Multi-dimensional Seismic Control with TLD in Vortex Effect Hao-Xiang HE 1,a, Kui CHEN 2,b,*, Rui-Feng LI

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1 Interntionl Conference on Mechnic nd Civil Engineering (ICMCE 2014) Multi-dimenionl Seimic Control with TLD in ortex Effect Ho-Xing HE 1,, Kui CHEN 2,b,*, Rui-Feng LI 1 Beijing Key Lbortory of Erthquke Engineering nd Structure Retrofit, Beijing Univerity of Technology, Beijing , Chin 2 Beijing Collbortive Innovtion Center for Metropolitn Trnporttion, Beijing , Chin hhx7856@163.com, b ck2013@emil.bjut.edu.cn *Correponding uthor Keyword: Tuned Liquid Dmper; ortex, Hydrulic Preure; Eccentric Structure, Structurl Control. Abtrct. Seimic control of the horizontl nd torionl repone by cylindricl tuned liquid dmper i tudied. The hydrulic preure model of the cylindricl TLD including pule preure nd convection preure i etblihed by membrne nlogy method. The torionl model of the cylindricl TLD i obtined bed on the qui-uniform vortex theorem. The eqution of motion for the control ytem conidering eccentric torion effect i derived nd the olving method i preented. A 10-tory eccentric tructure with TLD i nlyzed to verify the control performnce for the horizontl nd torionl coupled ytem under erthquke. The reult how tht the coupled repone cn be reduced effectively nd the vortex torionl force by TLD cnnot be ignored. Introduction In recent yer, trong erthquke occurred contntly nd the dynmic ction h brought gret detruction to build tructure, which mke the importnce of tructurl eimic nd diter prevention nd mitigtion more pprecited. A for the importnt eccentric tructure, only conidering ingle dimenionl erthquke will underetimte the ctul dynmic repone of the tructure, therefore, the effect of the multi-dimenionl eimic ction on tructure i necery [1]. In ddition to generting the trnltionl vibrtion, the tructure lo produce torionl vibrtion which cnnot be ignored. The cue of torionl vibrtion i follow: there re rottionl component exit in ground motion, or the ground motion t ech point h phe difference. On the other hnd, the tructure itelf i eccentric, nmely the center of m nd the center of tiffne re not coincident [2]. Erthquke dmge how tht the torionl effect will ggrvte the tructurl dmge nd in ome ce it will become the min fctor cuing the tructurl filure. In order to reduce the dmge degree of tructure under erthquke, the dmping control technology emerge the time require. At preent, be ioltion, energy diiption, hrmonic vibrtion, ctive control technology cn ply role in dmping control for the pecific tructure [3]. In recent yer, the tuned m or tuned liquid control technology which develop ft doe not need to tke trditionl meure to trengthen the tructure, the dmping effect i obviou nd it ey to implement [4-7]. Tuning ytem i dding inertil m to the top of the tructure or dding the flowing liquid inide the ubidiry tructure, uing the econd order ytem to borb the vibrtionl energy of mjor tructure o to control the vibrtion of the mjor tructure. TLD h the chrcteritic of low cot, imple nd prcticble, low mintennce cot nd good dmping effect, o it h good propect of engineering ppliction. But the tuned m or tuned liquid dmping ytem h the following problem: the nlyi method of multi-dimenionl dmping till need to be improved, nd multi-dimenionl dmping effect hould be enhnced etc. The eimic control of the horizontl nd torionl repone by cylindricl tuned liquid dmper i tudied in pper [6] nd the circultion torion effect of the liquid i conidered in the nlyi. But nti twited propertie of the generl cylindricl tuned liquid dmper epecilly the nti-twited bility formed by vortex inide the liquid w lwy ignored in previou tudie. The mechnic principle nd the dmping effect of uing cylindricl tuned liquid dmper to relize the The uthor - Publihed by Atlnti Pre 233

