# Correlation Coefficient of Simplified Neutrosophic Sets for Bearing Fault Diagnosis

Authors: Lilian Shi
Journal: Shock and Vibration
 (2016)

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## Abstract

In order to process the vagueness in vibration fault diagnosis of rolling bearing, a new correlation coefficient of simplified neutrosophic sets (SNSs) is proposed. Vibration signals of rolling bearings are acquired by an acceleration sensor, and a morphological filter is used to reduce the noise effect. Wavelet packet is applied to decompose the vibration signals into eight subfrequency bands, and the eigenvectors associated with energy eigenvalue of each frequency are extracted for fault features. The SNSs of each fault types are established according to energy eigenvectors. Finally, a correlation coefficient of two SNSs is proposed to diagnose the bearing fault types. The experimental results show that the proposed method can effectively diagnose the bearing faults.


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## Body

## 1. Introduction

A rolling bearing is an important rotating part in a mechanical equipment, and its quality decides the operation performance of the equipment. A faulty bearing may cause the whole equipment to operate abnormally. Bearing faults must be effectively diagnosed to avoid catastrophic mechanical failures and significant economic losses.The vibration signals of rolling bearings often indicate some fault information. When the fault occurs in rotating bearings, different characteristic frequencies of vibration signals can be generated periodically [1]. Actually, for the original vibration signals, many useful fault features are usually hidden in noise, and the relationship between fault symptoms and causations is very complex, so it is difficult to make accurate and quantitative analysis for fault types. In recent years, many studies have been devoted to the fault diagnosis of rolling bearing. There are two critical issues for diagnosing bearing faults from vibration signals. One issue is how to extract fault features from vibration signals. Another one is how to analyze fault features and recognize fault types according to these features.In order to extract useful fault features from vibration signals, many techniques such as time domain, frequency domain, and time-frequency domain methods are extensively investigated [2]. In the time domain method, key parameters can be extracted directly from the original vibration signals, such as root mean square (RMS), crest factor, peak, and probability density function [3]. In addition, time domain signals can be transformed into frequency domain by Fourier transform. However, Fourier analysis may cause information loss during the transformation, particularly for nonstationary signals. The vibration of a rolling bearing is typically nonstationary, so it is difficult to extract accurate and complete fault features when adopting the traditional analysis only in the time or frequency domain. In time-frequency domain, the wavelet can reveal more complete information for nonstationary signals [4]. Many research results show that a wavelet packet is an effective tool to extract features from vibration signals for bearing fault diagnosis [5–8].The next key issue is to recognize fault types of bearings according to the extracted fault features from vibration signals. To solve this problem, various approaches such as expert systems [9, 10], neural networks [3, 11, 12], and fuzzy approaches [13–15] have been developed for fault diagnosis over the past few years. Fuzzy theory has attracted increasing attention in bearing fault diagnosis, and many researches show that fuzzy theory is an effective tool to diagnose bearing faults.Fuzzy sets (FSs) theory was proposed by Zadeh (1965) for handling uncertain information using single membership degree function [16]. The fuzzy sets were extended to intuitionistic fuzzy sets (IFSs) [17] and interval valued intuitionistic fuzzy sets (IVIFSs) [18] by using membership degree function, non-membership degree function, and degree function of hesitation simultaneously. FSs, IFSs, and IVIFSs have been widely applied in various fields. However, FSs, IFSs, and IVIFSs cannot deal with some types of uncertainties such as the indeterminate information and inconsistent information in real physical problems. Furthermore, Smarandache [19] proposed neutrosophy theory from philosophical point of view. Neutrosophic sets (NSs) are characterized by a truth-membership function, an indeterminacy-membership function, and a falsity-membership function. The functions of NSs take the value from real standard or nonstandard subsets of ]0-,1+[ [19], and NSs are difficult to be applied in engineering areas. For the real engineering applications, neutrosophic sets (NSs) can be described as simplified neutrosophic sets (SNSs) [20] with the normal standard real unit interval [0,1]. One major advantage of SNSs is the ability to perform analysis problems involving imprecise, undetermined, and inconsistent data. Recently, SNSs have been applied in many different fuzzy problems, such as medical diagnosis problems [21, 22], decision making problems [20, 23], and image processing [24].For vibrational fault diagnosis of rolling bearing, there is no direct accurate and quantitative relationship between fault vibration characteristics and fault types. Therefore, the fault-diagnosis process has certain vagueness. This paper mainly focuses on the fault diagnosis of rolling bearings based on vibration signals and SNSs. In this work, a morphological filter and wavelet packet decomposition are applied to preprocess the original vibration signals, and the SNSs of each fault type will be established according to energy eigenvectors. The fault types will be diagnosed using a new correlation coefficient of SNSs.The rest of the paper is organized as follows. Section2 gives the experimental system. Section 3 gives the data preprocessing techniques including morphological opening-closing operation and wavelet packet decomposition. In Section 4, some basic concepts of SNSs and a new correlation coefficient are introduced firstly, and then the fault-diagnosis method is presented based on SNSs. Conclusions of this work are summarized in Section 5.


