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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.

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 [

To overcome the shortcomings of the traditional methods, the holospectrum was put forth for synthesizing the information of the phase, amplitude, and frequency [

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 [

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 [

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

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.

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

The closer the distance between the nodes, the smaller

To minimize

Calculate the neighborhood matrix of topology network in initial output layer

where

The initial weight matrix

The location coordinate of topology node is set to the element of weight

where

With the iterate minimal,

The nodes which are greater than

Set

Manifold learning aims at discovering the intrinsic structure of nonlinear date. The process of the manifold learning with SOM is shown in Figure

Given a set of inputs

Selecting neighborhood adaptively: each element of

Extracting local information: compute the

Constructing alignment matrix: form the matrix

Aligning global coordinates: compute the

Schematic diagram of the proposed manifold learning with SOM method.

From the viewpoint of geometry, the vibration data of the same operation state of rotor system has the same geometric property in space distribution or topological structure, its mapping points in the low-dimension embedding space can be distributed in embedded manifolds or in its neighbor. However the embedding dimension

Based on the adaptive neighborhood selection, the low-dimensional embedding can be extracted effectively. Since node neighborhood is divided adaptively by the grid of network for competition mechanism, SOM can overcome the limitation of fixed neighborhood algorithms.

The state space is constructed by a set of basis vectors which are composed of the dynamic variables of a system. 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 of the dynamics of the system. Suppose measurements obtained through sampling can be defined by

Roughly speaking,

However, the selection of time delay and embedding dimension in the phase space reconstruction is a question. Except for uniform time delay, the nonuniform time delay is also used to build the phase space [

For feature extraction in rotating machinery fault diagnosis, the manifold learning with SOM is adopted to explore the geometric distribution properties embedded in the high-dimensional space. On the basis of the principles above, a new approach of feature extraction method based on adaptive manifold learning is proposed. First, high-dimensional observation space is built with phase space reconstruction, and then map the space phase data into a feature space, and estimate the intrinsic distribution of samples to gain the embedding manifold structure. Finally, the feature is represented by two dimensional projections for the sake of intuitive analyses of equipment operating status. The schematic diagram of the feature extraction method based on manifold learning is shown in Figure

Schematic diagram of feature extraction strategy.

In application, it is should be noted that the holospectrum is drawn from the different harmonic components depending on manual intervention. Instead of by frequency component selection, signal is directly used to construct the dynamic trajectory with phase space reconstruction, and then through adaptive neighborhood selection strategy, embedding low-dimensional manifold can be extracted, therefore, reducing the dependence on human experience.

To verify the capability of feature extraction of the proposed method, the nonlinear Lorenz system was adopted for test and is described as

In manifold learning, the minimum embedding dimension is set to 3. The two dimensional projection of the phase space is shown in Figure

Phase space of Lorenz system with added noise.

Embedding projection of manifold learning with SOM.

With the EM iteration, the neighborhood sizes learned from the SOM are shown in Figure

Neighborhood size learned from SOM.

Embedding projection of manifold learning with adaptive neighborhood selection.

The proposed method is applied to feature extraction of rotating mechanical nonlinear faults. Firstly, the fluid excitation failure in a N2 compressor high-pressure cylinder of petrochemical plant is adopted to extract feature. Usually, for the normal operation of rotation machinery, due to the laminar flow state of fluid medium through flow of rotor, the vibration of machinery is smaller. However, inappropriate adjustment of process parameters can lead to the steady turbulent flow phenomenon, resulting in the impact on the rotor. The compressor rotating speed is 11 416 r/min. Displacement transducers were used to acquire vibration signals of the rotor at the corresponding measurement points on coupling end. The sampling frequency is 2 000 Hz, and vibration waveform and frequency spectrum are shown in Figures

Vibration waveform of fluid excitation fault.

Frequency spectrum of fluid excitation fault.

Raw axis center orbit of fluid excitation fault.

Filtering axis center orbit of fluid excitation fault.

As a comparison, the proposed method is adopted to extract embedding manifold from the collected vibration data. The

Frequency spectrum of the first embedding.

Frequency spectrum of the third embedding.

Frequency spectrum of the fifth embedding.

Embedding projection of manifold learning with SOM.

It is very convenient to identify that, due to fluid excitation through flow of rotor, the projection trajectory are an unstable ellipse trace characteristic that is different from other faults. So, according to the special curve of the embedded manifold, the fault can be identified effectively. Obviously, the axis center orbit shown in Figure

In order to verify the capability of the proposed method, different neighborhood sizes were adopted by LTSA to extract the low-dimensional embedding, where the neighborhood size

Embedding projection with LTSA in

Embedding projection with LTSA in

Projection of embedding manifold with adaptive neighborhood selection.

Next, the rotor-stator rub fault of turbines in refinery plant is also adopted to be analyzed. It should be noted that, due to the motion interference between stator and rotor, the axis orbit of rub fault is extremely complex and often shows a sudden change in axis center orbit. However, considering the influence of noise, only in the case of large fault degree, the characteristics can be observed effectively. Figures

Waveform of rotor-stator rub.

Frequency spectrum of rotor-stator rub.

Purified axis center orbit.

According to the evaluation, the minimum embedding dimension is set to 4. In corresponding four embedding vectors, there are two kinds of frequency spectrum structure shown in Figure

Frequency spectrum of embedding manifold extracted with proposed method.

Frequency spectrum of the first embedding

Frequency spectrum of the third embedding

Projection of embedding manifold of rotor-stator rub.

Projection space in

Projection space in

In order to further test the effectiveness of the proposed method, the fixed neighborhood size was adopted by LTSA to extract the low-dimensional embedding, where

Projection of embedding manifold with neighborhood size

Projection space in

Projection space in

Frequency spectrum of embedding manifold extracted with fixed neighborhood size.

Frequency spectrum of the first embedding

Frequency spectrum of the third embedding

In this paper, a novel method for feature extraction of rotating machinery based on nonlinear manifold learning with neighborhood selection adaptive is proposed. In order to detect the nonlinear faults with less manual intervention, a single-signal phase space reconstruction is adopted to construct the high-dimensional manifold, and the embedding is extracted by manifold learning with SOM.

The proposed method is applied to nonlinear system simulation and vibration data of rotating machinery collected with different nonlinear faults. The experimental results illustrate that the new method is superior to the holospectrum methods in fault diagnosis. Even though the behavior of faults is nonlinear and complicated, the manifold method with the adaptive neighborhood selection has demonstrated its reliability in extracting fault features that are not accessible by axis center orbit.

It should be noted that the existing feature extraction approaches in literatures also use holospectrum and axis center orbits, but they are unavoidable to select proper harmonic components for correct judgment. However, the proposed method can distinguish the faults with less manual intervention. Therefore, it has higher accuracy in fault diagnosis than the traditional methods.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the Shaanxi overall innovation project of science and technology with Grant no. 2013KTCQ01-06.