A Novel Bearing Fault Diagnosis Methodology Based on SVD ...

Research ArticleA Novel Bearing Fault Diagnosis Methodology Based on SVD andOne-Dimensional Convolutional Neural Network

Yangyang Wang 1 Shuzhan Huang 2 Juying Dai 3 and Jian Tang 3

1Xichang Satellite Launch Center Xichang 615000 China2School of Graduate Army Engineering University of PLA Nanjing 210000 China3School of Field Engineering Army Engineering University of PLA Nanjing 21000 China

Correspondence should be addressed to Shuzhan Huang hsz_221618163com

Received 12 June 2019 Revised 16 October 2019 Accepted 12 December 2019 Published 30 January 2020

Academic Editor Salvatore Russo

Copyright copy 2020 YangyangWang et al -is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

-is paper constructs a novel network structure (SVD-1DCNN) based on singular value decomposition (SVD) and one-di-mensional convolutional neural network (1DCNN) which takes the original signal as input to realize intelligent diagnosis ofbearing faults -e output of the first convolution layer was also analyzed from the perspectives of time domain and time-frequency domain in the simulation experiment -rough qualitative analysis and quantitative analysis it was found that theconvolution kernel not only extracted the classification features of signals but also gradually highlighted the learned features in thenetwork training process Moreover applying this network in fault diagnosis of bearing date provided by the Case WesternReserve University (CWRU) Bearing Data Center it was found that the convolution kernel could also achieve the above operation-e novel network of this paper achieved a good classification effect on both the simulated signals and the measured signals

1 Introduction

A small fault in a mechanical device often affects the stabilityand safety of the entire system and can even lead to cata-strophic consequences [1] As a key component of me-chanical equipment bearings are widely used in varioustypes of machinery Failure of the bearing can cause manyserious mechanical failures so the safe and smooth oper-ation of the bearing is critical to the mechanical equipmentTimely detection positioning and troubleshooting ofbearing faults can effectively improve the safety of industrialproduction -erefore it is of great significance to study thefault diagnosis of bearing

-e traditional fault diagnosis process generally consistsof three steps data acquisition feature extraction and se-lection and fault pattern recognition [2]

-e collected data include vibration signal acousticsignal and temperature signal and since the vibration signalcan directly characterize the state of the mechanicalequipment the vibration signal is most commonly collectedin fault diagnosis [3] In the fault diagnosis of bearing

commonly used signal processing methods include Fouriertransform (FT) [4] short-time Fourier transform (STFT)[5] wavelet transform (WT) [6] WignerndashVille distribution(WVD) [7] and empirical mode decomposition (EMD) [8]-e above methods can extract features that are conducivefor classification and diagnosis [9 10] and then pass theextracted features through various classifiers to realizepattern recognition of bearing faults Among the variouspattern recognition methods the machine learning-basedmethod is the most used Wang et al [11] used KPCA toextract features from bearing fault signal and used k-nearestneighbor (KNN) as a classifier to achieve diagnosis Fei et al[12] reconstructed the characteristics of bearing vibrationsignal after singular value decomposition based on waveletpacket transform phase space and established support vectormachine (SVM) model of bearing diagnosis Mahamad andHiyama [13] performed fast Fourier transform (FFT) andenvelope processing on the bearing vibration signalextracted time domain and frequency domain feature asinput and then used ANN to fulfill the diagnosis Howeverthe existing intelligent fault diagnosis methods based on the

HindawiShock and VibrationVolume 2020 Article ID 1850286 17 pageshttpsdoiorg10115520201850286

above feature extraction and classification still have threelimitations first the feature extraction methods often re-quire the operators to have professional prior knowledgeand rich experience As the research progresses the form ofinput signal becomes more diversified and its objectivityand accuracy may be affected if feature extraction is stillbased on past experience [14 15] second the feature ex-traction methods are poor in generality and often a methodonly has a good feature extraction result for a certain type ofsignal and third feature extraction and pattern recognitionare two independent processes and the diagnosis modelcannot be jointly optimized globally

In recent years the successful application of deeplearning in the fields of speech recognition [16] face rec-ognition [17] computer vision [18] and image processing[19] has made it a research hotspot Various deep learningmodels can extract abstract features directly from theoriginal signal avoiding manual extraction of feature [20]and they also have better universality [21] and can jointlyoptimize the two processes of feature extraction and patternrecognition in various classification problems [22] -anksto these advantages researchers have introduced a variety ofdeep learning models in bearing fault diagnosis for exampleDuong and Kim [23] constructed a DNN structure which isbased on the stacked denoising autoencoder (DAE) non-mutually exclusive classifier (NMEC) method for combinedmodes to realize bearing fault diagnosis Shao et al [24]developed a convolutional deep belief network withGaussian visible units to obtain an excellent accuracy rate ofbearing fault diagnosis Chen and Li [25] utilized the ac-celeration sensors to collect the vibration signal of thebearing and input the time domain and frequency domaincharacteristics of the signal into multiple two-layer sparseautoencoder (SAE) neural networks for feature fusion andthen the fused feature was further classified by DBN Lu et al[26] established a deep neural network model based onautoencoder (AE) and achieved good results in bearing faultdiagnosis Shao et al [27] proposed a novel optimizationdeep belief network (DBN) for bearing fault diagnosis whichis verified by the simulation signal and experimental signalof a rolling bearing

