Research Article Distributed Multiple Tuned Mass …downloads.hindawi.com/journals/je/2014/198719.pdfResearch Article Distributed Multiple Tuned Mass Dampers for Wind Vibration Response

Research ArticleDistributed Multiple Tuned Mass Dampers for Wind VibrationResponse Control of High-Rise Building

Said Elias and Vasant Matsagar

Department of Civil Engineering, Indian Institute of Technology (IIT) Delhi, Hauz Khas, New Delhi 110 016, India

Correspondence should be addressed to Said Elias; [email protected]

Received 31 August 2014; Accepted 25 October 2014; Published 23 November 2014

Academic Editor: Radhey S. Jangid

Copyright © 2014 S. Elias and V. Matsagar. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Multiple tuned mass dampers (MTMDs) distributed along height of a high-rise building are investigated for their effectiveness invibration response control. A 76-storey benchmark building is modeled as shear type structure with a lateral degree of freedomat each floor, and tuned mass dampers (TMDs) are installed at top/different floors. Suitable locations for installing the TMDs andtheir tuning frequencies are identified based, respectively, on the mode shapes and frequencies of the uncontrolled and controlledbuildings. Multimode control strategy has been adopted, wherein each TMD is placed where the mode shape amplitude of thebuilding is the largest or large in the particular mode being controlled and tuned with the corresponding modal frequency.Newmark’smethod is used to solve the governing equations ofmotion for the structure.Theperformance of the distributedMTMDs(d-MTMDs) is compared with single tuned mass damper (STMD) and all the MTMDs placed at top floor. The variations of topfloor acceleration and displacement under wind loads are computed to study the effectiveness of the MTMDs in vibration controlof the high-rise building. It is concluded that the d-MTMDs are more effective to control wind induced vibration than the STMDand the MTMDs placed at top floor.

1. Introduction

The buildings are built taller, lighter, and slender as per mod-ern world requirement, with the use of advanced technology,knowledge of new materials, and analysis software, whichhave assured safe constructions and comfort to human life.In the tall buildings, wind and earthquake borne vibrationsare typically controlled by the use of tuned mass dampers(TMDs). The well-established concept of TMDs was orig-inated since an attempt made by Frahm [1]. Much later,Randall et al. [2] have computationally investigated optimallinear vibration absorber for linear damped primary system.The studies on optimum control of absorbers continuedover the years and different approaches have been proposedby the researchers. Tsai and Lin [3] concluded that theoptimum absorber can reduce the peak response for inputfrequencies near the natural frequency of the main system.They also showed that, for lower input frequencies, responseamplitudesmay amplify.Moreover, they concluded thatwhenthe main system had high damping, vibration absorber was

less effective in reducing the system response. Soong andDargush [4] concluded that the TMDs are most effectivewhen the firstmode contribution to the response is dominant.This is generally the case for tall, slender structural systems.

Multiple tuned mass dampers (MTMDs) have also beeninvestigatedwidely for their effectiveness in vibration control.Iwanami and Seto [5] had shown that two TMDs are moreeffective than single TMD and, later, Xu and Igusa [6]proposed the use of multiple suboscillators with closelyspaced frequencies. Their study confirmed that the optimallydesignedMTMDs aremore effective and robust than an opti-mally designed single TMD of equal total mass. Yamaguchiand Harnpornchai [7] reported improved performance ofoptimum MTMDs as compared to optimum single TMD.In addition, they reported that MTMDs can be much morerobust than single TMD. Abe and Fujino [8] showed that theMTMDs are efficient when at least one of the oscillators isstrongly coupled with the structure in any mode. In addition,they showed that properly designed MTMDs are robustthan a conventional single TMD. Kareem and Kline [9] had

Hindawi Publishing CorporationJournal of EngineeringVolume 2014, Article ID 198719, 11 pageshttp://dx.doi.org/10.1155/2014/198719

2 Journal of Engineering

306.

1 m

34 fl

oors

@

4.5 m4.5 m

35 fl

oors

@

42 m

4.5 m4.5 m

4.5 m4.5 m10 m

L2 X2L1 X1

L3 X3

L40 X40L39 X39L38 X38

L76 X76L75 X75L74 X74

x1 x2 xnm1 m2 mn

k1 k2 kn

c1 c2 cn

3.9

m/fl

oor=

136.5

m3.9

m/fl

oor=

132.6

m

(a)

4.5 m4.5 m10 m

34 fl

oors

@

4.5 m4.5 m

35 fl

oors

@

42 m

4.5 m4.5 m

L3 X3L2 X2L1 X1

L40

L65

L61

X40

X65

X61

L39 X39L38 X38

L76 X76L75 X75L74 X74

x1 x2 x3

x4

xn

m1 m2 m3

m4

mn

k1 k2

kn

k4

k3c1 c2

c3

c4

cn

3.9

m/fl

oor=

136.5

m3.9

m/fl

oor=

132.6

m

(b)

