Rotatonal Machne Fault Detecton wth Enemble Emprcal Mode Decompoton Baed on a hree Orthogonal Channel Senor L an Purdue Unverty North Central lzhetan@pnc.edu Alexander Mua Purdue Unverty North Central Jutn Polng Purdue Unverty North Central Ka Jutce Purdue Unverty North Central Hongbo Xu Nanjng Unverty of Scence and echnology Abtract Rotatonal machne fault detecton and condtonal montorng can prevent harmful envronment and enure relable operaton of equpment. In order to acheve fault detecton and condtonal montorng, many gnal proceng technque, uch a hort-tme Fourer tranform (SFF), wavelet tranform and emprcal mode decompoton (EMD), have been developed. Among thee method, the EMD proce ha been a very promng and effectve technque. h paper propoe an EMD-baed algorthm that cont of two tage. he frt tage procee vbratonal gnal ung three orthogonal channel recordng to obtan a prncpal component gnal. At the econd tage, the enemble emprcal mode decompoton (EEMD) appled to the prncpal component gnal to obtan the ntrnc mode functon (IMF). he Hlbert-Huang tranform pectrum baed on IMF for varou operatng load condton examned for fault dagno. he propoed algorthm allevate the computatonal load by ung the prncpal component gnal ntead of three ndvdual x-, y-, and z-channel recordng. he expermental valdaton of the propoed algorthm are demontrated ung vbraton gnal acqured from a three-phae electrc nducton motor for healthy and fault condton under varou load. Introducton For pat decade, tme-frequency and tme-cale analy method uch a hort-tme Fourer tranform (SF) and wavelet tranform [1-3] have been nvetgated for analy of nontatonary or nonlnear gnal wth applcaton n rotatonal machne fault detecton and health montorng to prevent harmful envronment and enure relable operaton of equpment. Although thee technque are uccefully appled to machne health dagno Proceedng of he 14 IAJC-ISAM Internatonal Conference ISBN 978-1-6643-379-9
and fault detecton, the reult depend on the electon of wndow type or the ue of a bae wavelet. Recently, the Hlbert-Huang tranform (HH) [4,5] ha been propoed to decompoe a gnal nto a et of ntrnc mode functon (IMF) va the emprcal decompoton (EMD) proce [6-1]. he EMD an adaptve approach and effectve to decompoe the non-tatonary or nonlnear gnal (ee detal below). However, the EMD proce uffer from a problem of mode mxng due to gnal ntermttency [11,1]. he problem may caue the decompoed reult vague and napproprately nterpreted. o elmnate the mode mxng problem, the enemble emprcal mode compoton (EEMD) algorthm [11, 1] ha been propoed. h method eentally procee the orgnal gnal wth an added whte noe equence nto a et of IMF ung the tandard EMD repettvely. he enemble mean of the correpondng IMF, whch are obtaned from the tandard EMD, ued a the fnal EEMD decompoed IMF (ee detal n below). Although the EEMD effectve for removng the mxng mode problem, the computatonal load huge due to the enemble proce. Specally, when applyng mult-channel gnal uch a gnal from a enor that provde three-orthogonal channel nformaton, proceng each channel data equence va the EEMD proce requre even more computaton and hnder practcal applcaton. h paper frt decrbe the prncple of the tandard EMD, Hlbert-Huang tranform, and EEMD. o reduce the computatonal load by ung the EEMD for multchannel gnal, a prncpal component emprcal decompoton (PCEEMD) algorthm propoed. he PCEEMD proce cont of two tage: the frt tage contruct the prncpal component gnal baed on the data ened from three-orthogonal channel, and the econd tage apple the tandard EEMD proce to the prncpal component gnal. After obtanng the IMF n the prncpal drecton, they can be drectly employed for