Shock and Vibration

Volume 2016 (2016), Article ID 8361289, 20 pages

http://dx.doi.org/10.1155/2016/8361289

## A Novel Method for Adaptive Multiresonance Bands Detection Based on VMD and Using MTEO to Enhance Rolling Element Bearing Fault Diagnosis

College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received 27 March 2015; Revised 9 August 2015; Accepted 20 August 2015

Academic Editor: Marcello Vanali

Copyright © 2016 Xingxing Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Vibration signals of the defect rolling element bearings are usually immersed in strong background noise, which make it difficult to detect the incipient bearing defect. In our paper, the adaptive detection of the multiresonance bands in vibration signal is firstly considered based on variational mode decomposition (VMD). As a consequence, the novel method for enhancing rolling element bearing fault diagnosis is proposed. Specifically, the method is conducted by the following three steps. First, the VMD is introduced to decompose the raw vibration signal. Second, the one or more modes with the information of fault-related impulses are selected through the kurtosis index. Third, Multiresolution Teager Energy Operator (MTEO) is employed to extract the fault-related impulses hidden in the vibration signal and avoid the negative value phenomenon of Teager Energy Operator (TEO). Meanwhile, the physical meaning of MTEO is also discovered in this paper. In addition, an idea of combining the multiresonance bands is constructed to further enhance the fault-related impulses. The simulation studies and experimental verifications confirm that the proposed method is effective for identifying the multiresonance bands and enhancing rolling element bearing fault diagnosis by comparing with Hilbert transform, EMD-based demodulation, and fast Kurtogram analysis.

#### 1. Introduction

Rolling element bearings are widely used in rotating machinery to support rotating shafts, and the major cause of machinery breakdown is the bearing failure. Hence, it is necessary to detect bearing faults at an early stage. However, the rolling element bearing early incipient defect feature is very weak for reasons of being buried in the strong background noises and the interference of the rotating frequency. Besides, there exist the severe signal attenuation phenomenon between the fault source and the sensor collecting the fault signal if the sensor is placed far from the fault-related location. Currently, the fault diagnosis of rolling bearing early weak fault is not only a hot area but also a difficult area [1].

Rolling element bearings usually consist of an inner race, an outer race, several rollers, and a cage. When the surface of one or more of these components develops a localized fault, the impacts generated excite the resonant frequencies of the bearing and adjacent components and induce a modulating phenomenon [2, 3], which is the basis of bearing fault diagnosis. Vibration signals collected from bearings carry the rich information on machine health conditions. Therefore, the vibration-based methods have received intensive study during the past decades. It is possible to obtain vital characteristic information from the vibration signals through the use of signal processing techniques [4, 5]. In order to extract the transient features from the vibration signals, different signal processing techniques have been developed in the area of rotary machine fault diagnosis, such as the wavelet analysis [6], empirical mode decomposition (EMD) [7], time-frequency analysis (TFA) [8], sparse decomposition [9–11], manifold learning [12], spectral kurtosis (SK) [13, 14], cyclic spectral analysis [15, 16], and envelope analysis [17–20]. In fact, the essence of most methods for the weak fault diagnosis is to detect the resonance bands excited by bearing defect. However, by the optimal frequency band selection methods, only the noises outside the selected frequency band are removed from the original signal, while those inside the selected frequency band cannot be wiped off effectively. As a result, the performances of these methods may be poor in the presence of low signal-to-noise ratio and each of these methods leaves something to be desired and some even perform badly in analyzing weak vibration signal. For example, the same class decomposition models of EMD are mostly limited by their algorithmic ad hoc nature lacking mathematical theory, the recursive sifting which does not allow for backward error correction, and the inability to properly cope with noise [21]. As for the wavelet transform, if the wavelet function is selected properly, the defect-related features may be well extracted [5]. Therefore, the constructing wavelet function that adaptively matches the defect-related characteristics of vibration signal is one of the key issues. However, the way of constructing wavelet function needs the prior knowledge and the hard band-limits of wavelet transform is also another inevitable shortcoming of it. In addition, the resonance bands excited by bearing defect are often more than one, while the conventional methods may fail to uncover the accurate multiresonance bands.

