Rolling Element Fault Diagnosis Based on VMD and Sensitivity MCKD

In order to improve the diagnosis accuracy and solve the weak fault signal of rolling element of rolling bearings due to long transmission path, a novel fault diagnosis method based on variational mode decomposition (VMD) and maximum correlation kurtosis deconvolution (MCKD), namely VMD-MCKD-FD is proposed for rolling elements of rolling bearings in this paper. In the proposed VMD-MCKD-FD, the vibration signal of rolling element of rolling bearings is decomposed into a series of Intrinsic Mode Functions (IMFs) by using VMD method. Then the number of modes with outstanding fault information is determined by Kurtosis criterion in order to calculate the deconvolution period T. The periodic fault component of reconstructed signal is enhanced by using sensitivity MCKD method. Finally, the power spectrum of the reconstructed signal is analyzed in detail in order to obtain the fault frequency and diagnose the rolling element fault of rolling bearings. The simulation signal and actual vibration signal are selected to verify the effectiveness of the VMD-MCKD-FD method. The experimental results show that the VMD-MCKD-FD method can effectively diagnose the rolling element fault of rolling bearings and obtain better fault accuracy.


I. INTRODUCTION
As one of the core components in industrial production, rolling bearings are widely applied in various rotating machinery. Therefore, it is very necessary to accurately diagnose faults of rolling bearings [1]- [3]. Rolling bearing faults mainly occur in the inner race, outer race and rolling element [4]- [7]. In recent years, most of researches focus on the fault of the rolling bearing outer race and inner race, and these methods are very effective, while the faults of the rolling elements are less studied [8]- [13]. Because the rolling elements are located inside the rolling bearings, the fault signal of the rolling elements is easily interfered by the external environment during the transmission [14]- [18]. Therefore, it is necessary to diagnose the faults of rolling elements of rolling bearings.
To effectively extract fault features from vibration signal of rolling bearings, a lot of experts have proposed effective feature extraction methods, such as wavelet The associate editor coordinating the review of this manuscript and approving it for publication was Dazhong Ma . transform (WT) [19], Hilbert-Huang transform [20], empirical mode decomposition (EMD) [21], Ensemble EMD (EEMD) [22], and so on [23]- [28]. Although these methods can better extract fault features, they still exist deficiencies in processing vibration signals. The WT is not an adaptive signal analysis method, and it requires to choose wavelet basis function in advance [29]- [33]. The EMD is limited due to mode mixing. Although the EEMD can solve the mode mixing problem, the increase of iterations results in the increase of computation time [34]- [39]. Zhao and Ye [40] gave the concept of singular value decomposition packet to realize signal decomposition. Guo and Deng [41] proposed an improved EMD to effectively extract fault features of inner and outer race. Hou et al. [42] proposed a weak fault feature extraction method based on optimized sparse coding and approximate SVD. Liu et al. [43] proposed a feature extraction method using EEMD and curve code. Zheng et al. [44] proposed a fault diagnosis method based on feature extraction and bag-of-words. Kuncan et al. [45] proposed a one-dimensional ternary pattern method to extract the features of rolling bearings. VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ Li et al. [46] proposed a new feature extraction method using EWT and improved MCKD with grid search. Zhou et al. [47] proposed a hybrid fault diagnosis method based on parameter-adaptive VMD and multi-point optimal minimum entropy deconvolution. Li et al. [48] proposed a double-feedback cascaded monostable stochastic resonance system by signal-to-noise ratio. Mahgoun et al. [49] proposed a gear fault detection method using VMD. From these methods, it can see that the VMD can extract fault features from vibration signals. However, there are still residual noise components in IMFs under strong noise interference. The MCKD method is a new deconvolution method, which highlights the periodic impact of fault signals by designing the optimal FIR filter [50]. It is helpful to the fault diagnosis of rolling bearings. Ren et al. [51] combined MCKD with EEMD to diagnose