Characteristics of Tuned Liquid Column Dampers for a 76-Story Benchmark Building

Transcription

1 ctbuh.org/papers tle: Authors: Subjects: Keywords: Characterstcs of uned Lqud Column Dampers for a 76-Story Benchmark Buldng Kyung-Won Mn, Assocate Professor, Dankook Unversty Sang-Hoon Song, Graduate Research Assocate, Dankook Unversty Sang Kyung Ahn, Vstng Researcher, Seoul Natonal Unversty Hong-Jn Km, Senor Researcher, Research Insttute of Industral Scence and echnology (RIS) Hyoung-Seop Km, Senor Researcher, Samsung E&C Buldng Case Study Structural Engneerng uned Lqud Damper Wnd Loads Publcaton Date: 2004 Orgnal Publcaton: Paper ype: CBUH 2004 Seoul Conference. Book chapter/part chapter 2. Journal paper 3. Conference proceedng 4. Unpublshed conference paper 5. Magazne artcle 6. Unpublshed Councl on all Buldngs and Urban Habtat / Kyung-Won Mn; Sang-Hoon Song; Sang Kyung Ahn; Hong- Jn Km; Hyoung-Seop Km

2 Characterstcs of uned Lqud Column Dampers for a 76-Story Benchmark Buldng Kyung-Won Mn, Hyoung-Seop Km 2, Sang Hyun Lee 3, Hongjn Km 4, Sang Kyung Ahn 5 Assocate Professor, Dankook Unversty 2 Gaduate Research Assocate, Dankook Unversty 3 Vstng Researcher, Research Center of Engneerng Scence, Seoul Natonal Unversty 4 Senor Researcher, RIS 5 Senor Researcher, Samsung Engneerng and Constructon Abstract In ths study, the performance of tuned lqud column damper (LCD) for a 76-story benchmark buldng s evaluated. he optmal desgn parameters are numercally obtaned by usng the whte nose and harmonc load n addton to the determnstc wnd loads gven by benchmark problem as wnd data. Further, the robustness of multple LCD s nvestgated, and the effects of mass rato, the number of dampers, central tunng frequency rato, and the frequency range on the performance of multple LCD are evaluated. Sem-actve LCD usng lnear quadratc Gaussan control algorthm s appled and ts performance s compared wth that of passve LCD and actve MD. Keywords : tuned lqud column damper (LCD), optmal desgn parameters, wnd loads. Introducton In order to mprove the performance of flexble structures that are senstve to wnd or earthquake loads, technologes of dsspatng the structural energy by nstallng auxlary mass have been developed and appled to buldng structures. Recently, the tuned lqud column damper (LCD) has receved the attenton of researchers as a type of auxlary mass system ). LCD has the control characterstcs smlar to that of tuned mass damper (MD), whch s one of most frequently used dampers for vbraton control, but t has the followng advantages over MD; () the tunng frequency rato, whch s crtcal parameter n the desgn of mass dampers, can be easly controlled by changng the length of lqud column; (2) dampng can be changed by controllng the sze of the orfce; (3) any ntal operatng devce s not requred snce the gravty equatng the level of lqud functons as stffness; (4) the lqud of LCD can be used for buldng water supply whle the mass of MD smply ncreases the gravty load. Because of these advantages, a growng number of brdge and buldng structures have been bult wth the LCD system over the past decade or so. Examples nclude Random House at New York, USA and One Wall Center at Vancouver, Canada. Accordng to Wrght, the LCD system has saved about mllon dollars of constructon cost compared to the system wthout damper 2), and Fortner reported that LCD saved the about 200 mllon Contact Author: Kyung-Won Mn, Assocate Professor, Dankook Unversty, Hannam-dong, Yongsan-gu, Seoul, Korea el: Fax: e-mal: kwmn@dankook.ac.kr dollars compared to MD system 3) Snce the vscosty term n the governng equaton of moton of LCD s a functon of the absolute value of lqud velocty, the equaton s nonlnear and the dynamc characterstcs of LCD depend on the magntude and the characterstcs of exctaton forces and the correspondng structural responses of the floor at whch LCD s nstalled 4),5). Extensve analytcal studes have been conducted on the optmal desgn parameters, such