ALONG-WIND AERO-ELASTICITY OF PRISMS WITH DIFFERENT HEIGHT/WIDTH RATIOS BY INDIRECT FORCED ACTUATION TECHNIQUE

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1 The Seventh si-pifi Conferene on Wind Engineering, Noveber 8-, 9, Tipei, Tiwn LONG-WIND ERO-ELSTICITY OF PRISMS WITH DIFFERENT HEIGHT/WIDTH RTIOS BY INDIRECT FORCED CTTION TECHNIE ong-cheng Wu nd Ying-Chieh Chng Professor, Deprtent of Civil Engineering, Tng niversity, Tsui, Tipei County 5, Tiwn, Forer Grdute Student, Deprtent of Civil Engineering, Tng niversity, Tsui, Tipei County 5, Tiwn BSTRCT This pper investigted nd opred the frequeny-dependent erodyni dping nd stiffness of priss in the long-wind otion through wind tunnel tests. new identifition shee bsed on the indiret fored tution tehnique ws developed, whih only involves siple tehnique of urve-fitting on the frequeny response funtion indued by the tution. To ensure globl iniiztion in urve-fitting, the eployent of the geneti lgorith nd well s onventionl grdient serh were prtied in obtining the finl results. n lterntive derivtion of the frequeny response funtion ws lso derived vi the tie-doin stte spe eqution whih n be used to siulte the tie history of the response. To deonstrte the pproh presented, the squre ross-setion priss with height/width rtios of 4, 7 nd denoted s HB4, HB7 nd HB, whih re to odel three different high-rise buildings, were used for experientl identifition. The identified results show tht their erodyni dpings re lwys negtive nd onotonilly derese with the redued veloity inresing, exept for segent of the HB se t the redued veloity beyond 6. nder the se redued veloity, the bsolute vlues of erodyni dping follow the trend of HB>HB7>HB4. For the erodyni stiffness, s the redued veloity inreses, the erodyni stiffness for HB4 is onotonilly inresing fro zero while tht for HB is onotonilly deresing. However, the erodyni stiffness for HB7 is nerly insignifint on the overll stiffness. KEYWORDS: INDIRCET FORCED CTTION, PRISM, HIGH-RISE BILDING, ERODYNMIC DMPING, ERODYNMIC STIFFNESS, STTE ETION, GENETIC LGORITHM Introdution In the pst dede, prtiulrly in the si re where ny high-rise buildings were to be onstruted in the developed ities for liited lnd, the wind-indued effet on suh type of strutures hs beoe n inevitbly iportnt engineering issue. The newly opleted Tipei building 58 in Tipei is one of the typil exples. For the buildings s suh, the exessive response is very liely to our during wind disturbne. Hene, the iplied wind flow y utully interts with the building response nd beoe no longer n independent externl lod. This intertion between the struturl response nd wind lod is generlly lled ero-elstiity. The effet of ero-elstiity for high-rise buildings ight be n unpredited but deisive ftor tht should be ounted for in the struturl design. ero-elstiity on buildings hs been n ttrtive topi for reserhers in wind engineering. Mny erlier reserhes foused on the observtion of this effet on twodiensionl osillting odel by esuring the vrition of its surfe pressure using pressure tubes over the surfe. Fro the oprisons of the drg nd lift fore oeffiients