Figure 1: Model of 76-storey benchmark building installed with (a) MTMDs all at top floor, (b) d-MTMDs along height of the building.

investigated the dynamic characteristics and effectivenessof the MTMDs with distributed natural frequencies underrandom loading. They reported that the MTMDs were mosteffective in controlling the motion of the primary system. Inaddition, they reported that the MTMDs require less spacefor an individual damper than one massive STMD, whichimproves their constructability and maintenance. Jangid [10]investigated the optimum parameters of the MTMDs forundamped main system. He reported that by increasing thenumber of TMDs the optimumdamping ratio of theMTMDsdecreases and the damping increases with increase in themass ratio. In addition, he reported that optimum bandwidthof the MTMD system increases with the increase of both ofthe mass and number of MTMDs. Further, it was showedthat optimum tuning frequency increases with the increasein the number of MTMDs and decreases with the increasein the mass ratio. Li [11, 12] reported improved performanceof optimum MTMDs as compared to the optimum STMD.In addition, the researcher showed that the MTMDs aremore robust as compared to the STMD. Chen and Wu [13]studied the effects of a TMD on the modal responses of a six-storey building to demonstrate the damper’s ineffectiveness

in seismic applications. They reported that the MTMDsare more effective in suppressing the accelerations at lowerfloors than at upper floors. In addition, the researchersreported that the MTMDs do not appear advantageous overa conventional STMD for displacement control. Bakre andJangid [14] studied the optimum parameters of the MTMDsystem, wherein the damping ratio and tuning frequencybandwidth were obtained using the numerical searchingtechnique for different values of number and mass ratio ofthe MTMDs. Han and Li [15] had reported the effective-ness of the MTMDs with their natural frequencies beinguniformly distributed around their mean natural frequency.The study recommended using the MTMDs with identicalstiffness and damping coefficient, however with unequalmass and uniform distribution of natural frequencies. Lin etal. [16] showed the effectiveness of the optimum MTMDswith limited stroke length. They showed with the help ofexperimental results that the MTMDs were not only effectivein mitigating the building responses but also successful insuppressing its stroke. Moon [17] had concluded that loss ofeffectiveness of theMTMDs is minimal if they are distributedvertically based on mode shape. A study reported by Patil

Journal of Engineering 3

and Jangid [18] showed that optimum MTMDs are muchmore effective and robust as compared to a single TMD forthe wind excited benchmark building. However, hardly hasany study so far been conducted on wind response controlof buildings wherein placement and tuning of the MTMDsare made in accordance with the modal proprieties of thebuilding. The objective of this study, therefore, is to studyeffective placement of TMDs based on the mode shapes andfrequencies of the main structure. The TMDs are placedwhere themode shape amplitude of the building is the largestor large in the particular mode and the TMDs are tuned tohigher modal frequencies while controlling first five modesfor mitigation of building vibration under across-wind load.On a particular floor not more than one TMD is proposedto be installed. The vibration control strategy adopted here istermed as multimode control.

2. Mathematical Model ofBenchmark Building

For this study, a 76-storey benchmark building is considered,having 306.1m height and 42m × 42m plan dimension. Itis sensitive to wind induced loads because the aspect ratio(height to width ratio) is 7.3. The first storey is 10m high; sto-ries from 2 and 3, 38–40, and 74–76 are 4.5m high; all otherstories are having typical height of 3.9m. Yang et al. [19] havegiven detailed description of the benchmark building andits mathematical model. In the model, the rotational degreesof freedom have been removed by the static condensationprocedure, only translational degrees of freedom; one at eachfloor of the building is considered [19]. Figure 1(a) shows theelevation of benchmark building installed withMTMDs all attop floor, and Figure 1(b) shows the elevation of benchmarkbuilding installed with d-MTMDs on different floors. Inaddition, the heights of various floors and configuration ofthe MTMDs have also been depicted.

The governing equations of motion for the wind excitedbenchmark building installed with all MTMDs at top floorand installed with d-MTMDs are obtained by considering theequilibriumof forces at the location of each degree of freedomas follows:

[𝑀𝑠] {�̈�𝑠} + [𝐶

𝑠] {�̇�𝑠} + [𝐾

𝑠] {𝑥𝑠} = {𝐹

𝑡} , (1)

where [𝑀𝑠], [𝐶𝑠], and [𝐾

𝑠] are the mass, damping, and

stiffness matrices of the building, respectively, of order (𝑁 +

𝑛)× (𝑁+𝑛). Here,𝑁 indicates degrees of freedom (DOF) forthe benchmark building and 𝑛 indicates degrees of freedomfor MTMDs/STMD. {𝑥

𝑠} = {𝑋

1, 𝑋2, . . . , 𝑋

𝑁, . . . , 𝑥

𝑛}𝑇

=

{{𝑋𝑖}, {𝑥𝑗}}𝑇, {�̇�𝑠}, and {�̈�

𝑠} are the unknown relative floor

displacement, velocity, and acceleration vectors, respectively;and {𝐹

𝑡} is the wind load vector of order (𝑁 + 𝑛).

