Optimal Configurations of Composite Multiple Mass Dampers in Tall Buildings

Transcription

1 Optimal Configuration of Compoite ultiple a Damper in S Avila avila@unbbr Univerity of Brailia UnB Engineering School of Gama Brailia, DF, Brazil P B Gonçalve Emeritu ember, ABC paulo@civpuc-riobr Catholic Univerity - PUC-Rio Civil Engineering Department Rio de Janeiro, RJ, Brazil Optimal Configuration of Compoite ultiple a Damper in Tall Building The effectivene of multiple tuned ma damper (TD) for uppreing harmonically forced ocillation i tudied in the paper In particular the influence of poible connection between the mae of the damper on the main ytem performance i invetigated uing four different configuration of a double ma damper For thi, a minimax procedure, which conider all damper parameter and variable, i ued to optimize each configuration and compare their influence on the minimum value of the maximum magnification factor of the main ytem A parametric tudy how that mall variation in the TD parameter and the way in which the mae are connected have a marked influence on the main ytem repone Thi enitivity give the deigner more freedom in chooing the proper damper configuration in a practical ituation Keyword: tuned ma damper, multiple tuned ma damper, vibration control, tructural dynamic, damper optimization Introduction A tuned ma damper (TD) i a paive vibration control device that ha been ued in ome engineering tructure and machine Uually it conit of a ingle ma-pring-dahpot ytem connected to the main tructure In particular, TD have been ued in recent year in everal tall building and tower to reduce the energy diipation demand of thee tructure under the action of wind load (Holme, 995) In thee application the TD i placed on the top of the building to maximize it efficiency The baic concept for the deign of a damped TD for an undamped tructure were preented by Den Hartog (956) He howed that under a imple harmonic load the main tructure could be kept completely tationary when the attached aborber i choen to be tuned to the excitation frequency A a reult, the vibrational tructural energy of the building i tranferred to the TD One of the drawback when a ingle TD i ued i it enitivity to mall variation in ytem parameter, in particular the natural frequency of the tructure and/or the TD damping conidered in the deign Uncertaintie in the damper and particularly in the ytem are inherent to engineering contruction To improve the reliability and effectivene of the damper, experimental meaurement have to be made to determine the dynamic propertie of one tructure Alternatively the ue of more than one damper ha been propoed (Xu & Igua, 99; Park & Reed, ; Abé & Fujino, 994; Igua & Xu, 994; Jangid, 999; Jangid & Datta, 997; Yamaguchi & Harnpornchai, 993; Kareem & Kline, 995; Gu et al, ; Poovarodom, 3; Yau & Yang, 4) In particular, previou tudie by Abé & Fujino (994), Igua & Xu (994) and Jangid (999), among other, have hown that multiple tuned ma damper (TD) can be more effective and robut than a ingle TD and that, in thi cae, the repone of the main ytem i not much influenced by relatively mall change or error in the value of ytem parameter ued in the TD deign According to Janjid & Datta (997) there i a region around the optimum frequency of the damper where the optimized TD exhibit an almot contant effectivene Some practical retraint, however, mut be oberved in the deign of a TD (Abé & Fujino, 994; Soong & Darguh, 997) The amount of added ma placed on the top tory of a building, the TD excurion travel relative to the floor, the friction between the Paper accepted September 8 Technical Editor: arcelo A Savi liding ma and the bearing urface, the amount of pring force on the building and alo the encloure of pace occupied by the TD are ome of the iue to be addreed in it deign Now the ue of a conception of a ytem of TD can give the deigner more freedom in chooing the propertie and the bet configuration of the multiple damper There are everal propoition for the election of the damper parameter: one can vary the ma or tiffne of the damper (or both) to cover a certain frequency range encompaing the main tuning frequency According to Xu & Igua (99), it i eaier in practice to fix the tiffne of each pring and vary the ma of each damper to cover the deired frequency range To improve the TD effectivene, everal optimization procedure are found in literature (Jangid, 999; Tai & Lin, 993; Hoang & Warnitchai, 5; agluta et al, 3; Carneiro, 4; Li & Qu, 6) According to Jangid (999), the optimal parameter of a TD cannot be obtained by a procedure imilar to that employed by Warburton (98) for a ingle damped TD He determine the optimal parameter by the minimax procedure propoed by Tai & Lin (993) for a ingle damped TD An optimization procedure for a TD wa propoed by Li () for a tructure ubmitted to a bae excitation Hoang & Warnitchai (5) developed a new method to deign TD uing a numerical optimizer In ome practical application the mae of the damper are connected in different way (Soong & Darguh, 997), on the other hand the majority of the invetigation on TD conider no connection between the damper In thi paper the influence of poible connection between the mae of the damper i tudied in detail For thi, four different configuration of a double ma damper are compared In each cae the damper parameter are optimized uing a minimax procedure that conider ma, tuning frequency and damping ratio of each damper a free variable Nomenclature C c C d i D K k K F = damping matrix = TD damping = modal damping = total diplacement = location vector = tiffne matrix = TD tiffne = modal tiffne = external load vector J of the Braz Soc of ech Sci & Eng Copyright 9 by ABC January-arch 9, Vol XXXI, No / 75