as mass rato, tunng frequency rato, and headloss coeffcent, under varous types of exctatons. Won et al (996) nvestgated the sesmc performance of LCD for flexble structure usng the artfcal non-statonary earthquake data 6). Yalla and Kareem (2002) numercally estmated the optmal desgn parameters of LCD for controllng a structure excted by wnd loads, whch are generated usng the frst order and second order flters 7). In order to mprove the robustness of sngle LCD, the multple-lcd (MLCD) has been nvestgated by many researchers. he performance of LCD prmarly depends on the tunng frequency rato, and ths makes the performance senstve to the accuracy of estmated natural frequency of a structure to be controlled. Sadek et al (998) calculated the mean dsplacement and acceleraton responses usng the 72 earthquake data for determnng the optmal desgn parameters of MLCD 8). Gao et al (999) conducted a parameter study of MLCD for the structure subjected to harmonc loads, and nvestgated the robustness wth respect to the tunng frequency rato and mass rato 9). Xu and Shum (2003) utlzed MLCD to reduce the torsonal response nduced by harmonc load and whte nose 0). 66 CBUH 2004 October 0~3, Seoul, Korea

3 he dependence of the optmal parameters on the magntude of exctaton also promoted the development of sem-actve LCD n whch the headloss coeffcent s controlled n real tme based on the nformaton on exctaton force and structural responses. Yalla et al (200) evaluated the performance of sem-actve LCD and proposed a control algorthm based on clpped-optmal and fuzzy control theory ). Km et al (2003) showed that the sem-actve LCD control system reduces the responses of three dmensonal rregular buldngs subjected to earthquake ground motons effcently 2). he purpose of ths study s to evaluate the performance of LCD for controllng wnd nduced structural response of a 76-story benchmark buldng. he optmal parameters are numercally obtaned by usng the whte nose and harmonc loads n addton to the determnstc wnd loads gven by benchmark problem as wnd data. Optmal headloss coeffcent and tunng frequency rato are obtaned wth regard to the mass rato and the performance ndces, and the effects of rato of horzontal length to the total length of lqud column (length rato) on control performance are nvestgated. he robustness of MLCD s examned, and the effects of desgn parameters such as mass rato, the number of dampers, central tunng frequency rato, and the frequency range on the performance of MLCD are evaluated. Also, the performance of MLCD wth non-unform mass dstrbuton s evaluated comparng to that of MLCD wth unform mass dstrbuton. In order to mprove the performance of passve LCD, sem-actve LCD usng lnear quadratc Gaussan (LQG) algorthm s appled and ts performance s compared wth that of passve LCD and actve MD (AMD) Story Benchmark Buldng Modelng he 76-story benchmark buldng 3), a 306-m hgh offce tower wth a heght-to-wdth rato of 7.3 proposed for the cty of Melbourne, Australa, s used for the numercal smulaton n ths paper. he buldng s a renforced concrete buldng consstng of a concrete core and an exteror concrete frame. he buldng has a total mass ncludng heavy machnery n the plant rooms of 53,000 metrc tons. he dampng devces are desgned to consder only the across wnd drectonal responses, and the across wnd data provded by Yang et al. are used he frst fve natural frequences of the structure are 0.60 Hz, Hz,.992 Hz, Hz, and Hz. he dampng matrx s composed for the frst fve modes to have % dampng rato. Performance evaluaton crtera Eght out of twelve crtera proposed by Yang et al. 3), denoted by performance ndces J to J 2, to evaluate control systems are used n ths research for the purpose of comparson. For each crteron, smaller numbers represent a more effectve response control performance. he frst sx performance measures (J to J 6 ) are root-mean-square (RMS) responses of the selected floors of the structure and actuator. he next sx