faults. Zhao and Li [52] proposed a new method based on combining MCKD with EMD to diagnose the weak faults. But these methods did not improve the weaknesses of the EMD and EEMD. Wang et al. [53] proposed a new fault diagnosis method for rolling bearings. In addition, some latest fault diagnosis methods are proposed for or rolling bearings. Wang and He [54] proposed a wavelet packet envelope manifold (WPEM) approach to extract the intrinsic envelope structure for well identification of the specific characteristic frequency. Sun et al. [55] proposed a novel intelligent diagnosis method based on the idea of compressed sensing and deep learning for fault identification of rotating machines. Udmale and Singh [56] proposed a novel intelligent diagnosis method using spectral kurtosis and extreme learning machine for fault classification of rotating machines. Hu et al. [57] proposed a new and adaptive spectral kurtosis method for the bearing fault detection. Li et al. [58] proposed an enhanced FBE (EFBE) adopting WPT as the filter of FBE to overcome the shortcomings of the original FBE. Chen et al. [59] proposed a novel method to find formal languages, written with signal spectral logic (SSL), to describe the fault behaviors among frequency domain for fault diagnosis. The other some methods are also proposed to realize the fault diagnosis of rolling bearings [60]- [62].
To sum up, some experts and scholars have proposed EMD, EEMD, VMD, SVD, MCKD, and so on in order to realize the signal analysis, fault feature extraction and fault diagnosis. These methods can better obtain processing results. But these methods have many deficiencies in analyzing signal, extracting fault features and diagnosing faults for rolling elements of rolling bearings, such as the over decomposition of VMD by improper selection of k, modal aliasing of EMD, presetting parameters of MCKD and so on. In addition, it is very difficult to realize rolling element fault feature extraction and fault diagnosis due to weak fault signal of long transmission path by using the traditional methods. Therefore, in order to solve the weak fault signal of rolling elements of rolling bearings due to the long transmission path and improve the diagnosis accuracy for rolling elements of rolling bearings, the advantages of VMD and MCKD are fully integrated to propose a novel rolling element fault diagnosis method to obtain better performance in extracting fault features for rolling elements of rolling bearings in this paper. Firstly, the vibration signal is decomposed into some IMFs by using VMD with the robustness. The number of modes with outstanding fault information is determined by Kurtosis criterion in order to calculate the deconvolution period T and IMFs with more fault information are chosen to reconstruct the vibration signal. Then the MCKD with sensitivity is used to enhance the periodic fault component of reconstructed signal. The power spectrum of the reconstructed signal is analyzed in detail in order to obtain the fault frequency and diagnose the rolling element faults of rolling bearings. The proposed method doesn't only eliminate the noise interference, but also solves the problem of selecting k of the VMD. It takes on better effect in extracting weak fault features.
The innovations and main contributions of this paper are described as follows.
• A novel rolling element fault diagnosis method based on combining VMD with MCKD is proposed.
• The VMD with the robustness used to decompose fault vibration signal of rolling element into a series of IMFs.
• The number of modes with outstanding fault information is determined by Kurtosis criterion in order to calculate the deconvolution period T.
• The MCKD with the sensitivity is applied to enhance the periodic fault component of the reconstructed signal.

A. KURTOSIS CRITERION
Kurtosis is a dimensionless parameter that describes the sharpness of the waveform and reflects the distribution characteristic of the vibration signal. The expression of kurtosis is described as follow.
where, µ is the mean of x, x is vibration signal, σ is the standard deviation of x. The vibration signal is normally distributed, so its kurtosis is about 3. But when there are more impulse components in signal, the kurtosis value increases obviously. The fault information of rolling bearings is often included in these signals with more impulse components. The signal with higher kurtosis has more obvious impulse components and easier to extract fault information.