## 2. Experimental Setup

This study was carried out with the experimental apparatus shown in Figure1. The principal axis is driven by an AC motor, and the vibration signals of bearings are acquired by an acceleration sensor and a data acquiring card NI USB-6251. The vibration signals will be processed using a computer and displayed by an oscilloscope. Some vibration signals are acquired by the experimental device of Jiliang University in China [25], shown in Figure 2. In this experiment, the type of bearings is NSK 6202 deep groove ball bearing whose specifications are listed in Table 1.Table 1
Bearing parameters.
Parameters Values (mm) Outer race diameter 35 Inner race diameter 15 Ball diameterd 7.5 Thickness 11 Number of balls 7 Pitch diameterD 25 Contact angleα 0Figure 1
Diagram of experimental system.(1) Motor, (2) driver, (3) principal axis, (4) bearing, (5) core axes, (6) acceleration sensor, (7) acquisition data card, (8) oscilloscope, and (9) computer.Figure 2
Experimental device.In order to diagnose the fault of bearings, four types of bearings are used: healthy, outer race fault, inner race fault, and ball fault bearings. The core axis is driven at the rotational speed of 25 Hz. NI Labview Signal Express will be applied for data acquisition with 10 KHz sampling frequency and 0.2 s sample time.When a fault exists in a bearing, vibration impulses will happen at a specific frequency. Theoretically, when a bearing rotates at a constant speed, the fault frequencies can be calculated by the following [26]:(1)FO=NB21-dD·cos⁡α·Fr,FI=NB21+dD·cos⁡α·Fr,FB=Dd1-dD·cos⁡α2·Fr,where d is the diameter of the rolling elements, D is the pitch diameter, Fr is the rotational speed of the shaft, NB is the number of rolling elements, and FO, FI, and FB represent the fault frequencies of outer race fault, inner race fault, and ball fault of a bearing, respectively.According to (1), we can calculate the fault frequencies FO=61.25 Hz, FI=113.75 Hz, and FB=75.83 Hz in this experiment.


## 3. Vibration Signal Data Preprocessing

The framework of diagnosing process is shown in Figure3.Figure 3
Block diagram of diagnosis for fault bearing using neutrosophic sets.The original vibration signals of bearings are usually ridden with noise. It is difficult to extract the fault features directly from original vibration signals. In order to remove the strong noises and detect the effective signals for bearing faults diagnosis, data processing algorithms are necessary to be performed. In this experiment, a morphological filter is used to remove high frequencies noise from the original vibration signals firstly, and then wavelet packet is applied to decompose the signals into the individual frequencies.


### 3.1. Morphological Filter

A morphological filter is a nonlinear signal processing and analysis tool in time domain, and it can be composed of several morphological operations [27]. The basic morphological operators include dilation, erosion, opening, and closing. Assume that x(n) and a structural element y(n) are discrete signals defined in X={0,1,…,N-1} and B={0,1,…,M-1}, respectively, and N≥M, the four basic operators of x(n) on y(n), are defined as follows:Dilation (2) x ⊕ y = max m = 0,1 , … , M - 1 ⁡ x n + m + y m ,  n = 0,1 , … , N + MErosion (3) x ⊖ y = min m = 0,1 , … , M - 1 ⁡ x n + m - y m ,  n = 0,1 , … , N + MOpening (4) x ∘ y = x ⊖ y ⊕ yClosing (5) x • y = x ⊕ y ⊖ yMorphological opening-closing filter as follows:(6)Foc=x∘y•yIn this experiment, the morphological opening-closing filterFoc was used to remove the strong noises.Figures4–6 show the signals of rolling element bearings with outer race, inner race, and ball fault, respectively [25]. In these figures, the fault signals have distinguishing peak value features at the fault frequencies. The results in Figures 4–6 indicate that morphological filter is an effective denoising technique for vibration signals of ball bearings.Figure 4
Signals of rolling bearing in outer race fault.
(a)
Original vibration in time domain (b)
Fast Fourier transform after morphological filterFigure 5
Signals of rolling bearing in inner race fault.
(a)
Original vibration in time domain (b)
Fast Fourier transform after morphological filterFigure 6
Signals of rolling bearing in ball fault.
(a)
Original vibration in time domain (b)
Fast Fourier transform after morphological filter


