Shock and Vibration

Volume 2016, Article ID 1948029, 14 pages

http://dx.doi.org/10.1155/2016/1948029

## Fault Diagnosis for Rolling Bearing under Variable Conditions Based on Image Recognition

^{1}School of Reliability and Systems Engineering, Beihang University, No. 37, Xueyuan Road, Haidian District, Beijing 100191, China^{2}Science & Technology on Reliability and Environmental Engineering Laboratory, Beijing 100191, China

Received 25 May 2016; Accepted 14 July 2016

Academic Editor: Minvydas Ragulskis

Copyright © 2016 Bo Zhou and Yujie Cheng. 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

Rolling bearing faults often lead to electromechanical system failure due to its high speed and complex working conditions. Recently, a large amount of fault diagnosis studies for rolling bearing based on vibration data has been reported. However, few studies have focused on fault diagnosis for rolling bearings under variable conditions. This paper proposes a fault diagnosis method based on image recognition for rolling bearings to realize fault classification under variable working conditions. The proposed method includes the following steps. First, the vibration signal data are transformed into a two-dimensional image based on recurrence plot (RP) technique. Next, a popular feature extraction method which has been widely used in the image field, scale invariant feature transform (SIFT), is employed to extract fault features from the two-dimensional RP and subsequently generate a 128-dimensional feature vector. Third, due to the redundancy of the high-dimensional feature, kernel principal component analysis is utilized to reduce the feature dimensionality. Finally, a neural network classifier trained by probabilistic neural network is used to perform fault diagnosis. Verification experiment results demonstrate the effectiveness of the proposed fault diagnosis method for rolling bearings under variable conditions, thereby providing a promising approach to fault diagnosis for rolling bearings.

#### 1. Introduction

Rolling bearings are considered to be a critical mechanical component in industrial applications. The bearings’ defects usually lead to a considerable decline in plant productivity and may even cause huge economic losses [1, 2]. Thus, it is important to diagnose rolling bearing fault to keep the bearings in good technical state.

Vibration-based methods have garnered particular attention due to their noninvasive nature and their high reactivity to incipient fault. Therefore, vibration signal analysis is vital for rolling bearings fault diagnosis due to its connection to fault feature extraction accuracy [3]. Aiming to extract the fault features, many feature extraction methods, including Wigner-Ville distribution (WVD) [4], wavelet packet decomposition (WPT) [5, 6], and empirical mode decomposition (EMD) [7–9], have been proposed and have been demonstrated to be powerful. Additionally, many fault diagnosis methods also have been proposed, such as fast spectral kurtosis based on genetic algorithms [10], multiscale entropy and adaptive neurofuzzy inference system [11], and time varying singular value decomposition [12]. However, most of these methods are proposed based on the assumption that the rolling bearings operate under fixed conditions when performing fault diagnosis. Moreover, the application of these methods is limited because of the tough, complex, and particularly variable working environment of rolling bearings [13, 14]. Therefore, it is important to investigate the fault diagnosis method suitable for varying conditions.

Many studies have researched rolling bearings fault diagnosis. However, thus far, few researchers have studied fault diagnosis under variable conditions. In 1990, Potter [15] proposed a constant angular sampling method (i.e., order tracking) that utilized the electronic impulse angular encoder and solved the frequency smearing phenomenon of the spectrum caused by fluctuating rotating speeds and realized the fault diagnosis for rotating machines. Considering the special analog hardware whose function is sampling data increases the cost of equipment; Fyfe and Munck [16] developed the computed order tracking (COT) technique based on order tracking to realize fault diagnosis for rotating machines. However, the COT may make the carrier frequencies of the transient responses, which are caused by the faults at various speeds, expand to a wider scope because the natural characteristic of the bearing system hardly changes, which is not beneficial for extracting the fault characteristic. In addition, [13] has proposed a new method for rolling bearing fault diagnosis under variable conditions. This new method utilizes LMD-SVD to extract features, but LMD also has the problem of iterative calculation capacity, frequency aliasing, end effect and other issues. Because of problems associated with the above methods, we need to research a new method for bearing vibration signal feature extraction, a method based on the nonstationary and nonlinear bearing vibration signals, thereby achieving fault diagnosis under variable conditions.

