Research ArticleOptimisation Design and Damping Effect Analysis of LargeMass Ratio Tuned Mass Dampers

Ying-jie Kang and Ling-yun Peng

Beijing Key Laboratory of Earthquake Engineering and Structural Retrofit Beijing University of TechnologyBeijing 100124 China

Correspondence should be addressed to Ling-yun Peng plybjuteducn

Received 1 August 2018 Revised 10 December 2018 Accepted 3 January 2019 Published 27 January 2019

Academic Editor Vadim V Silberschmidt

Copyright copy 2019 Ying-jie Kang and Ling-yun Peng -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

Under harmonic load and random stationary white noise load the existing fitting formulas are not suitable for calculating theoptimal parameters of large mass ratio tuned mass dampers (TMDs) For this reason the optimal parameters of large mass ratioTMDs are determined by numerical optimisation methods and a revised fitting formula is proposed herein based on a curvefitting technique Finally the dynamic time history analysis method is used to study the control effect of large mass ratio TMDs-e results show that when the mass ratio is large the error between the existing fitting formula and the actual optimal value isquite large and the revised fitting formula is applicable to the parameter design of the traditional small mass ratio and large massratio (le1) TMDs When the ratio of local base soil predominant frequency to structure vibration frequency is greater than 4 theoptimal parameters of a TMD under white noise excitation can be calculated according to the revised fitting formula and theremaining conditions should be determined by numerical optimisation In addition a large mass ratio TMD reduces the dynamicresponse of the main structure effectively compared with a small mass ratio TMD and reduces the relative displacement betweenthe TMD and main structure

1 Introduction

In the field of civil engineering the tuned mass damper(TMD) has been widely studied and applied as a vibrationcontrol technology [1 2] -e scope of the technologyrsquos useinvolves eg super high-rise structures [3] toweringstructures [4 5] and long span structures [6] With regardto the development of TMD optimisation design theorythe TMD optimal parameters (optimal frequency ratio andoptimal damping ratio) of a single degree of freedom(SDOF) structure without damping under harmonic loadwere given by Den Hartog as early as 1956 [7] Sub-sequently with different dynamic responses as optimisa-tion objectives the formula for calculating TMD optimalparameters under different load excitation conditions wassupplemented and expanded [8] When considering thedamping ratio of the main structure the optimal param-eters of a TMD often need to be obtained by a numerical

optimisation method and the formula itself is obtained bya curve fitting technique [9 10] Under the excitation ofharmonic load or white noise random load the fittingformula for calculating the optimal frequency ratio and theoptimal damping ratio of a TMD contains only two var-iables (1) the mass ratio between the TMD and the mainstructure and (2) the damping ratio of the main structure[11ndash13] -e existing research results show that the greaterthe mass ratio of TMD the better the control effect onthe structure In a traditional TMD vibration dampingstructure the construction of the TMD often requiresadditional mass blocks For large structures and super high-rise structures the mass ratio can only reach 2ndash5 con-sidering the influence of installation and cost -ereforethe existing fitting formula often considers only the tra-ditional mass ratio of less than 01 which is seldom in-volved in the calculation of the optimal parameters of alarge mass ratio TMD

HindawiShock and VibrationVolume 2019 Article ID 8376781 16 pageshttpsdoiorg10115520198376781

From the control idea of tune frequency to reductionvibration on the premise of ensuring that the localstructure can satisfy its normal use function the localstructure is selected as the mass block of the TMD -enthe nontraditional large mass ratio (gt015) TMD is formedsuch as the structure local isolation [6] the mega-substructure configuration [14 15] and the interlayerisolation structure system [16 17] -e noncritical com-ponents in the structure such as filled walls and floors canalso be used as the mass blocks to form a large mass ratioTMD [18 19] In addition the heavy equipment in anindustrial plant can be constructed into a large mass ratioTMD in the form of suspension or isolation [20] Forexample in the typical coal-fired power plant non-structural coal buckets are used as the mass blocks of TMDorMTMD which are called the coal bucket dampers In theside-coal-bunker thermal power plant structure the massratio of coal buckets to the structure is nearly 05 [21ndash23]When the main mode is controlled in a multi degree offreedom structure the modal mass ratio of some TMDs caneven reach values greater than 1 [24] Some scholars havestudied the parameter optimisation design of large massratio TMDs and their damping control effect -e resultsshow that compared with the traditional small mass ratioTMD the large mass ratio TMDs are effective in improvingthe seismic performance of the structure and are signifi-cantly robust in relation to the change of system parameters[6 16 24]

Against the background of good vibration controleffects of large mass ratio TMDs it remains to be de-termined whether the existing optimal parameter fittingformula is suitable for calculating the optimal parametersof large mass ratio TMDs -erefore based on previousstudies the error analysis and revision of the fittingformula of TMD optimal parameters are carried out inthis study -e paper is organised as follows first theoptimisation objective function and the optimisationanalysis method of an SDOF structure with a TMD underseismic excitation are determined -en the optimalparameter fitting formula of the TMD is revised for ex-citations in the form of harmonic and white noise loadsHence the formula is also suitable for the design of largemass ratio TMDs Subsequently a detailed numericaloptimisation is performed for the excitation of filteredwhite noise loads Finally a time history analysis methodis used to investigate the effectiveness of large mass ratioTMDs on controlling the dynamic response of structuresunder seismic loading

2 Dynamic Equilibrium Equation andStatement of the Optimisation Problem

-e schematic diagram of an SDOF structure equipped witha TMD is shown in Figure 1 As a substructure the TMD isconnected to the main structure through the spring anddamper -e main structure is characterised by its mass m1stiffness k1 and damping c1 Similar to the main structure

the TMD also has the properties of mass m2 stiffness k2 anddamping c2

21 Dynamic Equilibrium Equation -e displacement ve-locity and acceleration of the main structure relative to theground are defined as x1 _x1 and eurox1 respectively Similarlythe displacement velocity and acceleration of the TMDrelative to the ground are defined as x2 _x2 and eurox2 re-spectively When the whole structure is subjected to baseacceleration euroxg the dynamic equilibrium equation of thewhole system is as follows

m1 eurox1 + c1 _x1 + k1x1 + c2 _x1 minus _x2( 1113857 + k2 x1 minus x2( 1113857 minusm1 euroxg

m2 eurox2 + c2 _x2 minus _x1( 1113857 + k2 x2 minusx1( 1113857 minusm2 euroxg

⎧⎨

⎩

(1)

Let euroxg eiωt and the displacement transfer function ofthe main structure be h1(ω) then x1 h1(ω)eiωt -e ex-pression for h1(ω) can be obtained as

h1(ω) minusm1 + Z1(ω)

minusω2m1 + iωc1 + k1( 1113857minusω2Z1(ω)

Z1(ω) m2 iωc2 + k2( 1113857

minusω2m2 + iωc2 + k2( 1113857

(2)

where ω1 k1m1

1113968and ζ1 c12m1ω1 denote the natural

frequency and viscous damping ratio of the main structurerespectively -e natural frequency and viscous dampingratio of the TMD are denoted as ω2

k2m2

1113968and

ζ2 c22m2ω2 respectively -e mass and tuning frequencyratio of the TMD are denoted as μ m2m1 and f ω2ω1respectively -e ratio of the excitation frequency to thenatural frequency of the main structure is defined asg ωω1 Finally the following expression can be obtained

h1(ω) 1ω21

middotminus(1 + μ) 2igfζ2 + f2 minus g2( 1113857minus μg2

minusg2 + 2igζ1 + 1minusg2μ( 1113857 minusg2 + 2igfζ2 + f2( 1113857minus μg4

(3)

(1) Under the excitation of harmonic load the dynamicamplification factor R1 |ω2

1x1euroxg| of the mainstructure is

k1

k2

m1

c1 c2

m2

Figure 1 SDOF structure equipped with a TMD

2 Shock and Vibration

R1

(1 + μ)f2 minusg21113858 11138592

+ 4g2f2ζ22(1 + μ)2

μg2f2 + 4g2fζ1ζ2 minus g2 minus 1( 1113857 g2 minusf2( 11138571113858 11138592

+ 4g2 ζ1 g2 minusf2( 1113857 + fζ2 g2 + μg2 minus 1( 11138571113858 11138592

11139741113972

(4)

(2) Under the excitation of random load the meansquare displacement response σ21 of the mainstructure is given by

σ21 1113946+infin

minusinfinh1(ω)

111386811138681113868111386811138681113868111386811138682S(ω) dω (5)

where S(ω) denotes the spectral density function of randomloads If the external force is modelled as a Gaussian whitenoise with constant power spectral density that isS(ω) S0 then σ21 is given by

σ21 2πS0

ω31

ζ2 1 + f2(μminus 2)(1 + μ)2 + f4(1 + μ)4 + 4f2(1 + μ)3ζ221113960 1113961 + 4f2(1 + μ)2ζ21ζ2 + μfζ1 μ + f2(1 + μ)21113872 1113873 + 4fζ1ζ22(1 + μ)2 1 + f2(1 + μ)( 1113857

4 ζ1ζ2 1minus 2f2 + f4(1 + μ)2 + 4f2ζ22(1 + μ)1113966 1113967 + μfζ22 + μf3ζ21 + 4f2ζ31ζ2 + 4fζ21ζ22 1 + f2(1 + μ)( 11138571113960 1113961

⎡⎢⎣ ⎤⎥⎦

(6)

22 Optimisation Analysis -e displacement response ofthe main structure is taken as the evaluation index of theTMD damping effect Under a certain mass ratio μ theoptimal objective function is as follows

(1) Under the excitation of harmonic load

minR1 f ζ2( 1113857 (7)

(2) Under the excitation of random load

min σ21 f ζ2( 1113857 (8)

For a main structure with no damping ie for ζ1 0theoretical solutions of the TMD optimal design parameterscan be obtained as suggested in [8]

(1) Under the excitation of harmonic load

fopt

1minus μ2

1113968

1 + μ

ζ2opt

3μ

4(2minus μ)(1 + μ)

1113971

(9)

(2) Under the excitation of random load

fopt

1minus μ2

1113968

1 + μ

ζ2opt

μ(1minus μ4)

4(1 + μ)(1minus μ2)

1113971

(10)

where fopt and ζopt denote the optimal frequency ratio andthe optimal damping ratio of the TMD respectively

For a damped main structure ie for ζ1 ne 0 the theo-retical derivation of the optimal design parameters of theTMD is very complicated -erefore numerical optimisa-tion is often used to obtain the optimal parameters -ecommon numerical optimisation methods include thefollowing

(1) -e discretisation of independent variables andderivation of their corresponding objective functionvalues followed by the selection of the optimal valuesand corresponding variables from them [11] -ismethod is called DO in short

(2) Genetic algorithm optimisation methods -ismethod is called GA in short

(3) Particle swarm optimisation methods -is methodis called PSO in short [13]

-e MATLAB software provides an optimisation tool-box (OT) which can also solve multiple nonlinear functionoptimisation problems Under the excitation of randomload three methods including DO PSO and OTare used tosolve the optimal parameters of TMD -e damping ratio ofthe main structure is set to 0 so as to compare with thetheoretical solution Figure 2 shows the calculation results ofthe TMD optimal parameters by DO PSO and OT -eresults calculated by the theoretical equation (10) are alsoplotted It can be seen that the optimal frequency ratio foptdecreases logarithmically with the increase of mass ratio μwhile the optimal damping ratio ζopt increases logarithmi-cally with the increase of mass ratio μ In addition it can alsobe seen that the optimal parameters obtained by the threeoptimisation methods are consistent with the theoreticalresults indicating the accuracy of the numerical optimisa-tion method Based on the consideration of simplicity and

Shock and Vibration 3

eciency the MATLAB OT is used for numericaloptimisation

3 Fitting Formula for Optimal TMDParameters under Harmonic Load andError Analysis

Under harmonic excitation the tting formula for the TMDoptimal frequency ratio and optimal damping ratio is givenin reference [10] as shown below

fopt 1minus μ2radic

1 + μ+1minus 2ζ21radic

minus 1( )

minus(2375minus 1034 μ

radic minus 0426μ) μradic ζ1

minus(3730minus 16903 μ

radic + 20496μ) μradic ζ21

ζ2opt

3μ

4(2minus μ)(1 + μ)

radic

+ 0151ζ1 minus 0170ζ21( )

+ 0163ζ1 + 4980ζ21( )μ

(11)

e above optimisation analysis method is used to obtainthe optimal parameters of dierent mass ratio TMDs asshown in Figure 3 e calculated values of the ttingformula corresponding to the main structure with dierentdamping ratios are also plotted It can be seen that when themass ratio is less than 02 the optimal parameters obtainedfrom the tting formula are in good agreement with theactual optimal parameters which can meet practical designrequirements With the increase of the mass ratio the ttingformula value of the optimal parameters deviates from theactual optimal value and the error increases e tting

formula value of the optimal frequency ratio is smaller thanthe actual value and the error increases with the increase ofmain structure damping When the damping ratio ζ1 is 01and the mass ratio μ is greater than 055 the error is greaterthan 55 At the same time when the damping ratio ζ1 is002 the error of the tting formula value for the optimaldamping ratio is the smallest When the damping ratio ζ1 is01 and the mass ratio μ is greater than 075 the error isgreater than 535 On the whole when the mass ratio μ islarge the error of the existing tting formula is very obvious

Based on the above analysis the tting formula inequation (11) is not suitable for calculating the optimalparameters of large mass ratio TMDs erefore a curvetting method is used to revise the formula and the newtting formula is obtained as follows

fopt 1minus μ2radic

1 + μ+1minus 2ζ21radic

minus 1( )

+(minus26976 + 25809μ

radic minus 09656μ) μradic ζ1

+(minus15547 + 53501μ

radic minus 43634μ) μradic ζ21

ζ2opt

3μ

4(2minus μ)(1 + μ)

radic

+(09614minus 05667 μ

radic minus 01277μ)

middot μ

radicζ1 +(205477minus 678430

μ

radic + 626722μ)μζ21(12)

Figure 4 shows the relationship between the calculatedvalue of the revised tting formula and the actual optimalvalue and the change curve of the error rate along with themass ratio is shown in Figure 5 e calculation method ofthe error rate is as follows

0 02 04 06 08 1

04

05

06

07

08

09

1

micro

f opt

DOPSO

OTEquation (10)

(a)

0 02 04 06 08 1μ

005

01

015

02

025

03

035

04

045

ζ 2op

t

DOPSO

OTEquation (10)

(b)

Figure 2 Comparison of numerical optimisation algorithms and theoretical results (a) Optimisation results of the frequency ratio (b)Optimisation results of the damping ratio

4 Shock and Vibration

Error fopt_Form minusfopt_Num( )

fopt_Numtimes 100

Error ζ2opt_Form minus ζ2opt_Num( )

ζ2opt_Numtimes 100

(13)

where fopt_Form and ζopt_Form are the TMD optimal valuescalculated using the revised tting formula in equation (12)e actual optimal values obtained by a numerical opti-misation method are fopt_Num and ζ2opt_Num

It can be seen that the error between the optimal fre-quency ratio calculated by the revised tting formula and theactual optimal value is smaller When the damping ratio ofthe main structure is 0sim01 the maximum error rate is lessthan 1 When the mass ratio is less than 002 the error isrelatively large is is mainly due to the relatively smallTMD damping ratio which leads to a relatively large errorrate Under the condition of dierent mass ratio TMDs theerror rate can be kept within 5

On the whole the error of the revised tting formula issmall which can meet actual design requirements of a largemass ratio TMD under harmonic excitation

0 02 04 06 08 101

02

03

04

05

06

07

08

09

1

μ

f opt

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004ζ1 = 005ζ1 = 0075ζ1 = 010

Optimal solutionEquation (12)

(a)

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004ζ1 = 005ζ1 = 0075ζ1 = 010

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

μ

ζ 2op

t

Optimal solutionEquation (12)

(b)

Figure 4e relationship between the revised tting formula value and the actual optimal value with the TMDmass ratio and the dampingratio of the main structure under the excitation of harmonic load (a) Optimal frequency ratio (b) Optimal damping ratio

0 02 04 06 08 101

02

03

04

05

06

07

08

09

1

μ

f opt

ζ1 = 00ζ1 = 002ζ1 = 005

ζ1 = 0075ζ1 = 010

Equation (11)

(a)

μ

ζ1 = 00ζ1 = 002ζ1 = 005

ζ1 = 0075ζ1 = 010

Equation (11)

0 02 04 06 08 101

02

03

04

05

06

07

08

09

ζ 2op

t

(b)

Figure 3 e curve of the calculated value of the existing tting formula and the actual optimal value varies with the mass ratio under theexcitation of harmonic load (a) Contrast results of the frequency ratio (b) Contrast results of the damping ratio

Shock and Vibration 5

4 Fitting Formula for Optimal TMDParameters under Random Load andError Analysis

41 StationaryWhite Noise Random Load Under stationarywhite noise excitations the tting formula of the TMDoptimal frequency ratio and optimal damping ratio is givenin reference [11] as shown below

fopt 1minus μ2radic

1 + μ+(minus379441 + 987259

μ

radic minus 152978μ)

middot μ

radicζ1 +(minus136731 + 191284

μ

radic + 217049μ) μradic ζ21

ζ2opt

μ(1minus μ4)

4(1 + μ)(1minus μ2)

radic

(14)

However the tting formula only considers the case thatthemass ratio is less than 01When themass ratio of TMD isgreater than 01 the relationship between the optimal valuecalculated by the tting formula in equation (14) and theactual optimal value is shown in Figure 6 In equation (14)the optimal value of the damping ratio is independent of ζ1consequently there is only a curve for all considered valuesof ζ1 in Figure 6(b) It can be seen that when the mass ratio issmaller than 02 the error between the optimal parameterscalculated by the tting formula and the actual optimalparameters is still relatively small However with the in-crease of the mass ratio the error increases signicantly Inaddition the error of the tting formula increases with theincrease of the damping ratio of the main structure From

the error results the tting formula in equation (14) is alsonot suitable for calculating the optimal parameters of largemass ratio TMDs e new formula obtained by the curvetting method is as follows

fopt 1minus μ2radic

1 + μ+(minus29089 + 23354

μ

radic minus 05790μ) μradic ζ1

+(minus131740 + 307222μ

radic minus 212318μ) μradic ζ21

ζ2opt

μ(1minus μ4)

4(1 + μ)(1minus μ2)

radic

+(minus05747 + 15424μ

radic minus 07846μ) μradic ζ1

+(314166minus 968253 μ

radic + 739404μ)μζ21(15)

Under stationary white noise excitations the relation-ship between the calculated value of the revised ttingformula and the actual optimal value is shown in Figure 7e curve of error rate change relative to the mass ratio isshown in Figure 8e calculationmethod of the error rate isthe same as in equation (13)

e error between the optimal frequency ratio calculatedby the revised tting formula and the actual optimal value issmall e damping ratio of the main structure has an eecton the error However when the damping ratio is 01 themaximum error rate of the optimal frequency ratio is stillbelow 12 In addition the error rate of the optimaldamping ratio varies with the TMD mass ratio but it canalways be kept within 5

0 02 04 06 08 1ndash04

ndash02

0

02

04

06

08

1Er

ror (

)

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004

ζ1 = 005ζ1 = 075ζ1 = 010

μ

(a)

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004

ζ1 = 005ζ1 = 075ζ1 = 010

0 02 04 06 08 1ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

Erro

r (

)

μ

(b)

Figure 5e error rate of the revised tting formula value with the TMDmass ratio and the damping ratio of the main structure under theexcitation of harmonic load (a) Error rate of the frequency ratio (b) Error rate of the damping ratio

6 Shock and Vibration

On the whole the error of the revised tting formula issmall which meets actual design requirements of large massratio TMDs Comparisons between the design method inreference [16] and that in this paper are illustrated in Table 1It can be seen that the optimisation parameters in this studyare obviously larger than those in reference [16] is ismainly due to the inconsistency of the optimisation ob-jectives of the two methods

42 Filtered White Noise Random Load e ltered whitenoise random load is modelled as proposed by Kanai [25]

S(ω) ω4g + 4ζ2gω2

gω2

ω2g minusω2( )

2 + 4ζ2gω2gω2

S0 (16)

where ωg and ζg are the predominant frequency anddamping ratio of foundation soil respectively

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004ζ1 = 005ζ1 = 0075ζ1 = 010

Optimal solutionEquation (15)

0 02 04 06 08 101

02

03

04

05

06

07

08

09

1

μ

f opt

(a)

Optimal solutionEquation (15)

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004ζ1 = 005ζ1 = 0075ζ1 = 010

0 02 04 06 08 10

01

02

03

04

05

06

07

μ

ζ 2op

t

(b)

Figure 7e relationship between the revised tting formula value and the actual optimal value with the TMDmass ratio and the dampingratio of the main structure under stationary white noise excitations (a) Optimal frequency ratio (b) Optimal damping ratio

0 02 04 06 08 1ndash04

ndash02

0

02

04

06

08

1f op

t

ζ1 = 00ζ1 = 002ζ1 = 005

ζ1 = 0075ζ1 = 010

Equation (14)

μ

(a)

ζ1 = 00ζ1 = 002ζ1 = 005

ζ1 = 0075ζ1 = 010

Equation (14)

0 02 04 06 08 101

015

02

025

03

035

04

045

05

055

06

μ

ζ 2op

t

(b)

Figure 6 e curve of the calculated value of the existing tting formula and the actual optimal value varies with the mass ratio understationary white noise excitations (a) Contrast results of the frequency ratio (b) Contrast results of the damping ratio

Shock and Vibration 7

Substituting S(ω) into equation (5) the mean square ofthe structural displacement response is obtained as

σ21 int+infin

minusinfinh1(ω)∣∣∣∣

∣∣∣∣2ω4g + 4ζ2gω

2gω

2

ω2g minusω2( )

2 + 4ζ2gω2gω2

S0 dω (17)

