STRUCTURAL SYSTEM IDENTIFICATION BASED ON SUBSYSTEM ANALYSES

Transcription

1 STRCTRA SYSTEM IDENTIFICATION BASED ON SBSYSTEM ANAYSES Sej YAMADA Ad Ara NISHITANI SMMARY I estatg the oto characterstcs of a buldg, the odal decoposto techque s frequetly eployed. However, t does't ft to the estato of a substructure. Ths paper focuses o the dapg estato of substructure. By eas of the trasfer atr ethod, a structure s dvded to a uber of substructures, whch the are to be detfed. A stadg wave arse sde a structure s decoposed to forward ad bacward ovg waves vrtue of the wave propagato theory. The, a spatal dapg factor s estated fro the forward ovg wave. Ths dapg factor s regardg the spatal forato, whch the odal aalyss caot deal wth. To deostrate the valdty the proposed ethod, uercal sulatos are coducted wth respect to a four-story buldg odels subjected to susodal ad whte ose ectatos duced by AMD. INTRODCTION Alog wth the rearable developet of sesg, easureet ad coputer techologes, syste detfcato of structures has recetly eteded ts role cvl egeerg feld. I addto to the usual case, syste detfcato s coducted for the case of gettg ore precse forato regardg the oto characterstcs of a cotrolled buldg [Nshta ad Yaada, 999] ad for the case of structural health otorg systes. I partcular, the recet creasg dead for health otorg techology wll epectedly accelerate ore practcal developet of syste detfcato. Despte that, certa dffcultes arse coductg syste detfcato practce of a large structure. They are, for stace, the crease of coputato effort ad te due to the hadlg of a large structure. Fro the above reaso, t would be advatageous to dvde a etre structure to a uber of substructures or subsystes ad the to coduct syste detfcato of each substructure. The dea of substructures s dealt wth the trasfer atr ethod [Fuuwa et al, 99; Pestel ad ece, 963; Yaaawa ad Ohsh, 98] cojucto wth the eployet of wave propagato theory [Doyle, 997; Flotow, 986; Fuuwa et al, 99; Fuuwa et al, 99; Mead, 986; Taaa ad Kusha 99; Touoa, 985]. Ths paper focuses o the dapg of each substructure a buldg stead of dealg wth a odal dapg rato. The dapg of each substructure s to be estated ters of spatal dapg factor, whch represets the trasto of the state fro oe substructure to other substructure. The spatal dapg factor s regardg the spatal forato, whch the odal aalyss caot deal wth. Such a dapg factor tells us how ay degree of dapg ay substructure has. Ths forato about the dapg of each substructure s used for structural health otorg systes. To deostrate how effectve ths techque s, ult-story buldg odel s eployed, whch s ected by AMD stalled o the top floor. Departet of Archtecture, Waseda versty, Toyo, Japa Eal: ara@st.arch.waseda.ac.jp Departet of Archtecture, Waseda versty, Toyo, Japa Eal: ara@st.arch.waseda.ac.jp

2 EVAATION OF SPATIA DAMPING FACTORS. Trasfer Matr Cosder a shear structure odel represetg a ult-story buldg show Fgure. Suppose the odel oscllates wth frequecy [rad/s]. The state vector Z for th story cossts of the dsplaceet relatve to the base ad the shear force. The substructure fro the upper pot of ( )th ass to the upper pot of th ass s defed as th story ad s utlzed as a substructure. Two sets of states wth respect to are dealt wth: oe s Z o the upper pot of the ass ad the other Z o the lower pot of the ass (Fgure ). ad represet the state copoets of Z ad the state ad N represet the state copoets of Z. The relatoshp betwee these four copoets are gve by the followg equato: () The coeffcet atr of Z the rght-had sde of the above equato s called the pot atr. The stffess ad dapg c of th story are show Fgure 3. Assug the Kelv odel that cossts of ad c the parallel posto, oe gets () The coeffcet atr of Z - the rght-had sde of the above equato s called the fled atr. Eqs. () ad () yeld the followg relatoshp betwee Z ad Z -. (3) whch, the coeffcet atr of Z - represets how Z s effected by Z - ad s referred to as the trasfer atr of th story fro the upper pot of to the upper pot of, deoted as. Eq. (3) s the trasfer atr for the case of o eteral ectato. Sce the structure s ected by AMD, oly the top story has a eteral put force. Whe ected by the oveet of AMD f appled to, Eq. (3) wll be f (4) If the trasfer atrces of the odel are ow, all the states ca be calculated the above forulatos wth boudary codtos. If the states wth respect to all the stores are obtaed by the actual easureet, all the story trasfer atrces ca be estated. These relatoshps (Eqs.(3) ad (4)) are utlzed to get the relatoshps betwee the waves fro story to story.

