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.

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 [

There are many sorts of methods for instantaneous frequency estimation, and the first class is based on the phase demodulation method. For example, Coats [

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. [

Considering these issues, inspired by SST and the theory of ideal time-frequency analysis, Yu et al. [

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. [

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

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:

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:

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:

The center frequency is estimated according to the updated modes’ power spectrum, the center frequency is updated by Equation (

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 [

The normalized expression is

The threshold value is set for

Kurtosis is a dimensionless parameter describing the waveform peak, which is sensitive to the impulse characteristics of signal. For the discrete variable

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 [

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

SET is based on STFT, and the STFT representation

According to (

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 as

According to (

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

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:

Time-frequency representation is obtained.

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:

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:

The squared error is as follows:

According to these restrictive conditions:

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:

Decompose the original signal into a number of IMFs by VMD. The VMD parameters are set as the default value.

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.

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.

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

VMD-SET analysis flowchart for instantaneous frequency estimation of rolling bearing.

The characteristics and advantages of the proposed method include (1) the application of VMD, which can realize the signal decomposing and denoising; (2) VMD’s effective components are selected by combing MI with kurtosis, in which the acquired components can guarantee accuracy and completeness to the utmost extent; and (3) the proposed method possesses strong noise resistance, excellent time-frequency resolution, and high estimation precision of instantaneous frequency.

Rolling bearing vibration signal under variable speed is complex multicomponent signal. In different working conditions, the signal can be demodulated by different frequency, and the estimation accuracy of instantaneous frequency will be greatly affected due to strong background noise. Thus, in this paper, we design two kinds of demodulated signal (linear frequency modulation and sinusoidal frequency modulation multicomponent signals) to simulate rolling bearing vibration signal under two working conditions, which can demonstrate the method’s effectiveness more convincingly.

First of all, let's discuss why we choose to preprocess signals with VMD. In order to illustrate the rationality of VMD used in this article, we compare VMD with EMD to test its noise immunity and signal decomposition accuracy. A multicomponent harmonic signal is constructed, and the center frequencies of three components are 2 Hz, 24 Hz, and 288 Hz, respectively, as shown in (

The signal is decomposed by

Signal decomposition comparison: (a) waveforms and spectrums of VMD; (b) waveforms and spectrums of EMD.

In order to be a suitable preprocessing method for SET, good noise immunity and high-precision decomposition capability are necessary. From the above analysis, it can be seen that VMD has advantages over EMD. So, we choose VMD as a preprocessing method for SET.

The simulated multicomponent vibration signal of line frequency modulation is established, and the instantaneous angular frequency is

So, the instantaneous frequency is

The simulated signal is

The signal’s sampling frequency is 100 Hz, and we added strong noise to the signal to get SNR (signal-to-noise ratio) of −12 dB, which can make the advantage of the proposed method in antinoise more prominent. The signal’s waveform is shown in Figure

Waveform of LFM signal.

Waveform and spectrum of IMFs by VMD for LFM signal.

The kurtosis values and mutual information (MI) of IMFs for LFM signal.

IMF | IMF1 | IMF2 | IMF3 |
---|---|---|---|

Kurtosis | 4.78 | 3.57 | 4.13 |

MI | 0.856 | 0.769 | 0.985 |

Time-frequency representation of LFM signal based on VMD-SET.

In order to highlight the advantage and necessity of the proposed method, we provided the time-frequency representation of SET alone for comparison, as Figure

Time-frequency representation of LFM signal based on SET.

Comparison between estimated instantaneous frequency and true result of LFM signal.

In order to illustrate quantitatively the estimation accuracy of the two methods, we calculated the percentage value of the estimation error using Equation (

The simulated multicomponent vibration signal of sinusoidal frequency modulation is established, and the instantaneous angular frequency is

So, the instantaneous frequency is

The simulated signal is

The signal’s sampling frequency is also 100 Hz. We still added strong noise to the signal, and the SNR is equal to −12 dB. The signal’s waveform is shown in Figure

Waveform of SFM signal.

Waveform and spectrum of IMFs by VMD for SFM signal.

The kurtosis values and mutual information (MI) of IMFs for SFM signal.

IMF | IMF1 | IMF2 | IMF3 |
---|---|---|---|

Kurtosis | 3.28 | 4.87 | 5.12 |

MI | 0.723 | 0.859 | 0.967 |

Time-frequency representation of SFM signal based on VMD-SET.

Time-frequency representation of SFM signal based on SET.

Comparison between estimated instantaneous frequency and true result of SFM signal.

In this paper, we further analyze the antinoise property and the precision of instantaneous frequency estimation for the proposed method. To be more persuasive, the authors made a lot of tests and added white noise to the LFM signal and SFM signal to make SNR vary from −20 dB to 10 dB. We performed VMD-SET time-frequency analysis on LFM signal and SFM signal under different SNR and calculated instantaneous frequency estimation error under different SNR.