Wind load is considered acting on the 𝑁 floors of thebuilding but not on the TMDs. The detailed description ofthe wind tunnel tests conducted at the University of Sydneyis given by Samali et al. [20, 21] and the time histories ofacross-wind loads are available at the website [22]. Powerspectral density function (PSDF) of the wind load applied onthe building is shown in Figure 2. Locations for installation

0 1 2 3

0

12

24

36

48

60

Frequency (Ηz)

Average PSDF of {Ft}

For first storey, (Ft)N=1

For topmost storey, (Ft)N=76

PSD

F of

win

d fo

rce (

kN2 /

Hz)

PSDF of {Ft}

Figure 2: The PSDF of the wind forces applied on building.

of the TMDs are identified based on the mode shapes ofthe uncontrolled building and subsequently based on thecontrolled building in a step-by-step manner. The TMDs areplaced where the mode shape amplitude of the building isthe largest in the particular mode and each of the TMDs istuned with the corresponding modal frequency. Next largeramplitude is preferred over the largest when already a TMDis installed on a particular floor.

Figure 3 shows the first five mode shapes of the uncon-trolled/controlled building and the placement of the fiveTMDs as follows: TMD-1 at 76th floor, that is, at the topmostfloor; TMD-2 at 75th floor; TMD-3 at 74th floor; TMD-4at 61st floor; and TMD-5 at 65th floor. Note that while theplacement of the TMDs is in accordance with the largest orlarge amplitude of themode shape, notmore than oneTMD isplaced on one floor, which would ease installation intricaciesof the TMDs. In addition, placement of subsequent TMD hasbeenmade taking in to account themodifiedmode shape dueto the addition of the TMD in the preceding step. The firstfive natural frequencies of the uncontrolled (NC) building are0.1600, 0.7651, 1.9921, 3.7899, and 6.3945Hz, which were thetuning frequencies for the TMD-1, TMD-2, TMD-3, TMD-4, and TMD-5, respectively, controlling the correspondingmodes. Efficacy of the d-MTMDs is established by comparingthree cases: (i) placing one TMDat the topmost floor denotedby STMD; (ii) placing five TMDs at the topmost floor denotedby MTMDs; and (iii) the abovementioned pattern of fivedistributed TMDs on different floors denoted by d-MTMDs.Only first five modes are controlled in this work as theypredominantly influence the total dynamic response, theirmodal mass [𝑀

𝑟] participation being 90% or greater. For

the building considered herein, 90% of mass of the buildingparticipated in the first five modes, which have been decidedto control.

Figure 4 shows the procedure followed for (a) placementof the d-MTMDs and (b) optimization of parameters of


0 10

10

20

30

40

50

60

70

Mode 1

Floo

r num

bers

TMD-1@top

NC T1 = 6.25 s

−1

STMD T1 = 6.15 s5d-MTMDs T1 = 6.24 s

Normalized amplitude

(a)

0

10

20

30

40

50

60

70

Floo

r num

bers

TMD-2@75th

0 1 Mode 2

−1

NC T2 = 1.3 sSTMD T2 = 1.3 s5d-MTMDs T2 = 1.25 s


(b)

0

10

20

30

40

50

60

70

Floo

r num

bers

TMD-3@74th

0 1 Mode 3−1



(c)

0

10

20

30

40

50

60

70

Floo

r num

bers

TMD-4@61st

0 1 Mode 4−1



(d)

0

10

20

30

40

50

60

70

Floo

r num

bers

TMD-5@65th

0 1

Mode 5

−1



(e)

Figure 3: First five mode shapes of uncontrolled and controlled 76-storey benchmark building.

the d-MTMDs. The modal analysis is conducted to find thenatural frequencies [Ω

𝑖, 𝜔𝑗], mode shapes {𝜙

𝑖,𝑗}, and modal

mass contribution [𝑀𝑟] of the uncontrolled and controlled

building using its stiffness [𝐾𝑠] and mass [𝑀

𝑠] matrices for

(𝑁+𝑛) degrees of freedom (DOF).The first TMD is located atthe largest amplitude in first mode shape, 𝜙

1: top floor in this

building.The number ofmodes to be controlled is based on atleast 90% of the total mass of the building contributing in themodal response. Subsequently, the parameters of the TMDs

are optimized by assuming their stiffness to be the same. Themasses of TMDs are calculated from the known frequenciesand stiffness of the TMDs. The mass ratio, 𝜇 = 𝑚

