Research ArticleLever-Type Tuned Mass Damper for AlleviatingDynamic Responses

Eun-Taik Lee1 and Hee-Chang Eun 2

1Department of Architectural Engineering, Chung-Ang University, Seoul, Republic of Korea2Department of Architectural Engineering, Kangwon National University, Chuncheon, Republic of Korea

Correspondence should be addressed to Hee-Chang Eun; heechang@kangwon.ac.kr

Received 9 June 2019; Revised 25 September 2019; Accepted 1 October 2019; Published 31 October 2019

Academic Editor: Rafael J. Bergillos

Copyright © 2019 Eun-Taik Lee and Hee-Chang Eun. -is is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work isproperly cited.

-is study considers the structural vibration control by a lever-type tuned mass damper (LTMD). -e LTMD has a constraintcondition to restrict the motion at both ends of the lever. -e LTMD controls the dynamic responses at two locations combiningthe tuned mass damper (TMD) and the constraint condition. -e parameters of the LTMD are firstly estimated from the TMDparameters and should be modified by them to obtain from numerical results. -e effectiveness of the LTMD is illustrated in twonumerical experiments, and the sensitivity of the parameters is numerically investigated. It is shown that the LTMD leads to theremarkable displacement reduction and exhibits more definite control than the TMD system because the LTMD controls thevibration responses at two DOFs. More displacement responses are reduced when the installation locations of the LTMD coincidewith the nodes to represent the largest modes’ values at the first and second modes. -e application of the LTMD at the dynamicsystem of a few degrees of freedom (DOFs) is more effective than the system of many DOFs.

1. Introduction

Many countries specify the seismic-resistant design toprotect the building structures and the residents inside dueto earthquakes. Bracing and shear wall must be seismic-resistant members to reduce the structural responses byimproving the lateral stiffness of a structure. -e existingmethods to improve the seismic performance of structuresconsist of controlling the plastic hinge occurrence, in-creasing the deformation capacity and dissipated energy,strengthening or changing the structural system, and en-hancing the lateral stiffness.

-e dynamic control systems can be considered for moredefinite vibration control.-e utilization of a seismic controlsystem to dissipate earthquake energy has been raised toreduce loss of lives and property caused by seismic disasters.-e dynamic control systems are divided into three differentsystems of passive, active, and hybrid controls.

A tuned mass damper (TMD) is a kind of passive controlsystem installed on a structure for reducing the dynamic

responses. -e frequency of the TMD is tuned to thestructural frequency, and the energy is dissipated. -eCitigroup Center in New York City, Chiba Port Tower inJapan, John Hancock Tower in Boston, Canadian NationalTower in Toronto, Crystal Tower in Japan, Taipei 101 inTaiwan, etc., are representative examples to install TMDs.-e design theory for the TMD was initiated by Ormon-droyd and Den Hartog [1] in 1928. Various theories havebeen developed for the designs of undamped and dampedTMD installed on the undamped and damped single-degree-of-freedom (SDOF) system subjected to harmonic excitationor seismic excitation.

Tsai and Lin [2] proposed the optimum tuning frequencyand damping ratio of the TMD through a sequence of curve-fitting schemes. Abdulsalam et al. [3] suggested the optimumfrequency ratio and damping ratio of the TMD installed on astructure subjected to an earthquake loading. Den Hartog[4] did not consider the damping effect of the primarystructure. Abubakar and Farid [5] presented the optimumdesign parameters for the TMD considering the damping of

HindawiAdvances in Civil EngineeringVolume 2019, Article ID 5824972, 11 pageshttps://doi.org/10.1155/2019/5824972

mailto:heechang@kangwon.ac.krhttps://orcid.org/0000-0003-0701-9233https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2019/5824972

the primary structure. Okhovat et al. [6] performed a para-metric study to evaluate the e�ectiveness for the TMD atTehran Tower through the nite element analysis. Murudi andMane [7] investigated the seismic e�ectiveness of TMD andfound that the TMD is not a�ected by the intensity of groundmotion. Warburton and Ayorinde [8] studied the e�ect on theoptimum parameter conditions of light damping in the pri-mary system. Farghaly and Ahmed [9] discussed the designprocedure and the applications through a case study of asymmetrical moment resistance frame twenty-story three-di-mensional model. Nigdeli and Bekdas [10] investigated thecontrol e�ect depending on the location of a TMD on a seismicstructure for an e�ective response reduction. Bakre and Jangid[11] derived the optimum parameters of TMD installed on aviscously damped SDOF system for various combinations ofexcitation and response parameters.

