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Mechanical Systems and Signal Processing 21 (2007) 2794–2813

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Enhanced eigenvector algorithm for recovering multiple sourcesof vibration signals in machine fault diagnosis

P.W. Tse�, S. Gontarz, X.J. Wang

Smart Engineering Asset Management Laboratory (SEAM), Department of Manufacturing Engineering & Engineering Management,

City University of Hong Kong, Hong Kong, China

Received 24 April 2006; received in revised form 26 February 2007; accepted 28 February 2007

Available online 7 March 2007

Abstract

Many advanced techniques have been developed for vibration-based machine fault diagnosis. One of the prerequisites to

use vibration for fault diagnosis is the vibration signal measured from a machine component must be well isolated from

other vibrations that are generated by adjacent components. Many machines have numerous and small components that

are closely packed together. Due to limited space or accessibility for installing sensors on the inspected machine

component, sometimes only one sensor is allowed to be installed. An aggregated source of vibrations could be collected

rather than just the vibration generated by the inspected component. Hence, an effective algorithm must be employed to

recover the desired vibration out of the aggregated source of vibrations. The blind equalization-(BE)based eigenvector

algorithm (EVA) has proven its effectiveness in recovering the overwhelmed vibration signal in the application of machine

fault diagnosis. However, the conventional type of EVA can recover only one dominant source from the aggregated

vibration. This dominant vibration may belong to the larger vibration generated by the inspected component or a nearby

component. Hence, the ability of EVA in recovering signals besides the dominant signal is deemed necessary. In this paper,

we proposed an enhanced EVA that consists of channel extension and a post-processing method to recover multiple

sources of vibrations. The post-processing method includes the use of correlation and higher order statistics. With the help

of these proposed algorithms, the enhanced EVA can recover other vibrations that are less dominant but highly relevant to

existing faults. To verify its effectiveness, the ability of recovering the overwhelmed bearing faulty vibration is

demonstrated. The results of the experiments using simulated signals and real machine vibrations have proven the

effectiveness of the method. Hence, the enhanced EVA is suitable for vibration-based fault diagnosis on machines that

have many closely packed components.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Blind equalization; Eigenvector algorithm; Multiple sources; Bearing vibration; Machine fault diagnosis

1. Introduction

Mechanical facilities play important roles in various industries. They become more and more complicateddue to the fast growing trends in modernization and automation. Almost all machines have rotational

e front matter r 2007 Elsevier Ltd. All rights reserved.

ssp.2007.02.010

ing author.

ess: [email protected] (P.W. Tse).

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dx.doi.org/10.1016/j.ymssp.2007.02.010

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ARTICLE IN PRESSP.W. Tse et al. / Mechanical Systems and Signal Processing 21 (2007) 2794–2813 2795

components, such as bearings and shafts. Ignoring any small failure may lead to inefficient operation orbreakdown of the entire machine. Hence, industries always demand reliable and efficient methods for machinehealth condition monitoring. Since defects in rotational components can cause substantial vibrations, theincrease of vibration becomes an effective indicator for the occurrence of fault in a machine. However, theperturbations caused by the existence of multiple rotational components may introduce interference and noise,which make the vibration-based machine fault diagnosis difficult to perform accurately.

The continuous research in diagnosis techniques makes the vibration-based machine fault diagnosis possibleto provide accurate results in machine health monitoring. There are various methods to analyze vibrationsignals generated from bearings. One of them is the discrete component analysis of spectrum [1]. Based on thespectrum analysis with high resolution, some defect parameters can be estimated, such as harmoniccomponents, type and volume of modulations, which are relevant to the health of inspected bearings. Someother methods make use of the changes in whole spectrum that are caused by a faulty bearing. It is impossiblefor these methods to identify which part of bearing is suffering [2]. Other methods ground themselves withresonance frequency analysis. The defects are identified through the changes of signal envelope in resonancefrequency range of components [3]. Statistical methods are also considered in vibration analysis. It is assumedin these methods that the changes of vibration hold probability distribution as the fault evolves. Steward [4]proposed an approach to diagnose bearings based on the variation of kurtosis coefficient in specific frequencyranges. All of these methods are effective in practical applications if the vibration collected from the inspectedcomponent have not been seriously interfered by other vibrations generated by nearby components. However,for machines that have numerous small components closely packed together, the signal of interest is easilycorrupted by other vibrations and noise that are generated from adjacent components. As a result, thecollected signal is usually the complex superposition of a number of sources. Hence, the ability to recover aparticular type of fault-related signal is vital to the successful application of the aforementioned analyses invibration-based machine fault diagnosis.

Numerous efforts have been put on finding efficient methods for recovering faulty vibration signalsgenerated by a defective machine. Some of the successful examples are the exact wavelet analysis [5], theprincipal component analysis [6], neural networks [7] and adaptive noise cancellation (ANC) [8]. However, theeffectiveness of most of these methods in extracting fault-related vibrations is not always satisfactory,especially when the fault symptom is small whereas the interference or noise is relatively large. ANC can workeffectively for recovery useful signal that is corrupted by severe background noise, but its application requiresmultiple sensor inputs and positioning of the sensors for optimum measurement of signal and noise paths.Wavelet analysis has the ability to detect anomalous signals by applying decomposition to the raw signal [9]. Itcan detect every singular point that is embedded in the signal, so the incipient impulsive vibration can beaccurately extracted. However, its effectiveness will decrease if high interference exists. The principalcomponent analysis can filter out the noise and whiten the observations. Unfortunately in practice, vibrationof rotating machine may be severely corrupted with spatially correlated noise, and therefore the signalsubspace cannot be correctly estimated. In the use of neural networks, it is necessary to satisfy the assumptionof source independence in its applications, and the knowledge about the number of signal sources is alsoneeded. However, such assumptions may not be valid in practice. The ANC possesses the ability to minimizenoise or undesired signal, but it can only filter a correlated or related reference to subtract undesired signalmaximally from the primary input. Therefore, more efficient signal recovery methods are definitely required inorder to make an accurate and reliable diagnosis of bearings.

