Multiple manifolds analysis and its application to fault diagnosis
Introduction
Vibration signal analysis has been widely used in diagnosing machine faults [1], [2], [3]. Using time and frequency domain signal processing techniques, it is possible to obtain diagnosis information from the vibration signals [4], [5], [6], [7], [8], [9]. However, due to instantaneous variations in friction, damping and load, the mechanical systems are often characterized by non-linear behaviors. Therefore, non-linear analysis methods provide a good alternative to extract defect-related features hidden in the measured signals, which may not be effectively identified using the conventional methods. A number of non-linear methods, such as correlation dimension [10], [11], Lyapunov exponent [12] and approximate entropy [13], have been investigated. These methods are suitable to reveal the variations of the dynamical system where it is in the noise-free or low noise conditions. In fact, the vibration signal measured from the mechanical system is inevitably contaminated by random noise. The three non-linear methods are conducted by averaging all points in the embedding space. This may lose significant information about the time evolution and make the results are sensitive to the noise. For example, the maximal Lyapunov exponent and approximate entropy are finite positive values, which do not necessarily mean that the dynamical system is in chaos. It is probably caused by noise. So using the mentioned non-linear methods to evaluate the state of the system does not work well as desired.
Recently, a lot of manifold learning methods have emerged in non-linear research fields to identify meaningful low-dimensional structures hidden in high-dimensional observations, such as locally linear embedding [14], isometric feature mapping [15] and local tangent space alignment [16]. These methods have been applied in computer vision and document analysis successfully, but have seen far less investigation in the fault diagnosis. Yang et al. [17] uses the manifold learning to de-noise the vibration signal of gearbox. By extending the manifold learning to the multiple manifolds analysis (MMA), a new MMA-based approach is proposed according to the fact that different conditions of a mechanical system have different manifold topological structures. The main advantages of the approach, compared with other non-linear analysis methods, include that (1) a time series is embedded into the high-dimensional space, which is more effective to discover the essential characteristic of the dynamical system, (2) the local geometric information is approximated by extracting the tangent direction within a neighborhood, which is similar to a de-noise process in principle and efficiently reducing the noise impact on the results and (3) inter-related multiple manifolds are taken into account simultaneously to obtain the information of the mechanical system more completely. In a word, the MMA-based approach is extracting the variation among the multiple manifolds to reflect the states of the mechanical system rather than abstracting a feature by averaging all points with the time evolution.
The paper is organized as follows. 2 A manifold reconstruction from a time series, 3 Multiple manifolds analysis address the manifold reconstruction theory and the proposed MMA algorithms, respectively. Section 4 describes the MMA-based fault clustering experiments, followed by the MMA-based trend analysis experiments in Section 5. Finally, the conclusions are presented in Section 6.
Section snippets
A manifold reconstruction from a time series
The state space is constructed by a set of basis vectors which are composed of the dynamic variables of a system (i.e. positions and velocities). For example, consider a dynamical system in a state space Γ∈Rd, where d denotes the number of system variables. But most commonly, not all the dynamic variables of the system are accessible for measure, an alternative form known as embedded phase space is convenient for research the dynamics of the system. Suppose scalar measurements obtained
Multiple manifolds analysis
Suppose L data sets Y={X1, X2,…,XL} are available, in which a time series Xp=[xp1, xp2,…,xpN], p∈[1, L], sampled from the same mechanical system represents one of the states of the dynamical system. Multiple manifolds are reconstructed by embedding each time series Xp into an m-dimensional space with the delay time τ. Then the differences among these manifolds are extracted and its procedure is as follows.
Experimental device and data sets
The bearing vibration signals with different faults are obtained from the website of the bearing data center [23] of Case Western Reserve University. The bearing data have been validated in many researches [24], [25], [26], and has become a standard data set of the roller bearings.
In this experiment, the bearing type is SKF 6205, a deep groove ball bearing. Single point fault is introduced to the test bearing inner race and outer race, respectively, using an electro-discharge machining with
MMA-based trend analysis
Another application of the MMA algorithm is in trend analysis of the mechanical system. As described in Section 3, after the large principle components of the matrix B are extracted, a weighted score can be computed as the feature index for trend analysis. Since each component makes different contributions to every state of the dynamical system, the feature index can be defined:where Fih denotes the hth element of the ith component, and the weight coefficient ηi is
Conclusions
In this work, a fault diagnosis method based on multiple manifolds analysis is presented. According to the fact that different conditions of a mechanical system have different manifolds, the MMA are used to extract the variation among the multiple manifolds to reflect the states of the mechanical system. The procedure of the proposed MMA involves the embedding a time series into the high-dimensional space to reconstruct a dynamical manifold; the analyzing the local information within the
Acknowledgements
This research project was supported by the National Natural Science Foundation of China (No. 50705069), National High-tech R&D Program of China (No. 2007AA04Z169), Doctoral Program of Higher Education of China (No. 20070008050), the project fund of the HuBei province key laboratory of mechanical transmission and manufacturing engineering WuHan university of science and technology (No. 2007A19). Thanks are also due to the Case Western Reserve University, Rockwell Science Office of Naval Research
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2015, NeurocomputingCitation Excerpt :Recently, a new theory of nonlinear feature reduction called manifold learning becomes the research focus. It aims to project the complex high-dimensional data into a low-dimensional topological space by preserving the local neighborhood structure to discover the intrinsic feature of nonlinear high-dimensional data [20,21]. Currently, typical manifold learning methods mainly include Linear Discriminate Analysis (LDA) [22], Orthogonal Neighborhood Preserving Embedding (ONPE) [23], Local Fisher Discriminant Analysis (LFDA) [24], etc.