Research on Bearing Fault Identification of Wind Turbines’ Transmission System Based on Wavelet Packet Decomposition and Probabilistic Neural Network

Abstract

In order to improve the reliability and life of the wind turbine, this paper takes the rolling bearing in the experimental platform of the wind turbine as the research object. In order to obtain the intrinsic mode function (IMF) of each fault type, the original signals of different fault states of the rolling bearing on the experimental platform are decomposed by using the overall average empirical mode decomposition method (EEMD) and the wavelet packet decomposition method (WPD), respectively. Then the energy ratio of the IMF component of the different types of faults to the total energy value is calculated and the eigenvectors of different types of faults are constructed. The extreme learning machine (ELM) and probabilistic neural network (PNN) are used to learn fault types and eigenvector samples to identify the faults of the rolling bearing. It is found that the bearing fault characteristics obtained by the WPD method are more obvious, and the results obtained by the same recognition method are ideal; and the PNN method is obviously superior to the extreme learning machine method in bearing fault recognition rate.

1. Introduction

With the continuous progress of the industrial level, the size of the wind turbine continues to increase, coupled with the relatively harsh working environment of the wind turbine, the corresponding faults will also increase. According to the survey, the failure rate of blades is 17%, the failure rate of the generator is 28%, and the failure rate of the gearbox is the highest, which is 57% [1]. Once the gearbox fails, the fault detection priority of the bearing is the highest, and the fault of other parts will be analyzed only after it is confirmed that there is no problem with the bearing. The problems of wind turbines’ bearing include contact fatigue, wear, defects, dents, and corrosion failures [2]. The continuous abrasion leading to the failure of the bearing to operate properly is referred to as wear failure [3]. Contact fatigue of bearings is the change in material structure due to repeated stress application. It begins as pitting with a slight shape and size, but with the expansion of pitting comes fatigue spalling [4]. Therefore, the accuracy of bearing fault detection and fault type identification of the wind turbine drive system is very important.

Fault diagnosis is divided into two steps. Firstly, signal processing is carried out using time domain, frequency domain and time–frequency domain analysis methods to extract features; then, the support vector machine and other artificial intelligence methods are used for fault diagnosis [5]. The time domain analysis method can only reflect the changes of signal waveform and reveal the internal characteristics of the signal. The frequency domain analysis method can reflect the different frequency components of the signal, but it is not suitable for processing non-stationary vibration signals. The time–frequency domain analysis method can analyze the vibration signal from two different aspects of time domain and frequency domain, so it can be used to analyze the non-stationary mechanical vibration signal. The time–frequency domain analysis method based on signal decomposition is also widely used in the field of rolling bearing fault diagnosis, such as empirical mode decomposition (EMD), vibrational mode decomposition (VMD), local mean decomposition (LMD) and wavelet packet decomposition (WPD). At present, the fault diagnosis methods of rolling bearing based on artificial intelligence mainly include support vector machine (SVM), extreme learning machine (ELM), convolutional neural network (CNN) and auto encoder (AE) [6]. Hoang et al. can accurately diagnose the fault of rolling bearing by converting the original vibration signal into a two-dimensional gray image and combining the strategy of the convolutional neural network [7]. Peng et al. proposed a one-dimensional convolutional neural network model containing residual blocks. The model uses wide convolution and dropout technology to enhance the generalization ability of the network [8]. Jin proposed a rolling bearing fault diagnosis method based on a two-dimensional image convolutional neural network (CNN) structure [9]. Zhao et al. proposed a rolling bearing diagnosis method with one-dimensional CNN [10]. Hao et al. proposed an end-to-end pipeline for bearing fault diagnosis based on 1D-CLSTM [11]. Chen et al. proposed a novel short-term electric load forecasting method EMD-ELM which is based on empirical mode decomposition (EMD) and extreme learning machine (ELM). And the forecasting results of the EMD-ELM were proved to be better than all the other methods [12]. Zhang et al. found that the arc fault diagnostic method with the combination of EMD and ELM identifies arc fault for various loads effectively [13]. To sum up, the above-mentioned fault diagnosis method has achieved great success at this stage. But most equipment works under normal operating conditions [14].