### 3.2. Wavelet Packet Decomposition

According to the structure of wavelet decomposition, the input vibration signal can be decomposed into low-frequency and high-frequency parts for each step. The selection of a suitable level for the hierarchy depends on the signal, experience, and actual needs [4, 7]. The wavelet packet was applied to decompose the vibration signals into eight subfrequency bands in the practical application [2, 6]. Based on review of earlier researcher, in this work, 3-level wavelet packet decomposition is considered for bearing fault diagnosis, and experimental results show that the bearing fault feature can be extracted effectively from the decomposed signals.In this experiment, the vibration signals of bearing are preprocessed firstly by a morphological filter and then are decomposed using 3-level wavelet packet.Assume thatx(t) is a vibration signal; L(·) and H(·) are quadrature mirror filters, representing low-pass and high-pass wavelet filters, respectively. These filters associate with the scaling function and wavelet function and satisfy the condition H(n)=-1nL(1-n). Then, the signal x(t) can be decomposed into a set of high- and low-frequency components by the following recursive relationships:(7)xj,2k=∑nLn·xj-1,k,xj,2k+1=∑nHn·xj-1,k,where xj,2k denotes the wavelet coefficients at thejth level and 2kth subband.The diagram of 3-level wavelet packet decomposition is shown in Figure7. In Figure 7, the frequency intervals of each band can be computed by ((k-1)fs/24, kfs/24], where fs is sampling frequency. In this work, fs=10 kHz and fs/24 = 625 Hz. The frequency intervals are given in Table 2.Table 2
Frequency intervals of eight subfrequency bands.
Signals Frequency (Hz) x 3 , 0 ( 0,625 ] x 3 , 1 ( 625,1250 ] x 3 , 2 ( 1250,1875 ] x 3 , 3 ( 1875,2500 ] x 3 , 4 ( 2500,3125 ] x 3 , 5 ( 3125,3750 ] x 3 , 6 ( 3750,4375 ] x 3 , 7 ( 4375,5000 ]Figure 7
Diagram of 3-level wavelet packet decomposition (L: low-pass filter, H: high-pass filter).The vibration signalx(t) can be expressed as follows:(8)xt=∑k=023-1x3,kt,k=0,1,…,7,where k represents eight subfrequency bands and x3,k(t) is the wavelet coefficient at the 3-level and kth subfrequency band. After the decomposition, the energy in each subfrequency band can be defined as(9)E3k=∫x3,kt2dt=∑i=0Nx3,ki2,k=0,1,…,7,where x3,k(i) is the ith discrete point amplitude of wavelet coefficient (x3,k(t)) and N is its discrete point number in each subfrequency.The faults of rolling bearings will greatly influence the wavelet packet energy of vibration signals, so it is very useful to extract the energy eigenvalue for diagnosing bearing faults. In this experiment, an eigenvector based on energy eigenvalue of each frequency can be constructed as follows:(10)T=E30,E31,E32,E33,E34,E35,E36,E37.Furthermore, assume thatE3max is the maximum value of the energy eigenvalue in the 3-level subfrequency band, and then the eigenvalues can be normalized as follows:(11)E3∗k=E3kE3max,k=0,1,…,7.By the above normalization, the energy eigenvalue of the wavelet packet energy of vibration signals was bounded to[0,1], and then the normalized eigenvector can be described as follows:(12)T∗=E3∗0,E3∗1,E3∗2,E3∗3,E3∗4,E3∗5,E3∗6,E3∗7.The normalized energy eigenvalues of vibration signals are shown in Figure8.Figure 8
Energy histogram of rolling bearing.
(a)
Healthy rolling bearing (b)
Rolling bearing in outer race fault (c)
Rolling bearing in inner race fault (d)
Rolling bearing in ball faultFor diffident type faults of bearing, the eigenvalue of the wavelet packet energy has the distinguishing distribution at the individual subfrequency band. According to a lot of experimentation and data comparison, we extract the lower bound and upper bound of the energy eigenvalues for typical faults of bearing and establish the energy interval ranges as shown in Table3, and the energy interval ranges can be used to diagnose fault types of rolling bearings in the next step.Table 3
Energy interval ranges at the eight subfrequency bands.
Fault types Energy in each frequency band E 3 ∗ 0 E 3 ∗ 1 E 3 ∗ 2 E 3 ∗ 3 E 3 ∗ 4 E 3 ∗ 5 E 3 ∗ 6 E 3 ∗ 7 A 1 (healthy) [0.76, 0.80] [0.95, 1.00] [0.11, 0.15] [0.64, 0.71] [0.04, 0.06] [ 0.00 ,   0.02 ] [0.01, 0.05] [ 0.00 ,    0.03 ] A 2 (outer race fault) [1.00, 1.00] [0.24, 0.39] [0.02, 0.03] [0.13, 0.22] [0.00, 0.01] [ 0.00 ,    0.01 ] [0.01, 0.01] [ 0.01 ,    0.01 ] A 3 (ball fault) [0.82, 0.93] [1.00, 1.00] [0.11, 0.16] [0.65, 0.74] [0.06, 0.08] [ 0.00 ,    0.04 ] [0.02, 0.06] [ 0.00 ,    0.01 ] A 4 (inner race fault) [1.00, 1.00] [0.49, 0.55] [0.06, 0.10] [0.20, 0.24] [0.00, 0.00] [ 0.02 ,    0.03 ] [0.03, 0.05] [ 0.03 ,    0.06 ]


## 3.1. Morphological Filter

A morphological filter is a nonlinear signal processing and analysis tool in time domain, and it can be composed of several morphological operations [27]. The basic morphological operators include dilation, erosion, opening, and closing. Assume that x(n) and a structural element y(n) are discrete signals defined in X={0,1,…,N-1} and B={0,1,…,M-1}, respectively, and N≥M, the four basic operators of x(n) on y(n), are defined as follows:Dilation (2) x ⊕ y = max m = 0,1 , … , M - 1 ⁡ x n + m + y m ,  n = 0,1 , … , N + MErosion (3) x ⊖ y = min m = 0,1 , … , M - 1 ⁡ x n + m - y m ,  n = 0,1 , … , N + MOpening (4) x ∘ y = x ⊖ y ⊕ yClosing (5) x • y = x ⊕ y ⊖ yMorphological opening-closing filter as follows:(6)Foc=x∘y•yIn this experiment, the morphological opening-closing filterFoc was used to remove the strong noises.Figures4–6 show the signals of rolling element bearings with outer race, inner race, and ball fault, respectively [25]. In these figures, the fault signals have distinguishing peak value features at the fault frequencies. The results in Figures 4–6 indicate that morphological filter is an effective denoising technique for vibration signals of ball bearings.Figure 4
Signals of rolling bearing in outer race fault.
(a)
Original vibration in time domain (b)
Fast Fourier transform after morphological filterFigure 5
Signals of rolling bearing in inner race fault.
(a)
Original vibration in time domain (b)
Fast Fourier transform after morphological filterFigure 6
Signals of rolling bearing in ball fault.
(a)
Original vibration in time domain (b)
Fast Fourier transform after morphological filter


