#### Abstract

The fault vibration signal of a bearing has nonstationary and nonlinear characteristics and can be regarded as the combination of multiple amplitude- and frequency-modulation components. The envelope of a single component contains the fault characteristics of a bearing. Local characteristic-scale decomposition (LCD) can decompose the vibration signal into a series of multiple intrinsic scale components. Some components can clearly reflect the running state of a bearing, and fault diagnosis is conducted according to the envelope spectrum. However, the conventional LCD takes a single-channel signal as the research object, which cannot fully reflect the characteristic information of the rotor, and the analysis results based on different channel signals of the same section will be inconsistent. To solve this problem, based on full vector spectrum technology, the homologous dual-channel information is fused. A vector LCD method based on cross-correlation coefficient component selection is given, and a simulation analysis is completed. The effectiveness of the proposed method is verified by simulated signals and experimental signals of a bearing, which provides a method for bearing feature extraction and fault diagnosis.

#### 1. Introduction

Rotating machinery is developing in the direction of high speeds, heavy loads, and high reliability, which places higher requirements on mechanical transmission equipment [1]. The operational state of mechanical equipment is changing, and its safe, stable, and reliable operation must be ensured. Rolling bearings are widely used in mechanical equipment, and their working condition greatly affects its operation [2]. Owing to complex operating conditions and changing external environment, rolling bearings are prone to failure [3]. It is of great significance to monitor their working status and diagnose their fault degree [4, 5]. Some studies have focused on the fault features of rotating machinery through modern signal processing methods [6–8].

Fault vibration signals of bearings are usually weak nonstationary signals with complex frequency components. The key to fault diagnosis of rolling bearings is to extract effective feature information from vibration signals containing complex frequencies [9]. Vibration analysis and fault diagnosis have received considerable attention [10–14] and have been adopted to process nonstationary and nonlinear vibration signals. Among them, signal decomposition methods contribute much. Tiwari [15] described a self-adaptive signal decomposition technique, concealed component decomposition (CCD), as the basis of a precise bearing fault diagnosis model. Ying [16] introduced a novel permutation entropy-based improved uniform phase empirical mode decomposition (PEUPEMD) method and obtained better analysis than comparative methods about empirical mode decomposition (EMD) in decomposing accuracy and mode mixing suppression. Patel [17] applied variational mode decomposition (VMD) to filter out non-stationarities due to variable speed conditions and provided a complete diagnostic solution for the spur gear systems. Li [18] presented a local mean decomposition (LMD) method based on an improved compound interpolation envelope, whose effect was comparable to or slightly better than that of other methods. Zheng [19] proposed local characteristic-scale decomposition (LCD), a nonstationary signal analysis method that adaptively decomposes a signal to a series of intrinsic scale components in different scales. With good compatibility, LCD methods have seen new applications, such as the local characteristic-scale decomposition-Teager energy operator (LCD-TEO) [20], improved local characteristic-scale decomposition (ILCD) [21], and piecewise cubic Hermite interpolating polynomial-local characteristic-scale decomposition (PCHIP-LCD) [22].

However, the conventional LCD methods focus on single-channel signals, which probably cause incomplete fault feature extraction. In the fault diagnosis of rotating machinery, the sensor information collected by a multi-sensor system is related to the same or different sides of rotating machinery in the same environment. There is an inevitable connection between various types of information, which are different in time, space, credibility, and expression, whose focuses and uses are not exactly the same, and whose requirements for information processing and management are different. If the information collected by each source is considered in isolation, then their internal connections and characteristics are lost. Multi-source information fusion can solve this problem [23]. Multi-sensor data fusion methodologies include the holospectrum [24–27], full spectrum [28–31], and full vector spectrum [32]. Proposed by Han [32], the full vector spectrum has been widely studied and applied in engineering [33–35] and has formed the basis of many compound methods. Chen [36] applied full-vector signal acquisition and information fusion to fault prediction. Gong [37] combined the full vector spectrum with ensemble empirical decomposition and applied it to the diagnosis of gear faults. Yu [38] introduced the empirical wavelet transform and variance contribution rate to the full vector spectrum, which improved the adaptability and accuracy of full vector information fusion.

