Shock and Vibration

Volume 2017 (2017), Article ID 5963239, 14 pages

https://doi.org/10.1155/2017/5963239

## Casing Vibration Fault Diagnosis Based on Variational Mode Decomposition, Local Linear Embedding, and Support Vector Machine

State Key Lab of Control and Simulation of Power Systems and Generation Equipment, Department of Thermal Engineering, Tsinghua University, Beijing 100084, China

Correspondence should be addressed to Yizhou Yang; nc.ude.auhgnist.sliam@61zygnay

Received 10 October 2016; Revised 17 January 2017; Accepted 16 February 2017; Published 12 March 2017

Academic Editor: Dario Di Maio

Copyright © 2017 Yizhou Yang and Dongxiang Jiang. 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

To diagnose mechanical faults of rotor-bearing-casing system by analyzing its casing vibration signal, this paper proposes a training procedure of a fault classifier based on variational mode decomposition (VMD), local linear embedding (LLE), and support vector machine (SVM). VMD is used first to decompose the casing signal into several modes, which are subsignals usually modulated by fault frequencies. Vibrational features are extracted from both VMD subsignals and the original one. LLE is employed here to reduce the dimensionality of these extracted features and make the samples more separable. Then low-dimensional data sets are used to train the multiclass SVM whose accuracy is tested by classifying the test samples. When the parameters of LLE and SVM are well optimized, this proposed method performs well on experimental data, showing its capacity of diagnosing casing vibration faults.

#### 1. Introduction

The fault diagnosis of gas turbine engine is a large challenge to nowadays’ mechanical engineers due to the complexity of its interior structure. Displacement sensors sometimes cannot be installed inside the machine. Vibration signals will be collected by the accelerometers installed on the casing. But it is difficult to analyze casing signals because of the complex surrounding structure and the long path of transmission from the inside to the outside [1]. Frequency components of casing acceleration signals could be various and complex. Hence, the feature extraction and fault classification of its vibration faults are not the same with those designed for traditional rotating machineries.

To deal with the complex vibration signal of the rotor-bearing-casing system, signal decomposition can be used to extract time or frequency characteristics before further processing. In addition to traditional filters of IIR and FIR, the wavelet transform is another widely used technique that could decompose and reconstruct signals. Wavelet transform can extract the time-frequency characteristics of the signal with variable resolution [2], but the selection of wavelets and their parameters is hard to decide [3]. Empirical mode decomposition (EMD) is an adaptive technique which has a high-frequency resolution [4] but still has disadvantages like end effect and modes mixing. Since 1998, the application of EMD has been well developed. Many other adaptive decomposition methods were put forward, like local mean decomposition (LMD) [5], intrinsic time-scale decomposition (ITD) [6], and variational mode decomposition (VMD) [7]. This paper is employing the VMD to do signal decomposition.

Features of signals reflect the vibrational condition of the rotor-bearing-casing system. But too many features would cause multiple dimensional difficulty during analysis. Dimensionality reduction is the way to handle multidimensional data, which could reduce the complexity of calculation [8]. Many nonlinear dimensionality reduction techniques have been proposed these years, which have advantages over the traditional linear techniques like principal components analysis (PCA) [9]. Local linear embedding (LLE) is a nonlinear technique proposed in 2000 [10] that has been widely used in many data processing problems. LLE is employed here to do dimensionality reduction, for its high efficiency in providing local properties of data.

The neural network has shown a strong capacity of solving nonlinear problems [11]. But it still has some disadvantages like existence of many local minima solutions and the difficulty in choosing the number of hidden units. Support vector machine (SVM) is a breakthrough of neural network based on the development of statistical learning theory and structural risk minimization [12]. The original SVM algorithm was invented in 1963. And in 1992, it was improved to be a nonlinear classifier by applying the kernel trick to maximum-margin hyperplanes [13].

In this paper, faults of rotor-bearing-casing system are recognized by analyzing the casing vibration signal. Vibrational features in time and frequency domain, along with these extracted by VMD method, are used as the database of fault classification. The experimental data are obtained from the rotor-bearing-casing test system, where several kinds of mechanical faults could be simulated. Since the vectors consisting of features are multidimensional and complicated, they are dimensionally reduced through LLE to get a low-dimensional data set. Then the low-dimensional data set plays as the training data of multiclass SVM. The parameters of LLE and SVM are optimized, so that a capable classifier for faults of rotor-bearing-casing system could be produced.

#### 2. Theoretical Background

##### 2.1. Variational Mode Decomposition

VMD defines Intrinsic Mode Function (IMF) as an amplitude-modulated-frequency-modulated signal:where is the instantaneous amplitude, is the instantaneous phase, and the instantaneous frequency is .

Three important signal processing tools are involved in the building of the VMD model: Wiener filtering, Hilbert transform, and frequency mixing.

The goal of VMD is to decompose a real valued input signal into subsignals (modes) written as , and each of them is mostly compact around a center pulsation . The constrained variational model VMD built [7] can be described as

To construct the model, first compute the associated analytic signal for each mode by means of the Hilbert transform to obtain a unilateral frequency spectrum. Multiplied by , then shift the mode’s frequency spectrum to baseband. At last the bandwidth is estimated by means of the squared -norm of the gradient.