## 3.2. Wavelet Packet Decomposition

According to the structure of wavelet decomposition, the input vibration signal can be decomposed into low-frequency and high-frequency parts for each step. The selection of a suitable level for the hierarchy depends on the signal, experience, and actual needs [4, 7]. The wavelet packet was applied to decompose the vibration signals into eight subfrequency bands in the practical application [2, 6]. Based on review of earlier researcher, in this work, 3-level wavelet packet decomposition is considered for bearing fault diagnosis, and experimental results show that the bearing fault feature can be extracted effectively from the decomposed signals.In this experiment, the vibration signals of bearing are preprocessed firstly by a morphological filter and then are decomposed using 3-level wavelet packet.Assume thatx(t) is a vibration signal; L(·) and H(·) are quadrature mirror filters, representing low-pass and high-pass wavelet filters, respectively. These filters associate with the scaling function and wavelet function and satisfy the condition H(n)=-1nL(1-n). Then, the signal x(t) can be decomposed into a set of high- and low-frequency components by the following recursive relationships:(7)xj,2k=∑nLn·xj-1,k,xj,2k+1=∑nHn·xj-1,k,where xj,2k denotes the wavelet coefficients at thejth level and 2kth subband.The diagram of 3-level wavelet packet decomposition is shown in Figure7. In Figure 7, the frequency intervals of each band can be computed by ((k-1)fs/24, kfs/24], where fs is sampling frequency. In this work, fs=10 kHz and fs/24 = 625 Hz. The frequency intervals are given in Table 2.Table 2
Frequency intervals of eight subfrequency bands.
Signals Frequency (Hz) x 3 , 0 ( 0,625 ] x 3 , 1 ( 625,1250 ] x 3 , 2 ( 1250,1875 ] x 3 , 3 ( 1875,2500 ] x 3 , 4 ( 2500,3125 ] x 3 , 5 ( 3125,3750 ] x 3 , 6 ( 3750,4375 ] x 3 , 7 ( 4375,5000 ]Figure 7
Diagram of 3-level wavelet packet decomposition (L: low-pass filter, H: high-pass filter).The vibration signalx(t) can be expressed as follows:(8)xt=∑k=023-1x3,kt,k=0,1,…,7,where k represents eight subfrequency bands and x3,k(t) is the wavelet coefficient at the 3-level and kth subfrequency band. After the decomposition, the energy in each subfrequency band can be defined as(9)E3k=∫x3,kt2dt=∑i=0Nx3,ki2,k=0,1,…,7,where x3,k(i) is the ith discrete point amplitude of wavelet coefficient (x3,k(t)) and N is its discrete point number in each subfrequency.The faults of rolling bearings will greatly influence the wavelet packet energy of vibration signals, so it is very useful to extract the energy eigenvalue for diagnosing bearing faults. In this experiment, an eigenvector based on energy eigenvalue of each frequency can be constructed as follows:(10)T=E30,E31,E32,E33,E34,E35,E36,E37.Furthermore, assume thatE3max is the maximum value of the energy eigenvalue in the 3-level subfrequency band, and then the eigenvalues can be normalized as follows:(11)E3∗k=E3kE3max,k=0,1,…,7.By the above normalization, the energy eigenvalue of the wavelet packet energy of vibration signals was bounded to[0,1], and then the normalized eigenvector can be described as follows:(12)T∗=E3∗0,E3∗1,E3∗2,E3∗3,E3∗4,E3∗5,E3∗6,E3∗7.The normalized energy eigenvalues of vibration signals are shown in Figure8.Figure 8
Energy histogram of rolling bearing.
(a)
Healthy rolling bearing (b)
Rolling bearing in outer race fault (c)
Rolling bearing in inner race fault (d)
Rolling bearing in ball faultFor diffident type faults of bearing, the eigenvalue of the wavelet packet energy has the distinguishing distribution at the individual subfrequency band. According to a lot of experimentation and data comparison, we extract the lower bound and upper bound of the energy eigenvalues for typical faults of bearing and establish the energy interval ranges as shown in Table3, and the energy interval ranges can be used to diagnose fault types of rolling bearings in the next step.Table 3
Energy interval ranges at the eight subfrequency bands.
Fault types Energy in each frequency band E 3 ∗ 0 E 3 ∗ 1 E 3 ∗ 2 E 3 ∗ 3 E 3 ∗ 4 E 3 ∗ 5 E 3 ∗ 6 E 3 ∗ 7 A 1 (healthy) [0.76, 0.80] [0.95, 1.00] [0.11, 0.15] [0.64, 0.71] [0.04, 0.06] [ 0.00 ,   0.02 ] [0.01, 0.05] [ 0.00 ,    0.03 ] A 2 (outer race fault) [1.00, 1.00] [0.24, 0.39] [0.02, 0.03] [0.13, 0.22] [0.00, 0.01] [ 0.00 ,    0.01 ] [0.01, 0.01] [ 0.01 ,    0.01 ] A 3 (ball fault) [0.82, 0.93] [1.00, 1.00] [0.11, 0.16] [0.65, 0.74] [0.06, 0.08] [ 0.00 ,    0.04 ] [0.02, 0.06] [ 0.00 ,    0.01 ] A 4 (inner race fault) [1.00, 1.00] [0.49, 0.55] [0.06, 0.10] [0.20, 0.24] [0.00, 0.00] [ 0.02 ,    0.03 ] [0.03, 0.05] [ 0.03 ,    0.06 ]


## 4. Fault Diagnosis of Rolling Bearing Based on SNSs

In this section, we briefly introduce basic concepts of simplified neutrosophic sets (SNSs) and propose a new correlation coefficient of two SNSs, which will be needed in the following analysis. Then, we establish the fault SNSs of bearings according to energy features. Finally, we present the method for fault diagnosis of rolling bearing according to the correlation coefficient of SNSs.


### 4.1. Simplified Neutrosophic Sets (SNSs)

Definition 1 (see [19]).
LetU be a universe of discourse; then, the neutrosophic set (NS) A is defined by(13)A=x,TAx,IAx,FAx,x∈U,where the functions TA(x), IA(x), and FA(x) represent a truth-membership function, an indeterminacy-membership function, and a falsity-membership function of the element x∈U to the set A, respectively, with the conditions TA(x),IA(x),FA(x):U→]0-,1+[ and 0-≤sup⁡TAx+sup⁡IAx+sup⁡FA(xi)≤3+.
The above concept of a neutrosophic set (NS) is presented from philosophical point of view, and it takes the value from real standard or nonstandard subsets of]0-,1+[. It will be difficult to apply ]0-,1+[ in scientific and engineering areas. For the real applications, a simplified neutrosophic set (SNS) is introduced by Ye [20] as the following definition.Definition 2 (see [20]).
LetU be a space of points (objects) with generic elements in U denoted by x. A simplified neutrosophic set (SNS) A in U is characterized by a truth-membership function TA(x), an indeterminacy-membership function IA(x), and a falsity-membership function FA(x). For each point x in U, TA(x), IA(x), and FA(x) are singleton subintervals/subsets in the real standard [0,1], such that TA(x),IA(x),FA(x)∈[0,1]. Then, a simplified neutrosophic set (SNS) is denoted by(14)A=x,TAx,IAx,FAx∣x∈U.
Obviously, a simplified neutrosophic set (SNS) is a subclass of the neutrosophic set (NS) and satisfies the conditionsTA(x),IA(x),FA(x):U→[0,1] and 0≤TA(x)+IA(x)+FA(x)≤3.


