Optimum properties of multiple tuned mass dampers for reduction of translational and torsional response of structures subject to ground acceleration

Engineering Structures 28 (2006) 472–494 www.elsevier.com/locate/engstruct

Optimum properties of multiple tuned mass dampers for reduction of translational and torsional response of structures subject to ground acceleration Chunxiang Li a,∗ , Weilian Qu b a Department of Civil Engineering, Shanghai University, No. 149 Yanchang Road, Shanghai 200072, PR China b Institute of Civil Engineering and Architecture, Wuhan University of Technology, Wuhan 430070, PR China

Received 6 April 2004; received in revised form 17 August 2005; accepted 1 September 2005 Available online 21 October 2005

Abstract The application of multiple tuned mass dampers (MTMD) with identical stiffness and damping coefficient but different mass for suppressing translational and torsional responses is discussed for a simplified two-degree-of-freedom (2DOF) structure, able to represent the dynamic characteristics of general asymmetric structures subject to ground motions. This 2DOF structure is a generalized 2DOF system of an asymmetric structure with predominant translational and torsional responses under earthquake excitations using the mode reduced-order method. Depending on the ratio of the torsional to the translational eigenfrequency, i.e. the torsional to translational frequency ratio (TTFR), of asymmetric structures, the following cases can be distinguished: (1) torsionally flexible structures (TTFR < 1.0), (2) torsionally intermediate stiff structures (TTFR = 1.0), and (3) torsionally stiff structures (TTFR > 1.0). Taking into account the even placement of the MTMD within the width of the asymmetric structure, a careful examination of the effects of the normalized eccentricity ratio (NER) on the performance of the MTMD are carried out with resort to the provided analytical expressions for the dynamic magnification factors (DMF) of both the translational and torsional responses of the asymmetric structure. Extensive numerical simulations have been performed to accurately estimate the dynamic characteristics of the MTMD for asymmetric structures subject to ground acceleration. In the simulations, the dimensionless DMF parameters, bounded between zero and unity, are used as the formal indexes estimating the effectiveness of the MTMD in reducing both the translational and torsional responses of the asymmetric structure. A new basic result is that the NER affects significantly the performance of the MTMD for both torsionally flexible and torsionally intermediate stiff structures; while the influence of the NER is rather negligible on the performance of the MTMD for torsionally stiff structures, thus implying that in such a case the MTMD may be designed by ignoring the effects of torsional coupling. Likewise, the effectiveness and robustness of the MTMD strategies with different layouts are also investigated and demonstrated for the case of mitigating the torsional response of asymmetric structures, thus providing valuable guidance for the MTMD design. Furthermore, the frequency response curves of asymmetric structures without and with both the optimum MTMD and TMD are plotted for the three cases of TTFR as well, consequently obtaining some very useful results. c 2005 Elsevier Ltd. All rights reserved.  Keywords: Vibration control; Damping; Multiple tuned mass dampers (MTMD); Asymmetric structures; Ground acceleration; Torsional to translational frequency ratio (TTFR); Normalized eccentricity ratio (NER); Frequency response curves

1. Introduction In recent years, mitigating the responses of civil engineering structures to environmental loads such as earthquakes and wind loads has drawn the interest of many researchers. Many ∗ Corresponding author. Tel.: +86 21 6282 0010; fax: +86 21 6294 4002.

E-mail addresses: [email protected], [email protected] (C. Li). c 2005 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter  doi:10.1016/j.engstruct.2005.09.003

control devices, passive, semi-active, as well as active, have been developed. Among the available devices, the tuned mass damper (TMD) is one of the simplest and the most reliable control devices, which consists of a mass, a spring, and a viscous damper attached to the structure. Its mechanism of attenuating undesirable vibration of a structure is to transfer the vibration energy of the structure to the TMD and to dissipate the energy through the damping of the TMD. In order to enlarge the dissipation energy in the TMD, it is

C. Li, W. Qu / Engineering Structures 28 (2006) 472–494

Nomenclature ADMF acceleration dynamic magnification factors of structures with the MTMD AMTMD active multiple tuned mass dampers b/r normalized width of asymmetric structures, set in the present paper equal to 1.0 CM center of mass of asymmetric structures CR center of resistance of asymmetric structures cj damping coefficient of the j th TMD in the MTMD mode-generalized damping coefficient of asymcs metric structures constant damping coefficient of the MTMD cT DDMF displacement dynamic magnification factors of structures with the MTMD DMF dynamic magnification factors of structures with the MTMD DMF(I) translational displacement dynamic magnification factors of asymmetric structures with the MTMD DMF(II) torsional displacement dynamic magnification factors of asymmetric structures with the MTMD DMF∗ (I) translational displacement dynamic magnification factors of asymmetric structures without the MTMD DMF∗ (II) torsional displacement dynamic magnification factors of asymmetric structures without the MTMD DOF degree of freedom ratio of the eccentricity to the radius of gyration ER of the deck, referred in this paper to as the normalized eccentricity ratio (NER) ey eccentricity between the CR and CM of asymmetric structures f tuning frequency ratio of the MTMD j number of dampers (TMD number in the MTMD) kj spring stiffness of the j th TMD in the MTMD mode-generalized lateral stiffness of asymmetric ks structures in the translational (x) direction constant spring stiffness of the MTMD kT mode-generalized torsional stiffness of asymmetkθ ric structures with respect to the CM MTMD multiple tuned mass dampers Min.Min.Max.ADMF minimization of the minimum values of the maximum acceleration dynamic magnification factors Min.Min.Max.DDMF minimization of the minimum values of the maximum displacement dynamic magnification factors mj mass of the j th TMD in the MTMD mode-generalized mass of asymmetric structures ms n total number of dampers (TMD total number in the MTMD)

RI

473

minimization of the minimum values of the maximum translational displacement dynamic magnification factors of asymmetric structures with the MTMD minimization of the minimum values of the maxRII imum torsional displacement dynamic magnification factors of asymmetric structures with the MTMD minimization of the minimum values of the RIII maximum translational displacement dynamic magnification factors, nondimensionalized by the maximum translational displacement dynamic magnification factors of asymmetric structures without the MTMD minimization of the minimum values of the maxRIV imum torsional displacement dynamic magnification factors, nondimensionalised by the maximum torsional displacement dynamic magnification factors of asymmetric structures without the MTMD r radius of gyration of the deck about the vertical axis through the CM ratio of the natural frequency of the j th TMD to rj the uncoupled translational natural frequency of asymmetric structures T Fxs (−i ω) transfer function for the translational displacement of asymmetric structures with the MTMD T Fθs (−i ω) transfer function for the torsional displacement of asymmetric structures with the MTMD TMD tuned mass damper ground acceleration x¨g (t) xj translational displacement of every TMD with reference to the ground translational displacement of asymmetric strucxs tures with respect to the ground y(n+1)/2 center of the MTMD, i.e. placement of the (n + 1)/2th TMD in the MTMD θs torsional displacement of asymmetric structures α adjustable nondimensional parameter (0 ≤ α ≤ 0.5) β frequency spacing of the MTMD λ ratio of the external excitation frequency to the uncoupled translational natural frequency corresponding to the vibration mode being controlled, which is set within the range from 0.4 to 3.4 uncoupled torsional to translational frequency λω ratio (TTFR) ξj damping ratio of the j th TMD in the MTMD structural damping ratio, which is set in this study ξs equal to 0.02 average damping ratio of the MTMD (while in the ξT introduction denoting the constant damping ratio of the MTMD)

474

µj µT ω ωj ωs ωs1 ωs2 ωθ ωT

C. Li, W. Qu / Engineering Structures 28 (2006) 472–494

mass ratio of the j th TMD in the MTMD total mass ratio of the MTMD external excitation frequency natural frequency of the j th TMD in the MTMD uncoupled translational natural frequency of asymmetric structures coupled fundamental natural frequency of asymmetric structures coupled second natural frequency of asymmetric structures uncoupled torsional natural frequency of asymmetric structures average natural frequency of the MTMD

very important to determine the optimum parameters (i.e. the optimum tuning frequency ratio and optimum damping ratio) of the TMD. The determination of the optimum parameters of the TMD for an undamped structure subjected to harmonic external excitation over a broad band of excitation frequencies is described in Brock [1] and Den Hartog [2]. On the basis of Den Hartog’s method, Warburton and Ayorinde [3] have obtained the optimum parameters of the TMD for the undamped structure under harmonic support excitation, where the acceleration amplitude is set to be constant for all input frequencies and other kinds of harmonic excitation. The explicit formulae for the optimum parameters and the effectiveness of a TMD to control structural oscillations caused by different types of external excitations is now well established [4–11]. The applications of the TMD for tall buildings and long-span bridges under wind loads have been extensively investigated [12–15]. While in reducing seismic responses of structures, Sadek et al. [16] extended the work of Villaverde [17] to find the optimum parameters of a TMD by making the modal damping ratio of the first two modes of vibration equal for the reduction of seismic responses of structures. The parametric study and simplified design of a TMD were also made by Rana and Soong [18] using steadystate harmonic excitation and time-history analysis. In addition, a study was also carried out to investigate the possibility of controlling multiple structural modes with the multi-tuned mass dampers through tuning each damper to the corresponding mode to be controlled. Taking into account the possibility of damage to the structure during high intensity earthquakes, Soto-Brito and Ruiz [19] have studied the influence of ground motion intensity on the effectiveness of the TMD. The response of a 22-storey four-bay reinforced concrete nonlinear frame with a TMD has been investigated under moderate and high intensities of the 1985 Mexico City (SCT) ground motions. The results show that the effectiveness of the TMD is higher for structures with small nonlinearity produced by small and moderate earthquakes, than for structures with high nonlinear behavior, generally associated with high-intensity motions. Estimating the effectiveness of the TMD using peak response reduction of the nonlinear structures alone has been further found to be insufficient [20]. Evidently, in such circumstances, the TMD is expected to be capable of effectively reducing not

