EMD Based on Independent Component Analysis and Its Application in Machinery Fault Diagnosis

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1 30 JOURNAL OF COMPUTERS, VOL. 6, NO. 7, JULY 0 EMD Based o Idepede Compoe Aalyss ad Is Applcao Machery Faul Dagoss Fegl Wag * College of Mare Egeerg, Dala Marme Uversy, Dala, Cha Emal: wagflsky997@sa.com Deyou Zhao School of Shp Egeerg, Dala Uversy of Techology, Dala, Cha Absrac Local rub-mpac s he commo faul roag machery ad resuls mpac ad frco bewee roor ad saor. The vbrao sgal due o mpac ad frco s always o-saoary whch cludes hree compoes, amely, he rub-mpac sgal, he backgroud sgal ad he ose sgal. EMD (Emprcal mode decomposo) s based upo he local characersc me scale of sgal ad could decompose he complcaed sgal o a umber of IMFs (rsc mode fucos). However, because he weak rub-mpac sgal s always submerged he backgroud sgal ad ose sgal. The EMD procedure wll geerae he compoes redudacy. I order o solve he problem, a ovel mehod combg wh depede compoe aalyss (ICA) ad EMD s proposed. ICA s roduced o he EMD procedure, so ha he compoes are orhogoal o each oher ad he compoes redudacy ca be cu dow. I he ed, a much beer decomposo performaces ca be obaed. Furhermore, egrao of EMD wh Hlber evelope aalyss s appled o compoe saaeous amplude order o oba evelope specra from whch he mechacal faul ca be dagosed. The aalyss resuls from he rub-mpac vbrao sgals show ha he proposed mehod ca be appled o he machery faul dagoss effecvely. Idex Terms emprcal mode decomposo), depede compoe aalyss, Hlber rasform, faul dagoss, roag machery I. INTRODUCTION Rub faul may resul broke mache pars, ad lead caasrophc breakdow of he roag machery []. I order o avod he occurrece of rub-mpac, vbrao sgal aalyss s wdely used roag machery codo moorg ad faul dagoss []. Usually, depedg o mache operag codos ad severy Mauscrp receved Augus 0,00; revsed November, 00; acceped December 5, 00. * Correspodg auhor. of defecs, he measured vbrao sgal s always osaoary ad modulaed. Furhermore, whe he local rub-mpac occurs he roor sysem, he vbrao sgal cludes he rub-mpac sgal, he backgroud sgal ad he ose sgal [3]. The key of he rub-mpac faul dagoss s o exrac he rub-mpac feaure from he vbrao sgal of he roor sysem [4]. Nolear ad o-saoary sgal aalyss ad processg s oe of he mos mpora research areas formao scece [5]. A o-saoary sgal s usually assumed o be local saoary ad s processg s based o he classcal heores ad echques for saoary sgals. Wdowed Fourer rasform s a ypcal example based o such a assumpo. Wavele rasform s a me-scale mehod, whch ca zoom or ou o he me ad frequecy scales of a sgal adapvely. However, he wavele bass fuco mus be chose defely before s used, so cao be chaged adapvely accordg o he oscllaos of he sgal a dffere me. Moreover, a approprae wavele wll overwhelm he local characersc of vbrao sgal, ad los some useful deal formao of orgal sgal. EMD (emprcal mode decomposo) s a powerful ool for aalyzg he compose, olear ad o-saoary sgal. EMD s based o he local characersc me scale of sgal ad ca decompose he complcaed sgal o a umber of IMFs (rsc mode fucos), each of whch s bad lmed ad ca represe he feaures of sgal ad reserve he local formao [6]. However, he decomposo algorhm has some mplc dffcules ad he procedures creae several drawbacks ha orgae srage decomposos [7]. EMD cao guaraee compleeess ad orhogoaly, ad may produce reduda compoes ad affec he accuracy of decomposos. Idepede compoe aalyss (ICA) s a mehod for fdg a lear represeao of o-gaussa so ha he compoes are sascally depede, whch s effecve removg compoes redudacy [8]. I hs paper, he ICA s roduced o he EMD procedure o overcome he above lmaos, hus he compoes are orhogoal o each oher ad compoes redudacy ca be cu dow effecvely. I he ed, he o-saoary do:0.4304/jcp