### 4.2. Correlation Coefficient for SNSs

Correlation coefficient is an important tool for determining the correlation degree between fuzzy sets. Therefore, a new correlation coefficient of two SNSs is proposed by the following definition.Definition 3.
Assume that there are two SNSsA=xi,TAxi,IAxi,FAxi∣xi∈U and B=xi,TBxi,IBxi,FBxi∣xi∈U in the universe of discourse U=x1,x2,…,xn, xi∈U. A correlation coefficient between SNSs is defined as follows:(15)MSNSA,B=1n∑i=1nmin⁡TAxi,TBxi+min⁡IAxi,IBxi+min⁡FAxi,FBxiTAxiTBxi+IAxiIBxi+FAxiFBxi,where the symbol “min” is the minimum operation.According to the above definition, the correlation coefficient of SNSsA and B satisfies the following properties:(P1)
0≤MSNS(A,B)≤1.(P2)
MSNS(A,B)=MSNS(B,A).(P3)
MSNS(A,B)=1 if and only if A=B.If we consider the weights ofxi, a weighted correlation coefficient between SNSs A and B is proposed as follows:(16)MSNSA,B=∑i=1nwimin⁡TAxi,TBxi+min⁡IAxi,IBxi+min⁡FAxi,FBxiTAxiTBxi+IAxiIBxi+FAxiFBxi,where wi∈[0,1] and ∑i=1nwi=1 for i=1,2,…,n.


### 4.3. Bearings Neutrosophic Sets Models Based on Energy Eigenvectors

The SNSs models of rolling bearings can be built according to the energy intervals of the eight subfrequency bands as shown in Table3.Assume that a set of bearing faults isA={A1(healthy),A2(outerracefault),A3(ballfault), A4(innerracefault)}, and a set of energy eigenvector is E={e1(E3∗0),e2(E3∗1),e3(E3∗2),e4(E3∗3),e5(E3∗4), e6(E3∗5),e7(E3∗6),e8(E3∗7)}. In Table 3, let TAk(ei) and UAk(ei)  (k=1,2,3,4;i=1,2,…,8) be the lower bound and upper bound of the characteristic value ei for Ak, respectively; then, the characteristic intervals of rolling bearing can be represented by(17)Ak=e1,TAkx1,UAke1,e2,TAke2,UAke2,e3,TAke3,UAke3,e4,TAke4,UAke4,e5,TAke5,UAke5,e6,TAke6,UAke6,e7,TAke7,UAke7,e8,TAke8,UAke8,k=1,2,3,4LetUAk(ei)=1-FAk(ei) and IAk(ei)=UAk(ei)-TAk(ei), for k=1,…,4 and i=1,…,8. If UAk(ei)-TAk(ei)≤0.01, then let IAk(ei)=0.01. In this case, the sets Ak can be extended to simplified neutrosophic sets (SNSs) and can be rewritten as (18)Ak=e1,TAke1,IAke1,FAke1,e2,TAke2,IAke2,FAke2,e3,TAke3,IAke3,FAke3,e4,TAke4,IAke4,FAke4,e5,TAke5,IAke5,FAke5,e6,TAke6,IAke6,FAke6,e7,TAke7,IAke7,FAke7,e8,TAke8,IAke8,FAke8,where TAk(ei):U→[0,1], IAk(ei):U→[0,1], FAk(ei):U→[0,1], and 0≥TAk(ei)+IAk(ei)+FAk(ei)≤3, for k=1,…,4 and i=1,…,8.According to the definition of neutrosophic sets, the numbersTAk(ei), IAk(ei), and FAk(ei) represent a truth-membership, an indeterminacy-membership, and a falsity-membership, respectively. The neutrosophic sets of bearing fault types are shown in Table 4. Here, A1, A2, A3, and A4 are healthy, outer race fault, ball fault, and inner race fault bearings, respectively.Table 4
Energy values of bearing fault types represented by the form of SNS.
Fault types Energy in each frequency band E 3 ∗ 0 E 3 ∗ 1 E 3 ∗ 2 E 3 ∗ 3 E 3 ∗ 4 E 3 ∗ 5 E 3 ∗ 6 E 3 ∗ 7 A 1 e 1 , 0.76,0.14,0.20 e 2 , 0.95,0.05,0.00 e 3 , 0.11,0.04,0.85 e 4 , 0.64,0.07,0.29 e 5 , 0.04,0.02,0.94 e 6 , 0.00,0.02,0.98 e 7 , 0.01,0.05,0.95 e 8 , 0.00,0.03,0.97 A2 e 1 , 1.00,0.01,0.00 e 2 , 0.24,0.15,0.61 e 3 , 0.02,0.01,0.97 e 4 , 0.13,0.09,0.78 e 5 , 0.00,0.01,1.00 e 6 , 0.00,0.01,0.99 e 7 , 0.01,0.01,0.99 e 8 , 0.01,0.01,0.99 A3 e 1 , 0.82,0.11,0.07 e 2 , 1.00,0.01,0.00 e 3 , 0.11,0.05,0.84 e 4 , 0.65,0.10,0.25 e 5 , 0.06,0.03,0.92 e 6 , 0.00,0.04,0.96 e 7 , 0.02,0.04,0.94 e 8 , 0.00,0.01,1.00 A4 e 1 , 1.00,0.01,0.00 e 2 , 0.49,0.07,0.45 e 3 , 0.06,0.04,0.90 e 4 , 0.20,0.04,0.76 e 5 , 0.00,0.01,1.00 e 6 , 0.02,0.01,0.97 e 7 , 0.03,0.03,0.95 e 8 , 0.03,0.03,0.94