Figure 1 shows the main differences between the tra-ditional fault diagnosis method and the deep learning-basedfault diagnosis method

Convolutional neural network [28] is a typical deeplearning model that has also attracted attention It extracts thecharacteristics of the signal layer by layer through convolutionpooling and nonlinear activation function mapping Com-pared with the fully connected deep learning model CNN hasstronger robustness and better generalization ability [29] Atthe same time CNN improves network performance and re-duces training costs by weight sharing and pooling operationand is less prone to overfitting problem than other deeplearning models [30] From the perspective of input theexisting CNN models include two types one-dimensionalconvolutional neural network (1DCNN) and two-dimensionalconvolutional neural network (2DCNN)

For 2DCNN its input is actually two-dimensionalmatrix In the fault diagnosis of bearing researchers used a

variety of methods to convert one-dimensional originalsignal into two-dimensional matrix and then used it as2DCNN input In [20] the one-dimensional signal wasconverted into two-dimensional gray map as the input of2DCNN and the input of 2DCNN in [31] was the root meansquare (RMS) map of the characteristics of the vibrationsignal after Fourier transform (FT) In [32] the continuouswavelet transform scale (CWTS) map was directly classifiedby 2DCNN

However in practice the bearing vibration signal is aone-dimensional time signal and the method of convertingthe original one-dimensional signal to two-dimensionalsignal also depends on experience -ese methods cannotguarantee whether there is torsion distortion or even lossof useful information in the conversion process which mayresult in insufficient characteristic learning and low ac-curacy -erefore if the original one-dimensional signal isused as input directly the input of the network will containall the feature information in the original signal and theabove problem can be avoided In addition compared with2DCNN 1DCNN has better interpretability and theconvolution kernel and its extracted feature are one-di-mensional vectors so that multiple signal processingmethods can be used to study the convolution kernel andits extracted feature conveniently which is conducive tofurther understand 1DCNN and its feature extractionmechanism

For 1DCNN its input is one-dimensional vector Inpractice the actual measured signal often contains a lot ofnoise which will greatly increase the difficulty inextracting fault features in a simple shallow CNN modelIn the case where the measured noisy signal is input thediagnosis accuracy can be improved by the following twomethods

Feature extraction and selection

Deep learning networks

Classifiers

(b)(a)

Signal

Diagnostic result

Figure 1 Block diagram of diagnosis methods (a) traditional faultdiagnosis method (b) deep learning-based fault diagnosis method

2 Shock and Vibration

One idea is to preprocess and denoise the signalCommon denoising methods with good performance in-clude wavelet transform [33] singular value decomposition(SVD) [34] and ARMED filtering [35] -e noise compo-nents in the signal are removed by an artificial method andthe denoised signal is used as the input of the 1DCNNHowever these methods also rely on experience -edenoised signal also loses some features It is impossible todetermine whether the removed signal components containthe classification features required by the network and theprocess of denoising and network extraction is also twoindependent processes

Another way of thinking is to reduce the influence ofman-made directly using the original signal as input andcomplete feature extraction and pattern recognition through1DCNN Previous studies have shown that for noisy signalincreasing the number of network layers allows the networkto learn higher-level richer signal classification featuresHowever there are two shortcomings in the network withdeeper layers First the error is calculated by the chain rulein the form of backpropagation which easily leads to theexponential decreasing or increasing of the gradient with theincrease of layers-erefore the deeper the CNN network isthe easier it is to encounter gradient disappearance orgradient explosion problem and the more difficult to train[29] Second the deeper the network layer the more likely tocause network degradation which leads to the increase ofsample error in the training process Similarly increasing thenumber of feature maps can also increase the contentlearned by the network enabling the network to learn moresignal features but it also brings overfitting problem to thenetwork

-ese problems have greatly limited the application ofCNN in fault diagnosis -erefore this paper proposes anetwork structure based on SVD and 1DCNN (SVD-1DCNN) which improves the pattern recognition accuracyrate of the network by embedding the SVD layer in thenetwork and its input is the original signal -e feasibility ofthe method was verified by the simulated signal and themeasured signal

