online cm

 

ARTIC TICLE LE IN PR PRESS ESS AR

Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 19 (2005) 939–954 www.elsevier.com/locate/jnlabr/ymssp

A method for the correlation dimension estimation for on-line condition monitoring of large rotating machinery ´ a-Elena Montesino-Otero Alberto Rolo-Naranjo, Marı  ´a-Elena Higher Institute of Nuclear Sciences and Technology, Ave Salvador Allende y Luaces, Quinta de los Molinos, Plaza de la Revolucio´ n, n, Ciudad Habana, CP 10600, AP 6163, Cuba Received 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 automatic way for its implementation in on-line condition monitoring of large rotating machinery. The method is called Automatic-Attra Automatic-Attractor ctor Dimension-Quantit Dimension-Quantitative ative Estimation Estimation (A-AD-QE). (A-AD-QE). It is based on a system systemic ic analysis of the second derivative of the correlation integral obtained by the Grassberger and Procaccia algorithm. The A-AD-QE method concentrates its attention on the scaling region definition and it also has the possibility to analyse the geometrical structure of the obtained multidimensional second derivative of  the correlation integral integral and its relation relation with the pseudo-pha pseudo-phase se portrait. portrait. The effectivenes effectivenesss of the intro introduced duced method was verified by means of the calculation of well-known analytic models as Lorenz attractor, van der Pol oscillator and Henon Map. Furthermore, the A-AD-QE method was applied to process real vibration signals sig nals of large large rotatin rotating g machines machines.. As a typical typical example example we analysed analysed four mea measur sureme ements nts,, rec record orded ed at different points. The obtained results demonstrate the applicability of the method in real vibration signal processing for this kind of machines. r 2004 Published by Elsevier Ltd.

1. Introducti Introduction on

The development of nonlinear dynamics theory has brought new methodologies to identify and foreca for ecast st comple complex x nonline nonlinear ar vibrat vibration ion behavi behaviour ours. s. Nowada Nowadays, ys, the applic applicati ation on of nonlin nonlinear ear 

Corresponding author. E-mail address: [email protected] address:  [email protected] (A. Rolo-Naranjo).

0888-327 0888 -3270/$ 0/$ - see front matter r 2004 Published by Elsevier Ltd. doi:10.1016/j.ymssp.2004.08.001

 

ARTIC LE IN PR PRESS ESS ARTICLE

940

A. Rolo-Naranjo, M.-E. Montesino-Otero / Mechanical Systems and Signal Processing 19 (2005) 939–954

Nomenclature

C ðr; mÞ  correlation integral Yð Þ   Heaviside function t   lag time m   embedding dimension D2   (m) correlation exponent estimated from  from   m-dimensional embedding space D2   correlation dimension N    data volume volume of time series series xi    time series X ii     embedding space A-AD-QE automatic-attractor dimension-quantitative estimation LPZ LP Z le leng ngth th of th thee pl plat atea eau u zo zone ne xy-band-pass BPFW resizable  resizable   xy -band-pass filters windows x   height height of the BPFW L   length length of the BPFW

procedures in condition monitoring is a very active research field. One of the main objectives of  the introduction of these techniques is to extract the maximum of the all-significant diagnostic information from the original signals. Many authors have treated the evaluations of the chaotic patterns that can appear in some mechanical systems   [1–5]. [1–5]. For large rotating machines the chaotic behaviour is related to the intera int eracti ctions ons in the rotor/ rotor/bea bearin ring/s g/stat tator or system system.. The system system nonlin nonlinear earity ity can be foun found d in the discon dis contin tinuou uouss sti stiffn ffness ess,, dampin damping, g, surfac surfacee fri fricti ction on and impact impact.. Mus Muszyn zynska ska and Goldma Goldman n   [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 can give give inappr inappropr opria iate te result results. s. These These behav behaviou iours rs are descri described bed and cha chara racte cteris rised ed by means means of  nonlinear tools such as correlation dimension, the pseudo-space portraits and others. Nowadays, the correlation dimension and has been used as a powerful for interpreting irregular signals in electrical, mechanical otherwidely engineering domains. Thetool correlation dimension as a diagnostics indicato indi catorr gives gives informati information on about about the dimension dimensionalit ality y and complex complexity ity of dyna dynamica micall system. system. Very recent studies applied this descriptor as a representative approach to the fault characterisation  [6–12].  [6–12]. Logan Log an and Mathew Mathew   [6,7]   relate related d the correl correlati ation on dimens dimension ion with with faults faults in rol rollin ling g bea bearin rings gs condition monitoring. They shown that the correlation dimension can classify three major rolling element-bearing faults: outer race fault, inner race fault and roller fault. Also, the methodology for the practical computation of the correlation dimension, based on the embedding procedure is described there. Based on this methodology, all the parameters for the embedding procedure are introduced manually by the user. Finally, the user fits the straight line of the correlation integral plot interactively (on screen). Furthermore, the user has to fix the qualitative and quantitative results of the calculation. Thus, the correlation dimension estimation is strongly influenced by the user. Better results can be expected if the embedding parameters were defined in correspondence with the studied dynamic system.

 

ARTIC LE IN PR PRESS ESS ARTICLE

A. Rolo-Naranjo, M.-E. Montesino-Otero / Mechanical Systems and Signal Processing 19 (2005) 939–954   941

In [8] In  [8],, the correlation dimension is applied for gearbox fault diagnosis and its great potentiality [10] used  used the correlation dimension in condition as a diagnostic indicator is shown. Craig et al.  al.   [10] monitoring of systems with clearance. One of the main results in this paper is related with the potentiality of chaos techniques, as an alternative approach, in health monitoring of plant or machinery. Wang et al. [10,11] al.  [10,11] introduced  introduced a very important contribution for the diagnostic process of large rotating 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 was shown. More studies related with the application of pseudo-phase portrait, as indicator can be found in [13,14] in  [13,14].. In  In   [14], [14], a qualitative approach to the sensitive evaluation of the pseudo-phase portraits is given. Instea Ins tead d of the correl correlati ation on dimens dimension ion va value lue,, Koizum Koizumii et al.   [12]   introduced introduced the corre correlatio lation n exponent as a fault descriptor, estimated with the modified version of the well-known Grassberger and Procaccia algorithm. algorithm. The modified modified version version was based on a re-em re-embeddin bedding g proced procedure ure reported by Fraedrich and Wang  [15].  [15]. These authors applied the correlation exponent to investigate the chattering vibration during the cutting process. Good results can be obtained with the correlation exponent of some previous defined embedding dimension and fixed 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 multidimensional analysis, is the scaling region definition of the correlation integral as a function of the embedding dimension. Quite frequently the derivative of the log–log plot of   of   C (r,m) vs. ln r  is used for its calculation   [6–12]. calculation [6–12]. Here, the correlation dimension is expressed as a function of the embedding dimension: D2 ðr; mÞ ffi

 d  d½½ln lnððC ðr; mÞ dðln ln   rÞ

 

(1)

which is approximated by D2 ðr; mÞ ¼

 D ½ln lnððC ðr; mÞÞ   ; Dðln ln   rÞ

 

(2)

  D  is  f ð  f ðrsÞ method  f hod  f ððes rÞ  [16]. where defined by,  Dthis ffi  f ð ðr þ  þ 1  1Þ Þ  f  vides  [16] theults. plateau toe the attrac attoperator ractor tor dimens dim ension ion value value, thi met provid pro good goo. dRelating vis visual ual result res s. Theappearance platea plateau u zone zon (sometimes fitted by eyes) defines the correlation dimension value. Regardless of their advantages, the correlation dimension calculated with this method introduces some errors associated to the obse ob serv rver er’s ’s subj subjec ecti tivi vity ty and and it al also so li limi mits ts the ve velo loci city ty of the the de deci cisi sion on in ac acco cord rdan ance ce wi with th its its introduction as an indicator in a real system condition monitoring. Thee autho authors rs A statis statistic tical al approac approach h to the correla correlatio tion n dimens dimension ion estima estimation tion is given given in   [17]. [17]. Th introduce a method based on the maximum-likelihood principle, by means of which, the explicit expres exp ression sionss for the maximu maximum-l m-likel ikelihoo ihood d estimat estimatee of correla correlation tion dim dimens ension ion and its asympt asymptotic otic 2 variance varia nce are derived. They show how the w test is used to find 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 a robu robust st meth method od for for the the corr correl elat atio ion n di dime mens nsiion esti estima mati tion on in an autom utoma atic tic wa way y for for its its

implem imp lement entati ation on in on-line on-line condit condition ion monito monitorin ring g of large large rot rotati ating ng machin machinery ery.. The method method is

 

AR ARTIC TICLE LE IN PR PRESS ESS

942

A. Rolo-Naranjo, M.-E. Montesino-Otero / Mechanical Systems and Signal Processing 19 (2005) 939–954

called Automatic-Attractor Dimension-Quantitative Estimation (A-AD-QE). It is based on the systemic analysis of the second derivative of the correlation integral defined by Grassberger and Procac Pro caccia cia algori algorithm thm.. The A-AD-QE A-AD-QE method method concen concentra trates tes its attent attention ion on the sca scalin ling g reg region ion definition and it also has the possibility to analyse the geometrical structure of the obtained multidimensional second derivative of the correlation integral and its relation with the pseudophase portrait. Some applications of this method to the monitoring and surveillance of Nuclear Power Plant can be found in [18,19] in  [18,19].. One of the most important properties of the correlation dimension is its high sensitivity to the dynamical changes of the analysed system. Its application to the real system condition monitoring improves the real signal characterisation. More than 180 measurements of different points of a large rotating machine were analysed in order to evaluate the applicability of the introduced AAD-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 about the correlation dimension. In the same section, the introduced method is tested by means of wellknown analytic models, such as Lorenz attractor, van der Pol oscillator and Henon Map. In Section 3, we show some applications of the A-AD-QE method to the correlation dimension estimation of real measurements of a large rotating machine. Finally, the main conclusions are given in Section 4.

