Mathematical Problems in Engineering

Volume 2018, Article ID 6513045, 12 pages

https://doi.org/10.1155/2018/6513045

## Compound Fault Diagnosis of Bearings Using an Improved Spectral Kurtosis by MCDK

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

Correspondence should be addressed to Xiong Zhang; nc.ude.upecn@xzxjdh

Received 8 December 2017; Revised 8 January 2018; Accepted 7 February 2018; Published 19 March 2018

Academic Editor: Aimé Lay-Ekuakille

Copyright © 2018 Shuting Wan et al. 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 fast spectrum kurtosis (FSK) algorithm can adaptively identify and select the resonant frequency band and extract the fault feature by the envelope demodulation method. However, in practical applications, the fault source may be located in different resonant frequency bands; plus in noise interference, the weak side of the compound fault is not easy to be identified by the FSK. In order to improve the accuracy of fast spectral kurtosis analysis method, a modified method based on maximum correlation kurtosis deconvolution (MCKD) is proposed. According to the possible fault characteristic frequencies, the period of MCKD is calculated, and the appropriate filter length is selected to filter the original compound fault signal. In this way, the compound fault located in different resonance bands is separated. Then, the signal after MCKD filtering is analyzed by FSK. Through the simulation and experimental analysis, the MCKD can separate the compound fault information in different frequency band and eliminate the noise interference; the FSK can accurately identify the resonance frequency and identify the weak fault characteristics of compound fault.

#### 1. Introduction

The fault characteristics of rotating machinery have certain regularity and periodicity. The application of vibration signal and acoustic signal to the fault feature extraction of rotating machinery fault is of great value [1]. Rolling bearing is one of the most widely used parts in mechanical equipment. Its operation condition affects the working state of the whole system [2–5]. The fault identification and diagnosis of rolling bearing are of great significance to ensure the safe and reliable operation of mechanical equipment. Under the actual conditions, the failure of rolling bearing usually manifests itself as compound failure, and due to the influence of operating environment, the interaction between multiple noise source and compound fault is often presented. The separation of compound fault components from strong background noise is a difficult problem in the field of mechanical fault diagnosis [6–8]. Hemmati et al. proposed a modified and effective signal processing algorithm to diagnose localized defects on rolling element bearings components under different operating speeds, loadings, and defect sizes [9]. The algorithm was based on optimizing the ratio of kurtosis and Shannon entropy to obtain the optimal band pass filter utilizing wavelet packet transform and envelope detection. The performance of ELMD often heavily depends on proper selection of its model parameters; to overcome this shortcoming, Zhang et al. propose an optimized ensemble local mean decomposition method to determinate an optimum set of ELMD parameters for vibration signal analysis [10].

The concept of spectrum kurtosis is presented by Dwyer. The basic principle is to calculate the kurtosis value of each spectral line, and the value of different kurtosis will respond to the transient impact size. Antoni and Randall used the spectral kurtosis value as a short time Fourier window function and selected the parameters of the bandpass filter through the spectral kurtosis diagram and proposed the spectral kurtosis discrete algorithm [11]. For improving the kurtosis figure in the engineering application of operational efficiency and real-time, Antoni and Randall propose a fast kurtograph algorithm based on binary finite impulse response filter group. The wavelet kurtograph algorithm is proposed based on the nonorthogonal complex wavelet analysis by Sawalhi et al. [12], and the frequency band partition in the rapid spectrum is optimized. Liu et al. propose an adaptive spectral kurtosis filtering technique to extract the signal transients based on Morlet wavelet. The Morlet wavelet is used as a filter bank whose center frequency is defined by the wavelet correlation filtering [13].

The maximum correlation kurtosis deconvolution (MCKD) is proposed by McDonald et al. [14]. It is a method to extract weak impact components by raising the kurtosis of signals to achieve the purpose of noise reduction under low signal-to-noise ratio. Compared with the minimum entropy deconvolution (MED), MCKD has obvious advantages in dealing with strong background noise. When a fault occurs, we can define the discrete signal as the response of the bearing excited by the faulty impulse signal . MCKD searches for a FIR filter to maximize the correlation kurtosis of the signal recovered from the input signal and the general expression for the inverse is given bywhere , and is the length of the FIR filter.

The correlation kurtosis is defined aswhere represents the length of the signal, is the deconvolution cycle, and is the number of conversions.

The optimization function of MCKD is defined as

The optimization function is to find the optimal filter which maximizes the correlation kurtosis . The calculation formula is defined as

The FIR filter result of is defined aswhere

Fast spectral kurtosis is sensitive to noise, and it is easy to cause misdiagnosis or missed diagnosis for compound fault in different resonance frequency bands.

When determining the resonance band, the FSK can only select the resonance band with the maximum kurtosis. The information of compound fault is often distributed in two different resonance bands. Only when the kurtosis values of the two resonance bands are very close to each other, the FSK can identify two resonance bands accurately. However, for most cases, the compound fault has a large difference in the kurtosis values of the two resonance band components. This results in omission of part of the fault information when dealing with compound failure problems with FSK. Moreover, when the noise is relatively large, in the interference of noise, the two resonance bands may lose. The simulation signal with the impact characteristics located at two different resonance bands is defined aswhere = 1500 Hz is the first resonance frequency, = 3000 Hz is the second resonance frequency, = 50 Hz and = 125 Hz are two characteristic frequencies, is the damping ratio, and are the amplitudes, , and is the Gauss white noise. The problems encountered by FSK in analyzing compound fault are verified by adjusting the numerical values of , , and .

Figure 1(a) shows the kurtosis values of the two impact characteristics are similar in magnitude and the noise interference is small. The impact information located in different resonance bands can be extracted by the FSK method. Filtering is carried out in the two resonant frequency bands, and then the envelope spectrum analysis is performed to extract the characteristic frequencies of the two impact characteristics. Then the proportion of two impacts is changed by changing the size of A1 and A2. Figure 1(b) shows that the resonance band information of a relatively weak impact characteristic is submerged because of the changing proportion of the two impacts. Figure 1(c) shows that the two resonance bands are submerged when the noise is large. The above analysis shows that noise and compound fault may cause inaccuracy when determining the resonant bands by FSK.