-e rest of the paper is organized as follows Section 2briefly describes SVD-DCNN Section 3 performs simula-tion experiment Section 4 uses the proposed method forbearing fault diagnosis and verifies the effectiveness andfeasibility of the method and Section 5 presents theconclusions

2 Materials and Methods

21 SignalDenoisingBased on SVD SVD is a classical matrixtransformation method Because of its zero phase offset noinitialization parameters and easy implementation it hasbeen widely used in signal denoising

For an arbitrary m times n matrix after SVD decomposition

A UΣVT (1)

where U is a matrix of m times m V is a matrix of n times n Σ is amatrix of m times n whose elements are 0 except those on the

principal diagonal line and the elements on the principaldiagonal line of Σ are called singular values of matrix A

Express U and V in matrix form as follows U

[u1 u2 um]mtimesm and V [v1 v2 vn]ntimesn whereui isin Rmtimes1 and vi isin Rntimes1

Express Σ in matrix form as followsWhen mlt n

1113944

σ1 0 middot middot middot 0 0 middot middot middot 0

⋮ σ2 middot middot middot 0 0 middot middot middot 0

0 ⋮ ⋱ ⋮ ⋮ middot middot middot ⋮

0 0 middot middot middot σm 0 middot middot middot 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(2)

When mgt n

1113944

σ1 0 middot middot middot 0

0 σ2 middot middot middot 0

⋮ ⋮ ⋱ ⋮

0 0 middot middot middot σn

0 0 middot middot middot 0

⋮ ⋮ ⋮

0 0 middot middot middot 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(3)

A is further rewritten into the form of ui and vi

Amtimesn = [u1 u2 umndash1 um]mtimesm

0

σ1

σk0

0

vT1

vT2

vTnndash1

vTn

mtimesnntimesn

(4)

where k min(m n) and A is further rewritten into theform of matrix sum

Amtimesn σ1u1vT1 + σ2u2v

T2 + middot middot middot + σkukv

Tk (5)

It can be seen that the essence of SVD is to decomposeany matrix A of m times n into linear superposition of severalsubmatrices of the same dimension -e weight of eachsubmatrix ie singular value σi reflects the importance ofthe matrix Singular values often imply potentially importantinformation in matrix Based on the above characteristics ofSVD singular values of signal matrix containing complexinformation can be conveniently selected to study so as toprovide the possibility of signal feature extraction

As mentioned earlier SVD is a decomposition methodfor matrix but the actual signal is one-dimensional-erefore the key to extracting signal features by SVD is totransform one-dimensional signal into two-dimensionalmatrix -e existing forms of matrix construction mainlyinclude Cycle matrix Toeplitz matrix and Hankel matrixAmong them SVD based on Hankel matrix can betterhighlight the useful features of signals [36] which is con-ducive to the separation of useful signal and noise

For a noisy signal X [x1 x2 x3 xN] with lengthN the Hankel matrix of the signal is constructed as follows

Shock and Vibration 3

Hx

x1 x2 middot middot middot xNminus n+1

x2 x3 middot middot middot xNminus n+2

⋮ ⋮ ⋮

xNminus n+1 xNminus n+2 middot middot middot xN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(6)

where 1lt nltN m N minus n + 1 Each matrix has multipleHankel matrices with different column combinations Whenconstructing Hankel matrix the product of row number m

and column number n of matrix should be maximized as faras possible and the best way to construct matrix should besquare matrix or near square matrix [37] According to theinequality principle when m and n are equal or close theproduct of the two numbers is the largest so the structure ofthe optimal Hankel matrix is determined as follows

m

N

2 N is even

(N + 1)

2 N is odd

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(7)

-e key of denoising noisy signal by SVD is how todetermine the singular value of useful signal and the singularvalue of noise signal Zhao et al [38] proposed a method todetermine the singular value of useful signal based onsingular value difference spectrum Assuming that the formof Hankel matrix of noisy signal X is shown in equation (5)and that there are q(qlt k) singular values of useful signaldetermined by singular value difference spectrum methodso the Hankel matrix of useful signal separated by thismethod can be expressed as

Aprime σ1u1v1 + σ2u2v2 + middot middot middot + σquqvq(qlt k) (8)

Furthermore the denoised signal can be obtained byreducing Aprime to one-dimensional signal It can be seen thatthe singular value difference spectrummethod is a denoisingmethod based on the characteristics of the data itself

22 Proposal of Diagnostic Model Figure 2 shows the SVD-1DCNN structure constructed in this paper -e networkembeds an SVD layer after C1 to realize further featuretransformation -e feature maps in the SVD layer areconnected to the corresponding feature maps in C1

