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Mechanical Systemsand

Signal ProcessingMechanical Systems and Signal Processing 19 (2005) 939954

A method for the correlation dimension estimation for on-linecondition monitoring of large rotating machinery

Alberto Rolo-Naranjo , Mar a-Elena Montesino-OteroHigher Institute of Nuclear Sciences and Technology, Ave Salvador Allende y Luaces, Quinta de los Molinos,

Plaza de la Revolucio n, Ciudad Habana, CP 10600, AP 6163, CubaReceived 19 August 2003; received in revised form 23 July 2004; accepted 4 August 2004

Available online 18 September 2004

Abstract

In this paper, we introduce a robust method for the correlation dimension estimation in an automaticway for its implementation in on-line condition monitoring of large rotating machinery. The method iscalled Automatic-Attractor Dimension-Quantitative Estimation (A-AD-QE). It is based on a systemicanalysis of the second derivative of the correlation integral obtained by the Grassberger and Procacciaalgorithm. The A-AD-QE method concentrates its attention on the scaling region denition and it also hasthe possibility to analyse the geometrical structure of the obtained multidimensional second derivative of the correlation integral and its relation with the pseudo-phase portrait. The effectiveness of the introducedmethod was veried by means of the calculation of well-known analytic models as Lorenz attractor, van derPol oscillator and Henon Map. Furthermore, the A-AD-QE method was applied to process real vibrationsignals of large rotating machines. As a typical example we analysed four measurements, recorded atdifferent points. The obtained results demonstrate the applicability of the method in real vibration signalprocessing for this kind of machines.r 2004 Published by Elsevier Ltd.

1. Introduction

The development of nonlinear dynamics theory has brought new methodologies to identify andforecast complex nonlinear vibration behaviours. Nowadays, the application of nonlinear

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0888-3270/$- see front matter r 2004 Published by Elsevier Ltd.doi:10.1016/j.ymssp.2004.08.001

Corresponding author.E-mail address: [email protected] (A. Rolo-Naranjo).
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procedures in condition monitoring is a very active research eld. One of the main objectives of the introduction of these techniques is to extract the maximum of the all-signicant diagnosticinformation from the original signals.

Many authors have treated the evaluations of the chaotic patterns that can appear in somemechanical systems [15]. For large rotating machines the chaotic behaviour is related to theinteractions in the rotor/bearing/stator system. The system nonlinearity can be found in thediscontinuous stiffness, damping, surface friction and impact. Muszynska and Goldman [2]reported an important theoretical and practical review of these phenomena.

The system nonlinearity evaluation by means of traditional descriptors, i.e. FFT-based methods,can give inappropriate results. These behaviours are described and characterised by means of nonlinear tools such as correlation dimension, the pseudo-space portraits and others. Nowadays, thecorrelation dimension has been widely used as a powerful tool for interpreting irregular signals inelectrical, mechanical and other engineering domains. The correlation dimension as a diagnosticsindicator gives information about the dimensionality and complexity of dynamical system. Veryrecent studies applied this descriptor as a representative approach to the fault characterisation [612].

Logan and Mathew [6,7] related the correlation dimension with faults in rolling bearingscondition monitoring. They shown that the correlation dimension can classify three major rollingelement-bearing faults: outer race fault, inner race fault and roller fault. Also, the methodologyfor the practical computation of the correlation dimension, based on the embedding procedure isdescribed there. Based on this methodology, all the parameters for the embedding procedure areintroduced manually by the user. Finally, the user ts the straight line of the correlation integralplot interactively (on screen). Furthermore, the user has to x the qualitative and quantitativeresults of the calculation. Thus, the correlation dimension estimation is strongly inuenced by theuser. Better results can be expected if the embedding parameters were dened in correspondencewith the studied dynamic system.