performance measures (J 7 to J 2 ) are the peak responses of the selected floors of the structure and actuator. Among the twelve crtera, only 8 crtera (J to J 4 and J 7 to J 0 ) are used n ths research, because the other four crtera (J 5, J 6, J, and J 2 ) represent the performance of actuator. Nether sem-actve LCD nor hybrd damper-lcd system requres any actuator to operate, thus makng the other four crtera rrelevant. he frst four crtera n terms of the RMS responses are J = max( x, x30, x50, x55, x60, x65, x70, x75) x75o () J = 2 ( σ & x σ & xo ) for = 50, 55, 60, 65,70 and 75 (2) 6 σ σ (3) J3 = x76 x76o J 4 = 7 ( σ x σ xo ) for =50, 55, 60, 65,70, 75 and 76 (4) where σx& s the RMS acceleraton of the th floor; x75o s the RMS acceleraton of the 75th floor wthout 2 control whch s equal to 9.42 cm / sec ; σ x& o s the RMS acceleraton of the th floor wthout control; σx and σxo are the RMS dsplacements of the th floor wth and wthout control, respectvely; σ x76o s the RMS dsplacement of the 76th floor of the uncontrolled buldng whch s equal to 0.37 cm. he next four crtera n terms of peak responses are (& x p, & x p30, & x p50 ) o, & x p55, & x p60, & x p65, & x p70, & x p75 x p (5) for =50, 55, 60, 65,70 and 75 (6) J = max 75 J 7 & 8 = 6 ( & x J x p 76 x p 76o 9 = J0 = 7 ( x & p x po ) p x po ) (7) for =50, 55, 60, 65,70, 75 and 76 (8) where x& p and x& po are the peak acceleraton of th floor wth and wthout control, respectvely; x p and x po are the peak dsplacements of th floor wth and wthout control, respectvely; x p 76 o s the peak dsplacement of the 76th floor wthout control whch s equal to cm and & x p 75 o s the peak acceleraton of the 75th floor wthout control whch s equal to cm/sec 2. he characterstcs of structural response to gven wnd load he consdered benchmark buldng s a long perod structure of whch frst natural perod s 6.25 second. Snce the moton of long perod structure s generally governed by the frst modal response, the frequency of LCD should be tuned to the frst modal frequency of the structure. Fgure shows the power spectral densty (PSD) of the determnstc wnd data obtaned by wnd tunnel test, and Fgure 2 shows the PSD of dsplacement and acceleraton responses of 76 th floor CBUH 2004 October 0~3, Seoul, Korea 67

4 nduced by the wnd load. It s observed that frst modal response domnates for both the dsplacement and acceleraton responses. Accordngly, the frequency of LCD needs to be tuned to the frst modal frequency of the structure as expected. Fg.. Power spectral densty of the wnd data at 76th floor Fg. 2. Power spectral densty of the structural responses nduced by the wnd load: (a) dsplacement of 76th floor, (b) absolute acceleraton of 76th floor LCD LCD LCDm u L 3. Equaton of Moton he n degrees of freedom system wth m LCD shown n Fgure 3 has followng equaton of moton. M + M' M x& ( t ) C [] 0 x& ( t ) + A M S M ( t ) [] 0 m n K [] 0 nxm x( t ) F( t ) = [] 0 K u ( t ) [] 0 mxn B (a) S u orfce Fg. 3. LCD system: (a)u-shaped lqud column (b) MLCD system + (9) n whch, M, C, and K are, respectvely, nxn mass, dampng, and stffness matrx of the structure. x ( t ), u ( t ), and F( t ) denote nx dsplacement vector of the structure, mx dsplacement vector of the LCD, and nx exctaton vector. 0 s a null matrx and M, M', M S, M S, K, and C have the followng forms. M [ ] (0) = dag mt mt2 Λ mtm [] 0 ( n )x( n ) [] 0 ( n )x M' = () [] 0 x( n ) m [] 0 n m M = ( S ) M d = [ αmt αmt2 Λ αmtm ] (2) Md mx nxm C (b) ( t ) M S = M S (3) K (4) = dag[ 2ρA g 2ρA2 g Λ 2ρAm g] ρa ρa2 ρam C = dag ξ ( ξ2 2 Λ ξm m( (5) where m s the total mass of LCD and g denotes acceleraton due to gravtaton. m t, ρ, A, ξ, and u are, respectvely, mass, lqud densty, cross-secton area, headloss coeffcent, and vertcal dsplacement of the th LCD. α and m are represented as follows B α = m = m t (6) L = where B and L, respectvely, are the horzontal length and total length of th lqud column. he equaton of moton of the th LCD s gven