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Sparse signal representations attract much attention in the community of signal processing because only a few coefficients are required to represent a signal and these coefficients make the signal understandable. For bearing faults’ diagnosis, bearing faults signals collected from transducers are often overwhelmed by strong low-frequency periodic signals and heavy noises. In this paper, a joint signal processing method is proposed to extract sparse envelope coefficients, which are the sparse signal representations of bearing fault signals. Firstly, to enhance bearing fault signals, particle swarm optimization is introduced to tune the parameters of wavelet transform and the optimal wavelet transform is used for retaining one of the resonant frequency bands. Thus, sparse wavelet coefficients are obtained. Secondly, to reduce the in-band noises existing in the sparse wavelet coefficients, an adaptive morphological analysis with an iterative local maximum detection method is developed to extract sparse envelope coefficients. Simulated and real bearing fault signals are investigated to illustrate how the sparse envelope coefficients are extracted. The results show that the sparse envelope coefficients can be used to represent bearing fault features and identify different localized bearing faults.

Rolling element bearings are commonly used in machines to support rotation shafts. Their failures may cause unexpected machine breakdown and lead to huge economic loss. A rolling element bearing consists of an outer race, an inner race, several rollers, and a cage. Once a defect is formed on the surface of either the outer race or the inner race, an impact is generated by each of the rollers striking the defect surface and thus it excites the resonant frequencies of the structures between bearings and transducers [

In this paper, a joint signal processing method for extraction of sparse envelope coefficients is proposed. Firstly, an optimal wavelet filter is tuned by particle swarm optimization (PSO). For the use of wavelet transform, the similarity between a signal and a wavelet is the most concerned and the high similarity can result in large wavelet coefficients so as to highlight hidden transients. Because the shape of a Morlet wavelet is similar with the transients caused by localized bearing faults and the Morlet wavelet has a band-pass property, which can be used to retain one of the resonant frequency bands and enhance the signal to noise ratio of bearing fault signals, the Morlet wavelet is chosen in this paper [

Secondly, even though the optimal wavelet filtering is conducted on bearing fault signals for enhancement of bearing fault signatures, in-band noises still exist because the optimal wavelet filtering cannot remove the noises existing in the retained resonant frequency band. In recent years, an attracting method called morphological analysis (MA) is widely investigated due to its simplicity and effectiveness in extracting envelope signals [

The rest of this paper is outlined as follows. Section

Wavelet transform aims to calculate the inner product between an artificial wavelet and a signal. The mathematical formula for wavelet transform is defined as follows [

From (

It is not difficult to verify that if

PSO is a population based stochastic optimization method, which optimizes a metric by iteratively moving a number of particles in a searching space, according to some simple mathematical formulas related to the positions and velocities of all particles. Each particle represents one potential solution to the optimization problem. The movements of the particles are guided by their local best positions and the best swarm position. The basic theory of PSO is described in the following [

Then, in order to prevent the velocities from getting out of control, the influence of friction is considered by introducing an inertia weight

At last, (

MA aims to extract the morphological shape of a temporal signal. It uses a structuring element to intersect with the temporal signal. Let

The erosion operator reduces the wave peaks and enlarges the signal minima. On the contrary, the dilation operator increases the wave valleys and enlarges signal maxima [

The ability of the average operator lies in flattening both the positive and negative impulsive features. On the contrary, the difference operator is used to extract both the positive and negative impulsive features. The Black Top-Hat and the White Top-Hat operators are employed to extract the negative and positive impulsive features, respectively.

It is not difficult to find that the morphological features extracted by using MA fully depend on the shape of a temporal signal. Because of the interruption from strong low-frequency periodic components and heavy noises, the morphological features of bearing fault signals are prone to be overwhelmed. Therefore, it is necessary to enhance weak bearing fault signals prior to the use of MA. As illustrated in the previous sections, the complex Morlet wavelet optimized by PSO is used to preprocess bearing fault signals and an adaptive MA is developed to postprocess bearing fault signals and to extract sparse envelope coefficients for exhibiting bearing fault features. The flowchart of the proposed method is shown in Figure

The flowchart of the developed method for extracting sparse envelope coefficients.

Load an original bearing fault vibration signal, which is collected by using a transducer attached to a bearing housing.

The parameters of the complex Morlet wavelet are tuned by PSO. First, the parameters of particle swarm optimization must be initialized. Each particle corresponds to two-dimensional coordinates, which are used to represent the center frequency and bandwidth of the complex Morlet wavelet, respectively. The searching range for the center frequency was set to the frequency band ranging from

Once the weak bearing fault signal is enhanced by the optimal complex Morlet wavelet, MA is performed to get sparse envelope coefficients. As introduced in Section

The signals obtained by using various morphological operators: (a) the signal filtered by the erosion operator; (b) the signal filtered by the dilation operator; (c) the signal filtered by opening operator; (d) the signal filtered by the closing operator; (e) the signal filtered by the AVG operator; (f) the signal filtered by the DIF operator; (g) the signal filtered by the BTH operator; and (h) the signal filtered by the WTH operator.

Reference [

The bearing fault period

If the difference between the local maxima number

Locate all local maxima and denote them by

Assume that all the calculated distances are subject to a normal distribution

Find the abnormal distances, which are denoted by

Repeat the iterative local maximum selection method until all distances are within two standard deviations. Finally, sparse envelope coefficients are extracted.