In this paper, the frequency of different fault types of rolling bearing in the wind turbine drive system is studied. Through the experimental platform of the wind turbine drive system, the original signal data of bearing in the normal state, cage fault, rolling element fault, inner ring fault and outer ring fault are collected. The eigenvectors of different fault types are constructed by using the EEMD and WPD method, and the recognition accuracy of fault types by using the ELM and PNN method is compared and analyzed. It is found that the WPD method is better than the EEMD method, and PNN is better than the ELM method.

2. Research Object and Experimental Platform

2.1. Research Object

This paper uses the rolling bearing in the gearbox of the dynamic test platform for the planetary gear transmission system as the research object. The rolling bearing model is ER-16 K. The number of rolling elements of the bearing is 9, the diameter of the rolling element is 7.9375 mm, and the bearing pitch diameter of the bearing is 38.5064 mm. The structural diagram is shown in Figure 1.

In Figure 1, $D$ is pitch diameter, $d$ is rolling element diameter, $r_1$ is inner race radius, $r_2$ is outer race radius, $\alpha$ is contact angle, $Z$ is number of rolling elements, $f_i$ is inner race rotation frequency, $f_o$ is outer race rotation frequency, $f_c$ is cage rotation frequency, and $f_b$ is rolling element rotation frequency.

The speed at point $i$ on the inner race is as follows [15]:

$$V_i=2\pi r_1{f}_i=\pi f_i(D-d\cos \alpha )$$

(1)

The speed at point $i$ on the outer race is as follows:

$$V_o=2\pi r_2{f}_o=\pi f_o(D+d\cos \alpha )$$

(2)

The speed at point $i$ on the cage is as follows:

$$V_c=\frac{1}{2}(V_i+V_o)=\pi f_{c}D$$

(3)

Therefore, the characteristic frequency of cage failure is as follows:

$$f_c=\frac{V_i+V_O}{2\pi D}=\frac{1}{2}\left[\left(1-\frac{d}{D}\cos \alpha \right)f_i+\left(1+\frac{d}{D}\cos \alpha \right)f_o\right]$$

(4)

For a single rolling element, when the outer ring fails, the frequency of the rolling element passing through the outer ring is as follows:

$$f_{oc}=f_c-f_o=\frac{1}{2}\left(f_i-f_o\right)\left(1-\frac{d}{D}\cos \alpha \right)$$

(5)

When the inner ring fails, the frequency of the rolling element passing through the inner ring is as follows:

$$f_{ic}=f_c-f_i=\frac{1}{2}\left(f_o-f_i\right)\left(1+\frac{d}{D}\cos \alpha \right)$$

(6)

Because $V=\theta \cdot r$, the conversion frequency is inversely proportional to the radius, and the scale factor is $d/2r_1$; therefore,

$$\frac{f_{bc}}{f_{ic}}=\frac{2r_1}{d}=\frac{D-d\cos \alpha }{d}=\frac{D}{d}\left(1-\frac{d}{D}\cos \alpha \right)$$

(7)

Thus, the rotation frequency of the rolling element relative to the cage is obtained, that is, the failure frequency of the rolling element is as follows:

$$f_{bc}=\frac{D}{2d}\left(f_i-f_o\right)\left[1-{\left(\frac{d}{D}\right)}^2{\cos }^2\alpha \right]$$

(8)

The inner and outer rings of the bearing have a relative rotation frequency which is $f_r=f_i-f_o$, because the outer ring is fixed, so $f_o=0$, $f_r=f_i$. When there are $Z$ bearing rolling element, the fault characteristic frequency of each part can be obtained as follows.

Inner ring fault frequency:

$$f_{ic}=\frac{1}{2}Z\left(1+\frac{d}{D}\cos \alpha \right)f_r$$

(9)

Outer ring fault frequency:

$$f_{oc}=\frac{1}{2}Z\left(1-\frac{d}{D}\cos \alpha \right)f_r$$

(10)

Rolling element failure frequency:

$$f_{bc}=\frac{D}{2d}\left[1-{\left(\frac{d}{D}\right)}^2{\cos }^2\alpha \right]f_r$$

(11)

Cage failure frequency:

$$f_c=\frac{1}{2}\left(1-\frac{d}{D}\cos \alpha \right)f_r$$

(12)

The frequency corresponding to different frequency spectrum of bearing fault types is different. The more serious the fault is, the greater the corresponding amplitude is.