Because solving equation (17) analytically is very tediousa numerical integration method can be used e results byBakre [11] show that the TMD optimal parameters arerelatively close when the structure is excited by white noiserandom load and ltered white noise random load but onlywhen a mass ratio below 01 is examined e analysis resultobtained by Hoang et al [6] shows that there is a certaindierence between the optimal parameters for the largermass ratio TMD

When solving the optimal parameters the eect of themain structure damping ratio should be taken into accountthe predominant frequency and damping ratio of the ground

soil that is ωg and ζg are also taken into considerationHowever it is not practical to construct a correspondingtting formula erefore the optimisation process of TMDoptimal parameters is described in detail and the corre-sponding tting formula is no longer determined e de-tailed optimisation currenow chart is shown in Figure 9 Firstly

Table 1 Optimal design of large mass ratio TMD according toreference [16] and this study

Structuralparameters Method fopt ζopt σ21(2πS0ω3

1)

μ 0431 ζ1 002Reference

[16] 06871 02729 23830

is study 05970 02939 22429

μ 05 ζ1 002Reference

[16] 06543 02900 23862

is study 05549 03135 22246

μ 05 ζ1 003Reference

[16] 06492 02861 22947

is study 05432 03144 21320

Main structure parametersvibration frequency and damping ratio

The initial iteration value of the optimal parametercalculated by fitting formula (15)

Objective function the root mean square of the displacement response of the main structure

MATLAB optimization toolboxfunction Fmincon

Mass ratio of TMD

Parameters of ground motion power spectrum modelpredominant frequency and damping ratio

Optimal parameters and the correspondingobjective function values

Figure 9 e numerical optimisation process of TMD parametersunder ltered white noise excitations

0 02 04 06μ

08 1ndash04

ndash02

0

02

04

06

08

1

12Er

ror (

)

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004

ζ1 = 005ζ1 = 075ζ1 = 010

(a)

μ

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004

ζ1 = 005ζ1 = 075ζ1 = 010

0 02 04 06 08 1ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

Erro

r (

)(b)

Figure 8 e error rate of the revised tting formula value with the TMD mass ratio and the damping ratio of the main structure understationary white noise excitations (a) Error rate of the frequency ratio (b) Error rate of the damping ratio

8 Shock and Vibration

the dynamic characteristics of the main structure includingnatural frequency and damping ratio are determined enthe mass ratio of TMD and site condition parameters in-cluding the predominant frequency and damping ratio areselected After that the objective of optimisation is de-termined that is the mean square of the structural dis-placement Finally taking the calculated value of equation (15)as the initial value the optimal parameters can be obtained byoptimum analysis using MATLAB optimisation toolbox

Taking class II sites [26] as an example the parameters ofthe Kanai-Tajimi model ωg 15708 and ζg 072 aredetermined S0 is always taken as a constant and it has noeect on TMD parameter optimisation At the same time

λ ωgω1 is dened as the ratio between the predominantfrequency of ltered white noise and the natural frequency ofthe structure In order to study the TMD optimal parametersand the relative relationship between the excitation of sta-tionary white noise random load and ltered white noiserandom load the following denitions are given

ηf fopt_Num

fopt_Form

ηζ2 ζ2opt_Numζ2opt_Form

(18)

0 02 04 06 08 102

03

04

05

06

07

08

09

1

μ

f opt

λ = 1λ = 2λ = 3λ = 4

λ = 5λ = 75Equation (15)

(a)

μ0 02 04 06 08 1

098

1

102

104

106

108

11

112

114

λ = 1λ = 2λ = 3

λ = 4λ = 5λ = 75

η f(b)

λ = 1λ = 2λ = 3λ = 4

λ = 5λ = 75Equation (15)

0 02 04 06 08 101

015

02

025

03

035

04

045

05

μ

ζ 2op

t

(c)

λ = 1λ = 2λ = 3

λ = 4λ = 5λ = 75

μ0 02 04 06 08 1

084

086

088

09

092

094

096

098

1

102

104η ζ

2

(d)

Figure 10 e relation between optimal parameters with λ and μ and its deviation rate with equation (15) under ltered white noiseexcitations (a) Optimal frequency ratio (b) ηf (c) Optimal damping ratio (d) ηζ2opt

Shock and Vibration 9

Under stationary white noise excitations fopt_Form andζopt_Form are the TMD optimal frequency ratios and theoptimal damping ratios calculated by the tting equation(15) respectively Under ltered white noise excitationsfopt_Num and ζopt_Num are the TMD optimal frequency ratioand the optimal damping ratio respectively obtained by thenumerical optimisation method

When the main structure with a damping ratio of 005 isexcited by ltered white noise load the relationship betweenthe optimal parameters of the TMD and μ and λ is shown inFigures 10(a) and 10(c) In order to facilitate comparisonthe optimal parameter calculated by the tting formula inequation (15) is also plotted Figures 10(b) and 10(d) showrespectively the trends of ηf and ηζ2opt with μ and for dif-ferent values of λ

From the above results the following results areobtained

When the mass ratio μ is smaller than 02 the optimalparameters of the stationary white noise random load andthat of the ltered white noise random load are more similarbut the dierence between the two is increased graduallywith the increase of the mass ratio which can be clearly seenfrom the relationship curve of ηf to μ

e larger the λ the more ηf and ηζ2opt become closer to1 When λge 4 even if the TMD mass ratio reaches 1 theerror of the optimal parameters of the TMD under the

stationary white noise random load and the ltered whitenoise random load is not more than 5

In addition the damping ratio of the main structurehas a relatively small impact on the error therefore theincurrenuence of the damping ratio of the main structure onthe error of the tting formula is no longer discussed inthis paper In conclusion for the optimal design of a largemass ratio TMD when λge 4 the tting formula inequation (15) is also suitable for ltered white noise ex-citations while in the other cases it is suggested to de-termine the optimal TMD parameters using a numericaloptimisation method

5 Damping Effect Analysis of a Large MassRatio TMD

Four SDOF structures with 05 10 20 and 30 s periods (T)are selected and the damping ratio of the four structures is005 e TMD mass ratios are 005 025 050 075 and 1and the TMD parameters are calculated by the tting for-mula in equation (15) Two far-eld seismic waves (ElCentro Hachinohe) and two near-eld seismic waves(Northridge Kobe) are used for load input [27] Seismicwave information is provided in Table 2 e seismic waveacceleration-time history curve is shown in Figure 11 and

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

Acce

lera

tion

(ms

2 )Ac

cele

ratio

n (m

s2 )

El Centro

Hachinohe

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

(a)

Acce

lera

tion

(ms

2 )Ac

cele

ratio

n (m

s2 )

Northridge

Kobe

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

(b)

Figure 11 Time history curve of seismic records (a) Far-eld seismic waves (b) Near-eld seismic waves

Table 2 Seismic wave information used in the analyses

Type Name Earthquake Station Component Year

Far eld El Centro Imperial valley El Centro N-S 1940Hachinohe Tokachi-Oki Hachinohe city N-S 1968

Near eld Northridge Northridge SCH N-S 1994Kobe Hyogo-ken KJMA N-S 1995

10 Shock and Vibration

the seismic wave peak value is 1ms2 In order to investigatethe eect of the TMD mass ratio on the control of thestructural displacement response four sets of seismic timehistory analyses were carried out for structures with andwithout a TMD respectively

e reduction rate of peak displacement Re(Peak) andthe reduction rate of root mean square of structural dis-placement Re(RMS) are used as the evaluation index re-spectively e two formulas for calculating the dampingrate are shown as follows

Re(Peak) 1minusMax xTMD( )Max(x)

Re(RMS) 1minusRMS xTMD( )RMS(x)

(19)

where x and xTMD are the time history record of the dis-placement response of the structure without and with aTMD respectively e formula for calculating the rootmean square of the displacement is as follows

0 02 04 06 08 1micro

0

10

20

30

40

50Re

(Pea

k) (

)

EI CentroHachinoheKobe

NorthridgeMean

(a)

0 02 04 06 08 1micro

0

10

20

30

40

50

Re(P

eak)

()

EI CentroHachinoheKobe

NorthridgeMean

(b)

0 02 04 06 08 1micro

0

10

20

30

40

50

Re(P

eak)

()

EI CentroHachinoheKobe

NorthridgeMean

(c)

0 02 04 06 08 1micro

0

10

20

30

40

50Re

(Pea

k) (

)

EI CentroHachinoheKobe

NorthridgeMean

(d)

Figure 12 e damping eect of dierent mass ratio (TMD) on the peak value of structural displacement response (a) T 05 s (b)T10 s (c) T 20 s (d) T 30 s

Shock and Vibration 11

RMS(x) sqrt1NsumN

i1x2i (20)

where xi is the structural displacement response corre-sponding to the ith time andN is the total number of pointscollected

Re(Peak) and Re(RMS) of the displacement responsewith dierent mass ratio TMDs are shown in Figures 12 and13 e following conclusions can be obtained

(1) TMD can eectively control the displacement re-sponse of the structure and the large mass ratio

(gt025) TMD is more eective than the conventionalsmall mass ratio (lt005) TMD But it can also befound that when the mass ratio of the TMD is greaterthan 05 the gain eect will diminish with increasingmass ratio

(2) e TMD with the same mass ratio shows certaindiscreteness for the structures with dierent naturalvibration periods and dierent seismic waves Forexample as shown in Figure 12(b) when thestructurersquos period is 10 s the damping rate of fourseismic waves is distinct When the mass ratio is 05the minimum damping rate is 1047 and the

0 02 04 06 08 10

10

20

30

40

50

60

micro

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(a)

0 02 04 06 08 1micro

0

10

20

30

40

50

60

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(b)

0 02 04 06 08 10

10

20

30

40

50

60

micro

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(c)

0 02 04 06 08 1micro

0

10

20

30

40

50

60Re

(RM

S)

EI CentroHachinoheKobe

NorthridgeMean

(d)

Figure 13e damping eect of dierent mass ratios (TMD) on the mean square root of structural displacement response (a) T 05 s (b)T10 s (c) T 20 s (d) T 30 s

12 Shock and Vibration

Table 3 Maximum peak value of relative displacement between a TMD with optimal parameters and the main structure (cm)

T (s) μ El Centro Hachinohe Northridge Kobe Mean

05

005 436 296 381 312 356010 268 214 298 22 25050 214 254 335 191 249075 183 26 275 279 249100 224 274 224 337 265

10

005 818 883 1039 698 859010 353 483 512 598 487050 341 535 376 588 46075 391 59 355 546 47100 356 494 358 502 428

20

005 1477 209 866 1834 1567010 794 1197 554 1042 897050 53 863 417 858 667075 394 795 345 705 56100 389 846 332 629 549

30

005 1509 3529 666 1993 1925010 682 13 378 889 812050 491 1128 329 654 651075 463 1037 293 553 586100 509 1125 262 536 608

minus002

0

002

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(a)

minus002

0

002

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

micro = 005micro = 075

wo

(b)

Figure 14 Continued

Shock and Vibration 13

maximum is 4622 which is usually mainly due tothe dierence in the frequency relationship betweenthe structure and the seismic waves

Table 3 shows the statistical results of the relative dis-placement between the TMD and main structure underdierent time history analysis conditions It can be seen thatthe relative displacement between the TMD and the mainstructure is inversely proportional to the mass ratio of theTMD that is when a large mass ratio TMD is used thedisplacement response of the structure is eectively con-trolled and the displacement stroke of the TMD is clearlyreduced which reduces the requirements for the elasticcomponents and the damping components used to constructthe TMD

e displacement time histories of the structures withperiods of 10 and 20 s and the relative displacement timehistories between the TMD and the structures are shown inFigures 14 and 15 respectively e schemes of mass ratiosof 005 and 075 are compared It can be clearly seen that theeect of a TMD in controlling the structural response andthe TMD displacement stroke is more obvious for the TMDwith a mass ratio of 075 than for the one with a mass ratio of005

In summary the large mass ratio TMD has a moresignicant eect in seismic control of the main structurethan the small mass ratio TMD

6 Conclusions

In order to control the dynamic response and improve theaseismic performance of a structure a large mass ratio TMDdamping system is formed by using the equipment in thebuilding structure or relying on new structural forms e

existing optimal parameter tting formula is not applicableto large mass ratio TMDs so it is revised by numericaloptimisation and curve tting and the dynamic time historyanalysis method is used to study the eect of vibrationdamping control of large mass ratio TMDs e followingconclusions are obtained

Compared with the traditional small mass ratio (lt005)TMD the large mass ratio (gt05) TMD has obvious ad-vantages in controlling the displacement response of themain structure e control eect is about 15sim325 timeshigher the damping eect of the structural displacementpeak can reach about 30 and the damping ratio of the rootmean square displacement can reach about 436 At thesame time the relative stroke between the TMD and themain structure can be reduced with up to 30sim65 which ishighly benecial to the practical engineering design of TMDstructures

When the mass ratio of a TMD is relatively large (gt02)the results calculated by the existing tting formula diersignicantly from the actual optimal value and the calcu-lated values of the revised formula proposed in this paper areshown to be in good agreement with the actual optimalvalue In general the revised formula can be applied to bothtraditional small mass ratio and large mass ratio (le1) TMDsWhen the mass ratio is greater than 1 the optimal pa-rameters of TMD can also be obtained by the methodpresented in this paper

When the mass ratio is greater than 02 the optimalparameters of the stationary white noise random load andthat of the ltered white noise random load are more similarbut the dierence between the two is gradually increasedwith the increase of the TMD mass ratio For the optimalparameters of large mass ratio TMDs (gt02) the error is lessthan 005 when the ratio of the predominant frequency of the

0 5 10 15 20 25 30

minus002minus001

0001002

t (s)

0 5 10 15 20 25 30t (s)

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(c)

0 5 10 15 20 25 30t (s)

0 5 10 15 20 25 30t (s)

minus002

0

002

004

Disp

lace

men

t (m

)

minus01

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(d)

Figure 14 e damping eect of dierent mass ratio TMDs on the displacement-time history of the main structure (T10 s) and therelative displacement-time history between the TMD and the structure (wo denotes the displacement of a structure without a TMD) (a) ElCentro (b) Hachinohe (c) Northridge (d) Kobe

14 Shock and Vibration

base soil and the vibration frequency of the structure isgreater than 4 and the optimal parameters of the TMD canbe calculated by the tting formula proposed in this paperUnder other conditions it is suggested to use an optimi-sation method to determine the optimal value of TMDparameters

At present the actual engineering projects with largemass ratio TMD damping systems are less prominent buttheir aseismic advantages will bring a broad range of benetsfor research and practice

Data Availability

e data used to support the ndings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no concurrenicts of interestregarding the publication of this article

Acknowledgments

is research was supported by Grant no 51478023 from theNational Natural Science Foundation of China

References

[1] M Gutierrez Soto and H Adeli ldquoTuned mass dampersrdquoArchives of Computational Methods in Engineering vol 20no 4 pp 419ndash431 2013

ndash005

0

005

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

ndash01ndash005

0005

01

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(a)

ndash02

ndash01

0

01

02

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

ndash005

005

0

micro = 005micro = 075

wo

(b)

0 5 10 15 20 25 30

minus005

0

005

Disp

lace

men

t (m

)

t (s)

0 5 10 15 20 25 30t (s)

minus01

0

01

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(c)

minus002

0

002

004D

ispla

cem

ent (

m)

0 5 10 15 20 25 30t (s)

0

minus005

005

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30t (s)

micro = 005micro = 075

wo

(d)

Figure 15 e damping eect of dierent mass ratio TMDs on the displacement-time history of the main structure (T 20 s) and therelative displacement-time history between the TMD and the structure (wo denotes the displacement of a structure without a TMD) (a) ElCentro (b) Hachinohe (c) Northridge (d) Kobe

Shock and Vibration 15

[2] S Elias and VMatsagar ldquoResearch developments in vibrationcontrol of structures using passive tuned mass dampersrdquoAnnual Reviews in Control vol 44 pp 129ndash156 2017

[3] D Wang T K T Tse Y Zhou and Q Li ldquoStructural per-formance and cost analysis of wind-induced vibration controlschemes for a real super-tall buildingrdquo Structure and In-frastructure Engineering vol 11 no 8 pp 990ndash1011 2014

[4] N Longarini and M Zucca ldquoA chimneyrsquos seismic assessmentby a tuned mass damperrdquo Engineering Structures vol 79pp 290ndash296 2014

[5] L Tian and Y Zeng ldquoParametric study of tunedmass dampersfor long span transmission tower-line system under windloadsrdquo Shock and Vibration vol 2016 Article ID 496505611 pages 2016

[6] N Hoang Y Fujino and PWarnitchai ldquoOptimal tuned massdamper for seismic applications and practical design for-mulasrdquo Engineering Structures vol 30 no 3 pp 707ndash7152008

[7] J P Den Hartog Mechanical vibrations McGraw-Hill NewYork NY USA 1956

[8] G B Warburton ldquoOptimum absorber parameters for variouscombinations of response and excitation parametersrdquoEarthquake Engineering amp Structural Dynamics vol 10 no 3pp 381ndash401 1982

[9] H-C Tsai and G-C Lin ldquoExplicit formulae for optimumabsorber parameters for force-excited and viscously dampedsystemsrdquo Journal of Sound and Vibration vol 176 no 5pp 585ndash596 1994

[10] H-C Tsai and G-C Lin ldquoOptimum tuned-mass dampers forminimizing steady-state response of support-excited anddamped systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 22 no 11 pp 957ndash973 1993

[11] S V Bakre and R S Jangid ldquoOptimum parameters of tunedmass damper for damped main systemrdquo Structural Controland Health Monitoring vol 14 no 3 pp 448ndash470 2007

[12] C C Lin C M Hu J F Wang and R Y Hu ldquoVibrationcontrol effectiveness of passive tuned mass dampersrdquo Journalof the Chinese Institute of Engineers vol 17 pp 367ndash376 1994

[13] A Y T Leung and H Zhang ldquoParticle swarm optimization oftuned mass dampersrdquo Engineering Structures vol 31 no 3pp 715ndash728 2009

[14] M Q Feng and A Mita ldquoVibration control of tall buildingsusing mega SubConfigurationrdquo Journal of Engineering Me-chanics vol 121 no 10 pp 1082ndash1088 1995

[15] X X Li P Tan X J Li and A W Liu ldquoMechanism analysisand parameter optimisation of mega-sub-isolation systemrdquoShock and Vibration vol 2016 p 12 2016

[16] A Reggio and M D Angelis ldquoOptimal energy-based seismicdesign of non-conventional tuned mass damper (TMD)implemented via inter-story isolationrdquo Earthquake Engi-neering amp Structural Dynamics vol 44 no 10 pp 1623ndash16422015

[17] S J Wang B H Lee W C Chuang and K C ChangldquoOptimal dynamic characteristic control approach forbuilding mass damper designrdquo Earthquake Engineering ampStructural Dynamics vol 47 no 3 2017

[18] H Anajafi and R A Medina ldquoPartial mass isolation systemfor seismic vibration control of buildingsrdquo Structural Controlamp Health Monitoring vol 25 no 2 article e2088 2017

[19] K Yuan M S He and Y M Li ldquoShaking table tests forenergy-dissipation steel frame structures with infilled wallMTMDrdquo Journal of Vibration and Shock vol 33 no 11pp 200ndash207 2014

[20] R Ding M X Tao M Zhou and J G Nie ldquoSeismic behaviorof RC structures with absence of floor slab constraints andlarge mass turbine as a non-conventional TMD a case studyrdquoBulletin of Earthquake Engineering vol 13 no 11 pp 3401ndash3422 2015

[21] L Y Peng Y J Kang Z R Lai and Y K Deng ldquoOptimisationand damping performance of a coal-fired power plantbuilding equipped with multiple coal bucket dampersrdquo Ad-vances in Civil Engineering vol 2018 p 19 2018

[22] Z Shu S Li J Zhang and M He ldquoOptimum seismic designof a power plant building with pendulum tuned mass dampersystem by its heavy suspended bucketsrdquo Engineering Struc-tures vol 136 pp 114ndash132 2017

[23] K Dai B Li J Wang et al ldquoOptimal probability-based partialmass isolation of elevated coal scuttle in thermal power plantbuildingrdquo Structural Design of Tall and Special Buildingsvol 27 no 11 article e1477 2018

[24] M De Angelis S Perno and A Reggio ldquoDynamic responseand optimal design of structures with large mass ratio TMDrdquoEarthquake Engineering amp Structural Dynamics vol 41 no 1pp 41ndash60 2015

[25] K Kanai ldquoSemi-empirical formula for the seismic charac-teristics of the groundrdquo Bulletin of the Earthquake ResearchInstitute gte University of Tokyo vol 35 pp 309ndash325 1957

[26] Code for seismic design of buildings GB50011-2010 BeijingChina 2010

[27] B F Spencer R E Christenson and S J Dyke ldquoNextgeneration benchmark control problem for seismically excitedbuildingsrdquo in Proceedings of the 2nd World Conference onStructural Control pp 1351ndash1360 Kyoto Japan June 1998

16 Shock and Vibration

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R1

(1 + μ)f2 minusg21113858 11138592

+ 4g2f2ζ22(1 + μ)2

μg2f2 + 4g2fζ1ζ2 minus g2 minus 1( 1113857 g2 minusf2( 11138571113858 11138592

+ 4g2 ζ1 g2 minusf2( 1113857 + fζ2 g2 + μg2 minus 1( 11138571113858 11138592

11139741113972

(4)

(2) Under the excitation of random load the meansquare displacement response σ21 of the mainstructure is given by

σ21 1113946+infin

minusinfinh1(ω)

111386811138681113868111386811138681113868111386811138682S(ω) dω (5)

where S(ω) denotes the spectral density function of randomloads If the external force is modelled as a Gaussian whitenoise with constant power spectral density that isS(ω) S0 then σ21 is given by

σ21 2πS0

ω31

ζ2 1 + f2(μminus 2)(1 + μ)2 + f4(1 + μ)4 + 4f2(1 + μ)3ζ221113960 1113961 + 4f2(1 + μ)2ζ21ζ2 + μfζ1 μ + f2(1 + μ)21113872 1113873 + 4fζ1ζ22(1 + μ)2 1 + f2(1 + μ)( 1113857

