Mathematical Problems in Engineering

Volume 2015, Article ID 873905, 11 pages

http://dx.doi.org/10.1155/2015/873905

## Manifold Learning with Self-Organizing Mapping for Feature Extraction of Nonlinear Faults in Rotating Machinery

^{1}School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an 710049, China^{2}Key Laboratory of Education Ministry for Modern Design and Rotor-Bearing System, Xi’an Jiaotong University, Xi’an 710049, China^{3}Engineering Workshop, Xi’an Jiaotong University, Xi’an 710049, China^{4}State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, China

Received 19 September 2014; Revised 27 December 2014; Accepted 4 January 2015

Academic Editor: Saeed Balochian

Copyright © 2015 Lin Liang 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

A new method for extracting the low-dimensional feature automatically with self-organization mapping manifold is proposed for the detection of rotating mechanical nonlinear faults (such as rubbing, pedestal looseness). Under the phase space reconstructed by single vibration signal, the self-organization mapping (SOM) with expectation maximization iteration algorithm is used to divide the local neighborhoods adaptively without manual intervention. After that, the local tangent space alignment algorithm is adopted to compress the high-dimensional phase space into low-dimensional feature space. The proposed method takes advantages of the manifold learning in low-dimensional feature extraction and adaptive neighborhood construction of SOM and can extract intrinsic fault features of interest in two dimensional projection space. To evaluate the performance of the proposed method, the Lorenz system was simulated and rotation machinery with nonlinear faults was obtained for test purposes. Compared with the holospectrum approaches, the results reveal that the proposed method is superior in identifying faults and effective for rotating machinery condition monitoring.

#### 1. Introduction

Rotating machinery covers a wide range of mechanical equipment and is of importance in industrial applications. Therefore, faults in rotating machinery may severely affect operations in industry and even safety. To minimize the number of breakdowns as well as to increase the reliability, rotating machinery condition should be monitored for symptoms and incipient fault detection. By this, the life of machinery could be prolonged and the catastrophic consequences of unplanned failure could be avoided. Traditionally, to monitor the conditions and diagnose the faults of rotating machinery, vibration signals are most selected due to its easy-to-measure characteristics and analysis [1–4]. The process technologies of vibration signals in common use are frequency spectrum, axis center orbit, time-frequency analysis, and so on. However, for the nonlinear faults of rotating machinery, such as rubbing, pedestal looseness fluid, and so forth, there are still some problems, for example, the amplitude and phase information of vibration signals are separated from each other, and the correlation of the vibration signals on the vertical and horizontal direction cannot be obtained. Hence, it is difficult to recognize the panorama of the rotor’s vibration.

To overcome the shortcomings of the traditional methods, the holospectrum was put forth for synthesizing the information of the phase, amplitude, and frequency [5]. As an effective fault diagnosis technology for the rotation machinery, it takes advantage of the improved Fourier transform algorithm to analyze the vibration displacement signals from one measuring section which can provide much more information about rotor vibration behavior. However, it needs artificial experience for better result. In case of holospectrum, it is unavoidable to select proper harmonic components for correct judgment. For example, for misalignment fault, the orbit of axis center of rotor is synthesized by 1x, 2x, and 4x frequency components, while the axis center orbit for oil whirl is built by 0.45x frequency component. That is to say, that the method failed to deal with the vibration data without manual intervention.

Due to instantaneous variations in friction, damping, and load, the mechanical systems are often characterized by nonlinear behaviors. Therefore, nonlinear analysis methods provide a good choice to extract defect-related features hidden in the measured signals, which may not be effectively identified using the conventional methods. Many nonlinear methods, such as correlation dimension, Lyapunov exponent, and approximate entropy [6], 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 signals obtained from the mechanical system are inevitably contaminated by noise. Thus the above methods are conducted by averaging all points in the embedding space, and this may lose significant information about the time domain.

As a new dimension reduction technique, manifold learning methods have emerged in nonlinear research fields to identify meaningful low-dimensional structures hidden in high-dimensional observations, such as locally linear embedding [7], isometric feature mapping [8], and local tangent space alignment [9]. These methods have been applied in computer vision, document analysis, and fault diagnosis [10–12]. Yang et al. [13] proposed a method for nonlinear time series noise reduction based on principal manifold learning applied to the analysis of gearbox vibration signal with tooth broken. Li et al. [14] proposed the multiple manifolds analysis approach to extract manifold information from the bearing vibration signals with different faults. As for rotor systems, Jiang et al. [15] recently proposed the supervised manifold learning algorithm for effective feature extraction. Based on the survey of methods above, it is found that the manifold learning is an effective method for feature extraction. However, the features are extracted usually in uniform distribution of sample data, ignoring the influence of neighborhood size.

