}}{{\sum }_{k\in \left\{IN,LN,BN\right\}}{e}^{{\lambda }_k}},k\in \left\{IN,LN,BN\right\}$$

(5)

The calculation methods of the mean and variance of IN, LN, and BN are as follows:

$${\mu }_{IN}=\frac{1}{HW}{\sum }_{i,j}^{H,W}{h}_{ncij},{\sigma }_{IN}{}^2=\frac{1}{HW}{\sum }_{i,j}^{H,W}{\left(h_{ncij}-{\mu }_{IN}\right)}^2$$

(6)

$${\mu }_{LN}=\frac{1}{C}{\sum }_{c=1}^C{\mu }_{IN},{\sigma }_{LN}{}^2=\frac{1}{C}{\sum }_{c=1}^C\left({\sigma }_{IN}{}^2+{\mu }_{IN}{}^2\right)-{\mu }_{LN}{}^2$$

(7)

$${\mu }_{BN}=\frac{1}{C}{\sum }_{n=1}^N{\mu }_{IN},{\sigma }_{BN}{}^2=\frac{1}{N}{\sum }_{n=1}^N\left({\sigma }_{IN}{}^2+{\mu }_{IN}{}^2\right)-{\mu }_{BN}{}^2$$

(8)

3.3. Adaptive Momentum Algorithm

To speed up the convergence speed and accuracy, the adaptive momentum (Adam) algorithm was used to update the parameters and optimize the model. Adam is by far the best-performing gradient descent model optimization algorithm. It can be seen as a combination of the momentum algorithm and the RMSProp algorithm. When updating parameters, Adam considers not only the gradient of the current iteration but also the gradient of all previous iterations and the square of the gradient. By accumulating the gradient index weights, Adam introduces the first- and second-order moment estimates of the parameters into the update operation. Thus, the adaptive change in the learning rate can be realized. The specific steps of the Adam parameter update are as follows:

Steps of Adam parameter update:

Set the learning rate α, the first-order moment estimate exponential decay rate ρ₁, the second-order moment estimate exponential decay rate ρ₂, and the minimum constant ε. The parameters in this paper are all default values, α = 0.001, ρ₁ = 0.9 = 0.9, ρ₂ = 0.999, ε = 1 × 10⁻⁸.

Require: Initialize the first-order moment estimate s₀ and second-order moment estimate r₀ to 0, and initialize the iteration step number t to 0.

Require: Calculate the gradient of the loss function ƒ(θ) for the parameter θ, and loop the following updates:

While θ is not converged do:

1. Update steps: $t\leftarrow t+1$

2. Computing the gradients:$g_t\leftarrow {\nabla }_{\theta }\cdot f\left(\theta \right)$

3. Update biased first moment estimates:$s_t\leftarrow {\rho }_1\cdot s_{t-1}+\left(1-{\rho }_1\right)\cdot g_t$

4. Update biased second-order moment estimates:$r_t\leftarrow {\rho }_2\cdot r_{t-1}+\left(1-{\rho }_2\right)\cdot g_t{}^2$

5. Correct the deviation of the first moment:${\widehat{s}}_t\leftarrow \frac{s_t}{1-{\rho }_1{}^t}$

6. Correct the deviation of the second moment:${\widehat{ r}}_t\leftarrow \frac{r_t}{1-{\rho }_2{}^t}$

7. Calculation update:$\mathsf{\Delta }\theta \leftarrow -\alpha \cdot \frac{{\widehat{s}}_t}{\sqrt{{\widehat{r}}_t+\epsilon }}$

8. App update:$\theta =\theta +\mathsf{\Delta }\theta$

End while

Return θ

3.4. Fault Diagnostic Process

The algorithm consisted of the following two stages, as shown in Figure 4. The detailed fault diagnosis process flow is as follows:

(1)

Training Stage:

Step 1—data preprocessing: The vibration signals of the rolling bearings were collected, and the original data were expanded by overlapping sampling; then, these were divided into samples of length N, and the training data were input into the SNDCNN.

Step 2—training network: After the training data were input into the first layer of wide convolution kernels, they underwent a non-linear transformation of the ReLU function to become a set of feature maps, which were then put through pooling layer 1 for further feature extraction. The processed data were then input into a continuous six-layer superimposed small convolution kernel layer, which could effectively extract data features. Between each convolutional layer and the activation layer above, a switchable normalization method was used for processing. Together, the dropout rate was 0.5. After the nonlinear transformation of the ReLU function, its output was to the final Softmax layer.

Step 3—well-trained SNDCNN: A well-trained SNDCNN was saved for rolling bearing fault diagnosis.

