Shock and Vibration

Volume 2015 (2015), Article ID 962793, 13 pages

http://dx.doi.org/10.1155/2015/962793

## An Accurate Integral Method for Vibration Signal Based on Feature Information Extraction

^{1}Hebei Provincial Key Laboratory of Heavy Machinery Fluid Power Transmission and Control, Yanshan University, Qinhuangdao, Hebei 066004, China^{2}Key Laboratory of Advanced Forging & Stamping Technology and Science, Yanshan University, Ministry of Education of China,
Qinhuangdao, Hebei 066004, China^{3}College of Mechanical Engineering, Yanshan University, Qinhuangdao, Hebei 066004, China

Received 16 June 2014; Revised 10 September 2014; Accepted 23 September 2014

Academic Editor: Nuno M. Maia

Copyright © 2015 Yong Zhu 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

After summarizing the advantages and disadvantages of current integral methods, a novel vibration signal integral method based on feature information extraction was proposed. This method took full advantage of the self-adaptive filter characteristic and waveform correction feature of ensemble empirical mode decomposition in dealing with nonlinear and nonstationary signals. This research merged the superiorities of kurtosis, mean square error, energy, and singular value decomposition on signal feature extraction. The values of the four indexes aforementioned were combined into a feature vector. Then, the connotative characteristic components in vibration signal were accurately extracted by Euclidean distance search, and the desired integral signals were precisely reconstructed. With this method, the interference problem of invalid signal such as trend item and noise which plague traditional methods is commendably solved. The great cumulative error from the traditional time-domain integral is effectively overcome. Moreover, the large low-frequency error from the traditional frequency-domain integral is successfully avoided. Comparing with the traditional integral methods, this method is outstanding at removing noise and retaining useful feature information and shows higher accuracy and superiority.

#### 1. Introduction

The vibration parameters such as acceleration, velocity, and displacement have been commonly utilized in the mechanical fault diagnosis and condition monitoring [1]. At present, the vibration velocity (VV) and vibration displacement (VD) can be measured by laser equipment or extensometer, but it is complicated in the actual testing [2]. By contrast, the vibration acceleration (VA) is widely used due to the desired advantages such as small size, convenient installation, and wide frequency range. Nevertheless, in the vibration system analysis, VV and VD are regularly required [3, 4]. They are also used to analyze the operating condition and health status of machinery. Thus, the integral of vibration signal is involved in many practical engineering and is worthy of further exploration.

The integral process can be realized via hardware or software. Now, many instruments can implement the conversion by using integral circuit. However, the performance parameters of electronic components possess large discreteness. The precision of results will decline and even produce distortion [5]. To ensure the precision and accuracy of hardware-based integration, the calibration and correction are necessary for the signals with different frequency and amplitude. Meanwhile, the integrator with high performance is quite expensive, and multichannels will greatly increase the cost [1, 3, 5]. Currently, with the development of computer and data processing technology, these deficiencies make software-based integration method more attractive.

At present, there are basically two software integral methods. One is time-domain integral (TDI), and the other is frequency-domain integral (FDI) [2, 3]. Thong et al. theoretically analyzed the determinants of root mean square error under the condition of noise, which pointed out the direction to reduce error [6, 7]. A baseline correction method was based on least squares fitting and could eliminate the trend item in time domain [8]. Zhou et al. described a waveform correction solution based on least squares fitting, which could well remove the primary trend item [9]. Zhang et al. analyzed the error reasons of the software-based integral and discussed the merit and demerit of the high-pass filtering and piecewise polynomial fitting when they were used to eliminate trend item [10]. Hong et al. proposed a TDI method which converted integral problem into a boundary value problem [11].

The FDI method also attracted wide attention. Han et al. pointed out that the error of FDI mainly came from spectral leakage and low-frequency component generated by noise. To reduce the influence of noise, the frequency components whose amplitude was less than direct current component by 6 dB or more were suggested to be set to zero [12, 13]. However, a larger error would exist in the integral initial position. On that basis, a frequency fitting method named vibration attenuation integral was further proposed, and the results had been significantly improved [14]. Ribeiro et al. put forward the FFT-DDI method which reduced the influence of the low-frequency component and was highly effective for the case with high signal-to-noise ratio (SNR) [15]. Duan et al. employed a waveform correction algorithm based on time-domain interpolation and fast Fourier transform (FFT) to eliminate trend item [16]. Wen et al. proposed an approach based on signal reconstruction from the perspective of threshold setting, and its effectiveness was laudably proved [17].

