793. Instantaneous frequency identification of a time varying structure using wavelet-based state-space method

Transcription

1 793. Insananeous fequency denfcaon of a me vayng sucue usng wavee-based sae-space mehod X. Xu 1, Z. Y. Sh, S. L. Long 3 Sae Key Laboaoy of Mechancs and Cono of Mechanca Sucues Nanng Unvesy of Aeonaucs and Asonaucs 51 Ma Box, 9 Yudao See, Nanng, 10016, Chna E-ma: 1 xnxu@nuaa.edu.cn, zysh@nuaa.edu.cn, 3 ongshuang@nuaa.edu.cn Te. / Fax: Receved 30 Mach 01; acceped 14 May 01 Absac. Ths pape pesens a mehod o denfy he nsananeous fequency of he me vayng sucues based on wavee and sae space mehods by usng fee and foced vbaon esponse daa. Fsy, he second-ode vbaon dffeena equaons ae ewen as he fs-ode sae equaons usng sae space heoy. Secondy, boh excaon and esponse sgnas ae poeced by he Daubeches wavee scang funcons. Thus, he fs-ode sae equaons ae ansfomed no nea agebac equaons usng he ohogonay of he scang funcons. Lasy, he equvaen me vayng sae space sysem maces of me vayng sucue ae exaced decy by sovng he nea equaons. The nsananeous fequences ae deemned va egenvaue decomposon of he sae space sysem maces. The poposed denfcaon agohm s nvesgaed wh a fou degees-of-feedom spng-mass-dampe mode. Numeca smuaons demonsae ha he poposed mehod s obus and effecve fo denfcaon of he abupy, smoohy and peodcay changng nsananeous fequences of me vayng sucues. Keywod: me vayng sucue, nsananeous fequency, denfcaon, wavee, sae space. 1. Inoducon A numbe of appoaches have been poposed fo he dynamc paamees denfcaon of nea sysems n he eaue ove he pas 30 yeas [1-4]. The npu-oupu moda denfcaon mehods ae deveoped based ehe on a se of Fequency Response Funcons o on he coespondng Impuse Response Funcons. Fo vey age and fexbe sucues e bdge o dam, he foced excaon eques exemey heavy and expensve equpmen vey sedom avaabe n mos dynamc expemens. The oupu-ony moda denfcaon appoaches have been deveoped by esmaon of Powe Speca Densy Funcons o Coeaon Funcons of amben excaon esponse daa. Refeence [5] pesens a bef evew of he evouon of Expemena Moda Anayss fom Inpu-Oupu o Oupu-Ony moda denfcaon echnques and he appcaons n cv engneeng. Some benchma pobems of cv engneeng sucues fo he sysem denfcaon and damage deecon of cv sucues have been esabshed [6-7]. Many nenaona pacpans have poposed dffeen appoaches o denfy sucua damage usng dffeen moda denfcaon echnques. The Foue ansfom FT can be consdeed as a decomposon of a sgna no a nea combnaon of vecos gven by Foue coeffcens. Howeve hs decomposon does no gve me-ocazed nfomaon abou he sgna. The echnques based on he wavee ansfom WT fo sucua damage deecon have eceved consdeabe aenon n ecen yeas [8-9] because of he aby o decompose he measued sgnas n he fequency me doman. The moda denfcaon mehods have been poposed o denfy moda paamees base on wavee anayss pocessng fee esponses of mechanca sucues. The mehods fo esmang moda fequency and dampng aos have been poposed based on he connuous wavee ansfom n 671 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE 01. VOLUME 14, ISSUE. ISSN

