Shock and Vibration

Volume 2016 (2016), Article ID 1835127, 12 pages

http://dx.doi.org/10.1155/2016/1835127

## Sparse Signal Representations of Bearing Fault Signals for Exhibiting Bearing Fault Features

^{1}Zhongshan Institute, University of Electronic Science and Technology of China, Zhongshan 528402, China^{2}Institute of Reliability Engineering, School of Mechatronics Engineering, University of Electronic Science and Technology of China, Chengdu 610051, China^{3}Department of Systems Engineering and Engineering Management, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong^{4}School of Mechanical and Electrical Engineering, Soochow University, Suzhou 215021, China

Received 5 May 2015; Revised 7 October 2015; Accepted 12 October 2015

Academic Editor: Peng Chen

Copyright © 2016 Wei Peng 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

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.

#### 1. Introduction

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 [1–4]. Therefore, to extract bearing fault features, envelope analysis is one of the most effective methods. To conduct envelope analysis, two steps are needed. The first step aims to use a band-pass filter to retain one of the resonant frequency bands for enhancing the signal to noise ratio of bearing fault signals because bearing fault signals are often overwhelmed by strong low-frequency vibration components and heavy noises. The second step is extracting the envelope of the signals filtered by the band-pass filter [5–7]. Moreover, if the envelope signals can be represented by a few coefficients, namely, sparse envelope coefficients [8], bearing fault signals are more understandable and easily interpreted.

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 [9–14]. To automatically tune the parameters of the Morlet wavelet, two aspects including a metric and an optimization algorithm must be determined. In the past years, some metrics, such as kurtosis, entropy, smoothness index, and sparsity measurement, have been investigated for optimization of the Morlet wavelet transform [9–14]. Their comparisons show that the sparsity measurement can generate better visual inspection performance and highlight bearing fault signatures, such as bearing fault characteristic frequency and its harmonics [15]. Therefore, the metric used in this paper is the sparsity measurement. To achieve the global optimal parameters of the Morlet wavelet transform, genetic algorithm, differential evolution, and stepwise scanning have been studied [9–15]. To explore the feasibility of an easy and simple optimization algorithm, particle swarm optimization is used in this paper to tune the sparsity measurement because the core of particle swarm optimization is based on the simple physical relationship among position, velocity, and acceleration. The use of PSO is simply introduced as follows. A number of particles move in a searching space. Then, a simple mathematic algorithm searches the best position by sharing the cognitive and social influences among all particles in the searching space [16–18]. For intelligent machine fault diagnosis, PSO was used to tune the parameters of support vector machine, artificial neural network, and proximal support vector machine, respectively [19–21]. The results show that these statistical prediction models combined with PSO have good prediction accuracies for identification of different machine faults.

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 [22–29]. MA aims to use Minkowski addition and subtraction to intersect the morphological features of bearing fault signals with a structuring element (SE). However, if the morphological features are overwhelmed by other strong vibration components and heavy noises, MA may fail to retain the morphological features of bearing fault signals. Therefore, MA can be used to postprocess bearing fault signals, if the signal to noise ratio of bearing fault signals is not high. In this paper, an adaptive MA with an iterative local maximum detection method is developed to automatically find the optimal parameter of MA and extract sparse envelope signals so as to exhibit bearing fault features.

The rest of this paper is outlined as follows. Section 2 introduces the fundamental algorithms used in this paper. These algorithms include wavelet transform, particle swarm optimization, and morphological analysis. In Section 3, extraction of sparse envelope coefficients from bearing fault signals is proposed. In Section 4, simulated and real bearing faults signals are analyzed by using the proposed method. Conclusions are drawn in Section 5.

#### 2. Fundamental Algorithms

##### 2.1. Fundamental Theory of Wavelet Transform

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 [30, 31]:where is the scale parameter and is the translation parameter. represents the convolution operator. takes the complex conjugate of the signal. According to the properties of Fourier transform, (1) is rewritten aswhere is the inverse Fourier transform and is the Fourier transform. As explained in Introduction, the complex Morlet wavelet is chosen in this paper. Its temporal waveform and the corresponding frequency spectrum are given as follows:

From (5), it is obvious that the complex Morlet wavelet has a band-pass property and its frequency support is constrained to the frequency band . Because any wavelet must satisfy the admission condition, which means that the integration of a wavelet over time must be equal to zero, the following equation should be satisfied:

It is not difficult to verify that if , .

##### 2.2. Fundamental Theory of Particle Swarm Optimization

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 [16]. Considering the physical relationship among position, velocity, and acceleration, the following basic physic principle is listed as follows:where and mean the th and the th positions of the th particle, respectively. is the th velocity of the th particle. is the th acceleration of the th particle. Then, the acceleration of the th particle is divided into a cognitive acceleration, which is proportional to the distance between the current position of the th particle and the personal best position of the th particle, and a social acceleration, which is proportional to the distance between the current position of the th particle and the global best position of the th particle. To make these two new parts more flexible, a cognitive coefficient and a social coefficient are used. Consequently, (7) is reformulated as

Then, in order to prevent the velocities from getting out of control, the influence of friction is considered by introducing an inertia weight , which is smaller than 1, to (8). Besides, the constant is replaced by two random numbers and , which are limited to the values between 0 and 1. Equation (8) is revised as follows:

At last, (9) consists of the following velocity and position update equations:

##### 2.3. Fundamental Theory of Morphological Analysis

MA aims to extract the morphological shape of a temporal signal. It uses a structuring element to intersect with the temporal signal. Let be the one-dimensional signal over a domain and let be the structuring element over a domain . The basic morphology operators include the dilation operator and the erosion operator, which are related to Minkowski addition and subtraction. The equations for the erosion operator and the dilation operator are defined as follows [22]:

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 [24]. Other morphological operators are constructed based on the combination of the above two operators. Some popular morphological operators used in bearing fault diagnosis are introduced in the following. The opening operator and the closing operator are defined as:where is the reflection of . The opening operator function is smoothing the signal from the bottom by cutting wave peaks and the closing operator function is smoothing the signal from the top by filling up its wave valleys [24]. The average operator (AVG), the difference operator (DIF), the Black Top-Hat operator (BTH), and the White Top-Hat operator (WTH) are defined as [32]

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.

#### 3. Extraction of Sparse Envelope Coefficients for Exhibiting Bearing Fault Features

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 1. Each step used in Figure 1 is detailed in the following paragraphs.