-e SVD-1DCNN mainly includes input layer convo-lution layers pooling layers fully connected layer outputlayer and SVD layer -e convolution layers the poolinglayers and the SVD layer are the core structures of the SVD-1DCNN and each of the convolution layers the poolinglayers and the SVD layer has several feature maps Eachfeature map connected to a corresponding feature map onthe adjacent layer and the output of the previous layerfeature map is the input of the next layer feature map

In this structure SVD layer denoises and reconstructsthe output of C1 (primary classification feature) to achievejoint optimization of feature extraction and denoising andreconstruction -e denoised and reconstructed features areused as input to the next layer In this network structure theconvolution kernels realize adaptive denoising of signals

and the useful feature components required by the networkare highlighted -e SVD layerrsquos denoising and recon-struction process further highlights the features so it is moreconducive to network extraction classification features

Since SVD-1DCNN includes a SVD layer in order toensure that the network can carry out backpropagation it isnecessary to ensure that the error can be backpropagatedfrom S2 layer to C1 layer and the weights and bias in C1 canbe updated -is process of SVD-1DCNN is describedbelow

Suppose the signal in the input layer isL0 [l01 l02 l0s ] the output of C1rsquos feature map isL1 [l11 l12 l1s ] and the output of SVD layerrsquos corre-sponding feature map is L2 [l21 l22 middot middot middot l2s ] -erefore theoutput of the i-th node of the feature map of C1 is l1i theoutput of the corresponding node on the SVD layerrsquos cor-responding feature map is l2i and the difference between thetwo is Δli l2i minus l1i -e singular value difference spectrummethod is based on the characteristics of the signal itself toachieve denoising so when given the input Δli l2i minus l1i is aconstant In the process of error backpropagation assumingthat the error from S2 layer to the i-th node on SVD layerrsquosfeature map is E2

i then the error from this node on SVDlayerrsquos feature map to the corresponding i-th node on C1rsquoscorresponding feature map is E1

i E2i + Δli Because Δli is a

constant E1i is differentiable to the weights and bias in the

backpropagation process In the process of network trainingthe weights and bias in C1 can be updated by E1

i Figure 3 shows the flowchart of the method and the

specific steps are as follows

Step 1 the original signals are taken as network inputAfter the feature transformation through the C1 layerthe primary classification features in the original signalsare extracted and the primary classification features areused as the input of the SVD layerStep 2 in each training process the SVD layer denoisesand reconstructs the output of C1 to further extracthigher-level classification featuresStep 3 the classification features extracted by the SVDlayer are used as the input of the next layer to achievedeeper feature transformation

In each training process the SVD layer denoises andreconstructs the output of C1 which is conducive to thenetwork to obtain more obvious classification characteristicsof signals under the background of noise and enhance thenetwork fault diagnosis ability

3 Performance Analysis Based onSimulated Signals

31 Construction of Simulation Signals -e following arefour types of simulation signals with a signal-to-noise ratioof 20 dB -e feasibility of the research is verified by theclassification of four types of signals Table 1 shows therelevant parameters of the signals where u(t) is theGaussian white noise and φij is the phase of the signalswhich is randomly generated between 1 and 100 -e time

4 Shock and Vibration

domain and time-frequency diagrams of the four types ofsignals are shown in Figure 4

In SVD-1DCNN since the original signals contain noisethe output of each feature extraction layer also contains

noise In order to quantify the feature extraction effect ofeach feature extraction layer eSNR is defined as an indexAssuming that the output of a layer is x and the denoisedsignal reconstructed by SVD is s the noise component can

C1 SVDlayer S2

Fullyconnected

layerS4C3 Output layer

Iteration doneWeight andbias update

N

Fault classify

Y

Data input

SVD-1DCNN

Result output

Figure 3 SVD-1DCNN diagnosis flowchart

Input Fully connectedlayer

S4C3

S2C1Output

layer

Downsampling

Con

volu

tion

Con

volu

tion

Downsampling

SVDlayer

Figure 2 Structure of SVD-1DCNN

Table 1 Four types of simulated signals

Signal Expression φij

Y1 sin(100πt + φ1j) middot cos(250πt + φ2j) + u(t)φ1j rand[1 100]

φ2j rand[1 100]

Y2 eminus t sin(400πt2 + φ3j) + u(t) φ3j rand[1 100]

Y3 2 cos(200πteminus t + φ4j) + u(t) φ4j rand[1 100]

Y4 t sin(150πt + φ5j) + t cos(550πt + φ6j) + u(t)φ5j rand[1 100]

φ6j rand[1 100]

Shock and Vibration 5

be expressed as x minus s -en the eSNR of the output of thislayer can be expressed by the following equation

eSNR 10 lgs

x minus s1113874 1113875 (9)

Further the eSNR of the simulation signals are calcu-lated by equation (9) and the results are shown in Table 2 Ascan be seen from Table 2 eSNR can accurately reflect theSNR of the signals

