PARAMETER IDENTIFICATION-BASED DAMAGE DETECTION FOR LINEAR TIME-VARYING SYSTEMS

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Imogene Sanders
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1 RMETER IDENTIFICTION-BSED DMGE DETECTION FOR INER TIME-VRING SSTEMS Z.. S Cvl ad Sruural Egeerg Depare, Hog Kog ole Uvers (a prese) Hog Kog College of erospae Egeerg, Nag Uvers of eroaus ad sroaus eople s Republ of Ca S. S. aw Cvl ad Sruural Egeerg Depare, Hog Kog ole Uvers Hog Kog ezs@polu.edu. or zs@uaa.edu. bsra Ma daage deeo eods assue e sse s lear before ad afer daage ourree. Ts assupo usuall does o aord w fa ad s ause dfful o e suessful applao of a exsg daage deeo approaes. Ts paper preses a eque for paraeer defao based daage deeo for lear e-varg dsree daal sses. Frs, e paraeer defao of lear e-varg sses s developed based o e Hlber-Huag rasfor ad e epral ode deoposo eod va free vbrao daa. Te e approa s used daage deeo of lear e-varg dsree sse w deosrao w ueral exaples w sgle- ad ul-degree-of-freedo dsree sses. Iroduo Daages of vl egeerg sruures, ludg e sffess degradao of olus, os ad beas, e ollapse of braes, e., resul a sudde age of sffess ad/or dapg rao, ad ee e da araerss of e sruures. Ma daage defao eques ave bee proposed based o lear e-vara (TI) odel e las wo deades [-3]. However, e assupo of TI usuall ao desrbe e araerss of e daage ourree sruures. Te vbraoal sgals reorded b e sruural eal oorg sse are usuall o-saoar. Terefore, e Fourer rasfor, used os of e above-eoed daage deeo eods, s o able o apure e e-depeda araerss a ool lude e daage forao [4]. Due o e e-freque ul-resoluo proper, e wavele rasfor (WT) as bee deosraed as a prosg ool for proessg e o-saoar sgals ad a be used o deere e daage e saes ad loaos [5-6]. ewse, e Hlber rasfor (HT) also as e apabl of deoposg sgals e freque e doa o apure e e-loalzed forao a ever e sa. suessful defao approa for a sgle-degree-of-freedo (SDOF) as bee developed based o a lear evarg (TV) odel [7]. Te, e epral ode deoposo obed w e Hlber rasfor, alled as Hlber-Huag rasfor (HHT), s proposed [8]. Reel, e eod of Epral Mode Deoposo (EMD) ad Hlber-Huag rasfor as bee used suessful for daage defao of lear sruures based o TI odel [9]. I s paper, a paraeer defao-based daage deeo approa for lear e-varg daal sses usg e Hlber rasfor ad e epral ode deoposo eod s

2 proposed w free vbrao respose sgals. Seo develops e defao eque. Seo 3 preses SDOF ad MDOF ueral exaples o deosrae e effeveess ad aura of e proposed eod. Nueral sulao resuls sow e proposed approa a suessfull be used o dee ad quaf e daage e lear e-varg daal sses. araeer Idefao Teque Te Hlber rasfor s well ow o ave good apabl of aalss of a sgle opoe sgal e e-freque doa o apure e e-loalzed forao su as e saaeous freque a ever e sa [8]. d e Epral Mode Deoposo (EMD) eque s reogzed as a good ool o exra Irs Mode Fuos (IMF) fro a geeral sgal. Te proedure of EMD s o osru e upper ad lower evelopes of e sgal b ub sples ad e ea of upper ad lower evelopes s opued. Te, s ea s subraed fro e orgal sgal. Ts s referred o as e sfg proess. B repeag e sfg proess ul e resulg sgal s e IFM,.e., e followg wo odos are sasfed: ) e uber of exrea ad e uber of zero rossgs us be equal or dffer a os b oe over e ere durao of e sgal; a ) a a po, e ea value of e evelope defed b e loal axa ad e evelope defed b e loal a s zero. Te IMF s a sgle opoe sgal suable for aalss w Hlber rasfor. Te sfg proess s repeaed o exra aoer IMF ul e resdual sgal beoes so sall a s less a a pre-deered value or e resdual sgal beoes a ooo fuo [9]. Suppose Irs Mode Fuos are obaed afer e sfg proess. Terefore, e orgal sgal a be expressed as w r,,, K, r s e resdue, w s a ooo fuo fro w o ore IMF a be exraed. For a lear e-varg MDOF sse, e goverg free vbrao equao s gve b are IMFs of e orgal sgal, ad ( ) M & C( ) & ( ) K( ) ( ) 0 & w [ ( ), ( ),, ( ) T K ] s e dsplaee veor, ad M, C ad K are e-varg ass, dapg ad sffess ares respevel. For e lear sse, e dsplaee veor a be expressed as e suao of IMFs aordg o e addo rule. d e elee of e dsplaee veor a be deoed as [ ψ ] [ ] osψ (,, K, ) (3) w os ( ) s e IMF exraed usg EMD eod for e elee (w s orrespodg o e DOF) of e dsplaee veor. ( ) s e saaeous aplude, ψ s e saaeous pase agle. For a da respose sgal ( ), Te Hlber rasfor of ( ) gve b, deoed b ~, s