2 The Seventh si-pifi Conferene on Wind Engineering, Noveber 8-, 9, Tipei, Tiwn thus lulted to those fro stti odel, signifint differenes were observed nd onfired e.g., Nur nd Mizot 975, Bern nd Obsju 98. Generlly speing, these reserh results hd onluded tht, without onsidering the effet of eroelstiity, the long-wind otion of high-rise buildings ppers to be onservtive, while the ross-wind otion is, on the ontrry, under-estited. In the lst dede, few ppers hve investigted building ero-elstiity by using three-diensionl osillting odels, prtiulrly for the ross-wind otion, nd the siilr onlusion ws found e.g., Soto nd Oiwe 984, Viery nd Steley 993, nd et.. For prediting the threediensionl building response, soe ppers foused on librting the erodyni dping fro nuerous response esureents Cheng et l. nd et.. In these onventionl pprohes, the flow pttern whih is ffeted by the intertion ws the jor onerns. Therefore, the esureent of wind pressure on the odel surfes ws required This pper foused on investigting the globl effet of ero-elstiity existing in the long-wind otion of high-rise buildings by introduing the ide of frequeny-dependent erodyni dping nd stiffness. nlie the onventionl pproh, the building responses re to be esured insted of the wind pressure by pplying indiret fored tution to the building, nd thus new identifition shee for erodyni dping nd stiffness ws developed. This pproh only involves siple tehnique of urve-fitting on the frequeny response funtion. n lterntive derivtion of the frequeny response funtion ws lso derived vi the tie-doin stte spe eqution for tht the stte eqution n be used to siulte the tie history of the response. To deonstrte the presented shee, three squre-shpe priss with the height/width rtios of 4, 7 nd to odel three different types of high-rise buildings were onstruted in the wind tunnel of Deprtent of Civil Engineering, Tng niversity, Tiwn for experientl identifition. The erodyni dpings nd stiffnesses were suessfully identified nd their differenes were opred. Forultion Eqution of Motion of Priss Subjeted to Indiret Fored tution nd Wind Lod Consider sheti digr of the experientl setup shown in Fig.. The pris odel is bse-pivoted rigid odel with onneting rod rigidly jointed t the botto. It is pled on the wind tunnel floor below whih shing devie tht n generte desirble exittion is lined to the rod through spring nd dshpot. In this wy, the pris n be displed by pivoted otion tht represents single-degree-of-freedo building with the swy response distributed in liner ode shpe. Suh response eultes the first ode response of building, whih is prtilly enough for nlysis in view of engineering purposes. s shown in Fig., when the pris odel is siultneously disturbed by the wind flow nd horizontl indiret fored tution fro the shing devie, the eqution of otion n be expressed s h d d x d d x z Wind x d x H Ground Level Pivoting xis Shing Devie Gliding Ril Fig.: Sheti Digr of Experientl Setup for long-wind ero-elstiity Identifition of Priss f z,t dz M t M b t

3 The Seventh si-pifi Conferene on Wind Engineering, Noveber 8-, 9, Tipei, Tiwn in whih is ss oent of inerti with respet to the pivoting xis; is rottionl ngle of the pris; nd re spring stiffness nd internl dping oeffiients, respetively; d is length of the onneting rod; h is building height; f z, t is distributed wind lod per unit length long the tributry height z; M t is otion-indued oent; M b t is externl oent generted fro buffeting wind gust, whih is onsidered to be trivil if no wind turbulene is present.; x is bsolute displeent of the shing devie. For onveniene of presenttion, the following nottions were substituted into the eqution of otion, Eq. : the exittion displeent = d, the syste stiffness d, the syste dping d ξ / x, the syste nturl frequeny ω ω nd the syste dping rtio /. The resulting eqution is ξ ω ω M t M t Identifition of Struturl Preters in Priss In bsene of the wind disturbne i.e., M t = M b t=, the building response is entirely indued by the indiret fored tution fro the shing devie. By ting the Fourier trnsfor on both sides of Eq., the frequeny response funtion of indued by, H s, n be expressed s s / / H 3 ξω ω in whih is the exittion frequeny in rd/se. By urve-fitting this theoretil expression to experientl dt in oplex doin through iniizing the weighted su of squre errors between eh other, the oeffiients ξ ω nd ω re deterined fro the denointors oeffiients nd so re nd Dennis et l. 983, Wu. In ddition, the stiffness oeffiient n be obtined by perforing the slope librtion on the fore-displeent reltion. Then, the ss oent of inerti n be oputed through the reltions. With the vlues of given, the vlues of nd / n be lso obtined fro the nuertor s oeffiients in Eq. 3. ero-elsti Frequeny Response Funtion in Priss By onsidering the se pris odel Fig. subjeted to sooth wind flow nd indiret fored tution, the eqution of otion n be redued by setting M b t = nd result in ξ ω ω M t 4 Siilr to the onept of flutter derivtives in the nlysis of bridge response, the erodyni oent with respet to the building bse indued by the pris rottion n be ssued to be of the for expressed by D t M t D H K B K K B K t 5 in whih is ir density; is en wind veloity; D is hrteristi tributry width; K Dω / B is non-diensionl frequeny; nd re two non-diensionl funtions of B B K. Physilly, nd n be interpreted s the frequeny-dependent erodyni dping nd stiffness, respetively. By ting Fourier trnsfor on Eq. 5, the erodyni B b