2. Theoretic Theoretical al backgroun background d

 2.1. Embedding procedure The embedding procedure is the first step of the phase space reconstruction of a dynamical system from the observation of a single variable. The most common phase space reconstruction technique is the method of delays (MOD) proposed by Takens   [20]. [20]. By means of MOD it is possible to build an attractor by replacing the derivatives with delayed repetitions of only one measured variable of the system. The time series  series   x1 ; x2 ; x3 ; . . . ; xN  is represented as a sequence of vectors X i   ¼ fxi ; xi þt ; . . . ; xi þ   þðm ðm1Þt g;

(3)

 N (m1) is the length of the reconstructed vector  X i i,   m  m is where i   ¼ where i  ¼  1 ; 2; . . . ; N     ðm    1  1ÞÞt; where  where N   is the embedd embedding ing dimens dimension ion of the recons reconstru tructe cted d phase phase space space and   t   is the lag time in units of  sampling interval.  m  is a question of  In recent investigations, the calculation of the embedding parameters  t   and and m special interest. The optimal choice of these embedding parameters depends of the respective  t  is of importance for the MOD to give sensible results. Two application [21] application  [21].. A good definition of  t major problems are described in [22] in  [22]:: redundancy and irrelevance. The first one is related to the choice of   t  as small as possible. In this case the consecutive measurements of the reconstructed vectors will give nearly the same results. Hence, the topological vectors constructed via the MOD, will be stretched along the diagonal in the m the  m-dimensional -dimensional embedding space and thus the analysis of the picture of the attractor will be very difficult difficult [23]  [23]..

 

AR ARTIC TICLE LE IN PR PRESS ESS

A. Rolo-Naranjo, M.-E. Montesino-Otero / Mechanical Systems and Signal Processing 19 (2005) 939–954   943

The second problem (irrelevance) appears with the choice of the   t  too large. In this case the reconstructed vectors become totally uncorrelated and the extraction of any information from this phase space picture becomes impossible.  R ðtÞ ¼  0 was used in order to avoid In this paper, the first zero of the autocorrelation function  Rð both mentio mentioned ned proble problems ms relate related d with with the   t   definit definition ion (see (see Eq. (4)). (4)). For all rec recons onstruc tructed ted dynamics defined with this value, the pseudo-phase portraits showed a regular behaviour and reflected clearly the main trajectories of the dynamical system

P

ðxði Þ    x Þ ðx ðxði þtÞ    x Þ

RðtÞ ¼

i 

P  2 xði Þ

 

;

 

(4)

i 

where,  x  is the arithmetic mean. The definition of the embedding dimension in this paper was taken according to the Takens’ +1). ). For the the th theo eore rem, m, wher wheree the the numb number er of   of   m-re -recon constr structe ucted d vector vectorss should should be   mX(2D (2D2+1 application of the A-AD-QE method to the unknown dynamical systems, we suggest to carry out a preliminary study where m where  m  should be calculate calculated d until 21. This value value was enough to characteri characterise se the low-dimensional systems studied in this paper. Therefore, an optimum value of   of   m  should be defined in order to decrease the processing time in on-line measurements.  2.2. Correlation dimension The correl correlati ation on dimens dimension ion is deriv derived ed from from the correl correlati ation on integr integral, al, whi which ch is a cumula cumulativ tivee correlation function that measures the fraction of points in the mthe  m-dimensional dimensional reconstructed space and is defined as  

2 C ðr; mÞ ¼ N m ðN m     1  1ÞÞ

N  X m

i ; j   ¼  1







 xðð j ; mÞ ;   xði ;  mÞ  x Y   r  

 

(5)

ð j 4i Þ where   Y   is the Heavis Heaviside ide functi function, on, such such that that   YðxÞ ¼  0 if   if   xp0 and   YðxÞ ¼  1 for   x40;   JyJ indicates the Euclidean norm of the vector, N  vector,  N m   ¼  N    1ÞÞt  is the length of the reconstructed   ðm    1 vectors and  and   r  is the correlation length  [24].  [24]. C (r) is related with the correlation dimension by means of the power law: C ðr; mÞ   r D2 ðmÞ ;

 

(6)

thee co corr rrel elat atio ion n ex expo pone nent nt and and va vari ries es with with the the incr increa ease se of  of    m. The partic particula ularr where   D2(m) is th correlation exponent can be found as the slope over the lineal region (scaling region) from the log–log plot of   C ðr; mÞ   vs. ln(r ln(r). The standard deviation of the fit is taken as the error of the obtained dimension value. A minimal value for the standard deviation directly leads to a good approach for the correlation exponent estimation. vs.   m  may converge for sufficiently For low-dimensional dynamic systems, the plot of   D2(m) vs.  large   m. The obtained value for the plateau zone, defines the correlation dimension   D2. large

 