B. VMD
The VMD is an adaptive signal decomposition method, which decomposes the signal into a series of IMFs according to different center frequencies. The decomposition process is mainly divided into the construction and variational solution.
In the VMD, each IMF is regarded as an AM-FM signal with limited bandwidth, which is described as follow.
where, A k (t) is instantaneous amplitude, ω k (t) is the instantaneous power of the signal, ω k (t) = dφ k (t) /dt. The variation problem is to decompose the input signal f into k IMFs, and each IMF signal is analyzed by Hilbert transform, then mix it by means of the estimated center frequency e −jω k t . The sum of components is equal to the input signal under the constraint, so the variational problems is described as follow.
The quadratic penalty operator α is used to ensure the reconstruction accuracy of the signal, and the Lagrange multiplication operator λ (t) is used to maintain the constraint strictness.
The alternate direction method of multipliers is used to alternatively updateû n+1 k ,ω n+1 k is used to get the optimal solution, that is the saddle point of the Lagrangian function of Eq. (4). The Fourier transform ofû n+1 k is the intrinsiĉ λ n+1 mode functions, so that the set of IMFs {u k } and central frequency {ω k } can be expressed as follows.
In order to restore the collected signal to the original input signal, the MCKD is used to find a finite impulse response filter according to the maximum correlation kurtosis.
where, y (n) is the collected signal by the sensor, x (n) is the periodic pulse signal, h (n) is attenuation response of the transmission path, e (n) is the noise component. The kurtosis is affected by a single or partial highamplitude pulse, so the periodicity and continuity of the faults signal cannot be fully considered. Therefore, the MCKD uses the correlation kurtosis to extract the pulse sequence.
The correlation kurtosis is described as follow.
Matrix form of optimal filter f is obtained as follow.
where, L is the filter length, T is the deconvolution period, and M is the shift-order.

III. DETERMINATION METHOD OF DECONVOLUTION PERIOD
The MCKD has a strong ability to decrease noise and enhance the signal features, but it needs to preset parameters to ensure the effect of signal processing, such as deconvolution period T , and these advantages depend on the correct selection of T . Only by finding a suitable deconvolution period can the periodic impulse component of the fault signal will be highlighted, and fault frequency will be found in the frequency domain. Conversely, if a wrong value of deconvolution period is determined, the fault frequency will be shielded, which cannot be effectively extracted. The deconvolution period is actual impulse period of the fault in the vibration signal, which can also be understood as the samples of each revolution. Therefore, in actual operation, all samples are used to calculate the deconvolution period. The value of T can be calculated according to the Eq. (10).
where, S is the number of samples, S 0 is the number of fault samples, t is the time of sampling, f s is the sampling frequency, and f 0 is fault frequency.

IV. A NOVEL FAULT DIAGNOSIS METHOD A. THE BASIC IDEA
For the problem that the weak fault features of rolling elements are difficult to be extracted, the other methods usually cannot effectively solve. Therefore, a novel rolling element fault diagnosis method based on VMD and MCKD, namely VMD-MCKD-FD is proposed in this paper. The function of the VMD is to decompose the original signal into k IMFs to extract effective information. Because the VMD needs to be preset a suitable value of k to avoid the under-decomposition problem or over-decomposition problem. The selection of k is a difficult problem, but it can be avoided by signal reconstruction. The improper value of k will directly affect the decomposition result of the signal. If the value of k is too small, the similar components are decomposed to the same time scale, which results in mode aliasing. If the value of k is too large, the false components will appear in the decomposition. Therefore, before the VMD is used to decompose the signal, the number of k is preset as a fixed value.  [66]. In most of the literatures, the value of k is 5. Therefore, it would be more beneficial to the VMD that the value of k is taken as 5 according to the experiment, some published papers and our experiences. Therefore, the VMD decomposes the signal into 5 IMFs, and 3 IMFs with larger kurtosis are selected for signal reconstruction according to the kurtosis criterion. Kurtosis value is used to evaluate signal pulse. In the bearing fault diagnosis, it is generally used to evaluate the proportion of fault impact component of signal.
The kurtosis value is greater, the more fault information is contained in IMF. There are two reasons for selecting three IMFs in here. Firstly, in the process of weak fault diagnosis, due to the interference of noise, the fault information may also appear in the IMF with the second or third largest kurtosis value. Secondly, three IMFs are selected for signal reconstruction in order to avoid the problem of selecting the K value of VMD. Therefore, we choose the 3 IMFs with the highest kurtosis as useful IMFs. Because it is difficult to use the VMD to extract the fault frequency. Therefore, the MCKD is used to further process the signal, and the maximizing correlation kurtosis is applied to enhance the periodic impulse component to achieve the fault frequency extraction of rolling elements for rolling bearings.