Based on the above analysis, the main contributions of this paper are as follows:(1)A signal processing method, vector LCD, is proposed, which fully considers homologous signals and intrinsic scale components (ISCs)(2)Vector LCD can simplify the analysis of ISCs by taking the cross-correlation coefficient in screening components(3)The fusion of optimal components can obtain more complete and accurate fault features

The remainder of this article is arranged as follows. Section 2 shows the calculation of LCD, presents the theory of the cross-correlation coefficient, describes the principles of the conventional full vector spectrum, and introduces a method for bearing feature extraction and fault diagnosis based on the correlation coefficient vector LCD. In Section 3, the proposed methodology is veriﬁed through application to the homologous simulation signals of a rolling bearing. In Section 4, the rolling bearing experimental data from two directions of sensors are used to validate vector LCD. Conclusions are given in Section 5.

#### 2. Theoretical Description of Vector LCD

The proposed fault diagnosis method using vector local characteristic-scale decomposition (Vector LCD) is presented in Figure 1. Vibration signals were obtained from a test stand by a signal acquisition module. Then, the vector LCD method was used to compute and analyze the data. Next, the fusion data were enveloped into the spectrum. Lastly, the fault frequency features were matched with the specific fault type and the failure reason was located.

##### 2.1. Local Characteristic-Scale Decomposition

According to the extreme value of a signal, the LCD can adaptively decompose nonlinear and nonstationary signals to a series of ISCs satisfying the following conditions:(1)The length between any two adjacent extreme points of the data sample is monotonic(2)If the extreme point in a data sample is (*k* = 1, 2, ..., M) and is the corresponding time, then any two maximum (or minimum) value points (, ), (, ) can be connected to form a line segment. is the corresponding time of the maximum (or minimum) value point (, ) in the middle of the line segment. The corresponding function value at this moment is

The ratio of the function value to the maximum (or minimum)remains unchanged, where is a constant, and for frequency modulated, amplitude modulated, amplitude-frequency modulated, and sine-cosine signals.

On the basis of the ISC, LCD can decompose any signal to a series of ISCs, as follows [39]:(1)Find all extreme points of and their corresponding moments (*k* = 1, 2, ..., M), set , and make a linear transformation for between any two extreme points, where .(2)Subtract from the original signal to get a new signal,(3)Ideally, can be used as the first ISC. At this moment is equal to zero; in practice, assuming a variable , the iteration ends when . If does not meet the two conditions of ISC, the above steps are repeated *k* times until satisfies the conditions and denote as the first ISC of .(4)Subtract from to get a new signal, . Take as the original data and repeat steps 1–3 to get the second ISC component, , of . Repeat *n* times to get *n* ISCs of the signal . The function does not terminate until is monotonic:

It can be seen from equation (5) that the signal can be reconstructed by *n* ISCs and a monotonic signal.

##### 2.2. Correlation Coefficient

Correlation is a kind of non-deterministic relationship, and the correlation coefficient measures the degree of linear correlation between variables. The correlation coefficient between sequences and can be calculated aswhere and are the mean values of sequences *x*-and *y*, respectively.

Suppose and are normal operating signals perpendicular to each other, and , are the homologous signals when faults occur. After the decomposition of LCD, we obtain four ISCs, , , , and , where *k* *=* 1, 2, *…*, *N* is the order of an ISC. The cross-correlation coefficient is calculated as follows:(1)Calculate the correlation coefficients between and , , and , and find their average, ;(2)Calculate the correlation coefficients between and , , and , and find their average, ;(3)Calculate the correlation coefficients between and , , and , and find their average, ;(4)Calculate the sensitivity factor,where and indicate the degree of correlation between the decomposed ISC and the initial signal. The larger the value, the more similar are ISC and the initial signal. The indicates the correlation between ISCs of the same order. The smaller the value, the greater the change in the ISC. Overall, the larger the , the more sensitive the ISC of this order, and the more it can reflect spectral changes.

##### 2.3. Full Vector Spectrum

To overcome limitations due to incomplete and inaccurate sensor information, two orthogonal sensors are usually fixed on the same section of the rotor in the field test of large rotating machinery. The full vector spectrum technology meets the accuracy and reliability requirements of condition monitoring and fault diagnosis. The vortex phenomenon of the rotor is the combined effect of each harmonic frequency, and the vortex intensity at each harmonic frequency is the basis for fault judgment and identification. The space rotation trajectory of each harmonic is an ellipse, and the maximum vibration vector is in its long-axis direction [40].