2 S Avila and P B Gonçalve f( = modal dynamic excitation force g( = external load acting on the tructure = ma matrix m = TD ma = modal ma p( = interaction force between the TD and the tructure q i = relative diplacement R d = magnitude of the firt element of the complex-frequency repone tranfer matrix y = diplacement vector y i ( = the abolute lateral diplacement of the i=th floor where the TD i intalled relative to the building bae Y(ω) = tranfer function matrix z( = relative diplacement Greek Symbol α = natural frequency ratio β = forced frequency ratio φ = firt mode hape ω = tructure natural frequency ω e = excitation force frequency ω TD = TD frequency ξ = damping ratio = ma ratio Sytem Decription and Equation of otion Sytem with TD The equation of motion of a building-td ytem can be expreed in matrix form a y( + C y( + Ky( = F ( + Dp( () m z( + c z( + kz( = m y N ( + g( () where, C and K are, repectively, the ma, damping and tiffne matrixe of the NDOF tructural ytem while m, c and k are the ma damper parameter; F( and g( are the external load acting on the tructure and TD, repectively; p( = cż (+kz(; y i ( i the abolute lateral diplacement of the i-th floor of the building relative to it bae; z( i the relative diplacement of the TD with repect to the floor where it i intalled; D i a localization vector whoe component d j are given by, j k d j =, j = k where k i the floor where the damper i intalled Let u conider a tall building where the natural frequencie are well paced o that the tructure ocillate around a predominant mode In thi cae, repone vector of the tructure can be approximately repreented by a ingle coordinate y N and a mode hape y = φ y N (4) Subtituting Eq (4) into Eq () and pre-multiplying Eq () by φ T, one obtain the following reduced ytem T + C y N + K y N = c z+ kz φ f ( ) (5) y N + t (3) where φ T = φ i the modal ma; C = ξω, K = ω and f( i the modal dynamic excitation force Here, ξ and ω are repectively the damping ratio and natural frequency of the tructure Sytem with TD Conider now the implified SDOF ytem with n tuned ma damper with different dynamic characteritic attached to the top floor, a hown in Fig The main ytem i characterized by K, and C, while the j-th TD i characterized by a ma m j, damping c j and tiffne k j So, the (n+) equation of the compoite ytem are given by y( + C y( + Ky( = F ( (6) where, C, and K are the ma, damping, and tiffne tructural matrice, repectively; y( i an (n+) vector which repreent the main ytem and mae diplacement relative to the main ytem; F( = [f (, f (,, f n (] T i the external excitation For analyi, the frequency domain approach will be adopted ince the dynamic behavior of the tructure can often be decribed more imply by a tranfer function in the frequency domain, and the excitation, uch a wind load, are often modeled a tochatic procee characterized by their pectral denity function in the frequency domain Adopting thi approach, f i ( = f i e -iωt and y( = Y(ω)e -iωt Subtituting thee expreion in Eq (6), one obtain Y(ω) = (ω -iωc+k) - F (7) C K m, k, c m, k, c m, k, c n n n Figure Structural model: multiple tuned ma damper (TD) attached to the main tructure R d i defined in thi work a the magnitude of the firt element of the complex-frequency repone tranfer matrix (ω -iωc+k) - in Eq (7) The ij element of thi matrix repreent the permanent repone of coordinate i due to a harmonic load applied to coordinate j The variation of the abolute value of the matrix firt element i oberved becaue it repreent the permanent repone of the main ytem due to a harmonic load applied to it To demontrate the effectivene of a TD, conider a tall building modeled a a SDOF ytem with modal parameter, relative to the firt lateral bending mode, = 8 x 4 t; K = 8 x 4 kn/m; ξ S = and ω S = rad/ Conider alo a TD with m = 36 x t; k = 33 x kn/m; ξ = 36; ω TD = 97 rad/ and α = 98 Thee dynamic characteritic are related to the 74 m tall Citicorp Center in New York (Soong & Darguh, 997)The wind load i approximately decribed by the periodic force: 76 / Vol XXXI, No, January-arch 9 ABC