B. IMPLEMENTATION STEPS
The flow of VMD-MCKD-FD is shown in FIGURE 1.
The steps of the VMD-MCKD-FD for rolling elements of rolling bearings are described in detail as follows.
Step 1. Firstly, the fault vibration signals are decomposed into k IMFs by using the VMD method.
Step 2. Then some IMFs with larger kurtosis are selected for reconstruction according to the kurtosis criterion.
Step 4. The MCKD is used to process reconstructed signals, the periodic impulse component is enhanced, and the noise interference is reduced.
Step 5. Finally, the power spectral density is calculated from the Hilbert envelope of the obtained signal, and the fault frequency is found by analyzing the power spectrum, so as to diagnose the rolling element faults.

V. SIMULATION RESULTS
In order to verify the validity of the VMD-MCKD-FD for fault diagnosis of rolling elements of rolling bearings, the simulation signals are formed according to the periodic feature faults of rolling elements.
where, x 1 (t) = 0.7 * sin (800πt) * sin (3000πt), fault signals x 2 (t) = 2A i * e −350t 1 sin (8000πt), and random noise signal x 3 (t). A i is a sequence of random numbers from 0 to 1. Due to the usual slip of rolling elements and cage, the repetition and the experience amplitude are a certain degree of randomness [56], [57], t 1 = mod t, 1 f 0 , the sampling frequency is 40kHz. After random noise is added, the signal-to-noise ratio (SNR) of the simulation signal is −13.2dB, −8.   components. Because each IMF contains different frequency components, it is impossible to distinguish which IMF contains more fault information. Therefore, Kurtosis criterion is used to evaluate each IMF, and three frequency components with the largest kurtosis are selected for reconstruction signal.  This selection can retain the fault information. The kurtosis indexes of 5 IMFs are shown in TABLE 1.
According to the Kurtosis criterion, IMF1, IMF3 and IMF5 are selected to reconstruct the signal for simulation signal with the noise (SNR = −13.2dB), and IMF2, IMF3 and IMF5 are selected to reconstruct the signal for simulation signal with the noise (SNR-8.3dB). Then the power spectrums of reconstructed signal are shown in FIGURE 7 and FIGURE 8.
Compared with the original signal in FIGURE 2(b) and FIGURE 3(b), the noise interference of reconstructed signal is effectively suppressed in FIGURE 7 and FIGURE 8. However, three most obvious frequency components have no relationship with the fault frequency, which shows that there is still noise interference in the signal. The VMD cannot extract the weak fault feature, so the reconstructed signal needs to be further processed.
According to Eq. (10), the deconvolution period T is calculated to 333. The filter size L is generally in the range of [100,500]. M is generally in the range of [1,7], because the iterative method can reduce the numerical precision if we select a lager M. From experience, L is selected as 300, and M is selected as 5. For the simulation signal with the noise (SNR = −13.2dB), the MCKD is further used to process VOLUME 9, 2021   the reconstructed signal, the power spectrum of processed signal is shown in FIGURE 9(c). In order to illustrate the advantages of the VMD to preprocess the vibration signal, the EMD and EEMD are selected to decompose the simulation signal, and 3 IMFs with larger kurtosis are selected to reconstruct the signal. These reconstructed signals are processed by using MCKD with same parameters, the power spectrum of processed signals is shown in FIGURE 9(a) and From FIGURE 9(a) and (b), it can be seen that the most obvious frequency component is 839.8Hz. From FIGURE 10 (b), it can be seen that the most obvious frequency component is 957Hz. In FIGURE 9(a), the most obvious frequency component is 839.8Hz, although the fault frequency and its harmonics can be found, but the frequency component with the largest amplitude of them is 3f 0 (351.6Hz), the fault frequency f 0 (117.2Hz) is not obvious. In FIGURE 9 (b), the frequency component with maximum amplitude is still 839.8Hz, the amplitude of the fault frequency is the second largest. In FIGURE 9(c), the frequency component with the largest amplitude is the fault frequency f 0 (117.2Hz), and the second harmonic is found, the amplitude of other frequency components is relatively small. In FIGURE 10(a), the fault frequency f 0 (117.2Hz) and 4 f 0 (468.8Hz) are found. In FIGURE 10(b), the fault frequency f 0 (117.2Hz) and 4 f 0 (468.8Hz) are found, the other amplitudes of frequency components also exist. In FIGURE 10(c), the fault frequencies f 0 (117.2Hz), 2f 0 (234.4Hz), 3f 0 (351.6Hz) and 4f 0 (468.8Hz) are found, and the other amplitude of frequency components don't exist. Because the collected vibration signal contains some interference signals, which will affect the feature extraction of vibration signal. As a result, the obtained fault frequency has some deviations for dominant frequencies, 117 Hz, 234 Hz, 351Hz and 468Hz. These phenomena show that the VMD has better robustness than EMD and EEMD, and the VMD-MCKD-FD has better performance in extracting the features of weak fault signals. In order to prove the value of deconvolution period T on the result of feature extraction, the value of T was changed to 250 without changing L and M, and then the reconstructed signal (−13.2dB) is reprocessed. The power spectrum of the simulated signal by using the VMD-MCKD-FD (T = 250) is shown in FIGURE11.
It can be seen from FIGURE 11, Three most obvious frequency components in the figure are 17.58Hz, 146.5Hz and 46.88Hz, they are not related to the fault frequency or its harmonics. This phenomenon is due to the fact that the MCKD method can shield the signal component whose impact period is not 250. Therefore, it can be known that the deconvolution  period T should be correctly selected when the MCKD is used to extract fault features, especially for the weak fault feature.