2 efeences [10-11]. Refeence [1] chose he Cauchy wavee and deveoped a pocedue o denfy naua fequences and vscous dampng aos as we as mode shapes. Refeence [13] maes use of he connuous wavee ansfom o deveop a compee pocedue fo denfcaon of naua fequency, vscous dampng ao and mode shapes fom fee esponses. The choce of he mohe wavee and s ocazaon popees s dscussed. The edge effec s hghghed by noducon of wo ea coeffcens o aow hs effec o be aen n o accoun dung he denfcaon pocess. The Hbe ansfom HT aso has he capaby of decomposng measued sgnas n he fequency me doman o exac he nsananeous fequency a evey me nsan. Theefoe sysem denfcaon mehods based on he HHT ae hen deveoped o esmae he moda paamees of mu-degee-of-feedom MDOF sysems ncudng no ony he naua fequences, dampng aos, bu aso he mode shapes as we as he mass, sffness and dampng maces of he nea sysems n whch he mode shapes may be ea o compex [14-15]. Fuhe, he Hbe Huang speca anayss has been nvesgaed fo he heah monong of sucues and he damage deecon of a benchma pobem. In gh of he above-menoned efeences, a moda denfcaon appoaches and he appcaon n damage deecon ae deveoped ony fo he nea me-nvaan LTI sysems. Howeve, sucua sysems accumuae damage unde sevce oad and envonmena excaons. In such case, a nea me-vayng LTV mode shoud be abe o capue he nsananeous chaacescs of he sysem and o denfy s damage [16-17]. The denfcaon of nea me-vayng sysems has eceved nceasng aenon nowadays. Some effos have been made n exendng he dscee me sae-space denfcaon agohm fom nea me-nvaan sysems o nea me-vayng sysems fom he 90s [18-19]. A subspace-based denfcaon agohm ha uses fee esponses o denfy successve dscee anson maces and use egenvaues of he esmaed anson maces o chaaceze popees of he nea me-vayng sysems has been poposed [0]. Expemena vefcaon sudes on an axay movng caneve beam wee addessed n efeence [1]. Refeence [] poposes a new adapve acng echnque based on he eas-squaes esmaon appoach o ac he abup change of he me-vayng sucua paamees. The wavee-based denfcaon appoach fo he anayss of nea me-vayng sysems was pesened [3]. The denfcaon agohm focuses on he denfcaon of he dampng and sffness paamees assocaed wh he dffeena equaon mode eang npu and he oupu measuemens, whch has been dscezed foowng a Gaen pocedue usng wavee based [4]. The Hbe ansfom echnque has been nvesgaed o denfy sysem nsananeous moda paamees fo nonnea sysems. Fo MDOF sucues, an denfcaon agohm of nea me-vayng sysem based on he Hbe Tansfomaon and empca mode decomposon has been deveoped usng fee and foced vbaon esponse daa [5]. The poposed echnque s capabe of acng sow, abup and peodc changes of dampng and sffness coeffcens of sysems. In hs pape, we popose a wavee-based sae-space denfcaon appoach fo me-vayng sucues. We mode he LTV sucue usng he sae space mehod and exac he nsananeous fequency by wavee anayss. The Daubeches wavee anayss s noduced o he sae space mode, hus he fs-ode sae equaons can be ansfomed no nea agebac equaons usng he ohogonay of he scang funcons and he me vayng dynamc paamees can be decy esmaed va sovng he nea agebac equaons. Moe mpoany, hs mpovemen of combned use of wo mehods wavee and sae-space successfuy avods compung he connecon coeffcen beween wavee scang funcons and he devave of wavee scang funcons boh of he fs and second ode. Ths mpovemen s abe o save he compuaon me fo he denfcaon pocedue. 67 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE 01. VOLUME 14, ISSUE. ISSN

3 . Sae-space mode of LTV sucues The vbaon equaon of a p degees-of-feedom LTV sucue can be expessed as: xɺ + E xɺ + K x bu M ɺ 1 p me-dependan mass, dampng and sffness x s he p 1 dspacemen veco, b s n npu foce veco, and n denoes he numbe of npu foce sgnas. The oupu equaon fo he same LTV sucue can be epesened as: n whch M, E and K ae p coeffcens maces of he sysem especvey, he p n npu shape max, u s he 1 y C xɺ + C xɺ C x whee ɺ a v + d y s he n 0 1 oupu veco, whch can be a combnaon of dffeen ypes of esponses, C a, C v, C d ae he oupu maces fo he measuemen of acceeaon, veocy and dspacemen, especvey, n 0 s he numbe of oupu sgnas. Equaons 1 and can be ansfomed no he sae space equaon as foows: A z B u z ɺ + 3 C z D u y + 4 n whch z s he n0 p sae veco, A s he p p sysem max, npu max, C s he n p oupu nfuence max and p n 0 dec ansmsson max, especvey. They ae expessed as: B s he D s he n 0 n M 0 Ι A 1 1 K M E 5 [ b] T 1 0 M B 6 C C C M 1 d a 1 Cv CaM K E T 7 D 1 C M b 8 a 3. Sgna poec usng he Daubeches wavee Daubeches wavee, wh he vanshng momens of N, coud be caed dbn wavee. The sae veco z can be poeced by usng scang funcons of dbn wavee a esouon : ~ / z α 9 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE 01. VOLUME 14, ISSUE. ISSN

