Shock and Vibration

Volume 2019, Article ID 7475868, 20 pages

https://doi.org/10.1155/2019/7475868

## Adaptive Asymmetric Real Laplace Wavelet Filtering and Its Application on Rolling Bearing Early Fault Diagnosis

Department of Mechanical Engineering, North China Electric Power University, Baoding 071003, China

Correspondence should be addressed to Bo Peng; moc.361@obgnepupecn

Received 13 November 2018; Accepted 19 December 2018; Published 14 January 2019

Academic Editor: Hamid Toopchi-Nezhad

Copyright © 2019 Shuting Wan and Bo Peng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The early fault of rolling bearing is weak and may not be readily detected. To overcome this issue, the present paper comes up with a rolling bearing fault-diagnosing approach based on adaptive asymmetric real Laplace wavelet (ARLW) filtering, which is on the strength of water cycle optimization algorithm (WCA). Firstly, ARLW is introduced to filter the initial vibration signal since its waveform has the same asymmetric structure as the fault impact. Secondly, the optimum center frequency and bandwidth of ARLW is found out adaptively by applying the WCA through the proposed square envelope fault energy ratio (SEFER). Finally, envelope analysis is conducted to the narrowband signal obtained by the optimum ARLW filtering, and its envelope spectrum presents the rolling bearing fault characteristic frequency apparently. The proposed approach and two existing approaches are all tested in four signal analysis cases. The results are analyzed, and the conclusion is that the approach proposed by the present paper can detect the early fault of rolling bearing more accurately. The present research is valuable for diagnosing the early fault of rolling bearing.

#### 1. Introduction

Rolling bearing is broadly used in rotating equipment, and its fault acts on the safe operation of the whole equipment [1–3]. At the beginning of the rolling bearing fault, the impact component of vibration signal collected by the sensor is weak and often submerged in strong background noise, bringing challenges to the diagnostic process [4]. Consequently, diagnosing the early fault of rolling bearing acts as the focus and difficulty among researchers and scholars.

The periodic impact produced by partial defects on the roiling bearing surface can arouse the resonance between the rolling bearing and its adjacent parts [5]. Using resonance demodulation to extract impact response characteristics from vibration signal is a fast and simple approach of rolling bearing fault diagnosis, the pivotal step of which is to precisely identify the resonance frequency band that contains plentiful fault information. The traditional resonance demodulation approach has the limitation of the request of artificially presetting the parameters of the bandpass filter (center frequency and bandwidth).

Antoni et al. creatively put forward the spectral kurtosis theory [6, 7] and the fast spectrum kurtosis (FSK) approach [8] that can automatically set the parameters of the bandpass filter, which firstly used short time Fourier transform (STFT) or finite impulse response (FIR) filters to divide frequency bands and then used the kurtosis of signal as an evaluating indicator to identify the frequency band containing the most fault information. However, there are two disadvantages in the FSK approach. The first one is that kurtosis as an evaluation index cannot distinguish the random impact and periodic impact of the signal, which easily leads to identifying the erroneous resonance band. The second one is that the optimal frequency band selected by FSK may not include the whole resonant frequency band region. For the first disadvantage, several new evaluating indicators are raised to improve the accuracy of selecting the resonant frequency band, such as correlated kurtosis [9], harmonic-to-noise ratio [10], Gini index [11], and spectral L2/L1 [12]. In addition, the noise reduction approach can be used as a preprocessing method of FSK to avoid interference, such as the improved FSK approach with the aid of EEMD [13], ICA [14], and ITD [15]. Although the above approach improves the robustness of FSK, more prior knowledge is needed, and the calculation process is complex. For the second shortcoming, many scholars have done useful work. Lei et al. used the wavelet packet transform (WPT) instead of the STFT or FIR filters in the FSK, which segments the frequency band more finely [16–18]. Comparing with WPT, continuous wavelet transform (CWT) can flexibly and accurately divide the frequency domain. CWT segments the signal through constructing a series of filters with the same property but different center frequency (CF) and bandwidth (BW) by translating and stretching the mother wavelet. Thus, designing wavelet parameters properly to obtain the resonance frequency band is key to diagnosing rolling bearing fault by CWT. Qiu et al. [19] combines the Shannon entropy and SVD theory to realize the optimal wavelet transform. Bozchalooi et al. [20] uses the smoothing index to select the CF and BW of Gabor wavelet. The variable step size of the CF is 50 Hz, and the search range is [200, 6000]. The variable step size of the BW is 0.01, and the search range is [0.01, 1]. That is to say, we need to calculate 11700 (117 × 110) times to determine the best solution. It is obvious that intelligent algorithms should be used to replace the huge computation process and to reduce the computation time and memory footprint. Su et al. [21] puts forward an approach based on the minimal entropy criterion, using genetic algorithm to obtain the optimum wavelet parameters. Wang et al. [22] obtains the parameters of Morlet wavelet using simulated annealing algorithm that adopted the maximum sparsity as the fitness function. Chen et al. [23] used a particle swarm optimization algorithm on the basis of the correlated kurtosis of squared envelope spectrum as fitness function to obtain the parameters of Morlet wavelet.

At present, there are three main issues in using wavelet filtering to determine the resonance frequency band of rolling bearing: the selection of the wavelet basic function, the selection of evaluation index, and fast determination of wavelet parameters. Accordingly, this paper proposed an adaptive asymmetric real Laplace wavelet (ARLW) filtering approach based on water cycle optimization algorithm (WCA), which has the following improvements: ARLW is used as wavelet basis function; square envelope fault energy ratio (SEFER) is chosen as a new evaluation index; WCA is used to choose the optimum wavelet parameters quickly. Four signal analysis cases are conducted to testify the validity of the proposed approach, the results of which are compared to that of the two existing approach.

The rest of the present paper is structured as follows: Section 2 reviews the theoretical background of wavelet filtering. Section 3 discusses the wavelet parameter optimization process. Section 4 proposes the rolling bearing fault diagnosis approach on the basis of adaptive wavelet filtering. Section 5 verifies the proposed approach through four signal analysis cases. Section 6 summarizes the full text.

#### 2. Wavelet Filtering

##### 2.1. Continuous Wavelet Transform

The CWT of one-dimensional signal is defined aswhere represents the wavelet coefficient, *a* represents the scale parameter, *b* represents the shift parameter, *φ*(·) represents the wavelet basis function, and represents the conjugate. In the frequency domain, Equation (1) is expressed aswhere refers to the Fourier transform of *x*(*t*), *ψ*(f) refers to the Fourier transform of *φ*(t), and IFT refers to the inverse Fourier transform.

##### 2.2. Asymmetric Real Laplace Wavelet

In the fault diagnosis of the mechanical system, more fault features can be extracted when the wavelet function matches the impact response in the dynamic signal. Among the existing wavelet basic functions, the time domain waveform (TW) of ARLW and Morlet wavelet is similar to that of impact response caused by local fault of rolling bearing. Figures 1(a)–1(c) depict the TW of ARLW, Morlet wavelet, and the actual rolling bearing signal, respectively. Compared with the Morlet wavelet with symmetrical structure, the waveform of ARLW with asymmetrical structure is more similar to fault waveform, so ARLW is more suitable for extracting rolling bearing fault features. The ARLW *φ*(t) can be expressed as the following equation [24, 25]: