Bearing fault damage degree identification method based on SSA-VMD and Shannon entropy

Abstract

Aiming at the problem that the weak fault signal of rolling bearing is affected by background noise and the weak fault signal itself leads to the difficulty in extracting fault features, a weak fault diagnosis method of rolling bearing based on sparrow search algorithm-variational mode decomposition (SSA-VMD) and Shannon entropy–exponential entropy decision is proposed. Firstly, the failure energy ratio of the original signal is acquired to judge the bearing failure. Secondly, the original time-domain signal is decomposed by the VMD optimized by SSA-VMD to obtain the Intrinsic Mode Function (IMF) component, and the kurtosis and correlation coefficient are normalized and fused. The fusion parameter ratio (R_V) is used to filter the IMF component, and the filtered component is reconstructed to achieve the noise reduction effect. The reconstructed signal is subjected to Hilbert transform to obtain the envelope spectrum of the vibration signal, and the fault type of the bearing can be judged. Finally, the entropy of the reconstructed signal is input into the model based on entropy-multilayer forward neural network (MFNN) to identify the degree of bearing fault damage. The effectiveness of the method is verified by using the experimental data of different fault types of intermediate shaft bearings in Shenyang Aerospace University and the self-built experimental data of outer ring fault detachment evolution. The results show that the fault energy ratio of the original signal is more conducive to judging whether the bearing has a fault than the reconstructed signal. The bearing fault type diagnosis method based on SSA-VMD and parameter fusion screening can effectively identify fault characteristic frequency and its frequency doubling of the inner and outer rings of rolling bearings. The entropy values of different bearing damage signals have different distribution regions, which verify the effectiveness of the bearing fault damage identification method based on entropy–MLP judgement.

Keywords

Rolling bearing weak fault unified fusion Shannon entropy exponent entropy damage degree faults diagnosis

Introduction

Rolling bearings are widely used in various fields of national economy and national defence, and play an important role in bearing and transmitting loads. Due to the high load and high speed working environment of bearings, they are prone to fault, which affects the performance of the entire rotating mechanical system. The weak fault diagnosis of rolling bearings can accurately determine the location and degree of faults in advance, which is conducive to reducing the cost of maintenance, improving the service life of the supporting transmission system, and thus improving the economy of rotating machinery. In the aspect of bearing fault diagnosis, the main methods are vibration detection, acoustic emission detection, temperature detection, gap detection and so on. Vibration detection has been widely used because of its easy measurement. For the early weak fault diagnosis of bearings, the main obstacle to obtaining fault signals from vibration signals is the interference of background noise. Therefore, it is particularly important to design a method to reduce background noise so as to effectively judge the fault type. In addition, on the basis of the known fault types of rolling bearings, how to further extract the feature information related to the degree of fault in the denoised signal to identify the degree of fault damage is very challenging and practical.

At present, abroad, Khakipour et al.¹ proposed a morphological gradient wavelet method based on the combination of gradient operator and morphological wavelet theory. The main advantages of this method are fast speed and simple implementation, which is suitable for real-time signal processing of on-line condition monitoring. Miao et al.² proposed a bearing fault diagnosis method based on short-time Fourier transform and deep learning of neural network and verified the method by the vibration signal obtained from the bearing test bench. The results show that the method can accurately classify various bearing faults under different working conditions. Leite et al.³ studied the characteristics based on 12 entropy. By using the vibration time waveform signal to detect the performance of bearing fault, the research shows that the entropy value has important value in bearing fault monitoring and detection. Maliuk et al.⁴ proposed a bearing fault diagnosis method based on the combination of wavelet packet transform (WPT) and Boruta algorithm and verified the effectiveness of the method through the rolling bearing fault data of Case Western Reserve University (CWRU) laboratory. Chegini et al.⁵ proposed a bearing fault diagnosis method based on Fisher discriminate analysis and F-score algorithm, binary particle swarm optimization and support vector machine (SVM). Singh et al.⁶ proposed a rolling bearing fault diagnosis method based on ensemble empirical mode decomposition (EMD) and Jensen Rényi divergence and then verified the proposed method on experimental data (seed defect data and accelerated bearing life test data). The results show that the proposed method can be used as a potential tool for bearing fault diagnosis. Gundewar Swapnil et al.⁷ proposed a bearing fault diagnosis software based on the combination of piecewise Fourier synchronous squeezing change and convolutional neural network (CNN). The proposed method achieved 100% bearing fault classification accuracy on the developed experimental device and CWRU bearing vibration dataset. Abdelkader et al.⁸ proposed a bearing fault diagnosis method based on improved EMD combined with kurtosis and envelope spectrum. The effectiveness of the proposed method was verified by different experimental data. The results show that the method is more effective than the traditional denoising method and is more sensitive to the early detection and diagnosis of rolling bearing faults. Al Raheem et al.⁹ proposed a rolling bearing fault diagnosis method based on Laplace-wavelet combined with artificial neural network and verified the effectiveness of the method through real and simulated bearing vibration data. Bouhalais et al.¹⁰ proposed a rolling bearing fault diagnosis method based on Complete Ensemble Empirical Mode Decomposition with Adaptive Noise Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) and Optimized Wavelet Multi-Resolution Analysis. Karnavas Yannis et al.¹¹ proposed a method based on CNN to judge bearing faults and verified it on CWRU bearing dataset and Paderborn University bearing dataset. The results show that the method has high accuracy. Rabah and Abdelhafid et al.¹² proposed a rolling bearing fault diagnosis based on the combination of improved adaptive noise complete ensemble empirical mode decomposition (ICEEMDAN) and minimum entropy deconvolution, which was verified by the CWRU database. The results show that the proposed method has a good effect on the early detection and diagnosis of defects and can efficiently extract the defect characteristics of rolling bearings. Golafshan et al.¹³ proposed the use of multi-body simulation to study the influence of structural dynamics on rolling bearing fault diagnosis. Piltan et al.¹⁴ proposed a hybrid fuzzy V-structure fault estimator scheme for bearing fault diagnosis. Moussaoui et al.¹⁵ proposed a bearing fault detection method under time-varying speed based on empirical wavelet transform and cultural family optimization algorithm and random forest classifier. Al Mamun et al.¹⁶ proposed a multi-channel sensor fusion real-time bearing fault diagnosis method based on frequency domain multi-linear principal component analysis. Vilma et al.¹⁷ proposed a method based on colour recurrence plot for bearing fault diagnosis. The vibration dataset of CWRU was used to verify the effectiveness of the method. Toma et al.¹⁸ proposed a bearing fault classification method that combines image coding technology with CNN. Muruganatham et al.¹⁹ proposed a bearing fault diagnosis method using vibration signal singular spectrum analysis (SSA) to extract bearing fault features. Georgoulas et al.²⁰ proposed a bearing fault detection method based on symbolic aggregate approximation framework and related intelligent icon representation.

Singh et al.²¹ proposed a rolling bearing fault diagnosis method based on over-complete rational dilation wavelet transform and autocorrelation of analytical energy operator.

Klausen et al.²² proposed a bearing fault diagnosis method based on whitened cross-correlation spectrum and verified it with the bearing dataset of Case University of Western Reserves.