AR ARTIC TICLE LE IN PR PRESS ESS

944

A. Rolo-Naranjo, M.-E. Montesino-Otero / Mechanical Systems and Signal Processing 19 (2005) 939–954

 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 way for its implementation in on-line condition monitoring of large rotating machinery. Its principal idea is to provide a   D2  definition taking as reference the second derivative application to the correlation integral (d2 ½ln lnððC ðr; mÞÞ=dðln rÞ2 vs. lnð lnðri =r0 Þ). xy-band-pass The implementation of the A-AD-QE method is based on the use of resizable   xy -band-pass filter windows (BPFW) with dimensions: height  x  and length L length  L,, over all points of the normalised second derivative of the correlation integral. This BPFW is moving in a zero-vicinity of  x of  x-axis -axis of  the second derivative function. Once On ce the se seco cond nd de deri riva vati tive ve to th thee co corr rrel elat atio ion n inte integr gral al is co comp mpute uted, d, the the foll follow owin ing g st step ep in the A-AD A-AD-Q -QE E metho method d is an iter iterat ativ ivee ch chan ange ge of the the size size de defin finit itio ion n of the the BP BPFW FW.. Init Initia iall parameter   x   shoul should d be ta take ken n as mini minimu mum m as possi possibl blee and and   L   increase increasess accordingl accordingly y with the number of analysed points, which should satisfy that their ordinate is smaller than the height of the specifi specificc window, window, accordin according g to the iterat iterative ive process. process. The necess necessary ary and suffici sufficient ent con con-ditions to stop this process are the existence of the appropriate number of continuous points that are within the specific BPFW dimensions. In our experience the number of point should be high higher er or eq equa uall to six, six, in or orde derr to obtai obtain n a good good ap appr proa oach ch to the proper proper sc scal alin ing g re regi gion on definition of the correlation integral. As an important result, the difference among the initial ln(rrEND/r0)) and final points of the BPFW (length of the plateau zone—LPZ=ln(r zone—LPZ=ln(rINITIAL/r0)ln( defin defines es the the le leng ngth th of the the best best pl plat atea eau u zo zone ne LPZ, LPZ, whic which h is loca locate ted d in a zero zero vici vicini nity ty.. Th Thee LPZ definition is in correspondence with the scaling region of the correlation integral, which allows the   D2   determination in an automatic way. This is a principal advantage of the introFig g. 1   show showss a simp simpli lifie fied d flo flowc wcha hart rt of the the duce du ced d meth method od in on-l on-lin inee moni monitor torin ing g syst system em..   Fi A-AD-QE method. Thee appe Th appear aran ance ce in the co corr rrel elat atio ion n in inte tegr gral al pl plot ot of more more than than on onee sc scal alin ing g re regi gion on is a re real al problem for some mechanical systems [7] systems  [7].. In this case, the A-AD-QE method defines the first LPZ as a criterion for the scaling region definition. The introduced method was tested by means of well-known analytic models, such as van der Pol oscillator   [25], [25], Henon Map   [26] and [26]  and Lorenz attractor  [27].  [27]. The obtained results for the   D2 estimation are comparable other reported models (see  Table 1 1). ). the A-AD-QE method. In 2 shows  shows a schematicwith representation of practical application of Fig. 2 this figure, the correlation integral as well as its first and second derivatives of the 12th embedding dimension of the Lorenz attractor are shown. The LPZ coincides with the plateau zone in the first derivative plot and with the scaling region of the correlation integral. The least square regression of the selected scaling region yields that the slope of the scaling region is 2.05 7(o0.01). The D The  D 2 for this model is 2.07.

3. Case Case study study

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

 

ARTIC LE IN PR PRESS ESS ARTICLE

A. Rolo-Naranjo, M.-E. Montesino-Otero / Mechanical Systems and Signal Processing 19 (2005) 939–954   945

Fig. 1. Simplified flowchart of the A-AD-QE A-AD-QE method. (*) An iterative procedure, procedure, where the size of BPFW x  and length L  changes (see details in Section 2.3). Table 1 Comparison between reported D reported D 2  value and obtained by A-AD-QE method for different well-known models Modela

D2 reported reported

Lorenz attractor van der Pol oscillator Henon map

2.07 1.00 1.26

 

D2A-AD-QE 2.05 1.08 1.25

 

Absolute error

Data length

0.02 0.08 0.01

2050 2050 2050

Lorentz attractor: dx dx=dt  ¼  s  sðð y  xÞ;   d y=dt  ¼ xz þ  rx  y;   dz=dt  ¼  xy    bz;  s  ¼  10 ;  r  ¼  28 and b and  b  ¼  8 =3: 2 2 van der Pol oscillator: ðq  ð q x=qtÞ  qx=qtÞ þ  kx  ¼  f    a  f   cos  O t;  a  ¼  5 ;  k  ¼  1 ;  f   ¼  1 and  O  ¼  2 :446:  aðð1  x Þ ððq 2 and   b  ¼  0 :3: Henon map:  map:  X nþ1  ¼  1    aX n  þ  Y n ;   Y nþ1  ¼  bX n ;  a  ¼  1 :4 and  a Characteristic equations of the analysed models.

analysed signals and parameters for the embedding procedure are shown in Table in  Table 2. 2. The choice of  th thee samp sampli ling ng ra rate te was was prec preced eded ed by a stud study y of the the sp spec ectr trum um for for fr freq eque uenc ncie iess up to 5.4 5.4 kH kHzz (90  1X).

 

ARTIC TICLE LE IN PR PRESS ESS AR

946

A. Rolo-Naranjo, M.-E. Montesino-Otero / Mechanical Systems and Signal Processing 19 (2005) 939–954 15

10

5

0  C (r,m )

-5

 d ln (C ( r,m ))/d ))/d ln(r)  d2 ln (C (r,m ))/d ))/d ln(r)2

-10

-1

0

1

2

3

4

ln(r/r0) Fig. 2. Schematic representation representation of the A-AD-QE method. The LPZ in the second derivative of the correlation integ integral ral coincides with the plateau zone in the first derivative of the correlation integral and with its scaling region.