3 Optimal Configuration of Compoite ultiple a Damper in F( = P(3enω t + 7enω t + 5en3ω t + 4en4ω (8) e e where P i the magnitude of the excitation and ω e i the excitation frequency Figure (a) how the time repone of the building diplacement with and without a TD for P = 4 kn and ω e = rad/ Thi how clearly the advantage of uing a TD in a practical cae If the frequency of the periodic force in Eq (8) i increaed to ω e = 35 rad/, lightly higher than the lowet natural frequency of the building, one can oberve in Fig (b) that the control ytem looe it efficiency and the amplitude of teady-tate repone of the controlled ytem i much higher than the uncontrolled one In fact, a hown in Fig 3, where the magnitude of R d i given a a function of the frequency parameter β = ω e / ω, the repone of the building attain it maximum value at ω e = 35 rad/ Thi confirm the limitation of a ingle damper when the excitation frequency i lightly different of that conidered in deign e e Rd TD TD 5 5 Figure 3 Frequency repone of the tructure β Diplacement (m) Diplacement (m) t() without control (a) TD t(),,, 3,,,,,, -, -, -, -, without control (b) TD Figure Controlled and uncontrolled time repone of the main ytem varying excitation frequency: (a) ω e = rad/; (b) ω e = 35 rad/ To tudy the efficiency of the TD concept, let u conider a TD with two equal mae (m = m = 8, each half of that of the original TD, with the following dynamic characteritic: ω = 9 rad/; ω = 9 rad/; ξ = ; ξ = 5 The repone of the tructure in the frequency domain with a ingle and two independent damper i hown in Fig 3 It i clear that the TD i le enitive to variation in the excitation frequency and i more efficient in the vicinity of the natural frequency of the tructure The efficiency of the TD can be even better by optimizing it parameter and/or by connecting the two mae in different way a hown in Fig 4 It i alo hown in Fig 4, for each of the four poible configuration tudied in thi paper, the total and relative diplacement of each ma, repectively d i, and q i The normalized ma, tiffne and damping matrice in Eq (6) for each configuration are: Configuration + + = ξ ω C = ξ α ω ξ α ω ω K = ω α (9) ωα Configuration + + = + ξ ω C = + + ξ α ω ξ α ω ω K = ω α () ωα J of the Braz Soc of ech Sci & Eng Copyright 9 by ABC January-arch 9, Vol XXXI, No / 77

4 S Avila and P B Gonçalve q q Configuration 3 k c k c m m q p + = + K, C q = d p p q = d - d p q = d - d p ξ ω + ξ α ω C = ξ α ω ξ α ω ξ α ω ξ α ω k c Configuration k c m q ω + ωα ωα K = ω α () ωα ωα m q Configuration 4 K, C q = d p p q = d - d p q = d - d Configuration q p + = + ξω + 4ξ α ω C = ξ α ω 4ξ α ω ξ α ω ξ αω + ξ α ω ξ α ω 4ξ α ω ξ α ω 4ξ α ω q q k c k c ω + ωα ωα ωα K = ω α ω α + ω α ω α () ωα ωα ωα K, C m m q = d p p q = d - d p q = d - d - d p q p where ω and ξ are, repectively, the natural frequency and damping ratio of the main ytem, ω α = (3) ω = ω (4) α ω Configuration 3 where ω and ω are the frequencie of the two mae of the damper, and k c k c q q k 3 c 3 m = (5) m m q p m = (6) K, C q = d p p q = d - d p q = d - d - d p The minimax procedure a employed by Tai & Lin (993) conit in earching numerically for the deign parameter that yield the lowet peak of R d The preent work i baed on thi procedure, but here the maximum value of the repone i obtained from the following equation Configuration 4 Figure 4 Configuration of the compoite damper analyzed here R d = β (7) 78 / Vol XXXI, No, January-arch 9 ABC