Sinitsin et al.²³ proposed a bearing fault diagnosis method based on CNN-multilayer forward neural network (MLP). In China, Hongkai et al.²⁴ proposed a bearing early fault diagnosis method based on the combination of the second-generation wavelet packet and the corresponding decibel value of the fault characteristic frequency, and the effectiveness of the proposed method was verified by the early damage experiment of the outer ring of the rolling bearing. Jirong et al.²⁵ proposed a rolling bearing fault diagnosis method based on EMD and Shock Pulse Method (SPM) and verified the effectiveness of the method through a bearing system consisting of a weakly outer ring fault bearing and a weakly inner ring fault bearing. Yulong et al.²⁶ proposed a method based on entropy, Holder coefficient, improved fractal box dimension and grey correlation theory to judge the type and severity of bearing faults. The advantage of this method is that it can judge the type and severity of faults at the same time. The correctness of the method is verified by the bearing fault experimental data of the CWRU bearing data centre. Xiaochi et al.²⁷ proposed a fault diagnosis method based on the difference between the sum of the characteristic frequency doubling energy of the fault bearing and the normal bearing and the ratio of the energy of the entire envelope spectrum to determine the bearing fault. This method can visually indicate whether the bearing is faulty through the energy ratio. Fenglin et al.²⁸ proposed a rolling bearing fault diagnosis method based on WPT, signal eigenvalue and Extreme learning machine (ELM), which can classify the bearing fault types with high accuracy. Lei et al.²⁹ proposed an engine fault diagnosis method based on Shannon entropy to select wavelet packet signal features and verified it by piston pin knock fault diagnosis experiments. Zerui et al.³⁰ proposed a rolling bearing fault feature extraction method based on the combination of variational mode decomposition (VMD) and fast spectral kurtosis. This method solves the problem that the rolling bearing fault signal is susceptible to environmental noise interference, which leads to the difficulty of obtaining fault feature information. The rationality of the method is verified by public data and experimental analysis. Nanyang et al.³¹ proposed a bearing fault diagnosis method based on CEEMDAN and fast Kurtogram algorithm and verified the rationality of the method through the Gearbox dynamics simulation experiment platform (GDS) gearbox fault prediction comprehensive simulation test bench. The measured data show that the method can effectively detect the bearing inner ring fault. Si et al.³² proposed a weak fault signal extraction method for gearboxes based on the combination of maximum correlated kurtosis deconvolution and wavelet packet entropy. The bearing vibration acceleration signal was collected on the subway special gearbox test bench produced by SpectraQuest company, and the effectiveness of the method was verified by the collected data. A rolling bearing fault diagnosis method based on adaptive VMD and modulated signal bispectrum analysis (AVMD-MSB) is proposed by Shaoning et al.³³ By analysing the fault cases of the inner ring of the motor bearing and the rolling element of the supporting bearing respectively, it is shown that the method has high sensitivity and effectiveness in rolling bearing fault diagnosis compared with AVMD-Envelope and conventional VMD-MSB. Ming et al.³⁴ proposed a rolling bearing fault diagnosis method based on ICEEMDAN, multi-scale permutation entropy, skyhawk algorithm and least squares SVM. The effectiveness of the method was verified by the rolling bearing data of CWRU. Weiwang et al.³⁵ proposed a rolling bearing fault diagnosis method based on VMD, Sparrow Search Algorithm (SSA) and SVM and verified it by the rolling bearing fault data of CWRU laboratory. The results show that the method has high accuracy in fault identification. Miao et al.³⁶ proposed a new bearing weak fault diagnosis method based on improved singular spectrum decomposition and frequency-weighted energy slice bispectrum. The analysis results of simulation and experiment prove the effectiveness of the proposed method in alleviating modal aliasing and extracting weak fault symptoms of rolling bearings. Xiaocheng et al.³⁷ proposed a fault diagnosis of rolling element bearing weak fault based on sparse decomposition and broad learning network. By comparing with the commonly used intelligent network diagnosis methods, the superiority of the proposed method is verified. Qiao et al.³⁸ proposed a noise-enhanced weak fault diagnosis method by increasing the width of the potential well to enhance the mechanical weak fault characteristics, in which the amplitude amplification factor is regarded as an indicator to quantify the weak signal detection performance. The effectiveness of the proposed method is verified by some simulations and bearing fault experiments with outer ring defects. In order to use the weak characteristics to diagnose the fault of the wind turbine transmission system, Xu et al.³⁹ proposed a method for fault diagnosis of the wind turbine transmission system based on a Caputo–Fabrizio fractional order derivative (CF) SR improved by ascending density outlier factor. The effectiveness is validated by a simulation and two experimentations based on two real vibration signals collected from the key components of wind turbines. Compared with traditional diagnosis methods such as deconvolution and Kurtogram, the proposed method is superior for fault diagnosis of wind turbines working in severe environments. Zhihui et al.⁴⁰ proposed a high-performance adaptive weak fault diagnosis method based on the global parameter optimization model of a cascaded stochastic resonance system. And two rolling bearing weak fault diagnosis experiments are performed, thus verifying the effectiveness of the proposed approach in high-performance adaptive weak fault diagnosis.

In summary, at present, most of the bearing fault diagnosis methods at home and abroad are to judge the fault and diagnose the type of bearing fault, while the research on the method of identifying the damage degree of bearing fault is relatively less. And most of the existing damage degree identification methods can only be identified after the bearing has a moderate fault or even a serious fault, but cannot identify some early weak faults. This study is based on the analysis and processing of vibration signals. Firstly, the Hilbert transform is performed on the original time-domain signal to obtain the signal envelope spectrum and calculate the fault energy ratio. The fault energy ratio of the known normal bearing can be compared to determine whether the bearing is faulty. Secondly, the original time-domain signal is decomposed by SSA-VMD, and the signal is screened and reconstructed by kurtosis value and correlation coefficient. The envelope spectrum analysis of the reconstructed signal is carried out to obtain the characteristic fault frequency and its frequency doubling, and the characteristic frequency of the known fault type is compared to determine the fault type. Finally, the Shannon entropy and exponential entropy of the reconstructed signal are input into the construction model based on Shannon entropy–value entropy–MLP to identify the different damage degrees of bearing faults. The method is applied to a variety of bearing test data to verify the effectiveness of the method. This study provides an effective and accurate method for weak fault diagnosis of rolling bearings and achieves the purpose of reducing the fault rate of rolling bearings, thereby improving the utilization rate of rotating machinery and reducing maintenance costs. It has certain practicability and economic value.

Research on fault feature extraction and damage degree method of rolling bearing

Bearing fault diagnosis method based on Hilbert transform and fault energy ratio

1. According to the bearing parameters and speed information, the characteristic fault frequency and its frequency doubling of the inner ring fault and the outer ring fault of the bearing can be calculated (see Equations (15) and (16)).

2. For the original time-domain signal, the fault energy ratio of the signal is calculated according to the characteristic fault frequency of the inner ring fault and the outer ring fault.

3. The energy ratio of the inner ring characteristic fault frequency and the outer ring fault characteristic frequency is calculated and compared with the maximum energy ratio of the normal bearing signal. As long as any of the energy ratios is greater than the maximum energy ratio of the normal signal, the bearing fault can be determined.

The flowchart of the method is shown in Figure 1:

Bearing fault type diagnosis method based on SSA-VMD and parameter fusion screening

Sparrow search algorithm

The sparrow algorithm is a swarm intelligence optimization algorithm proposed by Xue et al.⁴¹ in 2020, which is not affected by the derivability, differentiability and continuity of the objective function. Sparrows have three possible behaviours in the process of foraging: (1) as a discoverer to find food and provide foraging areas and squares for the population; (2) as a participant, use the discoverer to obtain food and (3) as an investigator, when faced with danger to make anti-predator behaviour. The above can be summarized as an abstract model of search-follow-warning, which imitates how to obtain the solution of the problem to be optimized in the sparrow foraging. The following is the mathematical model of SSA algorithm:

Location update of discoverer

The discoverer has a good fitness value and will preferentially obtain food during the exploration process, and has a larger search range than the joiner. During each iteration, the discoverer ’s location is updated according to the following formula:

x_{i, d}^{t + 1} = {\begin{matrix} x_{i, d}^{t} \cdot \exp (- \frac{i}{α \cdot t_{max}}) if R_{2} < ST \\ x_{i, d}^{t} + Q \cdot L if R_{2} < ST \end{matrix}

(1)

where t is the current number of iterations, and $t_{max}$ is a constant representing the maximum number of iterations. $d = 1, 2, 3, \cdot \cdot \cdot, D$ , D are the dimensions of the variables to be optimized, and $x_{i, d}$ denotes the position of the ith sparrow in the dth dimension. $α (α \in (0, 1])$ is a uniform random number, Q is a random number that obeys a normal distribution, and L is a $1 \times D$ matrix in which all elements are 1. $R_{2} (R_{2} \in (0, 1])$ represents the warning value, and $ST (ST \in (0.5, 1])$ represents the safety value. When $R_{2} < ST$ , it means that there is no danger in the foraging environment at this time, and the discoverer can search freely; when the $R_{2} \geq ST$ , indicating that a sparrow found the danger, and an alarm, then the discoverer will follow the normal distribution of random movement to the current position near.

Participant location update

Participants will change their location because of poor foraging location or search for the finder who provides the best food in order to obtain food. The location update description of the participant is as follows:

x_{i, d}^{t + 1} = {\begin{matrix} Q \cdot \exp (\frac{x ω_{d}^{t} - x_{i, d}^{t}}{i^{2}}) if i > n / 2 \\ {xb}_{d}^{t + 1} + | x_{i, d}^{t} - {xb}_{d}^{t + 1} | \cdot A^{+} \cdot L if R_{2} < ST \end{matrix}

(2)

Here, n represents the total number of sparrows, $x ω_{d}^{t}$ represents the current global worst position and ${xb}_{d}^{t}$ represents the optimal position occupied by the current discoverer. A is a $1 \times d$ -size matrix, and each element in the matrix is randomly 1 or −1, $A^{+} = A^{T} {(A A^{T})}^{- 1}$ . When $i > n / 2$ , it means that the ith participant with lower fitness fails to find food and needs to change the position again.

Investigation and early warning

When foraging, some sparrows are responsible for vigilance. When the danger is close, both the finder and the participant will change their positions. The location update formula is as follows:

x_{i, d}^{t + 1} = {\begin{matrix} {xb}_{d}^{t} + β \cdot | x_{i, d}^{t} - {xb}_{d}^{t} | if f_{i} > f_{g} \\ x_{i, d}^{t} + K_{1} \cdot (\frac{| x ω_{d}^{t} - x_{i, d}^{t} |}{(f_{i} - f_{ω}) + ε}) if f_{i} > f_{g} \end{matrix}

(3)

Among them, $β$ is a normal distribution random number, obeying the mean of 0 and the variance of 1, which is used to control the step size. K₁ is a random number including a value $[- 1, 1]$ , $ε$ is the minimum constant, which can avoid zero denominator. $f_{i}$ represents the fitness value of the current sparrow individual, $f_{g}$ represents the current global best fitness value and $f_{w}$ represents the current global worst fitness value.