2 multi-dimenionl coupled vibrtion control of the eccentric tructure i tudied in thi pper. The Hydrulic Preure Model Etblihed by Membrne Anlogy Method The principle of tuned liquid dmper i uing the internl inerti nd vicoity of liquid in the continer fixed on the tructure to borb nd conume kinetic energy of the tructure in order to chieve the purpoe of reducing tructure vibrtion. Mny dometic nd foreign reercher propoed different implified dynmic model to reflect the energy diiption chrcteritic of TLD. Houner h propoed equivlent m method [8], thi method clified the hydrulic preure produced by liquid lohing into two ctegorie: Pule preure nd Convection preure, nd imulted by vibrtion effect of two equivlent qulity connected with the box body in different form. Syr etc. ued the pendulum model to imulte the movement of continer prtilly filled with wter nd tudied the movement of the continer ytem nd tructure [9]. At preent, the mot widely ued i the voltility model etblihed by fluid wve dynmic theory, nd it minly include the deep wter model nd the hllow wter model [10]. Ju Rongchu propoed the model of equivlent qulity etblihed by membrne nlogy method [7]. For hllow wter continer, the hydrulic preure cn be divided into pule preure nd convection preure, pule preure i ocited with the inerti force cued by the movement of the pule of the veel wll, the mplitude i directly proportionl to the veel wll ccelertion. The convection preure i due to the hydrulic preure generted by liquid lohing. For the continer which h verticl ide wll nd the horizontl plte, if pply horizontl pule ccelertion in certin direction of the ide wll, nd the ccelertion which i orthogonl to the ide wll cn be ignored, equivlent to the liquid in the continer i limited by the number of nd the bottom plte nd film force direction re orthogonl. It i equivlent to the liquid in the continer which i limited by ome membrne nlog tht re orthogonl to the bebord nd the direction of force. Thu, the pule preure of liquid cn be clculted by only conidering the pule preure generted by the membrne nlog. When the ide wll of the continer i ubjected to the excittion, the liquid will produce hydrulic preure in the veel wll nd floor. If only conidering the firt vibrtion mode of liquid, it cn be regrded the rigidity membrne nlog which cn rotte freely in the horizontl direction. For cylindricl continer, the equivlent m of the hydrulic preure of the ide wll i m m / 3 tnh( 3 / ) c0 lh r r h. In thi formul, m l i the totl liquid qulity, h i the height of the liquid in the cylinder, r i the rdiu of the cylinder. The totl liquid dynmic pule preure cting on the wll of the continer i F c rh r h u pp 2 3 / 3 tnh( 3 / ) (1) Where, c i tructure coupling coefficient, vlue of 0.3 to 0.4, i the liquid denity, u i horizontl ccelertion the continer ber. The firt equivlent qulity of the cylindricl continer on ide wll for convection preure i m rm tnh(1.837 r / )/ h, the firt equivlent qulity correponding tiffne 1 c l 2 2 i k 5.4 m gh / mr c c1, g i the ccelertion of grvity. The totl liquid convection preure cting on the wll of the continer i in 4 2 Fpf c r 0 t Where, ω i the circulr frequency correponding to the firt equivlent qulity of liquid, θ 0 i the free urfce of the liquid t the corner. The mximum totl liquid convection preure i c1 F c r, where i the erthquke influence coefficient liquid torge of the pf mx c1 48 g nturl vibrtion period of liquid torge. The nturl vibrtion period of the liquid torge in 234 (2)

3 continer i T 2 / ( 27 / 8 g / r) tnh( 27 / 8 h / r) c1 Torionl Model of the Cylindricl TLD bed on Qui-uniform ortex Theorem Under the ctul multi-dimenionl erthquke, in ddition to energy conumption nd hock borption in two horizontl direction, the liquid of the TLD intlled in tructure will produce vortex phenomenon which cn uppre the tructure torionl to certin extent. But t preent the mechnim of vortex nd torionl effect i not involved in TLD dynmic model. Thi pper will tudy. The theory of fluid mechnic h proved ellipoidl cvity i the only geometry cvity which cn chieve homogeneou vortex motion [12]. Pfeiffer propoed n pproximte theoreticl clcultion of non ellipoidl cvity fluid flow. The theory ue the vorticity of ech point in the men vlue to decribe the ctul flow vorticity pproximtely [13-14]. The motion tht ue uniform vortex motion to decribe the liquid moving under the ene of the verge i clled the qui-uniform vortex motion. Under the ctul erthquke, the rotry of liquid in cylindricl TLD belong to non homogeneou vortex, o thi pper will etblih the cylindricl TLD torionl model bed on the qui-uniform vortex theorem. The motion of n idel nd not compreible liquid in ny cvity, if it exitence of the vortex motion which cnnot be ignored, then introduce the men vorticity Ω into the cvity liquid, it i the verge vlue of the vorticity in the flow field. Ω 1 Ω( t, x, x, x ) d Where, repreent the volume of the fluid domin. Suppoe the fluid motion cloe to uniform vortex motion, it velocity ditribution i written v v0 Ω r Where, v i the bolute velocity of the fluid t ny point in the P, v0 i liquid filled rigid coordinte velocity, r i the rdiu vector tht fluid prticle P reltive to the origin of the ytem, i the velocity potentil function of the liquid. According to the theory of fluid mechnic, we cn obtin v v0 ω r u Where, ω i the rottion ngulr velocity of inerti coordinte ytem compred to the liquid filled rigid body fixed coordinte ytem. U i the reltive velocity of the fluid prticle compred to the motion of the cvity. The reltionhip between reltive velocity nd potentil function i: u ( Ω ) r For uniform vortex motion, the Helmholtz eqution i 1/ 2 d Ω ( u ) Ω Ω ( Ω ) v dt (7). (3) (4) (5) (6) For n rbitrry xiymmetric liquid in cvity, it men vorticity Helmholtz eqution: Ω tifie the verged 235