4 ζ1ζ2 1minus 2f2 + f4(1 + μ)2 + 4f2ζ22(1 + μ)1113966 1113967 + μfζ22 + μf3ζ21 + 4f2ζ31ζ2 + 4fζ21ζ22 1 + f2(1 + μ)( 11138571113960 1113961

⎡⎢⎣ ⎤⎥⎦

(6)

22 Optimisation Analysis -e displacement response ofthe main structure is taken as the evaluation index of theTMD damping effect Under a certain mass ratio μ theoptimal objective function is as follows

(1) Under the excitation of harmonic load

minR1 f ζ2( 1113857 (7)

(2) Under the excitation of random load

min σ21 f ζ2( 1113857 (8)

For a main structure with no damping ie for ζ1 0theoretical solutions of the TMD optimal design parameterscan be obtained as suggested in [8]

(1) Under the excitation of harmonic load

fopt

1minus μ2

1113968

1 + μ

ζ2opt

3μ

4(2minus μ)(1 + μ)

1113971

(9)

(2) Under the excitation of random load

fopt

1minus μ2

1113968

1 + μ

ζ2opt

μ(1minus μ4)

4(1 + μ)(1minus μ2)

1113971

(10)

where fopt and ζopt denote the optimal frequency ratio andthe optimal damping ratio of the TMD respectively

For a damped main structure ie for ζ1 ne 0 the theo-retical derivation of the optimal design parameters of theTMD is very complicated -erefore numerical optimisa-tion is often used to obtain the optimal parameters -ecommon numerical optimisation methods include thefollowing

(1) -e discretisation of independent variables andderivation of their corresponding objective functionvalues followed by the selection of the optimal valuesand corresponding variables from them [11] -ismethod is called DO in short

(2) Genetic algorithm optimisation methods -ismethod is called GA in short

(3) Particle swarm optimisation methods -is methodis called PSO in short [13]

-e MATLAB software provides an optimisation tool-box (OT) which can also solve multiple nonlinear functionoptimisation problems Under the excitation of randomload three methods including DO PSO and OTare used tosolve the optimal parameters of TMD -e damping ratio ofthe main structure is set to 0 so as to compare with thetheoretical solution Figure 2 shows the calculation results ofthe TMD optimal parameters by DO PSO and OT -eresults calculated by the theoretical equation (10) are alsoplotted It can be seen that the optimal frequency ratio foptdecreases logarithmically with the increase of mass ratio μwhile the optimal damping ratio ζopt increases logarithmi-cally with the increase of mass ratio μ In addition it can alsobe seen that the optimal parameters obtained by the threeoptimisation methods are consistent with the theoreticalresults indicating the accuracy of the numerical optimisa-tion method Based on the consideration of simplicity and

Shock and Vibration 3

eciency the MATLAB OT is used for numericaloptimisation

3 Fitting Formula for Optimal TMDParameters under Harmonic Load andError Analysis

Under harmonic excitation the tting formula for the TMDoptimal frequency ratio and optimal damping ratio is givenin reference [10] as shown below

fopt 1minus μ2radic

1 + μ+1minus 2ζ21radic

minus 1( )

minus(2375minus 1034 μ

radic minus 0426μ) μradic ζ1

minus(3730minus 16903 μ

radic + 20496μ) μradic ζ21

ζ2opt

3μ

4(2minus μ)(1 + μ)

radic

+ 0151ζ1 minus 0170ζ21( )

+ 0163ζ1 + 4980ζ21( )μ

(11)

e above optimisation analysis method is used to obtainthe optimal parameters of dierent mass ratio TMDs asshown in Figure 3 e calculated values of the ttingformula corresponding to the main structure with dierentdamping ratios are also plotted It can be seen that when themass ratio is less than 02 the optimal parameters obtainedfrom the tting formula are in good agreement with theactual optimal parameters which can meet practical designrequirements With the increase of the mass ratio the ttingformula value of the optimal parameters deviates from theactual optimal value and the error increases e tting

formula value of the optimal frequency ratio is smaller thanthe actual value and the error increases with the increase ofmain structure damping When the damping ratio ζ1 is 01and the mass ratio μ is greater than 055 the error is greaterthan 55 At the same time when the damping ratio ζ1 is002 the error of the tting formula value for the optimaldamping ratio is the smallest When the damping ratio ζ1 is01 and the mass ratio μ is greater than 075 the error isgreater than 535 On the whole when the mass ratio μ islarge the error of the existing tting formula is very obvious

Based on the above analysis the tting formula inequation (11) is not suitable for calculating the optimalparameters of large mass ratio TMDs erefore a curvetting method is used to revise the formula and the newtting formula is obtained as follows

fopt 1minus μ2radic

1 + μ+1minus 2ζ21radic

minus 1( )

+(minus26976 + 25809μ

radic minus 09656μ) μradic ζ1

+(minus15547 + 53501μ

radic minus 43634μ) μradic ζ21

ζ2opt

3μ

4(2minus μ)(1 + μ)

radic

+(09614minus 05667 μ

radic minus 01277μ)

middot μ

radicζ1 +(205477minus 678430

μ

radic + 626722μ)μζ21(12)

Figure 4 shows the relationship between the calculatedvalue of the revised tting formula and the actual optimalvalue and the change curve of the error rate along with themass ratio is shown in Figure 5 e calculation method ofthe error rate is as follows

0 02 04 06 08 1

04

05

06

07

08

09

1

micro

f opt

DOPSO

OTEquation (10)

(a)

0 02 04 06 08 1μ

005

01

015

02

025

03

035

04

045

ζ 2op

t

DOPSO

OTEquation (10)

(b)

Figure 2 Comparison of numerical optimisation algorithms and theoretical results (a) Optimisation results of the frequency ratio (b)Optimisation results of the damping ratio

4 Shock and Vibration

Error fopt_Form minusfopt_Num( )

fopt_Numtimes 100

Error ζ2opt_Form minus ζ2opt_Num( )

ζ2opt_Numtimes 100

(13)

where fopt_Form and ζopt_Form are the TMD optimal valuescalculated using the revised tting formula in equation (12)e actual optimal values obtained by a numerical opti-misation method are fopt_Num and ζ2opt_Num

It can be seen that the error between the optimal fre-quency ratio calculated by the revised tting formula and theactual optimal value is smaller When the damping ratio ofthe main structure is 0sim01 the maximum error rate is lessthan 1 When the mass ratio is less than 002 the error isrelatively large is is mainly due to the relatively smallTMD damping ratio which leads to a relatively large errorrate Under the condition of dierent mass ratio TMDs theerror rate can be kept within 5

On the whole the error of the revised tting formula issmall which can meet actual design requirements of a largemass ratio TMD under harmonic excitation

0 02 04 06 08 101

02

03

04

05

06

07

08

09

1

μ

f opt

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004ζ1 = 005ζ1 = 0075ζ1 = 010

Optimal solutionEquation (12)

(a)

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004ζ1 = 005ζ1 = 0075ζ1 = 010

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

μ

ζ 2op

t

Optimal solutionEquation (12)

(b)

Figure 4e relationship between the revised tting formula value and the actual optimal value with the TMDmass ratio and the dampingratio of the main structure under the excitation of harmonic load (a) Optimal frequency ratio (b) Optimal damping ratio

0 02 04 06 08 101

02

03

04

05

06

07

08

09

1

μ

f opt

ζ1 = 00ζ1 = 002ζ1 = 005

ζ1 = 0075ζ1 = 010

Equation (11)

(a)

μ

ζ1 = 00ζ1 = 002ζ1 = 005

ζ1 = 0075ζ1 = 010

Equation (11)

0 02 04 06 08 101

02

03

04

05

06

07

08

09

ζ 2op

t

(b)

Figure 3 e curve of the calculated value of the existing tting formula and the actual optimal value varies with the mass ratio under theexcitation of harmonic load (a) Contrast results of the frequency ratio (b) Contrast results of the damping ratio

Shock and Vibration 5

4 Fitting Formula for Optimal TMDParameters under Random Load andError Analysis

41 StationaryWhite Noise Random Load Under stationarywhite noise excitations the tting formula of the TMDoptimal frequency ratio and optimal damping ratio is givenin reference [11] as shown below

fopt 1minus μ2radic

1 + μ+(minus379441 + 987259

μ

radic minus 152978μ)

middot μ

radicζ1 +(minus136731 + 191284

μ

radic + 217049μ) μradic ζ21

ζ2opt

μ(1minus μ4)

4(1 + μ)(1minus μ2)

radic

(14)

However the tting formula only considers the case thatthemass ratio is less than 01When themass ratio of TMD isgreater than 01 the relationship between the optimal valuecalculated by the tting formula in equation (14) and theactual optimal value is shown in Figure 6 In equation (14)the optimal value of the damping ratio is independent of ζ1consequently there is only a curve for all considered valuesof ζ1 in Figure 6(b) It can be seen that when the mass ratio issmaller than 02 the error between the optimal parameterscalculated by the tting formula and the actual optimalparameters is still relatively small However with the in-crease of the mass ratio the error increases signicantly Inaddition the error of the tting formula increases with theincrease of the damping ratio of the main structure From

the error results the tting formula in equation (14) is alsonot suitable for calculating the optimal parameters of largemass ratio TMDs e new formula obtained by the curvetting method is as follows

fopt 1minus μ2radic

1 + μ+(minus29089 + 23354

μ

radic minus 05790μ) μradic ζ1

+(minus131740 + 307222μ

radic minus 212318μ) μradic ζ21

ζ2opt

μ(1minus μ4)

4(1 + μ)(1minus μ2)

radic

+(minus05747 + 15424μ

radic minus 07846μ) μradic ζ1

+(314166minus 968253 μ

radic + 739404μ)μζ21(15)

Under stationary white noise excitations the relation-ship between the calculated value of the revised ttingformula and the actual optimal value is shown in Figure 7e curve of error rate change relative to the mass ratio isshown in Figure 8e calculationmethod of the error rate isthe same as in equation (13)

e error between the optimal frequency ratio calculatedby the revised tting formula and the actual optimal value issmall e damping ratio of the main structure has an eecton the error However when the damping ratio is 01 themaximum error rate of the optimal frequency ratio is stillbelow 12 In addition the error rate of the optimaldamping ratio varies with the TMD mass ratio but it canalways be kept within 5

0 02 04 06 08 1ndash04

ndash02

0

02

04

06

08

1Er

ror (

)

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004

ζ1 = 005ζ1 = 075ζ1 = 010

μ

(a)

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004

ζ1 = 005ζ1 = 075ζ1 = 010

0 02 04 06 08 1ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

Erro

r (

)

μ

(b)

Figure 5e error rate of the revised tting formula value with the TMDmass ratio and the damping ratio of the main structure under theexcitation of harmonic load (a) Error rate of the frequency ratio (b) Error rate of the damping ratio

6 Shock and Vibration

On the whole the error of the revised tting formula issmall which meets actual design requirements of large massratio TMDs Comparisons between the design method inreference [16] and that in this paper are illustrated in Table 1It can be seen that the optimisation parameters in this studyare obviously larger than those in reference [16] is ismainly due to the inconsistency of the optimisation ob-jectives of the two methods

42 Filtered White Noise Random Load e ltered whitenoise random load is modelled as proposed by Kanai [25]

S(ω) ω4g + 4ζ2gω2

gω2

ω2g minusω2( )

2 + 4ζ2gω2gω2

S0 (16)

where ωg and ζg are the predominant frequency anddamping ratio of foundation soil respectively

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004ζ1 = 005ζ1 = 0075ζ1 = 010

Optimal solutionEquation (15)

0 02 04 06 08 101

02

03

04

05

06

07

08

09

1

μ

f opt

(a)

Optimal solutionEquation (15)

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004ζ1 = 005ζ1 = 0075ζ1 = 010

0 02 04 06 08 10

01

02

03

04

05

06

07

μ

ζ 2op

t

(b)

Figure 7e relationship between the revised tting formula value and the actual optimal value with the TMDmass ratio and the dampingratio of the main structure under stationary white noise excitations (a) Optimal frequency ratio (b) Optimal damping ratio

0 02 04 06 08 1ndash04

ndash02

0

02

04

06

08

1f op

t

ζ1 = 00ζ1 = 002ζ1 = 005

ζ1 = 0075ζ1 = 010

Equation (14)

μ

(a)

ζ1 = 00ζ1 = 002ζ1 = 005

ζ1 = 0075ζ1 = 010

Equation (14)

0 02 04 06 08 101

015

02

025

03

035

04

045

05

055

06

μ

ζ 2op

t

(b)

Figure 6 e curve of the calculated value of the existing tting formula and the actual optimal value varies with the mass ratio understationary white noise excitations (a) Contrast results of the frequency ratio (b) Contrast results of the damping ratio

Shock and Vibration 7

Substituting S(ω) into equation (5) the mean square ofthe structural displacement response is obtained as

σ21 int+infin

minusinfinh1(ω)∣∣∣∣

∣∣∣∣2ω4g + 4ζ2gω

2gω

2

ω2g minusω2( )

2 + 4ζ2gω2gω2

S0 dω (17)

Because solving equation (17) analytically is very tediousa numerical integration method can be used e results byBakre [11] show that the TMD optimal parameters arerelatively close when the structure is excited by white noiserandom load and ltered white noise random load but onlywhen a mass ratio below 01 is examined e analysis resultobtained by Hoang et al [6] shows that there is a certaindierence between the optimal parameters for the largermass ratio TMD

When solving the optimal parameters the eect of themain structure damping ratio should be taken into accountthe predominant frequency and damping ratio of the ground

soil that is ωg and ζg are also taken into considerationHowever it is not practical to construct a correspondingtting formula erefore the optimisation process of TMDoptimal parameters is described in detail and the corre-sponding tting formula is no longer determined e de-tailed optimisation currenow chart is shown in Figure 9 Firstly

Table 1 Optimal design of large mass ratio TMD according toreference [16] and this study

Structuralparameters Method fopt ζopt σ21(2πS0ω3

1)

μ 0431 ζ1 002Reference

[16] 06871 02729 23830

is study 05970 02939 22429

μ 05 ζ1 002Reference

[16] 06543 02900 23862

is study 05549 03135 22246

μ 05 ζ1 003Reference

[16] 06492 02861 22947

is study 05432 03144 21320

Main structure parametersvibration frequency and damping ratio

The initial iteration value of the optimal parametercalculated by fitting formula (15)

Objective function the root mean square of the displacement response of the main structure

MATLAB optimization toolboxfunction Fmincon

Mass ratio of TMD

Parameters of ground motion power spectrum modelpredominant frequency and damping ratio

Optimal parameters and the correspondingobjective function values

Figure 9 e numerical optimisation process of TMD parametersunder ltered white noise excitations

0 02 04 06μ

08 1ndash04

ndash02

0

02

04

06

08

1

12Er

ror (

)

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004

ζ1 = 005ζ1 = 075ζ1 = 010

(a)

μ

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004

ζ1 = 005ζ1 = 075ζ1 = 010

0 02 04 06 08 1ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

Erro

r (

)(b)

Figure 8 e error rate of the revised tting formula value with the TMD mass ratio and the damping ratio of the main structure understationary white noise excitations (a) Error rate of the frequency ratio (b) Error rate of the damping ratio

8 Shock and Vibration

the dynamic characteristics of the main structure includingnatural frequency and damping ratio are determined enthe mass ratio of TMD and site condition parameters in-cluding the predominant frequency and damping ratio areselected After that the objective of optimisation is de-termined that is the mean square of the structural dis-placement Finally taking the calculated value of equation (15)as the initial value the optimal parameters can be obtained byoptimum analysis using MATLAB optimisation toolbox

Taking class II sites [26] as an example the parameters ofthe Kanai-Tajimi model ωg 15708 and ζg 072 aredetermined S0 is always taken as a constant and it has noeect on TMD parameter optimisation At the same time

λ ωgω1 is dened as the ratio between the predominantfrequency of ltered white noise and the natural frequency ofthe structure In order to study the TMD optimal parametersand the relative relationship between the excitation of sta-tionary white noise random load and ltered white noiserandom load the following denitions are given

ηf fopt_Num

fopt_Form

ηζ2 ζ2opt_Numζ2opt_Form

(18)

0 02 04 06 08 102

03

04

05

06

07

08

09

1

μ

f opt

λ = 1λ = 2λ = 3λ = 4

λ = 5λ = 75Equation (15)

(a)

μ0 02 04 06 08 1

098

1

102

104

106

108

11

112

114

λ = 1λ = 2λ = 3

λ = 4λ = 5λ = 75

η f(b)

λ = 1λ = 2λ = 3λ = 4

λ = 5λ = 75Equation (15)

0 02 04 06 08 101

015

02

025

03

035

04

045

05

μ

ζ 2op

t

(c)

λ = 1λ = 2λ = 3

λ = 4λ = 5λ = 75

μ0 02 04 06 08 1

084

086

088

09

092

094

096

098

1

102

104η ζ

2

(d)

Figure 10 e relation between optimal parameters with λ and μ and its deviation rate with equation (15) under ltered white noiseexcitations (a) Optimal frequency ratio (b) ηf (c) Optimal damping ratio (d) ηζ2opt

Shock and Vibration 9

Under stationary white noise excitations fopt_Form andζopt_Form are the TMD optimal frequency ratios and theoptimal damping ratios calculated by the tting equation(15) respectively Under ltered white noise excitationsfopt_Num and ζopt_Num are the TMD optimal frequency ratioand the optimal damping ratio respectively obtained by thenumerical optimisation method

When the main structure with a damping ratio of 005 isexcited by ltered white noise load the relationship betweenthe optimal parameters of the TMD and μ and λ is shown inFigures 10(a) and 10(c) In order to facilitate comparisonthe optimal parameter calculated by the tting formula inequation (15) is also plotted Figures 10(b) and 10(d) showrespectively the trends of ηf and ηζ2opt with μ and for dif-ferent values of λ

From the above results the following results areobtained

When the mass ratio μ is smaller than 02 the optimalparameters of the stationary white noise random load andthat of the ltered white noise random load are more similarbut the dierence between the two is increased graduallywith the increase of the mass ratio which can be clearly seenfrom the relationship curve of ηf to μ

e larger the λ the more ηf and ηζ2opt become closer to1 When λge 4 even if the TMD mass ratio reaches 1 theerror of the optimal parameters of the TMD under the

stationary white noise random load and the ltered whitenoise random load is not more than 5

In addition the damping ratio of the main structurehas a relatively small impact on the error therefore theincurrenuence of the damping ratio of the main structure onthe error of the tting formula is no longer discussed inthis paper In conclusion for the optimal design of a largemass ratio TMD when λge 4 the tting formula inequation (15) is also suitable for ltered white noise ex-citations while in the other cases it is suggested to de-termine the optimal TMD parameters using a numericaloptimisation method

5 Damping Effect Analysis of a Large MassRatio TMD

Four SDOF structures with 05 10 20 and 30 s periods (T)are selected and the damping ratio of the four structures is005 e TMD mass ratios are 005 025 050 075 and 1and the TMD parameters are calculated by the tting for-mula in equation (15) Two far-eld seismic waves (ElCentro Hachinohe) and two near-eld seismic waves(Northridge Kobe) are used for load input [27] Seismicwave information is provided in Table 2 e seismic waveacceleration-time history curve is shown in Figure 11 and

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

Acce

lera

tion

(ms

2 )Ac

cele

ratio

n (m

s2 )

El Centro

Hachinohe

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

(a)

Acce

lera

tion

(ms

2 )Ac

cele

ratio

n (m

s2 )

Northridge

Kobe

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

(b)

Figure 11 Time history curve of seismic records (a) Far-eld seismic waves (b) Near-eld seismic waves

Table 2 Seismic wave information used in the analyses

Type Name Earthquake Station Component Year

Far eld El Centro Imperial valley El Centro N-S 1940Hachinohe Tokachi-Oki Hachinohe city N-S 1968

Near eld Northridge Northridge SCH N-S 1994Kobe Hyogo-ken KJMA N-S 1995

10 Shock and Vibration

the seismic wave peak value is 1ms2 In order to investigatethe eect of the TMD mass ratio on the control of thestructural displacement response four sets of seismic timehistory analyses were carried out for structures with andwithout a TMD respectively

e reduction rate of peak displacement Re(Peak) andthe reduction rate of root mean square of structural dis-placement Re(RMS) are used as the evaluation index re-spectively e two formulas for calculating the dampingrate are shown as follows

Re(Peak) 1minusMax xTMD( )Max(x)

Re(RMS) 1minusRMS xTMD( )RMS(x)

(19)

where x and xTMD are the time history record of the dis-placement response of the structure without and with aTMD respectively e formula for calculating the rootmean square of the displacement is as follows

0 02 04 06 08 1micro

0

10

20

30

40

50Re

(Pea

k) (

)

EI CentroHachinoheKobe

NorthridgeMean

(a)

0 02 04 06 08 1micro

0

10

20

30

40

50

Re(P

eak)

()

EI CentroHachinoheKobe

NorthridgeMean

(b)

0 02 04 06 08 1micro

0

10

20

30

40

50

Re(P

eak)

()

EI CentroHachinoheKobe

NorthridgeMean

(c)

0 02 04 06 08 1micro

0

10

20

30

40

50Re

(Pea

k) (

)

EI CentroHachinoheKobe

NorthridgeMean

(d)

Figure 12 e damping eect of dierent mass ratio (TMD) on the peak value of structural displacement response (a) T 05 s (b)T10 s (c) T 20 s (d) T 30 s

Shock and Vibration 11

RMS(x) sqrt1NsumN

i1x2i (20)

where xi is the structural displacement response corre-sponding to the ith time andN is the total number of pointscollected

Re(Peak) and Re(RMS) of the displacement responsewith dierent mass ratio TMDs are shown in Figures 12 and13 e following conclusions can be obtained