4 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE 01. VOLUME 14, ISSUE. ISSN whee neges Z ae ansaon facos, α ~ ae he appoxmaon coeffcens,, ae Daubeches scang funcons. Le, / ~ α α and equaon 9 s ewen as foows: Z z α 10 The fs devave of scang funcon o me s: ɺ 11 Subsue equaon 11 no he sae space equaon 3: Z z ɺ ɺ ɺ α 1 4. Agohm fo nsananeous fequences denfcaon 4. 1 Tansfomng he sae equaons o nea agebac equaons Subsung equaons 10, 11 and 1 o sae equaons 3 and 4, we have: + + D C B A β α γ β α α ɺ 13 Hee e / ~ β β, / ~ γ γ, n whch β ~ and γ ~ ae he appoxmaon coeffcens of u and y, havng been muped by, and ang he nne poduc of boh sde of equaon 13: + + d D d C d d B d A d β α γ β α α ɺ 14 Usng he ansaon ohogonay of he scang funcons d δ, he equaon 14 s ewen as:

5 1 αγ Bβ Aα γ Dβ Cα 15 n whch 1 Γ ɺ d ae he connecon coeffcens. Whou oss of he geneay, s assumed ha he engh of he sgna x s Z, hence we have f n. f When he ansaon facos [ N+, + N ] and he neges 0,1,, n1, he connecon coeffcens 1 Γ ae nonzeo. If he sysemc mass has been nown, he sysem max A and he oupu nfuence max C can be denfed by sovng he nea equaons. 4. Fee vbaon case When he excaon foce em of he equaon 1 s zeo, he dffeena equaons can be smpfed. In sae equaons, sysem max A and oupu nfuence max C ae me-dependen. Howeve, npu max B and dec ansmsson max D do no exs any onge. Accodng o he secon 3, sae veco z and he oupu y can be poeced by usng scang funcons of dbn wavee a esouon, hen dong vaabe subsuon and dong fs devave o scang funcon. Foowng he pocedues of secon 4.1, noducng sgnas Daubeches wavee poecon o he fee vbaon sae equaons: α ɺ d A β d C α α d d 16 ang he nne poduc of boh sdes of equaon 16, afe havng been muped by, usng he ohogonay of he ansaes of he scang funcons, equaon 16 s ewen o: 1 α Γ β Cα Aα 1 whee he connecon coeffcens Γ, ansaon facos and he neges a have he same defnons o he secon 4.1. Sovng he nea equaons above, he me-vayng sysem max A and oupu C can be deemned. nfuence max 4. 3 Pocedues fo nsananeous fequency denfcaon Tang he egen decomposed o he sysemc sae max A a dffeen me nsan: A ψ Λ ψ VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE 01. VOLUME 14, ISSUE. ISSN

6 p p whee Λ dag λ, λ C p p s a dagona max, ψ C s a max of egenveco, λ, λ ζ πω ± πω 1ζ, whee ω, ζ ae naua ccua fequency and dampng aon especvey, whch can be defned as foowng, n whch 1,, p ndexes he moda ode: 1 ω λ + λ π Re Im 19 Re λ ζ 0 πω 5. Numeca smuaons A 4-DOF LTV mode Fg. 1 s used fo he case sudy. The umped mass coeffcens ae M1 M M 3 M 4 1g. The dampng coeffcens ae E1 E4 0.6Ns/m, E 0.65Ns/m, E 3 0.7Ns/m. The sffness coeffcens ae K1 K K3 K N/s. The fee and foced vbaon esponse sequences ae cacuaed fom numeca souon of he equaon of moon usng Newma-bea mehod. The fee vbaon esponse s smuaed wh an na dspacemen a he fou umped masses as x 0, x 0. 05m, whe he 1,, 3 na veocy and acceeaon ae a zeo. Two ypes of foced esponses ae cacuaed unde mu hamonc excaon and andom excaon wh an a-zeo na condon. The mu u 00* sn16π, sn 3π,sn 76π,sn104π and he andom hamonc excaon s T excaon acng a fou umped masses, whch s a noma dsbuon andom vaue wh mean zeo and sandad devaon ffy. In hs exampe, we choose db 3 Daubeches wavee, and he esouon 11, so he sampng fequency of he me sees daa s 048 Hz. 4 Fg DOF umped mass mode Dffeen nds of paamees wh dffeen nds of me-vayng cases abup, smooh and peodc ae nvesgaed. The nsananeous fequences ae denfed usng he poposed mehod fom he fee and foced esponses mu hamonc and andom excaon. Case A. Mass coeffcen and snge sffness coeffcen ae me vayng , , 3000, 0.375< 0.75, M ; K sn1.33 π, 0.75<.5, ,.5< VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE 01. VOLUME 14, ISSUE. ISSN

