Shock and Vibration

Volume 2015, Article ID 167902, 9 pages

http://dx.doi.org/10.1155/2015/167902

## Rolling Element Bearing Fault Diagnosis Based on Multiscale General Fractal Features

^{1}School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing 100044, China^{2}Department of Mechanical Engineering, University of Connecticut, Storrs, CT 06269, USA

Received 30 March 2015; Accepted 16 July 2015

Academic Editor: Chuan Li

Copyright © 2015 Weigang Wen 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

Nonlinear characteristics are ubiquitous in the vibration signals produced by rolling element bearings. Fractal dimensions are effective tools to illustrate nonlinearity. This paper proposes a new approach based on Multiscale General Fractal Dimensions (MGFDs) to realize fault diagnosis of rolling element bearings, which are robust to the effects of variation in operating conditions. The vibration signals of bearing are analyzed to extract the general fractal dimensions in multiscales, which are in turn utilized to construct a feature space to identify fault pattern. Finally, bearing faults are revealed by pattern recognition. Case studies are carried out to evaluate the validity and accuracy of the approach. It is verified that this approach is effective for fault diagnosis of rolling element bearings under various operating conditions via experiment and data analysis.

#### 1. Introduction

Rolling element bearings are common mechanical parts which are subject to damage. Faulty bearings often cause machine failure and even contribute to the disaster in industry. Therefore, fault diagnosis of rolling element bearing is necessary in condition monitoring of machines. It is critical that fault pattern identification of bearing is performed to prevent machine breakdown and reduce economic loss in early period.

Vibration signal analysis has been used extensively in various bearing condition monitoring techniques and has become one of the most important methods applied for bearing fault diagnostics. The vibration signals generated by faults in bearings have been widely studied. Many studies have developed sound theoretical bases and approaches to diagnose bearing failure [1–3]. Much research focuses on obtaining fault information through time and frequency domain signal processing techniques. It is well known that the impact vibration produced by rolling elements in bearing excites resonances of the surrounding structures. But analysis of the vibration signal is complicated due to the stochastic movement of rolling elements. A method based on envelope spectrum analysis becomes a primary way for bearing fault diagnosis which was stated systematically in [4]. Various faults can be diagnosed through fault characteristic frequencies according to the bearing structure parameters. However, when the bearing rotational speed varies over time, the characteristic frequencies cannot be obtained in frequency domain. Order tracking (OT), which may involve extra computation or auxiliary equipment such as speedometer and tachometer, is used to remove speed fluctuation. The method is complete and complicated, to which many studies are related [5, 6]. And this technique is successful for a wide range of cases.

The specific characteristics of rolling element bearing vibration signal are not periodic, especially under variable speed. It is inappropriate to diagnose fault of rolling element bearing only by adopting traditional diagnosis techniques. Time-frequency domain methods have been adopted to implement bearing fault diagnosis, such as Wigner-Ville distributions (WVD) [7], empirical mode decomposition (EMD) [8], and wavelet transform (WT) [9]. Many kinds of features are extracted to represent the characteristics of vibration signal in different domains, for example, statistics of Root Mean Square (RMS), kurtosis, crest factor, correlation coefficient and spectrum, and independent component analysis (ICA) [10]. Finally, detection of bearing fault can be implemented by intelligent learning methods, such as perceptron, artificial neural network (ANN), or support vector machines (SVM) [11–13]. There are increasing research works that combine traditional time-frequency domain methods and intelligent learning methods for varied conditions of speed and load [14]. This ensemble of methods is developed as advanced hybrid intelligent fault diagnosis for rolling element bearings.

The vibration signals of bearings, especially with faults, often show mutation and nonlinearity [15]. As we know, the vibration signals that are excited by impacts of rolling element present nonstationary characteristics. Moreover, the nonstationary characteristics caused by faults in bearing are often mixed with nonlinear factors due to the complexity of the structure and operating conditions of rolling bearing, such as instantaneous variations in rolling ball movement, changing speed, and various loads. The traditional signal analysis methods based on linear system fail to extract nonlinear features in vibration signal. In order to analyze the nonlinear signals of bearings, a series of advanced techniques have been applied to extract fault features. It has been discovered that nonlinear analysis could provide a great alternative way to extract fault features out of vibration signals. Many nonlinear methods, such as chaos, fractal dimension, Lyapunov exponent, and approximate entropy, have been investigated [16–18]. The results have shown that nonlinear method is an effective way for rolling bearing fault diagnosis.