(1) TMD can eectively control the displacement re-sponse of the structure and the large mass ratio

(gt025) TMD is more eective than the conventionalsmall mass ratio (lt005) TMD But it can also befound that when the mass ratio of the TMD is greaterthan 05 the gain eect will diminish with increasingmass ratio

(2) e TMD with the same mass ratio shows certaindiscreteness for the structures with dierent naturalvibration periods and dierent seismic waves Forexample as shown in Figure 12(b) when thestructurersquos period is 10 s the damping rate of fourseismic waves is distinct When the mass ratio is 05the minimum damping rate is 1047 and the

0 02 04 06 08 10

10

20

30

40

50

60

micro

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(a)

0 02 04 06 08 1micro

0

10

20

30

40

50

60

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(b)

0 02 04 06 08 10

10

20

30

40

50

60

micro

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(c)

0 02 04 06 08 1micro

0

10

20

30

40

50

60Re

(RM

S)

EI CentroHachinoheKobe

NorthridgeMean

(d)

Figure 13e damping eect of dierent mass ratios (TMD) on the mean square root of structural displacement response (a) T 05 s (b)T10 s (c) T 20 s (d) T 30 s

12 Shock and Vibration

Table 3 Maximum peak value of relative displacement between a TMD with optimal parameters and the main structure (cm)

T (s) μ El Centro Hachinohe Northridge Kobe Mean

05

005 436 296 381 312 356010 268 214 298 22 25050 214 254 335 191 249075 183 26 275 279 249100 224 274 224 337 265

10

005 818 883 1039 698 859010 353 483 512 598 487050 341 535 376 588 46075 391 59 355 546 47100 356 494 358 502 428

20

005 1477 209 866 1834 1567010 794 1197 554 1042 897050 53 863 417 858 667075 394 795 345 705 56100 389 846 332 629 549

30

005 1509 3529 666 1993 1925010 682 13 378 889 812050 491 1128 329 654 651075 463 1037 293 553 586100 509 1125 262 536 608

minus002

0

002

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(a)

minus002

0

002

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

micro = 005micro = 075

wo

(b)

Figure 14 Continued

Shock and Vibration 13

maximum is 4622 which is usually mainly due tothe dierence in the frequency relationship betweenthe structure and the seismic waves

Table 3 shows the statistical results of the relative dis-placement between the TMD and main structure underdierent time history analysis conditions It can be seen thatthe relative displacement between the TMD and the mainstructure is inversely proportional to the mass ratio of theTMD that is when a large mass ratio TMD is used thedisplacement response of the structure is eectively con-trolled and the displacement stroke of the TMD is clearlyreduced which reduces the requirements for the elasticcomponents and the damping components used to constructthe TMD

e displacement time histories of the structures withperiods of 10 and 20 s and the relative displacement timehistories between the TMD and the structures are shown inFigures 14 and 15 respectively e schemes of mass ratiosof 005 and 075 are compared It can be clearly seen that theeect of a TMD in controlling the structural response andthe TMD displacement stroke is more obvious for the TMDwith a mass ratio of 075 than for the one with a mass ratio of005

In summary the large mass ratio TMD has a moresignicant eect in seismic control of the main structurethan the small mass ratio TMD

6 Conclusions

In order to control the dynamic response and improve theaseismic performance of a structure a large mass ratio TMDdamping system is formed by using the equipment in thebuilding structure or relying on new structural forms e

existing optimal parameter tting formula is not applicableto large mass ratio TMDs so it is revised by numericaloptimisation and curve tting and the dynamic time historyanalysis method is used to study the eect of vibrationdamping control of large mass ratio TMDs e followingconclusions are obtained

Compared with the traditional small mass ratio (lt005)TMD the large mass ratio (gt05) TMD has obvious ad-vantages in controlling the displacement response of themain structure e control eect is about 15sim325 timeshigher the damping eect of the structural displacementpeak can reach about 30 and the damping ratio of the rootmean square displacement can reach about 436 At thesame time the relative stroke between the TMD and themain structure can be reduced with up to 30sim65 which ishighly benecial to the practical engineering design of TMDstructures

When the mass ratio of a TMD is relatively large (gt02)the results calculated by the existing tting formula diersignicantly from the actual optimal value and the calcu-lated values of the revised formula proposed in this paper areshown to be in good agreement with the actual optimalvalue In general the revised formula can be applied to bothtraditional small mass ratio and large mass ratio (le1) TMDsWhen the mass ratio is greater than 1 the optimal pa-rameters of TMD can also be obtained by the methodpresented in this paper

When the mass ratio is greater than 02 the optimalparameters of the stationary white noise random load andthat of the ltered white noise random load are more similarbut the dierence between the two is gradually increasedwith the increase of the TMD mass ratio For the optimalparameters of large mass ratio TMDs (gt02) the error is lessthan 005 when the ratio of the predominant frequency of the

0 5 10 15 20 25 30

minus002minus001

0001002

t (s)

0 5 10 15 20 25 30t (s)

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(c)

0 5 10 15 20 25 30t (s)

0 5 10 15 20 25 30t (s)

minus002

0

002

004

Disp

lace

men

t (m

)

minus01

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(d)

Figure 14 e damping eect of dierent mass ratio TMDs on the displacement-time history of the main structure (T10 s) and therelative displacement-time history between the TMD and the structure (wo denotes the displacement of a structure without a TMD) (a) ElCentro (b) Hachinohe (c) Northridge (d) Kobe

14 Shock and Vibration

base soil and the vibration frequency of the structure isgreater than 4 and the optimal parameters of the TMD canbe calculated by the tting formula proposed in this paperUnder other conditions it is suggested to use an optimi-sation method to determine the optimal value of TMDparameters

At present the actual engineering projects with largemass ratio TMD damping systems are less prominent buttheir aseismic advantages will bring a broad range of benetsfor research and practice

Data Availability

e data used to support the ndings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no concurrenicts of interestregarding the publication of this article

Acknowledgments

is research was supported by Grant no 51478023 from theNational Natural Science Foundation of China

References

[1] M Gutierrez Soto and H Adeli ldquoTuned mass dampersrdquoArchives of Computational Methods in Engineering vol 20no 4 pp 419ndash431 2013

ndash005

0

005

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

ndash01ndash005

0005

01

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(a)

ndash02

ndash01

0

01

02

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

ndash005

005

0

micro = 005micro = 075

wo

(b)

0 5 10 15 20 25 30

minus005

0

005

Disp

lace

men

t (m

)

t (s)

0 5 10 15 20 25 30t (s)

minus01

0

01

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(c)

minus002

0

002

004D

ispla

cem

ent (

m)

0 5 10 15 20 25 30t (s)

0

minus005

005

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30t (s)

micro = 005micro = 075

wo

(d)

Figure 15 e damping eect of dierent mass ratio TMDs on the displacement-time history of the main structure (T 20 s) and therelative displacement-time history between the TMD and the structure (wo denotes the displacement of a structure without a TMD) (a) ElCentro (b) Hachinohe (c) Northridge (d) Kobe

Shock and Vibration 15

[2] S Elias and VMatsagar ldquoResearch developments in vibrationcontrol of structures using passive tuned mass dampersrdquoAnnual Reviews in Control vol 44 pp 129ndash156 2017

[3] D Wang T K T Tse Y Zhou and Q Li ldquoStructural per-formance and cost analysis of wind-induced vibration controlschemes for a real super-tall buildingrdquo Structure and In-frastructure Engineering vol 11 no 8 pp 990ndash1011 2014

[4] N Longarini and M Zucca ldquoA chimneyrsquos seismic assessmentby a tuned mass damperrdquo Engineering Structures vol 79pp 290ndash296 2014

[5] L Tian and Y Zeng ldquoParametric study of tunedmass dampersfor long span transmission tower-line system under windloadsrdquo Shock and Vibration vol 2016 Article ID 496505611 pages 2016

[6] N Hoang Y Fujino and PWarnitchai ldquoOptimal tuned massdamper for seismic applications and practical design for-mulasrdquo Engineering Structures vol 30 no 3 pp 707ndash7152008

[7] J P Den Hartog Mechanical vibrations McGraw-Hill NewYork NY USA 1956

[8] G B Warburton ldquoOptimum absorber parameters for variouscombinations of response and excitation parametersrdquoEarthquake Engineering amp Structural Dynamics vol 10 no 3pp 381ndash401 1982

[9] H-C Tsai and G-C Lin ldquoExplicit formulae for optimumabsorber parameters for force-excited and viscously dampedsystemsrdquo Journal of Sound and Vibration vol 176 no 5pp 585ndash596 1994

[10] H-C Tsai and G-C Lin ldquoOptimum tuned-mass dampers forminimizing steady-state response of support-excited anddamped systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 22 no 11 pp 957ndash973 1993

[11] S V Bakre and R S Jangid ldquoOptimum parameters of tunedmass damper for damped main systemrdquo Structural Controland Health Monitoring vol 14 no 3 pp 448ndash470 2007

[12] C C Lin C M Hu J F Wang and R Y Hu ldquoVibrationcontrol effectiveness of passive tuned mass dampersrdquo Journalof the Chinese Institute of Engineers vol 17 pp 367ndash376 1994

[13] A Y T Leung and H Zhang ldquoParticle swarm optimization oftuned mass dampersrdquo Engineering Structures vol 31 no 3pp 715ndash728 2009

[14] M Q Feng and A Mita ldquoVibration control of tall buildingsusing mega SubConfigurationrdquo Journal of Engineering Me-chanics vol 121 no 10 pp 1082ndash1088 1995

[15] X X Li P Tan X J Li and A W Liu ldquoMechanism analysisand parameter optimisation of mega-sub-isolation systemrdquoShock and Vibration vol 2016 p 12 2016

[16] A Reggio and M D Angelis ldquoOptimal energy-based seismicdesign of non-conventional tuned mass damper (TMD)implemented via inter-story isolationrdquo Earthquake Engi-neering amp Structural Dynamics vol 44 no 10 pp 1623ndash16422015

[17] S J Wang B H Lee W C Chuang and K C ChangldquoOptimal dynamic characteristic control approach forbuilding mass damper designrdquo Earthquake Engineering ampStructural Dynamics vol 47 no 3 2017

[18] H Anajafi and R A Medina ldquoPartial mass isolation systemfor seismic vibration control of buildingsrdquo Structural Controlamp Health Monitoring vol 25 no 2 article e2088 2017

[19] K Yuan M S He and Y M Li ldquoShaking table tests forenergy-dissipation steel frame structures with infilled wallMTMDrdquo Journal of Vibration and Shock vol 33 no 11pp 200ndash207 2014

[20] R Ding M X Tao M Zhou and J G Nie ldquoSeismic behaviorof RC structures with absence of floor slab constraints andlarge mass turbine as a non-conventional TMD a case studyrdquoBulletin of Earthquake Engineering vol 13 no 11 pp 3401ndash3422 2015

[21] L Y Peng Y J Kang Z R Lai and Y K Deng ldquoOptimisationand damping performance of a coal-fired power plantbuilding equipped with multiple coal bucket dampersrdquo Ad-vances in Civil Engineering vol 2018 p 19 2018

[22] Z Shu S Li J Zhang and M He ldquoOptimum seismic designof a power plant building with pendulum tuned mass dampersystem by its heavy suspended bucketsrdquo Engineering Struc-tures vol 136 pp 114ndash132 2017

[23] K Dai B Li J Wang et al ldquoOptimal probability-based partialmass isolation of elevated coal scuttle in thermal power plantbuildingrdquo Structural Design of Tall and Special Buildingsvol 27 no 11 article e1477 2018

[24] M De Angelis S Perno and A Reggio ldquoDynamic responseand optimal design of structures with large mass ratio TMDrdquoEarthquake Engineering amp Structural Dynamics vol 41 no 1pp 41ndash60 2015

[25] K Kanai ldquoSemi-empirical formula for the seismic charac-teristics of the groundrdquo Bulletin of the Earthquake ResearchInstitute gte University of Tokyo vol 35 pp 309ndash325 1957

[26] Code for seismic design of buildings GB50011-2010 BeijingChina 2010

[27] B F Spencer R E Christenson and S J Dyke ldquoNextgeneration benchmark control problem for seismically excitedbuildingsrdquo in Proceedings of the 2nd World Conference onStructural Control pp 1351ndash1360 Kyoto Japan June 1998

16 Shock and Vibration

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eciency the MATLAB OT is used for numericaloptimisation

3 Fitting Formula for Optimal TMDParameters under Harmonic Load andError Analysis

Under harmonic excitation the tting formula for the TMDoptimal frequency ratio and optimal damping ratio is givenin reference [10] as shown below

fopt 1minus μ2radic

1 + μ+1minus 2ζ21radic

minus 1( )

minus(2375minus 1034 μ

radic minus 0426μ) μradic ζ1

minus(3730minus 16903 μ

radic + 20496μ) μradic ζ21

ζ2opt

3μ

4(2minus μ)(1 + μ)

radic

+ 0151ζ1 minus 0170ζ21( )

+ 0163ζ1 + 4980ζ21( )μ

(11)

e above optimisation analysis method is used to obtainthe optimal parameters of dierent mass ratio TMDs asshown in Figure 3 e calculated values of the ttingformula corresponding to the main structure with dierentdamping ratios are also plotted It can be seen that when themass ratio is less than 02 the optimal parameters obtainedfrom the tting formula are in good agreement with theactual optimal parameters which can meet practical designrequirements With the increase of the mass ratio the ttingformula value of the optimal parameters deviates from theactual optimal value and the error increases e tting

formula value of the optimal frequency ratio is smaller thanthe actual value and the error increases with the increase ofmain structure damping When the damping ratio ζ1 is 01and the mass ratio μ is greater than 055 the error is greaterthan 55 At the same time when the damping ratio ζ1 is002 the error of the tting formula value for the optimaldamping ratio is the smallest When the damping ratio ζ1 is01 and the mass ratio μ is greater than 075 the error isgreater than 535 On the whole when the mass ratio μ islarge the error of the existing tting formula is very obvious

Based on the above analysis the tting formula inequation (11) is not suitable for calculating the optimalparameters of large mass ratio TMDs erefore a curvetting method is used to revise the formula and the newtting formula is obtained as follows

fopt 1minus μ2radic

1 + μ+1minus 2ζ21radic

minus 1( )

+(minus26976 + 25809μ

radic minus 09656μ) μradic ζ1

+(minus15547 + 53501μ

radic minus 43634μ) μradic ζ21

ζ2opt

3μ

4(2minus μ)(1 + μ)

radic

+(09614minus 05667 μ

radic minus 01277μ)

middot μ

radicζ1 +(205477minus 678430

μ

radic + 626722μ)μζ21(12)

Figure 4 shows the relationship between the calculatedvalue of the revised tting formula and the actual optimalvalue and the change curve of the error rate along with themass ratio is shown in Figure 5 e calculation method ofthe error rate is as follows

0 02 04 06 08 1

04

05

06

07

08

09

1

micro

f opt

DOPSO

OTEquation (10)

(a)

0 02 04 06 08 1μ

005

01

015

02

025

03

035

04

045

ζ 2op

t

DOPSO

OTEquation (10)

(b)

Figure 2 Comparison of numerical optimisation algorithms and theoretical results (a) Optimisation results of the frequency ratio (b)Optimisation results of the damping ratio

4 Shock and Vibration

Error fopt_Form minusfopt_Num( )

fopt_Numtimes 100

Error ζ2opt_Form minus ζ2opt_Num( )

ζ2opt_Numtimes 100

(13)

where fopt_Form and ζopt_Form are the TMD optimal valuescalculated using the revised tting formula in equation (12)e actual optimal values obtained by a numerical opti-misation method are fopt_Num and ζ2opt_Num

It can be seen that the error between the optimal fre-quency ratio calculated by the revised tting formula and theactual optimal value is smaller When the damping ratio ofthe main structure is 0sim01 the maximum error rate is lessthan 1 When the mass ratio is less than 002 the error isrelatively large is is mainly due to the relatively smallTMD damping ratio which leads to a relatively large errorrate Under the condition of dierent mass ratio TMDs theerror rate can be kept within 5

On the whole the error of the revised tting formula issmall which can meet actual design requirements of a largemass ratio TMD under harmonic excitation

0 02 04 06 08 101

02

03

04

05

06

07

08

09

1

μ

f opt

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004ζ1 = 005ζ1 = 0075ζ1 = 010

Optimal solutionEquation (12)

(a)

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004ζ1 = 005ζ1 = 0075ζ1 = 010

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

μ

ζ 2op

t

Optimal solutionEquation (12)

(b)

Figure 4e relationship between the revised tting formula value and the actual optimal value with the TMDmass ratio and the dampingratio of the main structure under the excitation of harmonic load (a) Optimal frequency ratio (b) Optimal damping ratio

0 02 04 06 08 101

02

03

04

05

06

07

08

09

1

μ

f opt

ζ1 = 00ζ1 = 002ζ1 = 005

ζ1 = 0075ζ1 = 010

Equation (11)

(a)

μ

ζ1 = 00ζ1 = 002ζ1 = 005

ζ1 = 0075ζ1 = 010

Equation (11)

0 02 04 06 08 101

02

03

04

05

06

07

08

09

ζ 2op

t

(b)

Figure 3 e curve of the calculated value of the existing tting formula and the actual optimal value varies with the mass ratio under theexcitation of harmonic load (a) Contrast results of the frequency ratio (b) Contrast results of the damping ratio

Shock and Vibration 5

4 Fitting Formula for Optimal TMDParameters under Random Load andError Analysis

41 StationaryWhite Noise Random Load Under stationarywhite noise excitations the tting formula of the TMDoptimal frequency ratio and optimal damping ratio is givenin reference [11] as shown below

fopt 1minus μ2radic

1 + μ+(minus379441 + 987259

μ

radic minus 152978μ)

middot μ

radicζ1 +(minus136731 + 191284

μ

radic + 217049μ) μradic ζ21

ζ2opt

μ(1minus μ4)

4(1 + μ)(1minus μ2)

radic

(14)

However the tting formula only considers the case thatthemass ratio is less than 01When themass ratio of TMD isgreater than 01 the relationship between the optimal valuecalculated by the tting formula in equation (14) and theactual optimal value is shown in Figure 6 In equation (14)the optimal value of the damping ratio is independent of ζ1consequently there is only a curve for all considered valuesof ζ1 in Figure 6(b) It can be seen that when the mass ratio issmaller than 02 the error between the optimal parameterscalculated by the tting formula and the actual optimalparameters is still relatively small However with the in-crease of the mass ratio the error increases signicantly Inaddition the error of the tting formula increases with theincrease of the damping ratio of the main structure From

the error results the tting formula in equation (14) is alsonot suitable for calculating the optimal parameters of largemass ratio TMDs e new formula obtained by the curvetting method is as follows

fopt 1minus μ2radic

1 + μ+(minus29089 + 23354

μ

radic minus 05790μ) μradic ζ1

+(minus131740 + 307222μ

radic minus 212318μ) μradic ζ21

ζ2opt

μ(1minus μ4)

4(1 + μ)(1minus μ2)

radic

+(minus05747 + 15424μ

radic minus 07846μ) μradic ζ1

+(314166minus 968253 μ

radic + 739404μ)μζ21(15)

Under stationary white noise excitations the relation-ship between the calculated value of the revised ttingformula and the actual optimal value is shown in Figure 7e curve of error rate change relative to the mass ratio isshown in Figure 8e calculationmethod of the error rate isthe same as in equation (13)

e error between the optimal frequency ratio calculatedby the revised tting formula and the actual optimal value issmall e damping ratio of the main structure has an eecton the error However when the damping ratio is 01 themaximum error rate of the optimal frequency ratio is stillbelow 12 In addition the error rate of the optimaldamping ratio varies with the TMD mass ratio but it canalways be kept within 5

0 02 04 06 08 1ndash04

ndash02

0

02

04

06

08

1Er

ror (

)

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004

ζ1 = 005ζ1 = 075ζ1 = 010

μ

(a)

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004

ζ1 = 005ζ1 = 075ζ1 = 010

0 02 04 06 08 1ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

Erro

r (

)

μ

(b)

Figure 5e error rate of the revised tting formula value with the TMDmass ratio and the damping ratio of the main structure under theexcitation of harmonic load (a) Error rate of the frequency ratio (b) Error rate of the damping ratio

6 Shock and Vibration

On the whole the error of the revised tting formula issmall which meets actual design requirements of large massratio TMDs Comparisons between the design method inreference [16] and that in this paper are illustrated in Table 1It can be seen that the optimisation parameters in this studyare obviously larger than those in reference [16] is ismainly due to the inconsistency of the optimisation ob-jectives of the two methods

42 Filtered White Noise Random Load e ltered whitenoise random load is modelled as proposed by Kanai [25]

S(ω) ω4g + 4ζ2gω2

gω2

ω2g minusω2( )

2 + 4ζ2gω2gω2

S0 (16)

where ωg and ζg are the predominant frequency anddamping ratio of foundation soil respectively

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004ζ1 = 005ζ1 = 0075ζ1 = 010

Optimal solutionEquation (15)

0 02 04 06 08 101

02

03

04

05

06

07

08

09

1

μ

f opt

(a)

Optimal solutionEquation (15)

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004ζ1 = 005ζ1 = 0075ζ1 = 010

0 02 04 06 08 10

01

02

03

04

05

06

07

μ

ζ 2op

t

(b)

Figure 7e relationship between the revised tting formula value and the actual optimal value with the TMDmass ratio and the dampingratio of the main structure under stationary white noise excitations (a) Optimal frequency ratio (b) Optimal damping ratio

0 02 04 06 08 1ndash04

ndash02

0

02

04

06

08

1f op

t

ζ1 = 00ζ1 = 002ζ1 = 005

ζ1 = 0075ζ1 = 010

Equation (14)

μ

(a)

ζ1 = 00ζ1 = 002ζ1 = 005

ζ1 = 0075ζ1 = 010

Equation (14)