SSA optimizes VMD

SSA is used to optimize the decomposition layer K and the penalty factor $α$ in the VMD algorithm to obtain the optimal K and $α$ values. Among them, for SSA: the envelope entropy is selected as the fitness function value, and the entropy value can well reflect the sparse degree of the signal, which can reflect the probability distribution characteristics of the signal decomposition. The number of populations and the maximum number of iterations are set to $[N, M] = [10, 20]$ , the number of decomposition layers is set to K and the penalty factor $α$ is set to $[α_{min}, α_{max}] = [100, 3000]$ .

Kurtosis value-correlation coefficient fusion

Kurtosis value

The kurtosis value of the corresponding signal can directly reflect the degree of impact component of the equipment. The larger the kurtosis value is, the larger the impact component is Zerui et al.⁴² The fault characteristic signal generally accumulates in the frequency band with a large kurtosis value. When the rolling bearing is fault-free, the kurtosis value of the vibration signal changes within a stable range. When the rolling bearing has a fault, the impact component in the vibration signal will increase, and the kurtosis value will also dramatically increased.²⁷ The calculation formula of kurtosis index is:

K = \frac{E {(y - \bar{y})}^{4}}{S^{4}}

(4)

In the formula, E is expectation; y represents the amplitude of the vibration signal; $\bar{y}$ represents the mean value of the vibration signal amplitude y; S represents the standard deviation of the fault signal.

Correlation coefficient

The correlation coefficient is an index used to study the closeness between variables. The correlation coefficient is the value between $[- 1, 1]$ , and the larger the value, the greater the correlation between the variables. Let the two samples be X and Y, respectively, and the correlation coefficient is:

r (X, Y) = \frac{Cov (X, Y)}{\sqrt{D (X)} \sqrt{D (Y)}}

(5)

In the formula, r is the correlation coefficient of X and Y; $Cov (X, Y)$ is the covariance of X and Y; $\sqrt{D (X)}$ is the variance of X; $\sqrt{D (Y)}$ is the variance of Y.

Parameter fusion

The original time-domain signal is decomposed by SSA-VMD to obtain K IMF components, and the kurtosis value and correlation coefficient of each component are calculated. The two parameter values are normalized, and each parameter value is given a corresponding weight value. Then the two parameter values are added as a new parameter value (denoted as V parameter). The V parameter is defined as:

V = λ_{1} K + λ_{2} r

(6)

In the formula, K is the kurtosis value, r is the correlation coefficient, $λ_{1}$ is the weight value of the kurtosis value, $λ_{2}$ is the weight value of the correlation coefficient, and the two weight values satisfy:

λ_{1} + λ_{2} = 1

(7)

Reconstruction component screening principle

The original signal is decomposed by SSA-VMD to obtain K IMF components. The kurtosis, correlation coefficient of each component signal are calculated and the fusion parameter value V is obtained after normalization and fusion. The larger the V value is, the more fault information is contained in the sub-signal. The number of reconstructed components T used to reconstruct the signal is very important. If the T value is too small, the fault information in the reconstructed signal is too small, which can easily lead to distortion. The T value is too large, and the interference information in the reconstructed signal is too much, and the fault feature information is easily overwhelmed by the interference information. In this article, the V value based on the component signal is proposed as the screening index of T. Firstly, the V values of each component signal are sorted in descending order, and the sum of the V values of all components is calculated as $V_{sum} = \sum_{i = 1}^{K} V_{i}$ , and the sum of the V values of the first k $(k \in [1, K])$ components is calculated as $V_{k} = \sum_{i = 1}^{k} V_{i}$ . If $0.65 \leq V_{k} / V_{sum} \leq 0.75$ , T = k is taken, and the fusion parameter ratio $R_{V} = V_{k} / V_{sum}$ is defined. The specific method flowchart is shown in Figure 2:

Bearing fault type diagnosis method process

The bearing fault type diagnosis method based on SSA-VMD and parameter fusion screening is mainly composed of four modules, namely SSA-VMD optimization and decomposition module, IMF component parameter processing module, IMF component screening and reconstruction module, and fault type diagnosis module. The specific process is shown in Figure 3.

1. The original signal is input, and the bearing fault vibration signal is decomposed by SSA-VMD to obtain K IMF components in time domain.

2. Calculate the kurtosis value and correlation coefficient of each component, normalize the two parameters of K IMF components, assign corresponding weights to each parameter value, add the two parameter values of each IMF component as the V value of the component, and sort the V value in descending order.

3. Calculate the fusion parameter ratio $R_{V}$ of the component signal, select T IMF components at $0.65 \leq R_{V} \leq 0.75$ to reconstruct, and obtain the reconstructed time-domain signal.

4. The reconstructed time-domain signal is demodulated by Hilbert envelope to obtain the frequency-domain signal. Finally, the frequency-domain signal is compared with the characteristic frequency of the known fault type, and the statistical analysis is carried out to diagnose the bearing fault type.

Bearing fault degree diagnosis method based on Shannon entropy–exponential entropy–MLP

Shannon entropy

Guihua⁴³ proposed that Shannon entropy is a measure of the degree of signal sparsity, and its size can intuitively reflect the sparse characteristics of the signal. If the decomposed IMF component contains rich regular fault signals, the IMF component shows strong sparse characteristics and the Shannon entropy is small. If the decomposed IMF component contains a large number of noise signals or irregular signals, this component does not contain important fault information. It shows weak sparse characteristics and large Shannon entropy. The law of IMF component can be applied to the complete signal after reconstruction.

At present, the commonly used statistic is modified Shannon entropy, and its definition is:

H_{s} = - \frac{1}{\lg N_{obs}} \sum_{i} p_{i} \lg p_{i}

(8)

In the formula, $P_{i}$ is the probability of the ith symbol sequence, $N_{obs}$ is the number of possible sequences actually observed in the data and the range of modified Shannon entropy is $[0, 1]$ . In the case of pure random time series and equal probability partition, $H_{s} = 1$ ; for the determined time series, $H_{s} \in [0, 1)$ , the smaller the $H_{s}$ , the stronger the certainty of the data structure.

Exponent entropy

The definition of Shannon entropy effectively solves some uncertainty problems in the information neighbourhood, but there are still some problems in solving practical problems. Therefore, in order to solve this problem and make the theory more practical, an entropy value, namely exponential entropy, is redefined. The algorithm is as follows:

S (p (t)) = \sum_{t = 0}^{T} p (t) e^{1 - p (t)}

(9)

Multilayer forward neural network

In the classification of known midges, the antennae and wing length of midges are often used as classification indicators. Shuiming⁴⁴ classified midges by homotopy BP neural network and achieved good results. Based on the fact that entropy can be used to measure the uncertainty of signal distribution state and the complexity of signal, this article introduces exponential entropy and Shannon entropy to reflect the index of signal fault characteristics and classifies the fault degree of signals with the same fault type by MLP.

MLP is composed of input layer, hidden layer and output layer, as shown in Figure 4. The input of the nodes in the hidden layer and the output layer is the sum of the output values of all nodes in the previous layer. The excitation output value of each node is determined by the node input, excitation function and offset.

In layer j, the input value of the node is:

ne t_{j} = \sum w_{ji} o_{j} + θ_{j}

(10)

In the formula, $θ_{j}$ is the threshold value, and the action of the positive threshold value shifts the excitation function left along the x-axis. $w_{ji}$ refers to the weight given by the hidden unit j to the output unit i.

The output value of the node is:

o_{j} = f (ne t_{j})

(11)

In the formula: f is the activation function of the node, select the Sigmoid function:

f (x) = \frac{1}{1 + \exp (- x)}

(12)

The network node input at the k-th layer is:

ne t_{k} = \sum w_{kj} o_{j} + θ_{k}

(13)

The output of the network node at the k-th layer is:

o_{k} = f (ne t_{k})

(14)

Process of bearing fault damage degree identification method

The bearing fault damage degree identification method based on Shannon entropy–exponential entropy–MLP is mainly composed of three functional modules, namely entropy calculation module, neural network training module and fault degree identification module. As shown in Figure 5, the specific process is described as follows:

1. The Shannon entropy and exponential entropy of the reconstructed time-domain signal processed by SSA-VMD and parameter fusion screening method are calculated to obtain the entropy values of signals with different fault degrees.

2. The database is established by using Shannon entropy and exponential entropy of known fault degree data, and the corresponding target input values are set up for the entropy data of different fault degrees, and trained by neural network.

3. When the entropy value of the unknown fault degree signal is input, the entropy value is automatically classified by the neural network, and the output value of the neural network is obtained. The fault degree is judged by comparing with the existing target input value.

Validation of rolling bearing fault feature extraction method

Bearing data validation of Case Western Reserve University

Test description of deep groove ball bearing of Case Western Reserve University

In order to verify the effectiveness of the proposed bearing fault feature extraction method in bearing fault diagnosis under simple transfer path, the method is verified by the typical test of deep groove ball bearing of Case Western Reserve University. The test bench is shown in Figure 6. It consists of a fan end bearing, a 2-horsepower motor, a drive end bearing, a torque sensor, a power tester and a control electronic device. The test bearing supports the motor shaft, and the single point fault is introduced into the test bearing by EDM. The fault diameters are 7, 14 and 21 miller (mil) (7 mil is equal to 0.1778 mm, 14 mil is equal to 0.3556 mm, 21 mil is equal to 0.53334 mm, all of which are weak faults). The test bearing model is 6205-2RS JEM SKF (Svenska Kullager-Fabriken, Gothenburg, Sweden) deep groove ball bearing, and the parameters are shown in Table 1. The fault type of the bearing is selected as the inner/outer ring pitting fault. The sampling frequency is 12 kHz, the sampling time is 10 s and the test speed is 1730, 1750, 1772 and 1797 rpm. The vibration signals of the normal bearing and the fault bearing at different positions are collected.

