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The compressive sensing (CS) theory provides a new slight to the big-data problem led by the Shannon sampling theorem in rolling element bearings condition monitoring, where the measurement matrix of CS tends to be designed by the random matrix (RM) to preserve the integrity of signal roughly. However, when the signal to be analyzed is infected with strong noise, not only does the signal become insufficiently sparse, but the randomness of the measurement matrix will bring down the sensing efficiency, resulting in the loss of fault feature. Thus, a sensing-enhanced CS scheme based on a series of modes after VMD decomposition is proposed under this paper. The core of this scheme is as follows: (1) the principal mode of VMD with better sparsity replaces the raw signal for compressive sensing; (2) all these modes contain the time-frequency characteristics of the raw signal; (3) a new measurement matrix called mode-circulant matrix (MCM) is defined by circulating the mode matrix, and when the amount of samples is shrunk, the sensing efficiency can be enhanced greatly. Besides, considering the fault signal of rolling bearings under variable speed, there is a need to use order tracking to overcome the nonstationarity of the signal before applying CS theory. The analysis results of simulation and experiment prove that the VMD- and MCM-based CS can successfully extract the weak fault feature of rolling bearings with operating speed changing.

The condition monitoring of rolling element bearings is a crucial method used in a wide range of helping to prevent unpredictable failure. So far, vibration-based diagnosis is a valid and frequently used pattern of bearing health assessment [

The cited literature demonstrated that the fault diagnosis of rolling bearings is explored increasingly. However, since almost all of the above analysis methods are based on signal sampling without loss, high sampling frequencies must be obliged to ensure the accurate acquisition of bearing fault feature under the Shannon sampling theorem [

Compressive sensing (CS) has provided a fine insight for tackling this challenge. This method was primarily proposed by Candès et al. and Donoho [

Owing to the excellent effect of CS theory in the application, it has attracted extensive attention from scholars in the field of fault diagnosis. Li et al. [

The measurement matrix is at the heart of signal compression and reconstruction of CS. Current literature on measurement matrix construction mainly focuses on the deterministic measurement matrix. Depending on the energy of each part of the signal, Guo et al. [

Therefore, a sensing enhanced method named the mode-circulant matrix (MCM) is utilized in this study to increase sensing efficiency on vibration signals, which are advantageous to the weak fault feature of rolling bearings based on CS theory. Assisted by VMD [

The structure of this paper is arranged as follows. Section

VMD is an adaptive signal decomposition method. It can nonrecursively decompose the vibration signal into several quasiorthogonal modes. The optimal solution of the variational model can be used to obtain the frequency center and bandwidth of each mode and adaptively separate the mode components [

To estimate the bandwidth of each mode, the constrained variational problem is constructed as shown in

The constrained problem can be simplified to the unconstrained variational problem by introducing quadratic penalty factor

Equation (

The procedure of VMD can be described as follows:

Initialize

Repeat

Update

Update

Repeat step (2)∼(4) until the convergence

If a sparse signal or the signal is sparse in a transform domain, the original signal can be recovered with high precision through the signal of a small number of samples [

If an original signal can be expanded under an orthogonal base, it can be shown as

It is assumed that the sparse vector includes

Equation (

As shown in (

As dimension

The sparse vector

Figure

Schematic of the proposed bearing weak fault feature extraction method.

Referring to order tracking steps [

Since the Fourier dictionary is chosen in the following as an orthogonal dictionary, the input signal length of compressed sensing needs to match the processing capacity of the computer. When it does not fit well, order tracking may have also additional use of compressing signals without loss of bearing fault feature.

Before the vibration signal is decomposed, we should predefine the balancing parameter of the data-fidelity constraint _{SN}) is used as a basis for quantitative comparison in this paper. _{SN} represents the ratio of energy generated in the case of bearing component failure to the total energy of vibration, and its definition is as follows:_{it}), _{ot}_{bt}), and _{ct}) represent the energy of the characteristic order of bearing outer race, inner race, rolling element, and cage fault, respectively, and _{it}, _{ot}, _{bt,} and _{ct}, respectively, denote the serial number corresponding to the fault characteristic order of outer race, inner race, rolling element, and cage.