Recently, the blind equalization (BE), which was initially used to recover sound signals, has had a wideapplication in the field of digital communication. The fundamental idea of BE is to derive the characteristic ofequalizer from the received signal without any prior knowledge about the source and its propagation channels.It can recover the corrupted source even with time delay involved in the signal collection. According to thedefinition of BE, the signal of interest can be recovered by using only one sensor [10]. Another advantage ofBE is that it does not require a training process, which is necessary in many conventional signal equalizationmethods. Because of its advantages in signal recovery, BE has received increasing attention and also showngood prospect in machine fault diagnosis. It is expected to use the method of BE to obtain good result in therecovery of mixed vibration signals that are generated by various components of an operating machine. Thereare few studies on the relevant topics. Peled et al. [11] have automated the process of separating signals using

ARTICLE IN PRESSP.W. Tse et al. / Mechanical Systems and Signal Processing 21 (2007) 2794–28132796

blind deconvolution. They used the kurtosis of each separated signal as a measure for maximization through acost function. A derived Hessian matrix was used in the maximization process of kurtosis for obtainingmultiple bearing vibration sources. The method can work automatically without human interaction, and beeffective in separating impulsive signals generated by bearing faults. However, they do not provide thenecessary mechanism to further verify the separated signal is truly generated by a certain kind of bearing fault.In the work of Jelonnek et al. [12], they applied a BE-based eigenvector algorithm (EVA) to get the optimumcommunication data which is an aggregation of data from many unknown transmission channels. The EVAalso proves to be very useful in recovering the fault-related signal by minimizing noise that is often embeddedin the signal. However, like other methods, EVA also has its own merits and shortcomings when it is appliedto vibration recovery.

EVA possesses an excellent convergence in the equalization and only modest samples of the received signalare required. In addition, EVA uses only one sensor to collect data, which is uniquely suitable for vibration-based rotary machine fault diagnosis. In a rotary machine, such as a motor, the vibrations may be generatedfrom its stator, rotor, shaft, or bearings at both ends. However, due to its compact arrangement and externalcover, only one vibration sensor may be allowed to be installed either at the drive end or the non-drive end ofthe motor. As a result, the sensor is collecting an aggregated signal that is a mixture of several vibrationsources. Even if multiple orientation sensors, such as in radial or axial direction, are used to be mounted on aparticular end, they still can only collect vibrations from a single location. In such case, one may suggestapplying blind source separation (BSS), which depends on multiple sensors to separate several sources [13].Liu and Randall [14] gave an application example of BSS in separating vibration signals from an internalcombustion engine. They used three acceleration vibration signals to separate vibrations caused during theoperation of a cylinder such as fuel injection, combustion, and piston slap. The effectiveness of the separationmethod is partially based on the use of multiple sensors. However, due to various physical limitations, costs,and difficulty in access, the installation of multiple sensors may not be feasible. Hence, the technique of BSS isnot practical if only one sensor is allowed to be installed on the inspected component. In such case, the BE-based EVA that employs only one sensor can be considered. On the other hand, the conventional EVA canrecover only one dominant source from the aggregated signal collected by the single sensor at a time. If thereare two vibration source signals that are simultaneously generated by two different kinds of defects, such as adefective bearing and an eccentric rotor, the source with higher amplitude in vibration will be recovered. Insuch case, since the vibration generated from the eccentric rotor is usually much larger than the vibrationgenerated by the defective bearing, the vibration of the eccentric motor will be recovered as it is the dominantsource. The vibration generated by the faulty bearing will be difficult to be recovered. Hence, the faultoccurring on the bearing will not be known to the operator if conventional EVA is used.

In this research, a modified EVA based on channel extension is applied to the vibration-based bearing faultdiagnosis. The multiple equalized results can be obtained from the modified EVA and they will be furtheranalyzed by our proposed post-processing method so that other less dominant bearing vibration sources canalso be recovered. The purpose of the post-processing method is to determine the relationship between therecovered signals and the bearing component of the inspected machine. The post-processing method is basedon correlation and kurtosis as criteria to investigate each EVA’s equalized source. Hence, any vibration signalthat belongs to the inspected bearing can be correctly recovered.

The modified EVA and the post-processing method construct the framework of our enhanced EVA. Byusing the enhanced EVA, more than one kind of vibration signals can be recovered from the aggregatedvibration. Both vibration signals generated from an experimental platform and industrial machines have beenused to verify the effectiveness of the enhanced EVA. By using the enhanced EVA, the machine operators canbe benefited from the use of a single sensor as provided by the conventional EVA, as well as the ability ofrecovering multiple sources. Hence, our enhanced EVA is well suitable for vibration-based machine faultdiagnosis, especially for the machines that have many small components closely packed together but haveconstraints on installing multiple sensors due to their limited space, accessibility, and physical covers of theinspected components.

The rest of this paper is organized as follows. In Section 2, the basic models of BE are presented. InSection 3, the theory of the conventional EVA is introduced. Its key shortcomings that are related to practicalapplications are also identified. In Section 4, our proposed theories for enhancing the conventional EVA in


recovering vibration signals are presented. Section 5 describes the experiments on using industrial machines toverify the effectiveness of the enhanced EVA. Both the results generated by conventional EVA and ourenhanced EVA are presented for comparison purpose. Section 6 describes the set up of the experimentalplatform to further verify the effectiveness of the enhanced EVA by considering different kinds of componentsand their faults. In Section 7, a generic approach of the enhanced EVA is proposed for machines that havemany small and closely packed components. The approach allows the recovery of several signal sources byusing a single sensor. Finally, the conclusion remarks and future developments for the enhanced EVA arepresented in Section 8.

