Fault Frequency Identification of Rolling Bearing Using Reinforced Ensemble Local Mean Decomposition

School of Mechanical Engineering, Inner Mongolia University of Science and Technology, Baotou, China Mining Research Institute, Inner Mongolia University of Science and Technology, Baotou, China Baotou Vocational and Technical College, Baotou, China State Key Laboratory of Mechanical Transmissions, Chongqing University, Chongqing, China School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore


Introduction
With the merits of low skin friction drag, high rotation speed, and standardized size, rolling bearings, referred to as ball bearings, have been extensively deployed in mechanical transmission systems [1], including but not limited to highspeed trains, vehicles, and wind turbines. Usually, the rolling bearings are used in harsh conditions, such as frequent turn on and off, time-varying load, and so on. Consequently, coupled with multiple negative factors, these complex working conditions make the cages, inner raceways, and outer rollers of rolling bearings prone to fall into failure. Periodic damages caused by the contact force between parts of a rolling bearing lead to the evolution of failures, leading to fatigue damage of the entire transmission system, causing substantial economic losses and even significant safety accidents. According to AlShalalfeh and Shalalfeh [2], 30% of rotating machinery failures could be attributed to rolling bearing failures. erefore, it becomes an urgent demand for a timely and accurate health evaluation method for rolling bearing, providing safe and efficient operation of the concerned transmission system.
In practice, vibration test analysis has been widely used in identifying the possible faulty type of rolling bearings [3][4][5]. To determine the location and degree of rolling bearing failure, it extracts fault frequency from the abundant state information in the measured vibration signal. However, vibration signals of rolling bearings measured on industrial sites show energy attenuation and complex timevarying modulation characteristics after being transmitted through multiple interfaces and complex paths. As a result, the essential components of early failures are often submerged in strong background noise. When the signal-tonoise ratio (SNR) is low, the accuracy of subsequent fault diagnosis is often unsatisfactory. e key to improving its fault diagnosis accuracy is how to accurately extract and locate the components containing sensitive fault characteristic frequencies from the rolling bearing vibration signal.
At the early stage, the empirical mode decomposition (EMD) method proposed by Huang et al. [6] has been widely adopted for frequency decomposition. EMD decomposed the original signal into a limited number of components ranging from high frequency to low frequency. However, the performance of EMD may be discounted due to mode mixing and the end-point effect [7]. To solve the problems of modal mixing and end-point effect of EMD, Jonathan [8] proposed a local mean decomposition (LMD) method, which adaptively converted a nonstationary signal with time-varying frequencies into the sum of a finite number of instantaneous frequency product functions (PFs). Each PF component is the product of an envelope signal and a frequency modulation (FM) signal. Also, the envelope represents the instantaneous amplitude of the PF component, and the FM signal is the instantaneous frequency of the PF component. As extreme points and uneven distributions exist, LMD may lead to discontinuity of the signal, which may lead to the same PF component decomposed from LMD containing both high and low frequencies. As a result, LMD cannot thoroughly solve the modal mixing problem. Recently, the ensemble local mean decomposition (ELMD) algorithm-based fault diagnosis methods [9][10][11][12] are increasingly popular and have become the mainstream for vibration signal decomposition. Specifically, the measured vibration signal will be decomposed into a finite series of components with specific frequency and amplitude. ese components have complete time-frequency distributions and contribute to the further development of fault diagnosis models. Adding white noise to the original signal increases the continuity of LMD, and the essence of ELMD is to utilize the statistical characteristics of Gaussian white noise in both time space and frequency space. Since the white noise added at each time is random, the PF components obtained by ELMD will be averaged to derive the final decomposition results. Although ELMD could provide feature decomposition of the vibration signal, the decomposed components that are responsible for the real fault of rolling bearings have not been clearly revealed. If insensitive components related to failure have been directly adopted, they will deteriorate the fault diagnosis model. Especially, rolling bearings are used in variable working conditions, in which the speed and torque of the rolling bearing always vary frequently. Consequently, the vibration signal has the characteristics of time-varying and complex frequency and amplitude. is dramatically increases the complexity in decomposing faulty frequency from the concerned signal. It is necessary to filter out unimportant features and retain crucial components regarding early faults from the components obtained from ELMD. To alleviate this problem, some works [13,14] relied on human experience to select corresponding components decomposed by ELMD, which may be inaccurate. Alternatively, Wang et al. [15] selected PF components presenting fault features with the fast peak diagram. Zhang and Zhou [16] combined the kurtosis and energy information to construct the fault feature. He and Zhou [17] adopted the Kullback-Leibler divergence-based adaptive method to select the proper PF components. Wang et al. [18] applied singular value decomposition to determine the sensitive PF components. Although the methods mentioned above contribute to selecting valuable components, the influence of ambient noise has not yet been considered. Consequently, it is difficult to identify all crucial components containing faulty frequencies accurately.
To solve the challenges mentioned above, this paper proposes a reinforced ELMD method to accurately determine the crucial components. First, we decompose the vibration signal through ELMD to obtain a series of components corresponding to different frequencies. Further, to distinguish the unimportant components from important ones, a two-step screening strategy is proposed. e first step augments the energy of components above the average, highlighting the specific frequency components. In the second step, the fast spectral kurtosis diagram algorithm is adopted to calculate the center frequency and bandwidth corresponding to the maximum kurtosis, serving as a metric to filter out components irrelevant to the fault signal. Based on this, this method is capable of not only identifying components containing faulty frequencies but also enhancing their power. e main contributions of the article have been summarized as follows: (1) We have augmented the energy of decomposed components corresponding to specific frequency components.
(2) We have adopted the fast spectral kurtosis diagram to calculate the center frequency and bandwidth corresponding to the maximum kurtosis. e rest of this article is structured as follows. In Section 2, the detailed process of the reinforced ELMD approach is illustrated. Section 3 verifies the efficacy of the proposed method through a benchmark case and a practical experiment system. Finally, Section 4 draws the conclusion.

