Total variation on horizontal visibility graph and its application to rolling bearing fault diagnosis

Highlights

• The bearing vibration signal is converted into a new graph signal.

• The vertex domain index TV_HVG is extracted as the fault feature of rolling bearing.

• TV_HVG can better distinguish the states of rolling bearings than ApEn, PE, KU, RMS.

• A novel fault diagnosis method for rolling bearing using TV_HVG and MD is proposed.

Abstract

The total variation on graph (TV_G) is a powerful vertex domain index for measuring the smoothness of graph signals, but its performance is closely related to the underlying graph. Since the horizontal visibility graph can better reflect the dynamics characteristics of bearing vibration signals than the path graph, the underlying graph of TV_G is designated as horizontal visibility graph. The vertex domain index TV_G defined on horizontal visibility graph is called simply as TV_HVG in this paper. For better distinguishing the different states of rolling bearings, the bearing vibration signal is converted into the graph signal indexed by its horizontal visibility graph, and the vertex domain index TV_HVG is extracted as the single fault feature. Based on TV_HVG feature extraction and Mahalanobis distance classification, a novel fault diagnosis method for rolling bearings is proposed. The proposed method is applied to analyze two sets of experimental data containing normal and faulty rolling bearings. The results indicate that the proposed method can diagnose the bearing faults with different types and degrees effectively, and the vertex domain index TV_HVG is superior to some classical time domain indexes in distinguishing the different states of rolling bearings.

Introduction

Rolling bearings are the common parts of rotary machinery. When the inner race, outer race or ball element of rolling bearing is damaged, its vibration signal will contain a large number of information related to the fault. Hence, vibration signal analysis is widely used in the fault diagnosis of rolling bearings [1,2]. The fault diagnosis methods based on vibration signal analysis generally contain three steps: the first step is the acquisition of vibration signals; the second step is the extraction of fault features; the third step is the identification and diagnosis of fault patterns. The most important and key step is the second step. Reviewing the literatures related to rotary machinery fault diagnosis, the fault features are usually extracted from time domain and frequency domain by means of the classical signal processing methods [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. For instance, Samanta and Al-Balushi [3] used the five time-domain features of the preprocessed vibration signals as the input of artificial neural network to achieve fault diagnosis of rolling bearings. Hu et al. [5] selected the optimal time-domain features of raw vibration signals and wavelet package coefficient as the input of support vector machines ensemble to achieve fault diagnosis of rotary machinery. Liu et al. [9] decomposed the bearing vibration signals by empirical mode decomposition, and took the optimal time- and frequency-domain features of intrinsic mode functions as the input of wavelet support vector machine to realize multi-fault classification for rolling bearings. There is no doubt that the classical signal processing methods have made outstanding performance in extracting the fault features of rolling bearings. However, due to the increasing complexity of mechanical equipment, coupled with the non-linearity and non-stationary of bearing vibration signals, it is sometimes difficult for the classical signal processing methods to extract effective fault features. Therefore, it is necessary to further explore a new and more effective method for extracting the fault features of rolling bearings.

Graph signal processing [13,14] is a new research field that emerged with the development of spectral graph theory and algebraic signal processing theory. Its main content is to extend the traditional signal processing methods to the analysis and processing of graph signals. In recent ten years, the theory of graph signal processing has been expanded rapidly, and the graph Fourier transform (GFT) [13,14], graph short-time Fourier transform (GSTFT) [15,16], graph wavelet transform (GWT) [17], [18], [19], graph empirical mode decomposition (GEMD) [20] and graph Hilbert transform (GHT) [21] have been proposed. The GFT can transform the graph signals from the vertex domain to the graph spectral domain, and the GSTFT, GWT and GEMD can transform the graph signals from the vertex domain to the vertex-graph spectral domain. Therefore, graph signals can be analyzed and processed in the vertex domain, graph spectral domain and vertex-graph spectral domain, which are the analogues to the time domain, frequency domain and time-frequency domain, respectively. Different from the traditional signal processing methods, these graph signal processing methods analyze graphs that represent the relational structure of datasets, rather than the datasets themselves. Thus they are particularly suitable for analyzing and processing the large and complex datasets resided on the vertices of graphs, for example, the datasets obtained from sensor networks, transportation networks, social and economic networks [13–21]. All in all, the traditional methods mainly analyze signals from the time domain, frequency domain, and time-frequency domain, while the graph signal processing methods analyze signals from the vertex domain, graph spectral domain and vertex-graph spectral domain. The graph signal processing methods provide three new dimensions for signal analysis, and based on them, a series of new fault diagnosis methods can be developed. Ou et al. [22–24] have successfully applied the GFT to the fault diagnosis of rolling bearings and the compound fault diagnosis of gearboxes.