Scale invariant feature transform (SIFT), an image invariant feature extraction method, can recognize the same image when it is rotated, scaled, translational, and affine transformed. By extracting the 128-dimensional feature containing scale, orientation, and location information, SIFT can perform image recognition and matching under translation, rotation, scaling, and brightness changes [17]. Many studies have used SIFT to recognize images. For example, Montazer and Giveki [18] have utilized SIFT to extract image features and match them to a database (i.e., a content-based image retrieval system). Li et al. [19] have employed Robust SIFT to match remote sensing images, and a number of studies have also applied SIFT to such methods as facial expression recognition [20], ear recognition for a new biometric technology [21], and wheat grain classification [22]. Inspired by SIFT, the vibration signals of rolling bearings are considered to be transformed into images. The recurrence plot (RP) is a kind of method to describe the recursive behavior of dynamic orbit in the phase-space reconstruction; it is an important method to analyze the instability of time series. RPs of rolling bearing vibration signals under different conditions reveal translation and scaling characteristics, so RP is employed to transform the vibration signals under different conditions into images and SIFT is utilized to extract the features of transformed RPs, which is without interference of working conditions.

After the 128-dimensional invariant features are extracted, to reduce the data redundancy between the extracted features and the occupation of computer resources, a dimensionality reduction method is utilized to identify the low-dimensional structure hidden in high-dimensional data. Principal component analysis (PCA) is a widely utilized dimension reduction technique performed by linearly transforming a high-dimensional input space onto a lower dimensional one in which the components are uncorrelated. However, PCA will not perform well when the process exhibits nonlinearity. Hence, kernel principal component analysis (KPCA) was developed to overcome the limitations of PCA in dealing with the nonlinear system [23].

This paper is structured as follows. Section 2 first introduces the image transformation method, which generates images for the following recognition. Then, SIFT, the core of this paper, is described, which is utilized to extract the stable fault features under variable working conditions. Subsequently, KPCA is introduced for the dimensionality reduction. At last, probabilistic neural network (PNN) is described for the final fault classification. Section 3 describes the entire fault diagnosis method for rolling bearing under variable conditions, including descriptions of the experimental data, image transformation, feature extraction, and fault classification. Section 4 includes the results and discussion, and the conclusions are presented in Section 5.

#### 2. Related Theories

##### 2.1. Recurrence Plot

To achieve fault diagnosis under variable conditions, image transformation for SIFT is important to ensure success. Therefore, choosing a good image transformation method is particularly important. On account of the nonlinear and nonstationary characteristics of rolling bearing signals, detecting dynamical changes in complex systems is one of the most difficult problems. Recursiveness is one of the basic characteristics of a dynamic system, and the recurrence plot (RP) based on this characteristic is a good dynamic mainstream shape-description method. Through the black and white dots in the two-dimensional space, the recursive state can be visualized in the phase space [24]. This approach can uncover hidden periodicities in a signal in the recurrence domain. These periodicities are not easily noticeable, and it is an important method that analyzes the periodic, chaotic, and nonstationary of time series. The following theories are related.

The RP analysis is based on the phase-space reconstruction theory, which is described as follows.(1)For a time series, (), whose sample interval is , we chose the mutual information method and CAO algorithm to calculate the suitable embedding dimension and delay time , which could reconstitute the time series. The reconstructed time series is (2)Calculate the Euclidean norm of and in the reconstructed phase space [25]:(3)Calculate the recurrence value [26]: where is the threshold value and is the Heaviside function:(4)Utilize a coordinate graph whose abscissa is and whose ordinate is to draw , where and are the time series labels and the image is RP.

###### 2.1.1. Mutual Information Method

The mutual information method estimating the delay time has been proposed by Fraser and Swinney, based on the Shannon information theory, which is widely used in phase-space reconstruction [27].