0 02 04 06 08 101

015

02

025

03

035

04

045

05

055

06

μ

ζ 2op

t

(b)

Figure 6 e curve of the calculated value of the existing tting formula and the actual optimal value varies with the mass ratio understationary white noise excitations (a) Contrast results of the frequency ratio (b) Contrast results of the damping ratio

Shock and Vibration 7

Substituting S(ω) into equation (5) the mean square ofthe structural displacement response is obtained as

σ21 int+infin

minusinfinh1(ω)∣∣∣∣

∣∣∣∣2ω4g + 4ζ2gω

2gω

2

ω2g minusω2( )

2 + 4ζ2gω2gω2

S0 dω (17)

Because solving equation (17) analytically is very tediousa numerical integration method can be used e results byBakre [11] show that the TMD optimal parameters arerelatively close when the structure is excited by white noiserandom load and ltered white noise random load but onlywhen a mass ratio below 01 is examined e analysis resultobtained by Hoang et al [6] shows that there is a certaindierence between the optimal parameters for the largermass ratio TMD

When solving the optimal parameters the eect of themain structure damping ratio should be taken into accountthe predominant frequency and damping ratio of the ground

soil that is ωg and ζg are also taken into considerationHowever it is not practical to construct a correspondingtting formula erefore the optimisation process of TMDoptimal parameters is described in detail and the corre-sponding tting formula is no longer determined e de-tailed optimisation currenow chart is shown in Figure 9 Firstly

Table 1 Optimal design of large mass ratio TMD according toreference [16] and this study

Structuralparameters Method fopt ζopt σ21(2πS0ω3

1)

μ 0431 ζ1 002Reference

[16] 06871 02729 23830

is study 05970 02939 22429

μ 05 ζ1 002Reference

[16] 06543 02900 23862

is study 05549 03135 22246

μ 05 ζ1 003Reference

[16] 06492 02861 22947

is study 05432 03144 21320

Main structure parametersvibration frequency and damping ratio

The initial iteration value of the optimal parametercalculated by fitting formula (15)

Objective function the root mean square of the displacement response of the main structure

MATLAB optimization toolboxfunction Fmincon

Mass ratio of TMD

Parameters of ground motion power spectrum modelpredominant frequency and damping ratio

Optimal parameters and the correspondingobjective function values

Figure 9 e numerical optimisation process of TMD parametersunder ltered white noise excitations

0 02 04 06μ

08 1ndash04

ndash02

0

02

04

06

08

1

12Er

ror (

)

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004

ζ1 = 005ζ1 = 075ζ1 = 010

(a)

μ

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004

ζ1 = 005ζ1 = 075ζ1 = 010

0 02 04 06 08 1ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

Erro

r (

)(b)

Figure 8 e error rate of the revised tting formula value with the TMD mass ratio and the damping ratio of the main structure understationary white noise excitations (a) Error rate of the frequency ratio (b) Error rate of the damping ratio

8 Shock and Vibration

the dynamic characteristics of the main structure includingnatural frequency and damping ratio are determined enthe mass ratio of TMD and site condition parameters in-cluding the predominant frequency and damping ratio areselected After that the objective of optimisation is de-termined that is the mean square of the structural dis-placement Finally taking the calculated value of equation (15)as the initial value the optimal parameters can be obtained byoptimum analysis using MATLAB optimisation toolbox

Taking class II sites [26] as an example the parameters ofthe Kanai-Tajimi model ωg 15708 and ζg 072 aredetermined S0 is always taken as a constant and it has noeect on TMD parameter optimisation At the same time

λ ωgω1 is dened as the ratio between the predominantfrequency of ltered white noise and the natural frequency ofthe structure In order to study the TMD optimal parametersand the relative relationship between the excitation of sta-tionary white noise random load and ltered white noiserandom load the following denitions are given

ηf fopt_Num

fopt_Form

ηζ2 ζ2opt_Numζ2opt_Form

(18)

0 02 04 06 08 102

03

04

05

06

07

08

09

1

μ

f opt

λ = 1λ = 2λ = 3λ = 4

λ = 5λ = 75Equation (15)

(a)

μ0 02 04 06 08 1

098

1

102

104

106

108

11

112

114

λ = 1λ = 2λ = 3

λ = 4λ = 5λ = 75

η f(b)

λ = 1λ = 2λ = 3λ = 4

λ = 5λ = 75Equation (15)

0 02 04 06 08 101

015

02

025

03

035

04

045

05

μ

ζ 2op

t

(c)

λ = 1λ = 2λ = 3

λ = 4λ = 5λ = 75

μ0 02 04 06 08 1

084

086

088

09

092

094

096

098

1

102

104η ζ

2

(d)

Figure 10 e relation between optimal parameters with λ and μ and its deviation rate with equation (15) under ltered white noiseexcitations (a) Optimal frequency ratio (b) ηf (c) Optimal damping ratio (d) ηζ2opt

Shock and Vibration 9

Under stationary white noise excitations fopt_Form andζopt_Form are the TMD optimal frequency ratios and theoptimal damping ratios calculated by the tting equation(15) respectively Under ltered white noise excitationsfopt_Num and ζopt_Num are the TMD optimal frequency ratioand the optimal damping ratio respectively obtained by thenumerical optimisation method

When the main structure with a damping ratio of 005 isexcited by ltered white noise load the relationship betweenthe optimal parameters of the TMD and μ and λ is shown inFigures 10(a) and 10(c) In order to facilitate comparisonthe optimal parameter calculated by the tting formula inequation (15) is also plotted Figures 10(b) and 10(d) showrespectively the trends of ηf and ηζ2opt with μ and for dif-ferent values of λ

From the above results the following results areobtained

When the mass ratio μ is smaller than 02 the optimalparameters of the stationary white noise random load andthat of the ltered white noise random load are more similarbut the dierence between the two is increased graduallywith the increase of the mass ratio which can be clearly seenfrom the relationship curve of ηf to μ

e larger the λ the more ηf and ηζ2opt become closer to1 When λge 4 even if the TMD mass ratio reaches 1 theerror of the optimal parameters of the TMD under the

stationary white noise random load and the ltered whitenoise random load is not more than 5

In addition the damping ratio of the main structurehas a relatively small impact on the error therefore theincurrenuence of the damping ratio of the main structure onthe error of the tting formula is no longer discussed inthis paper In conclusion for the optimal design of a largemass ratio TMD when λge 4 the tting formula inequation (15) is also suitable for ltered white noise ex-citations while in the other cases it is suggested to de-termine the optimal TMD parameters using a numericaloptimisation method

5 Damping Effect Analysis of a Large MassRatio TMD

Four SDOF structures with 05 10 20 and 30 s periods (T)are selected and the damping ratio of the four structures is005 e TMD mass ratios are 005 025 050 075 and 1and the TMD parameters are calculated by the tting for-mula in equation (15) Two far-eld seismic waves (ElCentro Hachinohe) and two near-eld seismic waves(Northridge Kobe) are used for load input [27] Seismicwave information is provided in Table 2 e seismic waveacceleration-time history curve is shown in Figure 11 and

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

Acce

lera

tion

(ms

2 )Ac

cele

ratio

n (m

s2 )

El Centro

Hachinohe

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

(a)

Acce

lera

tion

(ms

2 )Ac

cele

ratio

n (m

s2 )

Northridge

Kobe

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

(b)

Figure 11 Time history curve of seismic records (a) Far-eld seismic waves (b) Near-eld seismic waves

Table 2 Seismic wave information used in the analyses

Type Name Earthquake Station Component Year

Far eld El Centro Imperial valley El Centro N-S 1940Hachinohe Tokachi-Oki Hachinohe city N-S 1968

Near eld Northridge Northridge SCH N-S 1994Kobe Hyogo-ken KJMA N-S 1995

10 Shock and Vibration

the seismic wave peak value is 1ms2 In order to investigatethe eect of the TMD mass ratio on the control of thestructural displacement response four sets of seismic timehistory analyses were carried out for structures with andwithout a TMD respectively

e reduction rate of peak displacement Re(Peak) andthe reduction rate of root mean square of structural dis-placement Re(RMS) are used as the evaluation index re-spectively e two formulas for calculating the dampingrate are shown as follows

Re(Peak) 1minusMax xTMD( )Max(x)

Re(RMS) 1minusRMS xTMD( )RMS(x)

(19)

where x and xTMD are the time history record of the dis-placement response of the structure without and with aTMD respectively e formula for calculating the rootmean square of the displacement is as follows

0 02 04 06 08 1micro

0

10

20

30

40

50Re

(Pea

k) (

)

EI CentroHachinoheKobe

NorthridgeMean

(a)

0 02 04 06 08 1micro

0

10

20

30

40

50

Re(P

eak)

()

EI CentroHachinoheKobe

NorthridgeMean

(b)

0 02 04 06 08 1micro

0

10

20

30

40

50

Re(P

eak)

()

EI CentroHachinoheKobe

NorthridgeMean

(c)

0 02 04 06 08 1micro

0

10

20

30

40

50Re

(Pea

k) (

)

EI CentroHachinoheKobe

NorthridgeMean

(d)

Figure 12 e damping eect of dierent mass ratio (TMD) on the peak value of structural displacement response (a) T 05 s (b)T10 s (c) T 20 s (d) T 30 s

Shock and Vibration 11

RMS(x) sqrt1NsumN

i1x2i (20)

where xi is the structural displacement response corre-sponding to the ith time andN is the total number of pointscollected

Re(Peak) and Re(RMS) of the displacement responsewith dierent mass ratio TMDs are shown in Figures 12 and13 e following conclusions can be obtained

(1) TMD can eectively control the displacement re-sponse of the structure and the large mass ratio

(gt025) TMD is more eective than the conventionalsmall mass ratio (lt005) TMD But it can also befound that when the mass ratio of the TMD is greaterthan 05 the gain eect will diminish with increasingmass ratio

(2) e TMD with the same mass ratio shows certaindiscreteness for the structures with dierent naturalvibration periods and dierent seismic waves Forexample as shown in Figure 12(b) when thestructurersquos period is 10 s the damping rate of fourseismic waves is distinct When the mass ratio is 05the minimum damping rate is 1047 and the

0 02 04 06 08 10

10

20

30

40

50

60

micro

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(a)

0 02 04 06 08 1micro

0

10

20

30

40

50

60

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(b)

0 02 04 06 08 10

10

20

30

40

50

60

micro

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(c)

0 02 04 06 08 1micro

0

10

20

30

40

50

60Re

(RM

S)

EI CentroHachinoheKobe

NorthridgeMean

(d)

Figure 13e damping eect of dierent mass ratios (TMD) on the mean square root of structural displacement response (a) T 05 s (b)T10 s (c) T 20 s (d) T 30 s

12 Shock and Vibration

Table 3 Maximum peak value of relative displacement between a TMD with optimal parameters and the main structure (cm)

T (s) μ El Centro Hachinohe Northridge Kobe Mean

05

005 436 296 381 312 356010 268 214 298 22 25050 214 254 335 191 249075 183 26 275 279 249100 224 274 224 337 265

10

005 818 883 1039 698 859010 353 483 512 598 487050 341 535 376 588 46075 391 59 355 546 47100 356 494 358 502 428

20

005 1477 209 866 1834 1567010 794 1197 554 1042 897050 53 863 417 858 667075 394 795 345 705 56100 389 846 332 629 549

30

005 1509 3529 666 1993 1925010 682 13 378 889 812050 491 1128 329 654 651075 463 1037 293 553 586100 509 1125 262 536 608

minus002

0

002

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(a)

minus002

0

002

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

micro = 005micro = 075

wo

(b)

Figure 14 Continued

Shock and Vibration 13

maximum is 4622 which is usually mainly due tothe dierence in the frequency relationship betweenthe structure and the seismic waves

Table 3 shows the statistical results of the relative dis-placement between the TMD and main structure underdierent time history analysis conditions It can be seen thatthe relative displacement between the TMD and the mainstructure is inversely proportional to the mass ratio of theTMD that is when a large mass ratio TMD is used thedisplacement response of the structure is eectively con-trolled and the displacement stroke of the TMD is clearlyreduced which reduces the requirements for the elasticcomponents and the damping components used to constructthe TMD

e displacement time histories of the structures withperiods of 10 and 20 s and the relative displacement timehistories between the TMD and the structures are shown inFigures 14 and 15 respectively e schemes of mass ratiosof 005 and 075 are compared It can be clearly seen that theeect of a TMD in controlling the structural response andthe TMD displacement stroke is more obvious for the TMDwith a mass ratio of 075 than for the one with a mass ratio of005

In summary the large mass ratio TMD has a moresignicant eect in seismic control of the main structurethan the small mass ratio TMD

6 Conclusions

In order to control the dynamic response and improve theaseismic performance of a structure a large mass ratio TMDdamping system is formed by using the equipment in thebuilding structure or relying on new structural forms e

existing optimal parameter tting formula is not applicableto large mass ratio TMDs so it is revised by numericaloptimisation and curve tting and the dynamic time historyanalysis method is used to study the eect of vibrationdamping control of large mass ratio TMDs e followingconclusions are obtained

Compared with the traditional small mass ratio (lt005)TMD the large mass ratio (gt05) TMD has obvious ad-vantages in controlling the displacement response of themain structure e control eect is about 15sim325 timeshigher the damping eect of the structural displacementpeak can reach about 30 and the damping ratio of the rootmean square displacement can reach about 436 At thesame time the relative stroke between the TMD and themain structure can be reduced with up to 30sim65 which ishighly benecial to the practical engineering design of TMDstructures

When the mass ratio of a TMD is relatively large (gt02)the results calculated by the existing tting formula diersignicantly from the actual optimal value and the calcu-lated values of the revised formula proposed in this paper areshown to be in good agreement with the actual optimalvalue In general the revised formula can be applied to bothtraditional small mass ratio and large mass ratio (le1) TMDsWhen the mass ratio is greater than 1 the optimal pa-rameters of TMD can also be obtained by the methodpresented in this paper

When the mass ratio is greater than 02 the optimalparameters of the stationary white noise random load andthat of the ltered white noise random load are more similarbut the dierence between the two is gradually increasedwith the increase of the TMD mass ratio For the optimalparameters of large mass ratio TMDs (gt02) the error is lessthan 005 when the ratio of the predominant frequency of the

0 5 10 15 20 25 30

minus002minus001

0001002

t (s)

0 5 10 15 20 25 30t (s)

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(c)

0 5 10 15 20 25 30t (s)

0 5 10 15 20 25 30t (s)

minus002

0

002

004

Disp

lace

men

t (m

)

minus01

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(d)

Figure 14 e damping eect of dierent mass ratio TMDs on the displacement-time history of the main structure (T10 s) and therelative displacement-time history between the TMD and the structure (wo denotes the displacement of a structure without a TMD) (a) ElCentro (b) Hachinohe (c) Northridge (d) Kobe

14 Shock and Vibration

base soil and the vibration frequency of the structure isgreater than 4 and the optimal parameters of the TMD canbe calculated by the tting formula proposed in this paperUnder other conditions it is suggested to use an optimi-sation method to determine the optimal value of TMDparameters

At present the actual engineering projects with largemass ratio TMD damping systems are less prominent buttheir aseismic advantages will bring a broad range of benetsfor research and practice

Data Availability

e data used to support the ndings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no concurrenicts of interestregarding the publication of this article

Acknowledgments

is research was supported by Grant no 51478023 from theNational Natural Science Foundation of China

References

[1] M Gutierrez Soto and H Adeli ldquoTuned mass dampersrdquoArchives of Computational Methods in Engineering vol 20no 4 pp 419ndash431 2013

ndash005

0

005

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

ndash01ndash005

0005

01

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(a)

ndash02

ndash01

0

01

02

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

ndash005

005

0

micro = 005micro = 075

wo

(b)

0 5 10 15 20 25 30

minus005

0

005

Disp

lace

men

t (m

)

t (s)

0 5 10 15 20 25 30t (s)

minus01

0

01

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(c)

minus002

0

002

004D

ispla

cem

ent (

m)

0 5 10 15 20 25 30t (s)

0

minus005

005

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30t (s)

micro = 005micro = 075

wo

(d)

Figure 15 e damping eect of dierent mass ratio TMDs on the displacement-time history of the main structure (T 20 s) and therelative displacement-time history between the TMD and the structure (wo denotes the displacement of a structure without a TMD) (a) ElCentro (b) Hachinohe (c) Northridge (d) Kobe

Shock and Vibration 15

[2] S Elias and VMatsagar ldquoResearch developments in vibrationcontrol of structures using passive tuned mass dampersrdquoAnnual Reviews in Control vol 44 pp 129ndash156 2017

[3] D Wang T K T Tse Y Zhou and Q Li ldquoStructural per-formance and cost analysis of wind-induced vibration controlschemes for a real super-tall buildingrdquo Structure and In-frastructure Engineering vol 11 no 8 pp 990ndash1011 2014

[4] N Longarini and M Zucca ldquoA chimneyrsquos seismic assessmentby a tuned mass damperrdquo Engineering Structures vol 79pp 290ndash296 2014

[5] L Tian and Y Zeng ldquoParametric study of tunedmass dampersfor long span transmission tower-line system under windloadsrdquo Shock and Vibration vol 2016 Article ID 496505611 pages 2016

[6] N Hoang Y Fujino and PWarnitchai ldquoOptimal tuned massdamper for seismic applications and practical design for-mulasrdquo Engineering Structures vol 30 no 3 pp 707ndash7152008

[7] J P Den Hartog Mechanical vibrations McGraw-Hill NewYork NY USA 1956

[8] G B Warburton ldquoOptimum absorber parameters for variouscombinations of response and excitation parametersrdquoEarthquake Engineering amp Structural Dynamics vol 10 no 3pp 381ndash401 1982

[9] H-C Tsai and G-C Lin ldquoExplicit formulae for optimumabsorber parameters for force-excited and viscously dampedsystemsrdquo Journal of Sound and Vibration vol 176 no 5pp 585ndash596 1994

[10] H-C Tsai and G-C Lin ldquoOptimum tuned-mass dampers forminimizing steady-state response of support-excited anddamped systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 22 no 11 pp 957ndash973 1993

[11] S V Bakre and R S Jangid ldquoOptimum parameters of tunedmass damper for damped main systemrdquo Structural Controland Health Monitoring vol 14 no 3 pp 448ndash470 2007

[12] C C Lin C M Hu J F Wang and R Y Hu ldquoVibrationcontrol effectiveness of passive tuned mass dampersrdquo Journalof the Chinese Institute of Engineers vol 17 pp 367ndash376 1994

[13] A Y T Leung and H Zhang ldquoParticle swarm optimization oftuned mass dampersrdquo Engineering Structures vol 31 no 3pp 715ndash728 2009

[14] M Q Feng and A Mita ldquoVibration control of tall buildingsusing mega SubConfigurationrdquo Journal of Engineering Me-chanics vol 121 no 10 pp 1082ndash1088 1995

[15] X X Li P Tan X J Li and A W Liu ldquoMechanism analysisand parameter optimisation of mega-sub-isolation systemrdquoShock and Vibration vol 2016 p 12 2016

[16] A Reggio and M D Angelis ldquoOptimal energy-based seismicdesign of non-conventional tuned mass damper (TMD)implemented via inter-story isolationrdquo Earthquake Engi-neering amp Structural Dynamics vol 44 no 10 pp 1623ndash16422015

[17] S J Wang B H Lee W C Chuang and K C ChangldquoOptimal dynamic characteristic control approach forbuilding mass damper designrdquo Earthquake Engineering ampStructural Dynamics vol 47 no 3 2017

[18] H Anajafi and R A Medina ldquoPartial mass isolation systemfor seismic vibration control of buildingsrdquo Structural Controlamp Health Monitoring vol 25 no 2 article e2088 2017

[19] K Yuan M S He and Y M Li ldquoShaking table tests forenergy-dissipation steel frame structures with infilled wallMTMDrdquo Journal of Vibration and Shock vol 33 no 11pp 200ndash207 2014

[20] R Ding M X Tao M Zhou and J G Nie ldquoSeismic behaviorof RC structures with absence of floor slab constraints andlarge mass turbine as a non-conventional TMD a case studyrdquoBulletin of Earthquake Engineering vol 13 no 11 pp 3401ndash3422 2015

[21] L Y Peng Y J Kang Z R Lai and Y K Deng ldquoOptimisationand damping performance of a coal-fired power plantbuilding equipped with multiple coal bucket dampersrdquo Ad-vances in Civil Engineering vol 2018 p 19 2018

[22] Z Shu S Li J Zhang and M He ldquoOptimum seismic designof a power plant building with pendulum tuned mass dampersystem by its heavy suspended bucketsrdquo Engineering Struc-tures vol 136 pp 114ndash132 2017

[23] K Dai B Li J Wang et al ldquoOptimal probability-based partialmass isolation of elevated coal scuttle in thermal power plantbuildingrdquo Structural Design of Tall and Special Buildingsvol 27 no 11 article e1477 2018

[24] M De Angelis S Perno and A Reggio ldquoDynamic responseand optimal design of structures with large mass ratio TMDrdquoEarthquake Engineering amp Structural Dynamics vol 41 no 1pp 41ndash60 2015

[25] K Kanai ldquoSemi-empirical formula for the seismic charac-teristics of the groundrdquo Bulletin of the Earthquake ResearchInstitute gte University of Tokyo vol 35 pp 309ndash325 1957

[26] Code for seismic design of buildings GB50011-2010 BeijingChina 2010

[27] B F Spencer R E Christenson and S J Dyke ldquoNextgeneration benchmark control problem for seismically excitedbuildingsrdquo in Proceedings of the 2nd World Conference onStructural Control pp 1351ndash1360 Kyoto Japan June 1998

16 Shock and Vibration

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Error fopt_Form minusfopt_Num( )

fopt_Numtimes 100

Error ζ2opt_Form minus ζ2opt_Num( )

ζ2opt_Numtimes 100

(13)

where fopt_Form and ζopt_Form are the TMD optimal valuescalculated using the revised tting formula in equation (12)e actual optimal values obtained by a numerical opti-misation method are fopt_Num and ζ2opt_Num