2 JOURNAL OF COMPUTERS, VOL. 6, NO. 7, JULY sgal ca be decomposed accuraely, ad a much beer decomposo performaces ca be obaed. II. EMD AND HILBERT TRANSFORM EMD mehod s developed from he smple assumpo ha ay sgal cosss of dffere smple rsc modes of oscllaos. Ay sgal ca be decomposed o a fe umber of IMFs, each of whch mus sasfy he followg defo [6]: ) I he whole daa se, he umber of exrema ad he umber of zero-crossgs mus eher equal or dffer a mos by oe. ) A ay po, he mea value of he evelope defed by local maxma ad he evelope defed by he local mma s zero. A IMF represes a smple oscllaory mode compared wh he smple harmoc fuco. Wh he defo, ay sgal x() ca be decomposed as follows: ) Idefy all he local maxma ad local mma, ad he coec all hese exrema by cubc sple les o form upper ad lower evelopes. ) The mea of upper ad lower evelope value s desgaed as m ), ad he dfferece bewee he sgal x () ad m ( ) s he frs compoe, h ( ),.e. x ) m ( ) = h ( ). () ( If h ) s a IMF, ake as he frs compoe of x (). If h ) s o a IMF, ake as he orgal sgal ad repea he seps ul h k ( ) s a IMF, ad desgae h k ( ) as c ). c ) = h ( ). () k 3) Separae c ) from x() by r ) = x( ) h ( ). (3) 4)Trea resdue r ) as he orgal sgal ad repea he procedure o exrac all possble IMFs. The decomposo process ca be sopped, whe resdue (.e. () ) becomes a moooc fuco, from whch r o more IMFs ca be exraced. The orgal sgal ca be represeed as, x( ) = c ( ) = + r ( ). (4) Thus, we ca acheve a decomposo of he sgal o IMFs c ( ), c ( ),..., (), ad a resdue r (), whch s he mea red of x () c. The IMFs clude dffere frequecy bads ragg from hgh o low. The frequecy compoes coaed each frequecy bad are dffere ad chage wh he varao of sgal x (). For each IMF c () (4), we ca always have s Hlber rasform as, x τ H c dτ π + ( ) [ ( )] =. (5) τ Wh hs defo, we ca have a aalyc sgal as, where x ( ) ( ) () = a ()e jϕ () = c + jh. (6) + a ( ) = c ( ) H [ c ( )]. (7) H[ c ( )] ϕ ( ) = arca. (8) c ( ) From (8), we ca have he saaeous frequecy as: ϕ ( ) ω ( ) =. (9) π Afer performg he Hlber rasform o each IMF compoe, he orgal sgal ca be expressed as he real par ( RP ) he followg form, x( ) = RP = jϕ () a ()e. (0) III. INDEPENDENT COMPONENT ANALYSIS Idepede compoe aalyss(ica) s very closely relaed o he mehod called bld source separao(bss). The ma purpose of ICA s o fd a lear represeao of o-gaussa daa so ha he compoes are sascally depede. Cosder sources s, s,, s, whch are sascally depede, m measuremes from sesors x, x,, x m, whch are represeed as a lear combao of sources s as follows, x = As. () where A ad s are ukow ad x s kow. A s he mxg marx; The source vecor T s = s, s,..., ] ; The observed vecor [ s T = [ x, x,..., x m. x ] Our am s o seek a demxg marx whch recovers he source vecor s from he observed vecor x. The elemes of y are esmaes of he observed vecor x whch ca be used o represe he observed vecor x as follows,