by ρa L + ρaξ + 2ρA gu = ρa B & x( 2 (7) where x& ( s the acceleraton of the floor at whch the LCD s nstalled. It s noted from Eq. (7) that acceleraton of the structure s the cause of the lqud moton n LCD. Moreover, vscosty of the LCD, the second term of the left sde n Eq (7), ncludes the absolute value of lqud velocty, ( t ), and represents the nonlnearty of LCD. Consderng that the lqud velocty vares wth the structural acceleraton and the structural acceleraton s governed by exctaton forces, t can be seen from Eq. (7) that the dampng force of LCD s not a constant but a varable dependng on the characterstcs of exctaton forces. Consequently, the optmal value of headloss coeffcent depends on the characterstcs of exctaton forces such as the frequency contents and the magntude. 4. Optmal Desgn Parameters for Sngle lcd he tunng frequency rato, the headloss coeffcent, and the length rato are crtcal parameters n the desgn of LCD. In ths secton, the optmal headloss coeffcent and tunng frequency rato are obtaned wth regard to the mass rato and the performance ndces for RMS response, where the mass rato, µ, s rato of the lqud mass to the mass of structure. Further, the effects of length rato on control performance are nvestgated. Optmal values of the desgn parameters are frst obtaned for the determnstc wnd data, and then compared to those obtaned usng the harmonc load and whte nose. For an undamped system wth MD, closed form solutons of optmal tunng frequency rato and dampng rato can be obtaned wth regard to mass rato. he optmal parameters of LCD for damped system lke the benchmark buldng, however, can not be gven n closed form, and they can be determned only numercally. Optmal unng Frequency Rato unng frequency rato s the rato of frequency of LCD to that of the structure, and expressed as m 68 CBUH 2004 October 0~3, Seoul, Korea

5 Fg. 4. Varaton of performance ndces wth respect to tunng frequency rato and mass rato Fg. 6. Varaton of performance ndces wth respect to tunng frequency rato and headloss coeffcent (mass rato = %) Fg. 5. Varaton of performance ndces wth respect to headloss coeffcent and mass rato f ωt = ω (8) where ω s s the natural frequency of the structure and ω t s the frequency of the lqud column. Fgure 4 shows the varaton of performance ndces J to J 4 wth respect to f and µ. he length rato, α s fxed to be 0.85, and headloss coeffcent, ξ, that mnmzes J s selected. It can be seen from Fgure 4 that the optmal value of f to mnmze the performance ndces s close to, and t becomes closer to as µ decreases. Performance s mproved and the senstvty of performance ndces to f s reduced wth ncreasng µ. However, the performance mprovement becomes neglgble f µ >.5%. Optmal Headloss Coeffcent In Fgure 5, the varaton of performance ndces wth respect to ξ and µ s presented. α s fxed to be 0.85, and f that mnmzes J s selected. It s observed that the optmal value of ξ ncreases wth ncreasng µ. he larger µ for a gven value of ξ, however, does not always guarantee the superor control performance. Specally, the control performance deterorates n spte of larger µ when ξ s small. Such tendency can be observed n both dsplacement and acceleraton responses. Accordngly, determnng ξ optmally n terms of the gven µ s mportant n desgn of LCD. s Fg. 7. Varaton of performance ndces wth respect to the length rato and mass rato he optmal values of f and ξ for a gven µ can not be determned ndependently snce the effect of f on the control performance s correlated wth that of ξ. In order to dentfy the correlaton of the effect of f and ξ on control performance, the varaton of performance ndces wth regard to these two parameters s obtaned and plotted n Fgure 6. For ths numercal analyss, µ = % s used. It can be noted from Fgure 6 that the optmal values of f and ξ have to be determned smultaneously n desgnng LCD. Optmal Length Rato Fgure 7 shows the varaton of performance ndces J to J 4 wth respect to the length rato, α, and µ. he optmal values of f and ξ that mnmze J are used. It s observed that the larger µ and α result n the better control performance n general. Accordngly, t s advantageous to ncrease α as large as possble. However, α more than a certan threshold value destroys the basc characterstcs of LCD and thereby the followng constrant condton on α should be satsfed for practcal applcaton. u max (9) α 2 L where umax denotes the maxmum dsplacement of lqud column. Snce u max depends on the exctaton forces, an teratve procedure s necessary for determnng α. CBUH 2004 October 0~3, Seoul, Korea 69