### 4.4. Rolling Bearing Fault Diagnosis Using Correlation Coefficient

In this section, we apply the correlation coefficient of SNSs to diagnose rolling bearing faults. Assume thatAk (k=1,2,3,4) are SNSs models of rolling bearing faults and At is a testing rolling bearing signal expressed by a SNS. Then we can calculate the correlation coefficient value MSNS(Ak,At) (k=1,2,3,4) using (16). Finally, the fault-diagnosis order of the fault-testing sample can be ranked according to the correlation coefficient value, and the proper diagnosis Ak∗ for the bearing fault At is derived by(19)k∗=arg⁡max1≤k≤4⁡MSNSAk,At,This paper considers the same importance of the energy values in each frequency band; therefore, the weights ofwi (i=1,2,…,8) are wi=1/8.


### 4.5. Results and Discussions

To demonstrate the effectiveness of the new diagnosis method, we now provide two examples for fault diagnosis of bearings. Let us consider two testing bearing samplesB1 and B2 described as neutrosophic sets:(20)B1=e1,1.00,0.01,0.00,e2,0.51,0.01,0.49,e3,0.08,0.01,0.92,e4,0.24,0.01,0.76,e5,0.00,0.01,1.00,e6,0.03,0.01,0.97,e7,0.05,0.01,0.95,e8,0.06,0.01,0.94,B2=e1,1.00,0.01,0.00,e2,0.39,0.01,0.61,e3,0.03,0.01,0.97,e4,0.22,0.01,0.78,e5,0.00,0.01,1.00,e6,0.01,0.01,1.00,e7,0.01,0.01,1.00,e8,0.01,0.01,1.00.The correlation coefficient values between SNSsBj (j=1,2) and Ak (k=1,2,3,4) can calculated by (17) as follows:(21)MSNSA1,B1=0.8787,MSNSA2,B1=0.9483,MSNSA3,B1=0.8746,MSNSA4,B1=0.9819.MSNSA1,B2=0.8587,MSNSA2,B2=0.9714,MSNSA3,B2=0.8566,MSNSA4,B2=0.9590.For the fault-testing sampleB1, MSNS(A4,B1) is the maximum correlation coefficient, and MSNS(A2,B1) is the second correlation coefficient. According to the principle of correlation coefficient, the fault-diagnosis order is as follows:(22)A4⟶A2⟶A1⟶A3.Therefore, we can determine that the testing bearing is an inner race fault bearing. By actual observing, the testing bearing inner race was covered with scratches, and therefore the diagnosis result is correct.Similarly, for the fault-testing sampleB2, the fault-diagnosis order is as follows:(23)A2⟶A4⟶A1⟶A3.By actual checking, the fault of the bearing firstly resulted from damage of outer race and then inner race. So the diagnosis results are consistent with the actual situation.In the experiment, 120 rolling bearings were used for testing samples. In order to verify the effectiveness of the fault-diagnosis method proposed in this paper, we extracted the energy eigenvalues of bearing vibration signals firstly and then diagnosed the bearing faults using the correlation coefficient of SNSs and the support vector machine (SVM) [25], respectively. The fault-diagnosis results of rolling bearings are shown in Table 5. By comparing the diagnosis results shown in Table 5, it is clear that the diagnosis accuracy rate based on the correlation coefficient of SNSs is much higher than the accuracy rate based on SVM.Table 5
Fault diagnosis results based on the correlation coefficient of SNSs and SVM.
Fault type Method Test sample Diagnosis result Diagnosis accuracy rate (%) Healthy Outer race fault Ball fault Inner race fault Healthy SNSsSVM 30 2827  23  93.390 Outer race fault SNSsSVM 30  2828  22 93.393.3 Ball fault SNSsSVM 30 57  2523  83.376.7 Inner race fault SNSsSVM 30  5  3025 10083.3For further comparison, Table6 lists the diagnosis results based on the correlation coefficient of SNSs, SVM, BP, and GA-BP [28] methods, respectively. Obviously, the method based on the correlation coefficient of SNSs can achieve the average accuracy rate of 92.5%, and it is higher than the ones based on the other methods.Table 6
Fault diagnosis results based on the correlation coefficient of SNSs, SVM, BP, and GA-BP.
Diagnosis method Diagnosis accuracy rate (%) Average accuracy (%) Healthy Outer race fault Ball fault Inner race fault SNS 93.3 93.3 83.3 100 92.5 SVM 90 93.3 76.7 83.3 85.8 GA-BP 100 80 70 90 85 BP 90 70 50 90 75The above comparisons demonstrate that the proposed method in this paper is effective in the bearing fault diagnosis.


## 4.1. Simplified Neutrosophic Sets (SNSs)

Definition 1 (see [19]).
LetU be a universe of discourse; then, the neutrosophic set (NS) A is defined by(13)A=x,TAx,IAx,FAx,x∈U,where the functions TA(x), IA(x), and FA(x) represent a truth-membership function, an indeterminacy-membership function, and a falsity-membership function of the element x∈U to the set A, respectively, with the conditions TA(x),IA(x),FA(x):U→]0-,1+[ and 0-≤sup⁡TAx+sup⁡IAx+sup⁡FA(xi)≤3+.
The above concept of a neutrosophic set (NS) is presented from philosophical point of view, and it takes the value from real standard or nonstandard subsets of]0-,1+[. It will be difficult to apply ]0-,1+[ in scientific and engineering areas. For the real applications, a simplified neutrosophic set (SNS) is introduced by Ye [20] as the following definition.Definition 2 (see [20]).
LetU be a space of points (objects) with generic elements in U denoted by x. A simplified neutrosophic set (SNS) A in U is characterized by a truth-membership function TA(x), an indeterminacy-membership function IA(x), and a falsity-membership function FA(x). For each point x in U, TA(x), IA(x), and FA(x) are singleton subintervals/subsets in the real standard [0,1], such that TA(x),IA(x),FA(x)∈[0,1]. Then, a simplified neutrosophic set (SNS) is denoted by(14)A=x,TAx,IAx,FAx∣x∈U.
Obviously, a simplified neutrosophic set (SNS) is a subclass of the neutrosophic set (NS) and satisfies the conditionsTA(x),IA(x),FA(x):U→[0,1] and 0≤TA(x)+IA(x)+FA(x)≤3.