𝑛/𝑀𝑁, is

assumed to be 0.0082 as recommended by Patil and Jangid[18], and this value is kept the same in all cases of 𝑛 TMDsfor comparison purpose. Thus, the effectiveness of TMDinstalled on a structure will depend onmass ratio (𝜇) betweenthe total mass of the TMDs,𝑚

𝑛= ∑𝑗=𝑛

𝑗=1𝑚𝑗, and the building,

𝑀𝑁= ∑𝑖=𝑁

𝑖=1𝑀𝑖. The total mass of the STMD, MTMDs, and


Determine

Start

End

If any TMD isalready placed

Yes

Yes No

No

Go for next higher

Location optimizer

Modal analysis to find

If d-MTMDs

Yes

Locate TMD based

Newmark’s 𝛽solution for

No

Yes

No

No

Yes

Data input Ks,Ms,

Ft , j = 0

Ks,Ms, ,Cs and

,Cs and

Ft

Ωi, 𝜔j , 𝜙(i,j) , Mr , j = 0 : n

n = count if ∑N

i=1Mr ≤ 0.9MN

If n ≥ 2

on 𝜙i,j , j = 2 : n

(i + j)

i = N, j = 1 : n

i = N, j = 1 : n

If j = ncj = 2𝜉dmj𝜔j, j = 1 : n

1 : n

mn = 𝜇MN

Parameter mj, kj, cj optimizer

(

(

(

(

(

(

(

(

[ [[ [{

{

{

{

mj = kj/𝜔2j , j =

kj = mn/ 1/𝜔21 + 1/𝜔2

2 + · · · 1/𝜔2n

Figure 4: Flowchart for optimizing location and design parameters of d-MTMDs for wind response control of 76-storey benchmark building.

d-MTMDs is kept the same for comparison purpose in all thecases investigated. In the three cases, STMD, MTMDs, and

d-MTMDs, the mass matrix is of order (𝑁 + 𝑛) × (𝑁 + 𝑛) asfollows:

[𝑀𝑠] =

[[[[[[[[[[[[[[[[[[[

[

𝑀1

0 0 ⋅ ⋅ ⋅ 0 0 0 0 ⋅ ⋅ ⋅ 0

0 𝑀2

0 ⋅ ⋅ ⋅ 0 0 0 0 ⋅ ⋅ ⋅ 0

0 0 𝑀3⋅ ⋅ ⋅ 0 0 0 0 ⋅ ⋅ ⋅ 0

.

.

.

.

.

.

.

.

. d...

.

.

.

.

.

.

.

.

. d...

0 0 0 ⋅ ⋅ ⋅ 𝑀𝑁−1

0 0 0 ⋅ ⋅ ⋅ 0

0 0 0 ⋅ ⋅ ⋅ 0 𝑀𝑁

0 0 ⋅ ⋅ ⋅ 0

.

.

.

.

.

.

.

.

. ⋅ ⋅ ⋅

.

.

. 0 𝑚1

0 ⋅ ⋅ ⋅ 0

0 0 0 ⋅ ⋅ ⋅

.

.

. 0 0 𝑚2⋅ ⋅ ⋅ 0

.

.

.

.

.

.

.

.

. d...

.

.

.

.

.

.

.

.

. d...

0 0 0 ⋅ ⋅ ⋅ 0 0 0 0 ⋅ ⋅ ⋅ 𝑚𝑛

]]]]]]]]]]]]]]]]]]]

]

. (2)


For the building installed with the STMD, MTMDs, or d-MTMDs, stiffness and damping of the TMDs were input in

the generic stiffness matrix [𝐾𝑠] and damping matrix [𝐶

𝑠] as

follows:

[𝐾𝑠] =

[[[[[[[[[[[[[[[[[[[[

[

𝐾1+ 𝐾2+ 𝑘𝑛

−𝐾2

0 ⋅ ⋅ ⋅ 0 0 0 0 ⋅ ⋅ ⋅ −𝑘𝑛

−𝐾2

𝐾2+ 𝐾3

−𝐾3

⋅ ⋅ ⋅ 0 0 0 0 ⋅ ⋅ ⋅ 0

0 −𝐾3

𝐾3+ 𝐾4⋅ ⋅ ⋅ 0 0 0 0 ⋅ ⋅ ⋅ 0

.

.

.

.

.

.

.

.

. d...

.

.

.

.

.

.

.

.

. d...

0 0 0 ⋅ ⋅ ⋅ 𝐾𝑁−1

+ 𝐾𝑁+ 𝑘2

−𝐾𝑁

0 −𝑘2⋅ ⋅ ⋅ 0

0 0 0 ⋅ ⋅ ⋅ −𝐾𝑁

𝐾𝑁+ 𝑘1−𝑘1

0 ⋅ ⋅ ⋅ 0

.