rotatonal machne fault detecton by the HH pectrum, or, a an opton, the obtaned IMF can be projected nto three orthogonal axe. Algorthm Development Prncple of EMD, H Spectrum, and EEMD he EMD an adaptve decompoton approach whch appled to decompoe nonlnear and non-tatonary gnal. he EMD proce extract a et of IMF from the orgnal gnal. Each IMF mut meet two condton [4]: the number of extrama and the number of zero crong mut be ether equal or dffer at mot by one; the mean value of the envelope defned by the local maxma and the envelope defned by the local mnma at any pont mut be zero. he tep of the EMD algorthm [4-6] to decompoe a gnal x( t ) decrbed below: Step 1: Intalze r = x and et = Step : Extract the th IMF wth the followng ftng procedure: a. Intalze h ( k 1) = r 1 wth k = 1 Proceedng of he 14 IAJC-ISAM Internatonal Conference ISBN 978-1-6643-379-9
b. Fnd the local maxma and local mnma of gnal h ( ) ( 1) t c. Interpolate the local maxma and local mnma by cubc plne to contruct the upper and lower envelope of h ( ) ( 1) t k d. Calculate the mean m ( ) ( k 1) t of the upper and lower envelope of h ( ) ( k 1) t e. Calculate hk = h ( k 1) m ( k 1) f. If the top crteron for the teraton k gven below atfed [ h ( k 1) hk ] SD (1) h t= ( k 1) where SD a predefned value and uually et to.1, that, f equaton 1 atfed, h ( ) ( k ) t an IMF and then et c = h ( k ) ; ele et k = k + 1 and then go to tep (b). Step 3: Calculate equence: r ( ) ( ) ( ) + 1 t = r t c t Step 4: If r ( ) + 1 t ha at leat extrema then et = + 1 and go to tep ; ele the decompoton completed and r ( ) + 1 t the redual gnal. he decompoton reult are lted below: x c = r By ummng them, t follow that 1 1 r c = r 1.. r ( ) ( ) ( ) N 1 t cn t = rn t N = 1 k x = c + rn () It can be een that gnal x( t ) decompoed nto N ntrnc mode functon and a redue gnal rn ( t ). Once gnal x( t ) decompoed to N IMF, the Hlbert tranform (H) [4,5] can be appled to each of IMF to obtan ntantaneou frequency, that, 1 c y = dτ π (3) t τ he analytcal gnal z ( t ), whch contructed ung both IMF c ( t ) and t H y ( t ), can be expreed a z = c + jy (4) hen the ntantaneou envelope of the analytc gnal for the th IMF found to be a t = c t + y t (5) ( ) ( ) ( ) and the correpondng phae angle can be determned by 1 y θ = tan c (6) Proceedng of he 14 IAJC-ISAM Internatonal Conference ISBN 978-1-6643-379-9
Notce that ntantaneou envelope a ( t ) ndcate the gnal energy varaton whle the phae angle θ the ntantaneou phae. he ntantaneou frequency for the th IMF can be found by takng dervatve of the phae angle, that, dθ ω = (7) dt he Hlbert tranform baed on the IMF referred to Hlbert-Huang tranform (HH); and the tme-frequency plot of HH referred to HH pectrum. Although the EMD demontrate the effectvene n decompong nonlnear and non-tatonary gnal, the method ha a problem of mode mxng; that, a ngle IMF may contan ocllaton of dramatcally dparate cale, or a component of a mlar cale rede n dfferent IMF due to gnal ntermttency. he ntermttence could caue erou gnal alang n tmefrequency dtrbuton a well a make the phycal meanng of the ndvdual IMF unclear. he noe-ated data analy [11, 1] by addng noe to the orgnal gnal propoed. h method referred to the enemble emprcal mode decompoton (EEMD). he prncple of the EEMD can mply be decrbed a follow. he EEMD add whte noe to the orgnal gnal before applyng the EMD. Snce the added whte noe n background populate the whole tme-frequency pace unformly, the gnal component at dfferent cale can automatcally be projected onto proper cale of the reference etablhed by the whle noe n the background. Although each tral may produce noy reult, the noe n the reult can be cancelled out by ung