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Nomenclature

C r; m correlation integralY Heaviside functiont lag timem embedding dimensionD 2 (m) correlation exponent estimated from m-dimensional embedding spaceD 2 correlation dimensionN data volume of time seriesx i time seriesX i embedding spaceA-AD-QE automatic-attractor dimension-quantitative estimationLPZ length of the plateau zoneBPFW resizable xy-band-pass lters windowsx height of the BPFWL length of the BPFW

A. Rolo-Naranjo, M.-E. Montesino-Otero / Mechanical Systems and Signal Processing 19 (2005) 939954940

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In [8], the correlation dimension is applied for gearbox fault diagnosis and its great potentialityas a diagnostic indicator is shown. Craig et al. [10] used the correlation dimension in conditionmonitoring of systems with clearance. One of the main results in this paper is related with thepotentiality of chaos techniques, as an alternative approach, in health monitoring of plant ormachinery.

Wang et al. [10,11] introduced a very important contribution for the diagnostic process of largerotating machinery and gearboxes by means of the use of nonlinear methods. A high sensitivity of the pseudo-phase portrait, the singular-spectrum analysis as well as the correlation dimension wasshown. More studies related with the application of pseudo-phase portrait, as indicator can befound in [13,14]. In [14], a qualitative approach to the sensitive evaluation of the pseudo-phaseportraits is given.

Instead of the correlation dimension value, Koizumi et al. [12] introduced the correlationexponent as a fault descriptor, estimated with the modied version of the well-known Grassberger

and Procaccia algorithm. The modied version was based on a re-embedding procedure reportedby Fraedrich and Wang [15]. These authors applied the correlation exponent to investigate thechattering vibration during the cutting process. Good results can be obtained with the correlationexponent of some previous dened embedding dimension and xed scaling region. Nevertheless,its value only gives a partial portrait of the real complexity of the studied dynamical system.

One of the problems derived for the correlation dimension determination in multidimensionalanalysis, is the scaling region denition of the correlation integral as a function of the embeddingdimension. Quite frequently the derivative of the loglog plot of C (r,m) vs. ln r is used for itscalculation [612]. Here, the correlation dimension is expressed as a function of the embeddingdimension:

D2r; m ffi dlnC r; mdln r (1)

which is approximated by

D2r; m DlnC r; m

Dln r ; (2)

where operator D is dened by D f r ffi f r 1 f r [16]. Relating the plateau appearance tothe attractor dimension value, this method provides good visual results. The plateau zone(sometimes tted by eyes) denes the correlation dimension value. Regardless of their advantages,the correlation dimension calculated with this method introduces some errors associated to the

observers subjectivity and it also limits the velocity of the decision in accordance with itsintroduction as an indicator in a real system condition monitoring.A statistical approach to the correlation dimension estimation is given in [17]. The authors

introduce a method based on the maximum-likelihood principle, by means of which, the explicitexpressions for the maximum-likelihood estimate of correlation dimension and its asymptoticvariance are derived. They show how the w2 test is used to nd the upper cutoff of the scaling region.

We can realise the correlation dimension has a high applicability to the fault diagnostic of mechanical system but its calculation in an automatic way is lacking. In this paper, we introduce arobust method for the correlation dimension estimation in an automatic way for itsimplementation in on-line condition monitoring of large rotating machinery. The method is

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called Automatic-Attractor Dimension-Quantitative Estimation (A-AD-QE). It is based on thesystemic analysis of the second derivative of the correlation integral dened by Grassberger andProcaccia algorithm. The A-AD-QE method concentrates its attention on the scaling regiondenition and it also has the possibility to analyse the geometrical structure of the obtainedmultidimensional second derivative of the correlation integral and its relation with the pseudo-phase portrait. Some applications of this method to the monitoring and surveillance of NuclearPower Plant can be found in [18,19].

One of the most important properties of the correlation dimension is its high sensitivity to thedynamical changes of the analysed system. Its application to the real system condition monitoringimproves the real signal characterisation. More than 180 measurements of different points of alarge rotating machine were analysed in order to evaluate the applicability of the introduced A-AD-QE method. Four of them are presented in this paper.

