International Journal of Rotating Machinery

Volume 2017 (2017), Article ID 2598169, 12 pages

https://doi.org/10.1155/2017/2598169

## Rolling Element Bearing Performance Degradation Assessment Using Variational Mode Decomposition and Gath-Geva Clustering Time Series Segmentation

^{1}Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China^{2}Shanghai Maritime University, Shanghai 200135, China

Correspondence should be addressed to Hongru Li; moc.uhos@861rhil

Received 28 April 2017; Revised 23 May 2017; Accepted 5 September 2017; Published 12 October 2017

Academic Editor: Grzegorz M. Krolczyk

Copyright © 2017 Yaolong Li 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

By focusing on the issue of rolling element bearing (REB) performance degradation assessment (PDA), a solution based on variational mode decomposition (VMD) and Gath-Geva clustering time series segmentation (GGCTSS) has been proposed. VMD is a new decomposition method. Since it is different from the recursive decomposition method, for example, empirical mode decomposition (EMD), local mean decomposition (LMD), and local characteristic-scale decomposition (LCD), VMD needs a priori parameters. In this paper, we will propose a method to optimize the parameters in VMD, namely, the number of decomposition modes and moderate bandwidth constraint, based on genetic algorithm. Executing VMD with the acquired parameters, the BLIMFs are obtained. By taking the envelope of the BLIMFs, the sensitive BLIMFs are selected. And then we take the amplitude of the defect frequency (ADF) as a degradative feature. To get the performance degradation assessment, we are going to use the method called Gath-Geva clustering time series segmentation. Afterwards, the method is carried out by two pieces of run-to-failure data. The results indicate that the extracted feature could depict the process of degradation precisely.

#### 1. Introduction

Rolling element bearings (REBs) are critical components of “rotor-bearing” system in rotating machinery. Since the cruel working condition, the REBs are vulnerable. So it is important to monitor their condition to avoid catastrophic accident in modern industry. There are many ways to monitor bearings, such as vibration [1], acoustic emission [2], oil-debris [3], and ultrasound [4]. Among them, the method based on vibration signal analysis is extensively used for a bearing.

Recently, the prediction of the residual (or remaining) useful life (RUL) is a hotspot issue. To get a better bearing fault prediction result, the so-called performance degradation assessment (PDA) is a premise. The PDA contains two important aspects. One is to extract proper features that can reflect the process of degradation. The other one is to use a method to assess the REB’s performance. Feature extraction is the fundamental of PDA. The feature must reveal the real performance of the REB and be sensitive to the degradation. The types of features are usually classified into three categories, time domain features, frequency domain features, and time-frequency domain features. The time-frequency domain features are always based on time-frequency analysis, combined with the concept of spectrum, entropy, and complexity, for example, the Rényi entropy [5], the permutation entropy [6], and the general mathematical morphology particle [7]. In general, mechanical equipment undergoes a complete degradation from normal stage to failure. With the increasing of the running time and deepening of degradation, the amplitude of defect frequency (ADF) is raising therewith. So the ADF can reflect the degradation of REB directly.

To get the ADF, the first thing is to locate the defect frequency. In extraction of ADF, the common method is to use EMD and select appropriate intrinsic mode function (IMF) components, taking the envelope spectrum, and finally get the ADF [8]. However, EMD remains an exclusively empirical algorithm, and it lacks a solid mathematical foundation. Despite numerous attempts to improve its performance, EMD is still of low efficiency and has endpoint effect and mode mixing problems. In 2014, Dragomiretskiy and Zosso [9] proposed a method called variational mode decomposition (VMD) as an alternative to EMD, which can adaptively decompose a multicomponent signal into a number of quasiorthogonal IMFs. It has been verified that VMD outperforms EMD with regard to tone detection and separation as well as noise robustness [10]. It has a good performance and high operation efficiency. Particularly, it has solid foundation for a mathematical theory. Now the method has been applied to the abnormal ECG signal detection [11], the stock market forecasting [12], wind power prediction [13], power quality classification [14], and so on.

However, the parameters of VMD are selected by experience. Jun Zhu presented a method to optimize the parameters based on the kurtosis index through artificial fish swarm algorithm [15]. Tang and Wang used Shannon entropy as an index to optimize parameters by panicle swarm algorithm [16]. In this paper, we proposed a method to optimize the parameters based on BLIMFs themselves using genetic algorithm. As to PDA, Pan et al. [17] proposed a bearing degradation assessment method based on lifting wavelet packet decomposition and fuzzy C-means clustering. Wang et al. [18] proposed a method for PDA based on mathematical morphology fractal dimension and fuzzy C-means clustering. However, when using clustering, they only considered the distance factor, and no time parameter was recommended. The clusters should be contiguous in time. From this point of view, after extraction of ADF from the raw signals, the assessment of bearing performance is carried out by GGCTSS.

The remainder of this paper is organized as follows. In Section 1, the theory of VMD is briefly introduced. The optimization of the parameters is proposed in Section 2. In Section 3, by carrying out two run-to-failure data, we have extracted the ADF and used GGCTSS for PDA. Our conclusions are presented in Section 4.

#### 2. VMD and Its Parameters

##### 2.1. Variational Mode Decomposition

VMD can nonrecursively decompose a real-valued multicomponent signal into a discrete number of quasiorthogonal band-limited subsignals with specific sparsity properties in the spectral domain. Each mode is compacted around a center pulsation . For convenience, let us call these modes obtained by VMD as band-limited IMFs (BLIMFs). The VMD technique is essentially written as a constrained variational problem in [9]where and are the decomposed BLIMFs for the set of all* K* modes and their estimated center frequencies, respectively. The VMD process can be briefly described as follows; for further details, refer to [9].

##### 2.2. Parameters Involved in VMD

There are six parameters for VMD, where represents the numbers of decomposition modes and represents moderate bandwidth constraint; they are two parameters that significant influence on the decomposition results. In general, the parameter is set up to zero, which means that the Lagrangian multiplier is effectively shut off. The parameter* init* is set to be one, which suggests that center frequencies of all the modes are initialized in the uniform distribution. The parameter* DC* is set to zero, for no DC part imposed. And the parameter* tol* for tolerance is set as default value .

##### 2.3. The Optimization of VMD Parameters

Figure 1 shows the impact of the parameter . It can be seen that the parameter seems to have an inverse ratio of the decomposition modes’ bandwidth. In Figure 1(a), there is no high frequency mode. It seems to be a better one with in Figure 1(b). Because the parameter* init* is set to be one, the bandwidth of each mode is almost the same. To optimize the parameter and , A criterion must be raised. We have proposed a criterion in formula (2).where is the supposed frequency band of each mode and* l* is the actual frequency band of each mode. For example, in Figure 1(a), is 0–2000 Hz, 2000–4000 Hz, 4000–6000 Hz, 6000–8000 Hz, and 8000–10000 Hz, and the actual frequency bands of BLIMFs are 0–1396 Hz, 1426–2549 Hz, 2900–4014 Hz, 3686–4620 Hz, and 3926–5376 Hz with the -axis threshold 0.001. is calculated to 0.4021, since* K* must be set as integer, but not for parameter . We can find the optimized using genetic algorithm of each mode. The settings of genetic algorithm are using default in MATLAB and finally get the best* K* and . The minimum of* K* is 2. EMD can be used to find the maximum value of* K*. There are 10 IMFs of the signal decomposed by EMD, so the maximum of* K* is set up to 10.