only the peak response of the structure but also the induced damage of the structure. Recently, Pinkaew et al. [21] have used the damage reduction as an indicator to evaluate seismic effectiveness of the TMD for inelastic structures. The numerical simulations have been performed on the equivalent inelastic single-degree-of-freedom (SDOF) structure able to represent the 20-storey reinforced concrete structure, subjected to both the harmonic and 1985 Mexico City (SCT) ground motions. The results show that although the TMD cannot reduce the peak displacement of the controlled structure after yielding, it can significantly reduce damage to the structure. Likewise, from the application of the TMD, certain degrees of damage protection and collapse prevention can be gained as well. It is well known that however, the main disadvantage of a single TMD is its sensitivity of the effectiveness to the fluctuation in the tuning of the natural frequency of the TMD to the natural frequency of the structure and/or that in the damping ratio of the TMD [22]. The effectiveness of a single TMD is decreased significantly by the off-tuning or the offoptimum damping in the TMD. That is, a single TMD is not robust at all. Furthermore, the dynamic characteristics of structures will change under strong earthquakes due to a degradation of the structure stiffness. This change will degrade the performance of a single TMD considerably owing to the offset in the tuning of the frequency and/or in the damping ratio. As a result, the utilization of more than one tuned mass damper with different dynamic characteristics has been proposed in order to improve the effectiveness and robustness of a single TMD. Iwanami and Seto [22] proposed dual tuned mass dampers (2TMD) and conducted a research on the optimum design of 2TMD for harmonically forced vibration of the structure. It was shown in their papers that 2TMD are more effective than a single TMD. However, the effectiveness was not significantly improved. Recently, multiple tuned mass dampers (MTMD) with distributed natural frequencies were proposed by Igusa and Xu [23,24]. They derived a simple formula of equivalent additional damping and an integral form for the impedance based on an asymptotic analysis technique [23,24]. The multiple tuned mass dampers (MTMD) with the distributed natural frequencies were also studied by, for example, Yamaguchi and Harnpornchai [25], Abe and Fujino [26], Kareem and Kline [27], Jangid [28], Li [29], Park and Reed [30], Gu et al. [31], Chen and Wu [32], Yau and Yang [33,34], Kwon and Park [35], Lin et al. [36], and Hoang and Warnitchai [37]. The MTMD is shown to be more effective in mitigating the oscillations of structures with respect to a single TMD. These research findings have also confirmed the merit of the MTMD in seismic applications. In terms of installation the merit of the MTMD with respect to a single TMD is that the MTMD consists of distributed dampers with small mass and generally does not require any devoted space to install them. Engineers can then make full use of the spare space at different floors of the buildings and thus design them in a cost-effective way. Here, it is worth pointing out what we would really see in a practical situation is probably threedimensional (3D) frames a few stories high. That would make the special distribution of the MTMD much harder. However, in

C. Li, W. Qu / Engineering Structures 28 (2006) 472–494

such a case, the MTMD with the total number equal to three or five generally is required to be placed on the top floor. Another advantage is that the malfunction of an individual damper, due to its light weight, will not cause detrimental effects on the structural responses so that the MTMD strategy is very robust. Recently, based on the various combinations of the stiffness, mass, damping coefficient, and damping ratio in the MTMD, five MTMD models have been presented by Li [38]. Through implementing the minimization of the minimum values of the maximum displacement dynamic magnification factors (i.e. Min.Min.Max.DDMF) and the minimization of the minimum values of the maximum acceleration dynamic magnification factors (i.e. Min.Min.Max.ADMF), it has been shown that the MTMD with the identical stiffness (k T 1 = k T 2 = · · · = k T n = k T ) and damping coefficient (cT 1 = cT 2 = · · · = cT n = cT ) but unequal mass (m T 1 = m T 2 = · · · = m T n ) and damping ratio (ξT 1 = ξT 2 = ··· = ξT n ) can provide better effectiveness and wider optimum frequency spacing (i.e. higher robustness against the change or the estimation error in the structural natural frequency) with respect to the rest of the MTMD models [38]. Likewise, the studies by Li and Liu [39] have disclosed further trends of both the optimum parameters and effectiveness and further provided suggestion on selecting the total mass ratio and total number of the MTMD with the identical stiffness and damping coefficient but unequal mass and damping ratio. More recently, in terms of the uniform distribution of system parameters, instead of the uniform distribution of natural frequencies, eight new MTMD models have been proposed to seek for the MTMD models without the near-zero optimum average damping ratio. Six MTMD models without the near-zero optimum average damping ratio have been found. The optimum MTMD with the identical damping coefficient (cT 1 = cT 2 = · · · = cT n = cT ) and damping ratio (ξT 1 = ξT 2 = · · · = ξT n = ξT ) but unequal stiffness (k T 1 = k T 2 = · · · = k T n ) and with the uniform distribution of masses has been found able to render better effectiveness and wider optimum frequency spacing with respect to the rest of the MTMD models [40]. Likewise it is interesting to know that the two abovementioned MTMD models can approximately reach the same effectiveness and robustness [40]. Evidently, much progress has been extended in recent years in terms of the studies on the MTMD for mitigating oscillations of structures. However, in most studies on both the TMD and MTMD, it is assumed that a structure vibrates in only one direction or in multiple directions independently with its fundamental modal properties to design the TMD or the MTMD. This assumption simplifies the analysis of a system and the synthesis of a controller. In real structures, however, this assumption is not always appropriate because structures generally possess multidirectional coupled vibration modes and the control performance of controllers will degrade due to parameter variation or spillover induced by the effect of coupling. Furthermore, there exist not only transverse vibrations but also torsional vibrations in real structures and they generally possess coupling; that is, a real structure is actually asymmetric to some degree even

475

with a nominally symmetric plan and will undergo lateral as well as torsional vibrations simultaneously under purely translational excitations. Consequently, the controllers have to be designed through taking into account the effect of transverse–torsional coupled vibration modes in such cases. Examination of the TMD and MTMD for structures which possess transverse–torsional coupled vibration modes has already been recently performed by, for example, Jangid and Datta [41], Lin et al. [42], Lin et al. [43], Pansare and Jangid [44], Arfiadi and Hadi [45], Singh et al. [46], Ahlawat and Ramaswamy [47], and Wang and Lin [48]. Notwithstanding this, rather limited work has been reported on the investigations of the MTMD for suppressing both the translational and torsional responses of asymmetric structures. To date, only Refs. [41,44,48] may be found in terms of examining the performance of the MTMD for asymmetric structures. With a view to practical applications of the MTMD, it is imperative and of practical interest to include the effects of torsional coupling into consideration in estimating the performance of the MTMD. It is well known that structures characterized by non-coincident center of mass and center of stiffness will develop a coupled lateral–torsional response when subjected to earthquake ground motions. For conceptual design purposes, the idealized two-degree-of-freedom (2DOF) system can be considered as a generalized 2DOF system of an asymmetric structure with predominant translational and torsional response under earthquake excitations. In such a case, rather than using the real mass, the generalized translational and torsional masses will be introduced in all derivations below. A careful examination of the governing equations of motion of such a generalized 2DOF system with the MTMD will shed insight into the effects of the coupled lateral–torsional dynamic behavior of asymmetric structures on the optimum performance of the MTMD. Consequently, the present work on the MTMD has mainly attempted to control only the translational and torsional responses of such a generalized 2DOF system; notwithstanding this, this concept can easily be extended to multistory asymmetric structures using the mode reduced-order method. Likewise, the effectiveness and robustness of the MTMD strategies with different layouts are also investigated and demonstrated for the case of mitigating the torsional response of asymmetric structures, thus providing valuable guidance for the MTMD design. Furthermore, the frequency response curves of asymmetric structures without and with both the optimum MTMD and TMD will be plotted for three cases of the uncoupled torsional to translational frequency ratio (TTFR) as well, consequently obtaining some very useful results. 2. Estimating damping of asymmetric structures The structure to be controlled with an MTMD is the asymmetric structure [i.e. the center of resistance (CR) of the structure does not coincide with the center of mass (CM)], as shown in Fig. 1. A simplified two-degree-of-freedom (2DOF) structure, able to represent the dynamic characteristics of general asymmetric structures subject to ground motions, is