6 able. Optmal values of the desgn parameters obtaned numercally for dfferent types of wnd loads J J 2 J 3 J 4 µ =0.5% µ =.0% µ =.5% µ =2.0% H W D H W D H W D H W D f ξ f ξ f ξ f ξ Harmonc Load and Whte Nose Many researchers have proposed formulas for obtanng optmal desgn parameters n desgn of sngle LCD. hese formulas are generally derved based on the assumptons that the exctaton load s harmonc or whte nose. Consequently, a LCD desgned wth parameters obtaned usng those formulas does not always perform optmally when the real wnd load s appled. In ths secton, optmal parameters are numercally obtaned usng the harmonc load and whte nose, and are compared to those obtaned n the prevous sectons usng the determnstc wnd data. Wnd load s proportonal to the square of the wnd velocty multpled by the actng area. Accordngly, the vertcal dstrbuton over heghts of harmonc load and whte nose s determned to satsfy ths condton n the numercal analyss. Velocty at each floor level s assumed to be proportonal to that of the determnstc wnd load of benchmark problem. And the magntudes of loads are scaled to produce the same peak acceleraton of 75th floor to that nduced by determnstc wnd load. he frequency of harmonc load s selected to be dentcal to the fundamental frequency of the benchmark buldng. able presents the optmal values of the desgn parameters obtaned numercally for dfferent types of wnd loads. It can be seen n able that optmal value of f for harmonc load s always regardless of µ and performance ndces. For whte nose, optmal value of f s ether slghtly larger or smaller than, and t becomes smaller wth ncreasng µ. For the determnstc wnd load, however, values smaller than yeld optmal performances. For the numercal analyses usng harmonc loads wth frequences dfferent to the fundamental frequency of structure, t was found that the optmal values of f are dentcal to the ratos of the exctaton frequency to the fundamental frequency. hs mples that the frequency of the LCD should be tuned to the domnatng frequency of the structural response. he optmal values of ξ are found to be smaller for harmonc load than for whte nose and those for determnstc wnd are n between. 5. Multple lcd he natural frequency of the structure may be dfferent from the value assumed for the desgn of LCD, and ths dscrepancy results n the performance degradaton of sngle LCD system. Accordngly, MLCD system was developed n order to compensate ths shortcomng of the sngle LCD. MLCD system s generally known to have more robustness than sngle LCD because t has the broader frequency dstrbuton bandwdth. he desgn parameters of MLCD are the number of dampers, the frequency range, and central tunng frequency rato. It s reported that tunng the frequences of every LCD to the fundamental mode n MLCD s more effectve than tunng t to dfferent modes. Accordngly, the frequency of MLCD s tuned to the fundamental mode n ths study. Only the performance ndces for RMS responses are consdered smlarly to the case for determnng the optmal parameters of sngle LCD. Snce large value of α may destroy the fundamental characterstcs of LCD as prevously descrbed, α s set to be 0.8 n the subsequent sectons. Optmal Number of Dampers he optmal number of dampers, N, should be determned to consder the control performance and economcal and constructonal effcency. Fgure 8 shows the varaton of performance ndces