After the parameters are optimized, according to (_{SN} of different modes for noise reduction and sparsity enhancement. If mode _{SN} value, then it is the main mode. Later mode _{d}(

In this paper, the measurement matrix, called the mode-circulant matrix (MCM), is constructed by the modes decomposition of VMD, as described in the following.

As is shown in (

The measurement matrix Φ can be acquired by circulating the mode matrix _{c} times [_{c} is the number of cycles:

The number of rows of Φ can be calculated according to (

According to (

In this study, the Fourier dictionary is selected for a sparse representation of signals. The reason is that not only is the VMD mode of bearing fault signal sparse in the Fourier domain, but also the sparsity of harmonic signal in the Fourier domain is fixed [

The Fourier dictionary Ψ and the MCM Φ make up the sensing matrix A. According to (

It is known that the typical fault features of rolling bearings are that the fault impact excites resonance of natural frequency, and this frequency is modulated by faulty feature frequency as the carrier frequency. Thus, when the natural frequency and feature frequency (sideband frequency) are detected simultaneously, that is, sparsity is set to 4, the faulty feature frequency can be extracted.

After the crucial frequencies detected, the signal recovery can be achieved with the Fourier dictionary Ψ and the sparse vector

Fault diagnosis for rolling bearings can be achieved by comparing the dominant components in the envelope spectrum with the calculation value of the fault characteristic frequency of the bearing.

To verify the properties of the Fourier dictionary and MCM in this method, the fault signal of the inner race of the rolling bearing is constructed under speed-up conditions. The simulations are as follows [

The simulated signal of rolling bearings’ fault model can be expressed strictly as follows:_{i} is the amplitude of the _{i}_{i}) is the _{n}. Generally, _{i} can be refined into

The bearing fault simulation signal can be concluded as follows:_{i} is the corresponding time or the _{i}, and

The calculation process of _{i} under variable speed condition is as follows.

In this simulation, the fault characteristic order of bearing inner race is _{i} = 14.9; that is, the fault impact caused by the rotor every rotation cycle is 14.9. Also, it is assumed that the variation equation of rotor speed is _{r}(_{i} _{i} (_{i} corresponding to the impact can be obtained by solving the above equation.

This study assumes _{A} = 0.5, Φ_{A} = 30^{°}, _{n} = 3000, _{i} = 0.01, _{i} = 14.9. Besides, the fault characteristic orders of bearing inner race, rolling elements, and cage are 9.92, 2.4, and 0.4, respectively. In this paper, the sampling frequency of the simulation signal is 12.8 kHz, and the sampling length is 8192. The simulation signal of inner race defect without noise is shown, respectively, in Figure

The inner race fault simulation signal.

The fault vibration signal of rolling bearing measured _{b}(

Gaussian noise is added to the simulated signal in Section _{s} denotes the RMS value of the signal _{n} represents the peak value of the noise. According to (_{b}(

SNR2 is defined as follows:_{max} is the maximum order to be analyzed and _{t}) represents the total energy of bearing fault characteristic order and its multiple in the range of 0 to _{max}. _{it}) is the energy which is in the vicinity of _{t} − _{t} + _{t}. _{it} is the _{it} is the serial number of _{it}. _{t}) denotes the energy of noise in the same range and it excludes _{t}). According to (_{b}(

The signal after being affected by noise is shown in Figure

Simulated signal of bearing fault with noise. (a) The waveform of vibration and (b) speed trend.

According to the steps of the proposed method in Section

The angular signal with noise.

Particle Swarm Optimization has been applied to analyze the noisy signal in angle domain. A fitness curve can be obtained, as being presented in Figure

The trend of local characteristic order amplitude with evolution algebra.

Parameters setting and optimization results of PSO.

Parameters | ||
---|---|---|

Range of values | [0, 2000] | [ |

Maximum evolution | 50 | |

Maximum population | 10 | |

Local search speed | 1 | |

Global search speed | 1.2 | |

Optimization results | 664.7 | 6 |

Subsequently, the modes are generated by the angular signal’s VMD decomposition in which

Fitness values obtained by analyzing simulation signal with VMD.