476

C. Li, W. Qu / Engineering Structures 28 (2006) 472–494

taken into consideration in this paper. This 2DOF structure is a generalized 2DOF system of an asymmetric structure with predominant translational and torsional responses under earthquake excitations using the mode reduced-order method. Likewise the roof diaphragm/slab is considered to be infinitely stiff in its own plane (i.e. rigid diaphragm assumption); the two degrees of freedom are attached to the center of mass of the system. The two uncoupled frequencies of the structure are then defined as below:
ωs = ks /m s ; (1a)  (1b) ωθ = kθ /m s r 2 in which m s is the mode-generalized mass of the structure; ks = ks1 + ks2 is the mode-generalized lateral stiffness of the structure in the x direction, where ks1 and ks2 are the stiffness 2 + k y2 of two resisting elements, respectively; kθ = ks1 ys1 s2 s2 is the mode-generalized torsional stiffness of the structure with respect to the CM, where ys1 and ys2 are the distances from the CM to the two resisting elements, respectively; r represents the radius of gyration of the deck about the vertical axis through the CM. The equations of motion of the asymmetric structure can be written in the matrix form       ms 0 x¨s cs csθ x˙s + csθ cθ θ˙s 0 m s r 2 θ¨s      m ks ks e y x s = − s x¨g (t) + (2) ks e y kθ θs 0 in which cs , csθ , and cθ denote the elements of the damping matrix to be determined next; e y = (ks1 ys1 −ks2 ys2 )/(ks1 +ks2) represents the eccentricity between the CR and CM; and x¨g (t) is the ground acceleration. Denoting the fundamental and second natural frequencies with ωs1 and ωs2 , respectively, they can be derived through solving the eigenvalue problem associated with (2) as follows:     1 + λ2 ∓ (λ2 − 1)2 + 4E 2  ω ω R ωs1,s2 = ωs 2 (3) (ωs2 > ωs1 ) in which E R = e y /r represents the ratio of the eccentricity to the radius of gyration of the deck, referred to as the normalized eccentricity ratio (NER); λω = ωθ /ωs denotes the uncoupled torsional to translational frequency ratio (TTFR). With the hypothesis of the same damping ratio ξs1 = ξs2 = ξs (letting ξs = 0.02 in this study) for the two modes and superposing the modal damping matrices, the damping matrix can be expressed in the following form     2(ξs2ωs1 − ξs1 ωs2 ) 0 ms cs csθ = ωs1 ωs2 2 − ω2 csθ cθ 0 msr 2 ωs1 s2   2(ξs1 ωs1 − ξs2 ωs2 ) ks ks e y + (4) ks e y kθ ω2 − ω2 s1

s2

Fig. 1. Generalized two-degree-of-freedom (2DOF) system of an asymmetric structure with predominant translational and torsional response set with the multiple tuned mass dampers (MTMD).

Rearranging the above Eq. (4) yields every element of the damping matrix as shown below cs /m s = 2as ξs ωs ;

(5a)

as = [(ωs1 /ωs )(ωs2 /ωs ) + 1]/[(ωs1/ωs ) + (ωs2 /ωs )]

(5b)

csθ /m s r = 2asθ ξs ωs ;

(6a)

asθ = E R /[(ωs1 /ωs ) + (ωs2 /ωs )]

(6b)

C. Li, W. Qu / Engineering Structures 28 (2006) 472–494

cθ /m s r 2 = 2aθ ξs ωs ; (7a) aθ = [(ωs1 /ωs )(ωs2 /ωs ) + λ2ω ]/[(ωs1 /ωs ) + (ωs2 /ωs )]. (7b) 3. Transfer functions of the asymmetric structure with the multiple tuned mass dampers An MTMD with different dynamic characteristics, which is evenly placed within the width b with the center at the CM and within the width [(1 − 2α)b/2] with the centers, respectively, at y(n+1)/2 = [∓b(1 + 2α)]/4, where α (0 ≤ α ≤ 0.5) is an adjustable nondimensional parameter, is utilized here for reducing both the translational and torsional responses of the asymmetric structure, as shown in Fig. 1(a)–(c). For the three cases above, the ordinate of every TMD in the MTMD can be, respectively, determined by j −1 b ( j = 1, 2, . . . , n) n−1 j − 1 1 − 2α · b ( j = 1, 2, . . . , n) n−1 2 j − 1 1 − 2α y j = αb + · b ( j = 1, 2, . . . , n) n−1 2

b yj = − + 2 b yj = − + 2

n 

(8c)

× [µ j (ω2j − i 2ξ j ω j ω)/(ω2j − ω2 − i 2ξ j ω j ω)] Tr2 (−i ω) = Er2 (ω) + i Fr2 (ω) = E R ωs2 − i 2asθ ξs ωs ω − ω2 × [µ j (ω2j

(15)

n  (y j /r )

− i 2ξ j ω j ω)/(ω2j

j =1

− ω2 − i 2ξ j ω j ω)]

Tr3 (−i ω) = Er3 (ω) + i Fr3 (ω) = 1 n  + [µ j (ω2j − i 2ξ j ω j ω)/(ω2j − ω2 − i 2ξ j ω j ω)]

(16)

(17)

j =1

= ωθ2 − ω2 − i 2aθ ξs ωs ω − ω2 − i 2ξ j ω j ω)/(ω2j

n 

(y j /r )2

j =1

− ω2 − i 2ξ j ω j ω)]

(18)

Tr5 (−i ω) = Er5 (ω) + i Fr5 (ω) = ωs2 − ω2 − i 2as ξs ωs ω − ω2 n  × [µ j (ω2j − i 2ξ j ω j ω)/(ω2j − ω2 − i 2ξ j ω j ω)]

(19)

j =1

with definitions of
ω j = k j /m j ;

(20a)

ξ j = c j /2m j ω j ;

(20b)

µ j = m j /m s

(20c)

where ω j , ξ j , and µ j ( j = 1, 2, . . . , n) are the natural frequency, damping ratio, and mass ratio, respectively, of the j th TMD in the MTMD.

(11)

4. Evaluation criteria of the multiple tuned mass dampers

(12)

Reflecting upon the simplest manufacturing, the MTMD with the identical stiffness and damping coefficient but the unequal mass (for requirement of the uniform distribution of natural frequencies) is used here for attenuating both the translational and torsional responses of asymmetric structures. In practical terms, this MTMD model may render better effectiveness and higher robustness with reference to the rest of the MTMD models for structures without considering the effects of torsional coupling [38]. The MTMD thus is manufactured by keeping the stiffness and damping constant and the mass unequal [i.e. k1 = k2 = · · · = kn = k T , c1 = c2 = · · · = cn = cT (but ξ1 = ξ2 = · · · = ξn ), m 1 = m 2 = · · · = m n ]. nDefining the average natural frequency of the MTMD ωT = j =1 ω j /n, the natural frequency of every TMD in the MTMD can then be calculated from

in which m j , c j , and k j are the mass, damping, and stiffness of the j th TMD in the MTMD, respectively. With the hypothesis of x¨g (t) = X¨ g e−iωt and setting x s = [Hxs (−i ω)] e−iωt , r θs = [Hθs (−i ω)] e−iωt , x j = [Hx j (−i ω)] e−iωt ( j = 1, 2, . . . , n) in Eqs. (9)–(12), the transfer functions for both the translational and torsional displacements of the asymmetric structure with the MTMD can then be expressed in the following form Hxs (−i ω) X¨ g [Tr1 (−i ω)][Tr2 (−i ω)] − [Tr3 (−i ω)][Tr4 (−i ω)] (13) = [Tr4 (−i ω)][Tr5 (−i ω)] − [Tr2 (−i ω)][Tr2 (−i ω)] Hθs (−i ω) T Fθs (−i ω) = X¨ g [Tr2 (−i ω)][Tr3 (−i ω)] − [Tr1 (−i ω)][Tr5 (−i ω)] (14) = [Tr4 (−i ω)][Tr5 (−i ω)] − [Tr2 (−i ω)][Tr2 (−i ω)]

T Fxs (−i ω) =

(y j /r )

(10)

j =1

m j [x¨g (t) + x¨ j ] + c j [x˙ j − (x˙s + y j θ˙s )] + k j × [x j − (x s + y j θs )] = 0 ( j = 1, 2, . . . , n) ˙ F j (t) = c j [x˙ j − (x˙s + y j θs )] + k j [x j − (x s + y j θs )] ( j = 1, 2, . . . , n)

n  j =1

× [µ j (ω2j

(9) y j F j (t)

Tr1 (−i ω) = Er1 (ω) + i Fr1 (ω) =

(8b)

j =1

m s r 2 θ¨s + cθ θ˙s + kθ θs + csθ x˙ s + ks e y x s =

in which

Tr4 (−i ω) = Er4 (ω) + i Fr4 (ω) (8a)

when the relative displacements of both the structure (x s ) and every TMD (x j ) with reference to the ground are introduced, the governing equations of motion for the generalized 2DOF structure–MTMD system under ground acceleration can be formulated as follows: m s [x¨g (t) + x¨s ] + cs x˙s + ks x s + csθ θ˙s + ks e y θs n  = F j (t)

477

ω j = ωT + ωT [ j − (1 + n)/2][β/(n − 1)] ( j = 1, 2, . . . , n)

(21)

478

C. Li, W. Qu / Engineering Structures 28 (2006) 472–494

where the nondimensional parameter β = (ωn − ω1 )/ωT is defined to be the frequency spacing of the MTMD. Making use of (21), the ratio of the natural frequency of every TMD in the MTMD to the controlled natural frequency of the structure can then be written explicitly as follows: r j = ω j /ωs = f + f [ j − (1 + n)/2][β/(n − 1)] ( j = 1, 2, . . . , n)

µT =

µj =

j =1

ξj =

r j ξT f

µ j r 2j

n  1 2 r j =1 j

( j = 1, 2, . . . , n)

( j = 1, 2, . . . , n).