wth respect to N and µ of MLCD. Other parameters such central tunng frequency rato, headloss coeffcent, and the frequency range are obtaned to mnmze J through the teratve procedure. It s observed that the larger mass rato provdes the better control performance smlar to sngle LCD. Further, t can be seen that the performances are generally enhanced sgnfcantly when 3 LCDs are used n MLCD compared to that of sngle LCD. However, ncreasng N over 5 does not provde sgnfcant response reducton. Accordngly, N=5 wll be used n the subsequent sectons. Optmal Frequency Range he frequences of MLCD are dstrbuted at the same nterval of frequency wth the reference to central tunng frequency rato. he frequency range s gven by 70 CBUH 2004 October 0~3, Seoul, Korea

7 Fg. 8. Varaton of performance ndces wth respect to the number of dampers of MLCD Fg. 0. Varaton of performance ndces wth respect to central tunng frequency rato Fg. 9. Varaton of performance ndces wth respect to frequency range of MLCD ω = ω tn ω t (20) where ω tn and ω t are frequences of the frst and Nth LCD, respectvely. Fgure 9 llustrates the varaton of performance ndces wth respect to frequency range of MLCD. Note that the case of ω = 0 n fgures corresponds to the sngle LCD. he central tunng frequency rato and the headloss coeffcent have the value same to the ones used n Fgure 8. It s obvously shown that there exsts an optmal value of ω dependng on µ and the value ncreases wth ncreasng µ. Also, performance ndces become nsenstve to the value of ω as µ ncreases. hs mples that ncreasng the value of ω over the optmal value for small µ causes the sgnfcant deteroraton of control performance. Central unng Frequency Rato he central tunng frequency rato denotes the frequency of the LCD at the center of frequency range, and s gven by ωtj f N s odd number ω f = s o ω + tj ωtj+ otherwse (2) 2ωs Fg.. Comparson of harmonc responses controlled by MLCD wth unform and non-unform mass dstrbuton: (a) peak dsplacement of 76th floor (b) peak acceleraton of 75th floor (c)rms dsplacement of 76th floor (d) rms acceleraton of 75th floor where j equal to (N+)/2 f N s odd number and to N/2 otherwse. In Fgure 0, the effects of f o on the control performance of MLCD are presented. he value of ξ s optmzed to mnmze J through an teratve procedure. It can be seen that the optmal value of f o s close to smlarly to that of f for sngle LCD. One can enhance the control performance and reduce ts senstvty by ncreasng µ. hs mples that µ has nfluence on the robustness of MLCD whch wll be dscussed later. able 2. he performance ndces by MLCD wth unform and non-unform mass dstrbuton J J 2 J 3 J 4 unform non-unform J 7 J 8 J 9 J 0 unform non-unform Non-unform Mass Dstrbuton Most studes on MLCD have been conducted settng the mass rato of each LCD to be dentcal. Kareem and Klne (995) nvestgated the performance CBUH 2004 October 0~3, Seoul, Korea 7

8 of MLCD wth non-unform mass dstrbuton (non-unform MLCD) and non-unform frequency nterval, and concluded that non-unform MLCD does not provde any partcular advantages over usual MLCD wth unform mass dstrbuton (unform MLCD) 4). In ths secton, the control performance of non-unform MLCD s nvestgated usng the harmonc load and the determnstc wnd data. he mass of each LCD s dstrbuted n a trangular form wth ts maxmum value at the center of ω. Numercal smulaton results for harmonc loads are presented n Fgure, where ω e s the exctaton frequency of the harmonc load. It can be seen that the non-unform MLCD performs equal to or better than the unform MLCD near resonance frequency whle t does not do outsde of resonance frequency. hese results are reasonable consderng that the non-unform MLCD has the largest mass rato at the central tunng frequency where the resonance occurs. able 2 summarzes the results of the performance ndces for non-unform and unform MLCD when the determnstc wnd load s appled. otal mass rato of % s used and the values of ξ, f o, and