Modes obtained by VMD and their order spectra | Fitness (%) | |
---|---|---|

3.7 | ||

4.26 | ||

4.2 | ||

14.75 | ||

4.18 | ||

4.33 |

Based on Section

In the sparse transform domain, only

In this paper, firstly, the exponential function is utilized to fit the attenuation curve. Secondly, we calculate the sum of squares error (SSE) and coefficient of determination (R-squared) between the original curve and the fitted curve to assess whether the attenuation of the sparse coefficient is close to the exponential attenuation. Here, SSE is the square sum of the errors of the corresponding points of the fitting data and the original data. R-square is used to represent the quality of the fitting by the change of data. The closer it is to 1, the better the data fitting is. The closer SSE is to 0, the more successful the fitting is.

The angular waveform without noise is shown in Figure

In Fourier dictionary, (a) sparse coefficients of pure simulation signal; (b) attenuation curve of (a).

The attenuation curve of Figure

The fitting result of coefficients attenuation curve in Figure

The existence of noise in the vibration signal will increase the difficulty of sparse representation in the Fourier domain. Sparse coefficients of the noisy signal and the sequencing result are shown in Figures

In Fourier dictionary, (a) sparse coefficients of the noisy signal; (b) attenuation curve of (a).

The fitting result of coefficients attenuation curve in Figure

Finally, the sparsity of mode 4 in Table

In Fourier dictionary, (a) sparse coefficients of the mode 3 and (b) attenuation curve of (a).

The fitting result of coefficient attenuation curve in Figure

Gaussian random matrix (GRM) is frequently used as a measurement matrix of CS for a typical one-dimensional vibration signal [

In this study, a new compressive sensing measurement matrix is constructed in the form of the circulant matrix by using the VMD modes. Its sensing efficiency is much higher than GRM, which means the extraction accuracy of the MCM can be achieved by fewer measurements than GRM. The following parts will demonstrate it.

This verification firstly used the modes in Table

Figure

The envelope order spectra obtained by using (a) MCM: the spectrum is clear, and SNR2 is 33.4 dB; (b) GRM: the spectrum is clear, but SNR2 is only −44.7 dB.

Moreover, the change in success rate is shown in Figure

Comparison of detection effect of two measuring matrices.

Section

Before analyzing the bearing fault signal to NEA, the most fundamental issue to be solved first is the setting of the center frequency and bandwidth of the band-pass filter. However, it fails to extract effective features based on experience frequently. In recent years, SK has been extensively used in the selection of demodulation band parameters and gained good reputation [

Even angle resampling in the speed domain, order tracking is applied to the original signal, achieving a stationary signal in the angle domain instead of a nonstationary signal in the time domain.

The optimal envelope center frequency and bandwidth are determined by SK, based on which a band-pass filter is designed to filter the resampled signal.

The envelope demodulation method based on the Hilbert transform is applied to achieve the envelope order spectrum.

Figure

The analysis results using NEA: (a) kurtogram; (b) envelope order spectrum: the spectrum is obscure, and SNR2 is only −9.07 dB.

Besides, according to (

The facts of Section

To further verify the effectiveness of the weak fault feature extraction method based on the sensing-enhanced CS, the weak fault signals of the rolling bearings under variable speed conditions are collected in the test rig described below.

To simulate the weak fault of rolling elements to a large extent, the dual-rotor test rig is selected, which is designed and built to simulate the working state of the intershaft bearing on the actual double rotor equipment. As is illustrated in Figure

Dual-rotor vibration test rig.

In actual dual-rotor equipment, since the inner and outer rings of intershaft bearings are rotating, there is no fixed bearing housing for the installation of sensors, which must require indirect measurement of vibration. The indirectly measured vibration response needs to be transmitted to the adjacent bearing via the shaft and then to sensor position through the elastic support and complicated thin-wall paths so that the intershaft bearing signal is buried in the strong vibration and interference.