2. Brief review of BE

The general discrete time model of BE is illustrated in Fig. 1. The original source signal sðkÞ is anindependent, identically distributed sequence of random variables with a mean of zero, a variance ofs2 ¼ EfðsðkÞÞ2g, a skewness of g3 ¼ EfðsðkÞÞ3g and a kurtosis of g4 ¼ EfjsðkÞj4g � 2s4 � jEfs2ðkÞgj2. The sourcein each channel siðkÞ is a random process and the probability density function of typical modulated signal iseven, so it is non-Gaussian with a kurtosis of non-zero ðg4a0Þ and a skewness that vanishes to zero ðg3 ¼ 0Þ.The mixing system contains unknown composite channel hðkÞ, which is assumed to be time-invariant at leastin a short time period. Theoretically, it can be described as a causal possibly mixed phase FIR (finite impulseresponse) hðkÞ ¼ hð0Þ; . . . ; hðlÞ, where l denotes the filter order. Besides linear distortion, the received sequencexðkÞ is corrupted by independent stationary zero mean additive white Gaussian noise vðkÞ. The result of modelunder the assumptions is the output sequence xðkÞ.

xðkÞ ¼ hðkÞnsðkÞ þ vðkÞ or xðkÞ ¼XL

l¼1

hðlÞsðk � lÞ þ vðkÞ, (1)

xðkÞ is observed as the received signal in practical measurement. An inverse filter (blind equalizer eðkÞÞ isrequired to reconstruct the source signal sðkÞ with the received signal xðkÞ: An output signal yðkÞ is obtainedfrom the equalization. Therefore, the main aim of BE is to find an optimal inverse filter eðkÞ which satisfies

yðkÞ ¼ eðkÞnxðkÞ or yðkÞ ¼XL

l¼1

eðlÞxðk � lÞ, (2)

so that yðkÞ represents the recovery of the original source signal sðkÞ. It will be fulfilled when signal yðkÞ

constructed by (FIR) blind equalizer is as close as possible to the delayed original signal sðk � k0Þ in the meansquare error (MSE) sense

MSEðe; k0Þ ¼ EfjyðkÞ � sðk � k0Þj2g ¼ min , (3)

where k0 is the delay of the source signal.Different from conventional adaptive filtering, the BE is applied without access to the original sources or

prior-known knowledge of training sequence. Thus, information about the source must be extracted from thechannel output xðkÞ and equalizer output yðkÞ. The BE achieves it through equalizing the statistics of inputand output of the equalizer. Hence, it is only the output sequence and the probability distribution of the inputsequence that are all the available preliminary information required to adjust the equalizer coefficients in BE.One of the advantages of BE is that it can recover corrupted signals containing with time delay [15]. It makesBE very useful because time delays are always involved in captured signals practically. Another benefit of BE

Fig. 1. The general model of blind equalization.


is that only a single sensor is required for data collection in BE. The use of one sensor is more cost efficientthan other recovery methods. However, in case if the desired source signal has been severely overwhelmed bynoise, then the effectiveness of BE may be questionable.

3. The BE-based EVA (conventional EVA)

One of the solutions for linear BE was found based on the theory of eigenvector [16]. The method is namedas EVA and becomes the most adaptable technique of BE. EVA takes over all assumptions from BE theory. Ageneral model of BE is shown in Fig. 1. From the model of BE, the EVA further adopts an additional virtualequalizer f ðkÞ that is shown in Fig. 2. The virtual equalizer f ðkÞ will have the same order as the equalizer eðkÞ

in Fig. 1. This virtual equalizer is used to generate an implicit sequence as reference data that are necessary insubsequent iterative process for estimation of the reference output zðkÞ,

zðkÞ ¼ xðkÞ � f ðkÞ. (4)

The basic model of EVA is shown in Fig. 2.The EVA requires the two-dimensional fourth-order cross-cumulant matrix for equalizer between the

samples of the received signal xðkÞ and the reference system output zðkÞ. The cross-cumulant matrix with sizeðl þ 1Þ � ðl þ 1Þ is defined as

Czx4 ¼ Efzznxxng � EfzzngEfxxng � EfxzgEfxnzng � EfxzngEfxnzg (5)

and

jzðkÞj2 ¼ zðkÞnzðkÞ; zðkÞzðkÞn ¼ ½zðkÞnzðkÞ�n ¼ ½jzðkÞj2�n ¼ jzðkÞj2, (6)

so finally,

Czx4 ¼ EfjzðkÞj2xxng � EfjzðkÞj2gEfxxng � EfxzðkÞgEfxnznðkÞg � EfxznðkÞgEfxnzðkÞg, (7)

where x and x� denote the vectors,

xn ¼ ½xðkÞ;xðk � 1Þ; . . . ;xðk � lÞ�, (8)

x ¼ ½xnðkÞ;xnðk � 1Þ; . . . ;xnðk � lÞ�T . (9)

Using the autocorrelation matrix of received data xðkÞ

Rxx ¼ Efxxng, (10)

and the vector of the equalizer impulse response

e ¼ ½eð0Þ; . . . ; eðlÞ�T , (11)

we finally get jenCzx4 ej as the criterion. Proper adjustment of equalizer coefficients could be achieved by

maximizing it on the condition that

EðxnxÞ ¼ Ef½ðvneÞn�ðvneÞg ¼ EðenvvneÞ ¼ enEðvvnÞe ¼ enRe, (12)

Fig. 2. The basic model of EVA.


so

tjenRexxj ¼ s2 ¼ const.

The optimization of this criterion leads to a generalized eigenvector problem,

Czx4 eEVA ¼ lRxxeEVA. (13)

The coefficients eEVA ¼ ½eEVAð0Þ; . . . ; eEVAðlÞ�T are obtained by choosing the eigenvectors of R�1xx Czx

4 that isassociated with the maximum eigenvalue jlj ¼ maxfjl1j; . . . ; jlljg,

eEVA ¼1

lR�1xx Czx

4 eEVA. (14)

The result is called the ‘EVA-(lÞ solution’.The wðkÞ is identified as a joint impulse response of whole system,

wðkÞ ¼ hðkÞ � f ðkÞ. (15)

As mentioned before, the criteria that adjust equalizer coefficients, wðkÞ must reach maximum value only onceto guarantee the uniqueness of the solution, that is,

jwðkÞj ¼ maxfjwðkÞjg ¼ wm. (16)

If k ¼ km, the EVA solution converges to optimum result with a minimum MSE (MMSE) under the idealcondition, or ideal equalization

jwðkÞj ¼ jwðkmÞjdðk � kmÞ. (17)

Otherwise, a solution with close MMSE result is achieved, that is, jwðkÞj � jwðkmÞjdðk � kmÞ.The EVA technique has been discussed as a promising tool for machine fault diagnosis in many literatures

[17–19]. Some researches have proven that the EVA as a solution of BE is effective in recovering the periodicand impulsive types of signals corrupted by interference and noise, and collected by only one sensor. Such caseoften occurs in the vibration signals collected from real industrial machines. However, there are still manyshortcomings that are associated with the application of EVA in vibration-based machine fault diagnosis. Forinstance, the vibration signals collected from manufacturing machines are always very complex and prone tonoise and interference generated by various components located adjacent to the sensor. It means that thesignal that is collected with one sensor is definitely a mixture of numerous sources. Although the conventionalEVA can recover the dominant source of the aggregated signal, it can only recover the largest vibrationgenerated from a component that may not even the component that has the sensor installed on it. That is, byusing EVA, the component with the highest amplitude of vibration will be recovered. Other faultycomponents, which have lower vibration amplitudes, may not be recovered. Hence, the machine operatorsmay miss the possible faulty components which may eventually cause serious damage to the inspectedmachine.