(2)

Testing Stage:

Step 1—testing data acquisition: New vibration signals were acquired, and segmented samples were obtained as the testing dataset D_test.

Step 2—testing: D_test was input into the trained SNDCNN obtained from the training stage, providing the corresponding diagnosis result.

4. Experimental Verification

4.1. Case 1—Western Reserve University Rolling Bearing Data Analysis

The experimental verification in this study was carried out based on the rolling bearing fault dataset of Case Western Reserve University (CWRU). The test rig is shown in Figure 5. The rolling bearing mode was SKF 6205, which contained three fault locations, on the ball, inner ring, and outer ring of the bearing. The diameters of the bearing faults were 0.007 inches, 0.014 inches, and 0.021 inches, respectively.

4.1.1. Dataset Division

During the experiment, to avoid the SNDCNN overfitting problem caused by the insufficient amount of data in the training set, the data were expanded by overlapping sampling, that is, starting from the initial sampling point, the data length of each sampling was N. The position of the next sampling was shifted by n from the previous sampling position. The overlapping sampling process is shown in Figure 6:

The dataset division is shown in

Table 2. Division of rolling bearing data.

Fault Location		None	Ball			Inner Race			Outer Race			Load (hp)
Labels		0	1	2	3	4	5	6	7	8	9
Fault Diameter (inches)		0	0.007	0.014	0.021	0.007	0.014	0.021	0.007	0.014	0.021
A	Training	800	800	800	800	800	800	800	800	800	800	1
	Testing	40	40	40	40	40	40	40	40	40	40
	Validation	20	20	20	20	20	20	20	20	20	20
B	Training	800	800	800	800	800	800	800	800	800	800	2
	Testing	40	40	40	40	40	40	40	40	40	40
	Validation	20	20	20	20	20	20	20	20	20	20
C	Training	800	800	800	800	800	800	800	800	800	800	3
	Testing	40	40	40	40	40	40	40	40	40	40
	Validation	20	20	20	20	20	20	20	20	20	20
D	Training	2400	2400	2400	2400	2400	2400	2400	2400	2400	2400	1,2,3
	Testing	120	120	120	120	120	120	120	120	120	120
	Validation	60	60	60	60	60	60	60	60	60	60

. According to the load, the rolling bearing data of Western Reserve University were divided into three situations, A, B, and C, and the corresponding loads were 1 hp, 2 hp, and 3 hp, respectively. Dataset D was the sum of datasets A, B, and C, that is, it contained three different loads at the same time. Take dataset A as an example: it included 10 types of labels, 0–9; each label included 800 training samples, 40 test samples, and 20 verification samples; and each sample had a length of 2048. Dataset D contained 24,000 training samples, 1200 test samples, and 600 verification samples.

4.1.2. Diagnostic Results under Steady Conditions

Using the above dataset, the SNDCNN was used to train and verify datasets A, B, C, and D. The number of iterations for each dataset was 100, and the recognition rate, loss function, and training time of the SNDCNN were output. The test results are shown in Figure 7 and

Table 3. SNDCNN diagnosis results.

	Detection Rate	Loss Function	Time (ms/step)	Iterations
Dataset A	99.27%	0.061	0.653	100
Dataset B	100.00%	0.054	0.749	100
Dataset C	100.00%	0.011	0.671	100
Dataset D	99.45%	0.063	0.687	100

. In the figure, accuracy represents the detection rate of the training set, and var accuracy represents the detection rate of the test set.

In order to verify the impact of switchable normalization on the model, we compared it with batch normalization, which is widely used in convolutional neural networks, and the experimental results are shown in Figure 8. The average fault detection rate of switchable normalization for the four datasets was 99.68%, which was better than the average fault detection rate of batch normalization of 97.78%.

To further verify the fault detection ability of the SNDCNN proposed in this paper, we compared it with several traditional machine-learning-based diagnostic methods and several deep-learning-based diagnostic methods. The following methods all used the CWRU bearing dataset, with the only differences being in the number of training and test sets. Therefore, the comparison could objectively reflect the advantages of the SNDCNN in terms of the fault detection rate. The specific comparison results are shown in

Table 4. Comparison of test results of different diagnostic methods.

Diagnosis Method	Accuracy (%)
	Dataset A	Dataset B	Dataset C	Dataset D
EEMD + SVM [51]	92.25	93.16	92.56	87.25
STFT + SVM [52]	94.76	93.55	95.05	86.39
WT + BP [53]	93.65	92.14	94.85	84.27
BPNN [54]	62.11	—	—	—
MSCNN [54]	98.46	—	—	—
DTS-CNN [54]	99.14	—	—	—
1D-DCNN [54]	98.43	—	—	—
SNDCNN	99.27	100.00	100.00	99.45

.