the enemble mean wth a gnfcant number of tral. Hence, each IMF obtaned ung EEMD the enemble mean of tral. he EEMD then ummarzed below: Step 1: Add a whte noe equence to the orgnal gnal (the tandard devaton of noe =1~% of the tandard devaton of the orgnal gnal). Step : Decompoe the gnal wth added whte noe equence nto IMF [ c, at mth tral] ung EMD. Step 3: Repeat Step 1- wth dfferent whte noe equence for M tme Step 4: Calculate the enemble mean for each IMF M 1 c = c, = 1,, L, N (8), m M m = 1 Obvouly, the EEMD algorthm ha a large computatonal load. Development of Prncpal Component EEMD In many applcaton, a enor [13] acqurng gnal may contan mult-component. Fgure 1 depct an accelerometer (ngle taton) contng of three orthogonal x-, y-, and z- channel. Channel x and y are degnated to record acceleraton n x and y drecton, repectvely. Channel z meaure vertcal acceleraton. An ncomng gnal ha angle of α, m Proceedng of he 14 IAJC-ISAM Internatonal Conference ISBN 978-1-6643-379-9
β, and γ relatve to x-, y-, and z-axe, repectvely. Applyng the EEMD algorthm for each channel requre a gnfcant amount of computatonal load. Aumng ( ) Fgure 1. A enor wth x-, y-, z- channel recodng n the ncomng gnal wth a drecton of [ co co co ] Proceedng of he 14 IAJC-ISAM Internatonal Conference ISBN 978-1-6643-379-9 α β γ whle x( n ), y( n ), and z( n ) are the enor recordng, the ncomng gnal can be etmated by a lnear combnaton x( n) ( n) = U X = [ u1 u u 3] y( n) (9) z( n) U u u u X = x( n) y( n) z( n). where = [ ] and [ ] 1 3 Agan, the enor gnal are aumed to be zero mean proce, that, E( x( n)) E ( X ) = E( y( n)) = (1) E( z( n)) then E ( n) = E U X = (11) ( ) ( ) where E ( ) the expectaton operator. Note that the power of the ncomng gnal can be expreed a ( = E ( n) ) = U E ( XX ) U = U CU (1) where C a 3x3 covarance matrx defned below: x xy xz C = E ( XX ) = xy y yz (13) xz yz z x E x ( n) y E y ( n) z = E z ( n), xy = E ( x( n) y( n) ), wth = ( ), = ( ), ( ) = E ( x( n) z( n) ), and E ( y( n) z( n) ) xz =. Baed on Fgure 1, t can be een that yz
x( n) coα X = y( n) co β = ( n) z( n) coγ U = coα co β coγ found, the etmated ncomng gnal become If [ ] ( α β γ ) ( n) = U X = co + co + co ( n) = 1 ( n) (15) Equaton 15 ndcate that U [ u u u ] = hould be a untary vector, that, U U = 1 1 3 and the bet untary vector the one to maxmze the power of the ncomng gnal wth the followng contrant: max (16) U, λ L ( = U CU λ U U 1) (17) akng dervatve of to U and ettng the reult to zero, t follow that L ( C λ I ) U = (18) akng dervatve of to the Lagrange multpler L λ and ettng the reult to zero, the untary vector contrant U U = 1 yelded. It clear that the Lagrange multpler L C C C eentally an egenvalue of covarance matrx C, that, λ λ1, λ, λ3. o enure be the maxmum value, takng econd-order dervatve of to U lead to the followng Proceedng of he 14 IAJC-ISAM Internatonal Conference ISBN 978-1-6643-379-9 L λ (14) to matrx: L ( C λ I) U L = = C λ I = H (19) U U H H and H mut be em-negatve defnte. Let λ and U an egenvalue and an egenvector of matrx H. Equaton 19) become H L H ( H λ I) U = [ C ( λ + λ ) I] U = () H L H Equaton ndcate that λ + λ an egenvalue of matrx C, that, C L H λ = λ + λ = 1,,3 (1) and U H = U. he egenvalue of matrx H can be determned a H H C L λ = λ λ, = 1,,3 () H L C o enure λ (H mut be em-negatve defnte), λ = λmax. he correpondng untary C vector U for λmax the optmal vector whch repreent the gnal prncpal drecton. Wth the obtaned optmal untary vector of U, the prncpal component gnal acheved a ( n) = U X. herefore, the propoed prncpal component enemble emprcal mode decompoton (PCEEMD) algorthm ummarzed below: Step 1: Compute covarance matrx C.