Table 2 Time series characteristics and embedding parameters Parameter Sampling time Data volume No. of records Embedding dimension

Nomenclature        

T s N  N rec rec m

Value        

1.0ms 1024 10 21

3.1. Experiment Experimental al procedure procedure transduce ucers rs were were attach attached ed to the fluid-fi fluid-film lm The monitorin monitoring g system system is shown shown in   Fig. 3. 3. All transd bearings of a large rotating machine and used to generate scalar time series for the embedding procedure. proced ure. The signals for all measuremen measurements ts (more than 180) were sampled at 1.0 ms. The data volume was taken as 10 records of 1024 points each. The compar compariso ison n betwee between n al alll measur measureme ements nts has the object objective ive of analys analysing ing the A-AD-Q A-AD-QE E method met hod sensit sensitivi ivity ty under under differ different ent dynami dynamical cal states states.. The recorde recorded d signal signalss were were sample sampled d for different bearings, at 1–5 with the same configuration of the data collection system (Fig. ( Fig. 3). 3). In this pape pa per, r, fo four ur meas measur urem emen ents ts ar aree ev eval alua uate ted d (o (one ne meas measur urem emen entt for for ea each ch be bear arin ing g at 1– 1–4) 4).. Th Thee

 

ARTIC TICLE LE IN PR PRESS ESS AR

A. Rolo-Naranjo, M.-E. Montesino-Otero / Mechanical Systems and Signal Processing 19 (2005) 939–954   947

Fig. 3. Monit Monitoring oring syste system m for data collection. collection.

nomenclature of the measurements is taken with the capital letter B and subsequently, the bearing number and its respective direction. The waveforms waveforms after normalisati normalisation on by its maximum amplitudes amplitudes are shown in Fig. in  Fig. 4. 4. Note how the signal for the B1V is more irregular irregular than others, with a high influence influence of a low frequency frequency (subrotational frequency). The oscillation behaviour for the B2V and B3V is more regular, taking as reference the pick value evolution of the main period of the signals. In addition, the B2V signal seems see ms to be less contamina contaminated ted by noise. noise. In the case of the B4 V measur measureme ement, nt, its wav wavefo eform rm is disp displa lace ced d upwa upward rdss and and it is ex expe pect cted ed th that at this this be beha havi viou ourr be mani manife fest sted ed in the the dy dyna nami mics cs reconstruction. 3.2. Results Results The5. recons rec onstru tructe cted d pseudo pse por traits its, , obtaine obta d phase from from raw sig nals, s, clear are dis played yedical inl Fi Fig. g. The trajec tra jectori tories es udo-pha of -phase the se recons recportra onstruc tructed ted pseudopseined udo-pha se the portra portraits itssignal show cle ardispla topolog topo logica distributions, giving the possibility to analyse the dynamical system complexity and their direct relation with the time series. 3.3. Discussion Discussion The preliminary study in the reconstruction of the phase space until   m  ¼  ¼ 21  21 of the analysed mechanical systems corroborated that the saturation value of   of   D2(m) vs.   m   plot for   m46 was reached. According to this result, the application of the A-AD-QE method was done for m for  m  ¼  ¼ 10  10 : r,m)) Fig. 6 6 shows  shows the correlation integral (ln(C  (ln(C (r,m )) vs. ln(r/r ln(r/r0). 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 is due to the noise influence and will be corroborated in this study. In fact, the measurement B2V is lesss corrup les corrupted ted by noise, noise, its pseudo pseudo-pha -phase se portra portrait it is more more regula regularr (in compar compariso ison n wit with h other other

 

ARTICLE ARTIC LE IN PR PRESS ESS

948

A. Rolo-Naranjo, M.-E. Montesino-Otero / Mechanical Systems and Signal Processing 19 (2005) 939–954

  e    d   u    t    i    l   p

1.0

1.0

0.5

0.5   e    d   u    t    i    l   p

0.0

   A   m

   A   m

-0.5 -1.0 0.0

0.0

-0.5

0.2

0.4

0 .6

0.8

 

1.0

 

-1.0 0.0

0.4

0.6

Time (s)

Time (s)

B1V

B2V

1.0

1.0

0.5

0.5

  e    d   u    t    i 0.0    l   p   m    A

  e    d   u    t    i 0.0    l   p   m    A

-0.5 -1.0 0.0

0.2

0.8

 

1.0

-0.5

0 .2

0.4

0.6

0 .8

 

1.0

-1.0 0.0

0.2

0.4

0.6

Time (s)

Time (s)

B3V

B3V

0.8   1.0

Fig. 4. Waveforms Waveforms after normalisat normalisation ion of all meas measurem urements ents by its maximum amplitudes. amplitudes.

analysed signals) and its correlation integral shows a higher degree of smoothness. The higher absc ab scis issa sass of al alll cu curv rves es ar aree si simi mila larr and and we ex expe pect ct to obta obtain in a clea clearr plat platea eau u zone zone in the the indicator for the existence existence of an attractor attractor,, even though the d½ln lnððC ðr; mÞÞ=dðln rÞ  vs. ln(r/r ln(r/r0) plot as indicator analysed signals are contaminated by noise. As expected, plots of the first derivative of the correlation integral (d½ (d ½lnð lnðC ðr; mÞÞ=dðln rÞ   vs. r/r0)) display a plateau zone with values among 1 and 2 for all measurements (Fig. ( Fig. 7). 7). This ln(r/r ln( indica ind icates tes that that the analys analysed ed oscill oscillati ating ng proces processes ses are domina dominated ted mainly mainly by a low-di low-dimen mensio sional nal dynamicc system dynami system.. More More detail detailed ed discus discussio sion n about about the slope slope zones zones cla classi ssifica ficatio tion n can be fou found nd in in [27]  [27].. Throughout the second derivative plot of the correlation integral, in addition to the definition in automatic way of the  the   D2, the A-AD-QE method provides the possibility of a very clear zone definition of the plot.   Fig. 8  shows a typical example of the zones definition using the second derivative plot of the B4V measurement (m (m  ¼  3 221). For a particular study, we suggest to analyse vs.   m. only one embedding dimension of the saturation region of   of   D2(m) vs.  Each zone in the second derivative plot has its practical value. In this paper we concentrated ourr at ou atte tenti ntion on to the seco second nd one, one, whic which h is rela relate ted d dire direct ctly ly to the the sc scal alin ing g re regi gion on de defini finiti tion on.. Nevertheless, it is important to give some remarks about the others. Zones I and III can be used for practical evaluation of the noise influence on time series. Furthermore, the minimum value for the ln(r/r ln(r/r0) is related to the complexity of the analysed dynamics. The geometric configuration of  zone III can be used to evaluate the quality of the reconstruction of the dynamics.

