Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 2954094, 12 pages

https://doi.org/10.1155/2018/2954094

## A Sparse Modulation Signal Bispectrum Analysis Method for Rolling Element Bearing Diagnosis

^{1}Health Maintenance for Mechanical Equipment Key Laboratory of Hunan Province, Hunan University of Science and Technology, Xiangtan 411201, China^{2}Center for Efficiency and Performance Engineering, University of Huddersfield, Queensgate, Huddersfield HD1 3HD, UK

Correspondence should be addressed to Guangbin Wang; moc.621@bgwxxj

Received 7 July 2017; Accepted 21 February 2018; Published 19 April 2018

Academic Editor: Jussi Sopanen

Copyright © 2018 Guangbin Wang 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

Modulation signal bispectrum (MSB) analysis is an effective method to obtain the fault frequency for rolling bearing, but harmonics make fault frequency dense and even frequency aliasing. Carrier frequency of bearing is generally determined by its structure and inherent characteristics and changes with the increase of the damage degree, so it is hard to be accurately found. To solve these problems, this paper proposes a sparse modulation signal bispectrum analysis method. Firstly the vibration signal is demodulated by MSB analysis and its bispectrum is obtained. After the frequency domain filtering, the carrier frequency is computed based on the characteristics of energy concentration at the carrier frequency on MSB. By shift-frequency MSB (SF-MSB), the carrier frequency is moved to the coordinate origin, the entire MSB is shifted for the same distance, and SF-MSB is obtained. At last, the bispectrum is shifted to the frequency zero point and diagonal slices are performed to obtain a sparse representation of MSB. Experimental results show that sparse MSB (S-MSB) method can not only eliminate the interference of harmonic frequency, but also make the extracted characteristic frequency of fault more obvious.

#### 1. Introduction

Bearings are the important parts and easily damaged parts of rotating machine and have received massive attention. Because rolling bearing fault vibration signal is generally weak, nonstationary, nonlinear signal, how to extract fault information is the key to rolling bearing fault diagnosis.

Bispectrum analysis is a high-order spectrum analysis method, which can effectively solve the phase blind problem of power spectrum analysis as second-order statistics; its biggest advantage is that it can preserve the phase characteristic information of the analyzed signal when detecting, estimating, and reconstructing a signal in a low SNR environment, so it is a powerful tool for dealing with nonlinear and non-Gaussian signals [1].

However, bispectrum analysis generally requires the signal to be steady-state; for unsteady or cyclostationary signals, the analysis results are not accurate enough. So many scholars had proposed some improved algorithms based on the bispectrum analysis for different research objects and specific questions, like wavelet domain bispectrum analysis [2–4], order bispectrum analysis [5], vector bispectrum analysis [6], cyclic bispectrum analysis [7, 8], and so on [9–11]. In 2004, a new AM detector and its normalized form are proposed and defined [12]; Gu et al. named this method as the modulation signal bispectrum (MSB) analysis and achieved fault diagnosis of downstream mechanical equipment using electrical motor current signal based on MSB in 2011 [13]. After that, their team continued to deeply study, analyze, and improve on the MSB method and also achieved a series of results [14–18].

For the fault diagnosis of the rolling bearing, MSB analysis is effective in obtaining the fault frequency, but when fault signal contains multiple harmonics, fault frequency and interference signal frequency of the bearing exist simultaneously, which may make fault frequency dense and even frequency aliasing or distorting. Carrier frequency of bearing is generally determined by its structure and inherent characteristics and changes with the increase of the damage degree, so it is hard to be accurately found. To solve these problems, this paper proposes a sparse MSB analysis method, which finds computing method of carrier frequency and sparse demodulation method on the rolling bearing. Simulation and experiments show that sparse MSB method can extract fault characteristic frequency effectively.

#### 2. Shift-Frequency Modulation Signal Bispectrum (SF-MSB)

##### 2.1. Conventional Bispectrum (CB)

Suppose is a zero-mean stationary signal; the second-order cumulant and third-order cumulant can be calculated aswhere is the expectation operator, denotes complex conjugate, and is time delay. Using one-dimensional and two-dimensional Fourier transforms of cumulant and cumulant , the power spectrum and the bispectrum of can be obtained as shown below:(a)Power spectrum(b)Bispectrum

In (3), indicates three individual frequency components achieved from Fourier series integral. From (2) and (3), it can be seen that bispectrum of contains both magnitude and phase information of signal, but power spectrum contains only energy information. So bispectrum can suppress random noise significantly if the frequency components at are independent components, but when these frequency components are nonlinearly coupled to each other, this nonlinear coupling is indicated by a peak in the bispectrum at the bifrequency , and the statistical averaging will not lead to a zero value in the bispectrum [13].

##### 2.2. Modulation Signal Bispectrum (MSB)

MSB has the capability to enhance nonlinear components and suppress random noise by detecting phase coupling in modulation signal. The definition of MSB can be described by

The magnitude and phase of MSB can be expressed as

Signals have quadratic phase coupling characteristics; that is, their phases are related as follows:

The total phase of MSB will be zero and MSB amplitude will be the product of the four magnitudes, which is the maximum of the complex product. Therefore, a bispectral peak will appear at [13]. If these components are not coupled but they have random distribution, the magnitude of MSB will be close to zero. So wideband noise in bearing vibration signals needs to be suppressed effectively so that the discrete frequency components can be obtained more accurately [18].

##### 2.3. Comparison between CB and MSB for Analyzing Modulation Signal

The modulation signal may be defined as follows:

In (7), is the carrier frequency and is the modulation frequency, and are arbitrary constant amplitudes, and and are arbitrary constant phases. Fourier transform of is

Theoretically, from (3) and (8), nonzero values can be only obtained at the intersection points of the three group lines shown in (9) and in Figure 1(a), and other points except for these intersection points are zero on the whole plane [11]. Similarly, from (4) and (8), the MSB’s intersection points may be obtained, described with the red points as shown in Figure 1(b)