It can be seen that the error between the optimal fre-quency ratio calculated by the revised tting formula and theactual optimal value is smaller When the damping ratio ofthe main structure is 0sim01 the maximum error rate is lessthan 1 When the mass ratio is less than 002 the error isrelatively large is is mainly due to the relatively smallTMD damping ratio which leads to a relatively large errorrate Under the condition of dierent mass ratio TMDs theerror rate can be kept within 5

On the whole the error of the revised tting formula issmall which can meet actual design requirements of a largemass ratio TMD under harmonic excitation

0 02 04 06 08 101

02

03

04

05

06

07

08

09

1

μ

f opt

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004ζ1 = 005ζ1 = 0075ζ1 = 010

Optimal solutionEquation (12)

(a)

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004ζ1 = 005ζ1 = 0075ζ1 = 010

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

μ

ζ 2op

t

Optimal solutionEquation (12)

(b)

Figure 4e relationship between the revised tting formula value and the actual optimal value with the TMDmass ratio and the dampingratio of the main structure under the excitation of harmonic load (a) Optimal frequency ratio (b) Optimal damping ratio

0 02 04 06 08 101

02

03

04

05

06

07

08

09

1

μ

f opt

ζ1 = 00ζ1 = 002ζ1 = 005

ζ1 = 0075ζ1 = 010

Equation (11)

(a)

μ

ζ1 = 00ζ1 = 002ζ1 = 005

ζ1 = 0075ζ1 = 010

Equation (11)

0 02 04 06 08 101

02

03

04

05

06

07

08

09

ζ 2op

t

(b)

Figure 3 e curve of the calculated value of the existing tting formula and the actual optimal value varies with the mass ratio under theexcitation of harmonic load (a) Contrast results of the frequency ratio (b) Contrast results of the damping ratio

Shock and Vibration 5

4 Fitting Formula for Optimal TMDParameters under Random Load andError Analysis

41 StationaryWhite Noise Random Load Under stationarywhite noise excitations the tting formula of the TMDoptimal frequency ratio and optimal damping ratio is givenin reference [11] as shown below

fopt 1minus μ2radic

1 + μ+(minus379441 + 987259

μ

radic minus 152978μ)

middot μ

radicζ1 +(minus136731 + 191284

μ

radic + 217049μ) μradic ζ21

ζ2opt

μ(1minus μ4)

4(1 + μ)(1minus μ2)

radic

(14)

However the tting formula only considers the case thatthemass ratio is less than 01When themass ratio of TMD isgreater than 01 the relationship between the optimal valuecalculated by the tting formula in equation (14) and theactual optimal value is shown in Figure 6 In equation (14)the optimal value of the damping ratio is independent of ζ1consequently there is only a curve for all considered valuesof ζ1 in Figure 6(b) It can be seen that when the mass ratio issmaller than 02 the error between the optimal parameterscalculated by the tting formula and the actual optimalparameters is still relatively small However with the in-crease of the mass ratio the error increases signicantly Inaddition the error of the tting formula increases with theincrease of the damping ratio of the main structure From

the error results the tting formula in equation (14) is alsonot suitable for calculating the optimal parameters of largemass ratio TMDs e new formula obtained by the curvetting method is as follows

fopt 1minus μ2radic

1 + μ+(minus29089 + 23354

μ

radic minus 05790μ) μradic ζ1

+(minus131740 + 307222μ

radic minus 212318μ) μradic ζ21

ζ2opt

μ(1minus μ4)

4(1 + μ)(1minus μ2)

radic

+(minus05747 + 15424μ

radic minus 07846μ) μradic ζ1

+(314166minus 968253 μ

radic + 739404μ)μζ21(15)

Under stationary white noise excitations the relation-ship between the calculated value of the revised ttingformula and the actual optimal value is shown in Figure 7e curve of error rate change relative to the mass ratio isshown in Figure 8e calculationmethod of the error rate isthe same as in equation (13)

e error between the optimal frequency ratio calculatedby the revised tting formula and the actual optimal value issmall e damping ratio of the main structure has an eecton the error However when the damping ratio is 01 themaximum error rate of the optimal frequency ratio is stillbelow 12 In addition the error rate of the optimaldamping ratio varies with the TMD mass ratio but it canalways be kept within 5

0 02 04 06 08 1ndash04

ndash02

0

02

04

06

08

1Er

ror (

)

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004

ζ1 = 005ζ1 = 075ζ1 = 010

μ

(a)

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004

ζ1 = 005ζ1 = 075ζ1 = 010

0 02 04 06 08 1ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

Erro

r (

)

μ

(b)

Figure 5e error rate of the revised tting formula value with the TMDmass ratio and the damping ratio of the main structure under theexcitation of harmonic load (a) Error rate of the frequency ratio (b) Error rate of the damping ratio

6 Shock and Vibration

On the whole the error of the revised tting formula issmall which meets actual design requirements of large massratio TMDs Comparisons between the design method inreference [16] and that in this paper are illustrated in Table 1It can be seen that the optimisation parameters in this studyare obviously larger than those in reference [16] is ismainly due to the inconsistency of the optimisation ob-jectives of the two methods

42 Filtered White Noise Random Load e ltered whitenoise random load is modelled as proposed by Kanai [25]

S(ω) ω4g + 4ζ2gω2

gω2

ω2g minusω2( )

2 + 4ζ2gω2gω2

S0 (16)

where ωg and ζg are the predominant frequency anddamping ratio of foundation soil respectively

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004ζ1 = 005ζ1 = 0075ζ1 = 010

Optimal solutionEquation (15)

0 02 04 06 08 101

02

03

04

05

06

07

08

09

1

μ

f opt

(a)

Optimal solutionEquation (15)

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004ζ1 = 005ζ1 = 0075ζ1 = 010

0 02 04 06 08 10

01

02

03

04

05

06

07

μ

ζ 2op

t

(b)

Figure 7e relationship between the revised tting formula value and the actual optimal value with the TMDmass ratio and the dampingratio of the main structure under stationary white noise excitations (a) Optimal frequency ratio (b) Optimal damping ratio

0 02 04 06 08 1ndash04

ndash02

0

02

04

06

08

1f op

t

ζ1 = 00ζ1 = 002ζ1 = 005

ζ1 = 0075ζ1 = 010

Equation (14)

μ

(a)

ζ1 = 00ζ1 = 002ζ1 = 005

ζ1 = 0075ζ1 = 010

Equation (14)

0 02 04 06 08 101

015

02

025

03

035

04

045

05

055

06

μ

ζ 2op

t

(b)

Figure 6 e curve of the calculated value of the existing tting formula and the actual optimal value varies with the mass ratio understationary white noise excitations (a) Contrast results of the frequency ratio (b) Contrast results of the damping ratio

Shock and Vibration 7

Substituting S(ω) into equation (5) the mean square ofthe structural displacement response is obtained as

σ21 int+infin

minusinfinh1(ω)∣∣∣∣

∣∣∣∣2ω4g + 4ζ2gω

2gω

2

ω2g minusω2( )

2 + 4ζ2gω2gω2

S0 dω (17)

Because solving equation (17) analytically is very tediousa numerical integration method can be used e results byBakre [11] show that the TMD optimal parameters arerelatively close when the structure is excited by white noiserandom load and ltered white noise random load but onlywhen a mass ratio below 01 is examined e analysis resultobtained by Hoang et al [6] shows that there is a certaindierence between the optimal parameters for the largermass ratio TMD

When solving the optimal parameters the eect of themain structure damping ratio should be taken into accountthe predominant frequency and damping ratio of the ground

soil that is ωg and ζg are also taken into considerationHowever it is not practical to construct a correspondingtting formula erefore the optimisation process of TMDoptimal parameters is described in detail and the corre-sponding tting formula is no longer determined e de-tailed optimisation currenow chart is shown in Figure 9 Firstly

Table 1 Optimal design of large mass ratio TMD according toreference [16] and this study

Structuralparameters Method fopt ζopt σ21(2πS0ω3

1)

μ 0431 ζ1 002Reference

[16] 06871 02729 23830

is study 05970 02939 22429

μ 05 ζ1 002Reference

[16] 06543 02900 23862

is study 05549 03135 22246

μ 05 ζ1 003Reference

[16] 06492 02861 22947

is study 05432 03144 21320

Main structure parametersvibration frequency and damping ratio

The initial iteration value of the optimal parametercalculated by fitting formula (15)

Objective function the root mean square of the displacement response of the main structure

MATLAB optimization toolboxfunction Fmincon

Mass ratio of TMD

Parameters of ground motion power spectrum modelpredominant frequency and damping ratio

Optimal parameters and the correspondingobjective function values

Figure 9 e numerical optimisation process of TMD parametersunder ltered white noise excitations

0 02 04 06μ

08 1ndash04

ndash02

0

02

04

06

08

1

12Er

ror (

)

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004

ζ1 = 005ζ1 = 075ζ1 = 010

(a)

μ

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004

ζ1 = 005ζ1 = 075ζ1 = 010

0 02 04 06 08 1ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

Erro

r (

)(b)

Figure 8 e error rate of the revised tting formula value with the TMD mass ratio and the damping ratio of the main structure understationary white noise excitations (a) Error rate of the frequency ratio (b) Error rate of the damping ratio

8 Shock and Vibration

the dynamic characteristics of the main structure includingnatural frequency and damping ratio are determined enthe mass ratio of TMD and site condition parameters in-cluding the predominant frequency and damping ratio areselected After that the objective of optimisation is de-termined that is the mean square of the structural dis-placement Finally taking the calculated value of equation (15)as the initial value the optimal parameters can be obtained byoptimum analysis using MATLAB optimisation toolbox

Taking class II sites [26] as an example the parameters ofthe Kanai-Tajimi model ωg 15708 and ζg 072 aredetermined S0 is always taken as a constant and it has noeect on TMD parameter optimisation At the same time

λ ωgω1 is dened as the ratio between the predominantfrequency of ltered white noise and the natural frequency ofthe structure In order to study the TMD optimal parametersand the relative relationship between the excitation of sta-tionary white noise random load and ltered white noiserandom load the following denitions are given

ηf fopt_Num

fopt_Form

ηζ2 ζ2opt_Numζ2opt_Form

(18)

0 02 04 06 08 102

03

04

05

06

07

08

09

1

μ

f opt

λ = 1λ = 2λ = 3λ = 4

λ = 5λ = 75Equation (15)

(a)

μ0 02 04 06 08 1

098

1

102

104

106

108

11

112

114

λ = 1λ = 2λ = 3

λ = 4λ = 5λ = 75

η f(b)

λ = 1λ = 2λ = 3λ = 4

λ = 5λ = 75Equation (15)

0 02 04 06 08 101

015

02

025

03

035

04

045

05

μ

ζ 2op

t

(c)

λ = 1λ = 2λ = 3

λ = 4λ = 5λ = 75

μ0 02 04 06 08 1

084

086

088

09

092

094

096

098

1

102

104η ζ

2

(d)

Figure 10 e relation between optimal parameters with λ and μ and its deviation rate with equation (15) under ltered white noiseexcitations (a) Optimal frequency ratio (b) ηf (c) Optimal damping ratio (d) ηζ2opt

Shock and Vibration 9

Under stationary white noise excitations fopt_Form andζopt_Form are the TMD optimal frequency ratios and theoptimal damping ratios calculated by the tting equation(15) respectively Under ltered white noise excitationsfopt_Num and ζopt_Num are the TMD optimal frequency ratioand the optimal damping ratio respectively obtained by thenumerical optimisation method

When the main structure with a damping ratio of 005 isexcited by ltered white noise load the relationship betweenthe optimal parameters of the TMD and μ and λ is shown inFigures 10(a) and 10(c) In order to facilitate comparisonthe optimal parameter calculated by the tting formula inequation (15) is also plotted Figures 10(b) and 10(d) showrespectively the trends of ηf and ηζ2opt with μ and for dif-ferent values of λ

From the above results the following results areobtained

When the mass ratio μ is smaller than 02 the optimalparameters of the stationary white noise random load andthat of the ltered white noise random load are more similarbut the dierence between the two is increased graduallywith the increase of the mass ratio which can be clearly seenfrom the relationship curve of ηf to μ

e larger the λ the more ηf and ηζ2opt become closer to1 When λge 4 even if the TMD mass ratio reaches 1 theerror of the optimal parameters of the TMD under the

stationary white noise random load and the ltered whitenoise random load is not more than 5

In addition the damping ratio of the main structurehas a relatively small impact on the error therefore theincurrenuence of the damping ratio of the main structure onthe error of the tting formula is no longer discussed inthis paper In conclusion for the optimal design of a largemass ratio TMD when λge 4 the tting formula inequation (15) is also suitable for ltered white noise ex-citations while in the other cases it is suggested to de-termine the optimal TMD parameters using a numericaloptimisation method

5 Damping Effect Analysis of a Large MassRatio TMD

Four SDOF structures with 05 10 20 and 30 s periods (T)are selected and the damping ratio of the four structures is005 e TMD mass ratios are 005 025 050 075 and 1and the TMD parameters are calculated by the tting for-mula in equation (15) Two far-eld seismic waves (ElCentro Hachinohe) and two near-eld seismic waves(Northridge Kobe) are used for load input [27] Seismicwave information is provided in Table 2 e seismic waveacceleration-time history curve is shown in Figure 11 and

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

Acce

lera

tion

(ms

2 )Ac

cele

ratio

n (m

s2 )

El Centro

Hachinohe

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

(a)

Acce

lera

tion

(ms

2 )Ac

cele

ratio

n (m

s2 )

Northridge

Kobe

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

(b)

Figure 11 Time history curve of seismic records (a) Far-eld seismic waves (b) Near-eld seismic waves

Table 2 Seismic wave information used in the analyses

Type Name Earthquake Station Component Year

Far eld El Centro Imperial valley El Centro N-S 1940Hachinohe Tokachi-Oki Hachinohe city N-S 1968

Near eld Northridge Northridge SCH N-S 1994Kobe Hyogo-ken KJMA N-S 1995

10 Shock and Vibration

the seismic wave peak value is 1ms2 In order to investigatethe eect of the TMD mass ratio on the control of thestructural displacement response four sets of seismic timehistory analyses were carried out for structures with andwithout a TMD respectively

e reduction rate of peak displacement Re(Peak) andthe reduction rate of root mean square of structural dis-placement Re(RMS) are used as the evaluation index re-spectively e two formulas for calculating the dampingrate are shown as follows

Re(Peak) 1minusMax xTMD( )Max(x)

Re(RMS) 1minusRMS xTMD( )RMS(x)

(19)

where x and xTMD are the time history record of the dis-placement response of the structure without and with aTMD respectively e formula for calculating the rootmean square of the displacement is as follows

0 02 04 06 08 1micro

0

10

20

30

40

50Re

(Pea

k) (

)

EI CentroHachinoheKobe

NorthridgeMean

(a)

0 02 04 06 08 1micro

0

10

20

30

40

50

Re(P

eak)

()

EI CentroHachinoheKobe

NorthridgeMean

(b)

0 02 04 06 08 1micro

0

10

20

30

40

50

Re(P

eak)

()

EI CentroHachinoheKobe

NorthridgeMean

(c)

0 02 04 06 08 1micro

0

10

20

30

40

50Re

(Pea

k) (

)

EI CentroHachinoheKobe

NorthridgeMean

(d)

Figure 12 e damping eect of dierent mass ratio (TMD) on the peak value of structural displacement response (a) T 05 s (b)T10 s (c) T 20 s (d) T 30 s

Shock and Vibration 11

RMS(x) sqrt1NsumN

i1x2i (20)

where xi is the structural displacement response corre-sponding to the ith time andN is the total number of pointscollected

Re(Peak) and Re(RMS) of the displacement responsewith dierent mass ratio TMDs are shown in Figures 12 and13 e following conclusions can be obtained

(1) TMD can eectively control the displacement re-sponse of the structure and the large mass ratio

(gt025) TMD is more eective than the conventionalsmall mass ratio (lt005) TMD But it can also befound that when the mass ratio of the TMD is greaterthan 05 the gain eect will diminish with increasingmass ratio

(2) e TMD with the same mass ratio shows certaindiscreteness for the structures with dierent naturalvibration periods and dierent seismic waves Forexample as shown in Figure 12(b) when thestructurersquos period is 10 s the damping rate of fourseismic waves is distinct When the mass ratio is 05the minimum damping rate is 1047 and the

0 02 04 06 08 10

10

20

30

40

50

60

micro

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(a)

0 02 04 06 08 1micro

0

10

20

30

40

50

60

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(b)

0 02 04 06 08 10

10

20

30

40

50

60

micro

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(c)

0 02 04 06 08 1micro

0

10

20

30

40

50

60Re

(RM

S)

EI CentroHachinoheKobe

NorthridgeMean

(d)

Figure 13e damping eect of dierent mass ratios (TMD) on the mean square root of structural displacement response (a) T 05 s (b)T10 s (c) T 20 s (d) T 30 s

12 Shock and Vibration

Table 3 Maximum peak value of relative displacement between a TMD with optimal parameters and the main structure (cm)

T (s) μ El Centro Hachinohe Northridge Kobe Mean

05

005 436 296 381 312 356010 268 214 298 22 25050 214 254 335 191 249075 183 26 275 279 249100 224 274 224 337 265

10

005 818 883 1039 698 859010 353 483 512 598 487050 341 535 376 588 46075 391 59 355 546 47100 356 494 358 502 428

20

005 1477 209 866 1834 1567010 794 1197 554 1042 897050 53 863 417 858 667075 394 795 345 705 56100 389 846 332 629 549

30

005 1509 3529 666 1993 1925010 682 13 378 889 812050 491 1128 329 654 651075 463 1037 293 553 586100 509 1125 262 536 608

minus002

0

002

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(a)

minus002

0

002

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

micro = 005micro = 075

wo

(b)

Figure 14 Continued

Shock and Vibration 13

maximum is 4622 which is usually mainly due tothe dierence in the frequency relationship betweenthe structure and the seismic waves

Table 3 shows the statistical results of the relative dis-placement between the TMD and main structure underdierent time history analysis conditions It can be seen thatthe relative displacement between the TMD and the mainstructure is inversely proportional to the mass ratio of theTMD that is when a large mass ratio TMD is used thedisplacement response of the structure is eectively con-trolled and the displacement stroke of the TMD is clearlyreduced which reduces the requirements for the elasticcomponents and the damping components used to constructthe TMD

e displacement time histories of the structures withperiods of 10 and 20 s and the relative displacement timehistories between the TMD and the structures are shown inFigures 14 and 15 respectively e schemes of mass ratiosof 005 and 075 are compared It can be clearly seen that theeect of a TMD in controlling the structural response andthe TMD displacement stroke is more obvious for the TMDwith a mass ratio of 075 than for the one with a mass ratio of005

In summary the large mass ratio TMD has a moresignicant eect in seismic control of the main structurethan the small mass ratio TMD

6 Conclusions

In order to control the dynamic response and improve theaseismic performance of a structure a large mass ratio TMDdamping system is formed by using the equipment in thebuilding structure or relying on new structural forms e

existing optimal parameter tting formula is not applicableto large mass ratio TMDs so it is revised by numericaloptimisation and curve tting and the dynamic time historyanalysis method is used to study the eect of vibrationdamping control of large mass ratio TMDs e followingconclusions are obtained

Compared with the traditional small mass ratio (lt005)TMD the large mass ratio (gt05) TMD has obvious ad-vantages in controlling the displacement response of themain structure e control eect is about 15sim325 timeshigher the damping eect of the structural displacementpeak can reach about 30 and the damping ratio of the rootmean square displacement can reach about 436 At thesame time the relative stroke between the TMD and themain structure can be reduced with up to 30sim65 which ishighly benecial to the practical engineering design of TMDstructures

When the mass ratio of a TMD is relatively large (gt02)the results calculated by the existing tting formula diersignicantly from the actual optimal value and the calcu-lated values of the revised formula proposed in this paper areshown to be in good agreement with the actual optimalvalue In general the revised formula can be applied to bothtraditional small mass ratio and large mass ratio (le1) TMDsWhen the mass ratio is greater than 1 the optimal pa-rameters of TMD can also be obtained by the methodpresented in this paper

When the mass ratio is greater than 02 the optimalparameters of the stationary white noise random load andthat of the ltered white noise random load are more similarbut the dierence between the two is gradually increasedwith the increase of the TMD mass ratio For the optimalparameters of large mass ratio TMDs (gt02) the error is lessthan 005 when the ratio of the predominant frequency of the

0 5 10 15 20 25 30

minus002minus001

0001002

t (s)

0 5 10 15 20 25 30t (s)

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(c)

0 5 10 15 20 25 30t (s)

0 5 10 15 20 25 30t (s)

minus002

0

002

004

Disp

lace

men

t (m

)

minus01

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(d)

Figure 14 e damping eect of dierent mass ratio TMDs on the displacement-time history of the main structure (T10 s) and therelative displacement-time history between the TMD and the structure (wo denotes the displacement of a structure without a TMD) (a) ElCentro (b) Hachinohe (c) Northridge (d) Kobe

14 Shock and Vibration

base soil and the vibration frequency of the structure isgreater than 4 and the optimal parameters of the TMD canbe calculated by the tting formula proposed in this paperUnder other conditions it is suggested to use an optimi-sation method to determine the optimal value of TMDparameters

At present the actual engineering projects with largemass ratio TMD damping systems are less prominent buttheir aseismic advantages will bring a broad range of benetsfor research and practice

Data Availability

e data used to support the ndings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no concurrenicts of interestregarding the publication of this article

Acknowledgments

is research was supported by Grant no 51478023 from theNational Natural Science Foundation of China

References

[1] M Gutierrez Soto and H Adeli ldquoTuned mass dampersrdquoArchives of Computational Methods in Engineering vol 20no 4 pp 419ndash431 2013

ndash005

0

005

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

ndash01ndash005

0005

01

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(a)

ndash02

ndash01

0

01

02

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

ndash005

005

0

micro = 005micro = 075

wo

(b)