C Step : Determne the maxmum egenvalue λ max and t correpondng untary egenvector U. Step 3: Compute the prncpal component gnal ( n) = U X. Step 4: Apply the EEMD algorthm to the prncpal component gnal ( n) = U X to obtan IMF. Step 5: (optonal) Project IMF to x-, y-, and z-axe, repectvely, that, IMFx = u1 IMF, IMFy = u IMF, IMFz = u3 IMF Experment and Valdaton o valdate the propoed PCEEMD method for non-tatonary or nonlnear gnal analy, vbratonal gnal from the accelerometer baed on three-orthogonal channel were acqured from the three-phae nducton motor wth an adjutable load (Fgure ). A hown n Fgure, an accelerometer wa attached on the three-phae nducton motor. he acceleraton meaured n x-, y-, z- axe were obtaned va LabVew data acquton platform at a amplng rate of 1 khz wth a 16-bt data reoluton. he acqured data equence from each channel wa preproceed to remove t mean (DC component). he adjutable load va the belt wa coupled to the motor va a rubber coupler. A fault n the rubber coupler wa ntroduced. Fgure 3 how healthy and fault coupler condton. he coupler n fault condton ha nner and outer worn-out teeth on the drvng de. he experment were carred out wth no load (% load), medum load (5% load), and full load (1% load) wth the motor runnng at 1,8 rpm. he peed of the haft wa montored by an optcal encoder. Fgure. Expermental etup Proceedng of he 14 IAJC-ISAM Internatonal Conference ISBN 978-1-6643-379-9
Fgure 3. Coupler healthy condton and fault condton Notce that for the healthy condton valdaton experment, the healthy coupler wa ntalled whle for the fault condton valdaton, the healthy coupler wa mply replaced by the fault coupler. It alo aumed that the condton for the healthy coupler or the fault coupler tay the ame durng tetng. he tranton condton between the healthy coupler and fault coupler not condered n th paper and wll be nvetgated n the future. Fgure 4 how x-y-z channel vbraton gnal meaured from the enor a well a the prncpal component gnal produced by the PCEEMD algorthm for the healthy coupler and fault coupler under the 5% load. Proceedng of he 14 IAJC-ISAM Internatonal Conference ISBN 978-1-6643-379-9
x (n) y (n) z (n) (n).5 -.5 1 3 4 5.5 -.5 1 3 4 5.5 -.5 1 3 4 5.5 -.5 1 3 4 5 me (m) x (n) y (n) z (n) (n).5 -.5 1 3 4 5.5 -.5 1 3 4 5.5 -.5 1 3 4 5.5 -.5 1 3 4 5 me (m) Fgure 4. Acceleraton meaurement and generated prncpal componenet gnal ( n ) (left: healthy condton; rght: fault condton) he econd tage appled the EEMD algorthm to the prncpal component gnal. he prncpal component gnal wa added wth a whte noe equence wth a Gauan dtrbuton ung % of the tandard devaton of the orgnal gnal. he enemble mean of each IMF calculated ung 5 tral. he acheved correpondng IMF for both healthy and fault condton are gven n Fgure 5 for comparon. It can be een that for both cae, there are no gnfcant evdence of mxng mode. he haft frequency of 3 rev/ can be een n IMF6 for both cae. For the healthy condton, there are 9 IMF but 1 IMF for the fault Proceedng of he 14 IAJC-ISAM Internatonal Conference ISBN 978-1-6643-379-9
