The Rolling Bearing Fault Feature Extraction Method Under Variable Conditions Based on Hilbert-Huang Transform and Singular Value Decomposition

Transcription

1 The Rollg Bearg Fault Feature Extracto Method Uder Varable Codtos Based o Hlbert-Huag Trasform ad Sgular Value Decomposto Hogme Lu, Xua Wag ad Che Lu THE ROLLING BEARING FAULT FEATURE EXTRACTION METHOD UNDER VARIABLE CONDITIONS BASED ON HILBERT-HUANG TRANSFORM AND SINGULAR VALUE Hogme Lu 1, Xua Wag 2 ad Che Lu 3 1, 2, 3 School of Relablty ad Systems Egeerg, Behag Uversty, Bejg, , Cha 3 Scece & Techology o Relablty & Evrometal Egeerg Laboratory, Bejg, , Cha E-mal: luche@buaa.edu.c Abstract. The fault dagoss precso for rollg beargs uder varable codtos has always bee usatsfactory. For solvg ths problem, a feature extracto method combg the Hlbert-Huag trasform wth sgular value decomposto was proposed ths paper. The method cludes three steps. Frstly, stataeous ampltude matrces were obtaed by Hlbert-Huag trasform from rollg bearg sgals. Secodly, as the fault feature vector, the sgular value vector was acqured by applyg sgular value decomposto to the stataeous ampltude matrces. Thrdly, the detfcato ad classfcato of rollg bearg were acheved by Elma eural etwork classfer. The expermet shows that ths method ca effectvely classfy the rollg bearg fault modes wth hgh precso uder dfferet operatg codtos. 1. Itroducto I the moder rapd developg dustry, as the vtal compoet of most mechasms, the rollg bearg s cofroted wth a creasg complex workg evromet. I the recet research, the rollg bearg fault dagoss uder varable workg codtos s less effectve. So t s of great sgfcace to seek a rollg bearg dagostc method applcable to dfferet workg codtos. As may covetoal methods are usatsfactory for dagoss uder varable codtos, researchers have bee payg more atteto to relevat studes. Yum SHAO preseted a fault dagoss system based o a smart bearg cludg several sesg devces order to track the tmevaryg parameter [1]. Qgbo He addressed mafold learg o geerated tme-frequecy dstrbutos for mache fault sgature aalyss [2]. I all, all these works play a pvotal role fault dagoss uder varable codtos. Whereas, some of these works stll perform usatsfactorly certfyg the robustess, ad the others sacrfce low-costg wth addg addtoal sesg devces. Thus, a method combg Hlbert-Huag trasform (HHT) wth sgular value decomposto (SVD) s proposed ths paper. Uder varable codtos, the fault characterstc frequecy of the rollg bearg wll chage wth tme, whch makes t ecessary to take tme-frequecy aalyss. Compared wth other tme-frequecy methods, HHT has a dstct advatage hadlg o-statoary ad olear data. Nevertheless the result obtaed from HHT s always too eormous to apply to varable workg codtos. I addto, t s complcated to extract the correspodg ampltude at the tmevaryg characterstc frequecy. I vew of ths stuato, the sgular value decomposto could be used for compesato. The SVD greatly compresses the scale of the fault feature vector the foudato of expressg the orgal characterstc. Moreover, the sgular value has great stablty, ad t chages lttle whe the matrx elemets chage, whch makes t possble to ehace robustess of the fault dagoss method uder varable codtos. 3 To whom ay correspodece should be addressed. 80 VIBROENGINEERING. VIBROENGINEERING PROCEDIA. NOVEMBER VOLUME 2. ISSN

2 2. The feature extracto method based o HHT ad SVD 2.1. The tme-frequecy sgal decomposto based o Hlbert-Huag trasform Hlbert-Huag trasform, combg the Emprcal Mode Decomposto (EMD) wth the Hlbert trasform, could be appled to acqure the stataeous ampltude ad stataeous frequecy. After the mplemetato of EMD, the orgal sgal x( t ) could be decomposed as follows: x( t) = c ( t) + r( t) (1) where c( t ) s the th IMF compoet, ad represets the sgal from hgh frequecy to low frequecy dfferet bads. The, settg IMF equal to c() t, the aalytc sgal h() t of c() t s as follows: = 1 r t s the resdue. The IMF compoet c ( = 1,, ) h() t = c t + jcɶ t = a() t e jϕ t (2) where cɶ ( t) s the outcome fucto after applyg Hlbert trasform to c( t ) : cɶ t + 1 c( t) = dτ π t τ a ( t ) s the ampltude fucto ad also the evelope sgal of the IMF compoet c( t ) : ϕ ( t) s the phase fucto a t c t c t 2 = + ɶ 2 () (3) 1 ϕ t = ta ( cɶ ( t) / c( t)) (4) Calculate the dervato of the phase fucto to obta the stataeous frequecy: ( t) dϕ ω ( t) = (5) dt Accordgly, the orgal sgal x( t ) could be expressed by a tme-depedet fucto cotag the stataeous ampltude ad the stataeous frequecy; thus, t s possble to hadle the rollg bearg fault dagoss uder tme-varyg codtos by applyg HHT: H w, t Re a ( t)exp( j w ( t) dt) = (6) = 1 The IMF s stataeous ampltude s ts evelope sgal, whch s smlar to the perodc sgal [3].The evelope sgal s beefcal for detfyg fault eve weak oes. Thus, calculate the a t, whch s the evelope curve of the IMF for ampltude of each aalytc sgal h( t ) to get further aalyss. Eve so, t s extremely complcated to detect tme-varyg parameters that represetg varable workg codtos, to calculate the correspodg fault characterstc frequecy ad to extract the correspodg ampltude. I ths regard, t s hard to acheve rollg bearg fault detfcato uder varable operatg codtos. To ths ed, ths paper further proposed SVD combato wth HHT to acheve the rollg bearg fault feature extracto. VIBROENGINEERING. VIBROENGINEERING PROCEDIA. NOVEMBER VOLUME 2. ISSN