0 5 10 15 20 25 30

minus005

0

005

Disp

lace

men

t (m

)

t (s)

0 5 10 15 20 25 30t (s)

minus01

0

01

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(c)

minus002

0

002

004D

ispla

cem

ent (

m)

0 5 10 15 20 25 30t (s)

0

minus005

005

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30t (s)

micro = 005micro = 075

wo

(d)

Figure 15 e damping eect of dierent mass ratio TMDs on the displacement-time history of the main structure (T 20 s) and therelative displacement-time history between the TMD and the structure (wo denotes the displacement of a structure without a TMD) (a) ElCentro (b) Hachinohe (c) Northridge (d) Kobe

Shock and Vibration 15

[2] S Elias and VMatsagar ldquoResearch developments in vibrationcontrol of structures using passive tuned mass dampersrdquoAnnual Reviews in Control vol 44 pp 129ndash156 2017

[3] D Wang T K T Tse Y Zhou and Q Li ldquoStructural per-formance and cost analysis of wind-induced vibration controlschemes for a real super-tall buildingrdquo Structure and In-frastructure Engineering vol 11 no 8 pp 990ndash1011 2014

[4] N Longarini and M Zucca ldquoA chimneyrsquos seismic assessmentby a tuned mass damperrdquo Engineering Structures vol 79pp 290ndash296 2014

[5] L Tian and Y Zeng ldquoParametric study of tunedmass dampersfor long span transmission tower-line system under windloadsrdquo Shock and Vibration vol 2016 Article ID 496505611 pages 2016

[6] N Hoang Y Fujino and PWarnitchai ldquoOptimal tuned massdamper for seismic applications and practical design for-mulasrdquo Engineering Structures vol 30 no 3 pp 707ndash7152008

[7] J P Den Hartog Mechanical vibrations McGraw-Hill NewYork NY USA 1956

[8] G B Warburton ldquoOptimum absorber parameters for variouscombinations of response and excitation parametersrdquoEarthquake Engineering amp Structural Dynamics vol 10 no 3pp 381ndash401 1982

[9] H-C Tsai and G-C Lin ldquoExplicit formulae for optimumabsorber parameters for force-excited and viscously dampedsystemsrdquo Journal of Sound and Vibration vol 176 no 5pp 585ndash596 1994

[10] H-C Tsai and G-C Lin ldquoOptimum tuned-mass dampers forminimizing steady-state response of support-excited anddamped systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 22 no 11 pp 957ndash973 1993

[11] S V Bakre and R S Jangid ldquoOptimum parameters of tunedmass damper for damped main systemrdquo Structural Controland Health Monitoring vol 14 no 3 pp 448ndash470 2007

[12] C C Lin C M Hu J F Wang and R Y Hu ldquoVibrationcontrol effectiveness of passive tuned mass dampersrdquo Journalof the Chinese Institute of Engineers vol 17 pp 367ndash376 1994

[13] A Y T Leung and H Zhang ldquoParticle swarm optimization oftuned mass dampersrdquo Engineering Structures vol 31 no 3pp 715ndash728 2009

[14] M Q Feng and A Mita ldquoVibration control of tall buildingsusing mega SubConfigurationrdquo Journal of Engineering Me-chanics vol 121 no 10 pp 1082ndash1088 1995

[15] X X Li P Tan X J Li and A W Liu ldquoMechanism analysisand parameter optimisation of mega-sub-isolation systemrdquoShock and Vibration vol 2016 p 12 2016

[16] A Reggio and M D Angelis ldquoOptimal energy-based seismicdesign of non-conventional tuned mass damper (TMD)implemented via inter-story isolationrdquo Earthquake Engi-neering amp Structural Dynamics vol 44 no 10 pp 1623ndash16422015

[17] S J Wang B H Lee W C Chuang and K C ChangldquoOptimal dynamic characteristic control approach forbuilding mass damper designrdquo Earthquake Engineering ampStructural Dynamics vol 47 no 3 2017

[18] H Anajafi and R A Medina ldquoPartial mass isolation systemfor seismic vibration control of buildingsrdquo Structural Controlamp Health Monitoring vol 25 no 2 article e2088 2017

[19] K Yuan M S He and Y M Li ldquoShaking table tests forenergy-dissipation steel frame structures with infilled wallMTMDrdquo Journal of Vibration and Shock vol 33 no 11pp 200ndash207 2014

[20] R Ding M X Tao M Zhou and J G Nie ldquoSeismic behaviorof RC structures with absence of floor slab constraints andlarge mass turbine as a non-conventional TMD a case studyrdquoBulletin of Earthquake Engineering vol 13 no 11 pp 3401ndash3422 2015

[21] L Y Peng Y J Kang Z R Lai and Y K Deng ldquoOptimisationand damping performance of a coal-fired power plantbuilding equipped with multiple coal bucket dampersrdquo Ad-vances in Civil Engineering vol 2018 p 19 2018

[22] Z Shu S Li J Zhang and M He ldquoOptimum seismic designof a power plant building with pendulum tuned mass dampersystem by its heavy suspended bucketsrdquo Engineering Struc-tures vol 136 pp 114ndash132 2017

[23] K Dai B Li J Wang et al ldquoOptimal probability-based partialmass isolation of elevated coal scuttle in thermal power plantbuildingrdquo Structural Design of Tall and Special Buildingsvol 27 no 11 article e1477 2018

[24] M De Angelis S Perno and A Reggio ldquoDynamic responseand optimal design of structures with large mass ratio TMDrdquoEarthquake Engineering amp Structural Dynamics vol 41 no 1pp 41ndash60 2015

[25] K Kanai ldquoSemi-empirical formula for the seismic charac-teristics of the groundrdquo Bulletin of the Earthquake ResearchInstitute gte University of Tokyo vol 35 pp 309ndash325 1957

[26] Code for seismic design of buildings GB50011-2010 BeijingChina 2010

[27] B F Spencer R E Christenson and S J Dyke ldquoNextgeneration benchmark control problem for seismically excitedbuildingsrdquo in Proceedings of the 2nd World Conference onStructural Control pp 1351ndash1360 Kyoto Japan June 1998

16 Shock and Vibration

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4 Fitting Formula for Optimal TMDParameters under Random Load andError Analysis

41 StationaryWhite Noise Random Load Under stationarywhite noise excitations the tting formula of the TMDoptimal frequency ratio and optimal damping ratio is givenin reference [11] as shown below

fopt 1minus μ2radic

1 + μ+(minus379441 + 987259

μ

radic minus 152978μ)

middot μ

radicζ1 +(minus136731 + 191284

μ

radic + 217049μ) μradic ζ21

ζ2opt

μ(1minus μ4)

4(1 + μ)(1minus μ2)

radic

(14)

However the tting formula only considers the case thatthemass ratio is less than 01When themass ratio of TMD isgreater than 01 the relationship between the optimal valuecalculated by the tting formula in equation (14) and theactual optimal value is shown in Figure 6 In equation (14)the optimal value of the damping ratio is independent of ζ1consequently there is only a curve for all considered valuesof ζ1 in Figure 6(b) It can be seen that when the mass ratio issmaller than 02 the error between the optimal parameterscalculated by the tting formula and the actual optimalparameters is still relatively small However with the in-crease of the mass ratio the error increases signicantly Inaddition the error of the tting formula increases with theincrease of the damping ratio of the main structure From

the error results the tting formula in equation (14) is alsonot suitable for calculating the optimal parameters of largemass ratio TMDs e new formula obtained by the curvetting method is as follows

fopt 1minus μ2radic

1 + μ+(minus29089 + 23354

μ

radic minus 05790μ) μradic ζ1

+(minus131740 + 307222μ

radic minus 212318μ) μradic ζ21

ζ2opt

μ(1minus μ4)

4(1 + μ)(1minus μ2)

radic

+(minus05747 + 15424μ

radic minus 07846μ) μradic ζ1

+(314166minus 968253 μ

radic + 739404μ)μζ21(15)

Under stationary white noise excitations the relation-ship between the calculated value of the revised ttingformula and the actual optimal value is shown in Figure 7e curve of error rate change relative to the mass ratio isshown in Figure 8e calculationmethod of the error rate isthe same as in equation (13)

e error between the optimal frequency ratio calculatedby the revised tting formula and the actual optimal value issmall e damping ratio of the main structure has an eecton the error However when the damping ratio is 01 themaximum error rate of the optimal frequency ratio is stillbelow 12 In addition the error rate of the optimaldamping ratio varies with the TMD mass ratio but it canalways be kept within 5

0 02 04 06 08 1ndash04

ndash02

0

02

04

06

08

1Er

ror (

)

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004

ζ1 = 005ζ1 = 075ζ1 = 010

μ

(a)

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004

ζ1 = 005ζ1 = 075ζ1 = 010

0 02 04 06 08 1ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

2

4

Erro

r (

)

μ

(b)

Figure 5e error rate of the revised tting formula value with the TMDmass ratio and the damping ratio of the main structure under theexcitation of harmonic load (a) Error rate of the frequency ratio (b) Error rate of the damping ratio

6 Shock and Vibration

On the whole the error of the revised tting formula issmall which meets actual design requirements of large massratio TMDs Comparisons between the design method inreference [16] and that in this paper are illustrated in Table 1It can be seen that the optimisation parameters in this studyare obviously larger than those in reference [16] is ismainly due to the inconsistency of the optimisation ob-jectives of the two methods

42 Filtered White Noise Random Load e ltered whitenoise random load is modelled as proposed by Kanai [25]

S(ω) ω4g + 4ζ2gω2

gω2

ω2g minusω2( )

2 + 4ζ2gω2gω2

S0 (16)

where ωg and ζg are the predominant frequency anddamping ratio of foundation soil respectively

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004ζ1 = 005ζ1 = 0075ζ1 = 010

Optimal solutionEquation (15)

0 02 04 06 08 101

02

03

04

05

06

07

08

09

1

μ

f opt

(a)

Optimal solutionEquation (15)

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004ζ1 = 005ζ1 = 0075ζ1 = 010

0 02 04 06 08 10

01

02

03

04

05

06

07

μ

ζ 2op

t

(b)

Figure 7e relationship between the revised tting formula value and the actual optimal value with the TMDmass ratio and the dampingratio of the main structure under stationary white noise excitations (a) Optimal frequency ratio (b) Optimal damping ratio

0 02 04 06 08 1ndash04

ndash02

0

02

04

06

08

1f op

t

ζ1 = 00ζ1 = 002ζ1 = 005

ζ1 = 0075ζ1 = 010

Equation (14)

μ

(a)

ζ1 = 00ζ1 = 002ζ1 = 005

ζ1 = 0075ζ1 = 010

Equation (14)

0 02 04 06 08 101

015

02

025

03

035

04

045

05

055

06

μ

ζ 2op

t

(b)

Figure 6 e curve of the calculated value of the existing tting formula and the actual optimal value varies with the mass ratio understationary white noise excitations (a) Contrast results of the frequency ratio (b) Contrast results of the damping ratio

Shock and Vibration 7

Substituting S(ω) into equation (5) the mean square ofthe structural displacement response is obtained as

σ21 int+infin

minusinfinh1(ω)∣∣∣∣

∣∣∣∣2ω4g + 4ζ2gω

2gω

2

ω2g minusω2( )

2 + 4ζ2gω2gω2

S0 dω (17)

Because solving equation (17) analytically is very tediousa numerical integration method can be used e results byBakre [11] show that the TMD optimal parameters arerelatively close when the structure is excited by white noiserandom load and ltered white noise random load but onlywhen a mass ratio below 01 is examined e analysis resultobtained by Hoang et al [6] shows that there is a certaindierence between the optimal parameters for the largermass ratio TMD

When solving the optimal parameters the eect of themain structure damping ratio should be taken into accountthe predominant frequency and damping ratio of the ground

soil that is ωg and ζg are also taken into considerationHowever it is not practical to construct a correspondingtting formula erefore the optimisation process of TMDoptimal parameters is described in detail and the corre-sponding tting formula is no longer determined e de-tailed optimisation currenow chart is shown in Figure 9 Firstly

Table 1 Optimal design of large mass ratio TMD according toreference [16] and this study

Structuralparameters Method fopt ζopt σ21(2πS0ω3

1)

μ 0431 ζ1 002Reference

[16] 06871 02729 23830

is study 05970 02939 22429

μ 05 ζ1 002Reference

[16] 06543 02900 23862

is study 05549 03135 22246

μ 05 ζ1 003Reference

[16] 06492 02861 22947

is study 05432 03144 21320

Main structure parametersvibration frequency and damping ratio

The initial iteration value of the optimal parametercalculated by fitting formula (15)

Objective function the root mean square of the displacement response of the main structure

MATLAB optimization toolboxfunction Fmincon

Mass ratio of TMD

Parameters of ground motion power spectrum modelpredominant frequency and damping ratio

Optimal parameters and the correspondingobjective function values

Figure 9 e numerical optimisation process of TMD parametersunder ltered white noise excitations

0 02 04 06μ

08 1ndash04

ndash02

0

02

04

06

08

1

12Er

ror (

)

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004

ζ1 = 005ζ1 = 075ζ1 = 010

(a)

μ

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004

ζ1 = 005ζ1 = 075ζ1 = 010

0 02 04 06 08 1ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

Erro

r (

)(b)

Figure 8 e error rate of the revised tting formula value with the TMD mass ratio and the damping ratio of the main structure understationary white noise excitations (a) Error rate of the frequency ratio (b) Error rate of the damping ratio

8 Shock and Vibration

the dynamic characteristics of the main structure includingnatural frequency and damping ratio are determined enthe mass ratio of TMD and site condition parameters in-cluding the predominant frequency and damping ratio areselected After that the objective of optimisation is de-termined that is the mean square of the structural dis-placement Finally taking the calculated value of equation (15)as the initial value the optimal parameters can be obtained byoptimum analysis using MATLAB optimisation toolbox

Taking class II sites [26] as an example the parameters ofthe Kanai-Tajimi model ωg 15708 and ζg 072 aredetermined S0 is always taken as a constant and it has noeect on TMD parameter optimisation At the same time

λ ωgω1 is dened as the ratio between the predominantfrequency of ltered white noise and the natural frequency ofthe structure In order to study the TMD optimal parametersand the relative relationship between the excitation of sta-tionary white noise random load and ltered white noiserandom load the following denitions are given

ηf fopt_Num

fopt_Form

ηζ2 ζ2opt_Numζ2opt_Form

(18)

0 02 04 06 08 102

03

04

05

06

07

08

09

1

μ

f opt

λ = 1λ = 2λ = 3λ = 4

λ = 5λ = 75Equation (15)

(a)

μ0 02 04 06 08 1

098

1

102

104

106

108

11

112

114

λ = 1λ = 2λ = 3

λ = 4λ = 5λ = 75

η f(b)

λ = 1λ = 2λ = 3λ = 4

λ = 5λ = 75Equation (15)

0 02 04 06 08 101

015

02

025

03

035

04

045

05

μ

ζ 2op

t

(c)

λ = 1λ = 2λ = 3

λ = 4λ = 5λ = 75

μ0 02 04 06 08 1

084

086

088

09

092

094

096

098

1

102

104η ζ

2

(d)

Figure 10 e relation between optimal parameters with λ and μ and its deviation rate with equation (15) under ltered white noiseexcitations (a) Optimal frequency ratio (b) ηf (c) Optimal damping ratio (d) ηζ2opt

Shock and Vibration 9

Under stationary white noise excitations fopt_Form andζopt_Form are the TMD optimal frequency ratios and theoptimal damping ratios calculated by the tting equation(15) respectively Under ltered white noise excitationsfopt_Num and ζopt_Num are the TMD optimal frequency ratioand the optimal damping ratio respectively obtained by thenumerical optimisation method

When the main structure with a damping ratio of 005 isexcited by ltered white noise load the relationship betweenthe optimal parameters of the TMD and μ and λ is shown inFigures 10(a) and 10(c) In order to facilitate comparisonthe optimal parameter calculated by the tting formula inequation (15) is also plotted Figures 10(b) and 10(d) showrespectively the trends of ηf and ηζ2opt with μ and for dif-ferent values of λ

From the above results the following results areobtained

When the mass ratio μ is smaller than 02 the optimalparameters of the stationary white noise random load andthat of the ltered white noise random load are more similarbut the dierence between the two is increased graduallywith the increase of the mass ratio which can be clearly seenfrom the relationship curve of ηf to μ

e larger the λ the more ηf and ηζ2opt become closer to1 When λge 4 even if the TMD mass ratio reaches 1 theerror of the optimal parameters of the TMD under the

stationary white noise random load and the ltered whitenoise random load is not more than 5

In addition the damping ratio of the main structurehas a relatively small impact on the error therefore theincurrenuence of the damping ratio of the main structure onthe error of the tting formula is no longer discussed inthis paper In conclusion for the optimal design of a largemass ratio TMD when λge 4 the tting formula inequation (15) is also suitable for ltered white noise ex-citations while in the other cases it is suggested to de-termine the optimal TMD parameters using a numericaloptimisation method

5 Damping Effect Analysis of a Large MassRatio TMD

Four SDOF structures with 05 10 20 and 30 s periods (T)are selected and the damping ratio of the four structures is005 e TMD mass ratios are 005 025 050 075 and 1and the TMD parameters are calculated by the tting for-mula in equation (15) Two far-eld seismic waves (ElCentro Hachinohe) and two near-eld seismic waves(Northridge Kobe) are used for load input [27] Seismicwave information is provided in Table 2 e seismic waveacceleration-time history curve is shown in Figure 11 and

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

Acce

lera

tion

(ms

2 )Ac

cele

ratio

n (m

s2 )

El Centro

Hachinohe

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

(a)

Acce

lera

tion

(ms

2 )Ac

cele

ratio

n (m

s2 )

Northridge

Kobe

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

(b)

Figure 11 Time history curve of seismic records (a) Far-eld seismic waves (b) Near-eld seismic waves

Table 2 Seismic wave information used in the analyses

Type Name Earthquake Station Component Year

Far eld El Centro Imperial valley El Centro N-S 1940Hachinohe Tokachi-Oki Hachinohe city N-S 1968

Near eld Northridge Northridge SCH N-S 1994Kobe Hyogo-ken KJMA N-S 1995

10 Shock and Vibration

the seismic wave peak value is 1ms2 In order to investigatethe eect of the TMD mass ratio on the control of thestructural displacement response four sets of seismic timehistory analyses were carried out for structures with andwithout a TMD respectively

e reduction rate of peak displacement Re(Peak) andthe reduction rate of root mean square of structural dis-placement Re(RMS) are used as the evaluation index re-spectively e two formulas for calculating the dampingrate are shown as follows

Re(Peak) 1minusMax xTMD( )Max(x)

Re(RMS) 1minusRMS xTMD( )RMS(x)

(19)

where x and xTMD are the time history record of the dis-placement response of the structure without and with aTMD respectively e formula for calculating the rootmean square of the displacement is as follows

0 02 04 06 08 1micro

0

10

20

30

40

50Re

(Pea

k) (

)

EI CentroHachinoheKobe

NorthridgeMean

(a)

0 02 04 06 08 1micro

0

10

20

30

40

50

Re(P

eak)

()

EI CentroHachinoheKobe

NorthridgeMean

(b)

0 02 04 06 08 1micro

0

10

20

30

40

50

Re(P

eak)

()

EI CentroHachinoheKobe

NorthridgeMean

(c)

0 02 04 06 08 1micro

0

10

20

30

40

50Re

(Pea

k) (

)

EI CentroHachinoheKobe

NorthridgeMean

(d)

Figure 12 e damping eect of dierent mass ratio (TMD) on the peak value of structural displacement response (a) T 05 s (b)T10 s (c) T 20 s (d) T 30 s

Shock and Vibration 11

RMS(x) sqrt1NsumN

i1x2i (20)

where xi is the structural displacement response corre-sponding to the ith time andN is the total number of pointscollected

Re(Peak) and Re(RMS) of the displacement responsewith dierent mass ratio TMDs are shown in Figures 12 and13 e following conclusions can be obtained

(1) TMD can eectively control the displacement re-sponse of the structure and the large mass ratio

(gt025) TMD is more eective than the conventionalsmall mass ratio (lt005) TMD But it can also befound that when the mass ratio of the TMD is greaterthan 05 the gain eect will diminish with increasingmass ratio

(2) e TMD with the same mass ratio shows certaindiscreteness for the structures with dierent naturalvibration periods and dierent seismic waves Forexample as shown in Figure 12(b) when thestructurersquos period is 10 s the damping rate of fourseismic waves is distinct When the mass ratio is 05the minimum damping rate is 1047 and the

0 02 04 06 08 10

10

20

30

40

50

60

micro

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(a)

0 02 04 06 08 1micro

0

10

20

30

40

50

60

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(b)

0 02 04 06 08 10

10

20

30

40

50

60

micro

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(c)

0 02 04 06 08 1micro

0

10

20

30

40

50

60Re

(RM

S)

EI CentroHachinoheKobe

NorthridgeMean

(d)

Figure 13e damping eect of dierent mass ratios (TMD) on the mean square root of structural displacement response (a) T 05 s (b)T10 s (c) T 20 s (d) T 30 s

12 Shock and Vibration

Table 3 Maximum peak value of relative displacement between a TMD with optimal parameters and the main structure (cm)

T (s) μ El Centro Hachinohe Northridge Kobe Mean

05

005 436 296 381 312 356010 268 214 298 22 25050 214 254 335 191 249075 183 26 275 279 249100 224 274 224 337 265

10

005 818 883 1039 698 859010 353 483 512 598 487050 341 535 376 588 46075 391 59 355 546 47100 356 494 358 502 428

20

005 1477 209 866 1834 1567010 794 1197 554 1042 897050 53 863 417 858 667075 394 795 345 705 56100 389 846 332 629 549