The structure of this paper is as follows: In Section 2 a theoretical background of the A-AD-QE

method is given, starting with a brief review of the embedding procedure and some details aboutthe correlation dimension. In the same section, the introduced method is tested by means of well-known analytic models, such as Lorenz attractor, van der Pol oscillator and Henon Map. InSection 3, we show some applications of the A-AD-QE method to the correlation dimensionestimation of real measurements of a large rotating machine. Finally, the main conclusions aregiven in Section 4.

2. Theoretical background

2.1. Embedding procedure

The embedding procedure is the rst step of the phase space reconstruction of a dynamicalsystem from the observation of a single variable. The most common phase space reconstructiontechnique is the method of delays (MOD) proposed by Takens [20]. By means of MOD it ispossible to build an attractor by replacing the derivatives with delayed repetitions of only onemeasured variable of the system.

The time series x1; x2; x3; . . . ;xN is represented as a sequence of vectorsX i fx i ; x i t ; . . . ;x i m 1t g; (3)

where i 1; 2; . . . ;N m 1t ; where N (m 1) is the length of the reconstructed vector X i , m is

the embedding dimension of the reconstructed phase space and t is the lag time in units of sampling interval.In recent investigations, the calculation of the embedding parameters t and m is a question of

special interest. The optimal choice of these embedding parameters depends of the respectiveapplication [21]. A good denition of t is of importance for the MOD to give sensible results. Twomajor problems are described in [22]: redundancy and irrelevance. The rst one is related to thechoice of t as small as possible. In this case the consecutive measurements of the reconstructedvectors will give nearly the same results. Hence, the topological vectors constructed via the MOD,will be stretched along the diagonal in the m-dimensional embedding space and thus the analysisof the picture of the attractor will be very difcult [23].

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The second problem (irrelevance) appears with the choice of the t too large. In this case thereconstructed vectors become totally uncorrelated and the extraction of any information from thisphase space picture becomes impossible.

In this paper, the rst zero of the autocorrelation function Rt 0 was used in order to avoidboth mentioned problems related with the t denition (see Eq. (4)). For all reconstructeddynamics dened with this value, the pseudo-phase portraits showed a regular behaviour andreected clearly the main trajectories of the dynamical system

Rt P

i xi x xi t x

Pi

xi 2 ; (4)

where, x is the arithmetic mean.The denition of the embedding dimension in this paper was taken according to the Takens

theorem, where the number of m-reconstructed vectors should be mX (2D 2+1). For theapplication of the A-AD-QE method to the unknown dynamical systems, we suggest to carry outa preliminary study where m should be calculated until 21. This value was enough to characterisethe low-dimensional systems studied in this paper. Therefore, an optimum value of m should bedened in order to decrease the processing time in on-line measurements.

2.2. Correlation dimension

The correlation dimension is derived from the correlation integral, which is a cumulative

correlation function that measures the fraction of points in the m-dimensional reconstructed spaceand is dened as

C r; m 2

N mN m 1XN m

i ; j 1 j 4 i

Y r xi ; m x j ; m ; (5)

where Y is the Heaviside function, such that Yx 0 if xp 0 and Yx 1 for x4 0; J y Jindicates the Euclidean norm of the vector, N m N m 1t is the length of the reconstructedvectors and r is the correlation length [24].

C (r) is related with the correlation dimension by means of the power law:C r; m rD2m; (6)

where D2(m) is the correlation exponent and varies with the increase of m. The particularcorrelation exponent can be found as the slope over the lineal region (scaling region) from theloglog plot of C r; m vs. ln( r). The standard deviation of the t is taken as the error of theobtained dimension value. A minimal value for the standard deviation directly leads to a goodapproach for the correlation exponent estimation.