e strategies to improve seismic performance may beestablished by the structural type and assessment results.Stoica [12] provided a seismic retrotting method to con-solidate conventional methods and seismic device such asTMD. Brendike and Petryna [13] studied the TMD tocontrol the dynamic responses as a seismic retrot device ofRC frame structures. Suzuki et al. [14] developed a seismiccontrol device to increase damping of an old bridge forseismic retrot. Nawrotzki et al. [15] introduced the e�ec-tiveness of the tuned mass control systems for the seismicretrotting of existing structures.

is study considers the e�ectiveness of LTMD installedbetween the two nodes in the structure. e LTMD controlsthe structural responses and is designed using a constraintcondition of the lever responses as well as the optimumparameters of the TMD. is work performs the numericalstudy according to the design parameters of the LTMD andcompares the seismic e�ect by the LTMD and TMD in thenumerical experiment. It is shown that the LTMD is moree�ective in controlling the dynamic responses than theTMD. More displacement responses are reduced when theinstallation locations of the LTMD coincide with the nodesto represent the largest modes values at the rst and secondmodes. It is shown that the application of the LTMD at thedynamic system of a few DOFs is more e�ective than thesystem of many DOFs.

2. Formulation

2.1.TMDDesignParameters. Aprimary structure described bynDOFs can be idealized as a SDOF structure. Figure 1 representsa SDOF system consisting of a primary structure and a TMD.Many researchers provided the optimum parameter values of theTMD for reducing the responses of the undamped or dampedsystem subjected to harmonic forces or earthquake load.

e dynamic equation of motion for the systems can bewritten by

mp 0

0 mT[ ]

€up

€uT[ ] +

cp + cT − cT− cT cT

[ ]_up

_uT[ ]

+kp + kT − kT− kT kT

[ ]up

uT[ ] �

Fp

FT[ ],

(1)

where the subscripts “p” and “T” indicate the primarystructure and the TMD, respectively, m, c, and k denote themass, damping, and sti�ness, respectively, and u and F arethe displacement response and external force, respectively.

e optimum design parameters for the TMD takedi�erent forms depending on the types of external forces andthe presence of the damping in the primary structure.Considering the harmonic forces and the earthquake load asthe external forces, Fp � F0eiΩt and FT � 0 and Fp � − mp €ugand FT � − mT €ug, respectively. F0 and €ug represent the forcemagnitude and the acceleration of earthquake, respectively.

Tables 1 and 2 denote the optimum parameters sug-gested by various researchers according to the undampedand damped primary structures, respectively. In the tables,

μ � mT/mp: mass ratioξp � cp/2ωpmp: damping ratio of the primary structureξT � cT/2ωTmT: damping ratio of the TMDωp �

������kp/mp√

: natural frequency of the primarystructureωT �

������kT/mT√

: natural frequency of the TMDf � ωT/ωp: natural frequency ratiofOPT: optimal frequency ratioξOPTT : optimal damping ratio of the TMD

It is observed that the optimal parameters in Tables 1 and2 are deeply a�ected by the mass ratio between the primarystructure and TMD and the damping ratio of the primarystructure. However, the numerical values of the parametersare very close despite the di�erent mathematical forms.From the parameters of the TMD, the LTMD parameterscan be designed, and their e�ectiveness is investigated.

2.2. LTMD. is section considers the parameter design of aLTMD based on the concept of the TMD.e LTMD shownin Figure 2 is installed between the adjacent two nodes in astructure; it is designed by modifying the TMD designparameters and controls the dynamic responses at the twonodes unlike the TMD. e LTMD consists of the masslesslever, the masses, springs, and dampers at both ends, and the

uTkT

CT

mTFT

Fp

kp

mp

Cp

up

Figure 1: A TMD installed on a SDOF primary structure.