## 4.2. Correlation Coefficient for SNSs

Correlation coefficient is an important tool for determining the correlation degree between fuzzy sets. Therefore, a new correlation coefficient of two SNSs is proposed by the following definition.Definition 3.
Assume that there are two SNSsA=xi,TAxi,IAxi,FAxi∣xi∈U and B=xi,TBxi,IBxi,FBxi∣xi∈U in the universe of discourse U=x1,x2,…,xn, xi∈U. A correlation coefficient between SNSs is defined as follows:(15)MSNSA,B=1n∑i=1nmin⁡TAxi,TBxi+min⁡IAxi,IBxi+min⁡FAxi,FBxiTAxiTBxi+IAxiIBxi+FAxiFBxi,where the symbol “min” is the minimum operation.According to the above definition, the correlation coefficient of SNSsA and B satisfies the following properties:(P1)
0≤MSNS(A,B)≤1.(P2)
MSNS(A,B)=MSNS(B,A).(P3)
MSNS(A,B)=1 if and only if A=B.If we consider the weights ofxi, a weighted correlation coefficient between SNSs A and B is proposed as follows:(16)MSNSA,B=∑i=1nwimin⁡TAxi,TBxi+min⁡IAxi,IBxi+min⁡FAxi,FBxiTAxiTBxi+IAxiIBxi+FAxiFBxi,where wi∈[0,1] and ∑i=1nwi=1 for i=1,2,…,n.


## 4.3. Bearings Neutrosophic Sets Models Based on Energy Eigenvectors

The SNSs models of rolling bearings can be built according to the energy intervals of the eight subfrequency bands as shown in Table3.Assume that a set of bearing faults isA={A1(healthy),A2(outerracefault),A3(ballfault), A4(innerracefault)}, and a set of energy eigenvector is E={e1(E3∗0),e2(E3∗1),e3(E3∗2),e4(E3∗3),e5(E3∗4), e6(E3∗5),e7(E3∗6),e8(E3∗7)}. In Table 3, let TAk(ei) and UAk(ei)  (k=1,2,3,4;i=1,2,…,8) be the lower bound and upper bound of the characteristic value ei for Ak, respectively; then, the characteristic intervals of rolling bearing can be represented by(17)Ak=e1,TAkx1,UAke1,e2,TAke2,UAke2,e3,TAke3,UAke3,e4,TAke4,UAke4,e5,TAke5,UAke5,e6,TAke6,UAke6,e7,TAke7,UAke7,e8,TAke8,UAke8,k=1,2,3,4LetUAk(ei)=1-FAk(ei) and IAk(ei)=UAk(ei)-TAk(ei), for k=1,…,4 and i=1,…,8. If UAk(ei)-TAk(ei)≤0.01, then let IAk(ei)=0.01. In this case, the sets Ak can be extended to simplified neutrosophic sets (SNSs) and can be rewritten as (18)Ak=e1,TAke1,IAke1,FAke1,e2,TAke2,IAke2,FAke2,e3,TAke3,IAke3,FAke3,e4,TAke4,IAke4,FAke4,e5,TAke5,IAke5,FAke5,e6,TAke6,IAke6,FAke6,e7,TAke7,IAke7,FAke7,e8,TAke8,IAke8,FAke8,where TAk(ei):U→[0,1], IAk(ei):U→[0,1], FAk(ei):U→[0,1], and 0≥TAk(ei)+IAk(ei)+FAk(ei)≤3, for k=1,…,4 and i=1,…,8.According to the definition of neutrosophic sets, the numbersTAk(ei), IAk(ei), and FAk(ei) represent a truth-membership, an indeterminacy-membership, and a falsity-membership, respectively. The neutrosophic sets of bearing fault types are shown in Table 4. Here, A1, A2, A3, and A4 are healthy, outer race fault, ball fault, and inner race fault bearings, respectively.Table 4
Energy values of bearing fault types represented by the form of SNS.
Fault types Energy in each frequency band E 3 ∗ 0 E 3 ∗ 1 E 3 ∗ 2 E 3 ∗ 3 E 3 ∗ 4 E 3 ∗ 5 E 3 ∗ 6 E 3 ∗ 7 A 1 e 1 , 0.76,0.14,0.20 e 2 , 0.95,0.05,0.00 e 3 , 0.11,0.04,0.85 e 4 , 0.64,0.07,0.29 e 5 , 0.04,0.02,0.94 e 6 , 0.00,0.02,0.98 e 7 , 0.01,0.05,0.95 e 8 , 0.00,0.03,0.97 A2 e 1 , 1.00,0.01,0.00 e 2 , 0.24,0.15,0.61 e 3 , 0.02,0.01,0.97 e 4 , 0.13,0.09,0.78 e 5 , 0.00,0.01,1.00 e 6 , 0.00,0.01,0.99 e 7 , 0.01,0.01,0.99 e 8 , 0.01,0.01,0.99 A3 e 1 , 0.82,0.11,0.07 e 2 , 1.00,0.01,0.00 e 3 , 0.11,0.05,0.84 e 4 , 0.65,0.10,0.25 e 5 , 0.06,0.03,0.92 e 6 , 0.00,0.04,0.96 e 7 , 0.02,0.04,0.94 e 8 , 0.00,0.01,1.00 A4 e 1 , 1.00,0.01,0.00 e 2 , 0.49,0.07,0.45 e 3 , 0.06,0.04,0.90 e 4 , 0.20,0.04,0.76 e 5 , 0.00,0.01,1.00 e 6 , 0.02,0.01,0.97 e 7 , 0.03,0.03,0.95 e 8 , 0.03,0.03,0.94