Table 1.

6205-2RS JEM SKF deep groove ball bearing geometric parameters.

Parameter	Numerical value
Ball diameter (d/mm)	7.94
Pitch diameter (D/mm)	39.04
Contact angle (α/°)	0
Number of balls Z	9

Verification of bearing fault judgement method

The vibration signal data of normal bearing, inner ring fault bearing and outer ring fault bearing with sampling frequency of 12 kHz and motor speed of 1730, 1750, 1772 and 1797 rpm are selected for analysis. Among them, there are four groups of normal data, corresponding to the above four speeds. The inner/outer ring faults are all generated by electric spark technology. There are 12 sets of outer ring fault data, and there are three fault degrees, which are 7, 14 and 21 mil, respectively. Each fault degree corresponds to four speeds. The inner ring fault data is also 12 sets of data corresponding to three fault degrees and four rotational speeds. The envelope spectrum is obtained by envelope demodulation of the original time-domain signal. Each signal calculates the corresponding fault energy ratio under the condition of the inner ring fault characteristic frequency and the outer ring fault characteristic frequency at the corresponding speed, and each signal can obtain two fault energy ratios. The results are as follows: the maximum fault energy ratio of normal bearing data is 3.155%, whereas the minimum fault energy ratio of fault bearing of fault inner ring data is 13.262, and the minimum fault energy ratio of outer ring fault bearing data is 4.027%. As shown in Figure 7, the fault energy ratio of the inner ring fault bearing and the outer ring fault bearing is greater than the energy ratio of the normal bearing under the same working condition.

Verification of bearing fault type diagnosis method

The vibration signal data of normal bearing, inner ring fault bearing and outer ring fault bearing with sampling frequency of 12 kHz and motor speed of 1730 rpm are selected. The vibration signal of the outer ring fault bearing is analysed in detail. The time-domain signal diagram of the original signal is shown in Figure 8. It can be seen from the time-domain diagram that the vibration amplitude is −3 to 3 m/s². The original time-domain signal is processed by global fast Fourier transform, and the frequency-domain signal is shown in Figure 9. The characteristic frequency and frequency doubling of the outer ring fault cannot be directly seen in the frequency-domain signal. Therefore, the spectral analysis of the original time-domain signal cannot effectively identify the fault characteristics of the outer ring of the bearing.

The VMD is optimized by SSA. For the SSA algorithm, the maximum number of iterations is set to 30, the population size is 10, the range of penalty factors is [100, 3000] and the range of decomposition layers is [3, 12]. In fault vibration signal of the outer ring of the input data, the optimized K = 9, $α = 100$ and the minimum envelope entropy is 8.976. SSA-VMD is used to decompose the original data of bearing outer ring fault, and nine IMF components are obtained. As shown in Figure 10, the time-domain signals of nine IMF components are shown. It can be seen from the figure that some component signals show obvious periodicity, and some signals are irregularly distributed.

The kurtosis and correlation coefficient index values of nine IMF components are calculated and normalized. Each parameter is multiplied by the corresponding weight value and then added (the weight value of kurtosis is 0.5, and the weight value of correlation coefficient is 0.5). The two parameters are merged into a V index, as shown in Equation (6), as shown in Table 2. The kurtosis, correlation coefficient and normalized fusion V index value of each IMF component. The V values of the nine IMF components are sorted in descending order, and the number of components satisfying $0.65 \leq R_{V} \leq 0.75$ is solved. When k = 5, $R_{V} = 0.65$ and the number of reconstructed components T = 5 are obtained. The five IMF components with the largest V value (IMF2, IMF3, IMF4, IMF5, IMF7) are used for signal reconstruction, and the reconstructed time-domain signal is shown in Figure 11. By screening and reconstructing the IMF, the interference signal in the reconstructed signal is reduced, so as to achieve the purpose of noise reduction.

Table 2.

V values of nine IMF components after VMD decomposition.

Quantity	Kurtosis value	Correlation coefficient	V value
IMF₁	3.347	0.308	0.255
IMF₂	2.169	0.540	0.344
IMF₃	2.607	0.435	0.299
IMF₄	1.976	0.536	0.332
IMF₅	2.737	0.421	0.296
IMF₆	3.479	0.343	0.283
IMF₇	10.086	0.118	0.453
IMF₈	2.114	0.422	0.268
IMF₉	2.955	0.131	0.127

VMD: variational mode decomposition.

Envelope demodulation of the reconstructed time-domain signal is performed to obtain the reconstructed signal envelope spectrum as shown in Figure 12. As shown in Figure 12, the characteristic fault frequency and its frequency doubling of the outer ring fault can be obtained. The frequencies from one time to seven times are 103, 207, 310, 414, 517, 621 and 724 Hz respectively. Through the calculation formula of bearing fault characteristic frequency, the theoretical fault characteristic frequency can be obtained. By comparing the actual fault characteristic frequency with the theoretical fault characteristic frequency, the fault type can be judged. The following is the bearing characteristic frequency calculation formula:

The calculation formula of bearing outer ring fault is:

f_{o} = \frac{1}{2} Z (1 - \frac{d}{D} \cos α) f_{s}

(15)

The calculation formula of bearing inner ring fault is:

f_{i} = \frac{1}{2} Z (1 + \frac{d}{D} \cos α) f_{s}

(16)

The calculation formula of bearing rolling element fault is:

f_{b} = \frac{1}{2} \frac{D}{d} f_{s} (1 - {(\frac{d}{D})}^{2} \cos^{2} α)

(17)

In the formula,: D is the rolling element bearing pitch diameter (mm); d is the diameter of the rolling element (mm); Z is the number of rolling elements; α is the contact angle; f_s is the rotation frequency (Hz).

Similarly, the original signal of the vibration signal data of the inner ring fault bearing and the normal bearing with the speed of 1730 rpm is decomposed by VMD, and the reconstructed signal is obtained after the two indexes of kurtosis value-correlation coefficient are screened, and the envelope spectrum of the reconstructed signal is drawn. Figures 13 and 14 are the envelope spectrum of the inner ring fault bearing signal and the normal bearing signal respectively. For the data of the inner ring fault bearing, there will be an amplitude jump at the frequency doubling of its characteristic fault. The frequency doubling to the seven frequency doubling are: 155, 310, 465, 620, 775, 929 and 1084 Hz. For normal bearing data, the amplitude distribution in the envelope spectrum is disorderly, and there is no fault characteristic frequency and its frequency doubling.

In summary, it can be seen that the envelope spectrum of normal bearing data fails to reflect the characteristic fault frequency and its frequency doubling, whereas for the inner and outer ring fault bearings, the amplitude in the envelope spectrum will jump at the frequency doubling of its fault characteristics. By comparing with the theoretical fault characteristic frequency, the fault type can be distinguished.

Bearing data validation of Shenyang Aerospace University

Intershaft bearing test of Shenyang Aerospace University

In order to further verify the effectiveness of the method in this article, a fault simulation test bench for an aero-engine intermediate bearing is built independently. The test bench is mainly composed of high-speed shaft, low-speed shaft, casing, load loading system, lubricating oil system, water cooling system, electrical control system, compressed air system and fog removal device. The maximum speed of the aero-engine rotor system test bench is 18,000 rpm, and the maximum radial load is 20 kN, which can control the inner and outer rings to rotate in the same direction or in the opposite direction. Figure 15 shows the installation position of the vibration sensor and the simulation test bench for the intershaft bearing of the aero-engine. Fault signalling path of the test bench is shown in Figure 16. The intermediate bearing is located between the high and low pressure rotors of the engine. The test bearing type is selected as the short cylindrical roller bearing (inter-shaft bearing). The fault type is the inner ring wire cutting fault and the outer ring wire cutting fault. The fault is a rectangular groove damage with a length of 7 mm, a width of 1 mm and a depth of 1 mm by wire cutting technology. The local picture of the fault bearing is shown in Figure 17. The vibration sensor is installed outside the casing. The engine bearing type and its parameters are shown in Table 3.

Table 3.

Types and parameters of engine bearings.

Parameter	Numerical value
Outer ring diameter (mm)	140
Inner ring diameter (mm)	110
Ball diameter (mm)	8
Contact angle (°)	0
Number of balls	34

In this experiment, a certain type of aero-engine five fulcrum spindle bearing is used to carry out the inter-shaft fault test experiment of aero-engine intermediate bearing. The test system consists of speed sensor, Integrated circuits piezoelectric (ICP) acceleration vibration sensor, INV3062S intelligent acquisition instrument, DASP V11 (Dongfang Zhice Technology Co., Ltd., Beijing, China) engineering platform software, data processing software and computer. In the experiment, the 333B30 ICP acceleration sensor of PCB (PCB Piezotronics,Inc., Buffalo) company in the United States was used to collect the vibration signal, and the sensitivity of the sensor was 10 mV/g. The vibration sensor is fixed on the outer surface of the casing of the bearing test bench. The speed sensor and the vibration sensor are connected to the acquisition instrument through the signal line. The acquisition instrument is connected to the computer equipped with DASP test software. The connection diagram of the whole test system is shown in Figure 18.