2.2. The Experimental Platform

This paper uses the bearing in the gearbox of the dynamic test platform for the planetary gear transmission system as the research object, and the experimental platform of the transmission system is shown in Figure 2. In Figure 2, the experimental platform is composed of an industrial computer, a data-acquisition system, a controller, a drive motor, a planetary-gear box and a magnetic power brake. The controller controls the rotation frequency of the drive motor. The frequency displayed by the controller is the theoretical rotation frequency of the motor. The torque is controlled by the magnetic power brake. The data-acquisition card of the data-acquisition system is a four-channel data acquisition card. And the acceleration sensors are installed in three mutually perpendicular directions of the gearbox.

In the experiment of the bearing fault signal acquisition, bearing with different fault parts are installed in the gearbox for data acquisition. Set the sampling frequency as 5000 Hz, the sampling time as 1 s, and the torque as 22 N·m. Collecting the original signals under five different working conditions: normal state, cage fault, rolling element fault, inner ring fault and outer ring fault of the bearing, and 10 groups of data for each state are collected.

3. The Fault Feature Extraction of Bearing

3.1. The Fault Feature Extraction of Bearing Based on the EEMD Method

Since EMD uses cubic spline interpolation to fit the curve, frequency aliasing will occur, and the error will become larger. Therefore, the EEMD method was adopted to extract fault features of the bearing which is a noise aided data analysis method [16].

The EEMD method adds a uniformly distributed Gaussian white noise background to the original signal, and the spectrum of Gaussian white noise is uniformly distributed. Therefore, the original signal can be decomposed into continuous IMF of different scales to achieve the effect of suppressing mode aliasing.

Firstly, the EEMD method needs to set a reasonable overall average decomposition number N and the ratio of Gaussian white noise to the standard deviation of the original signal ε. The larger the value of N is, the smaller the noise error is, and the better the decomposition effect is, but the calculation time is doubled. If the value of the ratio of Gaussian white noise to the standard deviation of the original signal ε is too large, the components with low frequency will overlap. If the value of ε is too small, the components with high frequency will overlap. Therefore, it is necessary to make reasonable requirements for the determination of N and ε.

When the expected error is fixed to e = 0.01, the EEMD method is used to decompose the original signal. After decomposition, the amplitude standard deviation of the first signal is ${\sigma }_1$, and the ratio of this value to the amplitude standard deviation ${\sigma }_0$ of the original signal is ζ [17].

$$\varsigma =\frac{{\sigma }_1}{{\sigma }_0}$$

(13)

Determine parameters ε according to white noise addition criteria. And then N obtained by using the viewpoint given by Norden E. Huang [17].

$$\begin{gathered} 0<\epsilon <0.5\varsigma \\ N={\left(\frac{\epsilon }{e}\right)}^2 \end{gathered}$$

(14)

Taking the rolling element fault signal as an example, the original signal is decomposed by the EEMD method, and the amplitude standard deviation of the original signal and the first signal after decomposition is calculated. The results are shown in

Table 1. The amplitude standard deviation of the rolling bearing.

Parameters	σ1	σ0	ζ	0.5ζ
Value	0.00173627	0.00038614	0.222397457	0.112

. Finally, ε is 0.12, N is 200.

Assuming that the original signal is x(t), the combined signal x’(T) can be obtained by adding Gaussian white noise g(T) to the original signal [18].

$$x^{\prime }\left(t\right)=x\left(t\right)+g\left(t\right)$$

(15)

Then, the IMF can be obtained through decomposition of the combined signal with the EEMD method. Decompose the cycle N times to obtain N IMF, and take its mean value as the final IMF:

$$IMF=\frac{1}{N}\sum _{n=1}^{N}IM{F}_{n,i}$$

(16)

where, i is the number of cycles.

The EEMD method is used to decompose the original signal of the rolling bearing under the rolling element fault, and the decomposition results are shown in Figure 3.

From Figure 3, it can be seen that there are a number of IMF after decomposition by the EEMD method, but there are still meaningless components in IMF components. The last components are low-frequency components and residual components. If these components are also used as feature elements, it will cause interference and reduce the accuracy of the fault identification algorithm.