For low-dimensional dynamic systems, the plot of D2(m) vs. m may converge for sufcientlylarge m. The obtained value for the plateau zone, denes the correlation dimension D2.

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2.3. The A-AD-QE method

The A-AD-QE method is called to be a robust tool for the D 2 estimation in an automatic wayfor its implementation in on-line condition monitoring of large rotating machinery. Its principalidea is to provide a D2 denition taking as reference the second derivative application to thecorrelation integral (d 2lnC r; m=dln r2 vs. ln r i =r0).

The implementation of the A-AD-QE method is based on the use of resizable xy-band-passlter windows (BPFW) with dimensions: height x and length L, over all points of the normalisedsecond derivative of the correlation integral. This BPFW is moving in a zero-vicinity of x-axis of the second derivative function.

Once the second derivative to the correlation integral is computed, the following stepin the A-AD-QE method is an iterative change of the size denition of the BPFW. Initialparameter x should be taken as minimum as possible and L increases accordingly with the

number of analysed points, which should satisfy that their ordinate is smaller than the heightof the specic window, according to the iterative process. The necessary and sufcient con-ditions to stop this process are the existence of the appropriate number of continuous pointsthat are within the specic BPFW dimensions. In our experience the number of point shouldbe higher or equal to six, in order to obtain a good approach to the proper scaling regiondenition of the correlation integral. As an important result, the difference among the initialand nal points of the BPFW (length of the plateau zoneLPZ=ln( r INITIAL /r 0) ln( rEND /r 0))denes the length of the best plateau zone LPZ, which is located in a zero vicinity. TheLPZ denition is in correspondence with the scaling region of the correlation integral, whichallows the D2 determination in an automatic way. This is a principal advantage of the intro-duced method in on-line monitoring system. Fig. 1 shows a simplied owchart of theA-AD-QE method.

The appearance in the correlation integral plot of more than one scaling region is a realproblem for some mechanical systems [7]. In this case, the A-AD-QE method denes the rst LPZas a criterion for the scaling region denition.

The introduced method was tested by means of well-known analytic models, such as van derPol oscillator [25], Henon Map [26] and Lorenz attractor [27]. The obtained results for the D2estimation are comparable with other reported models (see Table 1 ).

Fig. 2 shows a schematic representation of practical application of the A-AD-QE method. Inthis gure, the correlation integral as well as its rst and second derivatives of the 12th embeddingdimension of the Lorenz attractor are shown. The LPZ coincides with the plateau zone in the rst

derivative plot and with the scaling region of the correlation integral. The least square regressionof the selected scaling region yields that the slope of the scaling region is 2.05 7 (o 0.01). The D 2for this model is 2.07.

3. Case study

In order to test the applicability of the A-AD-QE method in on-line monitoring, vibrationmeasurements of large rotating machines have been analysed. The main characteristics of these

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analysed signals and parameters for the embedding procedure are shown in Table 2 . The choice of the sampling rate was preceded by a study of the spectrum for frequencies up to 5.4 kHz(90 1X).

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Fig. 1. Simplied owchart of the A-AD-QE method. (*) An iterative procedure, where the size of BPFW x and lengthL changes (see details in Section 2.3).

Table 1Comparison between reported D 2 value and obtained by A-AD-QE method for different well-known models

Model a D 2 reported D2A-AD-QE Absolute error Data length

Lorenz attractor 2.07 2.05 0.02 2050

van der Pol oscillator 1.00 1.08 0.08 2050Henon map 1.26 1.25 0.01 2050

Lorentz attractor: d x=dt s y x; d y=dt xz rx y; dz=dt xy bz; s 10; r 28 and b 8=3:van der Pol oscillator: q2x=qt a1 x2 qx=qt kx f cos O t; a 5; k 1; f 1 and O 2:446:Henon map: X n 1 1 aX 2n Y n; Y n 1 bX n; a 1:4 and b 0:3:

a Characteristic equations of the analysed models.