 

ARTICLE ARTIC LE IN PR PRESS ESS

A. Rolo-Naranjo, M.-E. Montesino-Otero / Mechanical Systems and Signal Processing 19 (2005) 939–954   949

   )   u   a    T

1.0

1.0

0.5

0.5    )   u   a    T

0.0

   t    (   +   x

0.0

   t    (   +   x

-0.5 -1.0 -1.0

-0.5

-0.5

0.0 x(t)

0.5

-1.0 -1.0

1.0

-0.5

B1V

   )   u   a    T   +    t    (   x

0.5

1.0

B2V

1.0

1.0

0.5

0.5    )   u   a    T   +    t    (   x

0.0

-0.5 -1.0 -1.0

0.0 x(t)

0.0

-0.5

-0.5

0.0 x(t) B3V

0.5

1.0

-1.0 -1.0

-0.5

0.0 x(t)

0.5

1.0

B4V

Fig. 5. Reconstru Reconstructed cted pseudo-ph pseudo-phase ase portraits portraits for each meas measureme urement. nt. The values of   t;  based on the first zero of the autocorrelation function (R (R(0)=t) for the analysed signals are:  tB 1V   ¼  8 ;   tB 2V   ¼  3 ;   tB 3V   ¼  3 and  t B 4V   ¼  3 :

9 shows  shows the second derivative of the correlation integral (d 2 ½lnð lnðC ðr; mÞÞ=dðln rÞ2 vs. ln(r/ ln(r/ Fig. 9 r0)) and the LPZ after application of the A-AD-QE method. It can be observed that an important qual qu alit itat ativ ivee appr approx oxim imat atio ion n ca can n be made made be betw twee een n the the loca locall co comp mple lexi xity ty of the ps pseu eudo do-p -pha hase se portraits and the minimum value ln(r/r ln(r/r0) of zone I. Here the minimum value is related to the smalle sma llerr comple complexit xity y of the analys analysed ed dynami dynamicc system system.. This This statem statement ent is rel relate ated d direct directly ly to the 5). ). pseudo-phase pseudo -phase portrait portrait behaviour behaviour (see Fig. (see  Fig. 5 An important parameter derived for the A-AD-QE method is the LPZ value.  value.   Table 3 3   shows that the maximum value for LPZ is obtained for the B2V, which shows a more regular pseudophase portrait (see  Fig. 5). 5). The analysis of this parameter represents an important quantitative indicator to the dynamic characterisation. The obtained results for the evaluated raw signals corroborate their low complexity. For a real estimatio tion, n, a nonlin nonlinear ear filter filter should should be applie applied. d. Nev Nevert erthel heless ess,, for the analys analysed ed value val ue of   of   D2   estima dynamics and in general, for normal condition monitoring of large rotating machinery, the filter application will facilitate the plateau zone definition and will improve the applicability of the AAD-QE method. Furthermore, for this kind of machinery, we would like to remark that noise is an additional diagnostic source.

 

ARTICLE ARTIC LE IN PR PRESS ESS

950

A. Rolo-Naranjo, M.-E. Montesino-Otero / Mechanical Systems and Signal Processing 19 (2005) 939–954

14 12 10    )    )   m  ,   r    (    C    (   n    l

8

 B 1  B 2  B 3  B 4

6 4

V V V V

2 0 -4

-3

-2 ln (r/r0)

-1

0

Fig. 6. Correlatio Correlation n integral (lnðC ðr; mÞÞ  vs. ln(r/r ln(r/r0)) for all analysed dynamics.

14 12

 B 1 V

   )   r 10    (   n    l    d    / 8    )    )   m  , 6   r    (    C   n    l 4    (    d

 B 2 V  B 3 V  B 4 V

2 0 -4

-3

-2

-1

0

ln (r/r0) Fig. 7. First derivativ derivativee of the correla correlation tion integra integrall (d½lnðC ðr; mÞÞ=dðln rÞ  vs. ln(r/r ln(r/r0)) for all analysed signals.

