Shock and Vibration

Volume 2018, Article ID 8710190, 18 pages

https://doi.org/10.1155/2018/8710190

## An Improved Time-Frequency Analysis Method for Instantaneous Frequency Estimation of Rolling Bearing

Department of Electrical and Electronics Engineering, Shijiazhuang Railway University, Shijiazhuang 050043, China

Correspondence should be addressed to Zengqiang Ma; moc.621@newnulqzm

Received 16 May 2018; Revised 16 August 2018; Accepted 23 August 2018; Published 18 September 2018

Academic Editor: Adam Glowacz

Copyright © 2018 Zengqiang Ma 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

Instantaneous frequency estimation of rolling bearing is a key step in order tracking without tachometers, and time-frequency analysis method is an effective solution. In this paper, a new method applying the variational mode decomposition (VMD) in association with the synchroextracting transform (SET), named VMD-SET, is proposed as an improved time-frequency analysis method for instantaneous frequency estimation of rolling bearing. The SET is a new time-frequency analysis method which belongs to a postprocessing procedure of the short-time Fourier transform (STFT) and has excellent performance in energy concentration. Considering nonstationary broadband fault vibration signals of rolling bearing under variable speed conditions, the time-frequency characteristics cannot be obtained accurately by SET alone. Thus, VMD-SET method is proposed. Firstly, the signal is decomposed into several intrinsic mode functions (IMFs) with different center frequency by VMD. Then, effective IMFs are selected by mutual information and kurtosis criteria and are reconstructed. Next, the SET method is applied to the reconstructed signal to generate the time-frequency representation with high resolution. Finally, instantaneous frequency trajectory can be accurately extracted by peak search from the time-frequency representation. The proposed method is free from time-varying sidebands and is robust to noise interference. It is proved by numerical simulated signal analysis and is further validated by lab experimental rolling bearing vibration signal analysis. The results show this method can estimate the instantaneous frequency with high precision without noise interference.

#### 1. Introduction

Rolling bearing is a key part and most widely used in rotating machinery. Its working status directly affects the operation efficiency and service life of mechanical system, therefore, it is very important to diagnose fault of rolling bearing [1–5]. In practice, rolling bearing often works under variable speed conditions when the vibration signal shows strong nonstationary characteristics, which increases difficulty of fault diagnosis. In the field of rolling bearing fault diagnosis under variable speed [6, 7], order tracking is one of the most effective method [8]. There are a lot of ways for order tracking. Among them, order tracking based on time-frequency representation is the most popular method because it does not need to install the tachometer, so that it saves space and costs, while instantaneous frequency estimation is the most preliminary and vital step. Therefore, how to effectively estimate instantaneous frequency from complex vibration signal with strong noise is a significant issue for further fault diagnosis of rolling bearing under variable speed conditions.

There are many sorts of methods for instantaneous frequency estimation, and the first class is based on the phase demodulation method. For example, Coats [9] presented the multistep iterative method to realize phase demodulation and extracted instantaneous frequency information. Feng et al. [10] put forward time-varying demodulation analysis method by applying ConceFT. Such methods are restricted because the cross terms between the adjacent harmonics will be generated when the rotating speed varies widely. The second class is based on the time-frequency analysis method, to name a few for example, Guo et al. [11] proposed the short-time Fourier transform (STFT) combined with peak search and applied the method to instantaneous frequency estimation for variable speed motor. But it shows poor property under strong noise interference. Zhao et al. [12] devised STFT combined with the Viterbi algorithm to estimate instantaneous frequency of variable speed vibration signal. But the Viterbi algorithm’s complexity results in this method’s low computational efficiency, so it is hard to apply to actual working conditions.