3 ~ [ ] HT π ( τ ) d τ τ were s e Cau prpal value. W s defo, ( ) ougae par, so e aalal sgal ( ) of ( ) s expressed as ad ~ (4) fors a oplex w exp[ ψ ( )] (5) ( ) ( ) ~ ( ), ( ) ara ( ) ~ ( ) [ ] ψ (6) Furerore, we also a oba e frs ad seod dervaves of e aalal sgal as w [ ] [ && ω ( & ω & ω )] ( ) ( ) & ( ) ( ) ω ( ) & (7) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) & ( ) ~ & ( ) & ( ) ~ ( ) & ω ψ& I (9a) & ( ) & ( ) ~ ( ) ~ & ( ) ( ) ω ( ) ( ) & Re && & & ω I (9) ( ) && & Re ω (9d) ordg o e Bedrosa s eore [0] o e Hlber rasfor of a produ of wo sgals, osder e ase of o-overlappg spera of e sgals f ( ) ad g ( ), f f s lowpass ad H f g f H g. Usg Hlber rasfor for bo sdes of equao g s gpass, e ( ) ( ) [ ] ( ) [ ( )]. Mulplg ea sde of e obaed ew equao b ad addg o e orrespodg sde of equao, we ge M ( ) & ( ) C( ) & ( ) K( ) ( ) 0 ( ) (8) (9b) & (0) Solve e free vbrao Equao (0) b subsug & ad & usg Equaos (7) ad (8), ogeer w Equaos (9a) o (9d). ssug e ass o be ow, e Equao (0) a be splfed ad wre opa arx oao for e IMF of e aal sgal. β β w β { } T, β { } T, ad

4 3 3 3 (3) { } T (4) e oeffes ad are deoed as ( ) ω & (5) ( ) ( ) ( ) ω ω ω & & & & (6) Te oplex Equao (0) a be separaed o wo groups of equaos aordg o s real ad agar pars, ad e e wo pars a be assebled e followg for. ( ) ( ) ( ) ( ) ( ) ( ) β β ag real ag ag real real (7) For a DOFs lear e-varg sse, equao (7) oas e-varg uow paraeers e-varg equaos for ea IMF of e orgal sgal. Terefore, ea IMF exraed fro e orgal sgal a be used equao (7) o solve oe se of defao resuls. 3 Exaple Sudes Two ueral exaples are suded s seo o deosrae e effeveess ad aura of e paraeer defao-based daage deeo approa of e lear e-varg sse. Te respose sgals used o def ad quaf daage are opued fro ueral soluos of e goverg dffere equao of e e-varg sses usg e Newo-Rapso eod []. Te saplg rae of e aalss s N T s.

5 3. Sgle-degree-of-freedo sses s a exaple, we refer o e sgle DOF ass-sprg-dapg daal sse. & ( ) ( ) & ( ) ( ) ( ) 0 w e ass oeffe s assued a osa ( ), e sffess oeffe ad e dapg oeffe are e-depede. Two daage ases are suded,.e. oe ase s a abrup 00π, 0. for <. 5, 60π, 0. 7 for age of sffess, w s gve b ( ) 7. oer ase s a sool varg age of ad ( ) 80π, ( ) 0. 7 for > 3. 5 sffess ad dapg, w are gve b ( ) 00 0π, ( ) π,. for 3 & 0 0. π for < 3,. Te al odos for e free vbrao ase are 0. 0 ad For daage ase, defao resuls are sow Fgures ad. Fgure s e efreque dsrbuo of e e-varg respose sgal of e abrup daage of sffess sse. Te e-freque dsrbuo,.e., saaeous freque vs e, s opued based o e Hlber rasfor. Fro Fgure, s easl ow a ere s a daage e sse, ad e e of ourree a also be ow a. 5s ad 3. 5s. Te daage pe ad sze a be defed usg e proposed eod. Resuls deosrae a ere are wo abrup ages of sffess as sow Fgure. Te defed resuls deoed w e dased le a e rue values deoed b e sold le. ewse, we a deere e e of daage ourree e sse fro e defed saaeous freque. Two daage pe ad defao resuls are sow Fgures 3 o 4 respevel. Fgure 3 sows e defed sffess vs e. d e defed dapg resuls are ploed w e Fgure Idefao of a -DOF buldg odel wo-sore sear-bea buldg odel s sow Fgure 5. Two daage ases are assued. ) brup sffess daage ours. Te sffess oeffes are gve b: 40053, 8755 for 3 ; 36048, 7004 for > 3. ) Sool varg daage ours e sffess oeffe : 8755 for < ; for 3 ; 7004 for > 3 ; a a e. Te dapg oeffes ad ass oeffes are assued as osas: 30, 0 ad 50. Te al odos for e free vbrao are , 0. 0 ad & 0 0, & 0 0. Fg DOF sear-bea buldg odel frs, e respose sgals of e -DOF e-varg sse based o e Newo-Rapso eod are deoposed o IMFs usg e EMD eod. Ea IMF s suable for aalss w e Hlber rasfor. Te e-freque dsrbuo of ea IMF a be used o deere weer ad we daage ours e sruure. Te, e sffess ad dapg oeffes are defed usg e paraeer defao-based eod of e lear e-varg sse.