.

.

.

.

.

.

.

. ⋅ ⋅ ⋅

.

.

. −𝑘1

𝑘1

0 ⋅ ⋅ ⋅ 0

0 0 0 ⋅ ⋅ ⋅ −𝑘2

0 0 𝑘2

⋅ ⋅ ⋅ 0

.

.

.

.

.

.

.

.

. d...

.

.

.

.

.

.

.

.

. d...

−𝑘𝑛

0 0 ⋅ ⋅ ⋅ 0 0 0 0 ⋅ ⋅ ⋅ 𝑘𝑛

]]]]]]]]]]]]]]]]]]]]

]

, (3)

[𝐶𝑠] =

[[[[[[[[[[[[[[[[[[[

[

𝐶1+ 𝐶2+ 𝑐𝑛

−𝐶2

0 ⋅ ⋅ ⋅ 0 0 0 0 ⋅ ⋅ ⋅ −𝑐𝑛

−𝐶2

𝐶2+ 𝐶3

−𝐶3

⋅ ⋅ ⋅ 0 0 0 0 ⋅ ⋅ ⋅ 0

0 −𝐶3

𝐶3+ 𝐶4⋅ ⋅ ⋅ 0 0 0 0 ⋅ ⋅ ⋅ 0

.

.

.

.

.

.

.

.

. d...

.

.

.

.

.

.

.

.

. d...

0 0 0 ⋅ ⋅ ⋅ 𝐶𝑁−1

+ 𝐶𝑁+ 𝑐2

−𝐶𝑁

0 −𝑐2⋅ ⋅ ⋅ 0

0 0 0 ⋅ ⋅ ⋅ −𝐶𝑁

𝐶𝑁+ 𝑐1−𝑐1

0 ⋅ ⋅ ⋅ 0

.

.

.

.

.

.

.

.

. ⋅ ⋅ ⋅

.

.

. −𝑐1

𝑐1

0 ⋅ ⋅ ⋅ 0

0 0 0 ⋅ ⋅ ⋅ −𝑐2

0 0 𝑐2

⋅ ⋅ ⋅ 0

.

.

.

.

.

.

.

.

. d...

.

.

.

.

.

.

.

.

. d...

−𝑐𝑛

0 0 ⋅ ⋅ ⋅ 0 0 0 0 ⋅ ⋅ ⋅ 𝑐𝑛

]]]]]]]]]]]]]]]]]]]

]

. (4)

The above mentioned [𝑀𝑠], [𝐶𝑠], and [𝐾

𝑠] matrices of the

building are obtained when rotational degrees of freedomat each floor level are ignored as presented earlier by Eliasand Matsagar [23]. However, in the tall building consideredhere it is recommendable to consider the rotational degreesof freedom at each floor level. Nevertheless, the rotationaldegrees of freedom can be condensed and only translationaldegrees of freedom can be retained in the wind responseanalysis. The condensed stiffness matrix in NC case [𝐾

𝑁] is

calculated by removing the rotation degrees of freedom bystatic condensation, whereas the dampingmatrix, [𝐶

𝑁] is not

explicitly knownbut can be defined usingRayleigh’s approachwith damping ratio (𝜁

𝑠= 0.01) for five modes. For the

building installed with the STMD, MTMDs, or d-MTMDs,the stiffness and damping of the TMDs are incorporatedin the generic stiffness matrix [𝐾

𝑠] and damping matrix

[𝐶𝑠] defined, respectively, in (3) and (4) with corresponding

displacement and velocity vectors, as follows:

[𝐾𝑠] {𝑥𝑠} = [

[𝐾𝑁]𝑁×𝑁

[0]𝑁×𝑛

[0]𝑛×𝑁

[0]𝑛×𝑛

]{{𝑋𝑖}𝑁×1

{0}𝑛×1

}

+ [

[𝐾𝑛]𝑁×𝑁

− [𝐾𝑛]𝑁×𝑛

− [𝐾𝑛]𝑛×𝑁

[𝐾𝑛]𝑛×𝑛

]{

{𝑋𝑖}𝑁×1

{𝑥𝑗}𝑛×1

} ,

[𝐶𝑠] {�̇�𝑠} = [

[𝐶𝑁]𝑁×𝑁

[0]𝑁×𝑛

[0]𝑛×𝑁

[0]𝑛×𝑛

]{{�̇�𝑖}𝑁×1

{0}𝑛×1

}

+ [

[𝐶𝑛]𝑁×𝑁

− [𝐶𝑛]𝑁×𝑛

− [𝐶𝑛]𝑛×𝑁

[𝐶𝑛]𝑛×𝑛

]{

{�̇�𝑖}𝑁×1

{�̇�𝑗}𝑛×1

} ,

(5)