32 Pattern Recognition of Simulated Signal -e networkstructure of SVD-1DCNN is as shown in Figure 2 Be-cause SVD-1DCNN is an improved network based on1DCNN its network structure and network parametersare also based on specific pattern recognition tasks andexperience In this pattern recognition task there are onlyfour types of simulated signals so according to theprevious experience the learning rate of the network isset to be 01 the training batch is set to be 10 and themaximum number of iterations is set to be 1500 thepooling method of the two pooling layers is averagepooling and the step size is set to be 2 For convenience of

representation (m n)-[p q] is used to represent therelevant parameters in the network where m and n re-spectively represent the size of the convolution kernels inthe two convolution layers and p and q represent thenumber of convolution kernels in the correspondingconvolution layer

SVD is usually used in the preprocessing in signalprocessing that is the original signals are denoised firstlyand then the denoised signals are used in the subsequentanalysis -erefore in order to compare the classificationeffects a network structure (SVD+ 1DCNN) is constructedIn the new network the original signals are denoised firstlyand the denoised signals are used as the input of the networkto realize pattern recognition as shown in Figure 5-e other

0 01 02 03 04 05 06 07 08ndash2

ndash1

0

1

2

Time (s)

Time (s)

Am

plitu

de

02 03 04 05 06 07 080

200

400

Freq

uenc

y (H

z)

(a)

Time (s)

Time (s)

Am

plitu

deFr

eque

ncy

(Hz)

0 01 02 03 04 05 06 07 08ndash2

ndash1

0

1

2

02 03 04 05 06 07 080

200

400

(b)

Time (s)

Time (s)

Am

plitu

deFr

eque

ncy

(Hz)

0 01 02 03 04 05 06 07 08ndash4

ndash2

0

2

4

02 03 04 05 06 07 080

200

400

(c)

Time (s)

Time (s)

Am

plitu

deFr

eque

ncy

(Hz)

0 01 02 03 04 05 06 07 08ndash2

ndash1

0

1

2

02 03 04 05 06 07 080

200

400

(d)

Figure 4 Time domain and time-frequency diagrams of the simulated signals (a) Y1 (b) Y2 (c) Y3 and (d) Y4

Table 2 SNR(dB) and eSNR(dB) of four types of simulatedsignals

SNR (dB) eSNR (dB)

Y1 20 19287Y2 20 19166Y3 20 19109Y4 20 19001

6 Shock and Vibration

parameters setting in the new network are the same as inSVD-1DCNN

Four types of simulation signals are used for the ex-periment Each signal contains 60 samples (50 trainingsamples and 10 test samples) In order to verify the stabilityof the networks 10 experiments were conducted for eachnetwork structure In addition the classification results ofeach experiment of the two networks were evaluated byconfusion matrix and accuracy Confusion matrix is cal-culated by the four parts composed of true label and pre-diction label which are true positive (TP) false negative(FN) false positive (FP) and true negative (TN) respec-tively -e confusion matrix is shown in Table 3

-e accuracy is the overall judgment of the classificationmodel and the proportion of correct prediction in the totalamount -e calculation method is as follows

A TP + TN

TP + FP + FN + TN (10)

Ai(i 1 2 10) A and Var were respectively used torepresent the classification accuracy of each experiment theaverage accuracy and the variance of the accuracy and thenAi A and Var satisfy the following equations

A 111394410

i1

Ai

10

Var 111394410

i1

Ai minus A( 11138572

10

(11)

In the classification of simulation signals differentnetwork structures need to be set up for multiple

experiments to determine the best network structureAmong various network structures the network whosestructure is (351 80)-[3 3] is taken as an example to show itsconfusion matrix in one experiment and the A and Var ofthe network after 10 experiments

Table 4 is the confusion matrix in one experiment whenthe structure of SVD-1DCNN is (351 80)-[3 3]

Table 5 is the confusion matrix in one experiment whenthe structure of SVD+ 1DCNN is (351 80)-[3 3]

As can be seen from Table 4 SVD-1DCNN can classifyevery type of signal correctly as can be seen from Table 5 forY1 the classification accuracy of SVD+ 1DCNN is 90 but10 is classified as Y4 For Y2 the classification accuracy ofSVD+ 1DCNN is 80 but 10 is classified as Y1 and theremaining 10 is classified as Y4 For Y3 the classificationaccuracy of SVD+ 1DCNN is 90 but 10 is classified asY4 For Y4 the classification accuracy of SVD+ 1DCNN is90 but 10 is classified as Y1 In general SVD-1DCNNhas a higher classification accuracy than SVD+ 1DCNN

10 experiments were carried out on both networksTable 6 shows Ai of each experiment A and Var of the twonetworks after 10 experiments