2 Advances in Civil Engineering

hinge to restrict the responses at both ends. e system issubjected to a constraint to restrict the interstory drift andprovides the control forces at both ends of the lever. econtrol forces indicate the constraint forces required forsatisfying the constraint condition.

e dynamic equation for the LTMD can be written by

mT1 0

0 mT2[ ]

€uT1

€uT2[ ] +

cT1 0

0 cT2[ ]

_uT1

_uT2[ ]

+kT1 0

0 kT2[ ]

uT1

uT2[ ] �

FT1

FT2[ ],

(2a)

or MT €uT + CT _uT + KTuT � F, (2b)

where the subscripts “T1” and “T2” denote the DOFs at theboth ends of the lever, respectively, and the correspondingdisplacement response and external force are represented byu and F. e system constrained by one constraint conditionbecomes a SDOF system.

e constraint condition from the relationship of laθ �uT1 and lbθ � uT2 can be written by

αuT1 � uT2, (3)

where α � lb/la and θ denotes the rotational angle at thehinge of the lever.

e dynamic equation of motion for the constrainedsystem was proposed by Udwadia and Kalaba [18] in 1992.e equation was derived by minimizing the Gaussianfunction as a function by the di�erence between the con-strained and unconstrained accelerations. e dynamicequation for the LTMD system can be expressed by

€uc � €ua +M− 1/2T AM

− 1/2T( )

+b − A€ua( ), (4)

where €ua � − M− 1T (CT _uT + KTuT − F), €uc and €ua denote theacceleration vector for the constrained and unconstraineddynamic system, respectively, and the matrix A and thevector b represent the coe¥cients in di�erentiating equation(3) twice with respect to the time as

A � α − 1[ ],b � 0.

(5)

Substituting equations (2a), (2b), and (5) into equation(4), utilizing the linear algebra, and arranging the result, thedynamic equation at the upper DOF of the lever yields

mT1 €uT1,c + cT1 _uT1,c + kT1uT1,c � FT1,c, (6)

where cT1 � cT(α2m− 1T1 +m− 1T2)/(m− 1T2 + α2cm− 1T2), cT2 � ccT1,kT1 � kT(α2m− 1T1 +m− 1T2)/(m− 1T2 + α2βm− 1T2), kT2 � βkT1, andFT1,c � m− 1T2(FT1 + αFT2)/(m− 1T2 + α2m− 1T1), mT2 � ηmT1.

e coe¥cients β, c, and η denote the sti�ness ratio,damping ratio, and mass ratio at the one end with respect tothe sti�ness, damping, and mass at the other end of the lever,respectively.

e dynamic equation at the other end of the lever can bederived by inserting the relation of equation (3) into equation(6). e control forces exerted by the LTMD act on thestructure and are obtained by multiplying the second term inthe right-hand side of equation (4) by the mass matrix M:

Fc � M1/2 AM− 1/2( )+b − A€ua( ). (7)

e dynamic responses of the LTMD are controlled bythe control forces estimated by equation (7), and the forcesa�ect the responses of the entire structure.

e LTMD is designed by the ¨owchart shown in Fig-ure 3. It is shown that the TMD parameters are rstly es-timated using assumed mass ratio μ and Tables 1 and 2. edesign parameters of the LTMD α, β, c, and η are selected by

Table 1: Optimal TMD parameters for undamped primary structures.

Loading/researchers fOPT ξOPTTH.F./Den Hartog [1] 1/(1 + μ)

��������������(3/8)(μ/(1 + μ))√

H.A./Warburton [16] (1/(1 + μ))��������(2 − μ)/2√ �����������������

3μ/(4(1 + μ)(2 − μ))√

W.N.F./Warburton [16] (1/(1 + μ))��������(2 + μ)/2√ �������������������������

(μ(4 + 3μ))/(8(1 + μ)(2 + μ))√

W.N.A./Warburton [16] (1/(1 + μ))��������(2 − μ)/2√ ������������������������

(μ(4 − μ))/(8(1 + μ)(2 − μ))√

H.F., harmonic force; H.A., harmonic accelerations; W.N.F., white noise force; W.N.A., white noise accelerations.

Table 2: Optimal TMD parameters for damped primary structures.