## 4.4. Rolling Bearing Fault Diagnosis Using Correlation Coefficient

In this section, we apply the correlation coefficient of SNSs to diagnose rolling bearing faults. Assume thatAk (k=1,2,3,4) are SNSs models of rolling bearing faults and At is a testing rolling bearing signal expressed by a SNS. Then we can calculate the correlation coefficient value MSNS(Ak,At) (k=1,2,3,4) using (16). Finally, the fault-diagnosis order of the fault-testing sample can be ranked according to the correlation coefficient value, and the proper diagnosis Ak∗ for the bearing fault At is derived by(19)k∗=arg⁡max1≤k≤4⁡MSNSAk,At,This paper considers the same importance of the energy values in each frequency band; therefore, the weights ofwi (i=1,2,…,8) are wi=1/8.


## 4.5. Results and Discussions

To demonstrate the effectiveness of the new diagnosis method, we now provide two examples for fault diagnosis of bearings. Let us consider two testing bearing samplesB1 and B2 described as neutrosophic sets:(20)B1=e1,1.00,0.01,0.00,e2,0.51,0.01,0.49,e3,0.08,0.01,0.92,e4,0.24,0.01,0.76,e5,0.00,0.01,1.00,e6,0.03,0.01,0.97,e7,0.05,0.01,0.95,e8,0.06,0.01,0.94,B2=e1,1.00,0.01,0.00,e2,0.39,0.01,0.61,e3,0.03,0.01,0.97,e4,0.22,0.01,0.78,e5,0.00,0.01,1.00,e6,0.01,0.01,1.00,e7,0.01,0.01,1.00,e8,0.01,0.01,1.00.The correlation coefficient values between SNSsBj (j=1,2) and Ak (k=1,2,3,4) can calculated by (17) as follows:(21)MSNSA1,B1=0.8787,MSNSA2,B1=0.9483,MSNSA3,B1=0.8746,MSNSA4,B1=0.9819.MSNSA1,B2=0.8587,MSNSA2,B2=0.9714,MSNSA3,B2=0.8566,MSNSA4,B2=0.9590.For the fault-testing sampleB1, MSNS(A4,B1) is the maximum correlation coefficient, and MSNS(A2,B1) is the second correlation coefficient. According to the principle of correlation coefficient, the fault-diagnosis order is as follows:(22)A4⟶A2⟶A1⟶A3.Therefore, we can determine that the testing bearing is an inner race fault bearing. By actual observing, the testing bearing inner race was covered with scratches, and therefore the diagnosis result is correct.Similarly, for the fault-testing sampleB2, the fault-diagnosis order is as follows:(23)A2⟶A4⟶A1⟶A3.By actual checking, the fault of the bearing firstly resulted from damage of outer race and then inner race. So the diagnosis results are consistent with the actual situation.In the experiment, 120 rolling bearings were used for testing samples. In order to verify the effectiveness of the fault-diagnosis method proposed in this paper, we extracted the energy eigenvalues of bearing vibration signals firstly and then diagnosed the bearing faults using the correlation coefficient of SNSs and the support vector machine (SVM) [25], respectively. The fault-diagnosis results of rolling bearings are shown in Table 5. By comparing the diagnosis results shown in Table 5, it is clear that the diagnosis accuracy rate based on the correlation coefficient of SNSs is much higher than the accuracy rate based on SVM.Table 5
Fault diagnosis results based on the correlation coefficient of SNSs and SVM.
Fault type Method Test sample Diagnosis result Diagnosis accuracy rate (%) Healthy Outer race fault Ball fault Inner race fault Healthy SNSsSVM 30 2827  23  93.390 Outer race fault SNSsSVM 30  2828  22 93.393.3 Ball fault SNSsSVM 30 57  2523  83.376.7 Inner race fault SNSsSVM 30  5  3025 10083.3For further comparison, Table6 lists the diagnosis results based on the correlation coefficient of SNSs, SVM, BP, and GA-BP [28] methods, respectively. Obviously, the method based on the correlation coefficient of SNSs can achieve the average accuracy rate of 92.5%, and it is higher than the ones based on the other methods.Table 6
Fault diagnosis results based on the correlation coefficient of SNSs, SVM, BP, and GA-BP.
Diagnosis method Diagnosis accuracy rate (%) Average accuracy (%) Healthy Outer race fault Ball fault Inner race fault SNS 93.3 93.3 83.3 100 92.5 SVM 90 93.3 76.7 83.3 85.8 GA-BP 100 80 70 90 85 BP 90 70 50 90 75The above comparisons demonstrate that the proposed method in this paper is effective in the bearing fault diagnosis.


## 5. Conclusion

To diagnose rolling bearing faults, a new fault-diagnosis method was developed by combining correlation coefficient of SNSs with wavelet packet decomposition. A series of experiments were conducted to diagnose rolling bearing faults, and the experimental results demonstrated that the proposed method can effectively identify the bearing faults. For the novel fault-diagnosis method, there exist two key issues:(1) extracting useful fault features by wavelet packet decomposition; (2) building the accurate SNSs models of bearing faults. In the future work, the two issues will be further improved based on the analysis of a large amount of experimental data since they will influence the accuracy of fault-diagnosis results.



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*Source: 5414361-2016-12-06.xml*