Shock and Vibration

Volume 2016, Article ID 3196465, 11 pages

http://dx.doi.org/10.1155/2016/3196465

## A Novel Method for Mechanical Fault Diagnosis Based on Variational Mode Decomposition and Multikernel Support Vector Machine

^{1}The State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400030, China^{2}College of Mechanical and Power Engineering, Chongqing University of Science and Technology, Chongqing 401331, China

Received 4 June 2015; Accepted 25 October 2015

Academic Editor: Gyuhae Park

Copyright © 2016 Zhongliang Lv 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 novel fault diagnosis method based on variational mode decomposition (VMD) and multikernel support vector machine (MKSVM) optimized by Immune Genetic Algorithm (IGA) is proposed to accurately and adaptively diagnose mechanical faults. First, mechanical fault vibration signals are decomposed into multiple Intrinsic Mode Functions (IMFs) by VMD. Then the features in time-frequency domain are extracted from IMFs to construct the feature sets of mixed domain. Next, Semisupervised Locally Linear Embedding (SS-LLE) is adopted for fusion and dimension reduction. The feature sets with reduced dimension are inputted to the IGA optimized MKSVM for failure mode identification. Theoretical analysis demonstrates that MKSVM can approximate any multivariable function. The global optimal parameter vector of MKSVM can be rapidly identified by IGA parameter optimization. The experiments of mechanical faults show that, compared to traditional fault diagnosis models, the proposed method significantly increases the diagnosis accuracy of mechanical faults and enhances the generalization of its application.

#### 1. Introduction

To ensure the safe and reliable operation of mechanical equipments, vibrations are usually analyzed to diagnose mechanical faults [1–3]. Accurate diagnosis helps to make reasonable maintenance decision. However, the diagnosis accuracy is generally low and manual intervention of diagnosis is usually needed [4, 5], due to the features of large rotating machinery, including multicomponent coupling vibration, strong vibration noise interference, and the instability and nonlinearity of signals, as well as the low noticeability of early fault signals.

As an adaptive method which processes signal in time-frequency domain, empirical mode decomposition (EMD) [6] can decompose the complicated and unstable signal into several nearly stable IMFs. Therefore, it has been widely applied to the diagnosis of mechanical fault [7]. For example, Loutridis [8] used EMD method for gear fault diagnosis. Cheng et al. used EMD method to diagnose bearing failures [9]. However, EMD method is essentially a binary filter bank. The frequency domain splitting feature of EMD makes it disadvantageous in dealing with fault signals [10]. Since the band center and bandwidth of fault signal are unknown, strong interference may be introduced if the fault signal falls in the broad bands of the first component. If the fault signal is in the bands of higher-order components, important information on the feature may be missed as the signal is filtered out by the narrow band of low-order components. Dragomiretskiy and Zosso, in 2014, proposed a new adaptive signal processing method called VMD [11]. This method is able to determine the frequency center and bandwidth of each component in the process of acquiring decomposed components by iteratively searching the optimal solution of variational models, thus adaptively realizing the frequency domain split and the effective separation of each component.

In pattern recognition, support vector machine (SVM) [12] is based on Vapnik-Chervonenkis dimension theory and structural risk minimization principle. SVM finds the optimal compromise between model complexity and learning ability using limited sample information. It overcomes the drawback that traditional machine learning models are easy to get trapped in local minima. It has enormous potential to accurately classify the faults into multiple levels. Therefore, SVM has been widely applied to all kinds of nonlinear pattern recognition problems [13]. However, in complicated cases, especially when the data are heterogeneous [14] and samples are unevenly distributed [15], or samples are in large scales [16], SVM [17] begins to lose its advantages in accomplishing the tasks. MKSVM is a new machine learning model which combines all individual kernels by weights based on traditional single kernel SVM. MKSVM inherits the generalization and learning ability of single kernel SVM. Meanwhile, it reasonably adjusts the weight of each individual kernel and improves the adaptability and robustness of single kernels [18]. However, in the fault identification process using MKSVM, the identification performance of MKSVM is directly influenced by the choice of the function parameters of individual kernels and their weights. Traditional methods, including trial and error or traversing optimization, are not only complicated in computation, but also unable to acquire global optimal solution. Therefore, the adaptive diagnosis ability of MKSVM still needs further improvements. IGA [19] finds optimal solutions by synthetically considering the information interaction between antibodies of populations. Based on genetic algorithm [20], it integrates a series of mechanisms of biological immune system, such as antigen recognition, antibody diversity, density control, and elitist strategy. It greatly helps to avoid immature convergences and meanwhile preserves the global stochastic parallel searching character of genetic algorithm at the same time [19, 21]. Taking computational efficiency, stability, and global optimality factors into consideration, IGA is used to optimize the penalty parameter, weight factor, and kernel parameters of MKSVM, in order to improve the accuracy and stability of fault diagnosis, as well as to enhance the applicability of MKSVM.