(23) (24)

In view of Eqs. (15)–(19), it is natural to define the frequency ratio λ = ω/ωs with reference to ωs instead of ωs1 and ωs2 . Employing the above transfer functions [(13) and (14)], the DMF(I) and DMF(II) of the asymmetric structure with the MTMD can then be calculated from DMF(I) = |ωs2 [T Fxs (−iλ)]| 1/2  [Er1 (λ)Er2 (λ) − Er3 (λ)Er4 (λ) − Fr1 (λ)Fr2 (λ) + Fr3 (λ)Fr4 (λ)]2  +[Er1 (λ)Fr2 (λ) + Er2 (λ)Fr1 (λ) − Er3 (λ)Fr4 (λ) − Er4 (λ)Fr3 (λ)]2   =  [E (λ)E (λ) − E (λ)E (λ) + F (λ)F (λ) − F (λ)F (λ)]2  r4 r5 r2 r2 r2 r2 r4 r5 2 + [Er4 (λ)Fr5 (λ) + Er5 (λ)Fr4 (λ) − 2Er2 (λ)Fr2 (λ)] DMF(II) = |ωs2 [T Fθs (−iλ)]| 1/2  [Er2 (λ)Er3 (λ) − Er1 (λ)Er5 (λ) − Fr2 (λ)Fr3 (λ) + Fr1 (λ)Fr5 (λ)]2  +[Er2 (λ)Fr3 (λ) + Er3 (λ)Fr2 (λ) − Er1 (λ)Fr5 (λ) − Er5 (λ)Fr1 (λ)]2   =  [E (λ)E (λ) − E (λ)E (λ) + F (λ)F (λ) − F (λ)F (λ)]2  r4 r5 r2 r2 r2 r2 r4 r5 2 + [Er4 (λ)Fr5 (λ) + Er5 (λ)Fr4 (λ) − 2Er2 (λ)Fr2 (λ)]

in which Er1 (λ) =

n  (y j /r )µ j j =1

× [r 2j (r 2j

− λ2 ) + 4ξ 2j r 2j λ2 ]/[(r 2j − λ2 )2 + 4ξ 2j r 2j λ2 ]

Er2 (λ) = E R − λ2

n 

(y j /r )µ j

j =1

× [r 2j (r 2j − λ2 ) + 4ξ 2j r 2j λ2 ]/[(r 2j − λ2 )2 + 4ξ 2j r 2j λ2 ] n  Er3 (λ) = 1 + µj j =1

× [r 2j (r 2j

− λ2 ) + 4ξ 2j r 2j λ2 ]/[(r 2j − λ2 )2 + 4ξ 2j r 2j λ2 ] n  Er4 (λ) = λ2ω − λ2 − λ2 (y j /r )2 µ j j =1

× [r 2j (r 2j

− λ ) + 4ξ 2j r 2j λ2 ]/[(r 2j − λ2 )2 + 4ξ 2j r 2j λ2 ] 2

Er5 (λ) = 1 − λ2 − λ2

n  j =1

µj

Fr1 (λ) =

(25)

(26)

n  j =1

(y j /r )µ j (2ξ j r j λ3 )/[(r 2j − λ2 )2 + 4ξ 2j r 2j λ2 ]

Fr2 (λ) = −2asθ ξs λ − λ2

(22)

in which f = ωT /ωs is defined to be the tuning frequency ratio of the MTMD. Taking into consideration the fact that the damping ratio of every TMD in the MTMD is unequal, it is necessary n to introduce the average damping ratio ξT = j =1 ξ j /n. Employing the above definitions and derived expressions, the total mass ratio and the damping ratio of every TMD in the MTMD can be, respectively, calculated by [29,38] n 

× [r 2j (r 2j − λ2 ) + 4ξ 2j r 2j λ2 ]/[(r 2j − λ2 )2 + 4ξ 2j r 2j λ2 ]

Fr3 (λ) =

n 

µj j =1 × (2ξ j r j λ3 )/[(r 2j − λ2 )2 + 4ξ 2j r 2j λ2 ] n  µ j (2ξ j r j λ3 )/[(r 2j − λ2 )2 + 4ξ 2j r 2j λ2 ] j =1

Fr4 (λ) = −2aθ ξs λ − λ2 n  × (y j /r )2 µ j (2ξ j r j λ3 )/[(r 2j − λ2 )2 + 4ξ 2j r 2j λ2 ] j =1

Fr5 (λ) = −2as ξs λ − λ2

n 

µj

j =1 × (2ξ j r j λ3 )/[(r 2j −

λ2 )2 + 4ξ 2j r 2j λ2 ].

The assessment can now be performed on the optimum parameters and effectiveness of the MTMD for asymmetric structures through the implementation of the following four particular criteria R I = Min.Max.[DMF(I)] RII = Min.Max.[DMF(II)] Min.Max.[DMF(I)] RIII = Max.[DMF∗ (I)] Min.Max.[DMF(II)] RIV = Max.[DMF∗ (II)]

(27) (28) (29) (30)

in which DMF∗ (I) and DMF∗ (II) denote the dynamic magnification factors of the asymmetric structure without the MTMD, corresponding respectively to both the translational and torsional responses. Eqs. (27) and (28) mean that the examination of the optimum parameters is conducted through minimization of the minimum values of the maximum translational and torsional displacement dynamic magnification factors of asymmetric structures with the MTMD. They can be explicitly explained in the following steps. First of all, for a fixed value of λ (set within the range from 0.4 to 3.4) and a fixed tuning frequency ratio, the maximum amplitudes for different average damping ratios and frequency spacings are found, and the minimum amplitudes are selected from the maximum amplitudes, which is the minimax amplitude for that tuning frequency ratio. Then the above procedure is repeated for different tuning frequency ratios to find the minimax of each tuning frequency ratio. Finally, the smallest minimaxes are selected and the corresponding tuning frequency ratio, average damping ratio, and frequency spacing are optimum values. Eqs. (29) and (30), i.e. minimization of the minimum values of the maximum translational and torsional displacement dynamic magnification factors, nondimensionalised respectively by the maximum translational and torsional displacement dynamic magnification factors of asymmetric structures without the MTMD, are used to measure the effectiveness of the MTMD in reducing

C. Li, W. Qu / Engineering Structures 28 (2006) 472–494

479

opt

Fig. 2. Variation of the optimum parameters and effectiveness [(a) ξT ; (b) β opt ; (c) f opt ; and (d) RIII ] of the MTMD for reducing the translational response with respect to total mass ratio with total number of dampers n = 5 in the case of TTFR = 0.5.

the translational and torsional displacements of asymmetric structures, respectively. 5. Numerical study Displayed in Figs. 2–13 are the numerical results of the present research, in which the normalized width (i.e. b/r ) is set equal to 1.0. The degree of asymmetry (i.e. torsional coupling) of the structure depends upon the normalized eccentricity ratio (NER) and the torsional to translational frequency ratio (TTFR). In terms of the TTFR, asymmetric structures may be classified as the torsionally flexible structures (TTFR < 1.0), torsionally intermediate stiff structures (TTFR = 1.0), and torsionally stiff structures (TTFR > 1.0). The NER and TTFR thus are the key parameters of assessing the performance of the MTMD for asymmetric structures. The superscript opt denotes the optimum values of system parameters. Fig. 2 shows the variation of the optimum parameters and effectiveness of the MTMD with respect to the total mass

ratio with n = 5 and TTFR = 0.5, namely the torsionally flexible structures. It is seen from Fig. 2(a) that the NER (with the exception of NER = 0.2) makes some difference in the optimum average damping ratio of MTMD. However, the optimum average damping ratio of the MTMD for NER = 0.2 is significantly larger than that for the other NER values. Likewise the optimum average damping ratio of the MTMD for NER = 0.2 increases approximately linearly with the increase of the total mass ratio. It is seen from Fig. 2(b) that for the case of NER = 0.2, the optimum frequency spacing (reflecting the robustness) of the MTMD takes zero when the total mass ratio is beyond 0.015, implying that the MTMD acts similarly to a single TMD. For smaller or greater NER (such as NER = 0.1 or NER = 0.4 and 0.5), the optimum frequency spacing of the MTMD for asymmetric structures is practically equal to that for symmetric structures (i.e. NER = 0.0). For intermediate NER (i.e. NER = 0.3), however, significantly higher optimum frequency spacing is required to make the MTMD attain the optimum status for larger total mass ratio

