Correlation Coefficient of Simplified Neutrosophic Sets for Bearing Fault Diagnosis

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.


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][6][7][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][14][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 truthmembership 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.Section 2 gives the experimental system.Section 3 gives the data preprocessing techniques including morphological openingclosing 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.

Experimental Setup
This study was carried out with the experimental apparatus shown in Figure 1.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    1.
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]: where  is the diameter of the rolling elements,  is the pitch diameter,   is the rotational speed of the shaft,   is the number of rolling elements, and   ,   , and   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   = 61.25 Hz,   = 113.75Hz, and   = 75.83Hz in this experiment.

Vibration Signal Data Preprocessing
The framework of diagnosing process is shown in Figure 3.
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.

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 () and a structural element () are discrete signals defined in  = {0, 1, . . ., − 1} and  = {0, 1, . . ., −1}, respectively, and  ≥ , the four basic operators of () on (), are defined as follows: Opening Closing Morphological opening-closing filter as follows: In this experiment, the morphological opening-closing filter  oc was used to remove the strong noises.
Figures 4-6 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.

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 that () is a vibration signal; (⋅) and (⋅) are quadrature mirror filters, representing low-pass and highpass wavelet filters, respectively.These filters associate with the scaling function and wavelet function and satisfy the condition () = (−1)  (1 − ).Then, the signal () can be decomposed into a set of high-and low-frequency components by the following recursive relationships: where  ,2 denotes the wavelet coefficients at theth level and 2th subband.
The diagram of 3-level wavelet packet decomposition is shown in Figure 7.In Figure 7, the frequency intervals of each band can be computed by (( − 1)  /2 4 ,   /2 4 ], where   is sampling frequency.In this work,   = 10 kHz and   /2 4 = 625 Hz.The frequency intervals are given in Table 2.
The vibration signal () can be expressed as follows: where  represents eight subfrequency bands and  3, () is the wavelet coefficient at the 3-level and th subfrequency band.After the decomposition, the energy in each subfrequency band can be defined as where  3, () is the th discrete point amplitude of wavelet coefficient ( 3, ()) and  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:  Furthermore, assume that  3max is the maximum value of the energy eigenvalue in the 3-level subfrequency band, and then the eigenvalues can be normalized as follows: 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: The normalized energy eigenvalues of vibration signals are shown in Figure 8.
For 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 Table 3, and the energy interval ranges can be used to diagnose fault types of rolling bearings in the next step.

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.

Simplified Neutrosophic Sets (SNSs)
Definition 1 (see [19]).Let  be a universe of discourse; then, the neutrosophic set (NS)  is defined by  where the functions   (),   (), and   () represent a truth-membership function, an indeterminacy-membership function, and a falsity-membership function of the element  ∈  to the set , respectively, with the conditions 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.

Correlation Coefficient for
where the symbol "min" is the minimum operation.
According to the above definition, the correlation coefficient of SNSs  and  satisfies the following properties: (P 1 ) 0 ≤  SNS (, ) ≤ 1.

Rolling Bearing Fault Diagnosis Using Correlation Coefficient.
In this section, we apply the correlation coefficient of SNSs to diagnose rolling bearing faults.Assume that   ( = 1, 2, 3, 4) are SNSs models of rolling bearing faults and   is a testing rolling bearing signal expressed by a SNS.Then we can calculate the correlation coefficient value  SNS (  ,   ) ( = 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   * for the bearing fault   is derived by This paper considers the same importance of the energy values in each frequency band; therefore, the weights of   ( = 1, 2, . . ., 8) are   = 1/8.
The correlation coefficient values between SNSs   ( = 1, 2) and   ( = 1, 2, 3, 4) can calculated by (17) as follows: For the fault-testing sample  1 ,  SNS ( 4 ,  1 ) is the maximum correlation coefficient, and  SNS ( 2 ,  1 ) is the second correlation coefficient.According to the principle of correlation coefficient, the fault-diagnosis order is as follows: 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 sample  2 , the faultdiagnosis order is as follows: 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.
For further comparison, Table 6 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.
The above comparisons demonstrate that the proposed method in this paper is effective in the bearing fault diagnosis.

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

Figure 4 :
Figure 4: Signals of rolling bearing in outer race fault.

Figure 5 :
Figure 5: Signals of rolling bearing in inner race fault.

Figure 6 :
Figure 6: Signals of rolling bearing in ball fault.
Rolling bearing in ball fault

Table 2 :
Frequency intervals of eight subfrequency bands.

Table 4 :
Energy values of bearing fault types represented by the form of SNS.

Table 5 :
Fault diagnosis results based on the correlation coefficient of SNSs and SVM.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 samples  1 and  2 described as neutrosophic sets:

Table 6 :
Fault diagnosis results based on the correlation coefficient of SNSs, SVM, BP, and GA-BP.