Verification of bearing fault judgement method

The data of the intermediate bearing of Shenyang Aerospace University are selected for further verification of the method, which are: (1) the normal fault data of the actual speed of the outer ring is 3527 rpm, the actual speed of the inner ring is 2958 rpm, and the relative speed is 569 rpm; (2) the actual speed of the outer ring is 5955 rpm, the actual speed of the inner ring is 5644 rpm, and the relative speed is 311 rpm and (3) the actual speed of the outer ring is 2031 rpm, the actual speed of the inner ring is 1492 rpm and the relative speed is 539 rpm.

Obtain the fault energy ratio of the three signals. As shown in Table 4, the energy ratio of the normal bearing is 3.367% according to the characteristic frequency of the outer ring fault, and the energy ratio calculated according to the characteristic frequency of the inner ring fault is 3.153%. The energy ratio calculated by the outer ring fault characteristic frequency of the outer ring fault bearing is 8.247%, and the energy ratio calculated by the inner ring fault characteristic frequency of the inner ring fault bearing is 5.757%, which is greater than the energy ratio of the normal bearing under the corresponding characteristic fault frequency. Accordingly, it can be judged whether the bearing is faulty.

Table 4.

Fault energy ratio of normal bearing, inner ring fault bearing and outer ring fault bearing.

Data	Normal bearings	Inner ring fault bearing	Outer ring fault bearing
Outer ring fault Energy ratio	3.367	2.633	8.247
Inner ring fault Energy ratio	3.153	5.757	4.285

Verification of bearing fault judgement method (comparison of original signal and reconstructed signal)

The data of the intermediate bearing of Shenyang Aerospace University in section ‘Verification of bearing fault judgement method’ are selected to verify the method again. Firstly, the reconstructed signal is obtained by SSA-VMD and parameter fusion screening method. The fault energy ratio formula is used to calculate the fault energy ratio of the three reconstructed signals, as shown in Table 5. Comparing the reconstructed signal energy ratio table (Table 5) and the original signal energy ratio table (Table 4), it can be obtained that for the normal bearing signal, the energy ratio difference between the original signal and the reconstructed signal is not large; for the fault bearing signal, the energy ratio of the original signal is greater than the energy ratio of the reconstructed signal, that is, the energy ratio of the original signal can more effectively determine whether the bearing is faulty.

Table 5.

(Reconstructed signal) Fault energy ratio of normal bearing, inner ring fault bearing and outer ring fault bearing.

Data	Normal bearings	Inner ring fault bearings	Outer ring fault bearings
Outer ring fault energy ratio (%)	3.132	2.250	5.496
Inner ring fault energy ratio (%)	3.294	4.738	3.460

Verification of bearing fault type diagnosis method

Based on SSA-VMD and parameter fusion screening method, the data of normal bearing, inner ring fault bearing and outer ring fault bearing are processed, and the data of inner ring fault bearing are explained in detail. Firstly, SSA optimizes the VMD and obtains the K and $α$ values of each data under the optimal envelope entropy. The optimized values of the three signals are shown in Table 6. Secondly, SSA-VMD is used to decompose the original signal to obtain K IMF components. For example, Figure 19 is the original time-domain signal diagram of the inner ring fault bearing signal, and Figure 20 is the IMF component time-domain diagram of the inner ring fault bearing signal. Then, the kurtosis and correlation coefficient values of each component are calculated, and the two parameters are normalized and fused to obtain the V value (The weight value of kurtosis is 0.5, and the weight value of correlation coefficient is 0.5). Table 7 is the kurtosis, correlation coefficient and V value of each IMF component signal of the inner ring fault bearing signal. The fusion parameter ratio $R_{V}$ $(0.65 \leq R_{V} \leq 0.75)$ is used to filter the reconstructed component number T. The $R_{V}$ and the corresponding reconstructed component number of different fault types are shown in Table 8. Finally, the T component signals are reconstructed to obtain the reconstructed time-domain signal of the inner ring fault bearing signal. The reconstructed time-domain signal envelope demodulation is used to obtain the reconstructed signal envelope spectrum (Figure 21). As shown in Figure 21, the obvious inner ring fault characteristic fault frequency and its frequency doubling can be obtained, one time frequency 94 Hz, two times frequency 189 Hz, three times frequency 282.3 Hz, four times frequency 377.3 Hz and six times frequency 564 Hz. Similarly, the reconstructed signal envelope spectrum of the outer ring fault bearing data and the normal bearing data can be obtained, as shown in Figures 22 and 23. The fault type can be judged by comparing the characteristic fault frequency in the fault bearing signal envelope spectrum.

Table 6.

Three sets of data were optimized by SSA for VMD parameter values.

Data	Optimal envelope entropy	K (decomposition level)	α (Chasten factor)
Normal bearings	7.685	7	589
Inner ring fault bearing	7.589	7	1744
Outer ring fault bearing	7.840	12	3000

SSA: sparrow search algorithm; VMD: variational mode decomposition.

Table 7.

Kurtosis, correlation coefficient and V value of nine IMF components after VMD decomposition of inner ring fault bearing data.

Component	Kurtosis value	Correlation coefficient	V value
IMF₁	3.054	0.399	0.247
IMF₂	3.195	0.695	0.456
IMF₃	3.637	0.575	0.460
IMF₄	3.781	0.245	0.281
IMF₅	3.184	0.297	0.207
IMF₆	5.197	0.267	0.546
IMF₇	3.566	0.226	0.231

VMD: variational mode decomposition.

Table 8.

Fusion parameter ratio and corresponding reconstruction component number T of normal bearing, inner ring fault bearing and outer ring fault bearing data.

Data	R_V (Fusion parameter ratio)	K (Reconstructed component) number T
Normal bearings	0.73	4
Inner ring fault bearing	0.72	4
Outer ring fault bearing	0.65	6

Noise reduction effect verification of bearing fault type diagnosis method (simulation signal)

The simulation signal is used to verify the effectiveness of the bearing fault type diagnosis method in noise reduction, and the strong white noise is added to simulate the environmental noise and other interference signals in the actual working conditions. The simulation signal is as follows:

{\begin{matrix} y (t) = & x (t) + n (t) = \sum_{κ} A_{κ} h (t - κ T_{1} - τ_{κ}) + n (t) \\ A_{κ} = & A_{0} \sin (2 π f_{r} t) + 1 \\ h (t) = & \exp (- Ct) \sin (2 π f_{n} t) \end{matrix}

(18)

In Equation (18), $A_{0} = 0.5$ ; the frequency conversion $f_{r}$ is 25 Hz; the attenuation coefficient C is 800; the resonance frequency $f_{n}$ is 4000 Hz; inner ring fault frequency $f_{i} = 1 / T_{i} = 120 Hz$ ; $τ_{κ}$ is the small fluctuation of the κth impact relative to the period T. The random fluctuation obeys 0 as the mean normal distribution, and the standard deviation is 0.5% of the rotation frequency. $n (t)$ is a Gaussian white noise component, and the signal-to-noise ratio of the noisy signal is set to −5 dB; the sampling frequency $f_{s}$ is 12,800 Hz; the number of analysis points is 12,800.

The optimal parameters are obtained by SSA optimization: the number of decomposition layers K = 4, the penalty factor α = 100, and the minimum envelope entropy is 6.833. When the fusion parameter ratio (R_V) = 0.66, the number of reconstructed components T = 2.

As shown in Figure 24, the time-domain signal diagram of the analogue signal, the noisy signal, and the reconstruction signal (the optimized noisy signal) is shown. As shown in Figure 25, the envelope spectrum of the noisy signal is shown. As shown in Figure 26, the envelope spectrum of the reconstruction signal is shown. Compared with the two diagrams, obvious noise reduction can be seen. The signal-to-noise ratio is defined as the ratio of the intensity of the received useful signal to the intensity of the received interference signal, which can be used as a parameter to measure the signal quality. The mathematical expression of signal-to-noise ratio is as follows:

SNR = 10 \lg \frac{P_{s}}{p_{n}}

(19)

In the formula, $P_{s}$ is the power of the signal and $P_{n}$ is the power of the noise.

The signal-to-noise ratio of the noise-added signal and the reconstruction signal is shown in Table 9. The optimized signal-to-noise ratio is reduced by 3.254 dB, and the reduction rate is 37.4%. The quality of the signal is significantly improved.

Table 9.

The signal-to-noise ratio of the noisy signal and the optimized noisy signal.