condton. I M F 1. -. 5 1 15 5 3 35 4 45 5.1 -.1 5 1 15 5 3 35 4 45 5.5 -.5 5 1 15 5 3 35 4 45 5.5 -.5 5 1 15 5 3 35 4 45 5.5 -.5 5 1 15 5 3 35 4 45 5. -. 5 1 15 5 3 35 4 45 5.1 -.1 5 1 15 5 3 35 4 45 5. -. 5 1 15 5 3 35 4 45 5.5 -.5 5 1 15 5 3 35 4 45 5 I M F I M F 3 I M F 4 I M F 5 I M F 6 I M F 7 I M F 8 I M F 9 I M F 1 -.1 5 1 15 5 3 35 4 45 5 I M F 1.1 -.1 5 1 15 5 3 35 4 45 5 I M F.1 -.1 5 1 15 5 3 35 4 45 5 I M F 3.1 -. 5 1 15 5 3 35 4 45 5 I M F 4. -.5 5 1 15 5 3 35 4 45 5 I M F 5.5 -.5 5 1 15 5 3 35 4 45 5 I M F 6.5 -. 5 1 15 5 3 35 4 45 5 I M F 7. -. 5 1 15 5 3 35 4 45 5 I M F 8. -. 5 1 15 5 3 35 4 45 5. -. 5 1 15 5 3 35 4 45 5 I M F 9. me (m) me (m) Fgure 5. Decompoed IMF from two coupler condton ung the PCEEMD algorthm (left: healthy condton; rght: fault condton) he correpondng HH pectra for healthy and fault condton are depcted n Fgure 6. A hown Fgure 6a, the domnant frequency component come from the haft rotaton, that, 3 Hz (rev/ec); and for the fault condton a hown n Fgure 6b, bede the domnant frequency component of the haft rotaton (3 Hz), there appear the fourth harmonc component, rregular pule and hgh frequency noe. A an addtonal valdaton, the dcrete-fourer tranform pectra were calculated and dplayed n Fgure 7. Clearly, the haft frequency component domnant for both healthy and fault condton. Smlarly, the fourth harmonc frequency component become gnfcant n the fault condton. However, the DF pectrum doe not how tme-cale nformaton. Proceedng of he 14 IAJC-ISAM Internatonal Conference ISBN 978-1-6643-379-9
1 1 1 1.9.9.8 1 1.8.7.7.6.5 6 Frequency (Hz) Frequency (Hz) 8 8.6.5 6.4.4 4th Harmonc 4 4.3.3...1.1 5 1 15 5 3 me (m) 35 4 45 5 1 15 5 3 me (m) 35 4 45 Fgure 6. he HH pectra from two coupler condton ung the PCEEMD algorthm (left: Healthy condton; rght: fault condton).8.1.7.8.5 DF pectrum DF pectrum.6.4.3.6.4...1 4 6 8 Frequency (Hz) 1 1 4 6 8 Frequency (Hz) 1 1 Fgure 7. DF pectra from two coupler condton ung the PCEEMD algorthm (left: healthy condton; rght: fault condton) Addtonal Reult he valdaton for the % and 1% load alo how the mlar reult a that of the 5% load. he reult are content and ummarzed n able 1. able 1. Reult from HH Spectra Operaton Condton Healthy, % load Faulty, % load Healthy, 5% load Faulty, 5% load Healthy, 1% load Faulty, 1% load Domnant frequency Hgh-order harmonc Not Not Not Irregular pule Not Not Not Proceedng of he 14 IAJC-ISAM Internatonal Conference ISBN 978-1-6643-379-9 Hgh frequency noe Not Not Not
Snce the EEMD proce only apple to the prncpal component gnal, the computatonal load gnfcantly reduced. Mot mportantly, nce the prncpal component gnal contan the ened vbraton gnal wth t algned drecton o that the obtaned IMF and HH pectra wll preent mot meanngful nformaton for fault detecton and condtonal montorng. Concluon h paper propoed a prncpal component enemble emprcal decompoton (PCEEMD) algorthm for rotatonal machne fault detecton and condtonal montorng. he algorthm very effectve for proceng data from a ngle taton enor wth x-, y-, and z- enng component. he PCEEMD cont of two tage. he frt tage perform vbratonal gnal enhancement to acheve the prncpal component gnal accordng to three orthogonal channel recordng. Wth the prncpal component gnal, the EEMD algorthm appled to obtan the IMF wth an advantage of mxng mode elmnaton. he Hlbert-Huang tranform pectra are then obtaned for rotatonal machne fault detecton and dagno. he algorthm gnfcantly allevate the computatonal load by proceng the prncpal component gnal ntead of three ndvdual channel recordng. he expermental valdaton of the propoed method are demontrated ung vbraton data acqured from the three-phae electrc motor for healthy and fault condton under varou load. Reference [1] Sath, L. (1998). Short-me Fourer and Wavelet ranform for Fault Detecton n Power ranformer durng Impule et. Proceedng of the Inttute of