Therefore, the correlation coefficient method is used to filter out unimportant IMF components. According to the threshold value of the set coefficient, if the coefficient of the IMF component is smaller than the threshold value, it will be eliminated as an unimportant component. The calculation formula is as follows:

$${\delta }_{x,IM{F}_i}=\frac{{\displaystyle \sum _{n=1}^N}x\left(n\right)\cdot IM{F}_{n,i}}{\sqrt{{\displaystyle \sum _{n=1}^N}{x}^2\left(n\right)\cdot {\displaystyle \sum _{n=1}^N}IM{F_{n,i}}^2}}$$

(17)

The correlation coefficient of the IMF component of the rolling bearing under the rolling element fault after the calculation is shown in

Table 2. The correlation coefficient of IMF.

IMF Component	Correlation Coefficient	IMF Component	Correlation Coefficient	IMF Component	Correlation Coefficient
IMF1	0.8616	IMF5	0.0600	IMF9	0.0056
IMF2	0.2983	IMF6	0.0420	IMF10	0.0028
IMF3	0.2542	IMF7	0.0254	IMF11	0.0015
IMF4	0.0785	IMF8	0.0201	IMF12	0.0017

.

The IMF components with a correlation coefficient less than 0.02 in Table 2 are eliminated, and the first eight IMF components are retained as useful components. Different fault states of rolling bearings have different frequency ranges, and different frequency ranges correspond to different frequency band energies. Therefore, the frequency band energies under different fault states are regarded as the characteristics of fault diagnosis.

The calculation formula of IMF band energy is as follows [19]:

$$E_i={\displaystyle \sum _{n=1}^N}IM{F_{n,i}}^2$$

(18)

where, N is the number of sampling points; n is the sequence of sampling points, n = 1, 2, 3 … n; and IMF_n,i is the i-th IMF component.

The total band energy of the signal is as follows:

$$E={\displaystyle \sum _{n=1}^M}{E}_i$$

(19)

where, M = 8.

Taking the energy ratio of IMF component to total energy as the feature element of the feature vector to construct the feature vector, the ratio of each IMF component to total energy is as follows:

$$p_i=E_i/E$$

(20)

The fault feature vectors of the rolling bearing under the rolling element fault are shown in

Table 3. The fault feature vectors of the rolling bearing under the rolling element fault.

E1/E	E2/E	E3/E	E4/E	E5/E	E6/E	E7/E	E8/E
0.3221	0.1885	0.1409	0.0950	0.0904	0.0675	0.0607	0.0349
0.3172	0.1888	0.1366	0.1055	0.0788	0.0627	0.0558	0.0547
0.3123	0.1904	0.1325	0.1075	0.0901	0.0661	0.0498	0.0514
0.3179	0.1878	0.1266	0.1062	0.0868	0.0781	0.0551	0.0416
0.3149	0.1867	0.1348	0.1007	0.0831	0.0678	0.0646	0.0473
0.3259	0.1947	0.1400	0.1025	0.0848	0.0640	0.0504	0.0378
0.3235	0.1872	0.1403	0.1098	0.0825	0.0665	0.0478	0.0423
0.3163	0.1821	0.1345	0.1076	0.0862	0.0715	0.0484	0.0533
0.3145	0.1888	0.1403	0.1036	0.0845	0.0682	0.0460	0.0542
0.3169	0.1876	0.1372	0.1111	0.0833	0.0636	0.0532	0.0471

.

In the same way, the fault feature vectors of the rolling bearing in normal state, cage fault state, inner ring fault state and outer ring fault state can be obtained. Each state contains 10 groups of feature vectors. Some results are shown in

Table 4. The partial fault feature vectors of the rolling bearing under different fault conditions.

Fault Status	E1/E	E2/E	E3/E	E4/E	E5/E	E6/E	E7/E	E8/E
Normal state	0.4320	0.2567	0.0962	0.0616	0.0430	0.0409	0.0492	0.0206
Normal state	0.4189	0.2819	0.0907	0.0646	0.0387	0.0360	0.0487	0.0205
Normal state	0.4178	0.2705	0.0948	0.0678	0.0401	0.0468	0.0501	0.0122
Cage fault	0.4402	0.2322	0.1064	0.0624	0.0309	0.0509	0.0554	0.0215
Cage fault	0.4870	0.2011	0.1059	0.0569	0.0358	0.0481	0.0545	0.0108
Cage fault	0.4818	0.2099	0.1025	0.0532	0.0338	0.0437	0.0507	0.0245
Inner ring fault	0.3760	0.2709	0.1713	0.0618	0.0428	0.0417	0.0250	0.0105
Inner ring fault	0.3926	0.2372	0.1590	0.0743	0.0518	0.0369	0.0277	0.0205
Inner ring fault	0.3848	0.2582	0.1694	0.0711	0.0461	0.0330	0.0240	0.0134
Outer ring fault	0.5242	0.1827	0.0998	0.0552	0.0333	0.0355	0.0528	0.0166
Outer ring fault	0.5088	0.1858	0.1014	0.0637	0.0321	0.0348	0.0549	0.0186
Outer ring fault	0.5096	0.1796	0.0962	0.0601	0.0366	0.0399	0.0577	0.0203

.

