Analysis and Preliminary Experimental Study on Central Difference Method for Real-time Substructure Testing

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1 Analyss and Prelmnary xpermenal Sudy on enral Dfference ehod for Real-me Subsrucure Tesng B. Wu Q. Wang H. Bao and J. Ou ABSTRAT enral dfference mehod (D) ha s explc for pseudo dynamc esng s also supposed o be explc for real-me subsrucure esng (RST). However o oban correc velocy dependen resorng force of he physcal subsrucure beng esed he arge velocy s requred o be calculaed as well as dsplacemen. The sandard D provdes only explc arge dsplacemen bu no explc arge velocy. Ths paper nvesgaes he necessary modfcaon of sandard cenral dfference mehod when appled o RST and analyzes he sably of he modfed D for RST (D-RST). The analyss shows ha he sably of he D-RST decreases wh ncreasng dampng rao of he physcal subsrucure. Then a prelmnary expermenal research s descrbed. The es shows ha he calculaed resul agrees well wh he esed one when he dampng rao of he specmen (.e. damper) s relavely low bu he dscrepancy beween he esed and calculaed responses ncreases wh he ncreasng dampng rao of he specmen. ey words: real-me subsrucure esng cenral dfference mehod sably. ITRODUTIO The pseudo-dynamc esng (PDT) s an expermenal echnque for smulang he earhquake response of srucures and srucural componens n he me doman. In hs es he srucural sysem s represened as a dscree sprng-mass sysem and s dynamc response o earhquakes s solved numercally usng drec negraon. Unlke convenonal drec negraon algorhms n he pseudo-dynamc es he resorng forces of he sysem are no modeled bu are drecly measured from a es conduced n parallel wh he drec negraon. In many srucures he unpredcable nonlnear behavor ha provdes he movaon for laboraory esng s que localzed. In hese crcumsances a far more economcal es can be performed usng he pseudo-dynamc esng wh subsrucurng approach or pseudo-dynamc subsrucure esng (PST). The algorhms and mplemenaons of PDT and PST are well documened e.g. ahn& Shng(985) Takanash & akashma (987). One of he crcal prerequses for conducng PDT s ha he effec of he loadng rae on he resorng force of he srucure should be of mnor consequence because he srucure s loaded quas-sacally n he PDT. Laely a varey of new ypes of srucural componens and devces have been nroduced n srucures parcularly n connecon wh her vbraon conrol (Soong and Spencer ). any of hem are very velocy dependen n vbraon characerscs such as B. Wu School of vl ngneerng Harbn Insue of Technology Harbn 59 People s Republc of hna Q. Wang School of vl ngneerng Harbn Insue of Technology Harbn 59 People s Republc of hna H. Bao School of vl ngneerng Harbn Insue of Technology Harbn 59 People s Republc of hna J. Ou School of vl ngneerng Harbn Insue of Technology Harbn 59 People s Republc of hna