In the first case study, a simulated signal is used for analyses. The simulated signal contains the impulses with an exponential decay, two sinusoidal signals, and noises. Here, the two sinusoidal signals can be regarded as two strong interruptions caused by two low-frequency components:

The simulated temporal signals: (a) the pure simulated signal consisting of impulses and two sinusoidal signals; (b) the noise signal; and (c) the pure simulated signal mixed with noises.

The proposed method is employed to analyze the mixed signal. First, the optimal parameters of the complex Morlet wavelet are automatically determined by using PSO with the maximum sparsity measurement. The optimal center frequency, the optimal bandwidth, and the frequency spectrum of the complex Morlet wavelet are shown in Figure

Frequency spectra: (a) the frequency spectrum of the mixed signal; (b) the frequency spectrum of the optimal complex Morlet wavelet; and (c) the frequency spectrum of the signal filtered by the optimal complex Morlet wavelet transform.

The signals obtained by the optimal complex Morlet filtering: (a) the real part of the filtered signal; (b) the power spectrum of the envelope of the filtered signal.

The global best values at different iterations by using PSO.

At last, the morphological analysis with the closing operator is employed to extract the cyclic bearing fault characteristics. The obtained temporal signal by morphological analysis is given in Figure

The results obtained by morphological analysis: (a) envelope extraction of the signal shown in Figure

In this paper, the real motor bearing data picked up with a sampling frequency of 12 k Hz by an accelerometer at the drive end of the motor housing are used to validate the proposed method. A single-point defect was, respectively, introduced to the outer race and inner race of a normal bearing using electrodischarge machining. The fault diameter is 0.007 inches and the fault depth is 0.0011 inches. The motor load was 0 HP and the motor speed was 1797 rpm [

The original outer race fault signal and its frequency spectrum are shown in Figures

The signals: (a) the real outer race fault signal; (b) the frequency spectrum of the original outer race fault signal; (c) the frequency spectrum of the optimal complex Morlet wavelet; and (d) the frequency spectrum of the filtered signal.

The signals obtained by the optimal complex Morlet filtering: (a) the real part of the filtered signal; (b) the power spectrum of envelope of the filtered signal.

The global best values at different iterations by using PSO.

Finally, morphological analysis is used to modify the shape of the signal shown in Figure

The results obtained by morphological filtering: (a) envelope extraction of the signal shown in Figure

The same procedure is applied to process the inner race fault signal shown in Figure

The signals: (a) the real inner race fault signal; (b) the frequency spectrum of the real inner race fault signal; (c) the frequency spectrum of the optimal complex Morlet wavelet; and (d) the frequency spectrum of the filtered signal.

The signals obtained by the optimal complex Morlet filtering: (a) the real part of the signal filtered by the optimal Morlet wavelet transform; (b) the power spectrum of the envelope of the filtered signal.

The global best values at every iteration by using particle swarm optimization.

The optimal process for the length selection of the flat structuring element is shown in Figure

The results obtained by morphological filtering: (a) envelope extraction of the signal shown in Figure

This paper reported a method which was used to extract sparse envelope coefficients for exhibiting bearing fault features. The proposed method consisted of two steps. Firstly, a Morlet wavelet was optimized by particle swarm optimization and then sparse wavelet coefficients were extracted from bearing fault signals. Even though sparse wavelet coefficients were useful to represent bearing fault signatures, in-band noises still existed and could not be removed by using the solely optimal wavelet transform. In this step, the sparse wavelet coefficients were not succinct enough. Secondly, to reduce in-band noises, an adaptive morphological analysis with an iterative local maximum detection method was developed to postprocess the signal obtained by the optimal wavelet filtering. The optimal flat length used in morphological analysis with the closing operator was automatically determined to retain sparse envelope coefficients and remove the pseudo- and abnormal peaks caused by unexpected noises. After the above two steps were conducted, the sparse signal representation of bearing fault signals was obtained; only a few coefficients were used to represent bearing fault features. These sparse coefficients made bearing fault signals succinct and understandable. The case studies were conducted to illustrate how the proposed method worked and the results showed that the proposed method can be used to extract the sparse representation of bearing fault signals and identify different localized bearing faults.

In the future, the following works will be conducted. Firstly, more statistical metrics will be used as objective functions so as to guide optimization of wavelet transform. Secondly, besides the popular Morlet wavelet, more wavelets that are highly similar with impulses caused by bearing defects will be investigated. Thirdly, different optimization methods will be used to find optimal wavelet parameters and their comparisons will be made. Fourthly, even though wavelet transform is effective in extracting bearing fault features, in-band noises still exist. Morphological analysis with different structure elements and operators will be studied to make bearing fault features sparse and succinct. At last, based on bearing fault features extracted by the suggested steps, intelligent bearing fault diagnosis methods will be developed to automatically identify different bearing faults without requirement of expertise.

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

This research was partially supported by the National Natural Science Foundation of China (Grants nos. 51505311 and 51505307), the Science and Technology Foundation of Zhongshan City (Grant no. 20123A338), and the Natural Science Foundation of Jiangsu Province (Grant no. BK20150339). The authors would like to thank Professor K. A. Loparo for his permission to use the bearing data to validate the proposed method.