3.2. The Fault Feature Extraction of Bearing Based on the WPD Method

Wavelet packet decomposition (WPD) is used to decompose the fault signal into three layers, and the ratio of the signal energy in the decomposition frequency band to the total signal energy is also taken as the fault feature vector element to construct the fault feature vector of each fault type of the bearing. The schematic diagram of signal decomposition frequency band division by the WPD method is shown in Figure 4.

Let sequence ${\left\{h_n\right\}}_{n\in Z}$ satisfy the following [20]:

$$\sum _n{h}_{n-2k}{h}_{n-2l}={\delta }_{k,l},\sum _n{h}_n=2^{1/2}$$

(21)

Define a set of recursive functions ω_n ∈ L²(R) (n = 1, 2, 3,…), which are generated by scaling function φ(t) and wavelet function ψ(t). The relationship is as follows:

$$\begin{gathered} \begin{array}{c}\begin{array}{ccc}{\omega }_0\left(t\right)=\varphi \left(t\right)& & {\omega }_1=\psi \left(t\right)\end{array}\hfill \\ \\ \left\{\begin{array}{c}{\omega }_{2n}\left(t\right)=2^{1/2}{\displaystyle \sum _k}{h}_k{\omega }_n\left(2t-k\right)\\ {\omega }_{2n+1}\left(t\right)=2^{1/2}{\displaystyle \sum _k}{g}_k{\omega }_n\left(2t-k\right)\end{array}\right.\hfill \end{array} \end{gathered}$$

(22)

where, $g_k={\left(-1\right)}^k{h}_{1-k}$. When n = 0, ${\omega }_0\left(t\right)$ and ${\omega }_1\left(t\right)$, respectively, correspond to φ(t) and ψ(t). The sequence ${\left\{{\omega }_n\left(t\right)\right\}}_{n\in Z}$ generated by the formula is called the wavelet packet determined by the basis function ${\omega }_0\left(t\right)=\varphi \left(t\right)$.

The fast orthogonal wavelet packet transform in multiscale analysis is introduced into the wavelet packet algorithm. The detail signal d in wavelet space is further decomposed, and the wavelet packet decomposition formula and reconstruction formula of signal x(t) are obtained. They are as follows:

$$\left\{\begin{array}{c}{d}_{2n}^{j-1,p}={\displaystyle \sum _{k\in Z}}{d}_n^{j,k}h\left(k-2p\right)\\ d_{2n+1}^{j-1,p}={\displaystyle \sum _{k\in Z}}{d}_n^{j,k}g\left(k-2p\right)\end{array}\right.$$

(23)

$$d_n^{j,p}={\displaystyle \sum _{k\in Z}}{d}_{2n}^{j-1,k}h\left(2p-k\right)+{\displaystyle \sum _{k\in Z}}{d}_{2n+1}^{j-1,k}g\left(2p-k\right)$$

(24)

According to the multiresolution analysis relation $L^2(R)=\underset{j\in Z}{\oplus }{W}_j,j\in Z$, the decomposition relation in wavelet packet subspace $W_j^n$ is obtained as follows:

$$W_j^n=W_{j-1}^{2n}\oplus W_{j-1}^{2n+1},j\in Z$$

(25)

The wavelet packet decomposes wavelet subspace $W_j\left(j=\mathrm{1,2},\cdot \cdot \cdot \right)$ step by step, and the decomposition expression is as follows:

$$W_j=W_j^1=W_{j-1}^2\oplus W_{j-1}^3=W_{j-2}^4\oplus W_{j-2}^5\oplus W_{j-2}^6\oplus W_{j-2}^7=W_0^{2^j}\oplus W_0^{2^j+1}\oplus \cdot \cdot \cdot \oplus W_0^{2^{j+1}-1}$$

(26)

It can be seen from the above formula that the wavelet packet realizes the further decomposition of the high-frequency sequence. If j = 0, it indicates the original signal x(t) itself at the resolution j level, and is recorded as x₁; if x₁ is decomposed once, the decomposed signal signals x₂ and x₃ will be obtained at the first layer of wavelet packet decomposition; and if x₁ is decomposed twice, the decomposed signals x₄, x₅, x₆ and x₇ will be obtained at the second layer of wavelet packet decomposition, and so on.

