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In order to solve the problem of underdetermined blind source separation (BSS) in the diagnosis of compound fault of roller bearings, an underdetermined BSS algorithm based on null-space pursuit (NSP) was proposed. In this algorithm, the signal model of faulty roller bearing is firstly used to construct an appropriate differential operator in null space. With the constructed differential operator, the mixed signals collected by the vibration sensor are decomposed into a series of stacks of narrow band signal containing the characteristics of faulty bearing. Finally, the underdetermined problem is transformed to an overdetermined problem by combining the narrow band signals and the original mixed signals into a new group of observed signals. In this way, the separation of the mixed signals can be realized. Experiments and engineering data analyses show that the problem of underdetermined BSS can be solved effectively by this approach, and then the compound fault of the roller bearing can be separated.

Rolling bearing is the most commonly used mechanical component in mechanical equipment and its operational status directly influences the performance of the entire machinery. In engineering practice, bearing faults may cause severe mechanical failures and huge economic losses; therefore, the fault monitoring and diagnosis of rolling bearing are of high significance. However, as a typical nonstationary signal, the vibration signal of rolling bearing contains a large amount of noises in engineering practice. The vibration signal analysis and treatment are a complex nonlinear problem. So far, many researches have been carried out on fault diagnosis of rolling bearing at home and abroad.

Fourier transform offers a bridge of the signals between the time domain and frequency domain. It is the dominant method of signal treatment. However, Fourier transform only provides the statistical average of the signals on the whole but not the local information; thus it cannot characterize the stochastic, nonstationary signals. Based on Fourier transform, the researchers suggest a great many of time-frequency analysis methods after strenuous efforts which include wavelet analysis [

On the basis of EMD algorithm, Peng and Hwang put forward a self-adaptive decomposition algorithm utilizing the local narrow band signals and operator theory that is the null-space pursuit algorithm (NSP) [

Peng and Hwang carried out an optimization of this algorithm so that the frequency-modulated local narrow band signals can effectively disappear in [

In recent years some researchers have introduced BSS into compound fault diagnosis of rolling bearings and gained some progress [

The paper is organized as follows. Section

As known from literature [

The vibration signals acquired by the acceleration sensor are expressed as

The basic valid component of the faulty bearing signals is the attenuated signal of impact as follows:

The vibration model can be approximated to a mass-spring-damping system. It is known that, for a mass-spring-damping system, its dynamic model is expressed as follows:

Through the above analysis, it can be obtained that the basic component

NSP algorithm is a self-adaptive signal decomposition method based on the operator. Due to the vibration model of valid signal components among all signals to be decomposed, an appropriate operator can be estimated. By locating the valid signals among the signals to be decomposed in the null space, the operator is used to extract the narrow band signals from the signals to be decomposed that are related to the vibration model of valid signals. Usually, the signal in the form of

It is seen from the above analysis that the construction of operator in NSP algorithm is the core. For the rolling bearing, the valid components of the fault signals of the rolling bearing are located in the null space of the following operator according to formula (

In the fault signal

Because

The procedures of NSP algorithm proposed for the operator are as follows.

The original signal

Let

The parameter

The leakage parameter

It is determined whether

Matrix

BSS algorithm is a method of signal processing used to estimate the unknown source signal and parameter of unknown hybrid channel through only the observed signals collected by the sensor. Currently, the most widely used methods of BSS include independent component analysis (ICA) and the approximate joint diagonalization algorithm based on eigenmatrix proposed by Cui et al. and Cardoso [

The procedure is shown as follows.

(1) Whitening process is performed on the observed signal

Here the superscript

The signal after whitening is the mixture of unitary matrix of the source signal. In this way, the problem of determining the mixing matrix

(2) The joint diagonalization process is performed on the whitened signal to determine the unitary matrix

Moreover, with the optimization, the matrix

In the calculating process, the quadratic sum of the diagonal components in matrix

The unitary matrix

According to the analysis of BSS and NSP, the mixed signals can be separated when the number of the original signals is larger than the observed signals, and they can be decomposed into a series of stacks of narrow band signal which contains the bearing fault information only if the operator of NSP is selected appropriately. Therefore, the mixed signals can be preprocessed by NSP algorithm and obtain several observed signals which are combined with the original observed signals to get a new group of observed signals. Thus, the number of the observed signals is larger than the original signals. Furthermore, the underdetermined problem is transformed into overdetermined problem to realize the separation of the bearing fault characteristics. The method is performed as follows.

The differential operator is constructed according to the vibration model of signal of faulty roller bearing.

With the constructed operator, the observed signals are decomposed into a series of stacks of narrow band signal which contains characteristics of the faulty bearing.

The above narrow band signals and the original observed signals are combined into a new group of observed signals.

The whitening matrix and the observed matrix after whitening are obtained by prewhitening the new observed signals.

The joint diagonalization based on eigenmatrix is performed on the signals obtained in step (4) to determine the unitary matrix

The source signal can be estimated on which the Hilbert process is conducted to obtain the characteristics of faulty bearings.

The algorithm flow is shown as Figure

Diagram of the underdetermined BSS algorithm based on NSP.

The experimental system consists of bearing test stand, HG3528A data acquisition instrument, and laptop. The test stand (shown as Figure

Test rig of fault bearing.

When there are faults on both the inner and outer ring of the bearing, the observed signal is shown as Figure

Signal waveform and frequency spectrum.

Narrow band signals after NSP processing.

Narrow band signals after NSP processing

The frequency spectra of narrow band signals after NSP processing

Signals after BSS and NSP processing.

Demodulation spectra after BSS and NSP processing.

In order to verify the effectiveness of the proposed method, it is applied to engineering practice.

Figure

Driving chain of gearbox.

The given condition is that the fault feature frequencies of outer ring, inner ring, and rolling element of the bearing measured (indicated by the arrow in Figure

First, the Hilbert demodulation is conducted to analyze the signal, and the obtained frequency spectrum is shown as Figure

Hilbert demodulation spectrum.

Spectrum with BSS based on NSP processing.

Pitting faults at the bearing of shaft I at the north of the speed-up gearbox were found as shown in Figure

Pitting faults at the bearing of shaft I at the north of the speed-up gearbox.

Inner-race pitting

Outer-race pitting

The mixed signals can be separated effectively by the traditional BSS only if the number of the observed signals is larger than the original signals. To solve the underdetermined problem in single channel bearing fault diagnosis, a new BSS algorithm based on null-space pursuit algorithm was proposed. The null-space pursuit algorithm is an adaptive signal decomposition algorithm which can separate mixed signals into several narrow band signals effectively with effective operator constructed. In this paper, firstly, the signal model of faulty bearing is analyzed and the appropriate operator is constructed according to the characteristics of faulty bearings. Secondly, the mixed signals are preprocessed by the operator constructed above to get a group of narrow band signals that contains fault features. Thirdly, a new group of observed signals are obtained by combining the narrow band signals with the original observed signals; thus the separation of the bearing fault characteristics is realized. Finally, the engineering signals and the experiment signals are used to validate the proposed method. The results show that the traditional underdetermined problem of BSS is solved and the bearing fault characteristics are separated successfully. Therefore, diagnosis of roller bearings compound fault using underdetermined blind source separation algorithm based on null-space pursuit is effective.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the National Natural Science Foundation of China (51175007 and 51075023).