The rest of this paper is organized as follows. In Section 2, the theories of EMD and VMD are introduced. In Section 3, the theories of SVM and MKSVM are firstly reviewed. Then, the IGA optimized MKSVM (IGA-MKSVM) method is presented. In Section 4, the fault diagnosis strategy based on VMD and IGA-MKSVM is discussed. Finally, the effectiveness of the proposed method is verified by experiments.

#### 2. VMD Method

##### 2.1. EMD

IMF components obtained by EMD should meet the following criteria: (1) In a data sequence, the number of extreme points and the number of zero crossing points are equal, or up to a difference of 1; (2) at any data point, the average of the local maximum envelope and the local minimum envelope is 0. The basic algorithm of EMD is as follows.

*Step 1. *First, determine the local extremes of signal . Then use a cubic spline to connect all the local maximum points to form the upper envelope. After that, use another cubic spline to connect all the local minimum points to form the lower envelope. The upper and lower envelopes should enclose all data points. The average of the upper and lower envelopes is denoted as . refers to the new signal, which is obtained by subtracting from :Repeat Step 1 by times until becomes the basic IMF component.

*Step 2. *Define , , which is the first mode component obtained by processing the raw data. It should contain the shortest periodic component of raw signals. The residual component can be calculated as follows:

*Step 3. *Since the residual component still contains information of long periodic components, is still treated as new signal data. Repeat the above steps for the residual component , and the following results can be obtained:

*Step 4. *The original signal is finally decomposed into the sum of several IMF components and a residual :

##### 2.2. VMD

In VMD algorithm [11, 22], an intrinsic mode function is redefined as an AM-FM signal, which is expressed bywhere is the instantaneous amplitude of ; is the instantaneous frequency of and . and are slowly varying with respect to the phase . That is, in the interval of (where ), can be viewed as a harmonic signal with amplitude and frequency .

To obtain IMF components, VMD algorithm does not use the cycled screening stripping signal processing mode of EMD. Instead, VMD moves the signal decomposition process into the variational framework. It realizes adaptive signal decomposition by searching the optimal solution of the constrained variational model. The frequency center and bandwidth of each IMF component are updated in the iterative solving process of the variation model. The signal band is adaptively split according to the frequency domain features of the signal. Finally the narrow band IMF components are obtained.

Assuming the original signal is decomposed into IMF components, the corresponding constrained variational model is expressed as follows:where represents the IMF components decomposed by VMD; represents the frequency centers of all IMF components.

To obtain the optimal solution of the above constrained variational problem, the following augmented Lagrange function is introduced:where is a penalty factor, and is the Lagrange multiplier.

The optimal solution of the constrained variational model is derived by using alternating direction multiplier algorithm, which solves the saddle point of the above augmented Lagrange function. The decomposition of the original signal is then obtained. The detailed implementation steps are as follows:(1)Initialize , , , and as 0.(2); perform the entire cycle.(3)Perform the first inner cycle and update according to:(4); repeat step (3) until and finish the first inner cycle.(5)Perform the second inner cycle and update according to:(6); repeat step (5) until . Finish the second inner cycle.(7)Update according to .(8)Repeat steps (2) to (7) until the criterion is satisfied. Finish the entire cycle and output the results of narrow band IMF components.

#### 3. IGA-MKSVM

##### 3.1. SVM

Assume two linearly separable sample sets, . The general form of decision functions is , and the decision surface equation is as follows:

Normalizing the decision equation so that the samples of both classes satisfy . The samples closest to the decision surface satisfy . The decision surface correctly classifies all samples; that is,

Support vectors are samples satisfying (11) and such that is minimum. Those samples are on the lines of and , as shown in Figure 1, where is the optimal classification surface.