4. The enhancement of EVA for the recovery of multiple sources of vibrations

Bearings are the most frequently failed component in rotary machines. Hence, having an effective bearingfault diagnosis is vital to machine health monitoring. To facilitate the conventional EVA for vibration-basedbearing fault diagnosis, and enable the recovery of not only the dominant or primary vibration source but alsosome other subdominant sources from the aggregated signals, the enhancement methods are proposed here.Our enhanced EVA is mainly constructed by the methods of channel extension and post-processing algorithm.Fractional tap spacing (FTS) equalizer is also considered to improve the performance of EVA. The

Fig. 3. The multirate baseband model of BE with FTS equalizer.


advantages of FTS have been discussed in some literatures [20,21]. Fig. 3 presents a general schema of FTSequalizer. The detailed description about FTS equalizer can be found in the works of Jelonnek et al. [12].

In Fig. 3, the symbols " 2 and # 2 describe the processes of up-sampling and down-sampling, respectively,with a factor M ¼ 2, and the time index k refers to the sampling rate, and n indicates a sequence that issampled at M times sampling rate. The received sequence is shown as follows:

~xðnÞ ¼X

l

~hl ~sðn� lÞ þ ~vðnÞ, (18)

where

~sðnÞ ¼sðn=MÞ for even_n;

0 for odd_n:

(

The derivation of the above equation based on the received signal is cyclostationary [22], hence, thegeneralized EVA equation (13) cannot be directly adopted here. Rather, the equalized sequence yðkÞ isdecomposed into a sum of M sequences as follows:

yiðkÞ ¼ eiðkÞ � xiðkÞ, (19)

yðkÞ ¼ ~xðnÞ � ~eðnÞjn¼km ¼X

c

~eðcÞ � ~xðkm� cÞ; c ¼ ZM þ i

)XMi¼1

XZ

~eðZM þ iÞ � ~xððk � ZÞM � iÞ ¼XMi¼1

XZ

eiðZÞ � xiðk � ZÞ ¼XMi¼1

eiðkÞ � xiðkÞ, ð20Þ

where xiðkÞ ¼ ~xðkM � iÞ and eiðkÞ ¼ ~eðkM þ iÞ denote the ith components of ~xðnÞ and ~eðnÞ. According to theliterature [23], xiðkÞ is constructed from sðkÞ by a linear filtering with the ith sub-channel siðkÞ, thus it isstationary and can be defined as,

xiðkÞ ¼ sðkÞ � hiðkÞ with hiðkÞ ¼ ~hðkM þ iÞ. (21)

According to Eqs. (20) and (21), the model that is described in Fig. 3 can be decomposed and extended tomultiple channels that are shown in Fig. 4. That is, the EVA can now be directly applied to x1ðkÞ; . . . ;xMðkÞ.

The main idea of channel extension comes from the literature [12]. We show an enhanced application, whichenables the recovery of multiple aggregated vibration sources. Compared to the model of the conventionalEVA, the enhanced EVA possesses more than one parallel channels, which correspond to the source signalsgenerated by different components of the inspected machine, e.g. ho

ðnÞ; . . . ; heðnÞ that refer to different

Fig. 4. The enhanced EVA for multirate baseband BE model for multiple transmission channels.


propagation paths. In order to recover the original signals, the equalizer and reference system must beconsequently divided into the same number of channels [24]: eoðnÞ; . . . ; eeðnÞ and f o

ðnÞ; . . . ; f eðnÞ. Each of them

is characterized by a specific order of filter li. Total number of equalizer coefficients will be finally adjusted byEVA as

~l þ 1 ¼XMi¼1

ðli þ 1Þ ¼M þX

i

li. (22)

In order to obtain a workable solution for the enhanced EVA, qðkÞ and rðkÞ are defined in the overall systemwith the input, sðkÞ, and the outputs as yðkÞ and zðkÞ.

qðnÞ ¼XMi¼1

hiðnÞneiðnÞ (23)

and

rðnÞ ¼XMi¼1

hiðnÞnf iðnÞ. (24)

The criterion that is based on fourth-order cumulants between the qðnÞ and rðnÞ is used to identify thequality of equalization [25],

maximize jcqr4 ð0; 0Þj subject to rqqð0Þ ¼ s2. (25)

Referring to Eq. (2), yðkÞ can be expressed as a sum of input sequences xðkÞ of the equalizers

yðkÞ ¼XMi¼1

xiðnÞneiðnÞ ¼ ~xn

n ~e, (26)

where the vectors ~xn and ~e with a length of l þ 1 are composed of M subvectors.