inwhich [𝐾𝑛] and [𝐶

𝑛] are the stiffness and dampingmatrices

corresponding to the degrees of freedom of the TMDs.The first five modal frequencies to be controlled and

frequency of each TMD are calculated as

𝑓1=

𝜔1

Ω1

, 𝑓2=

𝜔2

Ω2

, 𝑓3=

𝜔3

Ω3

,

𝑓4=

𝜔4

Ω4

, 𝑓5=

𝜔5

Ω5

,

(6)

where the tuning frequency ratios are 𝑓1= 𝑓2= ⋅ ⋅ ⋅ = 𝑓

5=

1. Moreover, 𝜔1to 𝜔5and Ω

1to Ω5are the frequencies of

the TMD and first five natural frequencies of the building,respectively. In the MTMD devices, it is more suitable todesign a set of TMD units with equal stiffness, 𝑘

1= 𝑘2= 𝑘3=

⋅ ⋅ ⋅ = 𝑘𝑛, rather than identical masses.Therefore, stiffness (𝑘

𝑗)

of the TMDs is calculated as

𝑘𝑗=

𝑚𝑛

(1/𝜔2

1+ 1/𝜔

2

2+ ⋅ ⋅ ⋅ (1/𝜔

2

𝑛))

for 𝑗 = 1 to 5. (7)


−0.6

−0.4

−0.2

0.0

0.2

0.4

100 200 300 400 500 600 700 800 900−0.4

−0.2

0.0

0.2

0.4

0.6

0.1930.279

0.323

0.167

0.179

0.1550.178

NCSTMD

5MTMDs-all.top5d-MTMDs

0.317

Time (s)

Top

floor

di

spla

cem

ent (

m)

Top

floor

acce

lera

tion

(m/s

2 )

Figure 5: Time variation of top floor displacement and top floor acceleration for 76-storey benchmark building under wind forces.

Here,𝑚𝑛is calculated for a particular mass ratio, 𝜇. Themass

(𝑚𝑗) is used for adjusting the frequency of each TMD unit

such that

𝑚𝑗=

𝑘𝑗

𝜔2

𝑗

for 𝑗 = 1 to 5. (8)

The damping ratios (𝜉𝑑= 𝜉1= 𝜉2= ⋅ ⋅ ⋅ 𝜉

𝑛) of the TMDs are

kept the same and the damping (𝑐𝑗) of the TMDs is calculated

as

𝑐𝑗= 2𝜉𝑑𝑚𝑗𝜔𝑗

for 𝑗 = 1 to 5. (9)

3. Solution of Equations of Motion

Classicalmodal superposition technique cannot be employedin the solution of equations of motion here because thesystem is nonclassically damped owing to the differencein the damping in system with TMDs as compared tothe damping in the system with no control. Therefore, theequations of motion are solved numerically using Newmark’smethod of step-by-step integration, adopting linear variationof acceleration over a small time interval of Δ𝑡. The timeinterval for solving the equations of motion is taken as0.1333/100 s.

4. Numerical Study

A comparison of wind responses is made for the linearmodel of the 76-storey benchmark building installed withthe STMD, MTMDs all on top floor, and d-MTMDs. InFigure 5, time variation of top floor displacement and topfloor acceleration for the 76-storey benchmark building

under wind forces are plotted for the cases of (i) uncon-trolled (NC), (ii) controlled by single-TMD (STMD), (iii)controlled by installing five MTMDs on top floor of thebenchmark building (5MTMDs-all.top), and (iv) controlledby five distributedMTMDs (5d-MTMDs).Themass ratio forthe STMD is assumed to be 0.0082 of the total mass of thebuilding, and total mass of the MTMD is taken equal to massof the STMD.The peak top floor displacements for these fourcases are 0.323m, 0.279m, 0.193m, and 0.167m, respectively;and the peak top floor accelerations are 0.317m/s2, 0.155m/s2,0.179m/s2, and 0.178m/s2, respectively. It is observed that themaximum reduction of top floor displacement is achievedwhen the d-MTMDs are installed as per the optimizedlocation and design parameters (Figure 4). As compared tothe uncontrolled structure (NC), the top floor displacementsare reduced by around 15%, 40%, and 50%, respectively, whenSTMD, all MTMDs at top floor, and d-MTMDs are installed.The top floor acceleration is also reduced in all controlledcases significantly as compared to the NC case; nevertheless,maximum reduction in the acceleration is achieved in caseof the STMD installation. As compared to the uncontrolledstructure (NC), the top floor accelerations are reduced byaround 50%, 45%, and 45%, respectively, when STMD, allMTMDs at top floor, and d-MTMDs are installed.