Obviously, neighborhood of high dimension constructed with vibration signal can not ensure uniform distribution. Same neighborhoods size can falsely estimate the relationships between the neighbors; it is therefore worthy of considering variable number of neighbors that are adaptively chosen. In order to distinguish the nonlinear fault of rotating machinery with vibration signals, a new low-dimensional embedding extraction method based on the local tangent space alignment combined with self-organization mapping is proposed. The main advantages of the approach, compared with other nonlinear analysis methods, are as follows: vibration signals are embedded into a high-dimensional space, which is more effective to discover the essential characteristics of the dynamical system, and it can distinguish the type of faults with less manual intervention. In a word, the new approach extracts the low-dimensional embedding from the manifolds to reflect the states of the mechanical system rather than extract a feature by averaging all points with the time waveform.

The organization of the rest paper is given as follows: a brief introduction of manifold learning with self-organizing mapping is given in Section 2. In Section 3, the details of feature extraction scheme are proposed. And Section 4 applies the method in detection for nonlinear fault of rotating machine. Finally, conclusions are drawn in Section 5.

#### 2. Manifold Learning with Self-Organizing Mapping

##### 2.1. Adaptive Selection of Neighborhood

Obviously, large neighborhoods cause confusions when dealing with the highly twisted manifold. In contrast, small neighborhoods can falsely estimate the relationships between the neighbors. Thinking to added noise, the distribution of samples in feature space is usually nonuniform. Thus, the fixed sizes of neighborhoods cannot satisfy the changing manifold structures. It is inevitable that the neighborhood size should be selected adaptively with the principle that all of subspaces should be connected to construct the topology structure of manifold. Meanwhile, there should be enough overlaps between adjacent neighbors, in order to transmit the local information.

From the view of network, self-organizing mapping (SOM) has the ability to divide nodes adaptively. Using competing-layer neurons to match the center of local neighbors of manifold structures, node grids are organized to cover the topological structures. Then with the learning of SOM, the local neighbors of high-dimensional manifolds are divided adaptively.

##### 2.2. Self-Organizing Mapping with EM

A SOM is a type of artificial neural network that is trained using unsupervised learning to produce a low-dimensional mapping space, discretized representation of the input space of training samples, and a self-organizing mapping consists of components called nodes. Associated with each node is a weight vector of the same dimension as the input data vectors, and a position in the mapping space.

Let denote the probability that input is assigned to the node with weight . It is constrained by and . There is a neighborhood function that corresponds to the control strength between node and node . Usually, it is a decreasing function of the distance between nodes and . Given the data , the optimal goal is to find the probability assignments and weights that minimizes where is the distance between nodes of network and is the distance between inputs and weights .

The closer the distance between the nodes, the smaller value. So the logarithm likelihood function is defined as follows:where is priori probability distribution and its initial value is usually set to uniform distribution. Then plusing (1) into (2), the free energy function is now the following:

To minimize of constraint conditions, the implementation process of SOM can be viewed as in the condition of known data , seeking and posterior distribution density function. Obviously, it is suitable to select neighborhood with EM iteration [16]. Therefore, the neighborhood selection algorithm is as follows.(1)Calculate the neighborhood matrix of topology network in initial output layer and normalize it. Neighborhood function is given by where is the distance between nodes of network. Set the initial value: and ; thus the sum of the relative entropy in initial competitive layer network is defined as The initial weight matrix is given by , where and is a random value between 1 and .(2)The location coordinate of topology node is set to the element of weight . Then function is calculated as where and is the sum of relative entropy in competition layer network. is the distance between inputs and weights , where . The global optimal unit is in which is set to the winner neurons index for input .(3)With the iterate minimal, is found by The nodes which are greater than can be found in and are labeled with . So the node distribution is adjusted with(4)Set , in which* rate* is learning ratio. Then new , , , **,** and are calculated, and return to Step (3) until the elements in the** H** are big enough. Finally, with the above iteration, the weight** W** is calculated and then fixed in the maximization for the new value.

##### 2.3. Manifold Learning with SOM

Manifold learning aims at discovering the intrinsic structure of nonlinear date. The process of the manifold learning with SOM is shown in Figure 1, and the implementation procedures are detailed as follows.(1)Given a set of inputs , the SOM network is adopted to optimize the weights . Including multifrequency components or noise in vibration signals, obviously, the performance of the trajectory in phase space reconstruction is complex. Therefore, to balance the calculation and efficiency, network size is usually set to a larger scale. Meanwhile, to keep the consistency optimal results, initial weight of can be set to unit matrix, and the learning ratio of is also set to 1.1 for a gradual learning process.(2)Selecting neighborhood adaptively: each element of is used to set the center node of local neighbors. To ensure enough overlap, the radius of neighbor is equal to the half of maximal distance between center nodes. According to radius of neighbor, the local neighbors are selected, where and is the number of topology grids.(3)Extracting local information: compute the largest eigenvectors of the correlation matrix , and set , where is the mean of .(4)Constructing alignment matrix: form the matrix by locally summing if a direct eigensolver will be used. Otherwise implement a routine that computes matrix-vector multiplication for an arbitrary vector .(5)Aligning global coordinates: compute the smallest eigenvectors of and pick up the eigenvector matrix corresponding to the 2nd to + 1st smallest eigenvalues, and set the global coordinates . With the global coordinates, the feature can be reflected in low-dimension spaces .