Fractal dimensions are widely used in nonlinearity analysis because it can quantitatively characterize nonlinear behavior. The correlation dimension is used to reveal the fault feature of rotating machinery in [19]. The combination of box-counting dimension, information dimension, and correlation dimension is applied to realize bearing fault diagnosis in [20]. Wavelet packet fractal technology is also utilized to diagnose rotating machinery in [21]. Even so, many fractal dimensions are sensitive to a flurry of factors besides bearing faults. It was proposed that the nonlinear feature of correlation dimension is related to the length of signal, the embedded dimension, the time delay, and so on in the bearing fault diagnosis experiment [22]. Moreover, in practical applications, how to decide the threshold of the nonlinear features is quite a problem. Multiscale fractal dimension can describe local nonlinear feature in different scales. To address these problems, multiscale nonlinear features were introduced for bearing fault diagnosis [23, 24]. This paper combines nonlinear analysis and intelligent diagnostics to implement bearing fault diagnosis. Specifically, a whole methodology based on Multiscale General Fractal Dimensions (MGFDs) of vibration signal and pattern recognition method is proposed. General fractal dimension is defined and utilized to reveal the approximation and detail of vibration signal in different scales. Then a feature space is constructed through MGFDs. Finally, intelligent pattern recognition method is utilized to implement classification of fault pattern in the feature space.

The rest of this paper is organized as follows. In the second section, the definition of general fractal dimension is introduced. The principle and methodology of multiscale general fractal dimensions are addressed. In the third section, the vibration signals of rolling element bearings under different conditions are collected and analyzed. The experimental parameters are optimized according to the effectiveness of the methodology. The feasibility and reliability of the methodology for different bearing faults in various conditions are also proven in this section. In the fourth section, the conclusions are presented in closing.

#### 2. Principle and Methodology

##### 2.1. Preliminaries on Fractal Dimension

Theoretical fractals are infinitely self-similar, iterated locally and globally which are not easily described in traditional Euclidean geometric language. Fractals are not limited to nonlinear geometric patterns but can also describe processes in time. So fractal properties in the vibration time series can be suggested because of its nonstationary and nonlinear characteristics.

Fractal patterns are characterized by fractal dimension that is a ratio providing a statistical index of complexity. Fractal dimension can describe the changing of pattern with scale at which it is measured. But in reality, fractal characteristics only exist in a certain scale. Fractal dimensions in different scale can be estimated, respectively.

There are many types of definition of fractal dimension and several methods available to estimate fractal dimension, such as box-counting dimension, correlation dimension, and information dimension. The fractal dimension method is essentially a sequence of approximation associated with decreasing scale that is a geometric factor of simple figure forming the approximation [25]. Here, a general fractal dimension based on the time series cover is introduced to approximate the signal in time domain. This kind of fractal dimension concentrates on the changing pattern of time series.

##### 2.2. Principle of General Fractal Dimension

In order to investigate fractal dimension of vibration signal in time domain, a two-dimensional graph can be made for vibration time series by way of the sample time as -coordinate and the signal amplitude as -coordinate. According to the principle of fractal dimension, there iswhere is approximation area of the sampled signal trajectory , is the scale, and is fractal dimension of the trajectory. Based on the principle of cover fractal dimension, the fractal dimension of sampled vibration signal can be calculated by minimal cover of series ichnography.