Optimum number of dampers, 𝑛, should be determinedfor improving control performance, economy, and construc-tional feasibility. To facilitate direct comparisons and toshow the performance of various devices a set of twelveperformance criteria are proposed by Yang et al. [19] for the76-storey benchmark building. To measure the reduction inroot mean square (RMS) response quantities of the windexcited benchmark building, the performance criteria 𝐽

1to 𝐽4


0.45

0.50

0.55

0.60

1 2 3 4 50.4

0.5

0.6

0.7

Maximum normalized

nd-MTMDsMTMDs-all.top




Tuning frequency

Average normalized

Total degrees of

Maximum normalized

1 2 3 4 5Numbers of dampers, n Numbers of dampers, n

1 2 3 4 5Numbers of dampers, n

1 2 3 4 5Numbers of dampers, n

Number of stories

Average normalized

Maximum normalized Average normalized

Maximum normalized Average normalized RMS displacement (J3) RMS displacement (J4)

Perfo

rman

ce cr

iterio

n (J1,J

2,J

7,J

8)

Perfo

rman

ce cr

iterio

n (J3,J

4,J

9,J

10)

peak displacement (J9) peak displacement (J10)

peak acceleration (J7) peak acceleration (J8)RMS acceleration (J1) RMS acceleration (J2)

, N = 76

Mass ratio , 𝜇 = 0.0082Damping ratio , 𝜉d = 0.05

freedom (DOF) = (N + n) ratio, f = 1

Figure 6: Variation of performance criteria 𝐽1to 𝐽4and 𝐽7to 𝐽10with number of d-MTMDs and MTMDs.

are defined. These response quantities with control measuresare normalized by the response quantities of the uncontrolledbuilding.

The first evaluation criterion for the controllers is theirability to reduce the maximum floor RMS acceleration. Anondimensional form of this performance criterion is givenby

𝐽1= max

(𝜎�̈�1, 𝜎�̈�30

, 𝜎�̈�50

, 𝜎�̈�55

, 𝜎�̈�60

, 𝜎�̈�65

, 𝜎�̈�70

, 𝜎�̈�75

)

𝜎�̈�75o

,

(10)

where 𝜎�̈�𝑖

= RMS acceleration of the 𝑖th floor and 𝜎�̈�75o =

0.091m/s2 = RMS acceleration of the 75th floor without con-trol.

The second criterion is the average performance of accel-eration for selected floors above the 49th floor:

𝐽2=

1

6

∑

𝑖

𝜎�̈�𝑖

𝜎�̈�𝑖𝑜

, for 𝑖 = 50, 55, 60, 65, 70, 75, (11)

where 𝜎�̈�𝑖𝑜

= RMS acceleration of the 𝑖th floor without con-trol.

The third and fourth evaluation criteria are the abilityof the controller to reduce the top floor displacements. Thenormalized forms of the criteria are given as follows:

𝐽3=

(𝜎𝑋76

)

(𝜎𝑋76o)

,

𝐽4=

1

7

∑

𝑖

(𝜎𝑋𝑖)

(𝜎𝑋𝑖𝑜

)

, for 𝑖 = 50, 55, 60, 65, 70, 75, 76,

(12)

where 𝜎𝑋𝑖

and 𝜎𝑋𝑖o = RMS displacements of the 𝑖th floor

with and without control, respectively; 𝜎𝑋76o = 0.101m is

the RMS displacement of the 76th floor of the uncontrolledbuilding.

To find the peak response of controlled structure normal-ized by the peak response of the uncontrolled building, theperformance criteria 𝐽

7to 𝐽10are defined. Because the present

study is related to passive system of control, there is no needto consider the other four performance criteria 𝐽

5, 𝐽6, 𝐽11, and

𝐽12which represent the performance of the actuator in active

control systems.


−1.0

−0.5

0.0

0.5

1.0

0.643

STMD

0 100 200 300 400 500 600 700 800 900Time (s)

Stro

ke o

f TM

D (m

)

(a)

TMD-1@topfloor

0 100 200 300 400 500 600 700 800 900Time (s)

0.6490.669

d-MTMDsMTMDs-all.top

−1.0

−0.5

0.0

0.5

1.0

Stro

ke o

f TM

D (m

)(b)

−0.6

−0.3

0.0

0.3

0.60.459

0.480

0 100 200 300 400 500 600 700 800 900Time (s)

Stro

ke o

f TM

D (m

)

TMD-2@75

(c)

0 100 200 300 400 500 600 700 800 900Time (s)

−0.6

−0.3

0.0

0.3

0.6

Stro

ke o

f TM

D (m

)

TMD-3@74

0.396 0.447

(d)

0 100 200 300 400 500 600 700 800 900−0.4

−0.2

0.0

0.2

0.4

TMD-4@61

0.375 0.360

Time (s)

Stro

ke o

f TM

D (m

)

(e)

TMD-5@65

0 100 200 300 400 500 600 700 800 900Time (s)

−0.4

−0.2

0.0

0.2

0.4

Stro

ke o

f TM

D (m

)

0.351 0.367

(f)

Figure 7: Time variation of the strokes of the STMD, MTMDs, and d-MTMDs.