Data	Signal-to-noise ratio (dB)	Optimization quantity (dB)	Optimization rate (%)
Noisy signal	−8.699	3.254	37.4
Reconstruction signal	−5.445	3.254	37.4

Effectiveness verification of rolling bearing fault damage degree identification method

Bearing data validation of Case Western Reserve University

The fault data of the outer ring of the drive end and the normal bearing data are selected, and the sampling frequency is 12 kHz. The fault degree is 7, 14 and 21 mil, respectively. Each fault degree corresponds to four different speeds, which are 1797, 1772, 1750 and 1730 rpm, respectively. The data of three fault degrees of the outer ring and the normal bearing data are selected, and 16 groups of data are taken for each data, a total of 64 groups of data. Firstly, based on SSA-VMD and parameter fusion screening method, the reconstructed time-domain signal is obtained. The Shannon entropy and exponential entropy of the reconstructed time-domain signal are calculated according to Equations (8) and (9), respectively. The entropy database is established with a signal (Shannon entropy, exponential entropy) as a data point, and the entropy value of the bearing outer ring fault database is drawn as shown in Figure 27.

It can be seen from Figure 27 that for the outer ring fault, the data signals of different fault degrees are distributed in a specific area in the coordinate system with exponential entropy and Shannon entropy as parameters. For example, when the fault degree of the outer ring is 7 mil, the range of Shannon entropy is 1.519–1.913, and the range of exponential entropy is 1.974–2.214; for the data with the outer ring fault degree of 14 mil, the range of Shannon entropy is 2.071–2.328, and the range of exponential entropy is 2.347–2.436. The range of Shannon entropy is 0.696–1.167, and the range of exponential entropy is 1.381–1.727 when the fault degree of the outer ring is 21 mil. Figure 28 is the database entropy diagram of the outer ring fault bearing signal and the normal bearing signal. In the diagram, the fault diameter 14 mil data points and the normal bearing data points are locally amplified. From Figure 28, it can be seen that the entropy value of the normal bearing signal overlaps with the entropy value of the 14 mil fault signal. At this time, it is impossible to judge the fault damage degree corresponding to the entropy range. Therefore, in this article, the normal bearing signal is not introduced in the entropy analysis, and only the entropy value is used to identify the damage degree of the faulty signal. However, before identifying the damage degree, the method based on the energy ratio has judged whether the bearing is faulty or not.

Similarly, using the known inner ring fault diameter of 7, 14 and 21 mil three different fault degree data, each of 16 groups, a total of 48 groups of data to calculate the entropy value and establish the entropy value database. The entropy value diagram of the bearing inner ring fault database is shown in Figure 29. Because the fault diameter 21 mil data and the fault diameter 7 mil data entropy value distribution range are similar, the fault diameter 7 mil data point and the fault diameter 7 mil data point are locally amplified in Figure 29. From the diagram, it can be seen that the entropy value of the signal under different degrees of bearing inner ring fault is distributed in different ranges.

The multi-layer forward neural network is used to train the data of the outer ring database. The fault diameter is set to 7 mil data target input as (0, 1), the fault diameter is 14 mil data target input as (1, 0) and the fault diameter is 21 mil data target input as (1, 1). Four groups of data with outer ring fault degree of 7, 14 and 21 mil are selected, a total of 12 groups of sample inputs to be tested, and the target output after neural network classification is as shown in Table 10. It can be seen from the table that 12 sets of data are correctly identified under three fault degrees of bearing outer ring fault. The entropy diagram of 48 groups of known data and 12 groups of samples to be tested is shown in Figure 30. In Figure 30, the data points of fault diameter 7 mil and fault diameter 14 mil are partially amplified. It can be seen from the figure that the signals of different fault degrees are in accordance with the corresponding entropy distribution range, and the recognition rate of fault degree reaches 100%. It shows that the neural network classification method based on Shannon entropy and exponential entropy as classification indexes has a good recognition effect on the fault degree identification of bearing outer ring. Similarly, the multi-layer forward neural network is used to train the data of the inner ring database, and the data of the inner ring fault degree of 7, 14 and 21 mil are selected. A total of 12 groups of samples to be tested are input. The target output after neural network classification is shown in Table 11. The results of neural network recognition are as follows: 12 groups of samples to be tested are correctly identified. The entropy diagram of 48 groups of known data and 12 groups of samples to be tested is shown in Figure 31. In Figure 31, the data points of fault diameter 7 mil and fault diameter 21 mil are locally amplified, which conforms to the law that the entropy values of signals with different fault degrees are distributed in different entropy intervals, and the recognition rate of fault degree reaches 100%.

Table 10.

Entropy values of different degrees of bearing outer ring fault are classified by neural network.

Training samples	Target output	Network output	Training samples	Target output	Network output
(1.712, 2.1)	(0, 1)	(0, 1)	(2.087, 2.354)	(1, 0)	(1, 0)
(1.606, 2.031)	(0, 1)	(0, 1)	(2.023, 2.333)	(1, 0)	(1, 0)
(1.588, 2.024)	(0, 1)	(0, 1)	(0.752, 1.429)	(1, 1)	(1, 1)
(1.518, 1.967)	(0, 1)	(0, 1)	(0.787, 1.477)	(1, 1)	(1, 1)
(2.13, 2.37)	(1, 0)	(1, 0)	(0.871, 1.547)	(1, 1)	(1, 1)
(2.113, 2.367)	(1, 0)	(1, 0)	(0.691, 1.382)	(1, 1)	(1, 1)

Table 11.

Entropy values of different degrees of bearing inner ring fault are classified by neural network.

Training samples	Target output	Network output	Training samples	Target output	Network output
(1.805, 2.198)	(0, 1)	(0, 1)	(0.869, 1.581)	(1, 0)	(1, 0)
(1.77, 2.183)	(0, 1)	(0, 1)	(0.971, 1.68)	(1, 0)	(1, 0)
(1.775, 2.183)	(0, 1)	(0, 1)	(1.575, 2.068)	(1, 1)	(1, 1)
(1.787, 2.189)	(0, 1)	(0, 1)	(1.646, 2.1)	(1, 1)	(1, 1)
(1.058, 1.787)	(1, 0)	(1, 0)	(1.538, 2.052)	(1, 1)	(1, 1)
(0.834, 1.598)	(1, 0)	(1, 0)	(1.518, 2.033)	(1, 1)	(1, 1)

Bearing data validation of Xi’an Jiaotong University

Xi’an Jiaotong University Spectra Quest bearing test instructions

The effectiveness of the fault diagnosis method proposed in this article is further verified by using the SQ (Spectra Quest) bearing data of Xi’an Jiaotong University.⁴⁵ The experiment uses the SQ company’s mechanical fault comprehensive simulation test bench to simulate the fault of the outer ring and inner ring of the motor bearing. The structure of the test bench is shown in Figure 32. The test bench is composed of motor, rotor, load, acceleration sensor (sensitivity of 50 mV/g), CoCo80 data acquisition (sampling frequency of 25.6 KHz) and so on. The signal has three kinds of rotation frequencies, which are 19.05, 29.05 and 39.05 Hz, respectively. The bearing fault degree is divided into two types: inner ring fault and outer ring fault. There are three kinds of fault degrees under each type of fault type, which are weak fault, moderate fault and severe fault. The experimental bearing model is NSK6203 deep groove ball bearing.

Verification of bearing fault damage degree identification method

The fault type is selected as the inner ring fault, and the data under three fault degrees (weak, moderate and severe) are analysed. The rotation frequency of the data is 29.0 Hz. The local diagram of different fault degrees of the inner ring fault bearing is shown in Figure 33, and the characteristic fault frequency is 143.25 Hz. The data of each fault degree are taken from 12 groups, a total of 36 groups. The SSA-VMD is used to decompose the original signal, the kurtosis value-correlation coefficient is used to calculate the fusion parameter value V of the component signal, the $R_{V}$ is used to filter the component signal and the reconstructed component signal is used to obtain the reconstructed time-domain signal. The Shannon entropy and exponential entropy of each signal are calculated. Nine signals of each fault degree are taken, a total of 27 are used to establish a database, and the (Shannon entropy, exponential entropy) of a signal is used as a data point. The target input of weak fault data is set as (0, 1), the target input of moderate fault data is set as (1, 0) and the target input of severe fault data is set as (1, 1). The entropy value diagram of the database is shown in Figure 34, and the obvious distribution law can be seen from the diagram, that is, the signals of different fault degrees are distributed in a specific entropy value area and show a certain linear relationship. The MLP is used to train the database, and three groups of data with weak fault, moderate fault and severe fault of the inner ring are selected. A total of 9 groups of data to be tested are input into the Shannon entropy–exponential entropy–MLP model. The target output after MLP classification is shown in Table 12. From Table 12, it can be seen that nine sets of data under three fault degrees of bearing inner ring fault are identified correctly. The entropy diagram of 27 sets of database data and nine sets of data to be tested is shown in Figure 35, which conforms to the law that the entropy values of different fault degrees are distributed in a specific entropy range.

Table 12.

Inner ring fault bearing test data entropy and its target output.