5 Optimal Configuration of Compoite ultiple a Damper in R d β < (8) which define the maximum of R d Here β i the ratio of frequency ω to the excitation frequency Since Eq (7) i highly non-linear, the Newton-Raphon method i employed to obtain the ytem parameter Numerical Example Conider the ten-tory building analyzed previouly by Villaverde & Koyama (993) The modal characteritic of the reduced SDOF ytem aociated with the fundamental mode are: = 589 t; K = 594 x 3 kn/m and C = 748 kn/m Initially the ratio of the damper ma to the tructural ma i taken a: = 6 and = 4 and the frequency ratio α e α and damping ratio ξ and ξ are obtained by the minimax procedure The optimal parameter for each configuration are preented in Tab The correponding optimum R d value are compared with that obtained for a ingle TD with an equivalent total ma = 3, uing Den Hartog procedure (α = 96; ξ = 4) and with the optimal value for a TD deigned according to Jangid equation (Jangid, 999), alo in Tab The optimization procedure a implemented here lead to more efficient damper, with configuration and 3 leading to the better reult The harmonic repone of the main ocillator i plotted a a function of the frequency parameter β in Fig 5 Rd Table TD optimum parameter and Rd maximum value α α ξ ξ R d máx Config Config Config Config Jangid (999) 39 TD 49 5 TD Conf 4 Conf Conf and β the double damper (d max and d max ) for each configuration The bet configuration in term of travel pace i configuration 3, but configuration could lead to le encloure pace aximum Dipacement (mm) 3,5,5,5 3 4 d máx,4499,453,478,856 d máx,99,94,33,59 Configuration Figure 6 aximum diplacement for each ma of the damper obtained from the time repone In the previou example the ma ratio i were kept contant in the optimization procedure Now the procedure i repeated conidering alo the parameter and a deign variable The optimum parameter are hown i Tab together with the correponding minimum value for the maximum magnification factor R d In each cae the total ma ratio i kept contant and equal to = Again configuration and 3 are more efficient In thi optimization proce the repone of configuration i independent from the ratio i The incluion of the ma ratio in the optimization proce lead to even better reult for R d máx Table Optimum parameter for = α α ξ ξ R d máx Config Config Config Config Again, the equation of motion of each optimized ytem conidering F( = 3 in (ω N with ω = 374 rad/ (β = ) were integrated numerically The maximum diplacement for each ma of the damper are preented in Fig 7 The repone for the ma m i practically the ame, independent of the type of connection between the mae For m a configuration and 4 lead to the bet reult, but i le efficient in term of the main ytem reult, while configuration and 3 are more effective in reducing the diplacement of the main ytem, but exhibit higher diplacement relative to the main ytem Figure 5 Frequency repone for harmonic load Two important iue in a TD deign are it travel relative to the floor and the encloure of pace occupied by the TD To undertand the influence of the damper configuration on thee iue, the repone of each damper i computed conidering a harmonic force F( = 3 in (ω N with ω = 8566 rad/ (β = 9) Figure 6 how the maximum diplacement for each ma of J of the Braz Soc of ech Sci & Eng Copyright 9 by ABC January-arch 9, Vol XXXI, No / 79

6 S Avila and P B Gonçalve aximum Dipacement (mm) d máx d máx Configuration Figure 7 aximum diplacement Rd máx / Configuration 5 Now conider a TD with total ma ratio = 5 The optimized parameter are preented in Tab 3 Figure 8 how the frequency repone for thi cae Here for the ma ratio i more evident than the reult for a TD, which are le enitive and more effective than the reult obtained for a ingle TD Rd máx 5 Table 3 Optimum parameter for = 5 α α ξ ξ R d máx Config Config Config Config / Configuration 5 Rd 9 TD 8 7 Conf 4 6 Conf Conf and Figure 8 Frequency repone for harmonic excitation with paive element ( = 5) The influence of the ratio between the ma of the two damper / on the value of R d máx i hown in Fig 9 for a total ma = 5 To obtain the optimal value for R d, for configuration, the bet option i to adopt = ; for configuration and 3 m mut be larger than m, while for configuration 4 one ma mut be approximately 5% higher than the other β Rd máx Rd máx / Configuration / Configuration 4 Figure 9 Influence of the damper ma ratio ( / ) on the minimum value of the maximum magnification factor R d max 8 / Vol XXXI, No, January-arch 9 ABC