ω are optmzed to mnmze J. Both cases provde the almost equvalent control performance. Accordngly, the unform MLCD s preferred over non-unform MLCD consderng the neffcency n constructon for the non-unform MLCD. Robustness of MLCD In addton to the control performance, the robustness s one of the most mportant features of dampng devces. Benchmark problem of 76-story buldng suggested that the robustness of the controller should be dscussed. Snce the value of f s crtcal n control performance of LCD, the varaton of the frequency of the structure due to the measurement or calculaton error and the varaton of mass or stffness may cause the sgnfcant performance deteroraton. Under ths background, MLCD system has been proposed for enhancng the robustness n LCD. In ths secton, the robustness of LCD s dscussed by evaluatng the control performance when uncertanty exsts n the stffness of the structure. able 3 compares the performance ndces of SLCD and MLCD. Although the performance of MLCD s almost equvalent to that of sngle LCD when there exsts no uncertanty ( K=0%), MLCD shows the superor performance to the sngle LCD when stffness uncertanty exsts ( K=±5%), only except for J 5. As expected, robustness could be guaranteed by usng MLCD. Sem-actve LCD system he dependence of the optmal desgn parameters on the magntude of exctaton force promoted the development of sem-actve control algorthms, whch controls ξ accordng to the nformaton on exctaton force and structural responses. If the value of ξ, whch s determned by the sze of orfce, can be changed by a controllable orfce, then the passve dampng force s transformed nto an actve force whch controls the response of the structure. he equaton of moton for the sem-actve sngle LCD s gven by M + M' M S & x( C [] 0 nxm x& + [] [] + MS M 0 m n 0 mxm K [] 0 nxm x( F [] 0 mx [] = + fc (22) 0 mxn K u [] 0 mx [] m where s the matrx of whch all the elements are and control force, f c ( t ), s gven by ρaξ fc = c( = 2 (23) he control force f c ( t ) n Eqs. (22) and (23) can be determned by usng any actve control algorthm, and the headloss coeffcent whch realze the determned f c ( t ) s gven by * 2 fc ξ = ρa (24) Snce the value of ξ s lmted n a certan range practcally n a sem-actve LCD system, the value of ξ s regulated n accordance wth the sem-actve control law expressed as * ξ mn ξ ( < ξ mn or fc 0 * ξ = ξ * ξ max ξ ( > ξ max (25) where ξ max and ξ mn are, respectvely, the maxmum and mnmum practcal lmts of headloss coeffcent. In ths secton, the mass of passve and sem-actve LCD are set to be 500ton whch s the mass of AMD of benchmark problem. For passve LCD, 0.98 and 30 are used for f and ξ, respectvely. For sem-actve LCD, ξ max =50 and ξ mn =0 are used and the control force s calculated by usng LQG algorthm. LQG. controller s desgned based on the example provded by the benchmark problem 3). able 4 presents the performance ndces of AMD, passve LCD, and sem-actve LCD systems. It s able 3. Performance ndces by sngle LCD and MLCD RMS response Peak response K =+5% K =0% K =-5% K =+5% K =0% K =-5% N= N =7 N = N =7 N = N =7 N =7 N = N =7 N =7 N = N =7 J J J J J J J J CBUH 2004 October 0~3, Seoul, Korea

9 able 4. Performance ndces by AMD, passve LCD, and sem-actve LCD RMS response Peak response AMD Passve LCD Semactve LCD AMD Passve LCD Semactve LCD J J J J J J J J that the performance of MLCD s almost equvalent to that of sngle LCD when there exsts no uncertanty n stffness, whereas MLCD shows the superor performance to the sngle LCD when stffness uncertanty exsts. o mprove the performance of passve LCD, sem-actve LCD usng LQG algorthm s appled and ts performance s compared wth that of passve LCD and AMD. It s found that sem-actve LCD can mprove the performance of passve LCD system and t provdes the almost equvalent performance to that of AMD. Acknowledgement he work presented n ths paper was partally supported by both Korea Insttute of Constructon & ransportaton echnology