4 The Seventh si-pifi Conferene on Wind Engineering, Noveber 8-, 9, Tipei, Tiwn oent n be written s M ik H M ik M indued by, H M ik, n be expressed s H M ik D H i K B K B By physil observtion, it is oneivble to ssue tht H ik, in whih the frequeny response funtion of 6 M n be further relized by n equivlent liner syste tht hs frequeny response funtion expressed by n n b ik b ik b ik b n n H M ik D H 7 ik ik ik in whih i, b i re onstnt oeffiients to be identified s the preters for deterining the frequeny-dependent erodyni dping nd stiffness nd. It n be lterntively written s funtion of i ω s H M B B n n bn bn b b by onverting the oeffiients following the reltions i D D i i i,,, ; bj D H bj j,,, n 9 To be physilly orret, the order of the nuertor is oneivbly ten s lrger thn tht of the denointor by one i.e., n=+, nd ll the roots of the denointor polynoil should hve negtive rel prts in order to gurntee stble dynis, i.e., Rel{ i } Beuse the order of the nuertor is lrger thn tht of the denointor y one, the rtio of two polynoils in 8 n be further rewritten in ters of the quotient nd residue s H M j Bsed on our experiene, hoosing the orders s low s n=3 nd = is firly stisftory. s suh, the oeffiients,,, nd, in Eq. re relted to, nd b, b, b, b 3 s D H D/ b3 D H b b 3 D H D/ b b3 b b3 8 ; D H D/ b b D/ ; D/ b 3 By trnsforing the eqution of otion, Eq. 4, to frequeny doin nd plugging in the expression of H ik in Eq. using the oeffiients denoted in Eq., the ero-elsti M frequeny response funtion of indued by, H 3 [ / H, n be rrnged into finl for of ] One gin, to gurntee stble dynis, ll the roots of the denointor polynoil in Eq. 3 should hve negtive rel prts, i.e., 3

5 The Seventh si-pifi Conferene on Wind Engineering, Noveber 8-, 9, Tipei, Tiwn } Rel{ 3 4 i + 4 lterntive Derivtion of ero-elsti Frequeny Response Funtion vi Stte Spe Eqution ording to the liner syste theore [Chen 984], stte spe eqution in the ontrollble nonil for n be eployed to represent the reliztion of the equivlent dyni syste of Eq. in the tie doin, i. e., H M B 5 M C 6 in whih ; ; ; B ] [ C 7 In Eq. 5, is the stte vetor nd is the syste trix. To ensure stble dynis, ll the eigenvlues of the syste trix should preserve negtive rel prts, i.e., } det { Rel I 8, whih n be shown identil to the ondition in Eq. [Chen 984]. By sting the otion-indued oent Mt expressed by Eqs. 5 nd 6 to the eqution 4 of otion, the overll stte spe eqution tht inorportes ero-elstiity n be fored nd expressed s B q q 9 C q in whih q ; ; ; B C B C Consequently, the ero-elsti frequeny response funtion of indued by n be given s B I C H If the order of n nd re ten s n=3 nd = for sipliity, the expression of Eq. n be shown to be extly equl to Eq. 3. To ensure its stble dynis, ll the eigenvlues of the syste trix should lso hve negtive rel prts, i.e., } det { Rel I 3, whih n be shown identil to the ondition of Eq. 4 [Chen 984]. lthough the ero-elsti frequeny response funtion expressed in Eqs. 3 nd re theoretilly identil, the dvntge of introduing the stte spe equtions s H