7 Optimal Configuration of Compoite ultiple a Damper in The ue of multiple tuned ma damper to control the global repone of a tructure ha been invetigated in thi paper In the preent tudy four different configuration of a double damper ytem are tudied and their optimal propertie are numerically obtained for minimum diplacement of the main tructure The minimax procedure a implemented here conider a deign variable the ma ratio, the damping ratio and tuning frequencie of the damper The performance of each control ytem i aeed by detailed parametric tudie Thee reult how that, independent from the damper configuration, TD can enhance the effectivene of the control ytem when compared with the performance of a ingle optimized TD Reult alo how that the ytem repone i enitive to mall variation in the ma, tiffne and damping parameter The conideration of all thee parameter a deign variable lead to the mallet peak poible of the repone in the frequency domain The connection between the mae of the damper ha a meaurable influence on the performance of the compoite damper giving the deigner a certain flexibility in important deign iue, uch a the amount of added ma placed on the top tory of a building, TD travel relative to the floor, the amount of pring force on the building and encloure of pace occupied by the TD Acknowledgment The reearch work in thi paper wa partially upported by CAPES, the human reource formation agency of the Brazilian initry of Education Thi upport i gratefully acknowledged Reference Abé, and Fujino, Y, 994, Dynamic characterization of multiple tuned ma damper and ome deign formula, Earthquake Engineering and Structural Dynamic, Vol3, pp Carneiro, RB, 4, Vibration control of high building uing multiple tuned ma damper (TD) (in Portuguee), Sc Thei, Univerity of Brailia, Brailia Den Hartog, JP, 956, echanical Vibration, cgraw-hill, New York Gu,, Chen, R and Chang, CC,, Parametric tudy on multiple tuned ma damper for buffeting control of Yangpu bridge, Journal of Wind Engineering and Indutrial Aerodynamic, Vol 89, pp 987- Hoang N and Warnitchai P, 5, Deign of multiple tuned ma damper by uing a numerical optimizer, Earthquake Engineering and Structural Dynamic, Vol 34, No, pp 5-44 Holme, JD, 995, Liting of intallation, Engineering Structure, Vol 7, No 9, pp Igua, T and Xu, K, 994, Vibration control uing multiple tuned ma damper, Journal of Sound and Vibration, Vol 75, No 4, pp Jangid, R S and Datta, TK, 997, Performance of multiple tuned ma damper for torionally coupled ytem, Earthquake Engineering and Structural Dynamic, Vol 6, pp Jangid, R S, 999, Optimum multiple tuned ma damper for baedexcited undamped ytem, Earthquake Engineering and Structural Dynamic, Vol 8, pp 4-49 Kareem, A and Kline, S, 995, Performance of multiple ma damper under random loading, Journal of Structural Engineering, Vol, No, pp Li, C,, Performance of multiple tuned ma damper for attenuating undeirable ocillation of tructure under the ground acceleration, Earthquake Engineering and Structural Dynamic, Vol 9, pp 45-4 Li, C and Qu, W, 6, Optimum propertie of multiple tuned ma damper for reduction of tranlational and torional repone of tructure ubject to ground acceleration, Engineering Structure, Vol 8, pp agluta, C Ainworth Jr, GO and Roitman, N, 3, Comparion between multiple vibration aborber and ingle aborber ytem In: Proc of COBE XVII Int Cong ech Eng, São Paulo Park, J and Reed, D,, Analyi of uniformly and linearly ditributed ma damper under harmonic and earthquake excitation, Engineering Structure, Vol 3, pp 8-84 Peteren, NR, 98, Deign of large cale TD In: Proceeding of Structural Control, North Holland, Leipholz, HHE ed, pp Poovarodom, N, Kanchanoot, S and Warnitchai, P, 3, Application of non-linear multiple tuned ma damper to uppre maninduced vibration of a pedetrian bridge, Earthquake Engineering and Structural Dynamic, Vol 3, pp 7-3 Soong TT and Darguh SF, 997, Paive Energy Diipation Sytem in Structural Engineering, John Wiley & Son, Chicheter Tai, H and Lin, G, 993, Optimum tuned-ma damper for minimizing teady-tate repone of upport-excited and damped ytem, Earthquake Engineering and Structural Dynamic, Vol, pp Villaverde, R and Koyama, LA, 993, Damped reonant appendage to increae inherent damping in building, Earthquake Engineering and Structural Dynamic, Vol, pp Warburton, GB, 98, Optimum aborber parameter for variou combination of repone and excitation parameter, Earthquake Engineering and Structural Dynamic, Vol, pp 38-4 Xu, K and Igua, T, 99, Dynamic characteritic of multiple ubtructure with cloely paced frequencie, Earthquake Engineering and Structural Dynamic, Vol, pp 59-7 Yamaguchi, H and Harnpornchai, N, 993, Fundamental characteritic of multiple tuned ma damper for upreing harmonically forced ocillation, Earthquake Engineering and Structural Dynamic, Vol, pp 5-6 Yau, J and Yang Y, 4, A Wideband TD Sytem for Reducing the Dynamic Repone of Continuou Tru Bridge to oving Train Load, Engineering Structure, Vol 6, pp J of the Braz Soc of ech Sci & Eng Copyright 9 by ABC January-arch 9, Vol XXXI, No / 8

8 S Avila and P B Gonçalve 8 / Vol XXXI, No, January-arch 9 ABC