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The vibration signal measured from the mechanical equipment is associated with the operation of key structure, such as the rolling bearing and gear. The effective signal processing method for early weak fault has attracted much attention and it is of vital importance in mechanical fault monitoring and diagnosis. The recently proposed atomic sparse decomposition algorithm is performed around overcomplete dictionary instead of the traditional signal analysis method using orthogonal basis operator. This algorithm has been proved to be effective in extracting useful components from complex signal by reducing influence of background noises. In this paper, an improved linear frequency-modulated (Ilfm) function as an atom is employed in the proposed enhanced orthogonal matching pursuit (EOMP) algorithm. Then, quantum genetic algorithm (QGA) with the OMP algorithm is integrated since the QGA can quickly obtain the global optimal solution of multiple parameters for rapidly and accurately extracting fault characteristic information from the vibration signal. The proposed method in this paper is superior to the traditional OMP algorithm in terms of accuracy and reducing the computation time through analyzing the simulation data and real world data. The experimental results based on the application of gear and bearing fault diagnosis indicate that it is more effective than traditional method in extracting fault characteristic information.

According to the statistics, the fault derived from the mechanical parts, such as the shaft and the gear, mainly causes mechanical accidents. Commonly, these faults are often accompanied by nonlinear vibration phenomenon and its vibration signals are often complicated as the strong background noise [

In view of nonstationary and nonlinear signal processing, many studies have been implemented. Gabor [

The most traditional signal analysis methods are based on the inner product operation using orthogonal basis. Thus, it tries to use a fixed base function or the same base function of the properties that represents arbitrary signals and ignores the characteristics of the signal itself. Moreover, the energy of the decomposed signal will be distributed on different bases because of the orthogonality. This characteristic is bad for the signal recognition and compression. In order to achieve the fact that the representation of signal is self-adaptive, more flexible, and concise, Mallat and Zhang [

In view of the sparse theory, scholars have done some researches of fault diagnosis. Candes et al. [

In this paper, the enhanced orthogonal matching pursuit (EOMP) algorithm firstly uses an improved linear frequency-modulated (Ilfm) function as atoms, which can improve the precision and computing efficiency of reconstructing signal. Then it combines quantum genetic algorithm (QGA) with the OMP algorithm, as the QGA can quickly solve multiple parameters of the global optimal solution problem. It can further increase the calculation efficiency of the OMP algorithm. Thus, fault features extraction from vibration signal can be recognized. The EOMP algorithm is superior to traditional OMP algorithms on the accuracy and computation efficiency in the analysis of simulation data and experimental signal. It has been proved that the EOMP algorithms are more feasible and effective in real mechanical equipment fault diagnosis.

The MP algorithm belongs to the greedy iterative algorithm. In each iteration, it selects atoms from overcomplete dictionary that best matches with the structure of source signal. Then it requires only a small part of the atoms to accurately represent the source signal until the convergence by multiple iterations. The OMP algorithm is proposed by Pati et al. [

Given a source signal

If the energy of the residual

For the MP algorithm, after

However, in the OMP algorithm, the selected atom

Thus, the signal

In essence, the OMP algorithm is also greedy iterative algorithm and requires a lot of inner product operations to obtain the best atom in each decomposition, so the computation burden is too heavy. The EOMP method introduces quantum genetic algorithm into the OMP algorithm. It firstly codes the parameters of dictionary of atoms in the way of quantum bit and normalizes these parameters of atoms. Secondly, it makes the inner product of the atom and the signal as the fitness function. Then, in each iteration, it makes the best solution in the process of population evolution instead of the best solution in the contemporary evolution to update quantum gate. Finally it gets the optimal value of each parameter and the best atoms. In the end, the required signal is obtained by repeating the above iterative algorithm.