6 Fgure 6 ad Fgure 7 sow e defao resuls for daage ase. Te abrup sffess daages ad are deeed ad ploed agas e Fgures 6 ad 7 respevel. ll of e wo defao sffess oeffes are ver lose o e orrespodg rue values. For e sool varg daage ase, e urve Fgure 7,.e., saaeous freque vs e, deosraes daage ours a aroud s beause e saaeous freque ages durg s perod. Te sffess oeffe s defed ad sow Fgure 8. Te defao resul deoed w blue dased le s ver ag up o e rue value deoed w red sold le roug e wole defed perod. 4 Coluso I s paper, e proess of e ourree of daage e sruure s assued o be a lear e-varg proess. Terefore, e defao of lear e-varg sses s developed a frs based o e Hlber rasfor ad e epral ode deoposo eod usg free vbrao respose sgals. Te, e applao of e defao eod daage deeo s suded. Nueral resuls sow e paraeer defao based daage deeo approa a be used o ol o def e e sae of daage ourree e SDOF or MDOF sruures bu also o loalze ad quaf e sruural daage. 5 owledgee Ts resear s suppored b e Naoal Naural See Foudao of Ca roug Gra No ad Hog Kog ole Uvers Gra No. G-X6. 6 Referees [] Doeblg, S. W., Farrar, C. R., ad re, M. B. suar revew of vbraobased daage defao eods, So ad Vbrao Dges, 998; 30: [] S, Z.., aw, S. S., ad Zag,. M. Sruural Daage Deeo fro Modal Sra Eerg Cage, Joural of Egeerg Meas SCE, 000; 6: 6-3. [3] S, Z.., aw, S. S., ad Zag,. M. Opu Sesor laee for Sruural Daage Deeo, Joural of Egeerg Meas SCE, 000; 6: [4] Gurle, K., ad Karee,. pplao of wavele rasfor arquae, wd ad oea egeerg, Egeerg ad Sruures, 999; : [5] Hou, Z., Noor, M., ad ad, R. S. Wavele based approa for sruural daage deeo, Joural of Egeerg Meas SCE, 000; 4(0): [6] Su, Z., ad Cag, C. C. Sasal wavele-based eod for sruural eal oorg, Joural of Egeerg Meas SCE, 004; 30(7): [7] Felda, M. No-lear sse vbrao aalss usg Hlber rasfor - I: free vbrao aalss eod FREEVIB, Meaal Sses ad Sgal roessg, 994; 8: 9 7.

7 [8] Huag, N. E., Se, Z., og, S. R. e al. Te epral ode deoposo ad Hlber speru for olear ad osaoar e seres aalss, roeedgs of e Roal Soe of odo - Seres, 998; 454: [9] ag, J. N., e,.,, S., ad Huag, N. Sse defao of lear sruures based o Hlber-Huag speral aalss. ar : Noral odes, Earquae Egeerg ad Sruural Das, 003; 3: [0] Sefa,. H. Hlber rasfor sgal proessg, re House I, 996; Caper - roperes of e Hlber rasfor dervaves ad applao: [] Ca S.. ad Cu.. T. No-lear sa ad l aalss of seel fraes w se-rgd oeos, Elsever See d, 000; Caper 3 Seod-order elas aalss b e Newo-Rapso eod:

8 Fgure - brup age of s.d.o.f. sse: saaeous freque vs e Fgure - brup age of s.d.o.f. sse: defed sffess oeffe Fgure 3 - Soo age of s.d.o.f. sse: defed sffess oeffe Fgure 4 - Soo age of s.d.o.f. sse: defed dapg oeffe Fgure 6 - brup age of -d.o.f. sse: defed sffess oeffe Fgure 7 - brup age of -d.o.f. sse: defed sffess oeffe Fgure 8 - brup age of -d.o.f. sse: saaeous freque vs e Fgure 9 - Soo age of -d.o.f. sse: defed sffess oeffe

Solution. The straightforward approach is surprisingly difficult because one has to be careful about the limits.

Solution. The straightforward approach is surprisingly difficult because one has to be careful about the limits. ose ad Varably Homewor # (8), aswers Q: Power spera of some smple oses A Posso ose A Posso ose () s a sequee of dela-fuo pulses, eah ourrg depedely, a some rae r (More formally, s a sum of pulses of wdh