4 dω dt 1 ( ) d Ω v Through the continuity eqution u 0 nd the condition tht liquid cnnot penetrble in the cvity wll un0, eqution nd boundry condition re derived to tify the potentil 2 0 P (9) (8) ( Ω ) r n P n (10) Leding into the Stoke-Rukovkii vector potentil Ψ to meet the need Ψ ( Ω ) Thu, qui-uniform velocity formul of vortex motion of liquid (6) cn be rewritten v v0 Ω r Ψ ( Ω) Averging the Helmholtz eqution in the cvity pce, nd put (12) into (8), conolidte it cn be obtined dω dt 1 ( ) d ( ) Ω Ψ Ω For the qui-homogeneou vortex motion of rottionlly ymmetric in the cvity liquid, leding into the Pfeiffer coefficient 1 ( rψ ) nz d r S The verge Helmholtz eqution only retin the liner become (11) (12) (13) (14) Ω (1 ) e ( Ω ) (15) For the cylindricl cvity tht rdiu i r nd the height i h, the Pfeiffer coefficient i expreed c 8r 1 h 1 tnh( ) h r n 2 n1 n( n 1) 2 (16) n i the root tht derivtive of the vrible in the firt cl of Beel function. Uing the method of eprtion of vrible to olve the prtil differentil eqution (8), the moment of inerti of the cylindricl cvity cn be obtined with it geometric center verticl pindle: r h 2 16r 1 nh J c mlr ml ( ) mlr 1 tnh( ) h n1 n( n 1) 2r (17) The moment of inerti conidered the internl liquid vortex effect of cylindricl TLD from the formul cn be obtined. 236

5 Multidimenionl Dynmic Model of Eccentric Structure with TLD Fig. 1 Eccentric Structure Model with TLD Fig. 2 Diplcement Hitory Comprion of Top Story On top of the eccentric tructure which h n lyer et TLD long the longitudinl nd trnvere, i hown in Fig.1, in the two horizontl nd round the verticl hft torionl, ech floor i equivlent to prticle of ingle degree of freedom repectively, thu the tructure i implified to be lyer model. The eqution of motion under multiple erthquke excittion i MU + CU + KU -MU g ( t ) F ( t ) (18) Where, M, C nd K re 3n 3n m mtrix, dmping mtrix nd tiffne mtrix of the tructure. T T x y x1 xn y1 yn 1 n U [ U, U, U ] u,, u, u,, u, u,, u U ( t) [ U ( t), U ( t), Φ ( t)] T g xg yg g i the diplcement vector of three dimenionl centroid. F t i the erthquke input. F t F t F t x y ( ) 0,, ( ),0,, ( ),0,, ( ) T control force vector of three dimenionl. Where the expreion of Fx(t), Fy(t) nd Fθ(t) re i the F () t m u m l u F x Ti xn Ti yi n px i1 i1 F () t m u m l u F y Ti yn Ti xi n py i1 i1 (19) (20) 2 () t mtil yiuxn mtil xiuyn mti Ri u n Fp i1 i1 i1 F In the formul, mti repreent the qulity of liquid in the firt i TLD uxn uxn uxg, uyn uyn uyg, u n u n u g R l l, i xi yi F. px nd Fpy repreent the hydrulic preure of X nd Y direction of TLD, repectively. Combined formul (1) nd (2), there i (21) F px ci[ rh i i tnh( 3 ri / hi ) ri i 0 in it] uxn i (22) Fpy h the imilr expreion, Combined Eq.(17), then Fp repreent the nti-torion vortex effect conidering TLD. F J u p ci n i1 Since the liquid dmping mtrix i introduced in the tructure nd the coupling ytem with TLD, (23) 237