7 The emanng paamees ae nvaan. Fgs. and 3 povde he denfed nsananeous fequency of he LTV sysem. Fou fequences can be aced we usng boh fee and foces esponses. Howeve, s noed ha hee exss age denfcaon eo a he me nsance when he sysem paamees have an abup change. Ths ndcaes ha he poposed agohm has poo capaby n acng abup vaaons because of combnng a numbe of me pons dung he cacuaon pocedue of sovng nea equaons. Fg.. Idenfed nsananeous fequences 1s and nd ode Fg. 3. Idenfed nsananeous fequences 3d and 4h ode Case B. Mass coeffcen and mu sffness coeffcens ae me vayng , , 3000, 0.375< 0.75, M ; K sn1.33 π, 0.75<.5, ,.5< 3, 40000, , 8000, 0.375< 0.75, K sn1.33 π, 0.75<.5, ,.5< 3. VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE 01. VOLUME 14, ISSUE. ISSN

8 The emanng paamees ae nvaan. Case C. Mass, dampng and sffness coeffcens ae me vayng , , 3000, 0.375< 0.75, M ; E ; K sn1.33 π, 0.75<.5, ,.5< 3. The emanng paamees ae nvaan. A compason s shown n Fgs. 4-7 beween he ue vaue and he denfed vaue. Resus ndcae ha he poposed agohm has a good capaby of acng he me-vayng fequency of he LTV sysem. The esmaed esus fom fee vbaon esponses ae moe accuae han hose fom he foced esponses dung he whoe denfcaon me peod. Fg. 4. Idenfed nsananeous fequences 1s and nd ode 6. Concusons 678 Fg. 5. Idenfed nsananeous fequences 3d and 4h ode We pesened a new echnque o denfy he nsananeous fequency of he LTV sysem fom fee o foced esponses daa usng wavee and sae space mehods. The poposed agohm no ony successfuy avods compung he connecon coeffcen beween wavee scang funcons and he second ode devave of wavee scang funcons, bu aso VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE 01. VOLUME 14, ISSUE. ISSN

9 sgnfcany educes he compuaon me n compason o many subspace agohms and me sees ecusve agohms. A 4-DOF dscee mode wh hee nds of cases abup, smooh and peodc was nvesgaed. Numeca esus demonsae ha he poposed agohm has a good capaby fo acng he changes of nsananeous fequency of he LTV sysem. Fg. 6. Idenfed nsananeous fequences 1s and nd ode Acnowedgemens Fg. 7. Idenfed nsananeous fequences 3d and 4h ode Ths eseach s suppoed by he Naona Naua Scence Foundaon of Chna Gan No , and by he Gaduae Cuvaon and Innovaon Foundaon of Jangsu Povnce Gan No. CX10B_088Z and CX10B_105Z. Refeences [1] Maa N., e a. Theoeca and Expemena Moda Anayss. Reseach Sudes Pess, 1997, UK. [] Ibahm S. R., Muc E. C. A mehod fo he dec denfcaon of vbaon paamees fom he fee esponse. The Shoc and Vbaon Buen, 474, 1977, p [3] Benda J. S., Pesso A. G. Engneeng Appcaons of Coeaon and Speca Anayss. A Wey-Inescence Pubcaon, [4] Bnce R., Ken S., Kegaad P. H., Rye A. Idenfcaon of he dynamca popees fom coeaon funcon esmaes. Bygnngssase Meddeese, Dansh Socey fo Sucua Scence and Engneeng, 631, 199, p [5] Cunha A., Caeano E. Fom npu-oupu o oupu-ony moda denfcaon of cv engneeng VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE 01. VOLUME 14, ISSUE. ISSN