~xn

n ¼ ½xn

n;1;xn

n;2; . . . ; xn

n;M �T where xn

n;i ¼ ½xiðnÞ . . . ;xiðn� liÞ�, (27)

~e ¼ ½eT1 ; e

T2 ; . . . ; e

TM �

T where ei ¼ ½eið0Þ; . . . ; eiðliÞ�T . (28)

Then quality criterion of cross kurtosis is applied to yðkÞ that is obtained in Eq. (26),

~Czx

4 ¼ EfjzðkÞj2 ~xn ~xn

ng � EfjzðkÞj2gEf ~xn ~xn

ng � Ef ~xnzðkÞgEf ~xn

nznðkÞg � Ef ~xnznðkÞgEf ~xn

nzðkÞg, (29)

maximize j~en ~Czx

4 ~ej subject to j~en ~Rxx ~ej ¼ s2 ¼ const, (30)

where

~Rxx ¼ Ef ~xn ~xn

ng. (31)

The optimization of this equation leads to the generalized eigenvector problem,

~Czx

4 ~eEVA ¼ l ~Rxx ~eEVA. (32)

Its solution is obtained by choosing the eigenvector with the largest eigenvalue. In this way coefficient vector~eEVA ¼ ½eT

EVA;i; . . . ; eTEVA;M �

T is acquired and applied to BE in FTS configuration with two channels. There is adifferent sequence of filter coefficients eEVA;i ¼ ½eEVA;ið0Þ; . . . ; eEVA;MðliÞ�

T ði ¼ 1; . . . ;MÞ for each channel. Inthe final enhanced EVA solution, the impulse response of equalizer ~eðnÞ ¼ eðkÞ is constructed by interleavingthe eEVA;i components.

To obtain the multiple outputs from the equalization process, different sets of filter coefficients are applied.The filters in the proposed method are set with different lengths. In the examples of this paper, the enhancedEVA will adjust the number of coefficients to 4, 8, 16, 32, 64 for l ¼ 2i ði ¼ 2; 3; . . . ; 6Þ, so that five differentcharacteristic of blind equalizers are obtained. After applying these filters to inspected signals

xðkÞneiðkÞ ¼ yiðkÞ, (33)


the results of the equalized signals yiðkÞ are obtained, each of which may relate to a different vibrationcomponent in the aggregated raw input signal. The algorithm needs further refinement to define the possiblebearing signals of interest that may be overwhelmed by other sources, so the correlation coefficient andkurtosis are used as adequate evaluation criteria to identify faulty bearing vibration signals correctly.Generally, the correlation as a function of the time shift is a measure of the similarity between two signals, andthus it reaches a maximum value when two signals are similar in shape and phase. The correlation function oftwo discrete time signals gðtÞ and hðtÞ is defined by

cðtÞ ¼X

k

gðkÞ � hðtþ kÞ. (34)

Its correlation coefficient (cc) is a number that meets �1 � cc � 1, which presents the level of linear relationbetween two variables. The correlation coefficient is 1 if two variables are perfectly linearly related to eachother positively; it is �1 if the line of their relationship has a negative slope. Zero correlation coefficient meansthat there is no linear relation between the variables.

For the application in vibration-based bearing fault diagnosis, our enhanced EVA introduces a post-processing method into the EVA in channel extension, which involves the evaluation criteria of correlationand kurtosis. After obtaining the multiple equalized signals based on the conventional EVA, first, our post-processing method will calculate the correlation coefficients (cc) between the collected raw vibration signal xðtÞ

and each of the obtained equalized signals yiðkÞ. The signal with the largest value of cc in existing equalizedresults is found for an identification process, and then it will be converted to its frequency domain. From itsfrequency spectrum, we will verify the existence of bearing characteristic frequencies (BCFs) to confirmwhether this signal is generated from a bearing. If it belongs to the desired bearing vibration and the sensor ismounted on the bearing’s housing, then the process is halted as the desired vibration is found. If it does notbelong to the bearing, then it is necessary to search the bearing vibration from the rest of the equalized signals.

Second, to avoid sequential searching just based on cc that is time intensive, a fast searching approach isintroduced here by using Kurtosis. Kurtosis is a statistical indicator to evaluate the distribution of impulsivecharacteristic of a vibration signal [26]. It can be used to measure the distribution of spikiness in the vibrationsignal within a specified time frame. The higher the spikiness, the larger the value of kurtosis. Hence, it isparticularly suited to monitor bearing’s faults as most of them will exhibit impulsive characteristic in itsvibration signal if the bearing is defective. The value of kurtosis for each remained equalized signals will becalculated. The equalized signal that has the largest kurtosis will then be identified. That is, this signal willagain pass through the verification of the BCFs in its frequency spectrum as aforementioned in the first step. Ifthe signal belongs to one of the BCFs, then the process is halted. If the signal that has the largest kurtosis doesnot belong to one of the BCFs, then further step will be processed. However, it is rare that the signal havingthe largest kurtosis is not the bearing vibration except it may be noise.

The identification process will continue if the last step has failed to identify the bearing vibration. Theremaining equalized signals will be ranked again according to their cc or kurtosis and the verification of BCFswill be undertaken. If it is the bearing vibration, the searching process is halted. If not, then the process will berepeated as shown in Fig. 5 until the bearing vibration can be identified or all of the equalized signals areinvestigated. Note that if an equalized signal that may have a large kurtosis value but a low cc value, andcannot be verified as one of the BCFs, then such signal is probably noise.

Using such post-processing identification process, the faulty bearing vibration could be identified with avery high possibility. That is, with the help of the enhanced EVA, both the dominant vibration and theinterested bearing vibration that has been overwhelmed by other larger vibrations can be recovered. Someuseful signals relevant to the components of the inspected machine can also be extracted in this process forfurther identification. The flow chart of our enhanced EVA is presented and shown in Fig. 5. To make it easyto be understood, five equalizers have been used here to recover the overwhelmed bearing vibration. In Fig. 5,L denotes the number of received data samples and J denotes a threshold value that stops the iterationprocedure. The length of blind FIR equalizer is fixed, which are expressed as [q, w, e, r, t] with the value of½8; 16; 32; 64; 96� and then repeated for each length of filter. A real example of recovering the vibration from afaulty bearing overwhelmed by the vibration of a motor which is supported by the faulty bearing is presentedin the next section.

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Measurement: x(k)

Oversampling with factor M = 2

Create two sequences xo(n) and xe(n)

Initialization for two sequences:

Preliminary coefficients f o(k)

Interaction counter i = 0

Total interaction number J

Set-up length of blind equalizers (q,u.e, r,t)

Estimate the autocorrelation matrix

(equation 31)

Determine z(k) and estimate

4th order cross-cumulant matrix

(equation 29)

i< = J

Find the signal with the largest cc in y(k)

Frequency verification for

bearing vibration

No

Yes

No

Yes

Yes

No

Yes

Yes

No

No

End

i = i +1

(k))i(

EVAe~(k)

1)(if =

+

(k))i(

EVAe~(k)x~y(k) +=

Equalized results from the equalization process(k)]y(k),y(k),y(k),y(k),[yy(k) =

All equalized signals investigated?