3 304 JOURNAL OF COMPUTERS, VOL. 6, NO. 7, JULY 0 y = Wx = WAs. () where W s he demxg marx. ICA s a wo-sep process. Frs sep s o choose a prcple, based o whch a cos fuco s obaed. Nex, a suable mehod for opmzg he cos fuco eeds o be adoped. Oe of he bes mehods s he FasICA algorhm, whch uses he egeropy as he cos fuco o esmae s. Eropy s a measure of he average uceray a radom varable. A Gaussa varable has he maxmum eropy amog all radom varables wh equal varace. For radom vecor y, he egeropy s defed as [9]. J ( y) = H ( ygauss ) H( y). (3) Where y gauss s a Gaussa radom vecor wh he same covarace as y. Hece, egeropy of a Gaussa radom vecor s he dfferece of eropy wh he correspodg Gaussa radom vecor. To esmae egeropy effcely, a smpler approxmao of egeropy as follows s used, J ( y) c{ E[ G( y)] E[ G( y gauss )]}. (4) Where y s assumed o be of zero mea ad u varace; G s ay o-quadrac fuco; c s a cosa. Usg a fxed-po erao scheme o fd drecos whch he egeropy s maxmzed, he demxg marx W ca be acheved [0]. IV ICA-EMD METHOD The EMD mehod ca decompose a gve sgal o a fe umber of IMFs ha adm well-behaved Hlber rasforms. The ecessary codos for he bass o represe o-saoary me seres are complee ad orhogoal, bu he orhogoaly s o guaraeed heorecally. By vrue of he decomposo, he compoes should all be locally orhogoal o each oher, for each compoe s obaed from he dfferece bewee he sgal x() ad s local mea x () hrough he maxmal ad mmal evelopes; herefore, ( x ( ) x( )) x( ) = 0. (5) Neverheless, equao (5) s o srcly rue, because he mea s compued va he evelopes, hece s o he rue mea for olear ad o-saoary sgal aalyss. Furhermore, because he weak rub-mpac sgal s always submerged he backgroud ad ose sgals pracce egeerg, EMD cao guaraee compleeess ad orhogoaly, ad may produce reduda compoes. ICA s a mehod for fdg a lear represeao of o-gaussa daa so ha he compoes are sascally depede, whch s effecve removg redudacy. To overcome he above lmaos, he heory of ICA was roduced o he EMD procedure ad a ovel mehod combg EMD ad ICA s proposed, whch ca mprove he decomposo algorhm, hus he compoes are orhogoal o each oher ad he decomposo redudacy ca be cu dow. The he sgal compoes ca be expressed accuraely ad he weak mpulsve feaure ca be exraced from orgal sgals. The ICA- EMD mehod follows four operaos: ) Decompose he measured sgal X by usg he EMD mehod, ad oba he frs compoe IMF. ) Le IMF ad X be he pu o he FasICA algorhm, afer performg ICA procedure, we ca oba wo depede compoes whch are ICA ad ICA respecvely. 3) Because he hgh frequecy compoe was decomposed frsly, he hgher frequecy ICA compoe amog he wo ICA compoes ca be regarded as IMF compoe, le he oher ICA compoe be a ew measured sgal X. 4) Repea he above processes ad oba he oher orders IMFs. V ICA-EMD METHOD I order o verfy he valdy of he proposed mehod, he local rub-mpac faul occurs oly oe poso was coduced o a roor es rg. The radal dsplaceme vbrao sgal wh local rub-mpac faul pcked up by he dsplaceme sesor s show Fg.. The roag frequecy s 47 Hz ad he samplg frequecy s 560 Hz. The FFT specrum of he rub-mpac vbrao sgal s show Fg.. I ca be demosraed from Fg. ha he doma frequecy compoes of he rub-mpac sgal are he roag frequecy 47 Hz ad s. However, he hgher frequecy compoes wh rubmpac formao are submerged he sroger backgroud sgal. The rub-mpac vbrao sgal show Fg. s decomposed by EMD mehod ad 4 IMF compoes are obaed show as Fg.3. I ca be see from he IMF compoes lsed Fg.3 ha here was o ampludemodulaed characersc. Because he decomposo redudaces appear whe performg he EMD mehod o he rub-mpac vbrao sgal show Fg., he sgal compoes cao be expressed accuraely. So he EMD procedure he decomposo redudaces carres grea subjecvy ha would brg accurae dagoss resul, whereas he ICA-EMD mehod ca overcome he lmao whch occurs he EMD procedure.

