#### Abstract

The vibration signal of rolling bearing exhibits the characteristics of energy attenuation and complex time-varying modulation caused by the transmission with multiple interfaces and complex paths. In view of this, strong ambient noise easily masks faulty signs of rolling bearings, resulting in inaccurate identification or even totally missing the real fault frequencies. To overcome this problem, we propose a reinforced ensemble local mean decomposition method to capture and screen the essential faulty frequencies of rolling bearing, further boosting fault diagnosis accuracy. Firstly, the vibration signal is decomposed into a series of preliminary features through ensemble local mean decomposition, and then the frequency components above the average level are energy-enhanced. In this way, principal frequency components related to rolling bearing failure can be identified with the fast spectral kurtosis algorithm. Finally, the efficacy of the proposed approach is verified through both a benchmark case and a practical platform. The results show that the selected fault characteristic components are accurate, and the identification and diagnosis of rolling bearing status are improved. Especially for the signals with strong noise, the proposed method still could accurately diagnose fault frequency.

#### 1. 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 high-speed 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. Therefore, 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 [3–5]. 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 time-varying 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-to-noise ratio (SNR) is low, the accuracy of subsequent fault diagnosis is often unsatisfactory. The 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 [9–12] 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. These 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. This 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. The 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. The 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.(3)We have verified the efficacy of the proposed method through extensive experiments, including both numerical and experimental cases.

The 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.

#### 2. 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. The energy of corresponding faulty components may be weak compared to the strong ambient noise in the early failure of the rolling bearing. That 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. The specifics of the algorithm are sketched in Figure 1, which mainly includes three parts. The 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 , 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 and a pure frequency modulation signal . Then, the complete time-frequency distribution of can be obtained by summing the above-decomposed components as follows:where is the PF component obtained after LMD and is the residue.

To alleviate the problem of modal mixing in LMD mentioned above, Gaussian white noise 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 based on ELMD is summarized below. Step 1: assuming that the signal is repeated times with different randomly sampled Gaussian white noise , the mixture signal is presented as follows: where is the mixed signal and has the same meaning as that given in equation (1). Step 2: perform ELMD on the mixed signal and obtain PF components and the residual : where is the PF component in the decomposition after adding white noise to . Step 3: average the same batch of PF components obtained under different noise backgrounds, as shown in the red dashed box in Figure 2. The final component is calculated as the mean value of all components across times:

##### 2.2. 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 [19–21]. The 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. The 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. The specific calculation procedure is shown below:where is the value of the sample with the point in the feature, is the number of sampling points in , and 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 to after enhancement, making the screening of components more precise.

##### 2.3. 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. The 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. Through iteratively updating, frequency bands with large kurtosis and crucial components will be located. The specific calculation is shown below:where calculates the kurtosis value of a specific frequency , is the output of the filter of the 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. The 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. The specific steps are summarized below: Step 1: calculation of FSK values of the original signal. Performing FSK on , the frequency range of the fault is obtained according to the calculated center frequency and bandwidth . Step 2: assessment of RPFs.

With the frequency band calculated in the last step, judge whether there are important components by analyzing the spectrum of RPFs. The RPF with concentrated energy is defined as a sensitive fault component.

#### 3. Results and Discussion

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

##### 3.1. Benchmark Case from Case Western Reserve University

Rolling bearing data from CWRU are open-access and freely available. The experimental data are measured from a deep groove ball bearing with the type JEM SKF 6205-2RS. The 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/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 and the frequency band corresponding to the maximum kurtosis (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 , , and 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.

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 is the bearing speed, stands for the number of rolling elements, is the rolling element diameter, is the bearing pitch diameter, and 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 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 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.

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. The detailed results are given in Tables 3 and 4.

##### 3.2. 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. The 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. The 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. The 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 and the bandwidth corresponding to the maximum kurtosis 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 and containing prominent impact components are selected.

The 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. Therefore, the estimation error of the proposed method is 3.33 Hz, which is about 2.73%.

The manual selection strategy and the ELMD + FSK strategy are further adopted for comparison. Only the component will be selected for manual selection, inevitability leading to the poor fault diagnosis of the following analysis. For the ELMD + FSK strategy, components and plotted in Figure 12 are selected as crucial ones because large amplitude is observed in the frequency range from 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.”

#### 4. Conclusion

Accurate identification of fault frequency for rolling bearing remarkably influences the overall performance of machinery processes. This 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 energy-enhanced through the proposed mechanism. After that, the decomposed faulty components could be identified with the fast spectral kurtosis algorithm. The experimental results indicate that the proposed model outperforms other baseline models.

#### Data Availability

The 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

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This research was supported by the National Natural Science Foundation of China (nos. 51865045 and 61903327), Natural Science Foundation of Inner Mongolia (no. 2017MS0509), and Inner Mongolia Scientific Research Projects of Colleges and Universities (no. NJZY19298).