6 The Seventh si-pifi Conferene on Wind Engineering, Noveber 8-, 9, Tipei, Tiwn interedite step for derivtion is stted s follows. s long s the oeffiients i s nd b i s n be obtined following the iniiztion proedures in the next setion, the stte spe eqution 9 n be diretly used to filitte the tie doin nlysis of the ero-elsti responses under buffeting oent disturbne by siply repling by the buffeting M b oent. This nlogy n be esily observed when the two ters involving in Eq., i.e.,, is repled by the buffeting oent M b. Identifition of ero-elstiity in Priss In order to identify the frequeny dependent ero-dyni dping B nd stiffness, experientl dt of the frequeny response funtion of indued by should be obtined under sooth wind flow of vrious en veloities while indiret fored tution is pplied. The oeffiients i s nd b i s n be deterined by iniizing perforne index tht represents the weighted squre error between the experientl dt nd the vlue oputed by Eq. 3 or, i.e., PI n N w H f / B, 4, whih is onstrined by the two stble onditions, i.e., Eq. or Eq. 8 nd Eq. 4 or Eq. 3. In Eq. 4, H nd represent the theoretil nd experientl f /, frequeny response funtions under en wind veloity t the -th frequeny; w is the orresponding weighting oeffiients; nd N is the totl points onsidered in the iniiztion. One i s nd b i s re deterined, by equting the H M in Eqs. 6 nd 7, it is obvious tht n n [ bn ik bn ik b ] i B B 5 [ ik ik ] K Thus, B nd B re the orresponding iginry nd rel prts, respetively. To ensure tht globl iniiztion in the nueril serhing proess be hieved, the geneti lgorith G Mn, Tng nd Kwong 999 nd onventionl grdient ethod Dennis nd Shnbel 983 re used in ollbortion. In priniple, the G ethod eultes three bsi fetures in the geneti evolution proess, i.e., seletion, rossover nd uttion, to reh the finl result. However, due to its evolving nture, the finl result in every single serh ould differ, even though the genertion nd popultion nubers re ten lrge enough. s suh, gret del of G serhes should be firstly perfored to pproxitely lote the solutions in the globl sense, nd then the best solution ong the will be pied s the initil guess for the onventionl grdient ethod to finely tune to the preise optil solution. Experientl Results For bsi oprisons, the squre ross-setion priss with height/width rtios of 4, 7 nd, whih were denoted s HB4, HB7 nd HB, respetively, were used to odel three sled building odels in the experient. The struturl preters of ω, ξ nd identified for eh pris re 6.5 Hz,.76,.33 g-, 3.5 Hz,.75,.793 g, nd. Hz,.39,.773 g-, respetively. To identify the ero-elstiity, bnd-liited white-noise of indiret fored tution fro the shing devie see Fig. ws used to exite the pris in the wind tunnel see Fig. 3 while the sooth wind flow ws siultneously ting on it. The experientl results of under the wind flow t seven H

The Seventh si-pifi Conferene on Wind Engineering, Noveber 8-, 9, Tipei, Tiwn different en wind veloities were plotted in Fig. 4,, e. s observed fro Fig.

Following the G iniiztion in the urve-fitting, eh experientl urve of ws urve-fitted with n=3 nd = nd the results were shown in Fig. 4 b, d, f.

H B B Rod Spring Shing Devie Conlusions This pper presented new pproh to identify the frequeny-dependent erodyni dping nd stiffness of priss in the long-wind otion.

To ensure globl iniiztion in urve-fitting, the eployent of the geneti lgorith nd well s onventionl grdient serh were prtied in obtining the finl results.