Recently, Dragomiretskiy and Zosso [22] proposed a new decomposition model called VMD that determined the relevant bands adaptively and estimated the corresponding modes concurrently, thus properly balancing errors between them. VMD is a fully intrinsic and adaptive variational method which is motivated by the narrow-band properties corresponding to the current common intrinsic mode function (IMF) definition and looks for an ensemble of modes that reconstruct the given input signal optimally (either exactly, or in a least-squares sense). Each mode being band-limited about a center frequency is estimated on-line. Specifically, the method can address the presence of noise in the input signal in which the tight relations of it to the Wiener filter actually suggest that it has some optimality in dealing with noise. Subsequently, Mohanty et al. [23] employed the VMD to decompose the vibration signal in several modes, extracted the energy feature of it to diagnose whether the bearing doped with sand or not, and compared the method with EMD to verify its performance. Wang et al. [24] used the VMD to detect the rub-impact fault of the rotor system and verify that its performance outperforms the conventional decomposition methods such as empirical wavelet transform (EWT) [25], EMD [21], and EEMD [26]. To our knowledge, there is no report in the literature so far on its applications to weak bearing vibration signal analysis and its property of detecting multiresonance bands is also not discovered up till now. Thus, the method of VMD is firstly applied to detect the resonance bands for taking full advantage of the resonance information in vibration signal of defective bearing in our paper.

After the resonance bands are determined in the bearing vibration signal where the fault-related periodic impulse is a modulator to the high natural frequencies of the machine, the demodulation techniques should be used to demodulate the impact impulses from the resonance modes obtained by VMD. Many demodulation methods have been studied, such as FFT-based Hilbert transform [21, 27], wavelet-based [28, 29], and TEO [17, 18]. Among them, the TEO method is an attractive demodulation method proposed and developed by Maragos et al. [30–32], is a sort of time-frequency analyzer, and has been used for many applications such as speech processing, image processing, and AM/FM demodulation. Compared with HT method, TEO method is totally based on the local differential operation without involving integral transform, so it has a better localization property and lower computational complexity. The TEO is also known to be sensitive to spikes, where a spike means that a signal is concentrated in a short time interval and at a high frequency band. With these advantages, the TEO method has been also introduced into machinery fault diagnosis. Lin et al. [33] utilized TEO for resonance demodulation analysis to extract fault characteristic of roller bearings. Junsheng et al. [34] presented a TEO demodulation approach based on EMD to diagnose machinery fault. Liang and Bozchalooi [17] introduced a repetitive application of TEO on detecting the fault characteristic frequency in the spectrum of the energy-transformed signal. However, under a low signal-to-noise ratio (SNR) or in background noise at high frequencies, the TEO gets more sensitive to high noisy peaks than to the true fault-related impulses, and the performance of TEO as an impulse detector degrades rapidly. Besides, the extracted envelope waveform by these demodulation methods is at a single scale [35]. Thus, Choi et al. [35, 36] proposed the MTEO method to detect the action potential of neural signals which could make up for the weakness by tuning the TEO to the frequency of action potentials with the resolution parameter. The analysis results of experimental data in [35, 36] have shown that the MTEO method outperforms TEO and the other conventional demodulation methods in handling both noise and interferences. However, the MTEO physical meaning is not given in paper [35, 36]. We will further develop and uncover the MTEO physical meaning and then firstly employ MTEO to enhance the fault-related impulses in bearing vibration signal. And yet, the enhanced vibration signal still confronts the contamination of in-band noise. The optimal smoothing window, hamming window, together with MTEO is used to further enhance the fault-related impulses.

Furthermore, we can consider that the noise components are less correlated with the different resonance bands while the impulses of envelope waveform are still correlated. Therefore, the combination of more possible resonance bands could be further beneficial to in-band noise removal. Eventually, we propose the novel method for enhancing fault-related impulses in bearing vibration signal by combining the VMD and MTEO. The simulation studies and experiment verification on an experimental rolling element bearing will also be conducted to test the improved performance of the proposed method and compare with the conventional FFT-based Hilbert transform, EMD-based demodulation [27], and fast Kurtogram analysis [2, 13].

This paper is organized as follows. Section 2 describes the details on the theoretical background of the proposed method for the enhanced fault diagnosis of rotating machines. The MTEO physical meaning is also given and uncovered in this section. In Section 3, the simulation studies of the proposed method are studied where there are two cases, single resonance band and double resonance bands. Then in Section 4, the practical applications to bearing defect identification are conducted to verify the effectiveness of the proposed method. A discussion is given in Section 5. The conclusions are finally drawn in Section 6.