The Shannon theory shows that we can obtain the information content of from the event :The relationship between and could be expressed with comentropy :

Apply the theory of the mutual information, and set isand set is

Then, (6) translates into

Usually at the beginning, is large; therefore, we can obtain an infinite amount of information in . and are completely independent for chaotic signals when is large; when , . Generally the first minimum of is selected as the delay time.

###### 2.1.2. CAO Algorithm

The CAO algorithm was proposed by CAO in 1997, and it has excellent properties to clearly distinguish real signal and noise, as well as high computational efficiency [28]. First, we calculated the distance of the points under the embedded dimensionality:where is the norm of the vector; is th vector after phase-space rebuilding, and the embedded dimension is ; is the nearest vector from .

Next, we calculated the average value of the distance change under the same dimension:where is the average value of all .

Finally, according to the discriminant equationwhen , stops changing or changes slowly, and is the minimum embedding dimension.

##### 2.2. SIFT Theory

Recognizing the images that are rotating, scaling, and translating refers to finding the stable points of the images. These points, such as the corners, blobs, T-junctions, and light spots in dark regions, do not disappear with the rotation, scale, translation, and brightness changes. SIFT was developed by Nurhaida et al. to extract distinctive invariant features from images that can be used to perform reliable matching between different views of an object or scene [29]. SIFT has five basic steps: constructing scale space, extreme points detection, precise location of key points, orientation assignment, and descriptor calculation [30].

###### 2.2.1. Gaussian Blur

SIFT finds key points in the different scale spaces, and the acquisition of scale space needs to be realized using Gaussian blur. Lindeberg has proved that Gaussian convolution kernel is the only kernel to achieve scale transformation, and it is the only linear kernel [31].

Gaussian blur is an image filter that utilizes normal distribution to calculate the fuzzy template, and the template is used to perform convolution operations with the original image to achieve the transition of fuzzy images.

The normal distribution equation of dimensional space iswhere is the standard deviation of the normal distribution; the larger is, the fuzzier image is. is the fuzzy radius that refers to the distance between the template element and the center of the template. If the two-dimensional template size is , then on the template corresponding to the Gauss equation is

According to the value of , the size of the Gaussian blur template matrix is . Equation (14) is used to calculate the value of the Gaussian template matrix; convolution is calculated with the original image, and the Gaussian blur image of the original image is obtained.

###### 2.2.2. Scale Space Construction

*(1) Scale Space Theory*. Scale space theory was first proposed by Iijima in 1962, and it was widely used in the field of computer vision after being promoted by Duits et al. [32].

The basic concept of scale space is as follows. A scale parameter is introduced in the image model, and the scale space sequence at multiple scales is obtained through the continuous change of scale parameter. The principal contours are extracted from the scale space of these sequences, and the principal contours are utilized as a feature vector to realize edge detection, corner detection, and feature extraction at different resolutions.

*(2) Representation of Scale Space*. The scale space of an image is defined as the convolution calculation between the Gauss function and the original image :where represents convolution: where and are the dimensionality of the Gaussian template determined by . is the pixel location in the image. is the scale space factor, the smaller value of which is the least amount of the smoothed image, and the corresponding scale is smaller.

*(3) Gaussian Pyramid*. The pyramid model of an image is as follows: the original image is constantly downsampling, and it generates a series of different sizes of images, ranging from large to small and from the bottom to the top, thereby constructing a tower-shaped model. The original image is the first stratum of the pyramid, and the new image obtained through downsampling is a stratum of the pyramid. The number of strata in the pyramid is jointly decided through the size of the original and top images. The equation is as follows:where and are the sizes of the original image and is logarithm of the minimum dimensionality of the top image.

To reflect the continuity of scale, the Gaussian pyramid introduces the Gaussian filter on the simple downsampling, as shown in Figure 1. The image in each stratum calculates the Gaussian blur using different parameters; thus, each stratum of the pyramid contains multiple Gaussian blur images. The images in each stratum are named octaves. The initial image (bottom image) of an octave in the Gaussian Pyramid is obtained by sampling from the last third image of the previous octave of images.