Researches above have made remarkable achievements and a lot of beneficial explorations in both TDI and FDI, which provides research basis and ideas for subsequent study. However, in practical application, as a result of the influences of noise and low-frequency component, the integral accuracy still needs to be further improved [17]. Among all of the aforementioned methods, researches mainly focus on eliminating the low-frequency trend item. However, most parts of rotating machinery often contain complex, nonlinear, and nonstationary characteristics. Such features may range from quasiperiodic to completely irregular behaviors. Specifically as a fault machine, the vibration characteristics are varied. It is worth noting that the most important step for implementing fault diagnosis is to find out an appropriate feature, which can accurately represent the variation of vibration signals as the health condition change. However, it was indicated that traditional FFT-based frequency domain methods are not quite suitable for nonstationary signal analysis since the inherent information of nonstationary signals might not be revealed and extracted [18]. Hence, the analysis of nonstationary signals requires specific techniques [19].

In recent years, many time-frequency analysis techniques have been employed in vibration signal analysis. The short-time Fourier transform (STFT) [20] utilizes a sliding window and applies a stationary signal spectrum analysis on each signal frame. However, restricted by the Heisenberg uncertainty principle (HUP) [21], the trade-off between time resolution and frequency resolution is inevitable. Besides, the window width is fixed and the STFT cannot be suitable for multiresolution analysis in many application fields. On this basis, the wavelet transform (WT) is proposed with a variable window [22]. Nevertheless, there are some disadvantages in WT, such as the appropriate selections of the base function and certain frequency bands with defect information [23]. Since the WT derives from STFT, it is also restricted by HUP and the variable window is not self-adaptive. As a kind of quadratic transform, the Wigner-Ville distribution [24] can achieve a high accurate estimation for monofrequency signals. However, when the signal is nonlinear or multicomponent, cross terms will arise and result in misinterpretation of the signal [25]. Excitedly, a novel technique named ensemble empirical mode decomposition (EEMD) has been proposed [26], which is developed from the popular empirical mode decomposition (EMD) [27]. Recently, it has been proved that EEMD is an effective analysis method for nonlinear and nonstationary signals [28, 29]. It can decompose any nonlinear and nonstationary signals to several intrinsic mode functions (IMF) with different vibration modes. Thus, it can availably separate the useful components and the noise components [26]. The EEMD algorithm is essentially a sifting process with two main effects. One is to remove the superimposed waves and the other is to make the waveform more symmetrical. Moreover, the decomposition is self-adaptive and each IMF has integrity and orthogonality, which can be utilized to amend waveform. Meanwhile, EEMD has efficiently solved the modal aliasing problem. Although EEMD possesses many virtues, so far few studies have used it to solve the integral problem for vibration signal.

This paper proposes a novel integral method based on feature information extraction (FIE). This method takes full advantage of the unique virtues of EEMD in dealing with nonlinear and nonstationary signals. Moreover, the method merges the superiorities of kurtosis, mean square error, energy, and singular value decomposition on signal feature extraction. Then, the feature information inherent in vibration signal is accurately extracted, and the desired integral signals are precisely reconstructed. With this method, the interference problem of invalid signal such as trend item and noise which plague traditional integral methods is commendably solved. The great cumulative error from the traditional TDI is effectively overcome. Moreover, the large low-frequency error from the traditional FDI is successfully avoided. Comparing with the traditional methods, this method is splendid at removing noise and retaining useful feature information and shows higher accuracy and superiority. The feasibility and effectiveness of the proposed method are verified by the simulation and measured signals.

#### 2. Basic Principle of Signal Integral

##### 2.1. Time-Domain Integral Principle

Define as the acceleration signal. Then, the velocity signal after integration can be expressed as follows: The displacement signal after integration can be given by where is dynamic velocity component after integration, is dynamic displacement component after integration, is static velocity component after integration, and is static displacement component after integration, is time, is differential about .

In numerical calculation, the time-domain continuous integral aforementioned is usually replaced with Simpson integral to perform approximate calculation [30]. The velocity calculation formula is

The displacement calculation formula is where is signal sampling points and is sampling time.