6 The coupling dmping mtrix i not orthogonl, belong to non- clicl dmping, o cnnot ue rel modl trnform to nlyze decoupling nd time hitory of Eq. (18). In order to olve the problem, we cn ue the tte pce method, by introducing the tte vector to trnform the higher order ordinry differentil eqution of the ytem into the firt order differentil eqution coniting of tte vrible. Firt of ll, define the eqution Then the Eq. (18) cn be rewritten () t U ( t) ( I F ) U ( t) ge g MU () t g, where I i the unit mtrix. MU + CU + KU -MU ge () t (24) Define X U U T,Y U U T the tte vector of the ytem, then Eq.(2) cn be expreed the form of tte eqution X = AX + BU ge(t) Y = CX + DU ge(t) (25) 0 I I 0 A= C= -M K -M C, -M K -M C, B = D = Where, -I. Thu, ccording to the tructurl ccelertion excittion vector, by olving the ytem tte pce eqution cn clculte the coupling dynmic tructurl repone with TLD. Clcultion Exmple A 10-tory eccentric tructure with TLD i nlyzed to verify the control performnce for the horizontl nd torionl coupled ytem under erthquke. The tructure plne ize i 36m 16.5m, ech lyer i 3m. Column ection ize i 600mm 600mm, the reinforcement rtio i 2%. Bem ection ize i 300mm 650mm, the reinforcement rtio i 1.5%. Slb thickne i 120mm, Concrete trength grde i C40, tructure eccentric in the X direction nd Y direction re repectively 6m nd 6m.A i hown in Figure 1, on the top of the building long two horizontl eccentric kew ymmetric intll 10 cylindricl TLD, ech TLD rtio i 2m,the liquid height i 1m. All TLD ccounted for the totl m of the tructure i 7.31%. 0 Fig. 3 Accelertion Hitory Comprion of Top Story Fig. 4 Torion Angle Hitory Comprion of Top Story El Centro wve nd ny other eimic wve re ued ground motion fter etblihing lyer model, clcultion the multi-dimenionl eimic repone nd eimic mitigtion effect. In the preence of El Centro wve, the dynmic repone time hitory curve whether to intll the TLD tructure how from Fig. 2 to figure 4,we cn ee tht fter the intlltion of TLD, the tructurl repone including the diplcement, ccelertion nd ngle of twit, hve certin degree of inhibition. The dmping effect of TLD i idel. Fig. 5 nd Figure 6 repreent the diplcement of the tructure nd the torionl dmping effect, the reult how tht TLD h the effect of dmping vibrtion on the other lyer, but the dmping rte grdully wekened with the number of floor decree. 238

7 Fig.7 i the curve of inerti force nd hydrulic preure tht TLD pplied to the top of tructure in the x direction, combined Eq.(22) we cn ee tht eccentric inerti force generted by TLD will produce tble nd reltively obviou dmping effect, lo pule preure nd convection preure provided by TLD cn ply n importnt role in tructure dmping. Figure 8 i the curve of twiting force time hitory pplied to the top by TLD, combined Eq.(22) we cn ee tht the torionl force provided to tructure by TLD i minly obtined by relying on the vortex torionl obtined by rrnged on the oppoite direction of eccentric tructure, the torionl force provided by the TLD vortex effect lo ccounted for certin proportion, it role i bigger thn the twiting force dded by the horizontl inertil force by TLD, nd cnnot be ignored for eriou eccentric tructure. Fig. 5 Diplcement Reduction of All the Storie Fig. 6 Torion Reduction of All the Storie Fig. 7 Inerti Force nd Hydrulic Preure Curve of TLD Fig. 8 Torion Force Provided by TLD Summry The hydrulic preure model of the cylindricl TLD including pule preure nd convection preure i etblihed by membrne nlogy method. Thi model cn be coupled with dynmic eqution directly, nd it i more uitble for tructure deign. The torionl model of the cylindricl TLD i obtined bed on the qui-uniform vortex theory, in order to conider the repective vortex effect of tuned liquid. The eqution of motion of the control ytem for conidering eccentric torion effect i derived nd the olving method i preented. The nlyzed reult how tht pule preure nd convection preure provided by TLD cn ply n importnt role in tructure dmping. The torionl force provided to tructure by TLD i minly obtined by relying on the vortex torionl obtined by rrnged on the oppoite direction of eccentric tructure. The torionl force provided by the TLD vortex effect lo ccounted for certin proportion, it role i bigger thn the twiting force dded by the horizontl inertil force, nd cnnot be ignored for eriou eccentric tructure. 239

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