To sum up, instantaneous frequency estimation method based on the time-frequency analysis is simple in principle and free from speed fluctuation which has broader practical value, so, much attention should be paid to it. For this kind of method, the key to success is time-frequency analysis method, which should possess high time-frequency resolution and good antinoise property. Widely used traditional time-frequency analysis methods include short-time Fourier transform (STFT), wavelet transform (WT), S transform, and Wigner-Ville distribution (WVD), etc. These methods have been applied in many fields and acquired some good achievements. But, restricted, respectively, by the Heisenberg uncertainty principle, ineradicable cross-terms, and large computational costs, these methods reveal some limitations for practical purposes. In recent years, some new time-frequency analysis methods have been put forward. Among these methods, synchrosqueezed wavelet transforms (SST) attract the most attention, which was proposed by Daubechies et al. [13] in 2011. This method’s essence is the combination of wavelet transform and time-frequency reassignment. Time-frequency aggregation is improved by compressing wavelet transform coefficient in frequency/scale direction. However, SST still has many problems, like poor antinoise property and deficiency in processing multicomponents signal. Therefore, a lot of scholars proposed various improved methods. For example, Feng et al. [14] proposed iterative generalized synchrosqueezing transform to address the issue of time-frequency blurs of multicomponent and time-variant frequency signals and applied it to fault diagnosis of wind turbine planetary gearbox under nonstationary conditions. Wang et al. [15] introduced a matching synchrosqueezing transform to process signals composed of multiple components with fast varying instantaneous frequency that achieved a highly concentrated time-frequency representation.

Considering these issues, inspired by SST and the theory of ideal time-frequency analysis, Yu et al. [16] proposed a novel time-frequency analysis method named SET in 2017, which belongs to a postprocessing procedure of the STFT. Different from the squeezing manner of SST, the main idea of SET is to only retain the time-frequency information of STFT results most related to time-varying features of the signal and to remove most smeared time-frequency energy, so the time-frequency energy concentration can be improved significantly. And Yu successfully applied it to analyze instantaneous frequency of bat signal and mechanical vibration signal. Chen et al. [17] dealt with seismic signals making use of SET and realized hydrocarbon detection.

SET is very suitable to analyze time-frequency information of nonstationary signals and can achieve high time-frequency resolution. However, rolling bearing fault vibration signal is nonstationary and multicomponent with complex amplitude modulation features, frequency modulation features, and strong noise interference. SET alone is not sufficient to accurately extract time-frequency information. Therefore, the authors consider using a preprocessing method to denoise and decompose multicomponent signal into monocomponent signal and then performing SET, in which time-frequency resolution and antinoise property both can be enhanced very well.

Recently, a new adaptive signal decomposition method named variational mode decomposition (VMD) was proposed by Dragomiretskiy et al. [18] in 2014, which can select the relevant frequency bands of fault signal suppressing noise interference and decompose the signal into several monocomponent signal with high precision. VMD can efficiently overcome the mode mixing and misclassification problem of empirical mode decomposition (EMD) and ensemble empirical mode decomposition (EEMD) due to noniterative decomposition. Thus, it has been widely used in signal decomposition and denoising field, and many scholars applied it to fault diagnosis and other fields in practice [19–21]. VMD can decompose signal into several intrinsic mode functions (IMFs); however, some IMFs are redundant components irrelevant to original signal, and so how to select efficient components is a key step. Given this, mutual information (MI) and kurtosis are introduced to select effective IMFs, which can guarantee that the selected components contain the useful information to the utmost extent and purify the signal.

In this paper, we propose an improved time-frequency analysis method combining VMD with SET to estimate instantaneous frequency of rolling bearing under variable speed conditions. First, the signal is decomposed into some IMFs by VMD. Next, the effective components are selected via MI and kurtosis and reconstructed. Then, SET is carried out for the reconstructed signal. Last, peak search is performed on SET time-frequency representation, and instantaneous frequency can be accurately extracted.