Methodology
Although ELMD could effectively alleviate the problem of modal mixing compared with LMD during the vibration signal decomposition, it cannot clearly explain the correlations between obtained PF components and fault features. As a result, partial PF components are noncrucial and may lead to performance degradation if they are directly applied without further filtering. e energy of corresponding faulty components may be weak compared to the strong ambient noise in the early failure of the rolling bearing. at is, it is challenging to select the fault frequency of the vibration signal accurately. To solve this difficulty, we propose an enhancement strategy to find the crucial and sensitive faulty components. e specifics of the algorithm are sketched in Figure 1, which mainly includes three parts. e first part is the initial feature decomposition using ELMD. Next, the extracted PF components are further enhanced with a novel mechanism. To select the proper PF components after enhancement, a comparison between center frequency bands calculated with FSK and the frequency of each PF component is conducted in the third part. More details of the proposed method will be specified in the following sections.
2.1. Initial Feature Decomposition Using ELMD. For a given collection of vibration signal X(t), a finite number of instantaneous frequencies could be decomposed by the LMD algorithm, as shown in equation (1). Each PF component is obtained by the product of an envelope signal a n (t) and a pure frequency modulation signal s n (t). en, the complete time-frequency distribution of X(t) can be obtained by summing the above-decomposed components as follows: where P n is the n th PF component obtained after LMD and u(t) is the residue.
To alleviate the problem of modal mixing in LMD mentioned above, Gaussian white noise e i (t) will be added in equation (1) to improve the generation ability, which is the basic idea of ELMD. As shown in Figure 2, the detailed process of the feature decomposition of the rolling bearing vibration signal X(t) based on ELMD is summarized below.
Step 1: assuming that the signal X(t) is repeated M times with different randomly sampled Gaussian white noise e m (t), the mixture signal is presented as follows: where X m (t) is the m th mixed signal and X(t) has the same meaning as that given in equation (1).
Step 2: perform ELMD on the mixed signal X m (t) and obtain N PF components and the residual u m (t): where P n,m is the n th PF component in the m th decomposition after adding white noise to X m (t).
Step 3: average the same batch of PF components obtained under different noise backgrounds, as shown in the red dashed box in Figure 2. e final n th component P n (t) is calculated as the mean value of all components across M times:

Enhancement of the Initial Components.
Generally, the mean value of the vibration signal reflects the average level of the corresponding energy. Many unexpected factors, including surface waviness, surface roughness, raceway misalignment, etc., will cause periodic changes of the contact force between parts of rolling bearing [19][20][21]. e periodic changes will lead to periodic shocks and make the instantaneous amplitude of the vibration signal significantly higher than its average. Taking advantage of these characteristics, an enhancement strategy is proposed here to reinforce the influential components of the vibration signal. e energy of the corresponding component is enlarged by squaring the weak impact component above the average of the vibration signal under strong background noise transmitted through multiple interfaces. e specific calculation procedure is shown below: where x i,n is the value of the sample with the i th point in the n th feature, I is the number of sampling points in X(t), and x i,n is the value after feature enhancement. Taking a period of sinusoidal vibration signal as shown in Figure 3 as an example, it is employed to illustrate the basic idea of feature enhancement. In Figure 3, the blue dotted curve represents the original signal, and the red dotted curve stands for the enhanced signal. Using equation (5) to square the samples above the mean in the original signal to enlarge the energy, we enhance the originally weak fault features. For instance, the value of peak changes from x i to x i after enhancement, making the screening of components more precise.

Selection of Crucial Components Using FSK.
Although kurtosis information is insensitive to the varying load, working speed, size, etc., it is susceptible to shock signals. Out of this consideration, the kurtosis index is suitable for detecting shock faults, widely observed in early rolling bearing faults. For the fault-free situation of a rolling bearing, the amplitude distribution of the vibration signal is close to the normal distribution, and the value of the kurtosis index is chosen as 3. With the varying working conditions and the increases of service time, the distribution of the vibration signal deviates from normal distribution once it steps into failure. Correspondingly, the normal curve is skewed or scattered, and the kurtosis value increases accordingly.
e larger the value of the kurtosis, the more serious the failure of the bearing from its normal state. Antoni and Randall [22] proposed the FSK method by dividing the original signal frequency band hierarchically and estimating the kurtosis value of each frequency band.
rough iteratively updating, frequency bands with large kurtosis and crucial components will be located. e specific calculation is shown below:

Journal of Control Science and Engineering
where K(f) calculates the kurtosis value of a specific frequency f, c i k (t) is the output of the i th filter of the k th layer, 〈·〉 denotes modulus value, and | · | stands for expectation.
To improve the screening efficiency of the frequency components of the faulty rolling bearing, FSK is utilized to divide the frequency band of the vibration signal with the enhanced RPF components. e kurtosis value of the vibration signal frequency band is used to characterize the impact degree. Consequently, sensitive impact components of the vibration signal are quickly and effectively screened out. e specific steps are summarized below:  Vibration signal X (t) Step 1: calculation of FSK values of the original signal. Performing FSK on X(t), the frequency range of the fault is obtained according to the calculated center frequency f c and bandwidth B w .
With the frequency band calculated in the last step, judge whether there are important components by analyzing the spectrum of RPFs.
e RPF with concentrated energy is defined as a sensitive fault component.

Results and Discussion
is section verifies the efficacy of the proposed method through two cases. e first one is a benchmark provided by the Case Western Reserve University (CWRU) [23][24][25]. e second one is a self-built test platform to measure the vibration signal of the rolling bearing. e detailed information about data processing, testing result, and conclusion will be carefully illustrated in the following sections.