2 vscous dampers or vscoelasc dampers. To es he velocy dependen componens ncorporaed n srucures real-me subsrucure esng (RST) was developed n 99s. The frs repored RST es (akashma e al. 99) was performed on a vscous damper locaed a he base of a mul-sory buldng. Only he damper was esed physcally wh he solaed buldng modeled numercally. A key elemen of he RST as well as PDT s he numercal algorhm ha s used o perform he sepwse negraon of he equaons of moon. any numercal algorhms have been used n RST such as cenral dfference mehod (akashma e al.99 akashma and asaoka 999 Darby e al. 999 Horuch e al. 999 Horuch and onno ) lnear acceleraon mehod (Horuch e al. ) backward uler mehod (Igarash ) Tusn s mehod (Blakeborough e al. ) and frs-order-hold dscrezaon mehod (Darby e al. ). enral dfference mehod (D) whch s explc for PDT s also beleved explc for RST (Wllams and Blakeborough ). However o oban correc velocy dependen resorng force of he physcal subsrucure beng esed he calculaed arge velocy s requred as well as dsplacemen. The sandard D provdes only explc arge dsplacemen bu no explc arge velocy. Ths paper wll nvesgae he requred modfcaon of sandard cenral dfference mehod when appled o RST and analyze he sably of he modfed cenral dfference mehod of RST (D-RST) and hen wll descrbe he prelmnary expermenal sudy on a srucure wh a vscous damper.. TRAL DIFFR THOD FOR RST (D-RST) For RST he equaons of moon may be wren n marx form as R ( ) R ( ) F () where s he mass marx of he numercal subsrucure R resorng force vecor of he numercal subsrucure R resorng force vecor of he physcal subsrucure (es specmen) he vecor of nodal dsplacemens F he vecor of exernal excaon forces and dos represen dfferenaon wh respec o me. In many subsrucure ess he mass of he specmens can be gnored and he properes of he specmens are no relaed o acceleraon so ha he resorng forces ake he form of R ( ). Then equaon () becomes R ( ) R ( ) F () We assume ha he numercal subsrucure s wh lnear dampng force and dsplacemen dependen resorng force.e. R ( ) R ( ) (3) where s he dampng coeffcen of numercal subsrucure. Subsung equaon (3) no () we ge R ( ) R ( ) F (4) Usng he D he velocy and acceleraon n sep are approxmaed by (5) (6) where s me nerval. Subsung equaons (5) and (6) no equaon (4) a h sep we oban

3 ( ) R R F ) ( (7) From he above equaon we see ha he calculaon of nvolves and. Wh known (gven and s calculaed usng equaon (4)) he relaons n equaons (5) and (6) can be used o oban as (Bahe. 996) (8) In convenonal PDT he calculaed arge dsplacemen s mposed upon he specmen and hen he rae ndependen resorng force can be measured. For RST he velocy of he nex sep (.e. sep ) mus also be calculaed and mposed on he specmen o oban correcly he resorng force dependen on velocy. However wh curren D represened by equaons (4)-(6) he velocy of sep canno be calculaed explcly. To he wrers knowledge he ssue of he velocy calculaon mehod n RST and parcularly s consequence on he sably and accuracy have no been dscussed heorecally by oher researchers. To acheve suffcen accuracy n boh dsplacemen and velocy conrol a dgal servo-mechansm was used n he RST by akashma e al. (99). The mechansm nerpolaes he arge dsplacemen sgnal lnearly no a se of dsplacemen sgnals: wh δ and. In oher words he followng addonal assumpon for he arge velocy was mpled n akashma e al. (99) s es: (9) Wh he above equaon D becomes explc for velocy as well as for dsplacemen. 3.STABILITY AD AURAY AALYSIS OF D-RST To analyze he sably and accuracy we consder sngle-degree-of-freedom (SDOF) sysem wh lnear numercal and physcal subsrucures.e. R ) ( () R ) ( () n whch s sffness of he numercal subsrucure and are dampng coeffcen and sffness of physcal subsrucure respecvely. Subsung equaons () and () no (7) and hen (7) no (9) we oban F () F (3) The sably and accuracy can be evaluaed wh he free vbraon soluon succncly wren n he recursve form (Shng and ahn 985) AY Y (4) where

4 [ ] Y (5) and he amplfcaon marx of D-RST A s expressed as A (6) n whch ω ( )/ /( ω) /( ω). Sably and accuracy of an algorhm depend upon he egenvalues of amplfcaon marx. 3. Sably The sably condon of an negraon mehod s (Shng and ahn 985) ρ ( A) (7) where ρ(a) s specral radus of A whch s defned as max λ and λ s egenvalues of A. For he marx A n equaon (6) we have λ A ± B and λ 3 (8) where when or where when A A B ( ) ( ) ( ) 4 8 (9) < () λ A ± B and λ () B ( ) (3) From nequales or equaons (7)-(3) he sably crera for D-RST can be obaned as < > 4 4 Unsable (4a ) Sable (wo real and one zero egenvalues) (4b) Sable (wo complex conugae and one zero egenvalues) (4c) From nequaly (4a) we see ha unlke D for PDT (D-PDT)(akashma 985) he upper lm of for a sable D-RST s no consan and ha he sably lm of decreases wh ncreasng dampng rao of he physcal subsrucure whch means he sably of D-RST ()