As can be seen from Table 6 in multiple experiments thevariance of SVD-1DCNN is 0 and the variance of SVD-1DCNN is 125times10minus 4 indicating that both networks have

C1

SVD

S2Fully

connected layer

S4C3 Output layer

Iteration doneWeight andbias update

N

Fault classify

Y

Data input

SVD + 1DCNN

Result output

Figure 5 SVD+ 1DCNN diagnosis flowchart

Table 3 Confusion matrix

Predicted labelTrue label

Positive class Negative classPositive class True positive (TP) False positive (FP)Negative class False negative (FN) True negative (TN)

Shock and Vibration 7

excellent stability According to the final experimental re-sults SVD-1DCNN has a higher A than SVD+ 1DCNN Inthe classification of simulation signals SVD-1DCNN has abetter classification effect

In addition A and Var of SVD-1DCNN andSVD+1DCNN were calculated with different networkstructures as shown in Table 7

As shown in Table 7 both networks have excellentstability and the classification effect of SVD-1DCNN isbetter than that of SVD+ 1DCNN It can be seen that thenumber of convolution kernels has a greater impact on theclassification results -e number of convolution kernels inthe network is too small to make the network fail to achievehigh classification accuracy but the number of convolutionkernels is not as good as possible Excessive convolutionkernels may even reduce the training effect of the network

In the above network structure the network structure of(351 80)-[3 3] has the best classification effect so it will betaken as the research object in the following part -ereforethe final parameters of SVD-1DCNN are as follows the firstconvolutional layer contains three convolution kernels eachof which has a size of 1times 351 the second convolutional layer

contains three convolution kernels each with a size of 1times 80the learning rate is 01 the training batch is 10 the maximumnumber of iterations is 1500 the pooling mode of the twopooling layers is average pooling and the step size is 2 -ecorresponding parameters in SVD+ 1DCNN are the same asthose in SVD-1DCNN

33 Analysis of the Role of Convolution Kernel

331 Qualitative Analysis In the SVD-1DCNN networkstructure each feature map in the convolution layercontains a convolution kernel and the convolution resultsof the convolution kernel with the signals are the output ofthe feature map In order to analyze the role of theconvolution kernels during training Y1 is taken as anexample During the training process the output of thefeature map of C1 of SVD-1DCNN is extracted and itstime domain and time-frequency diagrams are as shownin Figure 6

It can be seen from Figure 6 that during the trainingprocess the convolution kernel highlights part of the fre-quency characteristics and suppresses other frequencycharacteristics C1 highlights this portion of the frequencycharacteristics as primary classification features of the inputIt is worth noting that the convolution kernel only selectspart of the frequency features from the input as the primaryclassification features which adaptively realizes the di-mensionality reduction of the data and improves the clas-sification efficiency of the network

Figure 7 shows the time domain and frequency domaindiagrams of the output of the C1 feature map for the fourtypes of signals at the end of the network training It can beseen that the convolution kernel performs differentdenoising operations on the four types of signals and retainsthe main frequency components in the original signals

At the same time in order to more intuitively analyze thefeature extraction effect of the convolution kernel on thesignals the output of C1 is shown in Figure 8 Figure 8 showsthe time-frequency diagrams of the four original signals andthe time-frequency diagrams of the output of C1 duringnetwork training

It can be seen from Figures 7 and 8 that in the trainingprocess the convolution kernel realizes the feature extrac-tion of the original signals As the number of iteration in-creases the noise components are gradually eliminatedhighlighting the features learned by the network

332 Quantitative Analysis In order to analyze the featureextraction effect of convolution kernels on the originalsignal eSNR is used as the index for evaluation Table 8 is theeSNR of C1rsquos three feature mapsrsquo output when the iterationtimes are 200 500 and 1000 respectively

In order to visually explain the change in eSNR Y1 isused as an object Figure 9 shows the eSNR of three featuremapsrsquo output according to equation (9)

It can be seen from Figures 8 and 9 that in the trainingprocess the primary classification features of each convo-lution kernel extraction have a higher eSNR than the input

Table 6 Ai A and Var of SVD-1DCNN and SVD+ 1DCNN

SVD-1DCNN SVD+1DCNN1 100 852 100 8753 100 904 100 8755 100 8756 100 8757 100 8758 100 8759 100 87510 100 875A 100 875Var 0 125 times 10minus 4

Table 4 Confusion matrix of SVD-1DCNN for classification re-sults of simulated signals

Predicted labelTrue label ()

Y1 Y2 Y3 Y4

Y1 100 0 0 0Y2 0 100 0 0Y3 0 0 100 0Y4 0 0 0 100

Table 5 Confusion matrix of SVD+ 1DCNN for classificationresults of simulated signals

Predicted labelTrue label ()