Suppose is signal function in domain of closed time interval ; the domain is divided into sections of , the division is , and . To cover these sections, every minimal cover of each section is a rectangle of which the length is and the height is that is the difference between maximal value and minimal value of the signal in section . The minimal cover of the time series of closed interval is

Define variance as

According to the definition of time series fractal dimension, set as minimal cover dimension:

As we knowwhere denotes the average within the time domain, because is equal to the length of time series so , and there is

With increasing, the scale factor is decreasing. An approximation of vibration time series in decreasing scale is made by this way. For the signal time series when Sowhere is a kind of general fractal dimension of the sampled time series. It denotes statistical property and fractal property of the time series. can be calculated as

If same fractal characteristics exist in all the scales, a straight line in the graph of can be fitted by least square method, and the slope of this line gives us an approximate estimation of fractal dimension. It can represent the change of vibration time series pattern. The time series minimum covering method can be completely independent from affine scaling of signal amplitude range. The general fractal dimension method is robust to variations in operating conditions.

##### 2.3. Methodology of Multiscale Fractal Dimensions

For real-world signals with fractal structures, a single global fractal dimension at all scales is impossible. The practical fractal dimension of signal is also dependent on the used scale. Hence, a single noninteger number is not enough to represent entire complexity of a signal. In order to solve this deficiency in characterization of the signal, multiscale fractal dimensions methodology is developed. Unlike global fractal dimension estimated by the slope of log-log curve, the multiscale fractal dimension scheme estimates local fractal dimensions along the scales [26].

For the scales ranking from small to large, the local fractal dimension is estimated by calculating the slope of a line segment fitted by least squares over the adjacent scales in plane. In this way, MGFDs can describe a signal by a series of fractal dimensions along the scales. The computation process is listed as follows.(1)A section of accelerometer data are collected for computing. Here, a section of time series including sampled points is truncated for each evaluation in the experiments.(2)The value of and with increasing scale is computed, where.(3)The series of local fractal dimensions are estimated through the adjacent points on plane. A series of MGFDs are obtained by this way.(4)A fractal feature space is constructed through MGFDs as the input of pattern recognition to identify fault patterns.

Here, the most popular intelligent methods of* K*-nearest neighbor classifier (*K*NNC), back-propagation neural networks (BPNNs), and least squares support vector machines (LS-SVMs) are selected as pattern recognition for training and testing [27–29]. Among these three classifiers, the* K*NNC algorithm predicts the test sample’s category according to the training samples which are the nearest neighbors to the test sample, and judge it to the category which has the largest category probability. BPNNs are constructed by three layers of neurons: input layer, hidden layer, and output layer. BPNNs are able to represent any functional relationship between input and output if there are enough neurons in the hidden layers. LS-SVMs are a class of kernel-based learning methods, which are a set of related supervised learning methods that analyze data and recognize patterns. In the experiments, is set for* K*NNC. The number of input neurons equals the dimension of feature space, the output neurons equal the type of bearing faults, and the hidden layer units are set to 3 in three levels of BPNNs. And the kernel function of LS-SVMs is 2nd order polynomial kernel aswhere polynomial degree.

By this methodology, MGFDs are estimated on the signal amplitude-time plane. They describe quantitatively nonlinear features of vibration signal through minimum covers that represent transformation of the vibration signal contour in different scales. We do not need to take into account the influence of the embedded dimension, time delay, signal shift, and so forth. The only parameters that need to be considered in computation are the length of signal and division of scales. Real-time calculation can be performed in the engineering field. So MGFDs can reveal the approximate and detailed essence of vibration signal in different scales, which are seldom interfered with by external condition, for example, the variation of rotating speed and noise. The detailed verification of this methodology via experiments is described in the next section.

#### 3. Experiments and Discussion

##### 3.1. Experiments of Different Bearing Faults

A series of vibration signals of rolling element bearings with different faults were acquired from a rolling element bearing test rig. In the experiments, the rolling bearings were NSK-6000 deep groove ball bearings. A single point fault was introduced to the test bearings, respectively, by electrodischarge machining with fault diameters of 0.3 mm, 0.6 mm, and 1.0 mm. The rolling element bearing components with faults are shown in Figure 1. Four data sets of normal condition, ball fault, inner race fault, and outer race fault were sampled from the experimental system with a sampling frequency of 12 kHz. The motor rotating speed was set to 1500 rpm at first.