7 The Seventh si-pifi Conferene on Wind Engineering, Noveber 8-, 9, Tipei, Tiwn different en wind veloities were plotted in Fig. 4,, e. s observed fro Fig. 4, it ws found tht the long-wind flow suppresses the vibrtion nd the suppression effet beoes stronger s the wind veloity inreses. Following the G iniiztion in the urve-fitting, eh experientl urve of ws urve-fitted with n=3 nd = nd the results were shown in Fig. 4 b, d, f. Consequently, the vlues of nd versus the non-diensionl wind veloity / K were obtined nd plotted in Fig. 5. H B B Rod Spring Shing Devie Conlusions This pper presented new pproh to identify the frequeny-dependent erodyni dping nd stiffness of priss in the long-wind otion. By utilizing indiret fored tution tehnique, this pproh only involves siple tehnique of urve-fitting on the frequeny response funtion. To ensure globl iniiztion in urve-fitting, the eployent of the geneti lgorith nd well s onventionl grdient serh were prtied in obtining the finl results. n lterntive derivtion of the frequeny response funtion ws lso derived vi the stte spe eqution whih filittes the tie history siultion of the response. Three squre ross-setion priss with height/width rtios of 4, 7 nd to odel three high-rise buildings were identified for oprison. The results showed tht the wind flow suppresses the long-wind vibrtion nd the effet beoes stronger s the wind veloity inreses. The erodyni dping B is lwys negtive nd onotonilly deresing with inresing exept for segent of HB t > 6. nder the se, the bsolute vlues of erodyni dping follow the trend of HB>HB7>HB4. For the erodyni stiffness, s inreses, the vlue of B for HB4 is onotonilly inresing fro zero while tht for HB is onotonilly deresing. The vlue of B for HB7 is onotonilly deresing fro zero, however, its vlue is nerly insignifint on the overll stiffness. Referenes Fig. : Horizontl Shing Devie Fig. 3: Squre-shpe Pris in the Wind Tunnel Bern, P. W. nd Obsju, E. D. 98, n Experientl Study of Pressure Flututions on Fixed nd Osillting Squre-setion Cylinders, ournl of Fluid Mehnis, 9, Chen, C. T. 984, Liner Syste Theory nd Design, Holt, Rinehrt nd Winston, In. Cheng, C.M., Lu, P.C. nd Tsi, M.S., rosswind erodyni Dping of Isolted Squre Shped Buildings, ournl of Wind Engineering nd Industril erodynis, 9, Dennis,.E., r., nd R.B. Shnbel 983, Nueril Methods for nonstrined Optiiztion nd Nonliner Equtions, Englewood Cliffs, N: Prentie-Hll.

8 The Seventh si-pifi Conferene on Wind Engineering, Noveber 8-, 9, Tipei, Tiwn Mn, Tng nd Kwong 999, Geneti lgorith, Springer. Nur, Y. nd Mizot, T. 975, nstedy Lifts nd Wes of Osillting Retngulr Priss, SCE ournl of the Engineering Mehnis Division, EM, Soto, H. nd Oiwe, S. 984, Flututing Fores on Retngulr Pris nd Cirulr Cylinder Pled Vertilly in Turbulene Boundry Lyer, Trnstions of the SME, 6, Viery, B.. nd Steley,. 993, erodyni Dping nd Vortex Exittion on n Osillting Pris in Turbulent Sher Flow, ournl of Wind Engineering nd Industril erodynis, 49, pp. -4. Wu,.C., Modeling of n tively Bred Full-Sle Building Considering Control-Struture Intertion, Erthque Engineering nd Struturl Dynis, 9, 9, plitude = 5 /s = 6 /s = 7 /s = 8 /s = 9 /s = /s = /s inreses plitude = 5 /s = 6 /s = 7 /s = 8 /s = 9 /s = /s = /s inreses plitude HB4 = 5 /s = 6 /s Frequeny Hz = 7 /s inreses = 8 /s = 9 /s = /s = /s HB7 plitude b HB4 = 5 /s = 6 /s Frequeny Hz = 7 /s inreses = 8 /s = 9 /s = /s = /s d HB7 plitude e HB = 5 /s f HB Frequeny Hz = 6 /s = 7 /s inreses = 8 /s = 9 /s = /s = /s plitude = 5 /s Frequeny Hz = 6 /s = 7 /s = 8 /s inreses = 9 /s = /s = /s Frequeny Hz Frequeny Hz Fig 4: Frequeny Response Funtions of the Three Pris under Different Wind Speeds: Experientl Curves of Model HB4; b Fitted Curves of Model HB4; Experientl Curves of Model HB7; d Fitted Curves of Model HB7; e Experientl Curves of Model HB; f Fitted Curves of Model HB. HB4 HB7 HB 4 HB4 HB7 HB - B - B b Ū Fig 5: erodyni Dping B ; b erodyni Stiffness Ū B

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