Training samples	Target output	Network output
(1.903, 2.272)	(1, 0)	(1, 0)
(1.872, 2.259)	(1, 0)	(1, 0)
(1.848, 2.251)	(1, 0)	(1, 0)
(1.868, 2.244)	(0, 1)	(0, 1)
(1.873, 2.242)	(0, 1)	(0, 1)
(1.832, 2.228)	(0, 1)	(0, 1)
(1.68, 2.159)	(1, 1)	(1, 1)
(1.702, 2.171)	(1, 1)	(1, 1)
(1.711, 2.185)	(1, 1)	(1, 1)

Optimization effect comparison of bearing fault damage degree identification method

The same SQ bearing data of Xi’an Jiaotong University in section “Verification of bearing fault damage degree identification method” are selected for analysis, and the Shannon entropy and exponential entropy of the original time-domain signal are directly calculated to compare the identification methods of bearing fault damage degree proposed in this article. The database entropy diagram is shown in Figure 36. Compared with the database entropy diagram of the reconstructed signal (Figure 34), it can be seen that the entropy convergence effect of the original signal is worse than that of the reconstructed signal under the same fault degree, and it is difficult for individual points to distinguish which fault degree they belong to. Three groups of data with weak fault, moderate fault and severe fault in the inner ring are selected, and a total of nine groups of data to be tested are input into the Shannon entropy–exponential entropy–MLP model. The target output after MLP classification is shown in Table 13. From Table 13, it can be seen that one of the nine sets of data under the three fault degrees of bearing inner ring fault failed to identify. The data target output with (Shannon entropy, exponential entropy) of (1.803, 2.236) should be (1, 0), but the actual output is (0.114, 0.680) ≈ (0, 1). The entropy diagram of 27 sets of database data and 9 sets of data to be tested is shown in Figure 37. The distribution position of data with (Shannon entropy, exponential entropy) of (1.803, 2.236) on the entropy diagram is between severe fault and moderate fault, and MLP identification is wrong.In summary, the entropy distribution law of the reconstructed signal obtained by SSA-VMD and multi-parameter fusion screening method is more obvious than that of the original signal, and it is more effective in identifying the degree of fault damage.

Table 13.

(original signal) Inner ring fault bearing test data entropy and its target output.

Training samples	Target output	Network output
(1.923, 2.285)	(1, 0)	(1, 0)
(1.923, 2.281)	(1, 0)	(1, 0)
(1.803, 2.236)	(1, 0)	(0.114, 0.680)
(1.846, 2.24)	(0, 1)	(0, 1)
(1.891, 2.259)	(0, 1)	(0, 1)
(1.836, 2.235)	(0, 1)	(0, 1)
(1.737, 2.191)	(1, 1)	(1, 1)
(1.77, 2.206)	(1, 1)	(1, 1)
(1.792, 2.217)	(1, 1)	(1, 1)

Mechanism analysis of Shannon entropy–exponential entropy distribution law of bearings with different damage degrees

Theoretical analysis

According to the definition of entropy value, the more severe the fault degree is, the more regular information should be contained in the vibration signal, and the smaller the entropy value of the signal will be. The analysis of SQ bearing data of Xi’an Jiaotong University shows that the entropy value distribution is shown in Table 14. The entropy value of weak fault signal is the largest, followed by that of moderate fault signal, and the entropy value of severe fault is the smallest, which conforms to the theoretical law.

Table 14.

Xi’an Jiaotong University bearing data entropy value sorting.

Fault mode	Theoretical entropy ranking	The actual entropy value sorting
Inner ring	Severe fault < moderate fault < weak fault	Severe fault < moderate fault < weak fault

Based on the data analysis of Case University of Western Reserve (7, 14 and 21 mil faults are all weak faults), the entropy value of different fault degree signals of inner ring and outer ring is theoretically distributed. The entropy value of 7 mil fault signal is the largest, the entropy value of 14 mil fault signal is the second and the entropy value of 21 mil fault signal is the smallest. However, the distribution law of entropy value on the entropy value diagram of different fault degree signals of bearing outer ring fault is as follows: the entropy value of 21 mil fault signal is the largest, the entropy value of 7 mil fault signal is the second, the entropy value of 21 mil fault is the smallest and the entropy value of 14 mil fault signal increases suddenly. The distribution law of entropy value on the entropy value diagram of different fault degrees of bearing inner ring fault is that the entropy value of 7 mil fault signal is the largest, the entropy value of 21 mil fault entropy signal is the second and the entropy value of 14 mil fault signal is the smallest. The entropy value of 21 mil fault signal has increased sharply, as shown in Table 15. In this article, the entropy value of the outer ring fault of the bearing data of Case Western Reserve University is suddenly increased when the 14 mil fault occurs and the entropy value of the inner ring fault of the bearing is suddenly increased when the 21 mil fault occurs. A reasonable explanation will be given from the background noise of the time–frequency-domain signal, the fluctuation frequency and deformation of the contact strain inside the fault site.

Table 15.

Ranking of bearing data entropy of Case Western Reserve University.

Fault mode	Theoretical entropy ranking	The actual entropy value sorting
Outer race defect	21 mil < 14 mil < 7 mil	21 mil <7 mil < 14 mil
Outer race defect	21 mil < 14 mil < 7 mil	14 mil < 21 mil < 7 mil

Mechanism analysis

Explanation of entropy distribution law of outer ring fault

The data of different fault damage degrees of the outer ring fault bearing of Case Western Reserve University are analysed. During the operation of the bearing, the rolling element rotates and the outer ring does not move, which can be explained from the background noise.

1. From the analysis of the time-domain signal, as shown in Figure 38, comparing the time-domain signals of three different fault degrees, it can be seen that in the time period of 0.05 s, for the 7-mil fault and the 21-mil fault, obvious periodic components can be obtained. For the 14-mil fault, the background noise has too much influence, and the periodic component cannot be seen. The background noise reduces the order of the time-domain signal and increases the entropy value when the 14 mil fault occurs.

2. From the frequency-domain signal analysis, as shown in Figure 39, comparing the envelope spectra of three different fault degrees, it can be seen that the background noise of 7 mil fault is very small, and the influence on the signal can be almost ignored. The background noise of 21 mil fault is the strongest, but it can still extract the obvious bearing fault characteristic frequency and its multiple frequency. The background noise has a certain influence on the signal. For 14 mil faults, although the background noise is not the strongest, it has the strongest influence on the signal. The background noise almost covers the fault information, and the corresponding fault characteristic frequency and its frequency doubling cannot be extracted. At this time, the disorder degree of the signal increases and the entropy value increases.

3. In summary, in the time-domain and frequency-domain signals, due to the great influence of background noise, the degree of disorder in the actual signal increases, and the entropy value increases. Therefore, the entropy value at 14 mil fault is greater than that at 7 mil fault.

Interpretation of inner ring fault entropy distribution law

For the inner ring fault, if only the background noise is used to analyse the entropy law, it is impossible to explain the difference between the actual entropy distribution law and the theoretical law. The actual entropy value at 21 mil fault is greater than that at 14 mil fault. During the operation of the bearing, both the inner ring and the rolling element rotate. The fault of the inner ring is more complicated than that of the outer ring. It can be explained by the fluctuation frequency and deformation of the contact strain inside the fault location, as shown in Figure 40:

1. With the increase of the fault degree, the width of the fault location increases, and the time of the rolling element passing through the fault location increases. Therefore, the fluctuation frequency of the internal contact strain of the inner ring fault location decreases.⁴⁶ The decrease of the fluctuation frequency in the same time will reduce the periodic law information, so the order degree of the signal decreases and the entropy value increases.

2. With the increase of fault degree, the rolling element only contacts with the I and II sides and does not contact with the bottom of the fault when the 7-mil fault occurs. The rolling element contacts with the I and II sides and transmits the contact deformation to the inside of the fault each time it passes through the fault site, accompanied by a large contact strain. In the case of 14 mil fault, when the rolling element passes through the fault part, it contacts with the I–II arc section, and the contact force with the fault-free part of the inner ring is reduced to 0, but the instantaneous contact with the bottom of the fault will produce small contact strain. In the 21 mil fault, the rolling element will roll a short distance (III–IV) when it passes through the fault site. The rolling element and the inner ring fault-free part have basically no contact for a long time, and the fault depth is very small. The impact of the rolling element on the bottom of the fault is very small, and the internal unit of the fault basically does not produce contact strain.⁴⁶ Therefore, as the degree of fault increases, the deformation of the internal contact strain of the inner ring fault part decreases, and the deformation can be regarded as an important fault information. The decrease of the deformation can be regarded as the decrease of the fault information, so that the entropy of the signal increases.

3. In summary, with the increase of the fault degree, the fluctuation frequency and deformation of the internal contact strain of the inner ring fault part decrease, which will increase the entropy value of the signal. Therefore, the entropy value at 21 mil fault is greater than that at 14 mil fault.

The application of damage degree identification method in the experiment of outer ring fault shedding evolution

Outer ring fault spalling evolution experiment

In order to verify the value of this method in engineering application, the evolution experiment of aero-engine outer ring fault spalling is carried out. The structure of the test bench is composed of motor, coupling, outer casing, temperature sensor, lubricating oil system and so on. The main bearing spalling fault expansion test bench of aero-engine is shown in Figure 41. The INV3062S2 intelligent signal acquisition and processing analyser of Beijing Oriental Institute was selected as the acquisition instrument, and the sampling frequency was 51.2 kHz. The vibration sensor used in the test is the ICP star three-axis acceleration sensor (X, Y, Z direction), and the sensor sensitivity is 10 mV/g; the measuring point is located directly above the outer end cover of the fault bearing. The sensor is shown in Figure 42, and the structure diagram of the test bench is shown in Figure 43. The test uses a certain type of aero-engine three-fulcrum main bearing to carry out the expansion test of the main bearing spalling fault of the aero-engine. The bearing parameters are as shown in Table 16, the initial fault size is 8 × 6 mm (length × width), the high-pressure speed is 100% operating state, each cycle is 0.5 h, a total of 36 cycles. After the test, the bearing spalling size is shown in Table 17, and the bearing spalling result is shown in Figure 44.⁴⁷

Table 16.