Benchmark Case from Case Western Reserve University.
Rolling bearing data from CWRU are open-access and freely available. e experimental data are measured from a deep groove ball bearing with the type JEM SKF 6205-2RS. e detailed information about the ball bearing is listed in Table 1. To simulate the real fault, a damage point with a diameter of 0.36 mm is manufactured by the electric discharging machining technique into the inner ring of the bearing ball. Besides, the motor speed is set to be r � 1730 r/ minute, and the sampling frequency is 12000 Hz.
A total of 4096 sampling points of the ball bearing were collected to test the performance of the proposed method. Figure 4 shows the time-domain waveform of the measured signal for intuitive understanding. According to the processing procedure given in Sections 2.1 and 2.2, crucial components of the faults are determined as follows: (1) Perform ELMD on the time-domain signal collected from SKF 6205-2RS deep groove ball bearing and obtain six PF components. (2) Calculate the frequency spectrum of each PF component, as shown in Figure 5. (3) Enhance all the six PF components according to equation (5) and obtain the frequency spectrum of the enhanced RPF components, as shown in Figure 6. (4) Perform FSK on a time-domain signal with inner ring fault, as shown in Figure 7.
As shown in Figure 7, it is observed that the center frequency f c and the frequency band B w corresponding to the maximum kurtosis K max (13.30) are 800.00 Hz and 106.67 Hz, respectively. Correspondingly, the frequency band range is [693.33 Hz, 906.67 Hz] marked in the black dashed box as shown in Figure 7. By comparing the RPF components in Figure 6, three components RPF 1 , RPF 2 , and RPF 3 containing obvious influential components are selected. Finally, the original signal is judged by the method of envelope demodulation. In Figure 8, it is found that the frequency corresponding to the maximum peak value of 0.07 is 155.30 Hz by using envelop analysis.

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According to the way given in [26] by using the physical mechanism with known parameters listed in Table 1, the real fault frequency of the deep groove ball bearing of SKF 6205-2RS is estimated as follows: where r is the bearing speed, E stands for the number of rolling elements, d is the rolling element diameter, D is the bearing pitch diameter, and α presents the contact angle. According to the values given in Table 1, the real fault frequency is calculated as 156.00 Hz. Consequently, the estimation error between the proposed method and the real value is 0.70 Hz. For comparison, the traditional "manual selection," "LMD + FSK," and "ELMD + FSK" strategies are employed. For manual selection, only component PF 1 is selected, and the other five PF components will be missed, which will affect the accuracy of the final diagnosis result. For the ELMD + FSK strategy, component PF 3 plotted in Figure 5 is selected because a large amplitude is observed in the frequency band frequency range [693.33 Hz, 906.67 Hz]. Although other components also contain fault frequency, their vibration amplitudes are too small to be considered. Similar results could be observed for "LMD + FSK-based approach." To comprehensively evaluate the proposed method, more fault scenarios are compared in Table 2, including inner fault under other rotating speeds and the outer fault with different rotating speeds. Computing the error ratio through dividing the error from its real value, it is observed that the proposed method could give the accurate fault frequency with a large amplitude. Specifically, for the outer fault with high rotating speed, errors of the proposed method are much smaller in comparison with the other methods.

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To evaluate the performances of each method under noisy environment, two different scenarios are compared by adding Gaussian white noise in different degrees. Here, SNR is reduced to 1/5 and 1/6 of the original signal, respectively. Although the original signal has been largely weakened, the proposed method still could accurately identify fault frequency in most cases. Especially for the outer fault with a rotating speed of 1750 r/min, the performance of the proposed method remained the same and was not influenced by noise. e detailed results are given in Tables 3 and 4.