5 deeroraes wh hgher dampng rao of he specmen. When he dampng rao of he physcal subsrucure s zero he sably lm becomes ha s he same as he resul of he D-PDT. 3. Accuracy The deals he accuracy analyss of he algorhm s referred o Wu el al.(4). Only he man resul are summarzed here as follows: ()numercal dampng rao s posve and ncreases boh wh ncreasng ω and wh ncreasng dampng rao of physcal subsrucure for mos ω n he sable range; mnor negave numercal dampng rao occurs when ω s near sable lm for he cases wh relavely low dampng rao of physcal subsrucure; ()perod dsoron ncreases wh ncreasng ω and dampng rao of physcal subsrucure excep for very hgh dampng rao of physcal subsrucure; and (3)he nal velocy s wsed and he amoun of wsng ncreases wh ncreasng ω and dampng rao of physcal subsrucure. physcal subsrucure 4. PRLIIARY PRITAL STUDY 4.Tes Seup A Fgure Schemac of he whole srucure The ess were carred ou a echancal and Srucural Tesng ener Harbn Insue of Technology (HIT). The whole srucure s a sngle sory frame srucure ncorporaed wh a vscous damper of whch he bare frame whou he damper was he numercal subsrucure and he damper was he physcal subsrucure. The schemac of he srucure s shown n Fgure. The Schenck servohydraulc acuaor was managed and conrolled by TS sofware sysem. The es seup s shown n Fgure. 4. Tes Program Fgure Tesng Seup for he physcal subsrucure A seres of prelmnary ess usng RST echnque have been done a HIT. The parameers of wo cases are lsed n Table. The excaon s l enro (S 94) earhquake wave. In hs prelmnary expermenal research we ddn' dvde he arge dsplacemen no several pars and send hem successvely n order o acheve he arge velocy as

6 akashma e al. (99). The hyseress behavor of he damper was esed prevously by Long(4). The vscous dampng facor of he damper was 53ks/m and a lnear model very well agreed wh he es resuls (Long 4). The dampng raos of he damper n Table are calculaed by usng hs esed vscous dampng facor. Table. Tes ase ase ( 3 kg) (k/m) (s) Peak acc. of excaon (m/s ) 48 5% %.s % Tes Resuls The es resuls of case wh dampng rao % of he physcal subsrucure s shown n Fgure 3. The dsplacemen command calculaed based on esed damper force s desgnaed "esed" and he dsplacemen calculaed usng he prevously esed dampng rao.e. 53ks/m s desgnaed "calculaed". From Fgure 3 we see ha he calculaed dsplacemen maches very well wh he esed one; he calculaed damper force also agrees well wh he esed one excep a around 8 s and 9.5s when here are some measurng noses. The es resuls of case wh dampng rao % of he physcal subsrucure s shown n Fgure 4. The dscrepances beween calculaon and es ncrease due o he ncrease of he dampng rao of he physcal subsrucure. The arge velocy of he damper was no guaraneed because only arge dsplacemen was mposed on he specmen. Then an error beween he acual velocy and calculaed velocy was nevable. The larger he dampng rao of he specmen he larger he dsagreemen beween he calculaed and esed responses would be. 5 Tesed alculaed Dsp.(mm) Tme(s) (a) Dsplacemen

7 5 Tesed alculaed Force(k) Tme(s) (b) Damper force Fgure 3. Tes resuls of ase ( 5 %.68) Dsp.(mm) Tme(s) (a) Dsplacemen esed alculaed