3 . Wave Propagato Cosder the structure subjected to certa susodal ectato due to AMD. The ectato wave propagates fro the top floor to the boudary of each story, ad the wave partally trasts ad partally reflects. The trasttg ad reflectg waves are called forward ad bacward ovg waves. There est both forward ad bacward ovg waves sde the structure the steady-state susodal oscllato. The wave vector W s defed here as certa d of state vector cosstg of the cople apltudes ad w of forward ad bacward ovg waves wth respect to th story. I cooperato wth the egevalues λ ad λ of, the trasto of the wave vector s gve by w ( w e w whch α jβ ) e ( α jβ ) w w (5) α Re(l λ ) ad β I(l λ ) (6) Eq. (5) llustrates how the wave propagates to the vertcal drecto. Ths paper cosders about a forward ovg wave the followg. I the Eq. (5), as w ad w s cople apltude, the epresso of the wave j t as a fucto of the te t s represeted by ultplyg e. The forward ovg wave s etracted. w e j t ( α jβ ) e w e jt (7) w j t The relatoshp betwee e ad be estated the et subsecto. w j t e yelds the eergy loss factor. The spatal dapg factor wll.3 Spatal Dapg Factors Estated Based o Eergy oss I ths study, a spatal dapg factor betwee stores s dealt wth. Such a spatal dapg s estated by coparg the eergy dsspated by a forward ovg wave as t progresses dowwards wth the aout of eergy dsspated by the daped free vbrato of a sgle-degree-of-freedo oscllatory syste. By eas of the equato represetg the daped free vbrato of the SDOF syste duced by a tal dsplaceet, the eergy loss factor ψ of the vbrato ca be wrtte as f 4πh ψ f ep (8) h whch h s the vscous dapg rato of ths syste, descrbg how the free vbrato s dshed as te passes. O the other had, fro the equato for the forward ovg wave, Eq. (7), the eergy loss factor ψ w of the forward ovg wave s represeted by α ψ w ep 4π (9) β Equalzg the two ds of eergy loss factors ψ f ad ψ w gve by Eqs. (8) ad (9), the spatal dapg factor h of the forward ovg wave leads to p α / β h p () ( α / β ) wth α ad β gve Eq. (6). 3

4 NMERICA EXAMPES 3. Models for Nuercal Eaples To deostrate the valdty of the proposed ethodology, uercal eaples are coducted for four-story buldg odels. The data of odels are tabulated Table. The specfc procedure s: () to easure the respose dsplaceet ; () to calculate ; () to detere wth ad fro Eq. (3) or (4); (v) to obta α ad β fro the egevalues of fro Eq. (6); ad (v) to evaluate the spatal dapg factors through Eq. (). Repeatg the above procedure at the varous frequeces, the spatal dapg factors as a fucto of the frequecy are estated. Table : Paraeter of buldg odels for uercal eaples Model Story A K K C C [g] All (-%) [N/] 9 8(-%) 8(-%) (5%) 4.3(%) c [N//s] (5%) 47.3(%) Susodal Ectato Iput Susodal ectato s appled to Model A the frst place. Susodal ectato puts of apltude [c] are gve by AMD o the top floor durg the frequecy rage [Hz] to [Hz] wth the creet of.4 [Hz]. The dsplaceet resposes of all the stores are easured as the outputs wth the saplg te t equal to. [s] ad. [s]. The estated spatal dapg factors ad actual values obtaed fro, c ad gve Table. are plotted Fgures 4 ad 5. It s recogzed that shorter saplg te s as eeded for hgher frequecy. Although shorter saplg te would be better, the estato s ot always satsfactorly accurate wth. [s]. Moreover, the accuracy of estato would be worse wth sall dapg. Therefore, short saplg te s eeded to prove the estato accuracy. However, short saplg te s ot realstc the applcato of the susodal ectato based ethodology to a actual structure. 3.3 Rado Iput Accoutg for the above dscusso, whte ose put s used stead of susodal ectato. However, the buldg respose to whte ose ectato volves a huge uber of frequeces. The drect eployet of the preseted ethod for ths case s ot possble. I estatg for every frequecy Procedure (), the put/output data are forulated by the lear regresso odel ad the coplace G ( j) s estated by the least squares ethod. X ( j) G ( j) () F( j) whch, X ( j) : dsplaceet vector ad F ( j) : ectato force due to AMD (scalar). If F ( j) s ow, the cople apltude vector of each story s calculated fro G ( j). Eq. () leads to the estato of for every frequecy. By coductg Procedures (), (), (v) ad (v), spatal dapg factors are estated. I coductg the aalyss, t s assued that the accelerato resposes of all the stores are easured as outputs. The saplg te s set to be. [s], the uber of easured data s 5 wth the degree of the lear regresso odel equal to. The estated spatal dapg factors ad ther actual values are show Fgure 6. Ths fgure dcates that the estated results agree wth the actual values. 4