VI. EXPERIMENTAL RESULTS AND ANALYSIS A. EXPERIMENTAL ENVIRONMENT AND DATA
In order to verify the effectiveness of the VMD-MCKD-FD for the weak fault of rolling elements, the experiment QPZZ-II platform is used to measure the vibration signal of rolling bearings in FIGURE 12. It was measured at a motor speed of 1500r/min and no-load, and the sampling frequency is 12kHz. The experiment used N205 rolling bearings. This bearing contains 13 rolling elements with a diameter of 7.5mm. And the pitch diameter of bearings is 38.5mm. According to the above information, the shaft rotation frequency f r can be calculated to be about 25Hz, and rolling element fault frequency f 0 is 125.2Hz.
The   noise, the frequency components of the vibration signal are irregularly distributed in the power spectrum, and the invalid information can be extracted. To extract fault features from the weak fault information, the vibration signal needs to be denoised firstly.

B. EXPERIMENTAL RESULT AND ANALYSIS 1) THE RESULTS OF VMD-MCKD-FD
The VMD-MCKD-FD method is used to process the normal vibration signal and fault vibration signal of rolling elements  From FIGURE 15 and FIGURE 16, each IMF contains different frequency components. In order to obtain 3 IMFs with more fault information, it is necessary to calculate the Kurtosis value of each IMF. The calculated results are shown in TABLE 2 and TABLE 3. According to Kurtosis criterion, IMF3, IMF4 and IMF5 are selected to reconstruct the vibration signal. The power spectrum of the reconstructed fault vibration signal of rolling elements of rolling bearings is shown in FIGURE 17.
From FIGURE 17, the fault frequency f 0 (123Hz), and its harmonics 2/3f 0 (82.03Hz) can be obtained. The frequency component with the largest amplitude is 2/3f 0 , the amplitude of fault frequency is not obvious. This phenomenon shows that the VMD can effectively denoise weak fault signals,   but it is not enough to extract fault frequency. Therefore, the MCKD is used to further process the reconstructed signal.
In order to illustrate the advantages of the VMD-MCKD-FD for processing weak fault vibration signals, the fault vibration signal of rolling elements of rolling bearings is processed by using EMD-MCKD-FD and EEMD-MCKD-FD methods, and the obtained results are compared with the results of the VMD-MCKD-FD. FIGURE18(a) is fault signal power spectrum of the EMD-MCKD-FD, FIGURE18(b) is fault signal power spectrum of the EEMD-MCKD-FD, and FIGURE 18(c) is fault signal power spectrum of the VMD-MCKD-FD. According to Eq. (10), the deconvolution period T is 96. L is selected as 200, and M is selected as 5.
From FIGURE 18(a), 1/2f r (11.72Hz), 5.5f r (128.9Hz) and 4f 0 (492Hz) can obtained, the frequency component with the largest amplitude is 1/2f r . From FIGURE 18(b), the fault frequency and its harmonics can be obtained, but he obtained fault frequency f 0 (123.7Hz) is not very close to the theoretical value(123Hz) and the frequency components with the largest amplitude is also 1/2f r (11.72Hz), and the fault frequency cannot be found. From FIGURE 18(c), the fault frequency f 0 (123 Hz) has the largest amplitude, and its harmonics can also be seen clearly. Therefore, although the EMD-MCKD-FD method and EEMD-MCKD-FD method can also extract the fault frequency and its harmonics, but the amplitude of fault frequency is not obvious in our experiment. In contrast, the VMD-MCKD-FD can more effectively extract the features of weak faults. By comparison, it is found  that the VMD-MCKD-FD takes on better feature extraction performance for the weak faults.