Similar to the actual dual-rotor equipment, #1 intershaft bearing, in the dual-rotor test rig, has not directly linked bearing housing. Thus, although the inner and outer races of #1 intershaft bearing are machined defectively, the accelerometer is installed at #2 bearing housing. Figure

Transfer path of vibration signal of #1 intershaft bearing.

The faults of inner and outer races of the intershaft bearing were simulated as follows. As for the outer race defect, a 1 mm deep and 1 mm wide groove was machined across the axis direction on the inner surface of the outer race. The outer surface of the inner race was also machined with the same size defects to simulate inner race fault.

The vibration signal is acquired by the BK4519 acceleration sensor, a key phase sensor, charge amplifier, and LMS SCADAS [

The 1# intershaft bearing fault vibration data used in this study are divided into the following two groups: (1) outer race fault signal where the inner race speed is increased from 625 r/min to 766 r/min; the sampling frequency is 25.6 kHz, and the number of vibration signals sampled for analysis is 102400; (2) inner race fault signal where rotational speed of inner race increased from 738 r/min to 825 r/min; sampling frequency is 102.4 kHz, and the amount of sample is 204800. The vibration waveforms and speed trends are shown in Figures

Fault in the outer race: (a) waveform of vibration; (b) speed trend.

Fault in the inner race: (a) waveform of vibration; (b) speed trend.

In the experiment, geometrical parameters of the faulty bearing are listed in Table

Geometrical parameters of 1# intershaft bearing.

Pitch diameter D (mm) | Ball diameter d (mm) | Contact angle | Number of rolling elements |
---|---|---|---|

125 | 8 | 0 | 34 |

Fault characteristic orders of 1# intershaft bearing.

Component | Outer race | Inner race | Rolling elements | Cage |
---|---|---|---|---|

Order | 15.9 | 18.1 | 7.8 | 0.5 |

The proposed method is first utilized to extract the faulty feature of 1# intershaft bearings with an outer race fault. As presented in Figure

The angular signal of outer race fault.

PSO is utilized to analyze the outer race fault signal in Figure

The trend of local energy ratio with evolution algebra.

Parameters setting and optimization results of PSO.

Parameters | ||
---|---|---|

Range of values | [0, 2000] | [ |

Maximum evolution | 50 | |

Maximum population | 10 | |

Local search speed | 1 | |

Global search speed | 1.2 | |

Optimization results | 65 | 3 |

The angular signal is decomposed by VMD, where the penalty factor and decomposition numbers are artificially set to be 65 and 3, respectively. Table

Fitness values obtained by analyzing the outer race fault signal with VMD.

Modes obtained by VMD and their order spectra | Fitness (%) | |
---|---|---|

5.26 | ||

3.2 | ||

6.28 |

Therefore, mode 3, as depicted in Figure

The denoised signal of outer race fault.

Modes obtained by VMD are continued to be used to construct the MCM, in the detection of the outer-race faulty bearing based on CS. The study selects the number of measurements to be 1500. Next, the amount of the analyzed signal is compressed to 1500 sampling points assisted by MCM and GRM, respectively.

Based on the constructed MCM, the fault feature is detected by orthogonal matching pursuit with the help of the Fourier dictionary and setting sparsity

The extracted result of outer race fault using CS-based on MCM: (a) the detected sparse coefficients; (b) the visualized envelope order spectrum.

Furthermore, the Fourier dictionary is used to transform the detected result in Figure

To fully validate the effectiveness of the proposed fault feature extraction strategy, 1# intershaft bearing with a faulty inner race was employed. Firstly, the raw vibration signal of inner race fault in Figure

The angular signal of inner race fault.

PSO is utilized to analyze the inner race fault signal in Figure

The trend of local energy ratio with evolution algebra.

Parameters setting and optimization results of PSO.