Loading/researchers fOPT ξOPTTH.F./Abubakar [5] (1/(1 + μ))(1 − 1.5906ξp

��������μ/(1 + μ)√

)��������������(3/8)(μ/(1 + μ))√

+ 0.1616ξp/(1 + μ)Sadek [17] (1/(1 + μ))(1 − ξp

����������(μ/(1 + μ))√

) (ξp/(1 + μ)) +��������μ/(1 + μ)√

kT1CT1

kp

kT2CT2 uT2

mT2 lb

la

mT1uT1

mp

Cp

up

Figure 2: LTMD conguration.

Advances in Civil Engineering 3

the numerical values minimizing the dynamic responses ofthe structure subjected to external excitations. In the designprocess, the optimum values of the variables μ, α, β, c, and ηare estimated numerically. e e�ectiveness and superiorityof the LTMD are numerically illustrated in the following twoexamples.

3. Applications

3.1. Control of aree-Story Building Structure. Consider thedesign of the LTMD installed between the second and third¨oors in a three-story building structure, as shown in Fig-ure 4, and its dynamic control. e TMD is installed on the¨oor of the structure to exhibit the largest mode value in therst mode. If we add another TMD for more displacementcontrol, it should be installed at the location to exhibit thehighest mode value in the second mode. Utilizing theconcept of MTMD (multi-TMD), the LTMD is located at thethird and second ¨oor corresponding to the highest modevalues in the rst and second mode, respectively.

e parameters interdependently a�ect the dynamicresponses and control. e control by the LTMD is nu-merically evaluated by the design parameters and comparedwith the dynamic control by the TMD. e mechanicalproperties of the primary structure are assumed asm1 � m2 � m3 � 10 kg, c1 � c2 � c3 � 2N·sec/m, andk1 � k2 � k3 � 1, 000N/m.

e primary structure is transformed to a SDOF systemusing the rst natural frequency ω1 and the correspondingmode shape φ1:

ω1 � 4.45 rad./sec.,

φ1 � 0.445 0.8019 1.000[ ]T.

(8)

e modal mass can be calculated by

φT1Mφ1 � 18.4 kg. (9)

Utilizing the above modal and the optimal parameterspresented by Den Hartog and various researchers in Table 1,the optimum parameters of the TMD, fOPT and ξOPTT , arecalculated using the prescribed mass ratio μ. e TMDparameters are designed selecting the mass ratios of 0.02 and0.03 for this study.

e design parameters α, β, c, and η of the LTMDmay beestimated by the TMD optimum parameters and numericalanalysis. is study numerically investigates the dynamiccontrol and design values of those parameters. Firstly, as-suming numerical values of α and ηwith the prescribedmassratio, the other parameters β and c to minimize the squareroot of the sum of the squares (SRSS) by the dynamic re-sponses during an external excitation are determined. Andanother SRSS is calculated using the predetermined pa-rameters β and c, and the parameters α and η to minimizethe SRSS are also selected. We assume that the half-scaledN-S acceleration components of the 1940 El Centroearthquake acted on the structure during the rst 30 sec-onds. Figure 5(a) represents the N-S acceleration componentof the El Centro earthquake.

Figure 5(b) represents the SRSS responses according tothe parameters β and c in substituting the assumed values ofμ � 0.02, α � 0.8091, and η � 1.0 into equation (6).e SRSSresponses are calculated in the range of 0.7≤ β≤ 1.15 and0.7≤ c≤ 1.15 with the step of 0.05. e minimum values ofthe SRSS responses are taken at β � 0.75 and c � 0.70.Substituting these values into equation (6) and numericallyintegrating the second-order di�erential equation, the SRSSresponses are determined according to the parameters α andη and are plotted in Figure 5(c). ey exhibit the minimizedvalues at α � 0.7 and η � 0.95. It is observed from the SRSSplots that the design parameters for the LTMD in-terdependently a�ect the dynamic responses of the structure.

Figures 5(d)–5(f ) compare three dynamic responses ofthe structure without any dynamic control system, withTMD or LTMD. It is shown that the control system re-markably reduces the dynamic responses. And the LTMDsystem is more e�ective in controlling the dynamic re-sponses than the TMD. It is due to the control of the dy-namic responses of adjacent two ¨oors unlike the TMD.econtrol forces or constraint forces necessary to satisfy theconstraint condition of the lever act on the structure, and thedynamic responses are controlled by the forces. Figure 5(g)represents the forces acting on the upper mass of the lever,and the forces αFcT1 act on the lower mass. e dynamiccontrol is accomplished by those forces.