480

C. Li, W. Qu / Engineering Structures 28 (2006) 472–494

opt

Fig. 3. Variation of the optimum parameters and effectiveness [(a) ξT ; (b) β opt ; (c) f opt ; and (d) RIV ] of the MTMD for reducing the torsional response with respect to total mass ratio with total number of dampers n = 5 in the case of TTFR = 0.5.

above 0.025. It is seen from Fig. 2(c) that for smaller NER (i.e. NER = 0.1), the optimum tuning frequency ratio of the MTMD for asymmetric structures is nearly the same as that for symmetric structures. However, the optimum tuning frequency ratio of the MTMD generally increases with the increase of the NER above 0.1. It is seen from Fig. 2(d) that the effectiveness of the MTMD for asymmetric structures with NER = 0.1 is practically identical to that for symmetric structures with NER = 0.0. For the case of NER = 0.2, the MTMD acts similarly to a single TMD. Likewise the excessively high damping ratio will make the effectiveness of the MTMD diminished, as will be demonstrated next. Therefore, the MTMD provides unsatisfactory effectiveness. Also the MTMD has lesser effectiveness for the case of NER = 0.2. This is because the excessively wide optimum frequency spacing impairs the effectiveness of the MTMD. However, the effectiveness of the MTMD for higher NER such as NER = 0.4

and 0.5 is less than that for NER = 0.1 because of greater torsional coupling. In terms of the above results, the MTMD may be designed for reducing the translational response of the torsionally flexible structures (TTFR < 1.0) by ignoring the effects of torsional coupling in the case where the NER is less than or equal to 0.1. However, for the NER above 0.1, effects of torsional coupling need to be accounted for in designing the MTMD for asymmetric structures. Fig. 3 shows the variation of the optimum parameters and effectiveness of the MTMD in suppressing the torsional response with respect to total mass ratio with n = 5 and TTFR = 0.5, namely the flexible structures. It is seen from Fig. 3(a) and (c) that the influence of lesser NER (such as NER = 0.1 and 0.2) is rather negligible on the optimum average damping ratio and tuning frequency ratio of the

C. Li, W. Qu / Engineering Structures 28 (2006) 472–494

481

opt

Fig. 4. Variation of the optimum parameters and effectiveness [(a) ξT ; (b) β opt ; (c) f opt ; and (d) RIII ] of the MTMD for suppressing the translational response with respect to total mass ratio with total number of dampers n = 5 in the case of TTFR = 1.0.

opt

Fig. 5. Variation of the optimum parameters and effectiveness [(a) ξT ; (b) β opt ; (c) f opt ; and (d) RIV ] of the MTMD for suppressing the torsional response with respect to total mass ratio with total number of dampers n = 5 in the case of TTFR = 1.0.

MTMD. It is worth noting that for the case of NER = 0.3, the MTMD takes the near-zero optimum average damping ratio. The MTMD with the near-zero optimum average damping ratio

is not meritorious for practical applications due to large stroke displacement. However, larger NER (such as NER = 0.4 and 0.5) increases remarkably the optimum average damping

482

C. Li, W. Qu / Engineering Structures 28 (2006) 472–494

opt

Fig. 6. Variation of the optimum parameters and effectiveness [(a) ξT ; (b) β opt ; (c) f opt ; and (d) RIII ] of the MTMD for attenuating the translational response with respect to total mass ratio with total number of dampers n = 5 in the case of TTFR = 2.0.

opt

Fig. 7. Variation of the optimum parameters and effectiveness [(a) ξT ; (b) β opt ; (c) f opt ; and (d) RIV ] of the MTMD for attenuating the torsional response with respect to total mass ratio with total number of dampers n = 5 in the case of TTFR = 2.0.

ratio and tuning frequency ratio of the MTMD. It is seen from Fig. 3(b) that the optimum frequency spacing of the MTMD generally decreases with the increase of the NER.

Likewise when the NER is equal to or larger than 0.4 the optimum frequency spacing of the MTMD reduces to zero, implying the MTMD acts similarly to a single TMD. It is

C. Li, W. Qu / Engineering Structures 28 (2006) 472–494

483

opt

Fig. 8. Variation of the optimum parameters and effectiveness [(a) ξT ; (b) β opt ; (c) f opt ; and (d) RIII ] of the MTMD for reducing the translational response with respect to total number of dampers with total mass ratio µT = 0.01 in the case of TTFR = 0.5.

opt

Fig. 9. Variation of the optimum parameters and effectiveness [(a) ξT ; (b) β opt ; (c) f opt ; (d) RIV ] of the MTMD for suppressing the torsional response with respect to total number of dampers with total mass ratio µT = 0.01 in the case of TTFR = 0.5.

seen from Fig. 3(d) that the NER affects significantly the effectiveness of the MTMD for suppressing the torsional response of asymmetric structures. Likewise, the effectiveness

of the MTMD for NER = 0.4 is significantly lower than that for the other NER. The reason for this may be attributed to a high damping ratio of the MTMD [see Fig. 3(a)]. The response

484

C. Li, W. Qu / Engineering Structures 28 (2006) 472–494

opt

Fig. 10. Variation of the optimum parameters and effectiveness [(a) ξT ; (b) β opt ; (c) f opt ; and (d) RIII ] of the MTMD for reducing the translational response with respect to total number of dampers with total mass ratio µT = 0.01 in the case of TTFR = 1.0.

opt

Fig. 11. Variation of the optimum parameters and effectiveness [(a) ξT ; (b) β opt ; (c) f opt ; and (d) RIV ] of the MTMD for suppressing the torsional response with respect to total number of dampers with total mass ratio µT = 0.01 in the case of TTFR = 1.0.

of every TMD in the MTMD is not significant in the case of a high damping ratio and hence, the MTMD can not effectively dissipate the vibration energy of structures.

Therefore, the MTMD is capable of suppressing the torsional response of the torsionally flexible structures (TTFR < 1.0) with the NER less than or equal to 0.2.

C. Li, W. Qu / Engineering Structures 28 (2006) 472–494

485

opt

Fig. 12. Variation of the optimum parameters and effectiveness [(a) ξT ; (b) β opt ; (c) f opt ; and (d) RIII ] of the MTMD for reducing the translational response with respect to total number of dampers with total mass ratio µT = 0.01 in the case of TTFR = 2.0.

opt

Fig. 13. Variation of the optimum parameters and effectiveness [(a) ξT ; (b) β opt ; (c) f opt ; and (d) RIV ] of the MTMD for suppressing the torsional response with respect to total number of dampers with total mass ratio µT = 0.01 in the case of TTFR = 2.0.

Fig. 4 shows the variation of the optimum parameters and effectiveness of the MTMD with regard to the total mass ratio with n = 5 and TTFR = 1.0, namely the torsionally intermediate stiff structures. It is seen from Fig. 4 that the influence of the NER is prominent on the

optimum parameters and the effectiveness of the MTMD. The MTMD for the torsionally intermediate stiff structures possesses higher optimum average damping ratio and optimum frequency spacing but smaller optimum tuning frequency ratio and effectiveness with reference to the MTMD for

486

C. Li, W. Qu / Engineering Structures 28 (2006) 472–494

Table 1 Comparison of the robustness of the MTMD strategies with different layouts for suppressing the torsional response of asymmetric structures with total number of dampers n = 5 and with total mass ratio µT = 0.01 TTFR

α

j −1 y j = − b2 + n−1 b NER = 0.1 NER = 0.2

NER = 0.3

0.09

0.09

0.5

– 0.0 0.1 0.2 0.3 0.4 0.5

0.09

1.0

– 0.0 0.1 0.2 0.3 0.4 0.5

0.15

2.0

– 0.0 0.1 0.2 0.3 0.4 0.5

0.08

0.18

0.14

j −1 1−2α y j = − b2 + n−1 · 2 b NER = 0.1 NER = 0.2

NER = 0.3

j −1 1−2α y j = αb + n−1 · 2 b NER = 0.1 NER = 0.2

NER = 0.3

0.08 0.09 0.09 0.09 0.08 0.10

0.09 0.09 0.09 0.09 0.09 0.09

0.09 0.10 0.10 0.10 0.10 0.10

0.08 0.09 0.08 0.08 0.08 0.08

0.10 0.09 0.05 0.09 0.09 0.09

1.22 0.10 0.05 0.91 0.30 0.05

0.11 0.11 0.11 0.12 0.12 0.12

0.21 0.22 0.23 0.23 0.23 0.23

0.24 0.25 0.26 0.28 0.29 0.31

0.07 0.06 0.05 0.01 0.02 0.00

0.11 0.10 0.06 0.04 0.04 0.04

0.08 0.08 0.06 0.06 0.06 0.06

0.14 0.14 0.13 0.13 0.13 0.13

0.13 0.13 0.13 0.13 0.13 0.13

0.13 0.13 0.13 0.13 0.13 0.13

0.13 0.13 0.13 0.12 0.12 0.12

0.13 0.12 0.12 0.12 0.12 0.12

0.12 0.12 0.12 0.12 0.12 0.12

0.19

0.13

Table 2 Comparison of the effectiveness of the MTMD strategies with different layouts for suppressing the torsional response of asymmetric structures with total number of dampers n = 5 and with total mass ratio µT = 0.01 j −1 y j = − b2 + n−1 b NER = 0.1 NER = 0.2