Frequency verification for

bearing vibration

All equalized signals investigated?

Find the signal with the largest kurtosis in y(k)

For all equalized results:

Calculate their correlation coefficients (cc) and kurtosis values

Delete the signal that has not

passed the verification from y(k)(k)y

Obtaining the recovered signals from the identification process

passed the verification from y(k)

Delete the signal that has not (k)y

Determine equalizer coefficients eEVA (k)~

Fig. 5. The block diagram of the enhanced EVA for recovering sources based on five equalizers.

P.W. Tse et al. / Mechanical Systems and Signal Processing 21 (2007) 2794–2813 2803


5. Recovery results of real vibration in industrial machines

In our previous works [8], two fault-related vibrations that were generated by two faulty components of atraction motor were obtained from the data collections of multiple sensors using the method of ANC. Thediagnosis of the two faulty components was confirmed by the maintenance staff after an overhaul had beendone on the traction motor. To investigate the effectiveness of the conventional EVA and the enhanced EVAin the recovery of vibration signals, the experiments were conducted on the same traction motor using thesame collected signals for comparison purpose.

The real vibration signals from a traction motor in a subway train were simultaneously collected for faultdiagnosis to validate the proposed method in this research. The engine was uncoupled from the gearbox forthe experiments. The motor consists of a 250 kg rotor that was supported by two rolling element bearings, oneof which was located on the drive end and the other on the non-drive end of the motor. The vibrations in threeorthogonal directions of each bearing were, respectively, monitored by three accelerometers at the positionsP2X, P2Y and P2Z. An additional sensor labeled as P2XA was also mounted at the axial direction, but muchcloser to the bearing housing of the motor than the other sensors. The experiment schema with all componentsthat were involved in the vibration monitoring and analysis is shown in Fig. 6. The signals from the sensorswere amplified by a coupler (Kistler 5134), and a digital cassette recorder (Sony PC204Ax) was used to acquirethe signal simultaneously at the sampling rate of 48 kHz for each of the channels. The record length of datawas approximately 1.5min. Then the signals were transmitted to PC with a data acquisition card for furtherprocessing. The sampling rate of data acquisition was 32.8 kHz.

The conventional EVA and the enhanced EVA in this research were, respectively, applied to recover thebuilt-in signals generated by the motor and the bearing. Both accelerometers P2XA and P2X could acquire thevibrations from the motor and the bearing. However, the sensor P2XA was mounted closer to the bearinghousing, and so it would capture more vibration that was generated by the bearing than the sensor P2X.Therefore, the vibration that was obtained at the position of P2XA was analyzed to recover the overwhelmedbearing vibration from all of motor signals. Fig. 7 shows the raw vibration from P2XA and the result after theapplication of the conventional EVA.

There were two faults existing in the tested motor, one of which was a motor eccentric problem due to poorworkmanship on the clearance between the rotor and the stator. The second fault was an outer-race defect thatoccurred in the inspected bearing. The collected raw signal in the top diagram contains the fault-relatedvibrations that were generated by both the bearing and the motor. It is difficult to identify the bearing signalbecause it was overwhelmed by the dominant vibration from the faulty motor. The conventional EVA wasused to adjust the equalizer length to 8 to recover the motor vibration because of its dominant status, which isshown in the bottom diagram of Fig. 7. However, the bearing vibration still remained ambiguous. The resultsof proposed enhanced EVA are presented in Fig. 8, and it shows more efficient ability than the conventionalEVA in the recovery of mechanical vibrations.

P2Z

P2XA

P2X

P2Y

Z

XY

Fig. 6. The schema of the testing system, and the position of four accelerometers in the motor.

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Fig. 7. The raw signal and recovered signal after implementation of the conventional EVA.


As proposed in Section 4, the motor vibration was firstly recovered as a dominant component of thecollected mixture. A filter with the length of 8 was used in the experiment to recover the dominant signal, andthe result is shown in the middle diagram of Fig. 8. The recovered vibration signal had a correlation coefficientof approximately 0.8, which was the largest value in all equalized results. Besides the extracted motor vibrationthat can also be recovered with the method of the conventional EVA, the enhanced EVA enables the recoveryof the second signal component, which is shown in the bottom diagram of Fig. 8. The signal recovered by theequalizer with a length of 32 had the largest kurtosis in all equalized results, and was marked as the seconddominant signal. In addition, its impulsive characteristic further proved that the second recovered vibrationcame from the faulty bearing. With the sampling rate of 32.77 kHz, the period of the impulses can be roughlyestimated through dividing the time length of the signal by the number of periods in it. In the bottom diagramof Fig. 8, the result shows that periodic impacts can be identified and they occur at a time interval of around10ms. Based on the knowledge about the diagnosed bearing (SKF 6215) and its rotation speed (25Hz), it iscalculated that the characteristic frequency of defected outer race is 113.6Hz or a time interval of 9ms. Thisvalue matches closely with the interval of 10ms shown by the impacts in the bottom diagram of Fig. 8. Hence,the impacts are confirmed to be generated by the defective bearing. These conclusions also correspond to theexperimental results that were obtained by the ANC method on the same test [8]. From the above results, itshows that the enhanced EVA is effective in the recovery of multiple vibration sources. Even when thevibrations generated from various mechanical components are captured as a mixture of vibration by onesensor, it is still possible to extract the diagnostically useful signals. That is, machines that have physicallimitations on installing more than one sensor at a time can use such method to recover aggregated signals.

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Fig. 8. The raw signal and two recovered signals after implementation of the enhanced EVA.