10 sucues. Fs Inenaona Opeaona Moda Anayss Confeence, Copenhagen, Dema, 005, p [6] Yeh S. C., Cheng C. B., Loh C. H. Anayss Resus of Shang Tabe Tes of a Fve-Soey Scaed See Budng Mode. Technca Repo MCREE-99-00, Naona Cene fo Reseach on Eahquae Engneeng, [7] Johnson E. A., Lam H. F., Kaafygos L. S., Bec J. L. A benchma pobem fo sucua heah monong and damage deecon. Poceedngs of he 14h ASCE Engneeng Mechancs Confeence, CD ROM, Ausn, TX, 000. [8] Hou Z., Noo M., Amand R. S. Wavee-based appoach fo sucua damage deecon. Jouna of Engneeng Mechancs, ASCE, 167, 000, p [9] Sun Z., Chang C. C. Sasca wavee-based mehod fo sucua heah monong. Jouna of Engneeng Mechancs, ASCE, 1307, 004, p [10] Saszews W. J. Idenfcaon of dampng n MDOF sysems usng me-scae decomposon. Jouna of Sound and Vbaon, 03, 1997, p [11] Ruzzene M., Fasana A., Gabad L., Pombo B. Naua fequences and dampng denfcaon usng wavee ansfom: appcaon o ea daa. Mechanca Sysems and Sgna Pocessng, 11, 1997, p [1] Agou P., Le T. P. Insananeous ndcaos of sucua behavo based on connuous Cauchy wavee ansfom. Mechanca Sysems and Sgna Pocessng, 17, 003, p [13] Le T. P., Agou P. Connuous wavee ansfom fo moda denfcaon usng fee decay esponse. Jouna of Sound and Vbaon, 77, 004, p [14] Yang J. N., Le Y., Ln S., Huang N. Sysem denfcaon of nea sucues based on Hbe-Huang speca anayss. Pa 1: Noma modes. Eahquae Engneeng and Sucua Dynamcs, 3, 003, p [15] Yang J. N., Le Y., Pan S., Huang N. Sysem denfcaon of nea sucues based on Hbe-Huang speca anayss. Pa : Compex modes. Eahquae Engneeng and Sucua Dynamcs, 3, 003, p [16] Ln C. C., Soong T. T., Nae H. G. Rea-me sysem denfcaon of degadng sucues. Jouna of Engneeng Mechancs, 116, 1990, p [17] Udwada F. E., Jeah N. Tme vaaons of sucua popees dung song gound shang. Jouna of he Engneeng Mechancs Dvson, Poceedngs of he ASCE, 106, 1980, p [18] Shoooh S., Sveman L. Idenfcaon and mode educon of me-vayng dscee-me sysems. Auomaca, 3, 1987, p [19] Vehaegen M., Yu X. A cass of subspace mode denfcaon agohms o denfy peodcay and abay me-vayng sysems. Auomaca, 31, 1995, p [0] Lu K. Idenfcaon of nea me-vayng sysems. Jouna of Sound and Vbaon, 04, 1997, p [1] Lu K., Deng L. Expemena vefcaon of an agohm fo denfcaon of nea me-vayng sysems. Jouna of Sound and Vbaon, 79, 004, p [] Yang J. N., Ln S. Idenfcaon of paamec vaaons of sucues based on eas squaes esmaon and adapve acng echnque. ASCE Jouna of Engneeng Mechancs, 1313, 005, p [3] Ghanem R., Romeo F. A wavee-based appoach fo he denfcaon of nea me-vayng dynamca sysems. Jouna of Sound and Vbaon, 344, 000, p [4] Amaaunga K., Wams J. R., Qn S., Wess J. Wavee-Gaen souons fo one dmensona paa dffeena equaons. Inenaona Jouna fo Numeca Mehods n Engneeng, 37, 1994, p [5] Sh Z. Y., Law S. S., Xu X. Idenfcaon of nea me-vayng mdof dynamca sysems fom foced excaon usng Hbe ansfom and EMD mehod. Jouna of Sound and Vbaon, 313-5, 009, p VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE 01. VOLUME 14, ISSUE. ISSN