As it was stated before, before, the zone II is the most important one because because it defines defines the LPZ and the corres cor respon pondin ding g scalin scaling g region region as a functio function n of   of   m. The main results of the A-AD-QE method obtained  D 2  values have direct relation with the dynamical application are shown in Table in  Table 3. 3. The obtained D

 

ARTIC TICLE LE IN PR PRESS ESS AR

A. Rolo-Naranjo, M.-E. Montesino-Otero / Mechanical Systems and Signal Processing 19 (2005) 939–954   951 60

III

II

I 40    2

   )   r    (    l   n 20    d    /    )    )   m  , 0   r    (    C    (   n    l

   2

-20

   d

-40

-60

-3

-2

-1

0

ln (r/r0) Fig. 8. The zones in the second derivative derivative of the corre correlatio lation n integ integral ral obtained obtained by the applicatio application n of the A-AD-QE A-AD-QE method for the measurement B4V.

complexity. Note that the standard deviation for the least square regression analysis is less than 0.01 according to the defined scaling region. This result demonstrates the high accuracy of the method.

4. Conclusi Conclusions ons

In this paper, a robust method (A-AD-QE) for the correlation dimension estimation in an automatic way is introduced, for its implementation in on-line condition monitoring of large rotating machinery. The applicability of the introduced method for the scaling region definition is discussed. In addition, it is demonstrated that the A-AD-QE method provides the possibility to analyse the geometrical structure of the obtained second derivative of the correlation integral and its relation with the pseudo-phase portrait. An import important ant parame parameter ter derive derived d for the A-AD-Q A-AD-QE E method method is the LPZ LPZ.. This This quanti quantitat tative ive indicator is used to characterise the studied dynamics. The effectiveness of the introduced method is verified by means of the calculation of wellknown kno wn analyt analytic ic models models.. Furthe Furthermo rmore, re, the A-AD-Q A-AD-QE E method method is applie applied d to proces processin sing g of real real signals, recorded from different points of a large rotating machine. In both cases, the results show its high sensitivity, taking as reference the obtained values for the standard deviation of the least square regression. Further works in this thematic should be directed to the verification of the introduced method in more complex dynamics and in other mechanical systems.

 

ARTIC LE IN PR PRESS ESS ARTICLE

952

A. Rolo-Naranjo, M.-E. Montesino-Otero / Mechanical Systems and Signal Processing 19 (2005) 939–954    2

   )   r    (   n    l    d    /    )    )   m  ,   r    (    C    (   n    l

   2

   d  

   2

   )   r    (   n 0.5    l    d    /    )    )   m  , 0.0   r    (

0.5

0.0

-0.5 - 4. 0

   C    (   n    l    2

- 3 .5

-3.0

-2.5

- 2. 0

-1.5

-1.0

-0.5

0. 0

0.5

0.5

0.0

0.0

-0.5

-1.10

-1.15

-1.20

 

-1.25

 

-1. 30 -1 30

-0.5 -4.0

   d  

-0.5

-1. 35 35

-3.5

-2.0

-3.0

-1.8

- 2. 5

-1.6

 ln (r/r0)

   2

   d  

-1.4

-0.5

0. 0

-1.2

   2

   )   r    (   n 0.5    l    d    /    )    )   m  , 0.0   r    (

0.5

0.0

-0.5 - 4. 0

   C    (   n    l    2

-3.5

- 3 .0

-2.5

-2.0

- 1 .5

-1.0

-0.5

0 .0

0.5

0.0

0.0

-1.1

-1.0

- 0 .9

-0.8

- 0 .7

- 0 .6

-0.5 -4.0

   d  

0.5

-0.5

-1.0

B2V

B1V

   C    (   n    l

-1.5

 ln(r/r0)

   2

   )   r    (   n    l    d    /    )    )   m  ,   r    (

- 2. 0

-0.5

-3.5

-1.0

- 3 .0

-0.9

 ln (r/r0)

-2.5

- 2 .0

-0.8

-1.5

-1.0

-0.7

-0.5

0.0

-0.6

 ln (r/r0)

B3V

B3V

Fig. 9. Second deriv derivativ ativee plot of the corre correlatio lation n integ integral ral for all analys analysed ed dynamics based on A-AD-QE method method..

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

B1V B2V B3V B4V

LPZ by A-AD-QE method

Correlation dimension (D27s)

|ln(rr1/r0)| |ln(

|ln(r1/r0)|

LPZ

1.37 2.13 1.11 0.99

1.07 1.08 0.59 0.65

0.30 1.05 0.52 0.34

1.867(o0.01) 1.157(o0.01) 1.147(o0.01) 1.347(o0.01)

Acknowledgements

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

 