Electrcal Engneerng, Scence, Meaurement and echnology, 145, 77-84. [] Wang, C., & Gao, R. (3). Wavelet ranform wth Spectral Pot-Proceng for Enhanced Feature Extracton. IEEE ranacton on Intrumentaton and Meaurement, 5, 196-131. [3] an, L., & Jang, J. (13). Dgtal Sgnal Proceng: Fundamental and Applcaton. (nd ed.). Amterdam: Elever. [4] Huang, N. E., Zheng, S., & Steven, R. L. (1998). he Emprcal Mode Decompoton Method and the Hlbert Spectrum for Non-Statonary me Sere Analy. Proceedng of Royal Socety of London, A454, 93-995. [5] Feldman, M. (11). Hlbert ranform n Vbraton Analy. Mechancal Sytem and Sgnal Proceng, 5, 735-8. [6] Le, Y., Ln, J., He, Z., & Zuo, M. (13). A Revew on Emprcal Mode Decompoton n Fault Dagno of Rotatng Machnery. Mechancal Sytem and Sgnal Proceng, 35, 18-16. [7] Lu, B., Remenchneder, S., & Xu, Y. (6). Gearbox Fault Dagno Ung Emprcal Mode Decompoton and Hlbert Spectrum. Mechancal Sytem and Sgnal Proceng,, 718-734. [8] L, R., & He, D. (1, Aprl). Rotatonal Machne Health Montorng and Fault Detecton Ung EMD-Baed Acoutc Emon Feature Quantzaton. IEEE ranacton on Intrumentaton and Meaurement, 61(4), 99-11. Proceedng of he 14 IAJC-ISAM Internatonal Conference ISBN 978-1-6643-379-9
[9] Yang, W., Court, P. J., avner, & Crabtree, C. J. (11). Bvarate Emprcal Mode Decompoton and It Contrbuton to Wnd urbne Condton Montorng. Journal of Sound and Vbraton, 33, 3766-378. [1] Gao, Q., Duan, C., Fan, H., & Meng, Q. (8). Rotatng Machne Fault Dagno Ung Emprcal Mode Decompoton. Mechancal Sytem and Sgnal Proceng,, 17-181. [11] Zhang, J., Yan, R., Goa, R., & Feng, Z. (1). Performance Enhancement of Enemble Emprcal Mode Decompoton. Mechancal Sytem and Sgnal Proceng, 4, 14-13. [1] Yan, R., Zhao, R., & Gao, R. (1, December). Noe-Ated Data Proceng n Meaurement Scence: Part wo. IEEE Intrumentaton and Meaurement Magazne, 15(6), 3-35. [13] Magotra, N., Ahmed, N., & Chael, E. (1989, January). Sngle-Staton Semc Event Detecton and Locaton. IEEE ranacton on Geocence and Remote Senng, 7(1), 15-3. Bographe LI AN a profeor wth the College of Engneerng and echnology, Purdue Unverty North Central. He receved the B.S. degree from the Southeat Unverty, Nanjng, Chna, n 1984, and the M.S. degree n Structural Engneerng and the M.S. and Ph.D. degree n Electrcal Engneerng from the Unverty of New Mexco, Albuquerque, n 1987, 1989, and 199, repectvely. Dr. an an IEEE Senor Member nce 1. H reearch nteret nclude the area of dgtal gnal proceng, actve noe control and control ytem, and dgtal communcaton. He co-authored two textbook: Dgtal Sgnal Proceng: Fundamental and Applcaton, nd ed., Elever, 13; and Analog Sgnal Proceng and Flter Degn, Lnu Publcaton, 9. He hold a US patent. He ha erved a an aocate edtor for the Internatonal Journal of Engneerng Reearch and Innovaton, and Internatonal Journal of Modern Engneerng. ALEXANDER MUSSA a enor electrcal engneerng tudent at College of Engneerng and echnology, Purdue Unverty North Central. JUSIN POLING a enor electrcal engneerng tudent at College of Engneerng and echnology, Purdue Unverty North Central. KAI JUSICE a enor electrcal engneerng tudent at College of Engneerng and echnology, Purdue Unverty North Central. HONGBO XU a Ph.D. tudent at the School of Mechancal Engneerng, Nanjng Unverty of Scence and echnology, Nanjng, Chna. He a vtng cholar n prng 14, at College of Engneerng and echnology, Purdue Unverty North Central. Proceedng of he 14 IAJC-ISAM Internatonal Conference ISBN 978-1-6643-379-9