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3.1. Experimental procedure

The monitoring system is shown in Fig. 3 . All transducers were attached to the uid-lmbearings of a large rotating machine and used to generate scalar time series for the embeddingprocedure. The signals for all measurements (more than 180) were sampled at 1.0 ms. The datavolume was taken as 10 records of 1024 points each.

The comparison between all measurements has the objective of analysing the A-AD-QEmethod sensitivity under different dynamical states. The recorded signals were sampled fordifferent bearings, at 15 with the same conguration of the data collection system ( Fig. 3 ). In thispaper, four measurements are evaluated (one measurement for each bearing at 14). The

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C ( r,m )

d ln (C ( r,m ))/d ln(r)

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Fig. 2. Schematic representation of the A-AD-QE method. The LPZ in the second derivative of the correlation integralcoincides with the plateau zone in the rst derivative of the correlation integral and with its scaling region.

Table 2Time series characteristics and embedding parameters

Parameter Nomenclature Value

Sampling time T s 1.0msData volume N 1024No. of records N rec 10Embedding dimension m 21


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nomenclature of the measurements is taken with the capital letter B and subsequently, the bearingnumber and its respective direction.

The waveforms after normalisation by its maximum amplitudes are shown in Fig. 4 . Note howthe signal for the B1V is more irregular than others, with a high inuence of a low frequency (sub-rotational frequency). The oscillation behaviour for the B2V and B3V is more regular, taking asreference the pick value evolution of the main period of the signals. In addition, the B2V signalseems to be less contaminated by noise. In the case of the B4 V measurement, its waveform isdisplaced upwards and it is expected that this behaviour be manifested in the dynamicsreconstruction.

3.2. Results

The reconstructed pseudo-phase portraits, obtained from the raw signals, are displayed inFig. 5 . The trajectories of the reconstructed pseudo-phase portraits show clear topologicaldistributions, giving the possibility to analyse the dynamical system complexity and their directrelation with the time series.

3.3. Discussion

The preliminary study in the reconstruction of the phase space until m 21 of the analysedmechanical systems corroborated that the saturation value of D2(m) vs. m plot for m4 6 wasreached. According to this result, the application of the A-AD-QE method was done for m 10:

Fig. 6 shows the correlation integral (ln( C (r,m )) vs. ln( r/r 0). We observe that the plots of B1V,B3V as well as B4V are similar and displayed a high slope for the low abscissas. This behaviour isdue to the noise inuence and will be corroborated in this study. In fact, the measurement B2V isless corrupted by noise, its pseudo-phase portrait is more regular (in comparison with other

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Fig. 3. Monitoring system for data collection.


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analysed signals) and its correlation integral shows a higher degree of smoothness. The higherabscissas of all curves are similar and we expect to obtain a clear plateau zone in thedlnC r; m=dln r vs. ln( r/r 0) plot as indicator for the existence of an attractor, even though theanalysed signals are contaminated by noise.

As expected, plots of the rst derivative of the correlation integral (d lnC r; m=dln r vs.ln( r/r 0)) display a plateau zone with values among 1 and 2 for all measurements ( Fig. 7 ). Thisindicates that the analysed oscillating processes are dominated mainly by a low-dimensionaldynamic system. More detailed discussion about the slope zones classication can be foundin [27].

Throughout the second derivative plot of the correlation integral, in addition to the denition

in automatic way of the D2, the A-AD-QE method provides the possibility of a very clear zonedenition of the plot. Fig. 8 shows a typical example of the zones denition using the secondderivative plot of the B4V measurement ( m 32 21). For a particular study, we suggest to analyseonly one embedding dimension of the saturation region of D2(m) vs. m.

Each zone in the second derivative plot has its practical value. In this paper we concentratedour attention to the second one, which is related directly to the scaling region denition.Nevertheless, it is important to give some remarks about the others. Zones I and III can be usedfor practical evaluation of the noise inuence on time series. Furthermore, the minimum value forthe ln( r/r 0) is related to the complexity of the analysed dynamics. The geometric conguration of zone III can be used to evaluate the quality of the reconstruction of the dynamics.