Y1 Y2 Y3 Y4

Y1 90 0 0 10Y2 10 80 0 10Y3 0 0 90 10Y4 10 0 0 90

8 Shock and Vibration

which highlights the useful feature components in the sig-nals At the same time as the number of iterations increasesthe eSNR of the primary classification features is higher andthe useful feature components in the signal are moresignificant

-rough simulation experiment it can be found that inthe training process the convolution kernels can adaptivelyremove the noise components in the signals according to the

characteristics of the original signals and retain the learnedfeatures It can be said that the convolution kernels not onlyextract the characteristic components in the original signalsbut also achieve denoising

34 Analysis of Two Networksrsquo Classification EffectsSVD+ 1DCNN and SVD-1DCNN have different classifi-cation effects on the same dataset In the two network

0 01 02 03 04 05 06 07 08ndash2

ndash1

0

1

2

Time (s)

Time (s)

Am

plitu

de

02 03 04 05 06 07 080

200

400

Freq

uenc

y (H

z)

(a)

Time (s)

Time (s)

0 01 02 03 04 05ndash1

ndash05

0

05

1

015 02 025 03 035 04 045 050

200

400A

mpl

itude

Freq

uenc

y (H

z)

(b)

Time (s)

Time (s)015 02 025 03 035 04 045 05

0

200

400

0 01 02 03 04 05ndash1

ndash05

0

05

1

Am

plitu

deFr

eque

ncy

(Hz)

(c)

Time (s)

Time (s)

0 01 02 03 04 05ndash1

ndash05

0

05

1

015 02 025 03 035 04 045 050

200

400

Am

plitu

deFr

eque

ncy

(Hz)

(d)

Figure 6 Time domain and time-frequency diagrams of the output of C1 in different iteration stages (a) original signal (b) iteration is 200(c) iteration is 500 (d) iteration is 1000

Table 7 A and Var of different network structures

SVD-1DCNN SVD+1DCNNA Var A Var

(351 80)-[1 1] Divergency Divergency Divergency Divergency(351 80)-[2 2] 95 125 times 10minus 4 85 125 times 10minus 4

(351 80)-[3 3] 100 0 875 125 times 10minus 4

(351 80)-[4 4] 975 0 85 0(351 80)-[6 6] 90 125 times 10minus 4 825 125 times 10minus 4

(351 80)-[8 8] 90 0 825 0(351 80)-[10 10] 875 0 825 0

Shock and Vibration 9

0 01 02 03 04 05ndash1

ndash05

0

05

1

Time (s)

Am

plitu

deFr

eque

ncy

(Hz)

015 02 025 03 035 04 045 050

200

400

Time (s)

(a)

Am

plitu

deFr

eque

ncy

(Hz)

0 01 02 03 04 05ndash10

ndash5

0

5

10

Time (s)

Time (s)015 02 025 03 035 04 045 05

0

200

400

(b)

Am

plitu

deFr

eque

ncy

(Hz)

0 01 02 03 04 05Time (s)

015 02 025 03 035 04 045 05Time (s)

ndash10

ndash5

0

5

10

0

200

400

(c)

Am

plitu

deFr

eque

ncy

(Hz)

0 01 02 03 04 05Time (s)

Time (s)015 02 025 03 035 04 045 05

ndash4

ndash2

0

2

4

0

200

400

(d)

Figure 7 Time domain and time-frequency diagrams of the output of C1 of four types of signals at the end of training (a) Y1 (b) Y2 (c) Y3and (d) Y4

04 080

500

Freq

uenc

y (H

z)

Original signal

ndash80ndash60ndash40ndash20

02 040

500Iteration is 200

ndash80

ndash60

ndash40

02 040

500Iteration is 500

ndash100ndash80ndash60ndash40ndash20

02 040

500Iteration is 1000

ndash80ndash60ndash40ndash20

Time (s)

(a)

Freq

uenc

y (H

z)

Original signal Iteration is 200 Iteration is 500 Iteration is 1000

04 080

500

ndash80ndash60ndash40ndash20

02 040

500

ndash100

ndash50

02 040

500

ndash80

ndash60

ndash40

ndash20

02 040

500

ndash80ndash60ndash40ndash20

Time (s)

(b)

Figure 8 Continued

10 Shock and Vibration

structures except for the structure the other parameters arethe same By comparing the two network structures it can beseen that in SVD+ 1DCNN the S2rsquos input is the output ofC1 and in SVD-1DCNN that is the output of the SVD layer-erefore eSNR of the S2rsquos input is calculated to analyze thefeature extraction capabilities of the two networks Forconvenience of presentation map 1 map 2 and map 3 areused to represent the output of C1 of SVD+ 1DCNN andSmap 1 Smap 2 and Smap 3 are used to represent the outputof the SVD layer in SVD-1DCNN Figure 10 shows eSNR ofthe S2rsquos input in two networks