To verify whether the SNDCNN classification ability was improved by increasing the number of training sets, t-SNE dimensionality reduction technology was used to visualize the corresponding SNDCNN classification results when the number of training sets was 120, 1200, 3000, and 24,000, as shown in Figure 9. The results showed that the data enhancement of the training set could effectively improve the SNDCNN detection effect.

4.1.3. Diagnostic Results under Variable Load Conditions

In actual applications, machines often work under different operating loads. Such a complex working environment elicits higher requirements for fault diagnosis models. Neural networks trained under one load often cannot handle another load. To verify the fault detection rate of the SNDCNN under different loads, the SNDCNN was trained by the vibration data under 1 hp (dataset A), 2 hp (dataset B), and 3 hp (dataset C), where 1 hp, 2 hp, and 3 hp correspond to the speeds of 1772 r/min, 1750 r/min, and 1730 r/min, respectively. Figure 10 shows the data under different operating loads.

Figure 10 shows the ball fault signal with a fault level of 0.007 inches under different loads. It can be seen that the amplitude of the fault signal varied under different loads.

The experiment used the signals under one load as the training set and the signals under the other two loads as the test set for testing. The test results are shown in Figure 11 and

Table 5. Diagnostic results of the SNDCNN under different load conditions.

	Detection Rate	Loss Function	Time (ms/step)	Iterations
A→B	99.18%	0.101	0.674	100
A→C	95.72%	0.431	0.703	100
B→A	97.14%	0.372	0.689	100
B→C	90.00%	0.624	0.683	100
C→A	79.40%	3.195	0.714	100
C→B	84.15%	0.961	0.692	100
Average	90.93%	0.947	0.693	100

.

The results showed that the SNDCNN had a detection rate of more than 90% under different loads, and the average training time of each signal was about 0.693 ms. Therefore, the SNDCNN had strong adaptability across load domains and could complete fault diagnosis tasks in complex and changing operating environments.

4.1.4. Diagnostic Results under Different Noise Conditions

To show that the SNDCNN had a high anti-noise performance, we also studied the diagnostic effect of the SNDCNN under different noise environments by introducing the concept of the signal-to-noise ratio (SNR), which is the ratio of useful signal power to noise power. The SNR is an important criterion for evaluating the strength of noise for a signal. The calculation formula of the SNR is as follows:

$$SNR\left(\mathrm{dB}\right)=10{\mathit{log}}_{10}\left(\frac{P_{signal}}{P_{noise}}\right)=20{\mathit{log}}_{10}\left(\frac{A_{signal}}{A_{noise}}\right)$$

(9)

where P_signal is the signal power; P_noise is the noise power; A_signal is the signal amplitude; and A_noise is the noise amplitude.

Environmental noise of different intensities has different effects on the raw vibration signal and may cover up the fault information of the raw vibration signal, making the SNDCNN unable to diagnose the fault well. Figure 12 shows the results of adding SNRs of −4 dB, 0 dB, 4 dB, and 8 dB Gaussian white noise to the raw vibration signal.

At this stage, the SNDCNN was trained by four vibration datasets with noise to test its detection rate at different noise intensities, and the diagnosis results were compared with support vector machines, multi-layer perceptrons (MLPs), and deep neural networks in the same noise environment. The average detection rate under the four different noise environments was calculated. The results of the average detection rate are shown in

Table 6. Diagnostic results of SNDCNN under different noise environments.

	SVM	MLP	DNN	SNDCNN
−4 dB	67.35	30.48	42.24	92.05
0 dB	95.15	41.75	57.93	98.37
4 dB	97.62	76.93	83.45	99.03
8 dB	98.73	95.02	96.64	99.25
Average	89.7125	61.045	70.065	97.175

and Figure 13. The results showed that the detection rate of the SNDCNN model in the four different noise environments was significantly higher than that of the other three diagnostic models, and the average detection rate was 97.15%, demonstrating a high anti-noise performance.

4.2. Case 2—Rolling Bearing Test Platform Data Analysis

4.2.1. Test Bench Description

Figure 14 displays the rolling bearing test platform, which consisted of several parts. This test platform could conduct fault simulation tests of bearings under different operating conditions. The adjustable operating conditions of the test platform mainly included speed and load. The test bearing was a TPI6205 rolling bearing, and three fault forms of the ball, inner ring, and outer ring were processed by laser pitting. The specific parameters of the rolling bearings are shown in

Table 7. TPI6205 rolling bearing parameters.