NER = 0.3

0.50

0.47

0.5

– 0.0 0.1 0.2 0.3 0.4 0.5

0.47

1.0

– 0.0 0.1 0.2 0.3 0.4 0.5

0.30

2.0

– 0.0 0.1 0.2 0.3 0.4 0.5

TTFR

α

0.52

0.49

0.32

j −1 1−2α y j = − b2 + n−1 · 2 b NER = 0.1 NER = 0.2

NER = 0.3

j −1 1−2α y j = αb + n−1 · 2 b NER = 0.1 NER = 0.2

NER = 0.3

0.51 0.49 0.47 0.46 0.44 0.43

0.41 0.41 0.42 0.42 0.42 0.42

0.46 0.36 0.38 0.39 0.37 0.36

0.55 0.45 0.42 0.42 0.40 0.38

0.55 0.66 0.59 0.47 0.40 0.32

0.65 0.41 0.57 0.67 0.68 0.53

0.49 0.50 0.51 0.53 0.54 0.55

0.50 0.50 0.51 0.53 0.55 0.57

0.50 0.50 0.50 0.50 0.50 0.51

0.50 0.52 0.53 0.53 0.54 0.54

0.57 0.58 0.59 0.60 0.60 0.61

0.52 0.53 0.56 0.56 0.56 0.59

0.32 0.32 0.32 0.33 0.33 0.33

0.33 0.33 0.33 0.33 0.33 0.33

0.33 0.33 0.33 0.33 0.33 0.33

0.31 0.31 0.32 0.32 0.32 0.32

0.33 0.33 0.33 0.33 0.33 0.34

0.34 0.34 0.34 0.35 0.35 0.35

0.49

0.33

symmetric structures. The above observations indicate that the MTMD may reduce the translational response of the torsionally intermediate stiff structures (TTFR = 1.0) with various NER values. However it is noteworthy that the effectiveness of the MTMD in suppressing the translational response is less for the torsionally intermediate stiff structures (TTFR = 1.0) than the torsionally flexible structures (TTFR < 1.0).

Fig. 5 presents the variation of the optimum parameters and effectiveness of the MTMD with regard to the total mass ratio with n = 5 and TTFR = 1.0, namely the torsionally intermediate stiff structures. It can be seen from Fig. 5 that the NER affects significantly the optimum parameters and effectiveness of the MTMD for suppressing the torsional response of the torsionally intermediate stiff structures. It is

C. Li, W. Qu / Engineering Structures 28 (2006) 472–494

(a) NER = 0.0.

(b) NER = 0.1.

(c) NER = 0.2.

(d) NER = 0.25.

(e) NER = 0.3.

(f) NER = 0.4.

487

Fig. 14. Frequency response curves of asymmetric structures without and with both the optimum MTMD with total number of dampers n = 5 and the optimum TMD with equal total mass ratio µT = 0.01 in the case of TTFR = 0.5 and mitigating the translational response.

worth noting that the effectiveness of the MTMD for the torsionally intermediate stiff structures (TTFR = 1.0) is, greater than that for the torsionally flexible structures (TTFR = 0.5). Fig. 6 presents the variation of the optimum parameters and effectiveness of the MTMD with n = 5 and TTFR = 2.0, namely the torsionally stiff structures. It can be seen from Fig. 6 that the influence of the NER is rather negligible on the optimum average damping ratio, optimum frequency spacing, and effectiveness of the MTMD, but slightly greater on the optimum tuning frequency ratio of the MTMD. Therefore, for the torsionally stiff structures (TTFR = 2.0), the design of the

MTMD by ignoring the effects of torsional coupling is justified. Likewise, the effectiveness of the MTMD in suppressing the translational response is higher for the torsionally stiff structures (TTFR = 2.0) than both the torsionally flexible structures (TTFR = 0.5) and torsionally intermediate stiff structures (TTFR = 1.0). Fig. 7 shows the variation of the optimum parameters and effectiveness of the MTMD with n = 5 and TTFR = 2.0, namely the torsionally stiff structures. It can be seen from Fig. 7 that the influence of the NER is rather negligible for the optimum tuning frequency ratio and effectiveness of the MTMD, but not insignificant on the optimum average damping

488

C. Li, W. Qu / Engineering Structures 28 (2006) 472–494

(a) NER = 0.0.

(b) NER = 0.1.

(c) NER = 0.2.

(d) NER = 0.3.

(e) NER = 0.4. Fig. 15. Frequency response curves of asymmetric structures without and with both the optimum MTMD with total number of dampers n = 5 and the optimum TMD with equal total mass ratio µT = 0.01 in the case of TTFR = 1.0 and mitigating the translational response.

ratio and frequency spacing of the MTMD. It is noteworthy that the effectiveness of the MTMD in reducing the torsional response is much greater for the torsionally stiff structures (TTFR = 2.0) than for both the torsional intermediate stiff structures (TTFR = 1.0) and torsional flexible structures (TTFR = 0.5). Figs. 8 and 12 show the variations of the optimum parameters and effectiveness of the MTMD with regard to the total number with µT = 0.01, corresponding respectively to both the torsionally flexible structures (TTFR = 0.5) and torsionally stiff structures (TTFR = 2.0). It can be seen from Figs. 8 and 12 that the total number does

not amplify the influences of the NER on the optimum parameters and effectiveness of the MTMD for suppressing the translational response of both the torsionally flexible structures and torsionally stiff structures. It is also shown that the change trends of the optimum parameters and effectiveness of the MTMD for both the torsionally flexible structures and torsionally stiff structures are similar to those for symmetric structures. Figs. 9 and 11 show the variations of the optimum parameters and effectiveness of the MTMD with regard to the total number with µT = 0.01, corresponding respectively to the torsionally flexible structures (TTFR = 0.5) and torsionally

C. Li, W. Qu / Engineering Structures 28 (2006) 472–494

(a) NER = 0.0.

(b) NER = 0.1.

(c) NER = 0.2.

(d) NER = 0.3.

489

(e) NER = 0.4. Fig. 16. Frequency response curves of asymmetric structures without and with both the optimum MTMD with total number of dampers n = 5 and the optimum TMD with equal total mass ratio µT = 0.01 in the case of TTFR = 2.0 and mitigating the translational response.

intermediate stiff structures (TTFR = 1.0). It can be seen from Figs. 9 and 11 that the influences of the NER vary with the total number on the optimum parameters and effectiveness of the MTMD for suppressing the torsional response of both the torsionally flexible structures and torsionally intermediate stiff structures. Fig. 10 presents the variation of the optimum parameters and effectiveness of the MTMD with regard to the total number with µT = 0.01 and TTFR = 1.0, namely the torsionally intermediate stiff structures. It can be seen from Fig. 10 that the influences of the NER vary with the total number on the optimum parameters and effectiveness of the MTMD

for attenuating the translational response of the torsionally intermediate stiff structures. Fig. 13 presents the variation of the optimum parameters and effectiveness of the MTMD with regard to the total number with µT = 0.01 and TTFR = 2.0, namely the torsionally stiff structures. It can be seen from Fig. 13 that the total number does not amplify the influence of the NER on the optimum parameters and effectiveness of the MTMD for attenuating the torsional response of the torsionally stiff structures. Also Fig. 13 clearly indicates that the change trends of the optimum parameters and effectiveness of the MTMD for the torsionally stiff structures are similar to those for symmetric structures.

490

C. Li, W. Qu / Engineering Structures 28 (2006) 472–494

(a) NER = 0.1.

(b) NER = 0.2.

(c) NER = 0.3.

(d) NER = 0.4.

Fig. 17. Frequency response curves of asymmetric structures without and with both the optimum MTMD with total number of dampers n = 5 and the optimum TMD with equal total mass ratio µT = 0.01 in the case of TTFR = 0.5 and suppressing the torsional response.