It can be seen from Figure 10 that in the training processof the two networks the eSNR of S2rsquos input increases withthe increase of the number of iteration and the networksrsquofeature extraction ability is stronger With the same numberof iteration the eSNR of S2rsquos input in SVD-1DCNN is higherthan that in SVD+ 1DCNN -is indicates that the inputfeatures of S2 in SVD-1DCNN are more obvious In furtherfeature extraction features with high eSNR are more con-ducive to network learning high-level features

-erefore it can be said that SVD-1DCNN has strongerfeature extraction ability than SVD+1DCNN which isconducive to improving the accuracy of pattern recognition

4 Bearing Fault Diagnosis Based on SVD-1DCNN

41DataCollectionandProcessing -e experimental data inthis paper come from the bearing database of the CaseWestern Reserve University (CWRU) [38] -e experi-mental data are the acceleration data of the drive end at asampling frequency of 12KHz -e data include four typesdata with a fault diameter of 0007 mils on the rolling el-ement data with a fault diameter of 0007 mils on the innerring data with a fault diameter of 0007 mils on the outerring and normal data

-e length of each segment of the signal is about 120000In order to increase the randomness of the training set andthe test set a window of length 1024 is used to sample thesignal in random steps from the first node of each signal asshown in Figure 11 In the sampling process 60 samples areobtained from each signal and among the 60 samples 10samples are randomly selected as the test set and theremaining samples are used as samples of the training set Inthis way a training set containing 200 samples and a test setcontaining 40 samples are obtained

42 Fault Diagnosis ofMeasured Signals Figure 12 shows thesignals in the four states in the time domain and time-fre-quency diagrams It can be seen that the potential failuremodes are masked by noise and the signal characteristics arehidden in strong background noise and unrelated interference

As can be seen from Figure 12 the four measured signalscontain a large amount of noise which increases the diffi-culty of pattern recognition -e eSNR of the four types ofsignals is calculated according to equation (9) as shown inTable 9

Table 8 eSNR of C1rsquos three feature mapsrsquo output

Input C1 outputIterations

200 500 1000

Y1

Feature map 1 24090 37783 41507Feature map 2 27967 35906 40328Feature map 3 24672 37418 41003

Y2

Feature map 1 28414 33305 37819Feature map 2 27756 33910 38203Feature map 3 29455 34477 39551

Y3

Feature map 1 35443 40511 47872Feature map 2 35787 41335 46914Feature map 3 33706 40591 45370

Y4

Feature map 1 25801 36551 41434Feature map 2 25513 35857 38749Feature map 3 23790 35934 38023

Freq

uenc

y (H

z)Original signal Iteration is 200 Iteration is 500 Iteration is 1000

04 080

500

ndash100ndash80ndash60ndash40ndash20

02 040

500

ndash100ndash80ndash60ndash40ndash20

02 040

500

ndash100

ndash50

02 040

500

ndash80ndash60ndash40ndash200

Time (s)

(c)

Freq

uenc

y (H

z)

Original signal Iteration is 200 Iteration is 500 Iteration is 1000

04 080

500

ndash120ndash100ndash80ndash60ndash40ndash20

02 040

500

Time (s)

ndash100ndash80ndash60ndash40ndash20

02 040

500

ndash80ndash60ndash40ndash20

02 040

500

ndash80ndash60ndash40ndash20

(d)

Figure 8 Time-frequency diagrams of the output of C1 of four types of signals in different iteration stages (a) Y1 (b) Y2 (c) Y3 and (d) Y4

Shock and Vibration 11

Figure 13 shows the output of C1 of four types of signalsduring network training As can be seen from Figure 13 thecharacteristics of the original signals are submerged in alarge amount of noise but after C1 convolution the noise inthe original signal is gradually eliminated As the number ofiteration increases the characteristic components in theoriginal signal gradually become prominent

In order to visually reflect the change process of eSNRthe rolling element fault signal is selected for explanationFigure 14 shows the change of eSNR of the output of thethree feature maps of C1 during the training

It can be seen from Figure 14 that the measured signalhas a lower eSNR and the eSNR of the signal is improvedafter the C1 feature extraction In the pattern recognition of

20304050

20304050

0 500 1000 1500

20304050

Iteration

Iteration

Iteration

0 500 1000 1500

0 500 1000 1500

eSN

ReS

NR

eSN

R

Figure 9 eSNR of three feature mapsrsquo output

Smap 1Map 1

Smap 2Map 2

Smap 3Map 3

20406080

100

eSN

R

20406080

100

eSN

R

20406080

100

eSN

R

500 1000 15000Iteration

500 1000 15000Iteration

500 1000 15000Iteration

Figure 10 eSNR of the S2rsquos input of the two networks

12 Shock and Vibration