5 I the followg stage, ths ethodology s appled to structural health otorg systes. Model A s supposed to be a good codto, four dfferet ds of buldg odels are spected by the ethodology: they are Models K, K, C ad C. The data are show Table. Models K ad K are those odels, whch have saller stffess tha Model A, whch, ay result fro certa daages. O the other had, Models C ad C are those odels wth larger dapg. The coparso of estated spatal dapg factors of Model K, K, C ad C wth those of Model A are show Fgures 7 through. The spatal dapg factors of those stores wth certa chages characterstcs are dfferet fro Model A. The ethodology successfully dcates how ad where the chages stffess ad dapg occur. Table : Natural frequeces ad dapg ratos of buldg odels Natural frequecy [Hz] Dapg rato Model Mode A K K C C The atural frequeces ad dapg ratos obtaed by use of the odal aalyss are show Table. Each odel has dfferet values of both atural frequeces ad dapg ratos fro Model A. Nevertheless, t s hard to decde how ad where the local characterstcs chages have occurred. CONCSIONS The spatal dapg factor of the subsystes of a structure has bee dscussed. The dea of subsystes s dealt wth the trasfer atr ethod cojucto wth the eployet of wave propagato theory. The preseted ethodology s also appled to structural health otorg ssue. It s deostrated that:. The dapg of each story of a structure ca be detfed by eas of preseted ethodology:. Whe appled to a structural health otorg ssue, the ethodology satsfactorly detfes where the structure ad how the stffess ad dapg chages fro ther orgal or desred values. ACKNOWEDGEMENTS Ths research s partally supported by Waseda versty Research 999 Grat for Specal Research Projects (99A-8) ad JSPS Research for the Future Progra (96R57). REFERENCES Doyle, J. F. (997). Wave Propagato Structures : Spectral Aalyss sg Fast Dscrete Fourer Trasfors, Secod Edto, Sprger-Verlag New Yor, Ic. Flotow, A. H. vo (986). Dsturbace Propagato Structural Networs. Joural of Soud ad Vbrato, 6(3), pp Fuuwa, N., Katuura, H., Naa, S. ad Igusa, T. (99). A Study o the Dyacs Characterstcs of the Perodc Structure sg Trasfer Matr Method A Basc Study o the Wave Propagato the Perodc Structure Coposed of Oe-desoal Cotuu Body ad uped Masses- ( Japaese). Joural of Structural ad Costructo Egeerg, AIJ, No.4, pp.-8. Fuuwa, N., Katuura, H. ad Naa, S. (99). A Study o the Wave Dsperso the Dscrete Aalyss Model ad a Proposal of Optal Cosstet Mass Rato ( Japaese). Joural of Structural ad Costructo Egeerg, AIJ, No.433, pp

6 Mead, D.J. (986). A New Method of Aalyzg Wave Propagato Perodc Structures; Applcatos to Perodc Tosheo Beas ad Stffeed Plates. Joural of Soud ad Vbrato, Vol.4, pp.9-7. Nshta, A. ad Yaada, S. (999). H cotrol Syste Re-desg Based o Structural Syste Idetfcato wth AMD Provdg Iput ( Japaese). Joural of Structural ad Costructo Egeerg, AIJ, No.56, pp Pestel, E. C. ad ece, F. A. (963). Matr Methods Elastoechacs. McGraw-Hll Boo Copay, Ic. Taaa, N. ad Kusha, Y. (99). Fleural Wave Cotrol of a Fleble Bea (Proposto of the Actve S Method) ( Japaese). Trasactos of the Japa Socety of Mechacal Egeers (Seres C), Vol.56, No.5, pp Touoa, T. (985). Theory of Wave Propagato ( Japaese). Scece Publshg. Yaaawa, H. ad Ohsh, T. (98). Dyacs Respose Aalyss wth May Degrees of Freedo sg Step-by-Step Trasfer Matr Method ( Japaese). Trasactos of the Japa Socety of Mechacal Egeers (Seres C), Vol.48, No.49, pp.67-68, - Fgure : Dagra of state through passg, c Fgure : Shear structure odel ult-story buldg of a Fgure 3: Relatoshp asses betwee two 6

7 st story (actual) st story (estated) d story (actual) d story (estated) 3rd story (actual) 3rd story (estated) 4th story (actual) 4th story (estated) st story (actual) st story (estated) d story (actual) d story (estated) 3rd story (actual) 3rd story (estated) 4th story (actual) 4th story (estated) Fgure 4: Spatal dapg factors fro susodal put wth saplg te.s for Model A Fgure 6: Spatal dapg factors fro rado put wth saplg te.s for Model A st story (actual) st story (estated) d story (actual) d story (estated) 3rd story (actual) 3rd story (estated) 4th story (actual) 4th story (estated) st story (Model A) st story (Model K) d story (Model A) d story (Model K) 3rd story (Model A) 3rd story (Model K) 4th story (Model A) 4th story (Model K) Fgure 5: Spatal dapg factors fro susodal put wth saplg te.s for Model A Fgure 7: Coparso of Spatal dapg factors betwee Model A ad Model K 7

8 st story (Model A) st story (Model K) d story (Model A) d story (Model K) 3rd story (Model A) 3rd story (Model K) 4th story (Model A) 4th story (Model K) st story (Model A) st story (Model K) d story (Model A) d story (Model K) 3rd story (Model A) 3rd story (Model K) 4th story (Model A) 4th story (Model K) Fgure 8: Coparso of Spatal dapg factors betwee Model A ad Model K Fgure : Coparso of Spatal dapg factors betwee Model A ad Model C Fgure 9: Coparso of Spatal dapg factors betwee Model A ad Model C 8