Hereafter, this paper is structured as follows. In Section 1, we produce the relevant principles including VMD, mutual information, kurtosis, SET, and peak search and illustrate concrete steps of the proposed method. In Section 2, firstly, we compare VMD with EMD, then numerical simulated signals are analyzed to prove effectiveness of the proposed method and debate antinoise property and estimate accuracy. In Section 3, the lab experimental vibration signals of rolling bearing with rising speed and fluctuate speed are employed to further validate the practicability of the proposed method. At last, the conclusions are given in Section 4.

#### 2. Theory

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

The aim of VMD is to decompose multicomponent signal into a series of band-limited monocomponents with specific sparsity properties in the bandwidth, and all components are compact around a center pulsation, moreover, the decomposed components support reconstruction. It is carried out by working out the following constrained variational optimization problem:where is the component of the original signal, denotes a series of modes , is the center frequency corresponding to the component, denotes a series of center frequencies corresponding to which is represented as , is the original signal, and is the Dirac function.

For this model, first, the analytical signal and its single side spectrum are obtained via Hilbert transform. Then, it is multiplied with the exponential factor to modulate the all modes’ spectrum to the corresponding baseband. Last, the constraint variational problem is transformed into a nonconstrained variational problem by extending the Lagrange function and then solved. The expression is as follows:where denotes the balancing parameter of the data-fidelity constraint and is the Lagrange multiplier.

The Lagrange saddle point is acquired using alternate direction method of multipliers algorithm, which is the optimal solution of the original variational model. During the solution process, each mode is updated according to the following equation:where , respectively, denote the corresponding Fourier transform and is iterations.

The center frequency is estimated according to the updated modes’ power spectrum, the center frequency is updated by Equation (4). Further, also can be updated. The updating process ends until the iteration stop condition shown in Equation (5) is satisfied. Then K IMFs can be obtained.

##### 2.2. Mutual Information (MI)

MI derived from the entropy of information theory, which is the difference value of two random variables uncertainty represents the statistical correlation, and the larger the value, the greater the correlation. MI is often used to identify fake components of EMD, EEMD, and VMD. Some scholars compared MI with correlation coefficient, and the result shows that the MI is more accurate [22]. The expression of MI is as follows:where denotes of and , denotes the entropy of Y, and is the conditional entropy of at .

The normalized expression is

The threshold value is set for , and the correlation between the decomposed modes and original signal can be judged by the threshold value, and if is larger than the threshold value, the corresponding modes are effective components, otherwise, the corresponding modes are fake and should be removed.

##### 2.3. Kurtosis

Kurtosis is a dimensionless parameter describing the waveform peak, which is sensitive to the impulse characteristics of signal. For the discrete variable , the normalized fourth order central moment is called kurtosis, which is defined as follows:where is kurtosis value, denotes the fourth order mathematic expectation, is the mean value, and is the standard deviation.

When the rolling bearing is under normal working condition, the amplitude probability density of the vibration signal is close to the normal distribution, when the kurtosis value is about 3, which is a stationary or weak stationary process. However, when there is damage impulse due to the rolling bearing elements pitting or cracks, the amplitude probability density will deviate from the normal distribution, and the kurtosis value increases, that is, the more impulsive the signal is, the larger the kurtosis value becomes. Thus, the mode with larger kurtosis contains more abundant fault information [23].

##### 2.4. Synchroextracting Transform (SET)

SET is a novel time-frequency analysis method, and it is a postprocessing procedure of the STFT, which is a more energy concentrated time-frequency representation than classical time-frequency analysis methods and can effectively describe time-frequency characteristics.

For a multicomponent signal , which can be seen as the sum of nonstationary modes, its expression is as follows:where , respectively, denote the mode, the corresponding instantaneous amplitude, and instantaneous phase; is the first-order derivative of and denotes instantaneous frequency; and is the frequency support of window function.

SET is based on STFT, and the STFT representation of original signal is shown as the following form:where denotes the Fourier transform of the window function , .