The augmented Lagrangian is introduced to convert the constrained variational problem into an unconstrained variational problem:

The solution to the original minimization problem now becomes finding the saddle point of the augmented Lagrangian in a sequence of iterative subsignals and center pulsations. The algorithm is called alternate direction method of multipliers (ADMM) and the complete algorithm of VMD is shown as follows [14]:(1)Initialize , , , and as 0.(2); start iteration.(3)For , update for all : , , and are the Fourier transform of , , and .(4)For , update (5)Update for all :(6)Repeat step (2) to (5) until convergence: The IMF subsignals are the inverse Fourier transform of .

##### 2.2. Locally Linear Embedding

Local linear embedding (LLE) is a nonlinear dimensionality reduction method widely used these years. It constructs a neighborhood graph representation of the data points. In LLE, the local properties of the data manifold are constructed by turning the data points into a linear combination of their nearest neighbors. And by this way the global nonlinear structure of the data is recovered. The procedure of LLE could be summarized as follows [15].

Suppose the data , , . For arbitrary point , calculate the Euclidean distance between and other points and find the nearest points.

are linear coefficients that reconstruct each data point from its neighbor, as the weight of the th data point to the th reconstruction. The aim is to choose the weight to minimize the reconstruction error:with the constraint

Zero filling to -dimensional vector , so all the weights combine as matrix . Then the reconstruction error can be written as

Suppose , where is identity matrix. Then there arewhere is the trace of matrix.

To map the high-dimensional data into low-dimensional data representing global internal coordinates on the manifold, is selected to minimize reconstruction cost:where and . Write as and then

Because , is orthonormal. To minimize , the bottom nonzero eigenvalues and their corresponding eigenvectors are chosen. So the low-dimensional data can be obtained as .

##### 2.3. Support Vector Machine

The main purpose of support vector machine is to construct a hyperplane in space for the use of classification, regression, or other tasks. When used for classification, the largest distance between data points and the hyperplane means a good separation. If the samples are not linearly separated in the original space, it is proposed to map this space into a much higher-dimensional space, even an infinite-dimensional one, which may make it an easier separation [16].

Suppose the training data points , where is the input data and is the binary class labels. Define nonlinear function which maps the original space into the higher-dimensional feature space. When the hyperplane is and is its normal vector, the training data may be described as

Thus the classifier will be the form of

The problem of building a hyperplane then turns into an optimization problemwith the constraint conditionwhere are slack variables as the distance from the misclassified samples to the hyperplane, acting like the punishment. Positive constant is the penalty coefficient and the bigger is, the more the penalty of misclassification there will be.

It is a convex optimization problem and the Kuhn-Tucker theorem will be used to translate the problem into a dual problemwith the constraint condition

is not calculated in an explicit way but in the form of Mercer kernel

There are several kernels often used:(i)Linear: .(ii)Polynomial: .(iii)RBF: .(iv)MLP: .

So the nonlinear SVM classifier in dual space is turned into

Here is the decision function of the optimized hyperplane. The value without the “sign” function as the indicator may also be used in multiclass classification problems.

#### 3. Simulation Signal Analysis Using VMD

In a rotor-bearing-casing structure, resonances are often excited by the harmonic frequencies of the shaft and other forms of vibration. Large amount of information including fault features is buried in the resonances. High-Frequency Resonance Technique (HFRT) is a normally used method to extract vibrational parameters from modulated signals [17]. But HFRT selects resonant frequency bands in empirical ways. It is unpractical to select frequency band for each sample when samples are too many. VMD helps to decompose the signal into a fixed number of intrinsic modes automatically.

Here is an example of a simulation signal, showing the effect of VMD decomposing modes apart. Suppose the shaft vibration consists of rotational frequency and its subharmonic and harmonic frequencies. Two resonances of the casing are excited by the shaft vibration. Center frequencies of the resonances are set to 1000 Hz and 2400 Hz. So the casing signal can be expressed as below, where is white Gaussian noise.here , , and

Figure 1 is the time waveform of and its frequency spectrum where components of shaft vibration and structural resonances are clear to see. Thus there are 3 so-called modes in the signal. The purpose of VMD here is to separate these 3 modes apart. Figure 2 is the output IMF subsignals and their frequency spectrums, showing that main frequency components remained in the 3 IMFs and noise not close to these 3 modes is filtered out. For comparison, the signal is also decomposed by empirical mode decomposition (EMD). Figure 3 is time waveforms of the EMD subsignals (IMFs) and frequency spectrums of the first 7 IMFs. The stop criterion is set to recommend value by Rilling et al. [18], and 13 modes are decomposed out, most of which are not needed. Other stop criterions are tried but the mode number will not be decreased to 3. Figure 4 shows the center frequencies changing during VMD iteration. After convergence, these 3 center frequencies are 0.0076, 0.1248, and 0.3002 relative to the sampling frequency of 8000 Hz (60.8 Hz, 998.4 Hz, and 2401.6 Hz), very close to the center frequencies set before the simulation. If there is not white Gaussian noise in the simulated signal, the center frequencies from VMD will be 56 Hz, 1000 Hz, and 2400 Hz. This example can be a proof of VMD’s remarkable ability to deal with casing vibration signals.