#### 2. A Description of Theoretical Background

##### 2.1. Variational Mode Decomposition (VMD)

The VMD [22] is a recently developed methodology for adaptive signal decomposition which decomposes an input signal into discrete number of subsignals (modes) . These modes have specific sparsity properties while reproducing the input, where each mode has limited bandwidth in the spectral domain. Each mode is required to be mostly compact around a center pulsation determined along with the decomposition. The VMD to process the input signal is given as follows.

For each mode , compute the associated analytic signal by means of the Hilbert transform to obtain a unilateral frequency spectrum.

Shift the frequency spectrum of each mode to the baseband by mixing with an exponential tuned to the estimated center frequency, respectively.

Estimate the bandwidth through the Gaussian smoothness of the demodulated signal, that is, the squared -norm of the gradient. As a consequence, the constrained variational problem is given bywhere is the input signal, is the number of modes, is the th mode, is the central frequency of th mode, is the Dirac distribution, is the time script, and denotes convolution.

Both a quadratic penalty term and Lagrangian multipliers to render the problem unconstrained are used as the reconstruction constraints. The combination of both terms thus benefits greatly from the nice convergence properties of the quadratic penalty at finite weight and the strict enforcement of the constraint by the Lagrangian multiplier . Therefore, the solution to find the optimal with the inclusion of Lagrange multipliers and quadratic penalty is given bywhere denotes the balancing parameter of the data-fidelity constraint and the Lagrangian multiplier is a common way of enforcing constraints strictly.

Alternate direction method of multipliers (ADMM) optimization algorithm [37–39] is used to solve (2) to produce different decomposed modes and the center frequencies of these modes during each shifting operation. The procedures of these operations are shown in following.

(i) To update the modes , the subproblem is formulated as (3) which is rewritten as the equivalent minimization problem shown in (4):

Subsequently, the solution to find the optimal in spectral domain is given by (5). Moreover, the two regularized terms are written as half-space integrals over the nonnegative frequencies based on exploiting the Hermitian symmetry of the real signals in the reconstruction fidelity term. As a result, the solution of this quadratic optimization problem is readily found as (6) by forcing the first variation vanish for the positive frequencies which is identified as Wiener filter. The decomposed mode in time domain is eventually obtained by the inverse Fourier transform of the filtered analytic signal:

(ii) To update the central frequency , the other subproblem is formulated as (7). Because the center frequency does not appear in the reconstruction fidelity term, but only in the bandwidth prior, the relevant problem is thus read as (8):

As before, the optimization can also take place in Fourier domain, and the solution to find the optimal in spectral domain is given by (9). Therefore, this quadratic problem is easily solved as (10):which puts the new at the center of gravity of the corresponding mode’s power spectrum. The mean carrier frequency is the frequency of a least squares linear regression to the instantaneous phase observed in the decomposed mode. The complete algorithm of the VMD in detail can be found in [22].

##### 2.2. Teager Energy Operator (TEO)

###### 2.2.1. A Single TEO

The TEO method [31, 32] is an attractive demodulation method which is introduced in this subsection. It is applied on a continuous signal which is defined as

The definition of the original discrete-time TEO is then given by

The derived version of this operator has been used to separate the frequency and amplitude modulations of signals. It is logic to conjecture that the extraction of certain information relying on the combined amplitude and frequency demodulations could be carried out directly by this operator [17]. When we consider an arbitrary signal defined as (13), the corresponding discrete-time equivalent of the energy operator is given as (14):where and .

According to (14), the TEO extracts both the amplitude-modulation (AM) and frequency-modulation (FM) information of the signal. Although the energy operator has mainly been used to separate the amplitude and frequency modulations of a given signal, the separation of such information is not necessary in the context of machinery fault diagnosis [17]. The information of interest for machinery fault detection is the transient nature of the fault impulses resulting from both amplitude and frequency modulations. It is well known that TEO is sensitive to transient impact, where an impact means a signal that is concentrated in a short time interval and at a high frequency band, and valuable means of accentuating the transient fault characteristics relative to the other components of vibration signal such as gear meshing, shaft imbalance vibration, and noise component. However, the TEO gets more sensitive to high noisy peaks than to the true impact under a low SNR or background noise of the obtained signal at high frequencies, and the performance of the TEO as an impact detector degrades rapidly. Moreover, the negative value phenomenon which is nonphysical easily arises in the transformed signal by TEO. In the following subsection, the MTEO is introduced to avoid these drawbacks.