According to (10), we can calculate instantaneous frequency by

In order to enhance the time-frequency resolution, Yu designed an operator to only retain the time-frequency information most related to time-frequency characteristics of the target signal from STFT representation, which can remove the irrelevant interference and smeared time-frequency energy. The SET is formulated aswhere which is named the synchroextracting operator (SEO).

According to (10) and (12), the following expression can be deduced:

In this way, we can obtain a sharper time-frequency representation than STFT and extract instantaneous frequency with a highly precise degree.

##### 2.5. Peak Search

After obtaining the time-frequency representation, we need to extract the instantaneous frequency curve from it. For a time-frequency analysis method with high precision and high time-frequency aggregation, the peak search algorithm can extract the instantaneous frequency from the time-frequency diagram accurately, and the peak search principle is simple and the efficiency is high. Therefore, this paper uses the peak search method to extract the instantaneous frequency curve. The steps to extract instantaneous frequency curve from the time-frequency diagram by peak search are as follows:(1)Time-frequency representation is obtained.(2)Select the starting point of search. In the time frequency diagram, a point is selected as the starting point in the region where the peak value of the tracking component is prominent. After selecting the starting point, the peak value of the time frequency diagram is searched according to the following equation:where denotes the number of time lines in the time-frequency grid, denotes the number of frequency lines in the time-frequency grid, IFE is the peak search function, argmax is the parameter when the objective function takes the maximum value, and SPEC is the corresponding time-frequency representation. is the first instantaneous frequency coordinate obtained from peak search with as the starting point. denotes the range of peak search; is the instantaneous frequency coordinate corresponding to each time after peak search.(3)Instantaneous frequency curve fitting. The least squares fit is performed on the discrete instantaneous frequency obtained above. According to the trend of instantaneous frequency change of each point, the number of polynomials is selected. Normally, the rotation speed will not be abrupt, so we can choose low-order polynomial fitting. Take the second-order polynomial as an example. The fitting formula is as follows:where is time; denotes instantaneous frequency fitting function; and , , and are undetermined coefficients.

The squared error is as follows:

According to these restrictive conditions: , , , and can be determined.

##### 2.6. VMD-SET Analysis Procedure for Rolling Bearing Fault Signal

Considering rolling bearing fault signal shows strong nonstationary under variable speed condition and is complex multicomponent signal contaminated by strong noise, it is difficult to accurately estimate instantaneous frequency for time-frequency analysis alone even with high time-frequency resolution. Thus, it is necessary to preprocess signal to denoise and decompose the original. VMD has a solid theoretical foundation, which decomposes a multicomponent signal into a set of quasiorthogonal IMFs with different center frequency in nonrecursively way and is suitable to process the rolling bearing fault vibration signal. However, not all IMFs are valid, so we select the effective IMFs by MI and kurtosis, which not only removes the noise interference but also obtains the monocomponent signal containing the most useful information. After that, we perform SET and can eliminate the most-smeared time-frequency energy and get clear time-frequency representation.

For an actual rolling bearing vibration signal, the proposed method can be generated following the procedure listed below:(1)Decompose the original signal into a number of IMFs by VMD. The VMD parameters are set as the default value. .(2)Calculate the MI between each mode and the original signal and each IMF’s kurtosis value, and the components are removed that the MI is less than the threshold value and the kurtosis value is less than 3, and the other IMFs are selected. In this paper, the MI threshold value is determined as 0.1. Through a large number of experimental data analysis, the results show that it is the most appropriate to set the threshold value as 0.1, when the signal can retain the most useful information and eliminate noise effectively.(3)Add the selected IMFs to get the reconstructed signal, then apply SET to the reconstructed signal. When instantaneous frequency features can be shown clearly in the SET time-frequency representation.(4)Extract instantaneous frequency curve via peak search based on SET time-frequency spectrum.

The VMD-SET analysis flowchart for instantaneous frequency estimation of rolling bearing is shown in Figure 1.