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Fig. 4. Waveforms after normalisation of all measurements by its maximum amplitudes.


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Fig. 9 shows the second derivative of the correlation integral (d 2lnC r; m=dln r2 vs. ln( r/r0)) and the LPZ after application of the A-AD-QE method. It can be observed that an importantqualitative approximation can be made between the local complexity of the pseudo-phaseportraits and the minimum value ln( r/r 0) of zone I. Here the minimum value is related to thesmaller complexity of the analysed dynamic system. This statement is related directly to thepseudo-phase portrait behaviour (see Fig. 5 ).

An important parameter derived for the A-AD-QE method is the LPZ value. Table 3 showsthat the maximum value for LPZ is obtained for the B2V, which shows a more regular pseudo-phase portrait (see Fig. 5 ). The analysis of this parameter represents an important quantitativeindicator to the dynamic characterisation.

The obtained results for the evaluated raw signals corroborate their low complexity. For a realvalue of D2 estimation, a nonlinear lter should be applied. Nevertheless, for the analyseddynamics and in general, for normal condition monitoring of large rotating machinery, the lterapplication will facilitate the plateau zone denition and will improve the applicability of the A-AD-QE method. Furthermore, for this kind of machinery, we would like to remark that noise isan additional diagnostic source.

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As it was stated before, the zone II is the most important one because it denes the LPZ and thecorresponding scaling region as a function of m. The main results of the A-AD-QE methodapplication are shown in Table 3 . The obtained D2 values have direct relation with the dynamical

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complexity. Note that the standard deviation for the least square regression analysis is less than0.01 according to the dened scaling region. This result demonstrates the high accuracy of the

method.

4. Conclusions

In this paper, a robust method (A-AD-QE) for the correlation dimension estimation in anautomatic way is introduced, for its implementation in on-line condition monitoring of largerotating machinery. The applicability of the introduced method for the scaling region denition isdiscussed. In addition, it is demonstrated that the A-AD-QE method provides the possibility toanalyse the geometrical structure of the obtained second derivative of the correlation integral and

its relation with the pseudo-phase portrait.An important parameter derived for the A-AD-QE method is the LPZ. This quantitativeindicator is used to characterise the studied dynamics.

The effectiveness of the introduced method is veried by means of the calculation of well-known analytic models. Furthermore, the A-AD-QE method is applied to processing of realsignals, recorded from different points of a large rotating machine. In both cases, the results showits high sensitivity, taking as reference the obtained values for the standard deviation of the leastsquare regression.

Further works in this thematic should be directed to the verication of the introduced methodin more complex dynamics and in other mechanical systems.

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Acknowledgements

One of the authors (A.R-N) is very grateful for the nancial support of the German AcademicExchange Service (DAAD) for a 3 months research grant in the Technomathematic Center of the

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Fig. 9. Second derivative plot of the correlation integral for all analysed dynamics based on A-AD-QE method.

Table 3Results of A-AD-QE methods and D 2 for all analysed dynamics

Measurement LPZ by A-AD-QE method Correlation dimension ( D 27 s )

|ln( r1/r0)| |ln( r1/r0)| LPZ

B1V 1.37 1.07 0.30 1.86 7 (o 0.01)B2V 2.13 1.08 1.05 1.15 7 (o 0.01)B3V 1.11 0.59 0.52 1.14 7 (o 0.01)B4V 0.99 0.65 0.34 1.34 7 (o 0.01)


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University of Bremen. He would also like to express his gratefulness to Prof. Dr. Peter Maa b andto Dr. Torsten Ko hler for the interesting discussions and their invaluable scientic advice.

References

[1] A. Muszynska, Whirl and whiprotor bearing stability problems, Journal of Sound and Vibration 110 (1986)443462.

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