###### 2.2.2. Multiresolution TEO (MTEO)

In paper [35, 36], MTEO is proposed to accurately identify the action potentials in the neural signal by tuning TEO to the frequency range of the impacts with the multiresolution parameter (). The definition of the discrete-time MTEO is given in the following sense:

However, the physical meaning of (15) is not given clearly in [35, 36]. Hereby, we deduce the essential mechanism to deeply uncover (15). We define differentiation operator which is useful to suppress low frequency interferences and noise, integration operator which can enhance the signal in the presence of noise or increase signal-to-noise ratio (SNR) due to its smoothing effects, and composite operator as follows:where the , , represent the order differentiation operator, integration operator, and composite operator, respectively. It should be noted that the composite operator is only defined as the two formations in (18).

According to (11), a new transform operator is formed aswhere is regarded as the generalized form of (12) and is shown in the following:

As a result, (15) can be obtained by using (19). It can be concluded from (19) that MTEO is formed by the composite operator which consists of differentiation and integration operators. Therefore, MTEO takes advantages of two original operators for suppressing low frequency interferences and increasing SNR, besides the virtue of TEO.

Another important element that affects the impact detector performance is the smoothing window. When using the -TEO as a tool of enhancing impulses, it only uses three samples to calculate an output value at a time instance. One prominent noise sample can induce a peak at the output and disturb the accurate fault impact. When the SNR is low, such noisy peaks cause serious problems and must be removed by a smoothing window. As suggested by paper [35], the hamming window was suitable for the matched filter of impulses and paper [36] also gave a logical interpretation to the smoothing window using the matched filter theory, and both of them recommended the hamming window with length of as an optimal window for the -TEO. In addition, after the components obtained from VMD are processed by -TEO, they still contain some noises around the impulses. Therefore, we select the hamming window as the smoothing window to extract the more pure impulses. This optimal smoothing window, together with -TEO, is used to process the fault-related signal. We can adjust the -TEO to be sensitive to the frequency of the fault-related impacts. When we take the effect of the output window into account, the windowed output of a white Gaussian noise iswhere denotes the th coefficient of the smoothing window matched to the -TEO.

According to the central limit theorem, is approximated as a Gaussian random variable with mean and variance . Therefore, the mean-square value of the windowed noise is

The windowed output can be regarded as a signal of defect-related feature mixed in the background noise . However, the value of could change at different resolution parameters where some false impacts may be caused by the in-band noise. Therefore, let the window coefficients be (23) to normalize the output noise power to a constant value:

For MTEO, all these elements, that is, differentiation, integration and energy operator, and hamming window, are implemented by a simple formula in one step. The main attractive advantages of MTEO include its simplicity, computational efficiency, excellent time resolution, and the leaving-out of the band-pass filtering process. As such, it is suited to on-line bearing fault detection in a noisy environment with multiple vibration interference.

##### 2.3. Summary of the Proposed Method

In summary, the procedure of the proposed method can be described briefly as follows:(1)Conduct VMD on the measured raw vibration signal to transform it into several modes. Considering that fault-related resonance bands are identified preferentially by VMD, the number of the resonance bands may be more than one in the vibration signal of early stage defect generally. Therefore, the number of decomposed modes by VMD is set as at least 2, but should not be too large for saving the computing time.(2)Calculate the kurtosis value of these decomposed modes.(3)Employ MTEO and smoothing window to enhance the fault-related waveforms of the decomposed modes with a large kurtosis value on four scales with spacing 3. The four scales could cover the analysis frequency range as suggested in paper [35, 36].(4)Observe the enhanced time domain waveform and its frequency spectrum from the four enhanced waveforms to diagnose the faulty bearing. If there are multiresonance bands in the obtained modes, the combination of them will be used to further enhance the fault-related waveform.

Since the VMD could adaptively extract the modes in the resonance bands, the proposed method does not need more prior knowledge. In addition, the MTEO is a nonlinear method. Therefore, it can reveal the nonlinear transient envelope information of machinery systems by combining one or more decomposed modes. Thus the result analyzed by the proposed method is taken for further pure impulse and spectrum analysis to identify the fault-related characteristic and locate the defect position in a machine. The frame of the proposed method is shown in Figure 1.