The performance in terms of the peak response quantitiesis also important in design of the system. This set ofnondimensional performance criteria is defined as follows:

𝐽7=

max (�̈�𝑝1, �̈�𝑝30

, �̈�𝑝50

, �̈�𝑝55

, �̈�𝑝60

, �̈�𝑝65

, �̈�𝑝70

, �̈�̈𝑥75)

(�̈�𝑝75𝑜

)

,

𝐽8=

1

6

∑

𝑖

(�̈�𝑝𝑖)

(�̈�𝑝𝑖𝑜)

, for 𝑖 = 50, 55, 60, 65, 70, 75,

𝐽9=

(𝑋𝑝76

)

(𝑋𝑝76o)

,

𝐽10=

1

7

∑

𝑖

(𝑋𝑝𝑖)

(𝑋𝑝𝑖𝑜)

, for 𝑖 = 50, 55, 60, 65, 70, 75, 76,

(13)

where𝑋𝑝𝑖and𝑋

𝑝𝑖𝑜= peak displacements of the 𝑖th floor with

and without control; �̈�𝑝𝑖

and �̈�𝑝𝑖𝑜

= peak accelerations ofthe 𝑖th floor with and without control; for instance, 𝑋

𝑝76𝑜=

0.323m and �̈�𝑝75𝑜

= 0.317m/s2.The variations of the performance criteria with increased

number of TMDs for a chosen mass ratio are shown inFigure 6. It can be observed that the performance criteriafor normalized peak acceleration, 𝐽

1, 𝐽2, 𝐽7, and 𝐽

8in both

cases, MTMDs and d-MTMDs, have improved significantly.Improved performance is achieved by installing the d-MTMDs as compared to the STMD and MTMDs installa-tions; however, the variations of the performance criteria byincreasing the number of TMDs in two cases of MTMDsand d-MTMDs for acceleration are observed to be similar.Most significant advantage of installing the d-MTMDs inimproving performance is to control the RMS and peakdisplacement of the building as compared to when MTMDsand STMD are installed (Figure 6).

To study the performance of the d-MTMDs the numberof TMDs is increased up to five, with each controllingdifferent modal response. The improvement in the perfor-mance criteria 𝐽

3, 𝐽4, 𝐽9, and 𝐽

10is achieved when five

modes are controlled in the d-MTMDs with their optimizedlocations and parameters (Figure 4), as compared to that ofthe MTMDs all installed at the top floor.

From (7) and (8), it may be inferred that TMD-1 willbe more effective as compared to TMD-2, TMD-3, TMD-4,and TMD-5 in the systems, MTMDs-all.top and d-MTMDs.The effectiveness of each TMD can be studied by calculatingstroke, such that a TMD unit with higher stroke is moreeffective.Therefore, strokes of the TMDs in the three systems,STMD, 5MTMDs-all.top, and 5d-MTMDs, are calculated andshown in Figure 7. It is observed that STMD and TMD-1 inboth the systems exhibit largest strokes. Thereby, substantialamount of energy of the wind load will be dissipated inthe first mode control. In addition, it is also evident fromthe figure that the performance of the MTMDs-all.top andd-MTMDs is improved as compared to the STMD case.Moreover, the strokes in case of MTMDs-all.top are lowermarginally than those in case of the d-MTMDs, thereby

signifying effectiveness of the latter. It can therefore beconcluded that the TMDs installed to control higher modesare effective.

5. Conclusions

Wind response control of a 76-storey benchmark buildinginstalled with nondistributed and distributed MTMDs asper modal frequencies and mode shapes is investigated. Acomparison of the response of the buildings installed withthe TMDs all at top floor and distributed along the heightof the building (d-MTMDs) with optimized location andparameters is made. From the trends of the results of thepresent study, the following conclusions are drawn.

(1) The installation of d-MTMDs is effective in signif-icantly reducing the peak top floor displacement ofthe building under the wind excitation. The acceler-ation response is also controlled effectively by the d-MTMDs as compared to the STMD and MTMDs.

(2) The installation of d-MTMDs in accordance withthe modal properties, that is, modal frequencies andmode shapes, is more effective than the STMD and allTMDs installed on top floor.

(3) The peak displacement response reductions in case ofthe STMD, MTMDs all at top floor, and d-MTMDs,respectively, are 15%, 40%, and 50%. The peak accel-eration response reductions in case of the STMD,MTMDs all at top floor, and d-MTMDs, respectively,are 50%, 45%, and 45%.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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