8 Force() Tme(s) Tesed alculaed (b) Damper force Fgure 4. Tes resuls of ase ( %.68) 5. OLUSIOS To manan he explc form of D boh for velocy and for dsplacemen some modfcaon on he algorhm s requred when s appled o RST. The sably of a modfed D for RST wh addonal assumpon abou arge velocy s nvesgaed. The analyss resul shows ha he sably of D-RST deeroraes as he dampng rao of physcal subsrucure ncreases. A prelmnary expermenal research s carred ou on a sngle sory srucure ncorporang a vscous damper. The es resul shows ha he calculaed resul agrees well wh he esed one when he dampng rao of he specmen (.e. damper) s relavely low bu he dscrepancy beween he esed and calculaed responses ncreases wh ncreasng dampng rao of he specmen. AOWLDGTS Ths work was suppored by Gran 5338 from he aonal Scence Foundaon of hna and Gran AA65 from nsry of Scence and Technology of hna. The help of Professor S. Tan and r H. Zhang are graefully acknowledged. RFR Bahe.J. 996 Fne lemen Procedures Prence Hall: nglewood lffs ew Jersey 77. Blakeborough A. Wllams.S. Darby A.P. & Wllams D.. The developmen of real-me subsrucure esng Phl. Trans. R. Soc. Lond. A 359:

9 hopra A Dynamcs of Srucures Prence Hall: nglewood lffs ew Jersey 45. Darby A. P. Blakeborough A. & Wllams.S. 999 Real-me subsrucure ess usng hydraulc acuaor J. ngng ech 5: Darby A.P. Blakeborough A. & Wllams.S. Improved conrol algorhm for real-me subsrucure esng arhquake ngng Sruc. Dy 3: Horuch T. Inoue. onno T. & ama Y. 999 Real-me hybrd expermenal sysem wh acuaor delay compensaon and s applcaon o a ppng sysem wh energy absorber arhquake ngng Sruc. Dynam 8:-4. Horuch T. & onno T. A new mehod for compensang acuaor delay n real-me hybrd expermens Phl. Trans. R. Soc. Lond. A 359: Horuch T. Inoue. & onno T. Developmen of a real-me hybrd expermenal sysem usng a shakng able Proc. h World onf. arhquake ngneerng Auckland ew Zealand paper no Igarash A. Iemura H. & Tanaka H. Developmen of Subsrucure Hybrd Shake Table Tes ehod and Applcaon o Verfcaon Tess of Vbraon onrol Devces hna-japan Workshop on Vbraon onrol and Healh onorng of Srucures and Thrd hnese Symposum on Srucural Vbraon onrol Shangha hna. Long. 4 Tesng analyss and desgn of srucures wh velocy-dependen dampers (n hnese) Docoral Dsseraon Harbn Insue of Technology. ahn S.A. & Shng P.B. 985 Pseudodynamc mehod for sesmc esng J. Sruc. ngng: akashma ao H. & Takaoka. 99 Developmen of real-me pseudo dynamc esng arhquake ngng Sruc. Dynam :79-9. akashma. & asaoka. 999 Real-me on-lne es for DOF sysems. arhquake ngng Sruc. Dynam 8: akashma. 985 Par : relaonshp beween negraon me nerval and response sably n pseudo dynamc esng Journal of Srucural and onsrucon ngneerng 353:9-34 Soong T.T. Spencer Jr. B.F. Supplemenal energy dsspaon: sae-of-he-ar and sae-of-he-pracce ngneerng Srucures 4: Shng P.B. & ahn S.A. 985 ompuaonal aspecs of a sesmc performance es mehod usng onlne compuer conrol arhquake ng. Sruc. Dyn 3: Takanash & akashma 987 Japanese acves on onlne esng J. ngng ech 3: 4-3. Wllams.S. and Blakeborough A. Laboraory esng of srucures under dynamc loads: an nroducory revew Phl. Trans. R. Soc. Lond. A 359: Wu B. Bao H. Ou J. and Tan S. 4 Sably and Accuracy Analyss of enral Dfference ehod for Real-me Subsrucure Tesng submed o arhquake ngneerng and Srucural Dynamcs.