P.W. Tse et al. / Mechanical Systems and Signal Processing 21 (2007) 2794–28132806

It is worth to note here that the scales used in representing the amplitudes of the recovered signal are notequal or relatively proportional to the original scales of the amplitudes representing the raw aggregatedsignals. For instance, in the top diagram of Fig. 8, the scale of amplitude for the raw signal is from �4 to 4,whilst, in the bottom diagram of Fig. 8, the scale of amplitude for the recovered bearing faulty signal is from�6 to 6. The problem of such inconsistency on scales is a major discrepancy in separating or recovering signals


using BSS- or BE-based methods. This problem has already been identified by us, and we have proposedsolutions to overcome it in another paper [13]. Nevertheless, such inconsistency of scales does not affect theresults for fault diagnosis. We are not using the increase of vibration amplitude, but the characteristics of thebearing fault as an indicator of the occurrence of fault. As aforementioned, in the bottom diagram of Fig. 8,the period of impacts (10ms) was identified as the inverse of the bearing characteristic frequency of defectedouter race. That is, the appearance of the bearing defect characteristic frequency was used to verify the type ofbearing fault that had occurred. Here, we used the appearance of a certain kind of characteristic frequency asan indicator of the occurrence of a particular kind of bearing fault, but not the direct comparison on thechange of scales of amplitudes between the raw and recovered signals.

6. Recovery results of simulated signal on experimental platform

To further verify the performances of the proposed method for the recovery of mechanical signals, anexperimental platform was built as machinery fault simulator, and it is showed in Fig. 9. The platform wasequipped with an electric motor with changeable rotation speed, which was connected with main shaft of thesystem through a coupler. Two ball bearings were used to mount the shaft. Bearing I that is showed in Fig. 9had a defect in outer race, whereas Bearing II was mechanically healthy without any faults. Two flywheelswere introduced to generate misalignment. This platform was able to generate the required vibrations fortesting. These signals were collected by four sensors in different positions, which are shown in Fig. 9. The firstsensor was installed at the driving end of the motor (sensor 1). The second and third sensors were equipped inthe middle of bearing housing after the coupler of the shaft in the vertical and horizontal directions (sensors 2,3). The fourth sensor was installed on the other bearing housing at the end of the drive shaft (sensor 4). Theexperiment was executed at 70Hz rotation rate of shaft and the vibrations were captured at the samplingfrequency of 40 kHz.

More than one source signal were generated in the experimental system, one of which came from the faultybearing. The mixture of signals consisted of not only the vibration signal generated by the component that wasexpected to be measured by placing the sensor on the component, but also the other vibration signalsgenerated by nearby components placed on the same main shaft. Hence, the signal of the desired componentwas heavily interfered by other undesired signals. The main objective was to recover the vibration that wasrelated with the faulty bearing. The characteristic of this signal was predictable because of the available

3

II

I

21

4

Fig. 9. The experimental platform for the machinery fault simulation.


knowledge about the defect, bearing and relevant experiment parameters. The results will be confirmed basedon these informations.

The results showed that the signal from sensor 1 revealed the usual vibration behavior of a motor that drivesthe shaft. The signal from sensor 4 presented the characteristics of vibrations that were introduced by themisalignment of the rotating wheels. Sensors 2 and 3 were mounted on the defected bearing housing, so thevibration signal that was collected by them contained the defect-related information from the faulty bearing.The raw vibration that was collected by sensor 3 and its spectrum is presented in Fig. 10.

The enhanced EVA was adopted herein to recover the source signal from the faulty bearing and two signalswith different characteristics were obtained. The signals recovered by the equalizers with the lengths of 96 and16, respectively, were marked based on the values of their correlation and kurtosis according to the proposedidentification process. Their appearances in time and frequency domains are presented in Figs. 11 and 12.

According to the first recovered result that is displayed in the upper diagrams in both Figs. 11 and 12, it isobviously shown that the signal indicates the vibration from shaft misalignment because of the second andfifth harmonics (140 and 350Hz, respectively) of the fundamental rotational frequency of 70Hz. It was thefirst recovered signal, and the correlation coefficient between it and the original mixture that was collectedfrom sensor was the largest. The misalignment vibration is regarded as the dominant component in collectedsource in this situation, and thus it can be isolated correctly even if the conventional EVA is employed. Butwith the enhanced EVA, another signal can be recovered from the mixture of sources, which cannot beachieved in the conventional EVA. The second recovered result and its spectrums are shown in the bottomdiagrams of Figs. 11 and 12. In the experiment, the kurtosis of the second recovered signal was 3.1, which wasthe largest value in all equalized results. The investigation in frequency domain of this signal showed that itsmain frequency was 208Hz, so further calculation was made to validate if the isolated signal was generated bya faulty bearing. The schematic diagram of the bearing that was used in the experimental platform together

Fig. 10. The raw vibration signal from sensor 3 and its spectrum.

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Fig. 11. Two recovered signals from the collection of sensor 3.


with its dimension information is shown in Fig. 13. D is assumed as the pitch diameter, D1 and d2 as thediameters of outer race and internal race, d as ball diameter and Z as number of rolling elements. It is knownthat the bearing has a defect in the outer race, so the vibration frequency that was related with this fault iscalculated based on the equations shown in Fig. 13. The calculation result is: f od ¼ 209:8Hz.

The analysis result (208Hz) is very close to the calculated result (209.8Hz). It is concluded that the secondisolated signal belongs to the defected bearing. In addition, the investigation for the characteristic of this signalin time domain also shows that there are lots of impulsive components, which commonly occurs in thebearings that generate vibration. A great deal of energy appears at higher frequency range in the bottomdiagram of Fig. 12. It is a typical frequency spectrum of a faulty bearing with high excitation frequency fromthe faulty bearing housing.

As mentioned in Section 5, the phenomenon of inconsistency in amplitude scale also appears in Figs. 10 and11. Nevertheless, we used the appearance of the second and fifth harmonics (or higher harmonics) on thefrequency spectrum as an indicator of misalignment occurred (the upper diagram of Fig. 12). For determiningbearing faults, the authors used the amount of impulses that exist in the recovered signal, which is in term ofthe kurtosis’s value, as well as the appearance of the bearing defect characteristic frequency. Note that theidentification of bearing fault is based on the calculation of the bearing outer-race defect characteristicfrequency, which is 209.8Hz (the lower diagram of Fig. 12). Again, to avoid the problem of inconsistency ofamplitude scales, we were not using the increase of vibration amplitude, but the characteristic frequency ofeach kind of fault as an indicator of the occurrence of fault. In other words, the inconsistency of scaling theamplitudes does not affect the accuracy in fault diagnosis.

