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Bearing fault signal analysis is an important means of bearing fault diagnosis. To effectively eliminate noise in a fault signal, an adaptive multiscale combined morphological filter is proposed based on the theory of mathematical morphology. Both simulation and experimental results show that the adaptive multiscale combined morphological filter can remove noise more thoroughly and retain details of the fault signal better than the dual-tree complex wavelet filter, traditional morphological filter, adaptive singular value decomposition method (ASVD), and improved switching Kalman filter (ISKF). The adaptive multiscale combined morphological filter considers both positive and negative impulses in the signal; therefore, it has strong adaptability to complex noise in the environment, making it an effective new method for bearing fault diagnosis.

Bearings are important components in rotating mechanical equipment, and their operating condition affects the overall working state of mechanical equipment. A bearing is prone to defects after prolonged use, and these defects can be useful during bearing fault diagnosis [

The mathematical morphology filter, developed on the basis of mathematical morphology transformation, is an effective method of analyzing nonlinear signals. The mathematical morphology filter selects appropriate structuring elements to decompose the signal into several components, according to the geometric characteristics of the signal. Even if there is strong noise or distortion in the signal, the basic morphological characteristics of the signal can be preserved after the signal is filtered. Compared with other signal analysis methods, the mathematical morphology filter, which has been widely used in pattern recognition, image processing, computer vision, power signals, ECG and EEG signal processing, mechanical equipment fault diagnosis, and other fields, has the advantages of easy implementation, fast algorithms, and minimal calculations [

An adaptive multiscale combined morphological filter is developed based on the concept of multiscale overall filtering to overcome the shortcomings of the traditional morphological filter. Both simulation and experimental results show that the adaptive multiscale combined morphological filter has a better denoising effect and can retain useful signals better than the traditional filter.

Dilation and erosion are the two basic operators of mathematical morphology. The collected vibration signal of the bearing is a one-dimensional dispersed signal, so this paper only introduces a one-dimensional dispersed gray-value morphological transformation.

The dilation of

The erosion of

The dilation operation is equivalent to the maximum value filtering of the disperse function in the sliding filter window (structuring element), which widens the peak value as well as increases the valley value of the signal. The erosion operation is equivalent to the minimum value filtering of the disperse function in the sliding filter window (structuring element), which widens the valley value as well as increases the peak value of the signal. Dilation and erosion, as two basic operations, can be used to form opening operations and closing operations, and their expressions are as follows:

The opening operation

The closing operation

From Equation (

Selection of the structuring element used in morphological transformation is also very important because different structuring elements have different effects on signal processing. Several common types of structuring elements are flat, circular, cosine, triangular, and curved. The flat structuring element has the following advantages: the amplitude is zero, the calculation is simple and fast, the signal processing efficiency is high, and the processing effect can be guaranteed. Therefore, in this paper, a flat structuring element is selected for the morphological filtering of the bearing fault signal.

Because the dilation, erosion, opening, and closing operations can only filter one of the positive and negative impulse noise in the signal, in many cases, the opening and closing operations are combined to construct the morphological closing-opening filter and morphological opening-closing filter, which are as shown in Equation (

Morphological closing-opening and opening-closing filters combine two kinds of operations (opening and closing), which can weaken both positive and negative impulse noise in the signal, but they cannot avoid the statistical bias of the filtering results caused by the dilatability of the closing operation and the erodibility of the opening operation. To achieve a better filtering effect, the combined morphological filter is usually constructed by combining morphological closing-opening and opening-closing filters, as shown in Equation (

Originally, multiscale morphology was established to enrich the practicability of morphology in the field of image processing and shorten the operation time when the size of the structuring element is large, which is based on decomposing morphological structuring elements. When applied to the research of a one-dimensional vibration signal, the structuring element of the multiscale morphological filter can be obtained by corresponding time dilations of the unit structuring element itself.

If the structuring element of scale

The dilation and erosion operations of signal

Similarly, the multiscale morphological opening, closing, opening-closing, and closing-opening operations of signals are as shown in Equations (

The environment and working conditions of the bearing are often complex and changeable; consequently, noise in the collected signal of the bearing is also complex and diverse; thus, it is difficult to achieve an ideal filtering effect by using a single-scale structuring element. To enable the morphological filter to deal with noise in various situations, multiple groups of structuring elements with different scales are adopted to carry out morphological closing-opening and opening-closing combined filtering for a signal based on the concept of overall multiscale filtering. In addition, structuring elements with different scales are used successively in the same group. The final scale signal is obtained by adaptive weighting combination for each group of operation results. The principle is as follows:

In Equation (

The final signal filtered by

It can be seen that the morphological closing-opening (opening-closing) operation results from large-scale structuring elements have a large proportion of weight in the final signal, and the morphological closing-opening (opening-closing) operation results from small-scale structuring elements have a small proportion of weight, which considers denoising and protecting details under the premise of giving first priority to denoising.