4 JOURNAL OF COMPUTERS, VOL. 6, NO. 7, JULY Fgure. The radal dsplaceme vbrao sgal of he roor sysem wh local rub-mpac vbrao faul. Fgure. FFT specrum of rub-mpac Sgal. appled o he compoe C order o oba he evelop specra of he compoe C whch s show Fg.5. From Fg.5 ca be see ha he evelop specrums of he compoe Chere s he obvous specrum le a he roag frequecy 47 Hz, whch s produced by he perodc mpac bewee he roor ad he saor. Hece, he compoe C cludg he rubmpac formao s jus he rub-mpac characersc sgal. The oher compoes C ad C3 are he backgroud sgal relaed o he roag frequecy, he frequeces cluded he compoes C3 ad C are he roag frequecy ad s double, whch cocdes wh he FFT specrum show Fg., whle he resdue C4 s he ose sgal wh lower frequecy compoe. Therefore, by usg he ICA-EMD mehod, he rubmpac sgal, he backgroud sgal ad he ose sgal ca be separaed from he vbrao sgal of he roor sysem wh local rub-mpac faul, hus he rub-mpac formao he hgh frequecy bad ca be exraced from he srog backgroud sgal effecvely, ad he faul feaure of he rub-mpac vbrao sgal ca be obaed. Fgure 4. Decomposo resul of he rub-mpac sgal by usg ICA-EMD. Fgure 3. Decomposo resul of he rub-mpac sgal by usg EMD. To exrac he rub-mpac sgal, he ICA-EMD mehod was appled o deec he rub-mpac vbrao sgal show Fg.. Frsly, he vbrao sgal s deosed by usg lfg wavele, he s decomposed by he ICA-EMD mehod. The decomposo resul s show Fg.4. I ca be see from Fg.4 ha he compoe C has he obvous amplude-modulaed characerscs. The Hlber evelope aalyss s he Fgure 5. Hlber evelope specrum of he compoe C. I hs experme, we ulze he EMD mehod ad he ICA-EMD mehod o decompose he rub-mpac sgal respecvely. Fg. 3 ad 4 are he decomposo resuls

5 306 JOURNAL OF COMPUTERS, VOL. 6, NO. 7, JULY 0 respecvely. We cosder he frs IMF whch represes he ma feaures of he vbrao sgal. From he resuls, we ca fd ha he decomposo performace obaed by usg he ICA-EMD mehod s much beer ha ha of he sraghforward EMD mehod. Compared wh he sraghforward EMD procedure, he ICA-EMD mehod ca elmae compoes redudacy ad oba a much beer decomposo performace, so he sgal compoes ca be expressed accuraely. The rub-mpac formao he hgh frequecy bad ca be exraced from he srog backgroud sgal effecvely by usg he ICA-EMD mehod. Therefore, he faul feaure of he vbrao sgal ca be obaed effcely. VI. DISCUSSION EMD as a daa drve alerave approach o he aalyss of o-saoary sgals appears some drawbacks. Because he rub-mpac sgal s weak, s very dffcul o separae he rub-mpac sgal from he vbrao sgal cludg ose ad backgroud sgal by usg he sraghforward EMD mehod. Here we proposed soluos for he problem of compoes redudacy order o oba a algorhm wh beer performaces. The proposed mehod ca remove he compoes redudacy EMD mehod ad mprove he qualy of decomposo. Compare o he sraghforward EMD, he ICA-EMD mehod ca oba a much beer decomposo performace, ad he ampludemodulaed characersc formao of he rub-mpac sgal ca be exraced from he measured vbrao sgal. Expermeal aalyss resuls show ha he proposed mehod ca be appled o he feaure exraco of he roor sysems wh local rub-mpac faul effecvely. REFERENCES [] A. Muszyska, Roor-o-Saoary Par Rubbg Coac Roag Machery. Roor Dyamcs, CRC Press, Taylor ad Fracs Group, 005. [] A. Muszyska, Roor-o-Saoary Eleme Rub-Relaed Vbrao Pheomea Roag Machery-leraure Survey, Shock ad Vbrao Dges, vol., 989,.pp.3. [3] R.F. Beay, Dffereag roor respose due o radal rubbg, Tras ASME, Joural of Vbrao, Acouscs, Sress, ad Relably Desg, vol. 07,985, pp [4] Z.K. Peg, F.L. Chu, ad P.W. Tse, Deeco of he rubbg-caused mpacs for roor saor faul dagoss usg reassged scalogram, Mechacal Sysems ad Sgal Processg, 9(),005, pp [5] X.D. Zhag, Z. Bao, Aalyss ad Processg of Nolear Saoary Sgals, Naoal Defese ad Idusral Publsher, Bejg, Cha,998. [6] N.E. Huag, Z. She, S.R. Log, M.L. Wu, H.H. Shh, Q. Zheg, N.C. Ye, C.C. Tug, ad H.H. Lu, The emprcal mode decomposo ad he hlber specrum for olear ad osaoary me seres aalyss, Proceedgs of he Royal Socey of Lodo Seres, A 454,998, pp [7] R.T. Rao, M.D. Orguera, ad A.G. Basa, O he HHT, s problems, ad some soluos, Mechacal Sysems ad Sgal Processg, vol., 008, pp [8] Hyvare A, Oja E, Idepede compoe aalyss by geeral olear Hebba-Lke learg rules, sgal processg, 4(3),998, pp [9] Amar S, Cardoso J F, Bld source separao-semparamerc sascal approach, IEEE Tras. O Sgal Processg, 45(),997, pp [0] A Hyvare, E Oja., A fas fxed-po algorhm for depede compoe aalyss, Neural Compuao, 9(7),997, pp ACKNOWLEDGMENT The auhors wsh o hak he revewers for valuable commes ad recommedaos.