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Fig. 12. The spectrums of recovered signals from the collection of sensor 3.

odf = (1- ) . Z . fs2

1

where . cosα =D

d

and 2

21 dDD

+=

Number of ball bearing 8

Train Frequency FTF 0.375

Ball Pass Frequency Outer (BPFO) 2.998

Ball Pass Frequency Inner (BPFI) 5.002

Element Defect Frequency (pure rolling) 3.742

Ball Spin Frequency (BSF) 1.871

Ball Diameter 0.2813 inches

Pitch Diameter 1.1228 inches

γ

γ

Fig. 13. The diagram and parameters of the bearing used in the experiments.



The results of experiments show the new abilities of the enhanced EVA in machine fault diagnosis, whichallow more than one source signals to be recovered with only one sensor. The chances to identify fault-relatedsignals are increased greatly even when they are in the initial stage or overwhelmed by other dominant signals.

7. Potential of proposed method for complex machine fault diagnosis

It has been mentioned in the previous sections that the signal collected from the machine with one sensor isalways a mixture of numerous sources, and conventional EVA cannot meet the requirement of recoveringmultiple faulty vibrations from a machine. The results of the conducted experiments show our proposedmethod has great potential for complex machine fault diagnosis. Based on the algorithm of our enhancedEVA, a new model of monitoring system and its schema can be theoretically proposed here.

The general model of mechanical system of analysis is presented in Fig. 14 with multiple sources and thedata collection of a single sensor in consideration. Each source signal siðkÞ that represents the signal generatedfrom a different component of machine has its own propagation channel with an additive noise. Therefore, thesignal that is collected by a single sensor will be a composition of various sources and noises with different timedelays. For example in practice, if the objective is to discover the fault of bearing, it is always necessary torecover the actual bearing signal from the noises and vibrations that are generated by other components ingood or bad conditions. Only when all other signal components are small enough to keep the dominance levelof vibration in received signal, the conventional EVA can achieve successful result. Otherwise it is difficult torecover any overwhelmed signal of interest. This is the main predicament that must be concerned beforeapplying the conventional EVA.

Our proposed method can enhance the effectiveness of the conventional EVA in machine fault diagnosisthrough the construction of a single sensor input and multiple recovery outputs. The suggested schema isillustrated in Fig. 15. Only one sensor is used to collect the mixed signals, minimizing the requirement fornumbers of sensor in many practical applications. Compared with the basic model of EVA, the new enhancedEVA keeps the mixing model of multiple unknown channels in forward part of the conventional EVA, andextends the number of blind equalizer in output part from one channel to i parallel channels. The equalizersare, respectively, used to eliminate the distortion that is introduced in each propagation path between sourcesignal siðkÞ and the sensor signal xðkÞ. And the proposed post-processing method is used to recover andidentify possible existing vibrations of interest in the equalized results. The practice described in this papermainly focuses on bearing diagnosis. Selection and application of other effective identification criterion foridentification process can enable the recovery of different faults relevant to the other components of theinspected machine. The recovery of multiple sources based on this proposed structure will be very useful forthe practical machine fault diagnosis.

Fig. 14. The model of signal generation in machine and collection with a single sensor.

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Fig. 15. Basic schema of proposed method with single sensor input and multiple recovery output.


8. Conclusions

The EVA, which is one of the most efficient solutions to BE, is discussed in detail in the paper. Itsapplication in the recovery of vibrations for machines fault diagnosis is investigated. Based on the analyzedresults, the shortcomings of the conventional EVA for the processing of mechanical vibration signals areidentified. In order to solve these problems, an enhanced algorithm of BE based on EVA is proposed in thisresearch. The enhancement of EVA includes the extension and modification of the conventional EVA, so thatit can perform successfully in complicated mechanical applications. Experiments with simulated signals andreal vibrations from industrial machine are conducted to demonstrate the effectiveness of enhanced algorithmin the paper.

Through extensive analysis of relevant theories and experimental results, the advantages of the newapproach are presented herein. First, the enhanced EVA is efficient for recovering bearing vibration even if thebearing vibration is relatively smaller than other source vibrations aggregated in the raw signal. For example,in the measurement of vibrations through the use of a single accelerometer installed on the bearing housing,even several sources of vibrations generated by the bearing and its nearby components are measured together,more than one source signal could be recovered. Second, the application of the conventional EVA is oftenconstrained by the selection of parameters, such as filter length, sample number, and iteration number. Theproper selection of each parameter will affect the type of signal being recovered and its accuracy. Ourenhanced EVA provides the ability to adjust different equalizer lengths in order to produce different recoveredsignals so that the signal that has small signal-to-noise ratio or has been seriously interfered by other signalswill not be ignored in the recovering process.

Although the results generated by our enhanced EVA are satisfactory, there are some minor problemsneeded to be considered when applying this algorithm to practical vibration-based fault diagnosis. The minorproblems include the stability of the algorithm when it is subjected to a signal with too small signal-to-noiseratio, and the automation of the parameter selection process without human interaction. The verificationmechanism can be further enhanced to recover the signal of interest even when its signal-to-noise ratio isrelatively small. Currently, we are using the characteristic frequencies of different kinds of faults forverification. In future, we can use correlation or covariance to identify those interested signals. Forautomating the selection of equalizer length or other parameters, we can employ other high order statistics asevaluation indexes to automatically obtain the optimal parameters.

Compared with conventional EVAs, our enhanced EVA has more prospects for complicated vibrationrecovery. It provides the possibility to recover more than one source signal even only using a single sensor.Such kind of multiple sources recovery is impossible by using conventional EVAs except using BSS installedwith multiple sensors at least equal to the number of possible sources. As the industrial manufacturingmachines have been continuously becoming complicated, vibration-based fault diagnosis is more and moredifficult to be practically used. Hence, more sophisticated signal processing methods, such as our enhancedEVA, are required to be employed so that fatal breakdown of machine and interruption of production can beminimized.


Acknowledgements

The work that is described in this paper was fully supported by a grant from the City University of HongKong, Hong Kong, China (Project no. 7001808).

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