The vibration signal commonly possesses complex structure and contains multiple components. Before integration, the direct current component of signal must be eliminated to reduce the influence of trend item on result. However, it is impossible to synchronously sample all components with full period in actual test, which leads to an inevitable deviation in subsequently eliminating direct current operation and further the trend item cannot be completely removed [17]. Moreover, the constant terms and linear terms will be generated after integration. Specifically after the second integration, the amplitude of VD will result in large deviation. The cumulative errors from integrals have a considerable effect in the calculation accuracy. Furthermore, the complex noise components mixed in VA signal also cause the complex nonlinear items existing in VV and VD signals. Therefore, the VV and VD directly obtained by integration from VA will originate large deviation, even complete distortion. At present, the primary means to amend TDI is firstly to process the original signal by means of zero-mean processing, which can eliminate the direct current component in the signal. Then, TDI is performed to the preprocessed signal. Finally, the trend item produced in the integral is eliminated by means of polynomial fitting.

##### 2.2. Frequency-Domain Integral Principle

The principle of FDI is firstly to convert the time-domain signal into frequency domain via FFT. Then, the integral operation in time domain is changed into the algebraic operation of spectrum in frequency domain. Moreover, the integral results can be obtained by inverse fast Fourier transform (IFFT).

Consider that , , and are the discrete forms of the acceleration, velocity, and displacement, respectively. They can be described as follows: Among them, Then, the relationships of algebraic operations among them in frequency domain can be determined as follows: where , , and are Fourier transforms of the acceleration, velocity, and displacement, respectively; is the frequency resolution; is the imaginary unit; is the frequency characteristic of the band-pass filter; and are the lower and the upper cut-off frequency of the band-pass filter, respectively; is the number of the sampling points.

According to (7), FDI directly utilizes the algebraic operation relationship of spectrum in frequency domain as the principle. The cumulative error amplification effect can be efficiently avoided. But (7) also indicate that the amplitude of spectrum increases gradually when . Specifically the displacement is from the frequency-domain second integration, and the amplitude of spectrum is related to . When low frequency is close to zero, it will produce a large low-frequency oscillation and peak error, which exerts low-frequency sensitivity. However, the accuracy in the low band of acceleration sensor is unsatisfactory [3, 30]. As a result, the low band is an important error source of FDI. As a common approach, the components below a certain frequency in spectrum are directly set to zero to get rid of the trend item.

#### 3. Feature Extraction

For faulty rotating machinery, the effective components of signals mainly consist of rotational frequency and fault characteristic frequency [17]. In the signal spectrum, the frequency characteristic and the amplitude where spectrum peaks appear are commonly used to estimate the fault type and severity [31, 32]. This means that, in the vibration signal analysis of faulty rotating machinery, we can focus on a limited number of signal components, that is, rotational frequency, fault characteristic frequency, and its frequency multiplication which includes superharmonic and subharmonic frequency components. Furthermore, other components can be seen as noise to be eliminated.

Kurtosis is a numerical statistic that reflects distribution features of vibration signal. It is usually utilized to quantitatively describe the impact level of vibration signal due to its sensitivity to impact signal [33]. As for discrete signal, its expression can be expressed by where is sampling signal, is the mean of signal, and is standard deviation.

Mean square error (MSE) is usually utilized to quantitatively describe calculation or measurement accuracy [34]. As for discrete signal, its expression can be described as

Energy can reflect the strength degree of the whole data set and the energy difference of different waveforms [35]. Hence, the energy value can be used to quantificationally measure the overall effect of different movement types. As for discrete signal, it can be expressed as

Singular value reflects the energy distribution of useful signal and noise mixed in the signal. After the signal is processed by singular value decomposition (SVD), a set of singular values will be obtained. Among them, the larger mainly reflects the useful signal, and the smaller primarily reflects the noise. SVD can realize signal denoising. Moreover, the denoising results possess the advantages of zero phase shift, low waveform distortion, and high signal-to-noise ratio [36]. Meanwhile, SVD also does well in feature information separation and weak signal extraction. It can effectively extract the feature information from the signal mixed with noise and can accurately extract weak amplitude-modulation features implied in the vibration signal [37].

In order to effectively extract the useful features of vibration signal, this paper takes full use of the superiorities of kurtosis value, MSE value, energy value, and the largest singular value on the feature extraction and combines the four indexes into a feature vector .

#### 4. Integral Principle of FIE and Error Evaluation

##### 4.1. Integral Principle Based on FIE

Based on the aforementioned analysis, this paper proposes a novel integral method based on FIE. The dominant idea of the method is mainly to consider the effective vibration components that reflect signal features during integral process. The feature information of fault characteristics frequency and its superharmonic or subharmonic frequency components are extracted by search and correction. Then, the desired and accurate integral signals are reconstructed with the feature information.

The flow chart of the proposed FIE-based integral method is shown in Figure 1.