3 2.2. Sgular value decomposto based o Hlbert-Huag trasform The sgular value has great stablty, ad t chages lttle whe the matrx elemets chage [4]. Ths paper uses the sgular value decomposto to obta the trsc characterstc of the feature matrx obtaed from HHT. Sgular values are able to express the feature of orgal matrx the form of several values, whch s coductve to compress the scale of the feature vector. m* Sgular value decomposto s defed as: Suppose M s a m matrx. Ay matrx M R ca be factored to sgular value decomposto [5]: * M= U V (7) where r U mm * R ad V * R are orthogoal matrces ad = rak ( M ) prcpal postve dagoal etres. The p dagoal etres of m * R s dagoal matrx wth are usually deoted by σ for = 1,, p, where p= m{ m, } ad σ are called the sgular values of M. The sgular values are the square roots of the ozero egevalues of both values descedg order as commoly suggested. 3. Fault detfcato ad classfcato T MM ad T M M. Lst the sgular Fgure 1. Flowchart of obtag the feature vector. Fgure 2. The test bed. As show Fgure 1, the orgal vbrato sgal s decomposed by EMD, ad IMFs are obtaed. The pror k IMFs are the extracted for aalyss because the frst few IMFs have the hghest frequecy. The the aalytc sgal h s obtaed by applyg the hlbert trasform to each IMF. Subsequetly, the evelope sgal a s obtaed after calculatg the ampltude of each h. The a = 1,, k. After applyg the SVD to M, the sgular value feature matrx M s costructed by vector ad also the fault feature vector σ ( M ) of the feature matrx M could be obtaed: σ = σ σ σ (8) 1 2 k ( M ) [ M, M,, M ] The, the status wll be detfed ad classfed by utlzg Elma eural etworks. 4. Expermetal result ad dscusso I order to valdate the effectveess of proposed fault dagoss method, the RS deep groove ball bearg s used the expermet. The test bed of rollg bearg s show Fgure 2. The vbrato data s sampled uder dfferet fault modes ad operatg codtos. The hlbert spectrums of dfferet states are llustrated Fgure 3. The hlbert spectrum of ormal sgal Fgure 3(a) depcts that the eergy dstrbuto s uform ad the peak presets the frequecy bad aroud 1000Hz. Its maxmum ampltude s about The hlbert spectrum Fgure 3(b) dcates that the domat eergy dstrbutes the frequecy bad aroud 3000Hz ad the tme-frequecy dstrbuto 82 VIBROENGINEERING. VIBROENGINEERING PROCEDIA. NOVEMBER VOLUME 2. ISSN

4 s complcated the er-race fault mode. Its maxmum ampltude s about 0.9. For outer-race fault mode, the hlbert spectrum Fgure 3(c) shows that the domat eergy dstrbuto s smlar to that of er-race fault mode. Nevertheless, the maxmum ampltude s about 3, whch dcates the vbrato eergy s relatvely hgh. The hlbert spectrum of rollg elemet fault llustrates that eergy rarely dstrbutes the frequecy bad from Hz. Besdes, the vbrato eergy s low ad ts maxmum ampltude s oly about It could be see that the tme-frequecy dstrbuto dffers uder dfferet states. Therefore, the feature matrx could be costructed for fault detfcato. (a) (b) (c) Fgure 3. Hlbert spectrum of (a) ormal (b) er-race fault (c) outer-race fault (d) rollg elemet fault. After obtag stataeous ampltude matrx, the sgular value vector could be acqured by coductg SVD. Uder each fault mode, we collected data uder four operatg codtos correspodg to motor speed of 1730, 1750, 1772, 1797r/m. 25 sets of data are sampled uder each operatg codto each fault mode, ad 400 sets are obtaed total. I the fgure, the sgular value vector of each sample s preseted as a le. All the sgular value les are preseted the same graph to be observed effectvely. The sgular value clusters of er-race fault sgal, outer-race fault sgal ad rollg elemet fault sgal are show Fgure 4, Fgure 5 ad Fgure 6 respectvely. The sgular values clusters obtaed by EMD ad SVD [6, 7] are also llustrated smultaeously for comparso. As these three dagrams depct, the feature extracto method combato wth the HHT ad SVD has the promet advatage over the oe wth the combato of EMD ad SVD. Eve uder dfferet workg codtos, the sgular value vector of the same falure mode stll rema a hgh degree of cocdece. The sgular value clusters correspodg to the three fault modes metoed above ad the ormal state are placed Fgure 7 to observe fault mode separablty. The gaps betwee the four regos used to dcate four fault modes are huge. I other words, t s feasble to use the sgular value as the put of the eural etwork classfer, because the sgular value vector of dfferet fault modes retas well separablty. To further verfy the effectveess of the proposed method, the Elma eural etwork s appled for the fault patter classfcato, ad the precso wll be aalyzed. (d) VIBROENGINEERING. VIBROENGINEERING PROCEDIA. NOVEMBER VOLUME 2. ISSN