Type	Diameter of the Ball	Pitch Diameter	Ball Number	Contact Angle
TPI6205	0.3126 inches	1.537 inches	9	0

. Figure 15 shows photographs of the normal and faulted bearings. To collect vibration signals from the tested bearings, as shown in Figure 13, the accelerometer of type HD-YD-232 was positioned on the housing of the tested bearings, and the sampling frequency was set to 12 kHz.

The rolling bearing test bench was designed for experiments with different bearing failure degrees, different loads, different speeds, and a constant speed acceleration. We saved the experimental data to a series of files and filled in the file names with sequential numbers. The non-constant speed acceleration experiment stopped when the file size exceeded 16,384 KB, and the constant speed acceleration experiment stopped when the specified speed was reached. Taking the minor fault of the inner ring as an example, the details of the data records are shown in

Table 8. Details of the data records.

Filename	Load	Rotational Speed (rpm)	Filename	Load	Rotational Speed (rpm)	Comment
Test_001	0	800	Test_013	3	800
Test_002	0	1600	Test_014	3	1600
Test_003	0	2400	Test_015	3	2400
Test_004	0	3200	Test_016	3	3200
Test_005	1	800	Test_017	0	raising speed 40 rpm/s	800 to 3200 rpm
Test_006	1	1600	Test_018	0	raising speed 80 rpm/s	800 to 3200 rpm
Test_007	1	2400	Test_019	1	raising speed 40 rpm/s	800 to 3200 rpm
Test_008	1	3200	Test_020	1	raising speed 80 rpm/s	800 to 3200 rpm
Test_009	2	800	Test_021	2	raising speed 40 rpm/s	800 to 3200 rpm
Test_010	2	1600	Test_022	2	raising speed 80 rpm/s	800 to 3200 rpm
Test_011	2	2400	Test_023	3	raising speed 40 rpm/s	800 to 3200 rpm
Test_012	2	3200	Test_024	3	raising speed 80 rpm/s	800 to 3200 rpm

. Load 0 corresponded to operation under the no-load state, load 1 corresponded to a loading of 0.5 N∙m, load 2 corresponded to a loading of 1 N∙m, and load 3 corresponded to a loading of 1.5 N∙m.

4.2.2. Bearing Fault Forms and Vibration Signal Analysis

We selected four bearing types under the conditions of rotating speed n = 2400 r/min and 0 loads and drew their original time domain signal, frequency spectrum, and envelope spectrum, as shown in Figure 16.

As shown in Figure 15, in addition to 40 Hz and the frequency component of doubling frequency, frequency components of 186.3 Hz, 214.9 Hz, and 144.5 Hz appeared in the spectrum and envelope spectrum, which were close to the theoretical values of the corresponding fault characteristic frequencies of 188.54 Hz, 216.6 Hz, and 143.39 Hz, respectively.

Through the observation and analysis of the original time domain signals, spectra, and envelope spectra of the above four conditions, obvious frequency doubling characteristics could be seen in the time spectra of the four types of signals, with the inner ring faults, outer ring faults, and ball faults all having obvious characteristic frequencies. Therefore, the overall design of the rolling bearing fault diagnosis test bed at this stage of the study was reasonable and could effectively simulate the damage of rolling bearings in different positions.

4.2.3. Diagnostic Results under Steady Conditions

In the case of a single load, the bearing was diagnosed with a total of three damage positions, on the rolling body of the bearing, the inner ring, and the outer ring. The damage degree was divided into three types, mild, moderate, and severe, and with the data of the bearing in a healthy state, this made up a total of 10 operating states. The speed was 2400 rpm/min.

At this stage, we expanded upon the data of the training set by employing overlapping sampling, that is, starting from the initial sampling point, the data length of each sample was N, and the location of the next sample was offset by N from that of the previous one. During the neural network training, N data points were trained each time. In this section, the length of the sampled data was 2048, and the offset length of the sampling position was 64. The specific division of the experimental dataset is shown in

Table 9. The specific division of the experimental dataset.

Fault Location		None	Ball			Inner Race			Outer Race			Load
Labels		0	1	2	3	4	5	6	7	8	9
Fault Degree		None	Mild	Moderate	Severe	Mild	Moderate	Severe	Mild	Moderate	Severe
A	Training	800	800	800	800	800	800	800	800	800	800	0
	Testing	40	40	40	40	40	40	40	40	40	40
B	Training	800	800	800	800	800	800	800	800	800	800	1
	Testing	40	40	40	40	40	40	40	40	40	40
C	Training	800	800	800	800	800	800	800	800	800	800	2
	Testing	40	40	40	40	40	40	40	40	40	40
D	Training	800	800	800	800	800	800	800	800	800	800	3
	Testing	40	40	40	40	40	40	40	40	40	40

.