In the field of graph signal processing, the total variation on graph (TV_G) [25] is a vertex domain index for measuring the smoothness of graph signals. Generally, the value of TV_G is small when the graph signal has similar values at the adjacent vertices connected by an edge; i.e., when it is smooth. Vibration signals are typical time series signals, so they can be converted into graph signals by means of the mapping algorithms between time series and graph structure [26,27]. Naturally, the vertex domain index TV_G can also be used to measure the smoothness of vibration signals. Taking advantage of the vertex domain index TV_G, a new data classifier has been proposed, which can yield high classification accuracy even for a few known samples and outperforms two widely used classification methods, support vector machine and neural network [28]; the signal recovery methods for graph signals have been presented, whose effectiveness is fully verified by the real-world recovery problems, including online blog classification, bridge condition identification and temperature estimation [29]; the signal denoising algorithm for graph signals has also been put forward, which was applied to measurement denoising for temperature sensors and opinion combination for multiple experts [30]. All these applications of the vertex domain index TV_G indicate that it is a powerful index for measuring the smoothness of graph signals, and its value is very sensitive to the change of graph signals. In order to introduce the GFT into the field of mechanical fault diagnosis, Ou et al. [22] have directly regarded vibration signals as the path graph signals in a manifold perspective, and Gao et al. [31] have converted vibration signals into the graph signals indexed by their own horizontal visibility graphs. In addition, Gao et al. [31] have demonstrated that the horizontal visibility graphs can better reflect the dynamics characteristics of bearing vibration signals than the path graphs. Therefore, we also convert the bearing vibration signal into the graph signal indexed by its horizontal visibility graph, and the vertex domain index TV_G defined on horizontal visibility graph is called as TV_HVG for simplicity in this paper.

Generally, the vibration graph signals in normal state are relatively smooth, while the vibration graph signals in faulty states are relatively rough. Of course, the roughness of vibration graph signals are also different under different fault states. Taking into account that the vertex domain index TV_HVG is not only simple to calculate, but also powerful in measuring the smoothness of graph signals, the vertex domain index TV_HVG of vibration graph signals is extracted as the single fault feature of rolling bearings. Since the TV_HVG is a single feature that only contains one value, the Mahalanobis distance (MD) [32] is selected for pattern recognition, which is a simple but effective distance metric. Xiang et al. [33] calculated the MD instead of Euclidean distance in one-class support vector machine for improving classification performance. Ou and Yu [34] demonstrated that when the Laplacian energy is extracted as a single fault feature, the MD can accurately and effectively identify the rolling bearing faults only with a small number of sampling points and training samples. Based on TV_HVG feature extraction and MD classification, a novel fault diagnosis method for rolling bearings is proposed. In the proposed method, the bearing vibration signal is first converted into the graph signal indexed by its horizontal visibility graph, and then the vertex domain index TV_HVG is extracted as the single fault feature. Finally, the MD is used to identify the working states and fault patterns of rolling bearings. The analysis results of two sets of experimental data indicate that the proposed method can diagnose the bearing faults with different types and degrees effectively, and the vertex domain index TV_HVG is superior to some classical time domain indexes in distinguishing the different states of rolling bearings.

The remaining parts of this paper are arranged as follows. In Sections 2 and 3, the definitions of horizontal visibility graph and TV_HVG are given. In Section 4, the novel fault diagnosis method for rolling bearings based on TV_HVG feature extraction and MD classification is proposed. The experimental validations are performed in Section 5, and the conclusions of this paper are provided in Section 6.