Similar process is performed using the mass ratio ofμ � 0.03, and Figure 6 shows the numerical results. eminimum SRSS responses in Figures 6(a) and 6(b) appear atthe parameters of α � 1.15, β � 1.15, c � 1.15, and η � 1.0. It

mT1

mT2

X2

mT1mT2 m2

m3

m1

k1, C1

u1

k2, C2

u2

k3, C3

u3

Figure 4: A three-story building structure.

Transform primary structureinto a SDOF by model analysis

Design the TMD usingtables 1 and 2

Design the LTMD usingequation (6)Select α, β, γ, η

Assume μ

Figure 3: Flowchart for designing the LTMD.

4 Advances in Civil Engineering

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BetaGamma

SRSS

(m)

(b)

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1.131.141.151.161.171.18

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Figure 5: Continued.

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0 5 10 15 20 25 30–0.5

0

0.5

Time (sec.)

Cont

rol f

orce

s (N

)

(g)

Figure 5: Numerical results of dynamic responses using the mass ratio μ � 0.02. (a) Earthquake accelerations; (b) SRSS according to β and cat xed values of α � 0.8093 and η � 1.0; (c) SRSS according to α and η at xed values of β � 1.15 and c � 1.15; (d) dynamic responses (3rd¨oor) at α � 1.15, β � 1.15, c � 1.15, and η � 1.0; (e) dynamic responses (2nd ¨oor); (f ) dynamic responses (1st ¨oor); (g) control forces atthe upper mass of the lever.

0.81

1.21.4

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1.07

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SRSS

(m)

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1.51.05

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(m)

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t res

pons

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W/LTMD

W/TMD

(d)

Figure 6: Continued.

6 Advances in Civil Engineering

indicates that the LTMD to be installed between two ¨oorscorresponding to the highest mode values of the rst andsecondmodes is e�ective in reducing the dynamic responses.It more denitely controls the dynamic responses than theTMD system as shown in Figures 6(c)–6(e). And the LTMDis very e�ective in reducing the story drift by the controlforces as shown in Figure 6(f).

Figure 7 compares the dynamic responses and thecontrol forces in utilizing the coe¥cient values to minimizethe SRSS shown in Figure 5 (μ � 0.02) and Figure 6(μ � 0.03). It is shown that the dynamic responses are re-duced with the increase in the mass ratio.e increase in themass ratio also leads to the increase in the control forces asshown in Figure 7(c). ough the optimum parameters ofthe LTMD cannot be explicitly established, it is shown thatthey can be obtained by numerical experiments and theLTMD system can more denitely control the dynamicresponses than the TMD system.

3.2. Control of a Simply Supported Beam. e distance be-tween the locations to represent the largest mode values atthe rst and second modes increases with the increase in thenumber of DOFs. In the case of a xed-end beam shown inFigure 8, one end of the lever should be installed at themidspan to represent the largest mode value at the rstmode. is example investigates the vibration controlaccording to the location of the other end of the LTMD andcompares the e�ectiveness with the TMD system.

Assume that the external excitations of 10% magnitudeof the earthquake accelerations in Figure 5(a) vertically act atboth ends of the beam.e dynamic responses for the rst 30seconds are calculated by integrating the second-orderdi�erential equation with the time step of 0.02 seconds. enodal points and the elements are numbered as shown ingure. A beam with a length of 1m is modeled using 20beam elements. e beam has an elastic modulus of 1.95 ×105 MPa and a unit mass of 7, 860 kg/m3. e beam’s gross

cross section is b × h � 75mm × 9mm.e damping matrixis assumed as a Rayleigh damping to be expressed by thesti�ness and mass matrices with proportionality constants of0.001 and 0.002, respectively.