The performance assessment has been made above on the MTMD which is evenly housed within the width b with the center at the CM for reducing the torsional response of asymmetric structures. For the purpose of comparison, the two MTMD strategies which are evenly housed within the width [(1−2α)b/2] with the center respectively at y(n+1)/2 = −b(1+ 2α)/4 and y(n+1)/2 = b(1 + 2α)/4 are also considered here to suppress the torsional response of asymmetric structures. Their key results are presented in Tables 1 and 2, respectively. It can be seen in the comparison that the MTMD strategy which is evenly housed within the width [(1 − 2α)b/2] with the center at y(n+1)/2 = −b(1 + 2α)/4 is preferably suitable for reducing the torsional response of the torsionally flexible structures (TTFR < 1.0). It is noted that in mitigating the torsional response of the torsionally intermediate stiff structures (TTFR = 1.0), the MTMD strategy which is evenly housed within the width b with the center at the CM can render better effectiveness, while the MTMD strategy which is evenly housed within the width [(1 − 2α)b/2] with the center at y(n+1)/2 = −b(1 + 2α)/4 can present higher robustness. It is interesting to see from Tables 1 and 2 that the three MTMD strategies corresponding to the three layout schemes approximately reach the same robustness and effectiveness in reducing the torsional response of the torsionally stiff structures (TTFR > 1.0). Figs. 14–16 plot the frequency response curves of asymmetric structures without and with both the optimum MTMD with

total number of dampers n = 5 and the optimum TMD with equal total mass ratio µT = 0.01 in mitigating the translational response for the three cases of TTFR, each of which corresponds to several NER values. It is seen in Figs. 14–16 that the MTMD can render better effectiveness and higher robustness in comparison to the TMD. Notwithstanding this, for torsionally flexible structures (TTFR < 1.0) when the NER equals 0.25, the MTMD and the TMD have approximately same effectiveness and robustness, while when the NER reaches 0.4 both the MTMD and TMD will lose their effectiveness, thus meaning that the MTMD is only applicable to the cases of NER < 0.25. For torsionally intermediate stiff structures (TTFR = 1.0), the frequency response curve of the asymmetric structure with the optimum MTMD is in good agreement with that with the optimum TMD when the NER is equal to or larger than 0.3. Figs. 17–19 present the frequency response curves of asymmetric structures without and with both the optimum MTMD with total number of dampers n = 5 and the optimum TMD with equal total mass ratio µT = 0.01 in suppressing the torsional response for the three cases of TTFR, each of which corresponds to several NER values. From Fig. 17, it is important to note that the MTMD cannot offer any significant superiority in comparison to the TMD, thus indicating that the MTMD is not meritorious in reducing the torsional response of torsionally flexible structures (TTFR < 1.0). For torsionally intermediate stiff structures (TTFR = 1.0), when the NER is equal to or greater than 0.3 (see Fig. 18) the MTMD and

C. Li, W. Qu / Engineering Structures 28 (2006) 472–494

(a) NER = 0.1.

(b) NER = 0.2.

(c) NER = 0.3.

(d) NER = 0.4.

491

Fig. 18. Frequency response curves of asymmetric structures without and with both the optimum MTMD with total number of dampers n = 5 and the optimum TMD with equal total mass ratio µT = 0.01 in the case of TTFR = 1.0 and suppressing the torsional response.

the TMD are shown to have practically same effectiveness and robustness, and thus meaning that the MTMD will not render any significant superiority with respect to the TMD. But for torsionally stiff structures (TTFR > 1.0), the MTMD can always provide a better performance than the TMD. 6. Concluding remarks The main contributions of the present research to the MTMD can be drawn as follows: (1) The analytical expressions for the translational and torsional displacement dynamic magnification factors (DMF) are provided for the generalized 2DOF structure–MTMD system. (2) The dimensionless DMF parameters are used as the formal indexes estimating the effectiveness of the MTMD in reducing both the translational and torsional responses of asymmetric structures. (3) A new basic result is that the NER affects significantly the performance of the MTMD for both torsionally flexible and torsionally intermediate stiff structures; while the influence of the NER is rather negligible on the performance of the MTMD for torsionally stiff structures, thus implying that in such a case the MTMD may be designed by ignoring the effects of torsional coupling. (4) The effectiveness and robustness of the MTMD strategies with different layouts are also investigated and demonstrated for the case of mitigating the torsional response of asymmetric structures, thus providing valuable guidance for the MTMD design. (5) The frequency response

curves of asymmetric structures without and with both the optimum MTMD and TMD are plotted for the three cases of TTFR, consequently obtaining some very useful results. From the simulation results presented, the following main conclusions can be drawn: (1) The MTMD may be designed for suppressing the translational response of the torsionally flexible structures (TTFR < 1.0) by ignoring the effects of torsional coupling in the situation where the NER is less than or equal to 0.1. For the NER above 0.1, however, the effects of torsional coupling need to be accounted for in designing the MTMD. Further, the total number does not amplify the influences of the NER on the optimum parameters and effectiveness of the MTMD. Likewise, the change trends of the optimum parameters and effectiveness of the MTMD with the total number for the asymmetric structures (TTFR < 1.0) are similar to those for symmetric structures. (2) The MTMD is capable of attenuating the torsional response of the torsionally flexible structures (TTFR < 1.0) with the NER less than or equal to 0.2. However, the influences of the NER on the optimum parameters and effectiveness of the MTMD vary with the total number. (3) The MTMD may be employed to reduce both the translational and torsional responses of the torsionally intermediate stiff structures (TTFR = 1.0) with various NER values. However, the influences of the NER on the optimum

492

C. Li, W. Qu / Engineering Structures 28 (2006) 472–494

(a) NER = 0.1.

(b) NER = 0.2.

(c) NER = 0.3.

(d) NER = 0.4.

Fig. 19. Frequency response curves of asymmetric structures without and with both the optimum MTMD with total number of dampers n = 5 and the optimum TMD with equal total mass ratio µT = 0.01 in the case of TTFR = 2.0 and suppressing the torsional response.

parameters and effectiveness of the MTMD vary with the total number. (4) The MTMD can effectively suppress both the translational and torsional responses of the torsionally stiff structures (TTFR > 1.0). It is important to emphasize that the MTMD may be designed by ignoring the effects of torsional coupling when used to attenuate the translational response of the torsionally stiff structures. It is also found in this case that the total number does not amplify the influences of the NER on the optimum parameters and effectiveness of the MTMD. Likewise, the change trends of the optimum parameters and effectiveness of the MTMD with the total number for the torsionally stiff structures are similar to those for symmetric structures. (5) The MTMD strategy which is evenly housed within the width [(1−2α)b/2] with the center at y(n+1)/2 = −b(1+2α)/4 is preferably suitable for reducing the torsional response of the torsionally flexible structures (TTFR < 1.0). In mitigating the torsional response of the torsionally intermediate stiff structures (TTFR = 1.0), the MTMD strategy which is evenly housed within the width b with the center at the CM can render better effectiveness, while the MTMD strategy which is evenly housed within the width [(1 − 2α)b/2] with the center at y(n+1)/2 = −b(1 + 2α)/4 can present higher robustness. Likewise, the three MTMD strategies corresponding to the three layout schemes approximately reach the same robustness

and effectiveness in reducing the torsional response of the torsionally stiff structures (TTFR > 1.0). (6) The MTMD can render better effectiveness and higher robustness in comparison to the TMD in mitigating the translational response of asymmetric structures. But for torsionally flexible and torsionally intermediate stiff structures, the NER parameter has the respective upper bound, thus physically limiting the application of the MTMD. (7) In reducing the torsional response of torsionally stiff structures (TTFR > 1.0), the MTMD can always provide a better performance than the TMD; for torsionally intermediate stiff structures (TTFR = 1.0), the MTMD is conditionally superior to the TMD; while for torsionally flexible structures (TTFR < 1.0), the MTMD is not meritorious only in terms of both the effectiveness and robustness. It is well known that both the TMD and MTMD are very efficient for suppressing the wind-induced vibrations in buildings or structural components with very small damping. In the present study, a linear analysis is conducted to demonstrate the performance of the MTMD, thus implying the applicability to small and moderate earthquakes. However, in strong earthquakes, well designed structures are expected to exhibit some yielding, the non-linear analysis taking into consideration the yielding in structures is thus required to replace the linear analysis used in this paper. On the other hand, under strong earthquakes, the structural damping is generally rather high

C. Li, W. Qu / Engineering Structures 28 (2006) 472–494

(the damping in most structures is between say 4% to 6% of critical), so that the effectiveness of the MTMD is rather limited. For instance, according to the Chinese Code-Technical specification for steel structure of tall buildings, the damping ratio of steel structures is taken as 0.02 in small and moderate earthquakes; while under strong earthquakes the structures will be in the inelastic state and then this damping ratio, in such a case, may be set equal to 0.05. The benefits of both the TMD and MTMD thus are greatly reduced as compared to structures with damping ratios of say 2% or less. Another fact to be kept in mind is that for earthquake excitations there is no real resonance response of the structures as the dominant frequencies of the earthquake cover a certain frequency range, i.e. the resonance (maximum dynamic) response in a slightly damped structure subjected to earthquake ground motions is significantly less than that obtained by a steady-state wind excitation with constant (locked-in) frequency. Therefore, both the TMD and MTMD will provide less effectiveness in strong earthquake applications; in such case, the active multiple tuned mass dampers (AMTMD) [49] may be a better means of reducing translational and torsional responses of asymmetric structures with respect to the MTMD. This will be discussed in forthcoming papers by the authors. For the purpose of manufacturing the MTMD, three important parameters, the stiffness, damping, and mass, need calculating out beforehand with resorting to optimization. In the present paper, it is assumed that the stiffness and damping coefficient of each TMD in the MTMD are same and the natural frequencies of the MTMD are uniformly distributed. As a result, the MTMD is manufactured by keeping the stiffness, damping constant and mass variation, thus indicating that the manufacture is simplest. Estimating the structural properties accurately then is very important in that the optimization to be performed does tie in with structural natural frequency corresponding to the vibration mode being controlled. But in fact, the structural properties are only known with a degree of uncertainty. Likewise, they can vary with time, due to both the deterioration of structures, leading to a decrease in the natural frequency and an increase in the damping and the variation of the mass, related to the variation of the carried loads. For instant, under moderate and strong earthquakes, in RC structures due to the formation and progressive development of cracks the stiffness and thus the eigenfrequencies will be reduced. Evidently, in this case it is very beneficial to have the MTMD, which are also able to cover the progressive reduction in the dominant eigenfrequencies of the RC structures. There being a wide range of natural frequencies controlled by the MTMD, the effectiveness in the case where there are uncertainties in the structural natural frequency thus is better than that of a single TMD. References [1] Brock JE. A note on the damped vibration absorber. J Appl Mech ASME 1946;13:A-284. [2] Den Hartog JP. Mechanical vibrations. 4th ed. New York: McGraw-Hill; 1956.