In order to save space, we give VMD-SET time-frequency representation for LFM signal and SFM signal under four groups of SNR. The representation for LFM signal is shown in Figure

VMD-SET time-frequency representation of LFM signal under different SNR: (a) 10 dB, (b) 0 dB, (c) −10 dB, and (d) −20 dB.

VMD-SET time-frequency representation of SFM signal under different SNR: (a) 10 dB, (b) 0 dB, (c) −10 dB, and (d) −20 dB.

In order to validate the performance of the proposed method strictly, we adopt the quantified indicator to illustrate the effectiveness. The more energy-concentrated time-frequency representation manifests the better the ability of the time-frequency location and the better the characterization of time-varying feature. The Rényi entropy can evaluate the energy concentration of time-frequency representation, so we adopt it to validate the performance of VMD-SET and SET. The smaller the Rényi entropy is, the higher the time-frequency resolution, and its definition equation is shown in the following equation:

Under different SNR, the Rényi entropies of SET and VMD-SET time-frequency representations are respectively shown in Figures

The Rényi entropies of the time-frequency representation based on SET and VMD-SET under different SNR (LFM).

The Rényi entropies of the time-frequency representation based on SET and VMD-SET under different SNR (SFM).

Moreover, we calculate the estimation error of LFM signal and SFM signal under different SNR based on VMD-SET. It is listed in Figure

Instantaneous frequency estimation errors based on the proposed method under different SNR.

In this section, we apply the proposed method to the real rolling bearing vibration signal to further validate the practicability and effectiveness. The real data are from the mechanical fault diagnosis laboratory in Shijiazhuang Railway University, and the experimental setup is displayed in Figure

Experimental setup: (a) QPZZ-II fault simulation platform, (b) test rolling bearing, (c) acceleration sensor, and (d) experimental installation.

The sampling frequency of vibration signal is 25600 Hz, and the sampling frequency of the laser tachometer is 1 kHz. And the true rotational frequency is obtained by five-point formula according to the signal from laser tachometer, which is used to compare to estimated frequency. In order to amply demonstrate the effectiveness of the proposed method, we make analysis under two conditions, respectively, the condition with rising speed and complex fluctuated speed.

The vibration signal waveform is shown in Figure

Waveform of rolling bearing vibration.

Waveform and spectrum of IMFs by VMD for rolling.

The kurtosis values and mutual information (MI) of IMFs for real signal with rising speed.

IMF | IMF1 | IMF2 | IMF3 | IMF4 | IMF5 | IMF6 |
---|---|---|---|---|---|---|

Kurtosis | 3.78 | 4.13 | 4.87 | 2.84 | 2.65 | 3.02 |

MI | 0.856 | 0.923 | 0.985 | 0.076 | 0.069 | 0.085 |

Time-frequency representation of rolling bearing vibration signal with rising speed based on VMD-SET.

Comparison between estimated instantaneous rotational frequency and true result of rolling bearing vibration signal with rising speed.

The vibration signal waveform is shown in Figure

Waveform of rolling bearing vibration.

Waveform and spectrum of IMFs by VMD for rolling.

The kurtosis values and mutual information (MI) of IMFs for real signal with fluctuated speed.

IMF | IMF1 | IMF2 | IMF3 | IMF4 | IMF5 | IMF6 |
---|---|---|---|---|---|---|

Kurtosis | 4.58 | 4.93 | 4.12 | 2.93 | 2.26 | 2.69 |

MI | 0.905 | 0.986 | 0.725 | 0.086 | 0.048 | 0.067 |

Time-frequency representation of rolling bearing vibration signal with fluctuated speed based on VMD-SET.

Comparison between estimated instantaneous rotational frequency and true result of rolling bearing vibration signal with fluctuated speed.

In this paper, we propose an improved time-frequency analysis method named VMD-SET for instantaneous frequency estimation of rolling bearing. The proposed method uses VMD and mutual information and kurtosis indicators to select the effective components and then performs SET on the effective components, which not only retains the merits of VMD and SET, but also improves the signal decomposition accuracy and enhances the antinoise ability and time-frequency resolution. Compared with the SET time-frequency analysis alone, the advantages of the proposed method are obvious. Two simulation signal models with different SNR and two groups of real signals separately under rising speed and fluctuated speed conditions have been used to prove the effectiveness of the proposed method, and the results show that the proposed method can be successfully applied to the instantaneous frequency estimation of rolling bearing vibration signal under different complex working conditions, and the estimation accuracy is enough to meet actual requirement, which is an instantaneous frequency estimation method with practical application value.

The authors declare that they have no conflicts of interest.

This work was supported by National Natural Science Foundation of China (Grant nos. U1534202, 11372199, 11572206, and 51405313).