In this example, we consider two cases depending on theinstallation location of the other end of the LTMD: (a) case 1to position at the node 15 to represent the highest modevalue in the second mode and (b) case 2 to position at thenode 13 adjacent to the nodes 10 and 15. e sensitivity ofthe design parameters of the LTMD is numerically in-vestigated. Figure 9(a) represents the SRSS plot of case 1according to the variation of the parameters β and c at xedvalues of μ � 0.03, α � 1.414, and η � 1.0. e SRSS usingthe displacement responses at all 19 nodes is calculated withthe increase of 0.05 in the ranges of 0.7≤ β≤ 1.15 and0.7≤ c≤ 1.15. It is shown that the minimum SRSS is locatedat β � 1.15 and c � 0.95. In next stage, the SRSS of thedisplacements is determined with the increase of 0.05 in theranges of 0.7≤ α≤ 1.15 and 0.7≤ η≤ 1.15 at the xed pa-rameter values of β � 1.15 and c � 0.95.e optimum valuesof the parameters in these ranges can be estimated by α �1.15 and η � 0.85. It is shown in Figure 9(b) that the vi-bration can be more explicitly controlled with the increase inthe parameters α and β in the given ranges rather than theparameters c and η. It indicates that the vibration is moresensitive to the length ratio of the lever and the sti�ness ratioat both ends of the lever.

e SRSS of the displacements is shown in Figures 9(c)and 9(d) when the lever is installed at nodes 10 and 13.Minimum SRSS is obtained when the values of parametersare as follows: α � 1.15, β � 1.15, c � 0.9, and η � 0.8. eseplots also show that the length ratio and the sti�ness ratio aresensitive to the vibration control of the beam.

Figure 10 compares the displacement responses at nodes10 and 15 of case 1. e parameter values of μ � 0.03,α � 1.15, β � 1.15, c � 0.95, and η � 0.85 are utilized. It isobserved in Figures 10(a) and 10(b) that the dynamic re-sponses are remarkably reduced by the TMD or LTMD. And

0 5 10 15 20 25 30–0.05

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)W/TMD

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Figure 6: Numerical results of dynamic responses using the mass ratio μ � 0.03. (a) SRSS according to β and c at xed values of α � 0.8093and η � 1.0; (b) SRSS according to α and η at xed values of β � 0.75 and c � 0.7; (c) dynamic responses (3rd ¨oor) at α � 0.7, β � 0.75,c � 0.7, and η � 0.95; (d) dynamic responses (2nd ¨oor); (e) dynamic responses (1st ¨oor); (f ) control forces at the upper mass of the lever.

Advances in Civil Engineering 7

it is found in those plots that the LTMD system can reduce alittle more dynamic responses than the TMD system becausethe LTMD makes an additional control at another node 15unlike the TMD. -us, it is shown in Figure 10(c) that thedisplacement difference between nodes 10 and 15 can bereduced owing to the control force in the satisfaction of theconstraint condition of the lever motion.

Figure 11 compares the displacement responses at nodes10 and 15 depending on the installation locations of theLTMD of two cases. -e response difference at two casescannot explicitly be recognized but the LTMD of case 1 is alittle more effective than that of case 2. It is shown that theLTMD is a little more effective when it is installed at thelocation of the highest mode value of the second mode.

Figure 12 compares the control forces exerted by theLTMD and TMD. -e constraint forces at node 10 of the

LTMD corresponding to two cases are shown inFigure 12(a). It is shown that the constraint forces in case 1are a little higher than those in case 2. And, it is observed thatthe control forces exhibited by the TMD are larger than theconstraint forces by the LTMD. -us, it is found that thedisplacement responses can be more explicitly controlled bydistributing the control effect of the TMD system into twonodes of the LTMD.

It can be expected from the above two applications thatthe LTMD can be more effective in controlling the dynamicresponses of a low-rise building structure with a few DOFsthan those of a high-rise building structure with manyDOFs.

4. Conclusions

-is study illustrates the effectiveness of the vibrationcontrol by the LTMD. -e LTMD controls the dynamicresponses combining the TMD parameters and the con-straint condition. -ough the optimum parameter values ofthe LTMD cannot be explicitly established, they can beestimated by numerical experiments. -e numerical appli-cations exhibit that the LTMD leads to remarkable

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0

0.2

0.4

0.6

Time (sec.)

Cont

rol f

orce

s (N

)

Mass ratio = 0.02

Mass ratio = 0.03

(c)

Figure 7: Displacement responses and control forces depending on the mass ratio μ. (a) Dynamic responses (2nd floor); (b) dynamicresponses (3rd floor); (c) control forces at the upper mass of the lever.