Quantum Genetic Algorithm (QGA), proposed by Han and Kim [

The rotation angle of quantum gate is generally fixed and can be obtained from the table. To address this problem, a novel quantum genetic algorithm is proposed by Zhang et al. [

In addition to improving the process of the OMP algorithm, researchers have been the subjects of intense research for selection of the dictionary. Studies show that the dictionary has a great influence on the performance of the OMP algorithm [

A novel dictionary is recently proposed by Nagaraj et al. in the analysis of EEG signals [

This function of Ilfm dictionary is defined as

In this paper, the simulation platform is listed as follows: CPU (i7-4770k), Memory (8 G), and SSD (128 G). The simulation software is MATLAB. In order to test the effectiveness and accuracy of the EOMP method, the following evaluation indicators are introduced in this paper.

signal-to-error ratio (SER)

comparability index (

mean square error (MSE)

The simulated signal model is described in (

The waveform of simulated signal: (a) pure signal waveform

In order to achieve a comprehensive analysis of EOMP method, Gabor dictionary and Ilfm dictionary are used to decompose and reconstruct the same simulated signal in (

Gabor dictionary and Ilfm dictionary are used to decompose and reconstruct the simulated signal of (

Gabor dictionary and Ilfm dictionary are used to decompose and reconstruct the simulated signal of (

Gabor dictionary and Ilfm dictionary are used to decompose and reconstruct the simulated signal of (

It should also be noted that the simulated noisy signal of (

In Figure

Simulated signal analysis results: Gabor dictionary and Ilfm dictionary are used to match the simulated signal 100 times in the OMP algorithm, the combinations of the QGA and the OMP algorithm, and the combinations of the IQGA and the OMP algorithm: (a) SER is changed with the iterations (the higher value shows the faster decomposition), (b)

The above simulation results show the EOMP method has a high degree of accuracy in reconstruction of signal. In order to further explore the performance of this algorithm, it is used to reconstruct the useful signal from the noisy signal under the condition of different strength of the noise, namely, the influence of noise. Adopting the simulated signal of (

In Figures

Simulated signal analysis results from the proposed method: (a) waveform of the optimum reconstructed signal under the noise amplitude of 0.3, (b) waveform of the optimum reconstructed signal under the noise amplitude of 0.5, (c) waveform of the optimum reconstructed signal under the noise amplitude of 0.7, (d) waveform of the optimum reconstructed signal under the noise amplitude of 0.9, (e)

In the previous section, the EOMP method can extract the useful signal from the noisy signal. Furthermore, we explore whether or not it extracts the useful signal from the complicated signal for feature extraction in mechanical fault diagnosis.

In mechanical fault, effective extraction is critical to fault diagnose, and the impact signal has a high proportion. The accuracy of the feature extraction depends on whether or not we can find an appropriate form of basis functions (atoms). However, the actual mechanical fault signals are usually consisted by mixed sine signals and the transient impulse signal, and the effect of only using the single basis function for feature extraction is not ideal. If we can find a kind of basis functions (atoms) that can match sine component and transient impulse component, this problem can be solved.

Theoretically, the basis function (atom) of (

The simulated signal model is described in (

Simulated signal analysis results from the EOMP method: (a) waveform and FFT spectrum of the simulated signal

In Figure

In order to test effectiveness of the EOMP method in the actual mechanical equipment fault diagnosis, the gear fault signal and the rolling bearing fault signal from the fault diagnosis test-bed are used to decompose and reconstruct the fault signal for obtaining the fault feature frequency components. FFT spectrum and Hilbert spectrum as traditional analysis method are used in this part.

The bearing fault data are obtained from the bearing fault test-bed in

Rotating frequency (

Defect frequency on inner ring (

Defect frequency on outer ring (

Defect frequency on rolling element (

Defect frequency on cage train (

According to (

Rotating | Inner ring | Outer ring | Rolling element | Cage train |
---|---|---|---|---|

29.2 Hz | 158.12 Hz | 104.68 Hz | 137.63 Hz | 11.63 Hz |

In Figure

Bearing fault signal analysis results in the EOMP method: (a) waveform of the source signal and waveform of the reconstructed signal, (b) FFT spectrum of the source signal and FFT spectrum of the reconstructed signal, and (c) Hilbert spectrum of the source signal and Hilbert spectrum of the reconstructed signal.

The gear fault data are obtained from the gear diagnosis fault test-bed, and this equipment is shown in Figure

Rotating frequency (

Natural frequency of gear (

According to (

In Figure

Gear fault signal analysis results from the EOMP method: (a) waveform of the source signal and waveform of the reconstructed signal, (b) FFT spectrum of the source signal and FFT spectrum of the reconstructed signal, and (c) Hilbert spectrum of the source signal and Hilbert spectrum of the reconstructed signal.

An improved orthogonal matching pursuit algorithm is proposed in this paper. The main research work can be summarized as the following aspects:

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (no. 51475339) and the Natural Science Foundation of Hubei province (no. 2016CFA042).