Taking the rolling element fault signal as an example, the rolling element fault signal is decomposed by the WPD method. The decomposition diagrams of the third level are shown in Figure 5.

After the rolling element fault signal is decomposed by the WPD method, the fault feature vectors are obtained by taking the ratio of each frequency band energy to the total energy as an element. As shown in

Table 5. The fault feature vectors of the rolling bearing under the rolling element fault.

E1/E	E2/E	E3/E	E4/E	E5/E	E6/E	E7/E	E8/E
0.2063	0.1042	0.1004	0.1127	0.1253	0.1155	0.1138	0.1218
0.2109	0.0968	0.1153	0.1062	0.1252	0.1310	0.1002	0.1146
0.2427	0.0945	0.1001	0.1027	0.1315	0.1172	0.1032	0.1081
0.1803	0.1106	0.1042	0.1170	0.1236	0.1324	0.1178	0.1140
0.1762	0.0958	0.1144	0.1146	0.1275	0.1357	0.1192	0.1166
0.2505	0.1015	0.1106	0.0982	0.1231	0.1129	0.0942	0.1090
0.2089	0.1025	0.1034	0.1075	0.1311	0.1249	0.1024	0.1194
0.2004	0.1068	0.0971	0.1106	0.1227	0.1383	0.1107	0.1133
0.2203	0.1094	0.1063	0.1020	0.1276	0.1127	0.1120	0.1098
0.1880	0.1124	0.1019	0.1021	0.1273	0.1351	0.1039	0.1293

.

Similarly, the fault feature vectors of the inner ring, outer ring, cage fault signal and normal signal are obtained, and some calculation results are shown in

Table 6. The partial fault feature vectors of the rolling bearing under different fault conditions.

Fault Status	E1/E	E2/E	E3/E	E4/E	E5/E	E6/E	E7/E	E8/E
Normal state	0.4320	0.2567	0.0962	0.0616	0.0430	0.0409	0.0492	0.0206
Normal state	0.4189	0.2819	0.0907	0.0646	0.0387	0.0360	0.0487	0.0205
Normal state	0.4178	0.2705	0.0948	0.0678	0.0401	0.0468	0.0501	0.0122
Cage fault	0.4402	0.2322	0.1064	0.0624	0.0309	0.0509	0.0554	0.0215
Cage fault	0.4870	0.2011	0.1059	0.0569	0.0358	0.0481	0.0545	0.0108
Cage fault	0.4818	0.2099	0.1025	0.0532	0.0338	0.0437	0.0507	0.0245
Inner ring fault	0.3760	0.2709	0.1713	0.0618	0.0428	0.0417	0.0250	0.0105
Inner ring fault	0.3926	0.2372	0.1590	0.0743	0.0518	0.0369	0.0277	0.0205
Inner ring fault	0.3848	0.2582	0.1694	0.0711	0.0461	0.0330	0.0240	0.0134
Outer ring fault	0.5242	0.1827	0.0998	0.0552	0.0333	0.0355	0.0528	0.0166
Outer ring fault	0.5088	0.1858	0.1014	0.0637	0.0321	0.0348	0.0549	0.0186
Outer ring fault	0.5096	0.1796	0.0962	0.0601	0.0366	0.0399	0.0577	0.0203

.

After obtaining the fault feature vectors of the five states of the bearing, these fault feature vectors can be used as samples to identify the fault state.

4. The Fault Pattern Recognition Based on ELM

4.1. The Extreme Learning Machine (ELM)

The ELM can solve the problems of slow training speed, the difficulty reaching the global minimum, and the large impact of learning rate on the network. The structure of a typical single hidden layer feedforward neural network is shown in Figure 6.