20@50

1 2 3 18 19 201 2 3 18 19 20

Figure 8: A fixed-ended beam model.

8 Advances in Civil Engineering

0.81

1.21.4

0.81

1.21.4

2.4

2.42

2.44

2.46

2.48×10–4

BetaGamma

SRSS

(m)

(a)

×10–4

0.81

1.21.4

0.81

1.21.4

2.41

2.42

2.43

2.44

AlphaEta

SRSS

(m)

(b)

×10–4

0.81

1.21.4

0.5

1

1.52.425

2.43

2.435

2.44

2.445

Beta

SRSS

(m)

(c)

×10–4

0.81

1.21.4

0.81

1.21.4

2.4

2.42

2.44

2.46

AlphaEta

SRSS

(m)

(d)

Figure 9: Comparison of numerical results depending on installation location (μ � 0.03) of nodes 10 and 15 and 10 and 13. (a) SRSS of case1 when α � 1.414, β � 1.15, c � 0.95, and η � 1.0; (b) SRSS of case 2 when α � 1.15, β � 1.15, c � 0.95, and η � 0.85; (c) SRSS of case 1 whenα � 1.122, β � 1.15, c � 0.9, and η � 1.0; (d) SRSS of case 2 when α � 1.15, β � 1.15, c � 0.9, and η � 0.8.

×10–5

W/LTMD

W/O

W/TMD

5 10 15 20 25 30 350Time (sec.)

–2

–1.5

–1

–0.5

0

0.5

1

Disp

lace

men

t res

pons

es (m

)

(a)

×10–6

W/OW/LTMD

W/TMD

5 10 15 20 25 30 350Time (sec.)

–12

–10

–8

–6

–4

–2

0

2

4

6

8

Disp

lace

men

t res

pons

es (m

)

(b)

Figure 10: Continued.

Advances in Civil Engineering 9

×10–3

–8

–6

–4

–2

0

2

4

6

8

Con

trol f

orce

s (N

)

5 10 15 20 25 30 350Time (sec.)

(a)

25 30 3520105 150Time (sec.)

–0.02

–0.015

–0.01

–0.005

0

0.005

0.01

0.015

0.02

0.025

Con

trol f

orce

s (N

)

(b)

Figure 12: Comparison of control forces by the TMD and LTMD at node 10. (a) Constraint forces exhibited by the LTMD; (b) control forcesexhibited by the TMD.

×10–5

0 105 20 2515 3530Time (sec.)

–1.5

–1

–0.5

0

0.5

1

Disp

lace

men

t res

pons

es (m

)

(a)

×10–6

5 10 15 20 25 30 350Time (sec.)

–10

–8

–6

–4

–2

0

2

4

6

8D

ispla

cem

ent r

espo

nses

(m)

(b)

Figure 11: Comparison of displacement responses of cases 1 and 2. (a) Displacement responses at node 10; (b) displacement responses atnode 15.

×10–6

W/LTMD

W/O W/TMD

5 10 15 20 25 30 350Time (sec.)

–5

–4

–3

–2

–1

0

1

2

3

Disp

lace

men

t diff

eren

ce (m

)

(c)

Figure 10: Displacement responses at nodes 10 and 15 (μ � 0.03) of case 1 using α � 1.15, β � 1.15, c � 0.95, and η � 0.85. (a) Displacementresponses at node 10; (b) displacement responses at node 15; (c) displacement difference between nodes 10 and 15.

10 Advances in Civil Engineering

displacement reduction. And, the LTMD is more effectivecontrol system than the TMD because the LTMD controlsthe displacements between adjacent floors.-e control effectby the LTMD is more sensitive to the length ratio of the leverand the stiffness ratio at both ends of the lever than the otherparameters. -e LTMD is a little more effective when it isinstalled at the location of the highest mode value of thesecond mode. It is found that the displacement responsescan be more explicitly controlled by distributing the controleffect of the TMD system into two nodes of the LTMD.

Data Availability

-e data used to support the findings of this study are in-cluded within the article.

Conflicts of Interest

-e authors declare that they have no conflicts of interest.

Acknowledgments

-is study was supported by the 2018 Research Grant(PoINT) from Kangwon National University.