Evaluaton and Plan (KICEP, Project No. 03 Industry and Unversty Cooperatve Research C03A A ) and echnology n Korea and Samsung Engneerng & Constructon. Fg. 2. Structural responses controlled by passve LCD and sem-actve LCD: (a) dsplacement of 76th floor (b) acceleraton of 75th obvously observed that sem-actve LCD can mprove the performance of passve LCD system and t provdes the almost equvalent performance to that of AMD. Fgure 2 shows the tme hstores of the dsplacement and acceleraton of the 76th floor. It s also shown that sem-actve LCD s more effectve than the passve LCD for response control. Concluson In ths study, the performance of sngle and multple-lcd for controllng wnd nduced structural response of a 76-story benchmark buldng s evaluated. he optmal parameters of sngle LCD such as headloss coeffcent, tunng rato are numercally obtaned by usng the whte nose and harmonc load n addton to the determnstc wnd loads. It s found that the optmal value of tunng frequency rato s close to, and the value becomes closer to as the mass rato decreases. he senstvty of performance to the tunng frequency rato s reduced wth ncreasng mass rato. However, the larger mass rato does not always guarantee the superor control performance, when the headloss coeffcent s small. Consequenctly, determnng the headloss coeffcent optmally n terms of the mass rato s mportant for better performance. he robustness of MLCD s nvestgated, and the effects of desgn parameters such as mass rato, the number of dampers, central tunng frequency rato, and the frequency range are evaluated. Also, the performance of MLCD wth non-unform mass dstrbuton s evaluated wth the comparson to that of MLCD wth unform mass dstrbuton. It s found Reference ) Samal, B., Kwok, K., and Gao, H. (998) Wnd Induced Moton Control of a 76 Storey Buldng by Lqud Dampers, Proceedngs of Second World Conference on Structural Control, Vol., pp ) Wrght, G., Steadyng Influence - Damper systems save mllons n cost, whle reducng lateral movement n tall buldngs, Buldng Desgn & Constructon, Nov ) Fortner, B., Water anks Damp Moton n Vancouver Hgh-Rse, Cvl Engneerng, June, p.8., ) Chang, C. C. (999) Mass dampers and ther optmal desgns for buldng vbraton control, Engneerng Structures, Vol. 2, pp ) Balendra,., Wang, C. M., and Rakesh, G. (995) Vbraton control of varous type of buldngs usng LCD, Journal of Wnd Engneerng and Industral Aerodynamcs, Vol. 83, pp ) Won, A. Y. J., Pres, J. A., and Haroun, M. A. (996) Stochastc Sesmc Performance Evaluaton of uned Lqud Column Dampers, Earthquake Engneerng and Structural Dynamcs, Vol. 25, pp ) Yalla, S. K., and Kareem, A. (2000) Optmum Absorber Parameters for uned Lqud Column Dampers, Journal of Structural Engneerng, Vol. 26, No. 8, pp ) Sadek, F., Mohraz, B., and Lew, H. S. (998) Sngle-and Multple-uned Lqud Column Dampers for Sesmc Applcatons, Earthquake Engneerng and Structural Dynamcs, Vol. 27, pp ) Gao, H., Kwok, K. C. S., and Samal, B. (999) Characterstcs of Multple uned Lqud Column Dampers n Suppressng Structural Vbraton, Engneerng Structures, Vol. 2, pp ) Xu, Y. L., and Shum, K. M. (2003) Multple-uned Lqud Column Dampers for orsonal Vbraton Control of Structures: heoretcal Investgaton, Earthquake Engneerng and Structural Dynamcs, Vol. 32, pp ) Yalla, S. K., Kareem, A., and Kantor, J. C. (200) Sem-Actve uned Lqud Column Dampers for Vbraton Control of Structures, Engneerng Structures, Vol. 23, pp ) Km, H., Mn, K. W., and Lee, S. H. (2003) Sem-Actve Control of 3D Coupled Response of Buldng Structures, KSCE Journal of Cvl Engneerng, Vol. 7, No. 6, pp ) Yang, J. N., Wu, J. C., Samal, B., and Agrawal, A. K. (998), A Benchmark Problem for Response Control of Wnd-Excted all Buldngs, Proceedngs Of Second World Conference on Structural Control, Vol. 2, pp ) Kareem, A. and Klne, S. (995), Performance of Multple Mass Dampers under Random Loadng, Journal of Structural Engneerng, Vol. 2, No. 2, pp CBUH 2004 October 0~3, Seoul, Korea 73