Suppose the cross section channel signals ( and ) are perpendicular to each other and form them into a complex signal (), only a single Fourier transform (FT) of the complex signal is needed to obtain the characteristic information required by the full vector spectrum under each harmonic frequency. The algorithm is robust, it greatly reduces calculation, and it is compatible with conventional analysis methods. When processing a single-channel signal, the algorithm is still valid and can meet real-time requirements. The characteristic information includes the main vibration vector , assistant vibration vector , angle between the main vibration vector and the *x*-axis, and the elliptical trajectory’s initial phase angle , which are described bywhere *k* = 1, 2, …, N/2–1. The characteristic information of each harmonic trace from equation (8) is the main characteristic information under each harmonic trace of full vector spectrum technology.

Figure 2 is a flowchart of the proposed method. At first, the LCD is applied to the homology information acquired from two orthogonal sensors. The cross-correlation coefficient is selected to choose the ISCs of orthogonal signals, the next optimal ISC of orthogonal signals can be obtained, the vector ISCs are formed through the optimal ISC component fusion, and the vector ISCs are enveloped and demodulated. The proposed method can identify the nonlinear characteristics of fault signals for fault diagnosis.

#### 3. Simulation Analysis

Analog signals were analyzed to validate the effectiveness of vector local characteristic-scale decomposition in processing homologous signals. For the rolling bearing signal, the vibration signal at the time of failure presents a modulation phenomenon. The vibration signal of a rolling bearing with an outer ring fixed structure iswhere *f*_{b} is the passing frequency of the inner ring of the rolling bearing and *f*_{r} is the rotation frequency of the rotor.

Based on this, the following analog acceleration signal is constructed:where and are two initial vibration signals that are perpendicular to each other under normal operating conditions, and are two initial vibration signals that are perpendicular to each other under abnormal operating conditions, the sampling frequency , and there are sampling points.

The LCD is applied to decompose the two-channel signals corresponding to normal operating conditions. The time-domain waveforms and the ISCs are shown in Figure 3 for x-channel signals and Figure 4 for y-channel signals.

The LCD is then applied to decompose the two-channel signals corresponding to abnormal operating conditions, for which the time-domain waveforms and the ISCs of x- and y-channel signals are shown in Figures 5 and 6, respectively.

The correlation coefficients between ISCs and initial signals can be calculated, i.e., between ISC_{xnori} and x_{nor,} ISC_{ynori} and y_{nor,} ISC_{xabni} and x_{abn}, and ISC_{yabni} and y_{abn}, as shown in Table 1.

Figures 5 and 6 show that the initial signals are decomposed to six ISCs with different frequency bands. There is a great difference in amplitude between the x- and y-directions. If the signal is analyzed in a certain direction alone, the analysis results will also be quite different. Therefore, information fusion is necessary. The full vector fusion is introduced. Considering the value of the correlation coefficient, the first-order ISC_{xnor} and third-order ISC_{ynor} are closer to the initial signals than the others. The second-order ISC_{xabn} and second-order ISC_{yabn} have more fault information than others. Thus, we apply full vector fusion to the first-order ISC_{xnor}, third-order ISC_{ynor}, second-order ISC_{xabn}, and second-order ISC_{yabn}. However, the fault frequency information cannot be acquired just from time-domain waveforms. Finally, envelope and demodulation are applied to fusion signals, as shown in Figure 7. It is clear in Figure 7(a) that the rotation frequencies (20 Hz and 300 Hz) are consistent with the preset value under the normal operating condition. In Figure 7(b), the amplitudes of the fault frequency (100 Hz) and sidebands (20 Hz) show the existence of modulation under the abnormal operating condition.

**(a)**

**(b)**

From the simulation analysis, the vector LCD has a good analysis result when applied to fault signals with frequency or amplitude modulation, which can enhance the accuracy of fault diagnosis. The adoption of the cross-correlation coefficient avoids repeated analysis between multiple components and simplifies the analysis, for a unique and accurate conclusion.