Experimental Results with a Practical Platform.
Based on the SIEMENS-LMS Test Lab multichannel data acquisition instrument and the diagnostic comprehensive test platform of Spectra Quest, we developed a testing system as shown in Figure 9 to collect vibration data of the rolling bearing. e experimental data were further used to verify the efficacy of the proposed algorithm.
First of all, the American ER-12K rolling bearing with outer ring failure was installed near the input end of the coupling [27,28], as shown in Figure 10. e basic parameters of the rolling bearing are listed in Table 5. In the experiment, a PCB608A11 accelerometer was used to collect the vibration signals in the vertical direction of the bearing seat with a sampling frequency of 25600 Hz and a motor speed of 3100 r/min. One group of time-domain signals is shown in Figure 11.
Next, ELMD is performed on the time-domain waveform of the ER-12K rolling bearing to obtain six PF components, and then their frequency spectrum is calculated as shown in Figure 12. e proposed enhancement algorithm is used to enhance the six PF components mentioned above, and the corresponding spectrum is illustrated in Figure 13. Figure 14 shows the FSK diagram of the time-domain waveform of the ER-12K rolling bearing. It is observed that the center frequency f c and the bandwidth B w corresponding to the maximum kurtosis K max are 4800 Hz and 3200 Hz, respectively. According to the frequency band ranging from 3200 Hz to 6400 Hz in the black dashed box, two components APF 1 and APF 2 containing prominent impact components are selected. e real fault frequency is calculated by the envelope demodulation method, which is 122.08 Hz for the outer ring fault frequency with the parameter listed in Table 6. In Figure 15, it is found that the frequency corresponding to the maximum peak value of 0.002 is 118.75 Hz. erefore, the estimation error of the proposed method is 3.33 Hz, which is about 2.73%. e manual selection strategy and the ELMD + FSK strategy are further adopted for comparison. Only the component PF 1 will be selected for manual selection, inevitability leading to the poor fault diagnosis of the following analysis. For the ELMD + FSK strategy, components PF 1 and PF 2 plotted in Figure 12 are selected as crucial ones because large amplitude is observed in the frequency range from      Figure 11: e time-domain trend of the employed data collected from ER-12K deep groove ball bearing with outer ring fault.  2000  3000  4000  5000  6000  0  1000  2000  3000  4000  5000  6000   0  1000  2000  3000  4000  5000  6000  0  1000  2000  3000  4000  5000  6000   0  1000  2000  3000  4000  5000  6000  0  1000  2000  Amplitude (m·s -2 ) Figure 12: Calculated PF components with the ELMD algorithm for the experimental dataset.  2000  3000  4000  5000  6000  0  1000  2000  3000  4000  5000  6000   0  1000  2000  3000  4000  5000  6000  0  1000  2000  3000  4000  5000  6000   0  1000  2000  3000  4000  5000  6000  0  1000  2000  3000  4000  5000   3200 Hz to 6000 Hz. Although other components also contain fault frequency, their vibration amplitudes are very weak. As a result, some crucial components will be missed for the manual selection and the ELMD + FSK strategy. Similar results could be observed for "LMD + FSK-based approach."

Conclusion
Accurate identification of fault frequency for rolling bearing remarkably influences the overall performance of machinery processes. is study presents a reinforced ensemble local mean decomposition method to address the drawbacks of the traditional ensemble local mean decomposition method in handling the fault signals. Especially, strong ambient noise easily masked faulty signs of rolling bearings, resulting in inaccurate identification or even totally missing the real fault frequencies. We use ELMD to decompose the vibration signal into a series of preliminary features, and then the frequency components above the average level are energyenhanced through the proposed mechanism. After that, the decomposed faulty components could be identified with the fast spectral kurtosis algorithm. e experimental results indicate that the proposed model outperforms other baseline models.

Data Availability
e time-series data used to support the findings of this study are currently under embargo while the research findings are commercialized. Requests for data, six months after publication of this article, will be considered by the corresponding author.

Conflicts of Interest
e authors declare that they have no conflicts of interest.  Figure 15: Envelop analysis of the reconstructed signal only using the selected components.