ARTIC LE IN PR PRESS ESS ARTICLE

A. Rolo-Naranjo, M.-E. Montesino-Otero / Mechanical Systems and Signal Processing 19 (2005) 939–954   953

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

References [1] A. Muszynska, Muszynska, Whirl and whip—rotor whip—rotor bearing stabi stability lity problems, problems, Journal Journal of Sound and Vibration Vibration 110 (1986) (1986) 443–462. [2] A. Muszynska, P. Goldman, Chaotic responses responses of unbalanced rotor/bearing/stator rotor/bearing/stator systems with looseness or rubs, Chaos, Solitons and Fractals 5 (1995) 1683–1704. [3] F.L. Chu, Z.S. Zhang, Periodic, quasi-periodic quasi-periodic and chaotic vibrations of a rub-impact rub-impact system support supported ed on oil film bearings, International Journal of Engineering Science 35 (1997) 963–973. [4] J. Zapomel, E. Malenovsky, Approaches to investigation of the characte characterr of vibration of rotor systems supported by fluid-film bearings, International Conference on Engineering Mechanics, Avractka, Czech Republic, 15–18 May 2000. [5] J. Zapomel, E. Malenovsk Malenovsky, y, Approach Approaches es to numer numerical ical investi investigati gation on of the characte characterr and stability stability of forced and self-excited vibration of flexible rotors with non-linear supports, C576/016/2000, 2000. [6] D. Logan, J. Mathew, Using de correlation dimension for vibration vibration fault diagnosis of rolling element bearings—I. Basic concepts, Mechanical Systems and Signal Processing 10 (1996) 241–250. [7] D. Logan, J. Mathew, Using de correlation dimension for vibration fault diagnosis diagnosis of rolling element bearings—II. Selection of experimental parameters, Mechanical Systems and Signal Processing 10 (1996) 251–264. [8] J.D. Jiang, J. Chen, L.S. Qu, The application of correlation correlation dimension in gearbox condition monitoring, Journal of  Sound and Vibration 23 (1999) 529–542. [9] C. Craig, R.D. Neilson, J. Penman, The use of correlat correlation ion dimension dimension in condition monitoring monitoring of systems systems with clearance, Journal of Sound and Vibration 231 (2000) 1–17. [10] W.J. Wang, J. Chen, X. K, Z.T. Wu, The applic application ation of some non-linear non-linear methods methods in rotating rotating machinery machinery fault diagnosis, Mechanical Systems and Signal Processing 15 (2001) 697–705. [11] W.J. Wang, R.M. Lin, The applica application tion of pseud pseudo-pha o-phase se portrait in machine machine condition monitoring, monitoring, Journal Journal of  Sound and Vibration 259 (2003) 1–16. [12] T. Koiz Koizumi, umi, N. Tsujiuchi, Tsujiuchi, Y. Mats Matsumura umura,, Diag Diagnosis nosis with the correlation correlation integral in time domain, Mechanical Mechanical Systems and Signal Processing 14 (2000) 1003–1010. [13] Liang Liangsheng sheng Qu, Ailin Xie, Xiao Li, Study and perfo performan rmance ce evaluat evaluation ion of some nonlinear nonlinear diagnostic diagnostic methods methods for large rotating machinery, Mechanism and Machine Theory 28 (1993) 699–713. [14] A. Rolo, M. Piedra, Piedra, M.E. Monte Montesino, sino, A. Companion Companiony, y, Integra Integrall sensitivit sensitivity y of the phase-space phase-space and the lag time as

[15] [16] [17] [18] [19] [20] [21]

indicators of mechanical faults, III International Workshop of Management Sciences, III Maintenance Congress, Villa Clara, Cuba, 2000. K. Fraedrich, R. Wang, Estimating the correlation dimension of an attractor from noisy and small datasets based on re-embedding, Physica D 65 (1993) 373–398. J. Thiler, Spurious dimension from correla correlation tion algorithms algorithms applied to limited limited time-ser time-series ies data, Physical Physical Review A 34 (1986) 2427–2432. D.V. Pisarenko, V.F. Pisarenko, Statistical estima estimation tion of the correlation dimension, Physics Letters A 197 (1995) 31–39. A. Rolo, M.E. Montesino, A. Companioni, A. Azoy, A method for practical determination of attractor attractor dimension applied to the monitoring and surveillance in Nuclear Power Plants, ENFIR XIII, Brazil, 2002. M.E. Montesino, A. Rolo, A. Azoy, The attra attractor ctor dimension determination applied to monitoring monitoring and surveillance in nuclear power plants, Progress in Nuclear Energy 43 (2003) 389–395. F. Takens, Detectin Detecting g Strange Strange Attract Attractors ors in Turbu Turbulence lence,, Lect Lecture ure Notes in Mathemati Mathematics, cs, vol. 898, Springer, Springer, New York, 1981, pp. 366–381. R. Hegger, Hegger, H. Kantz, Kantz, T. Schreiber, Schreiber, Practic Practical al imple implement mentation ation of nonlinear nonlinear time series series methods: methods: the TISEAN TISEAN package, Chaos 9 (1999) 413.

 

ARTIC LE IN PR PRESS ESS ARTICLE

954

A. Rolo-Naranjo, M.-E. Montesino-Otero / Mechanical Systems and Signal Processing 19 (2005) 939–954

[22] M. Casdagli, S. Eubank, J.D. Farm Farmer, er, J. Gibson, State space reconstruction reconstruction in the presence of noise, Physica D 51 (1991) 52. [23] D. Kugiumt Kugiumtzis, zis, State space reconstructi reconstruction on in the analy analysis sis of chaotic chaotic time series—th series—thee role of the time window length, Physica D 95 (1996) 13–28. [24] P. Grassber Grassberger, ger, I. Procaccia Procaccia,, Meas Measuring uring the stra strangene ngeness ss of stra strange nge attract attractors, ors, Physica D 9 (1983) (1983) 189–208 189–208.. [25] Y. Ashkenazy, The use of genera generalized lized information dimension in measuring fractal fractal dimension of time series, Physics Letters A 269 (1999) 427–447. [26] J. Argyris, I. Andreadis, G. Pavlos Pavlos,, M. Athanasiou, On the influence of noise on the largest Lyapunov exponent on the geometric structure of attractors, Chaos, Solitons and Fractals 9 (1998) 947–958. [27] P. Sekar, Sekar, S. Narayana Narayanan, n, Chaos Chaos in mechanic mechanical al systems. A revi review, ew, Sadhana 20 (1995) (1995) 529–582. 529–582.