The process of the proposed morphological filter is as follows:

Read the signal

Calculate the value of

Obtain the width

According to Equation (

Implement a fast Fourier transform of

Output

To test the effect of the proposed adaptive multiscale combined morphological filter, a simulation signal is created for filtering research, which is shown in Equation (

In Equation (

The amplitude of the flat structuring element is 0. According to [

The waveform and frequency spectrum of the simulated signal are shown in Figure

Simulated signal and its frequency spectrum. (a) The waveform of the simulated signal. (b) The frequency spectrum of the simulated signal.

The waveform and frequency spectrum of the simulated signal denoised by the dual-tree complex wavelet are shown in Figure

Waveform and frequency spectrum of the simulated signal denoised by the dual-tree complex wavelet. (a) Waveform. (b) Frequency spectrum.

The waveform and frequency spectrum of the simulated signal denoised by the traditional morphological filter with small-scale structuring elements are shown in Figure

Waveform and frequency spectrum of simulated signal denoised by traditional morphological filter with small-scale structuring elements. (a) Waveform. (b) Frequency spectrum.

The waveform and frequency spectrum of the simulated signal denoised by the traditional morphological filter with large-scale structuring elements are shown in Figure

Waveform and frequency spectrum of simulated signal denoised by traditional morphological filter with large-scale structuring elements. (a) Waveform. (b) Frequency spectrum.

The waveform and frequency spectrum of the simulated signal denoised by the adaptive multiscale combined morphological filter are shown in Figure

Waveform and frequency spectrum of the simulated signal denoised by the adaptive multiscale combined morphological filter. (a) Waveform. (b) Frequency spectrum.

The bearing fault signal is collected through an experiment involving a two-stage transmission gearbox whose structure is shown in Figure

Structure diagram of the tested gearbox.

The width of the structuring element is chosen as

The waveform and frequency spectrum of bearing

Waveform and frequency spectrum of bearing

The waveform and frequency spectrum of bearing

Waveform and frequency spectrum of bearing

The waveform and frequency spectrum of bearing

Waveform and frequency spectrum of bearing

The waveform and frequency spectrum of bearing

Waveform and frequency spectrum of bearing

The waveform and frequency spectrum of bearing

Waveform and frequency spectrum of bearing

In recent years, Qin et al. [

Waveform and frequency spectrum of bearing

Waveform and frequency spectrum of bearing

To address the insufficient denoising effects of traditional filters, an adaptive multiscale combined morphological filter is proposed, which is composed of morphological closing-opening and opening-closing operations. The advantages of the proposed filter are verified by simulations and experiments, as specifically manifested in the following three aspects:

The morphological filter proposed in this paper uses multiple groups of structuring elements with different scales to carry out morphological closing-opening and opening-closing combined filtering for signals. In addition, structuring elements with different scales are used successively in the same group. The final scale signal is obtained by a self-adaptive weighting combination of each group of operation results. The morphological closing-opening (opening-closing) operation results by large-scale structuring elements have a large proportion of weight in the final signal, and the morphological closing-opening (opening-closing) operation results by small-scale structuring elements have a small proportion of weight, which can take into account the denoising and protecting details of the signal under the premise of giving first priority to denoising

The proposed filter can filter out noise more thoroughly and extract pulse signals more effectively than dual-tree complex wavelets, ASVD and ISKF. Compared with the traditional morphological filter with small-scale structuring elements, which has an incomplete denoising effect, the proposed filter can filter out noise more effectively; compared with the traditional morphological filter with large-scale structuring elements, which loses part of the useful signal, the proposed filter can preserve the useful signal much better

The proposed filter takes into account both positive and negative pulses in the signal; therefore, it has strong adaptability to complex and variable noise in actual environments. Thus, the filter provides a new effective method for rotating machinery fault diagnosis

The data used to support this research can be obtained from the corresponding author on request.

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

The research has been supported by the National Natural Science Foundation of China (grant number 51305454).