#### 4. Application

The validity and advantage of vector LCD in fault diagnosis of a rolling bearing were examined through experimental data from the Case Western Reserve University bearing data center. Figure 8 shows the layout of the test stand, which consisted of a 2-hp motor, torque transducer/encoder, dynamometer, and control electronics. The test bearings supported the motor shaft, and they had single-point faults from electro-discharge machining. Accelerometers collected vibration data and were attached to the support of the drive end bearing, fan end bearing, and motor supporting base. The technical parameters of fault diagnosis are listed in Table 2.

Normal baseline data, outer race fault data, and inner race fault data were adopted from the experimental signals. The ball fault data frequency is not matched with the usual value for the slip adjusting to lock onto a dominant frequency [41]. Therefore, ball fault data are not discussed here.

##### 4.1. Normal Baseline Data Analysis

Because the normal baseline data only have one channel of drive end section, processing the single-sensor data is just a special case of a full vector spectrum. According to the flowchart of vector LCD, we first applied LCD to the drive-end single-channel signal to obtain 10 ISCs. Considering that the higher the order of ISC, the smaller the amplitude, only the first six orders of ISCs are displayed in Figure 9. We compare the values of ISC cross-correlation coefficients in Table 3, where the first is the optimal component. Finally, the envelope and demodulation are applied to the first ISC. From Figure 10, the rotation frequency *f*_{r} (29.95 Hz) and 2*f*_{r} (59.9 Hz) account for the main components without other fault frequencies, which is a normal operating condition.

##### 4.2. Outer Race Fault Data Analysis

Outer race fault data were obtained by three homologous sensors in the same section, and channels of centered 6 : 00 clock and orthogonal 3 : 00 clocks were employed to vector LCD analysis. Similar to normal baseline data processing, ISC components were acquired after LCD. The ISC components of the x-and y-channels are shown in Figures 11 and 12, respectively. As seen in Table 4, the first ISC components of the x-and y-signals are optimal. From Figures 11 and 12, the time-domain waves of the first ISC components of the x- and y-signals have different amplitudes and frequencies. If only one direction of the x- or y-signal is analyzed, then different results will be obtained, which is contrary to fault diagnosis. Therefore, we apply full vector fusion to the two optimal ISC components to obtain a new combined signal. Finally, the envelope and demodulation are applied to the combined signal. The outer race fault frequency f_{OR} (107.3 Hz) can be easily distinguished from other spectra, and the amplitude of f_{OR} is the highest among all spectra (as seen in Figure 13).

##### 4.3. Inner Race Fault Data Analysis

Inner race fault data were recorded in a single channel for the drive end, and the inner race fault signal was treated in the same way as normal baseline data. The vector LCD spectra can be displayed after the optimal ISC is screened out, with decomposition results as shown in Figure 14. From Table 5, the optimal ISC (the first ISC component) can be easily obtained. From the vector LCD spectra in Figure 15, the inner race fault frequency f_{IR} (162.1 Hz) and rotation frequency harmonics *f*_{r} (29.95 Hz), 2*f*_{r} (59.9 Hz), and 4*f*_{r} (119.8 Hz) can be acquired from the spectra, which agrees with the inner race fault features.

This bearing experimental application shows that the proposed method can be applied to rolling bearing fault diagnosis. The screening of the optimal ISC can simplify fault diagnosis and clearly display the typical features. The full vector fusion between optimal ISCs of the x- and y-signals gives an accurate and unique conclusion for fault diagnosis. The vector LCD provides an easy way to extract fault features.

#### 5. Conclusion

A multiple signal processing method of rolling bearing fault diagnosis was described in this paper. By combining the full vector spectrum with local characteristic-scale decomposition, the vector LCD can fully consider homologous signals and ISCs. This method is used to synchronously handle multiple signals. The cross-correlation coefficient was introduced to choose ISCs, which simplifies the analysis of ISCs. The vector ISC displays a more complete and precise fault frequency than a single ISC. The simulation analysis and rolling bearing experimental fault diagnosis verified the effectiveness of vector LCD. With the good compatibility of vector LCD, various types of faults and machines could be diagnosed in our future work. Moreover, the signals from different types of sensors will be combined to improve the accuracy of fault diagnosis.

#### Data Availability

The experimental data were taken from the Case Western Reserve University bearing data center.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

The authors would like to thank the National Key Research and Development Program of China (2016YFF0203100) and the Henan Provincial Key Science and Technology Research Project of China (202102210075). The authors also thank the Case Western Reserve University bearing data center for the bearing fault test data.