References

[1] J. Ormondroyd and J. P. Den Hartog, “-e theory of thevibration absorber,” Transactions of the American Society ofMechanical Engineers, vol. 49, pp. A9–A22, 1928.

[2] H.-C. Tsai and G.-C. Lin, “Optimum tuned-mass dampers forminimizing steady-state response of support-excited anddamped systems,” Earthquake Engineering & Structural Dy-namics, vol. 22, no. 11, pp. 957–973, 1993.

[3] I. Abdulsalam, M. Al-Janabi, andM. G. Al-Taweel, “Optimumdesign of tuned mass damper systems for seismic structures,”Seismic Control Systems, vol. 59, pp. 175–184, 2012.

[4] J. P. Den Hartog, Dynamic Vibration Absorbers, MechanicalEngineering Publications University of California, Oakland,CA, USA, 1979.

[5] I. M. Abubakar and B. J. M. Farid, “Generalized Den Hartogtuned mass damper system for control of vibrations instructures,” WIT Transactions on 1e Built Environment,vol. 104, pp. 185–193, 2009.

[6] M. R. Okhovat, M. Rahimian, and A. K. Ghorbani-Tanha,“Tunedmass damper for seismic response reduction of TehranTower,” in Proceedings of the 4th International Conference onEarthquake Engineering, Taipei, Taiwan, 2006.

[7] M.M.Murudi and S. M.Mane, “Seismic effectiveness of tunedmass damper for differenct ground motion parameters,” inProceedings of the 13th World Conference on EarthquakeEngineering, Vancouver, BC, Canada, August 2004.

[8] G. B. Warburton and E. O. Ayorinde, “Optimum absorberparameters for simple systems,” Earthquake Engineering &Structural Dynamics, vol. 8, no. 3, pp. 197–217, 1980.

[9] A. A. Farghaly and M. S. Ahmed, “Optimum design of TMDsystem for tall buildings,” ISRN Civil Engineering, vol. 2012,Article ID 716469, 13 pages, 2012.

[10] S. M. Nigdeli and G. Bekdas, “Performance comparison oflocation of optimum TMD on seismic structures,” In-ternational Journal of 1eoretical and Applied Mechanics,vol. 3, pp. 99–106, 2018.

[11] S. V. Bakre and R. S. Jangid, “Optimum parameters of tunedmass damper for damped main system,” Structural Controland Health Monitoring, vol. 14, no. 3, pp. 448–470, 2007.

[12] N. D. Stoica, “Using modern methods of retrofitting existingseismic vulnerable buildings,” in Proceedings of the 13thSGEM GeoConference on Science and Technologies in GeologyExploration and Mining, Sofia, Bulgaria, 2013.

[13] A. Brendike and Y. Petryna, “Seismic retrofitting of reinforcedconcrete frame structures by tuned mass dampers,” in Pro-ceedings of the 8th International Conference on StructuralDynamics EURODYN, Leuven, Belgium, July 2011.

[14] T. Suzuki, I. Kaneki, and T. Katsukawa, “Application of thepowerful TMD as a measure for seismic retrofit of oldbridges,” in Proceedings of the 12th World Conference onEarthquake Engineering, Auckland, New Zeland, February2000.

[15] P. Nawrotzki, T. Popp, and D. Siepe, “Seismic retrofitting ofstructures with tuned-mass systems,” in Proceedings of the15th World Conference on Earthquake Engineering, Lisbon,Portugal, September 2012.

[16] G. B. Warburton, “Optimum absorber parameters for variouscombinations of response and excitation parameters,”Earthquake Engineering & Structural Dynamics, vol. 10, no. 3,pp. 381–401, 1982.

[17] F. Sadek, B. Mohraz, A. W. Taylor, and R. M. Chung, “Amethod of estimating the parameters of tuned mass dampersfor seismic applications,” Earthquake Engineering & Struc-tural Dynamics, vol. 26, no. 6, pp. 617–635, 1997.

[18] F. E Udwadia and R. E. Kalaba, “A new perspective onconstrained motion,” Proceedings of the Royal Society A:Mathematical, Physical and Engineering Sciences, vol. 439,no. 1906, pp. 407–410, 1992.

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