Figure 6 shows that the network is composed of the input layer, hidden layer and output layer. Suppose there are n neurons, there are n input characteristic variables, and there are m target characteristic variables in the output layer, and there are l neurons in the middle hidden layer. When there is a training set with Q samples, the matrix is n * Q, and the rows represent features, that is, suppose each sample has n features, and the columns represent samples. For the specific calculation principle, refer to [21]. H is the output matrix of the hidden layer.

$$\begin{array}{c}H\beta =T^{\prime }\\ H\left(w_1,w_2,\cdot \cdot \cdot ,w_l,b_1,b_2,\cdot \cdot \cdot ,b_l,x_1,x_2,\cdot \cdot \cdot ,x_Q\right)={\left[\begin{array}{c}g\left(w_1\cdot x_1+b_1\right)g\left(w_2\cdot x_1+b_2\right)g\left(w_l\cdot x_1+b_l\right)\\ g\left(w_1\cdot x_2+b_1\right)g\left(w_2\cdot x_2+b_2\right)g\left(w_l\cdot x_2+b_l\right)\\ \cdot \cdot \cdot \\ g\left(w_1\cdot x_Q+b_1\right)g\left(w_2\cdot x_Q+b_2\right)g\left(w_l\cdot x_Q+b_l\right)\end{array}\right]}_{Q\times l}\end{array}$$

(27)

where T′ is the transpose of matrix T. H is the matrix of Q * l, that is, Q samples multiplied by l neurons. Each row represents the output of a sample on l neurons, and each column represents the output of each neuron to Q samples.

4.2. The Experimental Analysis

The input part of the fault feature vectors of the bearing under different fault conditions into the ELM randomly select six groups of data for each state as training samples for learning, and the remaining samples as test samples for identification. Since the training samples and test samples are randomly selected each time, the results of each test are not necessarily the same. The results of 10 tests of the fault feature vectors extracted by EEMD are shown in

Table 7. Accuracy of fault type identification.

Test Serial Number	1	2	3	4	5	6	7	8	9	10
Accuracy rate	95%	90%	100%	80%	100%	95%	85%	95%	100%	95%

.

It can be seen that the average recognition accuracy of the ELM for the fault feature vectors extracted by EEMD is 93.5% from Table 7. This is obtained in the case of a small number of experimental data sets. In practical applications, because the collected data is more complex, it is likely to be lower than this accuracy. Several fault type identification errors are shown in Figure 7.

From Figure 7, it can be found that the fault type identification errors of the fault feature vectors extracted by the ELM is concentrated between 3 and 5, that is, between cage fault and outer ring fault, while the rolling element fault, inner ring fault and normal state can be identified more accurately. However, this method did not achieve the expected effect. In 10 tests, there are only three times when the samples of the test set are accurately classified, which is obviously not suitable for application to actual conditions which are more complex.

The ELM method is used to test the fault feature vectors obtained by WPD. The results of 10 tests are shown in

Table 8. Test results of wavelet packet decomposition feature extraction in the ELM recognition.

Test Serial Number	1	2	3	4	5	6	7	8	9	10
Accuracy rate	100%	95%	100%	100%	95%	100%	100%	100%	95%	100%

. It can be seen that the average recognition accuracy of the ELM for the fault feature vectors extracted by WPD is 98.5%. The recognition accuracy of these data is 5% higher than that of the EEMD extracted by the fault feature vectors, and the lowest accuracy is 95% in these 10 tests. Therefore, it can be explained that the effect of the fault feature vectors obtained by the WPD method is better than that obtained by the EEMD method.

Among Table 8, there are three fault-type identification errors, as shown in Figure 8. It can be seen that the three misclassification situations are 4 pairs of 3, 5 and 2 misclassifications, that is, the inner ring fault is incorrectly identified as the cage fault, outer ring fault and the rolling element fault, respectively, in the three processes. It shows that this method is not ideal for fault identification.

5. The Fault Pattern Recognition Based on the PNN

5.1. The Probabilistic Neural Network (PNN)

The PNN was introduced by Specht in the early 1990s which is the probabilistic neural network [22]. The PNN can be regarded as a radial basis neural network, which is a completely forward calculation process without reverse error calculation. And the PNN has the following advantages: fast operation, good stability, fault tolerance, fast convergence, will not fall into the local optimal situation, can use the linear learning algorithm to achieve the function of the nonlinear learning algorithm, and very suitable for pattern recognition [23]. The PNN method is a kind of feedforward neural network. It is generally composed of an input layer, mode layer, summation layer and output layer [24], and its structure is shown in Figure 9.