493

[3] Warburton GB, Ayorinde EO. Optimum absorber parameters for simple systems. Earthq Eng Struct Dyn 1980;8:197–217. [4] Ayorinde EO, Warburton GB. Minimizing structural vibration with absorbers. Earthq Eng Struct Dyn 1980;8:219–36. [5] Warburton GB. Optimum absorber parameters for minimizing vibration response. Earthq Eng Struct Dyn 1981;9:251–62. [6] Warburton GB. Optimum absorber parameters for various combinations of response and excitation parameters. Earthq Eng Struct Dyn 1982;10: 381–401. [7] Tsai HC, Lin GC. Optimum tuned mass dampers for minimizing steadystate response of support excited and damped system. Earthq Eng Struct Dyn 1993;22:957–73. [8] Tsai HC, Lin GC. Explicit formulae for optimum absorber parameters for force excited and viscously damped systems. J Sound Vib 1993;89: 385–96. [9] Thompson AG. Optimum tuning and damping of a dynamic vibration absorber applied to a force excited and damped primary system. J Sound Vib 1981;77:403–15. [10] Fujino Y, Abe M. Design formulas for tuned mass dampers based on a perturbation technique. Earthq Eng Struct Dyn 1993;22:833–54. [11] Macinante JA. Seismic mountings for vibration isolation. New York: John Wiley and Sons, Inc.; 1984. [12] Xu YL, Kwok KCS, Samali B. Control of wind-induced tall building response by tuned mass dampers. J Wind Eng Ind Aerodyn 1992;40:1–32. [13] Xu YL, Kwok KCS. Semianalytical method for parametric study of tuned mass dampers. J Struct Eng ASCE 1994;120(3):747–64. [14] Li QS, Cao H, Li GQ, Li SJ, Liu D. Optimal design of wind-induced vibration control of tall buildings and high-rise structures. Wind Struct 1999;2(1):69–83. [15] Gu M, Xiang HF. Optimization of TMD for suppressing buffting response of long-span bridges. J Wind Eng Ind Aerodyn 1992;41–44:1383–92. [16] Sadek F, Mohraz B, Taylor AW, Chung RM. A method of estimating the parameters of tuned mass dampers for seismic applications. Earthq Eng Struct Dyn 1997;26:617–35. [17] Villaverde R. Reduction in seismic response with heavily-damped vibration absorber. Earthq Eng Struct Dyn 1985;13:33–42. [18] Rana R, Soong TT. Parametric study and simplified design of tuned mass dampers. Eng Struct 1998;20(3):193–204. [19] Soto-Brito R, Ruiz SE. Influence of ground motion intensity on the effectiveness of tuned mass dampers. Earthq Eng Struct Dyn 1999;28(11): 1255–71. [20] Lukkunaprasit P, Wanitkorkul A. Inelastic buildings with tuned mass dampers under moderate ground motions from distant earthquakes. Earthq Eng Struct Dyn 2001;30(4):537–51. [21] Pinkaew T, Lukkunaprasit P, Chatupote P. Seismic effectiveness of tuned mass dampers for damage reduction of structures. Eng Struct 2003;25(1): 39–46. [22] Iwanami K, Seto K. Optimum design of dual tuned mass dampers and their effectiveness. Proc JSME(C) 1984;50(449):44–52 [in Japanese]. [23] Xu K, Igusa T. Dynamic characteristics of multiple substructures with closely spaced frequencies. Earthq Eng Struct Dyn 1992;21:1059–70. [24] Igusa T, Xu K. Vibration control using multiple tuned mass dampers. J Sound Vib 1994;175(4):491–503. [25] Yamaguchi H, Harnpornchai N. Fundamental characteristics of multiple tuned mass dampers for suppressing harmonically forced oscillations. Earthq Eng Struct Dyn 1993;22(1):51–62. [26] Abe M, Fujino Y. Dynamic characterization of multiple tuned mass dampers and some design formulas. Earthq Eng Struct Dyn 1994;23(8): 813–35. [27] Kareem A, Kline S. Performance of multiple mass dampers under random loading. J Struct Eng ASCE 1995;121(2):348–61. [28] Jangid RS. Optimum multiple tuned mass dampers for base-excited undamped system. Earthq Eng Struct Dyn 1999;28(9):1041–9. [29] Li C. Performance of multiple tuned mass dampers for attenuating undesirable oscillations of structures under the ground acceleration. Earthq Eng Struct Dyn 2000;29(9):1405–21. [30] Park J, Reed D. Analysis of uniformly and linearly distributed mass dampers under harmonic and earthquake excitation. Eng Struct 2001;23: 802–14.

494

C. Li, W. Qu / Engineering Structures 28 (2006) 472–494

[31] Gu M, Chen SR, Chang CC. Parametric study on multiple tuned mass dampers for buffeting control of Yangpu Bridge. J Wind Eng Ind Aerodyn 2001;89:987–1000. [32] Chen G, Wu J. Experimental study on multiple tuned mass dampers to reduce seismic responses of a three-storey building structure. Earthq Eng Struct Dyn 2003;32(5):793–810. [33] Yau JD, Yang YB. A wideband MTMD system for reducing the dynamic response of continuous truss bridges to moving train loads. Eng Struct 2004;26(12):1795–807. [34] Yau JD, Yang YB. Vibration reduction for cable-stayed bridges traveled by high-speed trains. Finite Elem Anal Des 2004;40(3):341–59. [35] Kwon S-D, Park K-S. Suppression of bridge flutter using tuned mass dampers based on robust performance design. J Wind Eng Ind Aerodyn 2004;92(11):919–34. [36] Lin CC, Wang JF, Chen BL. Train-induced vibration control of high-speed railway bridges equipped with multiple tuned mass dampers. J Bridge Eng ASCE 2005;10(4):398–414. [37] Hoang N, Warnitchai P. Design of multiple tuned mass dampers by using a numerical optimizer. Earthq Eng Struct Dyn 2005;34(2):125–44. [38] Li C. Optimum multiple tuned mass dampers for structures under the ground acceleration based on DDMF and ADMF. Earthq Eng Struct Dyn 2002;31:897–919. [39] Li C, Liu Y. Further characteristics for multiple tuned mass dampers. J Struct Eng ASCE 2002;128(10):1362–5.

[40] Li C, Liu Y. Optimum multiple tuned mass dampers for structures under the ground acceleration based on the uniform distribution of system parameters. Earthq Eng Struct Dyn 2003;32:671–90. [41] Jangid RS, Datta TK. Performance of multiple tuned mass dampers for torsionally coupled system. Earthq Eng Struct Dyn 1997;26:307–17. [42] Lin CC, Ueng JM, Huang TC. Seismic response reduction of irregular buildings using passive tuned mass dampers. Eng Struct 1999;21:513–24. [43] Lin YY, Cheng CM, Lee CH. A tuned mass damper for suppressing the coupled flexural and torsional buffeting response of long-span bridges. Eng Struct 2000;22:1195–204. [44] Pansare AP, Jangid RS. Tuned mass dampers for torsionally coupled systems. Wind Struct 2003;6(1):23–40. [45] Arfiadi Y, Hadi MNS. Passive and active control of three-dimensional buildings. Earthq Eng Struct Dyn 2000;29(3):377–96. [46] Singh MP, Singh S, Moreschi LM. Tuned mass dampers for response control of torsional buildings. Earthq Eng Struct Dyn 2002;31(4):749–69. [47] Ahlawat AS, Ramaswamy A. Multiobjective optimal absorber system for torsionally coupled seismically excited structures. Eng Struct 2003;25(7): 941–50. [48] Wang JF, Lin CC. Seismic performance of multiple tuned mass dampers for soil–irregular building interaction systems. Int J Solids Struct 2005; 42(20):5536–54. [49] Li C, Liu Y. Active multiple tuned mass dampers for structures under the ground acceleration. Earthq Eng Struct Dyn 2002;31(5):1041–52.