Bearing parameter.

Parameter	Value
Nodal diameter (D/mm)	167.5375
Roller diameter (d/mm)	22.225
Contact angle α/(°)	32–48
The number of rolling elements Z	20

Table 17.

Test bearing spalling size.

Driving cycle	Spalling size(length × width, mm)
2 h	15 × 11
6 h	25 × 17
10 h	25 × 20
14 h	31 × 20
18 h	70 × 20

The bearing fault is evaluated by time-domain parameter statistics (effective value, peak value, peak factor and kurtosis) and characteristic energy parameters, and the variation law of time-domain statistical parameters and characteristic energy with bearing operation time is analysed. The vibration signal data of the measuring points in the Z-axis (vertical) direction are selected, and the data of ±1 s at the maximum state position are analysed. The time-domain statistical parameters (effective value and peak value) are calculated by using the vibration time-domain parameter statistics, as shown in Figure 45. From the analysis, it can be seen that the effective value and peak value show an upward trend with the bearing damage throughout the operating time. After 12 cycles (6 h) of bearing operation, the time-domain parameters are in a significant upward trend, the effective value reaches 32.24 g, the peak value reaches 93.62 g and the bearing spalling is obvious. After 28 cycles (14 h), the time-domain parameters increase, the effective value reaches 52.91 g, and the peak value reaches 165.24 g. It can be judged that the bearing fault is very serious at this moment, and there is a risk of destroying the operation system.

The damage degree range of 31 × 20–70 × 20 (mm) is defined as first-level fault, the damage degree range of 25 × 20–31 × 20 (mm) is defined as second-level fault, and the damage degree range of 12 × 12–25 × 17 (mm) is defined as third-level fault. Select 12 groups of data for each fault degree, a total of 36 groups. The Shannon entropy and exponential entropy of each signal are obtained by using the method in this article. The entropy value of the database is shown in Figure 46, and the obvious distribution law can be seen from the figure, that is, the signals of different fault degrees are distributed in a specific entropy value area and show a certain linear relationship. The target input of third-level fault data is set as (0, 1), the target input of second-level fault data is set as (1, 0) and the target input of first-level fault data is set as (1, 1). The MLP is used to train the database, and 12 groups of data of three different fault degrees are selected. In the Shannon entropy–exponential entropy–MLP model, 12 groups of data under three fault degrees of bearing outer ring fault are identified correctly. The target output after neural network recognition is shown in Table 18, and the recognition is correct. The entropy diagram of 48 sets of database data and 12 sets of data to be tested is shown in Figure 47, which conforms to the law that the entropy values of different fault degrees are distributed in a specific entropy range.

Table 18.

Outer ring fault bearing test data entropy and its target output.

Training samples	Target output	Network output
(0.256, 1.177)	(1, 0)	(1, 0)
(0.309, 1.225)	(1, 0)	(1, 0)
(0.367, 1.281)	(1, 0)	(1, 0)
(0.340, 1.255)	(1, 0)	(1, 0)
(0.549, 1.477)	(0, 1)	(0, 1)
(0.461, 1.381)	(0, 1)	(0, 1)
(0.475, 1.396)	(0, 1)	(0, 1)
(0.540, 1.467)	(0, 1)	(0, 1)
(0.672, 1.623)	(1, 1)	(1, 1)
(0.670, 1.620)	(1, 1)	(1, 1)
(0.678, 1.630)	(1, 1)	(1, 1)
(0.662, 1.611)	(1, 1)	(1, 1)

Conclusion

This article focuses on the weak fault diagnosis of bearings and proposes a bearing fault damage degree identification method based on SSA-VMD and Shannon entropy–exponential entropy judgement. The conclusions are as follows:

1. Bearing fault diagnosis method based on Hilbert transform and fault energy ratio: The fault energy ratio of the original signal is more conducive to determining whether the bearing has malfunctioned than the reconstructed signal.

2. Bearing fault type diagnosis method based on SSA-VMD and multi-parameter fusion screening: Using SSA-VMD decomposition and reconstruction, kurtosis value and correlation coefficient fusion, fusion parameter ratio screening reconstruction component as signal pre-processing method, the original time-domain signal can be denoised, and then the envelope spectrum of the reconstructed signal can be obtained. The envelope spectrum can obtain the rotation frequency, fault characteristic frequency, modulation component, etc., and the fault type can be diagnosed according to the characteristic fault frequency and its frequency multiplication.

3. The identification method of bearing fault damage degree based on Shannon entropy index entropy MLP: According to the difference in entropy value of vibration signals under different fault degrees, and classified through neural networks, the accuracy of identifying outer ring fault degree and inner ring fault degree is 100%. In addition, we independently built an outer circle fault detachment evolution experiment to further verify the effectiveness of this method.

4. The explanation of the inconsistency between the entropy law and the theory under slight faults is given : For the outer ring, the sudden increase of entropy value during the 14-mil fault gives an explanation based on background noise; in the time–frequency-domain signal of the 14-mil fault, the influence of background noise is greater, which increases the degree of disorder in the actual signal and thus increases the entropy value; for the inner ring, the sudden increase of entropy value at 21 mil fault gives the explanation of the fluctuation frequency and deformation based on the internal contact strain of the fault; with the increase of the fault degree, the fluctuation frequency and deformation of the internal contact strain of the fault part decrease, which makes the entropy value increase.

Footnotes

Appendix. Symbol table.

Parameter	Interpretation
$t_{max}$	Maximum number of iterations
D	Variable dimension
$x_{i, d}$	The position of the ith sparrow in the dth dimension
$α$	Symmetrical random number
$x ω_{d}^{t}$	The optimal position occupied by the current discoverer
$β$	Normally distributed random numbers
Q	Random numbers obeying normal distribution
$R_{2}$	Warning value
ST	Safety values
n	Total sparrow number
$K_{1}$	Random values in [−1, 1]
$f_{i}$	The fitness value of the current sparrow individual
$f_{g}$	The current global optimal fitness value
$f_{w}$	The current global worst fitness value
K	Kurtosis value
r	Correlation coefficient
V	Fusion parameter index
$H_{s}$	Shannon entropy
$P_{i}$	The probability of the ith symbol sequence
$N_{obs}$	The number of possible sequences actually observed in the data
S	Exponent entropy
$θ_{j}$	Threshold value
$w_{ji}$	The weight given by the hidden unit j to the output unit i
f	The excitation function of the node

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors would like to acknowledge the financial support of National Natural Science Foundation of China (NSFC) (Grant No. 51579051); Scientific Research Fund of Liaoning Education Department (Grant No. JYT2020010).

ORCID iD

Xiaochi Luan

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Bearing fault damage degree identification method based on SSA-VMD and Shannon entropy–exponential entropy decision

Abstract

Keywords

Introduction

Research on fault feature extraction and damage degree method of rolling bearing

Bearing fault diagnosis method based on Hilbert transform and fault energy ratio

Bearing fault type diagnosis method based on SSA-VMD and parameter fusion screening

Sparrow search algorithm

Location update of discoverer

Participant location update

Investigation and early warning

SSA optimizes VMD

Kurtosis value-correlation coefficient fusion

Kurtosis value

Correlation coefficient

Parameter fusion

Reconstruction component screening principle

Bearing fault type diagnosis method process

Bearing fault degree diagnosis method based on Shannon entropy–exponential entropy–MLP

Shannon entropy

Exponent entropy

Multilayer forward neural network

Process of bearing fault damage degree identification method

Validation of rolling bearing fault feature extraction method

Bearing data validation of Case Western Reserve University

Test description of deep groove ball bearing of Case Western Reserve University

Verification of bearing fault judgement method

Verification of bearing fault type diagnosis method

Bearing data validation of Shenyang Aerospace University

Intershaft bearing test of Shenyang Aerospace University

Verification of bearing fault judgement method

Verification of bearing fault judgement method (comparison of original signal and reconstructed signal)

Verification of bearing fault type diagnosis method

Noise reduction effect verification of bearing fault type diagnosis method (simulation signal)

Effectiveness verification of rolling bearing fault damage degree identification method

Bearing data validation of Case Western Reserve University

Bearing data validation of Xi’an Jiaotong University

Xi’an Jiaotong University Spectra Quest bearing test instructions

Verification of bearing fault damage degree identification method

Optimization effect comparison of bearing fault damage degree identification method

Mechanism analysis of Shannon entropy–exponential entropy distribution law of bearings with different damage degrees

Theoretical analysis

Mechanism analysis

Explanation of entropy distribution law of outer ring fault

Interpretation of inner ring fault entropy distribution law

The application of damage degree identification method in the experiment of outer ring fault shedding evolution

Outer ring fault spalling evolution experiment

Conclusion

Footnotes

Appendix. Symbol table.

Declaration of conflicting interests

Funding

ORCID iD

References