The value from the training sample is input to the input layer, which transfers the eigenvector to the network [25]. The input layer is connected with the pattern layer through the weight ${\omega }_{ij}$, and the pattern layer is calculated to obtain the matching relationship between the eigenvector and different patterns. The sum of all the training samples is equal to the number of neurons in the pattern layer. The input samples are weighted and summed in the mode layer, and then transferred to the summation layer after the nonlinear operator $g\left(z_j\right)$ operation.

$$g\left(z_j\right)=\exp \left(\frac{z_j-1}{{\sigma }^2}\right)$$

(28)

The output of each mode unit in this layer is as follows:

$$f\left(X,W_i\right)=\exp \left(-\frac{{\left(X-W_i\right)}^T\left(X-W\right)}{2{\delta }^2}\right)$$

(29)

where X is the input sample; $W_i$ is the connection weight matrix between the input layer and the mode layer; and δ is the smoothing factor [26].

The summation layer accumulates the probabilities belonging to a certain class.

$$f_A=\sum _{i=1}^{m}g\left(z_i\right)$$

(30)

where m is the number of samples. The different states of the rolling bearing correspond to a summing layer unit, respectively, and the outputs of the corresponding mode layer units are added [26].

The output layer is a competitive neuron. Different neurons correspond to different states of bearings. They receive various density function outputs by the summation layer. The number of neurons in the output layer is equal to the number of types of fault types. The neuron with the largest probability density function outputs 1, and the other neurons output 0.

5.2. The Experimental Analysis

Part of the fault feature vectors of the bearing under different fault conditions are input into the PNN model. Six groups of data are randomly selected for each state as training samples for learning, and the remaining samples are used as test samples for identification. Since the training samples and test samples are randomly selected each time, the results of each test are not necessarily the same. The results of 10 tests of the fault feature vectors extracted by EEMD are shown in

Table 9. Test results of features extracted by the PNN recognition EEMD.

Test Serial Number	1	2	3	4	5	6	7	8	9	10
Accuracy rate	100%	95%	100%	100%	100%	95%	100%	100%	100%	100%

. The two inaccurate identifications are shown in Figure 10.

In the Figure 10, it can be seen that the misclassification of the features extracted by the PNN for EEMD is also concentrated between 3 and 5, that is, between the cage fault and the outer ring fault. The rolling element fault, inner ring fault and normal state can be accurately identified. However, it can be seen from Table 9 that the average recognition accuracy rate of the PNN for the fault feature vectors extracted by EEMD is 99%. Among the 10 times of recognition, there are 8 times that can accurately identify the fault state, and 2 times can also reach 95%.

The PNN model is used to test the fault feature vectors obtained by WPD. The results of 10 tests are shown in

Table 10. Test results of features extracted by the PNN recognition WPD.

Test Serial Number	1	2	3	4	5	6	7	8	9	10
Accuracy rate	100%	100%	100%	100%	100%	100%	100%	100%	100%	100%

.

From Table 10, it can be found that the test results of the PNN identifying the fault feature vectors obtained by WPD have reached 100% 10 times, that is, when the identification samples are collected in the laboratory, this method can accurately identify the fault type of the rolling bearing. Although the collected data will be more complex in the actual working condition, the effect of this method is expected to achieve the effect of preliminary judgment of the fault type.

6. Conclusions

In this paper, the rolling bearing in the experimental platform of the wind turbine gear transmission system is taken as the research object. The EEMD method and the WPD method are used to decompose the original signals of different fault states of the experimental rolling bearing and construct the fault feature vectors. Then, the ELM and PNN are used to learn fault types and the fault feature vector samples to identify fault types.

(1)

Using the same recognition method, it is found that the fault recognition accuracy of the fault feature vectors after the WPD method is higher, that is, the fault characteristics obtained after the WPD method are more obvious.

(2)

The recognition accuracy of the features extracted by EEMD and WPD using the ELM method were 93.5% and 98.5%, respectively; the recognition accuracy of the PNN method for the bearing fault is 99% and 100%, respectively. The PNN method is obviously superior to the ELM method.

(3)

Based on the results, the accuracy of the method of combining WPD and PNN to predict bearing fault types is relatively high.