The trained fault diagnosis model was used to verify the divided single-load datasets. The number of iterations was 100, and the experiment was repeated 10 times. The diagnostic results are shown in Figure 17.

It can be seen from Figure 17 that the model could quickly converge, and the diagnosis rate under each load reached more than 97%, with the diagnosis rate for the dataset with load 3 being the highest at 98.71%. The experimental results showed that the model had high diagnostic accuracy under a single load. The results of the experiment are shown in

Table 10. The results of the experiment.

	Detection Rate	Loss Function	Time (ms/step)	Iterations
Dataset A	97.14%	0.083	0.638	100
Dataset B	98.57%	0.069	0.726	100
Dataset C	98.71%	0.067	0.645	100
Dataset D	99.52%	0.069	0.633	100

.

4.2.4. Diagnosis Results under Variable Speed Conditions

In real industrial activities, bearings often work by accelerating or, in some cases, decelerating; the bearing fault diagnosis experimental platform designed in this study also simulated this situation. The control unit of the experimental platform was used to set the constant speed acceleration of the bearings. Two accelerations were set, 40 rpm/s and 80 rpm/s, to increase the running speed of the bearing from 800 rpm/min to 3200 rpm/min, respectively, and the recording times were about 1 min and 30 s.

The experimental data for a constant speed acceleration of 40 rpm/s were used in the experiment described in this section. Under these conditions, there were three fault types among the bearings, namely rolling body faults, inner ring faults, and outer ring faults. Each fault type comprised three degrees of damage. Including the running data of the bearings in a healthy state, there were a total of 10 operating states, and each damage degree comprised 300 training samples and 70 test samples. The training set adopted the method of overlapping sampling, with a sampling length of 2048 and a sampling location offset length of 128, while the test set did not adopt the method of overlapping sampling. Dataset A is the data collected under load 0, and datasets B, C, and D correspond to the data collected under loads 1, 2, and 3, respectively. The division of the experimental dataset is shown in

Table 11. The division of the experimental dataset.

Fault Location		None	Ball			Inner Race			Outer Race			Load
Labels		0	1	2	3	4	5	6	7	8	9
Fault Degree		None	Mild	Moderate	Severe	Mild	Moderate	Severe	Mild	Moderate	Severe
A	Training	300	300	300	300	300	300	300	300	300	300	0
	Testing	70	70	70	70	70	70	70	70	70	70
B	Training	300	300	300	300	300	300	300	300	300	300	1
	Testing	70	70	70	70	70	70	70	70	70	70
C	Training	300	300	300	300	300	300	300	300	300	300	2
	Testing	70	70	70	70	70	70	70	70	70	70
D	Training	300	300	300	300	300	300	300	300	300	300	3
	Testing	70	70	70	70	70	70	70	70	70	70

.

The trained fault diagnosis model was verified using the fixed and rising speed dataset with 100 iterations. The diagnostic results are shown in Figure 18.

It can be observed from Figure 18 that the model could quickly converge, and the diagnostic rate of the constant speed and acceleration datasets under each load reached more than 95%, with high diagnostic accuracy. The experimental results showed that the proposed method could achieve high diagnostic accuracy under the conditions of a constant and increasing bearing speed. The results of the experiment are shown in

Table 12. The results of the experiment.

	Detection Rate	Loss Function	Time (ms/step)	Iterations
Dataset A	95.79%	0.089	0.652	100
Dataset B	95.33%	0.092	0.701	100
Dataset C	97.69%	0.081	0.633	100
Dataset D	98.52%	0.079	0.655	100

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5. Conclusions

In this paper, a fault diagnosis method for rolling bearings based on the SNDCNN method was studied, and the following conclusions were drawn:

• The SNDCNN model, applicable to complex operating conditions, could directly input the raw vibration signal, and the fault detection rate reached 99.45% under multiple operating conditions.
• The method of increasing the convolution kernel width of the first layer and multi-layer convolution kernel stacking could effectively extract fault features and suppress high-frequency noise.
• The pooling operation of K-max pooling was used in the pooling layer, which could effectively retain the strong feature information.
• Each convolutional layer and fully connected layer adopted a switchable normalization method, which could effectively suppress overfitting and improve the model’s generalization performance.