Parameters | ||
---|---|---|

Range of values | [0, 2000] | [ |

Maximum evolution | 50 | |

Maximum population | 10 | |

Local search speed | 1 | |

Global search speed | 1.2 | |

Optimization results | 109 | 5 |

Five modes are obtained by VMD decomposition with penalty factor 109 and the decomposition numbers 5, which are prepared for denoising and constructing the MCM. Among the modes, mode 2 is the best component corresponding to the maximum energy ratio, so mode 2 is the main mode (Table

Fitness values obtained by analyzing inner race fault signal with VMD.

Modes obtained by VMD and their order spectra | Fitness (%) | |
---|---|---|

0.53 | ||

4.29 | ||

1.35 | ||

1.37 | ||

0.77 |

The denoised signal of inner race fault.

Next, the MCM and GRM are used as a measurement matrix to acquire low-dimensional signals with the measurement numbers set to 1500, respectively. When the sparsity is determined as 4, the coefficients corresponding to the faulty feature can be detected from the compressed signal by orthogonal matching pursuit with the sensing matrix constructed by MCM and Fourier dictionary. Next, the detected result, as shown in Figure

The extracted result of inner race fault using CS-based on MCM: (a) the detected sparse coefficients and (b) the visualized envelope order spectrum.

To further highlight the high sensing efficiency of MCM and the advantages of MCM-CS in signal compression and feature extraction, the GRM-based CS (GRM-CS) is selected as a comparison. The following steps can be used:

Signal preprocessing and sparsity enhancement: the same is as in Sections

Compressive sensing: the GRM is utilized to reduce the dimensions of the denoised signal, and the Fourier dictionary and GRM make up the sensing matrix; finally, the sparse vector is searched by orthogonal matching pursuit.

Fault feature extraction: the signal recovery can be achieved using the Fourier dictionary and the sparse vector. Because of amplitude modulation of the recovered signal

Thus, the envelope order spectrum is obtained, as shown in Figures

The envelope order spectrum of outer race fault using GRM-CS.

The analysis results for fault data of outer race by using NEA based on SK: (a) kurtogram; (b) envelope order spectrum.

Figure

Figure

The envelope order spectrum of inner race fault using GRM-CS.

The analysis of experimental data proves that the proposed MCM-based CS method has higher sensing efficiency than GRM-based CS again. Under the same measurement number, the former can successfully extract the weak fault feature of rolling bearings.

To assess the value of the proposed method properly and show that its novelty can be compensated by performance, following the steps in Section

Figure

Similarly, Figure

The analysis results for fault data of inner race by using NEA based on SK: (a) kurtogram; (b) envelope order spectrum.

It can be seen that the order of prominence in Figures

In this paper, a VMD-based CS method is proposed to extract the weak fault feature of rolling element bearings under variable speeds to overcome the inadequate sensing efficiency. First of all, order tracking is employed to eliminate the nonstationarity of the signal caused by the speed change. Secondly, through VMD, the angular vibration signal from the previous step can be decomposed into a series of modes, before which Particle Swarm Optimization algorithm is involved in the optimization of VMD parameters to retain bearing fault signals as much as possible. The maximum energy ratio mode is a great substitute for the next compression detection, because of the enhanced sparsity. In the application of CS, the mode matrix is used to construct MCM as the measurement matrix to compress the denoised signal. Based on the compressed signal, the orthogonal matching pursuit is employed to detect the sparse coefficients corresponding to fault feature where the Fourier dictionary is selected to construct sensing matrix and the sparsity is determined as 4 by signal feature. Finally, for feature visualization, the recovered signal is obtained from the detected result using the Fourier dictionary, and the demodulation is employed to extract the fault feature which characterizes the performance of rolling element bearings. The simulated signals and experimental data are utilized to validate that the sensing efficiency of the proposed method is significantly promoted. Despite the strong background noise, the proposed scheme can successfully extract the weak fault feature of rolling element bearings operating at variable speed.

The MAT data used to support the findings of this study are available from the corresponding author upon request.

The authors declare no conflicts of interest.

This work was supported by the NSFC-Liaoning Joint fund under Grant no. U1708257; Postdoctoral Innovation Talent Support Program (Grant no. BX20180031); and Fundamental Research Fund for the Central Universities (Grant no. JD1913).