2) THE EFFECT OF DECONVOLUTION PERIOD T ON THE MCKD
To illustrate the effect of the deconvolution period T on the MCKD, the value of T is randomly selected as 150, but the values of L and M have not changed, and then the same method is used to process the reconstructed vibration signal. The power spectrum of reconstructed vibration signal with a wrong deconvolution period is shown in FIGURE 19.
From FIGURE 19, there does not occur fault frequency or harmonic, and the frequency component with the largest amplitude is 2f r (46.88Hz). Compared with the processed signal power spectrum by using VMD-MCKD-FD, the incorrect selection of deconvolution period T makes the MCKD unable to find the impact period of the fault signal, the fault information and noise are processed by the filter together, so the fault frequency and its harmonics are obviously shielded. This phenomenon further illustrates the importance of a suitable deconvolution period T of the MCKD.

3) THE EFFECT OF VMD-MCKD-FD UNDER DIFFERENT SPEED
In this section, the speed is selected as 1000r/min, and then the experiment is carried out to verify the diagnosis effect of the VMD-MCKD-FD for rolling elements under the different speed. According to the information, the shaft rotation frequency f r can be calculated to be about 16.7Hz, and the fault frequency f 0 of rolling elements is about 83.5Hz. According to Eq. (10), the deconvolution period T is 144.
From FIGURE 20, the fault frequency f 0 (82.03Hz) and its harmonics 3f 0 (246.1Hz) can be found. The VMD-MCKD-FD can successfully extract the fault features under different speed.

VII. CONCLUSION
In this paper, a novel rolling element fault diagnosis method (VMD-MCKD-FD) is proposed. In addition, the influence of different selections of the deconvolution period T on MCKD also is analyzed and discussed. The VMD method is mainly used to decompose non-linear and non-stationary vibration signals of rolling elements of rolling bearings. After that, the IMFs with larger kurtosis values are selected for reconstructing the vibration signal. It not only avoids the selection of k in the VMD, but also fully retains the information of fault feature. The MCKD enhances the weak fault feature of rolling elements. In here, the diagnosis results of the EMD-MCKD-FD, EEMD-MCKD-FD and VMD-MCKD-FD with different deconvolution periods T are compared. The simulated and experimental results do not only prove the effectiveness of the VMD-MCKD-FD, but also prove the importance of the deconvolution period T .
The value of T is calculated through theory, but the parameters in real working conditions are slightly different from the theory. Therefore, it should consider how to adaptively select the parameters, and further improve the effect of feature extraction for the weak faults of rolling elements of rolling bearings in the future.