5 Fgure 4. The sgular value clusters of er-race fault sgal. Fgure 5. The sgular value clusters of outer-race fault sgal. Fgure 6. The sgular value clusters of rollg elemet fault sgal. Sgular Value Outer-race fault Ier-race fault Rollg elemet fault Normal Fgure 7. The sgular value clusters of dfferet fault model. To extract fault feature, sgular value feature vector are obtaed by the proposed HHT-SVD method, the fault are classfed by Elma eural etwork. The put of Elma eural etwork s the sgular value feature, ad the output s set as ( ), ( ), ( ), ( ), represetg the ormal, the er-race fault, the outer-race fault ad the rollg elemet fault sequece. The data were dvded to a trag dataset comprsg 25 cases (the eural etwork trag set s ot lsted here, due to the space lmtato) ad a testg dataset compromsg 375 cases. The partal results of the etwork eural recogto are lsted Table 1. Uder varable codtos, the actual output of eural etwork extremely agrees wth the target output. I addto, t s ot ecessary to chage parameters whle operatg codtos varyg to a certa pot because of the respectve merts of the HHT ad SVD. Cosequetly, the proposed method combed wth HHT-SVD ad eural etwork ca effectvely realze the fault dagoss of rollg bearg uder varable workg codtos. 84 VIBROENGINEERING. VIBROENGINEERING PROCEDIA. NOVEMBER VOLUME 2. ISSN

6 Sequece State Table 1. The eural etwork recogto results. Operatg codto Target output Actual output of etwork 1 ormal (1750r/m,2HP) ( ) E (1730r/m,3HP) ( ) errace 3 (1750r/m,2HP) ( ) (1772r/m,1HP) ( ) 2.94E E fault 5 (1797r/m,0HP) ( ) (1730r/m,3HP) ( ) E-11 outerrace 7 (1750r/m,2HP) ( ) 7.241e (1772r/m,1HP) ( ) fault 9 (1797r/m,0HP) ( ) 5.880e (1730r/m,3HP) ( ) rollg 11 (1750r/m,2HP) ( ) elemet 12 (1772r/m,1HP) ( ) fault 13 (1797r/m,0HP) ( ) Cocluso Ths paper has preseted a ew feature extracto method wth the combato of the HHT ad SVD. Frstly, the IMFs were obtaed by the EMD. Secodly, the stataeous ampltude matrx was calculated by applyg Hlbert trasform to each IMF ad the was decomposed by SVD. The expermetal results demostrated the effectveess of the sgular value for dfferet operatg codtos, whch acheves rollg bearg fault dagoss uder tme-varyg codtos. Takg the exstg EMD-SVD as a comparso ths paper, the HHT-SVD method proposed ths paper demostrates ts robustess. Fally, the effectveess s further verfed by usg the Elma eural etwork to classfy. Based o the dscusso above, the superorty of the proposed method hadlg the fault detecto uder the tme-varyg codto ca be verfed. Wthout the eed for huma terveto ad addtoal cost, ths method s a excellet ad worth advocatg automatc method for fault dagoss uder varable codtos. Ackowledgmet Ths research s supported by the Natoal Natural Scece Foudato of Cha (Grat No , , ), as well as the Techology Foudato Program of Natoal Defese (Grat No. Z132010B004). 6. Referece [1] lotta B, Pap R, Pug L, To P ad Vol AG 2001 Proc. It. Cof. o Advaced Itellget Mechatrocs (Como/IEEE) [2] Shao Y, Ge L ad Fag J 2008 Proc. It. Cof. o Automato ad Systems (Seoul/IEEE) [3] He Q, Lu Y, Log Q, et al J. IEEE Tras Istrum Meas [4] Wedog X, Ybg L, Yg H, et al 2012 World Cof. o Itellectual cotrol ad Automato (Bejg/IEEE) [5] Reju Z ad Husheg W 2010 Proc. It. Cof. o Computer Desg ad Applcatos(ICCDA) (Qghuagdao/IEEE) [6] Mart C D ad Porter M A 2012 J. Am Math Mo [7] Zhu Z ad Su Y 2007 Proc. It. Cof. o Power Egeerg (Sgapore/ IEEE) [8] Cheg J, Yu D, Tag J, et al 2009 J. Shock Vb VIBROENGINEERING. VIBROENGINEERING PROCEDIA. NOVEMBER VOLUME 2. ISSN