30

005 1509 3529 666 1993 1925010 682 13 378 889 812050 491 1128 329 654 651075 463 1037 293 553 586100 509 1125 262 536 608

minus002

0

002

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(a)

minus002

0

002

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

micro = 005micro = 075

wo

(b)

Figure 14 Continued

Shock and Vibration 13

maximum is 4622 which is usually mainly due tothe dierence in the frequency relationship betweenthe structure and the seismic waves

Table 3 shows the statistical results of the relative dis-placement between the TMD and main structure underdierent time history analysis conditions It can be seen thatthe relative displacement between the TMD and the mainstructure is inversely proportional to the mass ratio of theTMD that is when a large mass ratio TMD is used thedisplacement response of the structure is eectively con-trolled and the displacement stroke of the TMD is clearlyreduced which reduces the requirements for the elasticcomponents and the damping components used to constructthe TMD

e displacement time histories of the structures withperiods of 10 and 20 s and the relative displacement timehistories between the TMD and the structures are shown inFigures 14 and 15 respectively e schemes of mass ratiosof 005 and 075 are compared It can be clearly seen that theeect of a TMD in controlling the structural response andthe TMD displacement stroke is more obvious for the TMDwith a mass ratio of 075 than for the one with a mass ratio of005

In summary the large mass ratio TMD has a moresignicant eect in seismic control of the main structurethan the small mass ratio TMD

6 Conclusions

In order to control the dynamic response and improve theaseismic performance of a structure a large mass ratio TMDdamping system is formed by using the equipment in thebuilding structure or relying on new structural forms e

existing optimal parameter tting formula is not applicableto large mass ratio TMDs so it is revised by numericaloptimisation and curve tting and the dynamic time historyanalysis method is used to study the eect of vibrationdamping control of large mass ratio TMDs e followingconclusions are obtained

Compared with the traditional small mass ratio (lt005)TMD the large mass ratio (gt05) TMD has obvious ad-vantages in controlling the displacement response of themain structure e control eect is about 15sim325 timeshigher the damping eect of the structural displacementpeak can reach about 30 and the damping ratio of the rootmean square displacement can reach about 436 At thesame time the relative stroke between the TMD and themain structure can be reduced with up to 30sim65 which ishighly benecial to the practical engineering design of TMDstructures

When the mass ratio of a TMD is relatively large (gt02)the results calculated by the existing tting formula diersignicantly from the actual optimal value and the calcu-lated values of the revised formula proposed in this paper areshown to be in good agreement with the actual optimalvalue In general the revised formula can be applied to bothtraditional small mass ratio and large mass ratio (le1) TMDsWhen the mass ratio is greater than 1 the optimal pa-rameters of TMD can also be obtained by the methodpresented in this paper

When the mass ratio is greater than 02 the optimalparameters of the stationary white noise random load andthat of the ltered white noise random load are more similarbut the dierence between the two is gradually increasedwith the increase of the TMD mass ratio For the optimalparameters of large mass ratio TMDs (gt02) the error is lessthan 005 when the ratio of the predominant frequency of the

0 5 10 15 20 25 30

minus002minus001

0001002

t (s)

0 5 10 15 20 25 30t (s)

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(c)

0 5 10 15 20 25 30t (s)

0 5 10 15 20 25 30t (s)

minus002

0

002

004

Disp

lace

men

t (m

)

minus01

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(d)

Figure 14 e damping eect of dierent mass ratio TMDs on the displacement-time history of the main structure (T10 s) and therelative displacement-time history between the TMD and the structure (wo denotes the displacement of a structure without a TMD) (a) ElCentro (b) Hachinohe (c) Northridge (d) Kobe

14 Shock and Vibration

base soil and the vibration frequency of the structure isgreater than 4 and the optimal parameters of the TMD canbe calculated by the tting formula proposed in this paperUnder other conditions it is suggested to use an optimi-sation method to determine the optimal value of TMDparameters

At present the actual engineering projects with largemass ratio TMD damping systems are less prominent buttheir aseismic advantages will bring a broad range of benetsfor research and practice

Data Availability

e data used to support the ndings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no concurrenicts of interestregarding the publication of this article

Acknowledgments

is research was supported by Grant no 51478023 from theNational Natural Science Foundation of China

References

[1] M Gutierrez Soto and H Adeli ldquoTuned mass dampersrdquoArchives of Computational Methods in Engineering vol 20no 4 pp 419ndash431 2013

ndash005

0

005

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

ndash01ndash005

0005

01

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(a)

ndash02

ndash01

0

01

02

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

ndash005

005

0

micro = 005micro = 075

wo

(b)

0 5 10 15 20 25 30

minus005

0

005

Disp

lace

men

t (m

)

t (s)

0 5 10 15 20 25 30t (s)

minus01

0

01

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(c)

minus002

0

002

004D

ispla

cem

ent (

m)

0 5 10 15 20 25 30t (s)

0

minus005

005

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30t (s)

micro = 005micro = 075

wo

(d)

Figure 15 e damping eect of dierent mass ratio TMDs on the displacement-time history of the main structure (T 20 s) and therelative displacement-time history between the TMD and the structure (wo denotes the displacement of a structure without a TMD) (a) ElCentro (b) Hachinohe (c) Northridge (d) Kobe

Shock and Vibration 15

[2] S Elias and VMatsagar ldquoResearch developments in vibrationcontrol of structures using passive tuned mass dampersrdquoAnnual Reviews in Control vol 44 pp 129ndash156 2017

[3] D Wang T K T Tse Y Zhou and Q Li ldquoStructural per-formance and cost analysis of wind-induced vibration controlschemes for a real super-tall buildingrdquo Structure and In-frastructure Engineering vol 11 no 8 pp 990ndash1011 2014

[4] N Longarini and M Zucca ldquoA chimneyrsquos seismic assessmentby a tuned mass damperrdquo Engineering Structures vol 79pp 290ndash296 2014

[5] L Tian and Y Zeng ldquoParametric study of tunedmass dampersfor long span transmission tower-line system under windloadsrdquo Shock and Vibration vol 2016 Article ID 496505611 pages 2016

[6] N Hoang Y Fujino and PWarnitchai ldquoOptimal tuned massdamper for seismic applications and practical design for-mulasrdquo Engineering Structures vol 30 no 3 pp 707ndash7152008

[7] J P Den Hartog Mechanical vibrations McGraw-Hill NewYork NY USA 1956

[8] G B Warburton ldquoOptimum absorber parameters for variouscombinations of response and excitation parametersrdquoEarthquake Engineering amp Structural Dynamics vol 10 no 3pp 381ndash401 1982

[9] H-C Tsai and G-C Lin ldquoExplicit formulae for optimumabsorber parameters for force-excited and viscously dampedsystemsrdquo Journal of Sound and Vibration vol 176 no 5pp 585ndash596 1994

[10] H-C Tsai and G-C Lin ldquoOptimum tuned-mass dampers forminimizing steady-state response of support-excited anddamped systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 22 no 11 pp 957ndash973 1993

[11] S V Bakre and R S Jangid ldquoOptimum parameters of tunedmass damper for damped main systemrdquo Structural Controland Health Monitoring vol 14 no 3 pp 448ndash470 2007

[12] C C Lin C M Hu J F Wang and R Y Hu ldquoVibrationcontrol effectiveness of passive tuned mass dampersrdquo Journalof the Chinese Institute of Engineers vol 17 pp 367ndash376 1994

[13] A Y T Leung and H Zhang ldquoParticle swarm optimization oftuned mass dampersrdquo Engineering Structures vol 31 no 3pp 715ndash728 2009

[14] M Q Feng and A Mita ldquoVibration control of tall buildingsusing mega SubConfigurationrdquo Journal of Engineering Me-chanics vol 121 no 10 pp 1082ndash1088 1995

[15] X X Li P Tan X J Li and A W Liu ldquoMechanism analysisand parameter optimisation of mega-sub-isolation systemrdquoShock and Vibration vol 2016 p 12 2016

[16] A Reggio and M D Angelis ldquoOptimal energy-based seismicdesign of non-conventional tuned mass damper (TMD)implemented via inter-story isolationrdquo Earthquake Engi-neering amp Structural Dynamics vol 44 no 10 pp 1623ndash16422015

[17] S J Wang B H Lee W C Chuang and K C ChangldquoOptimal dynamic characteristic control approach forbuilding mass damper designrdquo Earthquake Engineering ampStructural Dynamics vol 47 no 3 2017

[18] H Anajafi and R A Medina ldquoPartial mass isolation systemfor seismic vibration control of buildingsrdquo Structural Controlamp Health Monitoring vol 25 no 2 article e2088 2017

[19] K Yuan M S He and Y M Li ldquoShaking table tests forenergy-dissipation steel frame structures with infilled wallMTMDrdquo Journal of Vibration and Shock vol 33 no 11pp 200ndash207 2014

[20] R Ding M X Tao M Zhou and J G Nie ldquoSeismic behaviorof RC structures with absence of floor slab constraints andlarge mass turbine as a non-conventional TMD a case studyrdquoBulletin of Earthquake Engineering vol 13 no 11 pp 3401ndash3422 2015

[21] L Y Peng Y J Kang Z R Lai and Y K Deng ldquoOptimisationand damping performance of a coal-fired power plantbuilding equipped with multiple coal bucket dampersrdquo Ad-vances in Civil Engineering vol 2018 p 19 2018

[22] Z Shu S Li J Zhang and M He ldquoOptimum seismic designof a power plant building with pendulum tuned mass dampersystem by its heavy suspended bucketsrdquo Engineering Struc-tures vol 136 pp 114ndash132 2017

[23] K Dai B Li J Wang et al ldquoOptimal probability-based partialmass isolation of elevated coal scuttle in thermal power plantbuildingrdquo Structural Design of Tall and Special Buildingsvol 27 no 11 article e1477 2018

[24] M De Angelis S Perno and A Reggio ldquoDynamic responseand optimal design of structures with large mass ratio TMDrdquoEarthquake Engineering amp Structural Dynamics vol 41 no 1pp 41ndash60 2015

[25] K Kanai ldquoSemi-empirical formula for the seismic charac-teristics of the groundrdquo Bulletin of the Earthquake ResearchInstitute gte University of Tokyo vol 35 pp 309ndash325 1957

[26] Code for seismic design of buildings GB50011-2010 BeijingChina 2010

[27] B F Spencer R E Christenson and S J Dyke ldquoNextgeneration benchmark control problem for seismically excitedbuildingsrdquo in Proceedings of the 2nd World Conference onStructural Control pp 1351ndash1360 Kyoto Japan June 1998

16 Shock and Vibration

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On the whole the error of the revised tting formula issmall which meets actual design requirements of large massratio TMDs Comparisons between the design method inreference [16] and that in this paper are illustrated in Table 1It can be seen that the optimisation parameters in this studyare obviously larger than those in reference [16] is ismainly due to the inconsistency of the optimisation ob-jectives of the two methods

42 Filtered White Noise Random Load e ltered whitenoise random load is modelled as proposed by Kanai [25]

S(ω) ω4g + 4ζ2gω2

gω2

ω2g minusω2( )

2 + 4ζ2gω2gω2

S0 (16)

where ωg and ζg are the predominant frequency anddamping ratio of foundation soil respectively

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004ζ1 = 005ζ1 = 0075ζ1 = 010

Optimal solutionEquation (15)

0 02 04 06 08 101

02

03

04

05

06

07

08

09

1

μ

f opt

(a)

Optimal solutionEquation (15)

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004ζ1 = 005ζ1 = 0075ζ1 = 010

0 02 04 06 08 10

01

02

03

04

05

06

07

μ

ζ 2op

t

(b)

Figure 7e relationship between the revised tting formula value and the actual optimal value with the TMDmass ratio and the dampingratio of the main structure under stationary white noise excitations (a) Optimal frequency ratio (b) Optimal damping ratio

0 02 04 06 08 1ndash04

ndash02

0

02

04

06

08

1f op

t

ζ1 = 00ζ1 = 002ζ1 = 005

ζ1 = 0075ζ1 = 010

Equation (14)

μ

(a)

ζ1 = 00ζ1 = 002ζ1 = 005

ζ1 = 0075ζ1 = 010

Equation (14)

0 02 04 06 08 101

015

02

025

03

035

04

045

05

055

06

μ

ζ 2op

t

(b)

Figure 6 e curve of the calculated value of the existing tting formula and the actual optimal value varies with the mass ratio understationary white noise excitations (a) Contrast results of the frequency ratio (b) Contrast results of the damping ratio

Shock and Vibration 7

Substituting S(ω) into equation (5) the mean square ofthe structural displacement response is obtained as

σ21 int+infin

minusinfinh1(ω)∣∣∣∣

∣∣∣∣2ω4g + 4ζ2gω

2gω

2

ω2g minusω2( )

2 + 4ζ2gω2gω2

S0 dω (17)

Because solving equation (17) analytically is very tediousa numerical integration method can be used e results byBakre [11] show that the TMD optimal parameters arerelatively close when the structure is excited by white noiserandom load and ltered white noise random load but onlywhen a mass ratio below 01 is examined e analysis resultobtained by Hoang et al [6] shows that there is a certaindierence between the optimal parameters for the largermass ratio TMD

When solving the optimal parameters the eect of themain structure damping ratio should be taken into accountthe predominant frequency and damping ratio of the ground

soil that is ωg and ζg are also taken into considerationHowever it is not practical to construct a correspondingtting formula erefore the optimisation process of TMDoptimal parameters is described in detail and the corre-sponding tting formula is no longer determined e de-tailed optimisation currenow chart is shown in Figure 9 Firstly

Table 1 Optimal design of large mass ratio TMD according toreference [16] and this study

Structuralparameters Method fopt ζopt σ21(2πS0ω3

1)

μ 0431 ζ1 002Reference

[16] 06871 02729 23830

is study 05970 02939 22429

μ 05 ζ1 002Reference

[16] 06543 02900 23862

is study 05549 03135 22246

μ 05 ζ1 003Reference

[16] 06492 02861 22947

is study 05432 03144 21320

Main structure parametersvibration frequency and damping ratio

The initial iteration value of the optimal parametercalculated by fitting formula (15)

Objective function the root mean square of the displacement response of the main structure

MATLAB optimization toolboxfunction Fmincon

Mass ratio of TMD

Parameters of ground motion power spectrum modelpredominant frequency and damping ratio

Optimal parameters and the correspondingobjective function values

Figure 9 e numerical optimisation process of TMD parametersunder ltered white noise excitations

0 02 04 06μ

08 1ndash04

ndash02

0

02

04

06

08

1

12Er

ror (

)

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004

ζ1 = 005ζ1 = 075ζ1 = 010

(a)

μ

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004

ζ1 = 005ζ1 = 075ζ1 = 010

0 02 04 06 08 1ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

Erro

r (

)(b)

Figure 8 e error rate of the revised tting formula value with the TMD mass ratio and the damping ratio of the main structure understationary white noise excitations (a) Error rate of the frequency ratio (b) Error rate of the damping ratio

8 Shock and Vibration

the dynamic characteristics of the main structure includingnatural frequency and damping ratio are determined enthe mass ratio of TMD and site condition parameters in-cluding the predominant frequency and damping ratio areselected After that the objective of optimisation is de-termined that is the mean square of the structural dis-placement Finally taking the calculated value of equation (15)as the initial value the optimal parameters can be obtained byoptimum analysis using MATLAB optimisation toolbox

Taking class II sites [26] as an example the parameters ofthe Kanai-Tajimi model ωg 15708 and ζg 072 aredetermined S0 is always taken as a constant and it has noeect on TMD parameter optimisation At the same time

λ ωgω1 is dened as the ratio between the predominantfrequency of ltered white noise and the natural frequency ofthe structure In order to study the TMD optimal parametersand the relative relationship between the excitation of sta-tionary white noise random load and ltered white noiserandom load the following denitions are given

ηf fopt_Num

fopt_Form

ηζ2 ζ2opt_Numζ2opt_Form

(18)

0 02 04 06 08 102

03

04

05

06

07

08

09

1

μ

f opt

λ = 1λ = 2λ = 3λ = 4

λ = 5λ = 75Equation (15)

(a)

μ0 02 04 06 08 1

098

1

102

104

106

108

11

112

114

λ = 1λ = 2λ = 3

λ = 4λ = 5λ = 75

η f(b)

λ = 1λ = 2λ = 3λ = 4

λ = 5λ = 75Equation (15)

0 02 04 06 08 101

015

02

025

03

035

04

045

05

μ

ζ 2op

t

(c)

λ = 1λ = 2λ = 3

λ = 4λ = 5λ = 75

μ0 02 04 06 08 1

084

086

088

09

092

094

096

098

1

102

104η ζ

2

(d)

Figure 10 e relation between optimal parameters with λ and μ and its deviation rate with equation (15) under ltered white noiseexcitations (a) Optimal frequency ratio (b) ηf (c) Optimal damping ratio (d) ηζ2opt

Shock and Vibration 9

Under stationary white noise excitations fopt_Form andζopt_Form are the TMD optimal frequency ratios and theoptimal damping ratios calculated by the tting equation(15) respectively Under ltered white noise excitationsfopt_Num and ζopt_Num are the TMD optimal frequency ratioand the optimal damping ratio respectively obtained by thenumerical optimisation method

When the main structure with a damping ratio of 005 isexcited by ltered white noise load the relationship betweenthe optimal parameters of the TMD and μ and λ is shown inFigures 10(a) and 10(c) In order to facilitate comparisonthe optimal parameter calculated by the tting formula inequation (15) is also plotted Figures 10(b) and 10(d) showrespectively the trends of ηf and ηζ2opt with μ and for dif-ferent values of λ

From the above results the following results areobtained

When the mass ratio μ is smaller than 02 the optimalparameters of the stationary white noise random load andthat of the ltered white noise random load are more similarbut the dierence between the two is increased graduallywith the increase of the mass ratio which can be clearly seenfrom the relationship curve of ηf to μ

e larger the λ the more ηf and ηζ2opt become closer to1 When λge 4 even if the TMD mass ratio reaches 1 theerror of the optimal parameters of the TMD under the

stationary white noise random load and the ltered whitenoise random load is not more than 5

In addition the damping ratio of the main structurehas a relatively small impact on the error therefore theincurrenuence of the damping ratio of the main structure onthe error of the tting formula is no longer discussed inthis paper In conclusion for the optimal design of a largemass ratio TMD when λge 4 the tting formula inequation (15) is also suitable for ltered white noise ex-citations while in the other cases it is suggested to de-termine the optimal TMD parameters using a numericaloptimisation method

5 Damping Effect Analysis of a Large MassRatio TMD

Four SDOF structures with 05 10 20 and 30 s periods (T)are selected and the damping ratio of the four structures is005 e TMD mass ratios are 005 025 050 075 and 1and the TMD parameters are calculated by the tting for-mula in equation (15) Two far-eld seismic waves (ElCentro Hachinohe) and two near-eld seismic waves(Northridge Kobe) are used for load input [27] Seismicwave information is provided in Table 2 e seismic waveacceleration-time history curve is shown in Figure 11 and

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

Acce

lera

tion

(ms

2 )Ac

cele

ratio

n (m

s2 )

El Centro

Hachinohe

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

(a)

Acce

lera

tion

(ms

2 )Ac

cele

ratio

n (m

s2 )

Northridge

Kobe

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

(b)

Figure 11 Time history curve of seismic records (a) Far-eld seismic waves (b) Near-eld seismic waves

Table 2 Seismic wave information used in the analyses

Type Name Earthquake Station Component Year

Far eld El Centro Imperial valley El Centro N-S 1940Hachinohe Tokachi-Oki Hachinohe city N-S 1968

Near eld Northridge Northridge SCH N-S 1994Kobe Hyogo-ken KJMA N-S 1995

10 Shock and Vibration

the seismic wave peak value is 1ms2 In order to investigatethe eect of the TMD mass ratio on the control of thestructural displacement response four sets of seismic timehistory analyses were carried out for structures with andwithout a TMD respectively

e reduction rate of peak displacement Re(Peak) andthe reduction rate of root mean square of structural dis-placement Re(RMS) are used as the evaluation index re-spectively e two formulas for calculating the dampingrate are shown as follows

Re(Peak) 1minusMax xTMD( )Max(x)

Re(RMS) 1minusRMS xTMD( )RMS(x)

(19)

where x and xTMD are the time history record of the dis-placement response of the structure without and with aTMD respectively e formula for calculating the rootmean square of the displacement is as follows

0 02 04 06 08 1micro

0

10

20

30

40

50Re

(Pea

k) (

)

EI CentroHachinoheKobe

NorthridgeMean

(a)

0 02 04 06 08 1micro

0

10

20

30

40

50

Re(P

eak)

()

EI CentroHachinoheKobe

NorthridgeMean

(b)

0 02 04 06 08 1micro

0

10

20

30

40

50

Re(P

eak)

()

EI CentroHachinoheKobe

NorthridgeMean

(c)

0 02 04 06 08 1micro

0

10

20

30

40

50Re

(Pea

k) (

)

EI CentroHachinoheKobe

NorthridgeMean

(d)

Figure 12 e damping eect of dierent mass ratio (TMD) on the peak value of structural displacement response (a) T 05 s (b)T10 s (c) T 20 s (d) T 30 s

Shock and Vibration 11

RMS(x) sqrt1NsumN

i1x2i (20)

where xi is the structural displacement response corre-sponding to the ith time andN is the total number of pointscollected

Re(Peak) and Re(RMS) of the displacement responsewith dierent mass ratio TMDs are shown in Figures 12 and13 e following conclusions can be obtained

(1) TMD can eectively control the displacement re-sponse of the structure and the large mass ratio

(gt025) TMD is more eective than the conventionalsmall mass ratio (lt005) TMD But it can also befound that when the mass ratio of the TMD is greaterthan 05 the gain eect will diminish with increasingmass ratio

(2) e TMD with the same mass ratio shows certaindiscreteness for the structures with dierent naturalvibration periods and dierent seismic waves Forexample as shown in Figure 12(b) when thestructurersquos period is 10 s the damping rate of fourseismic waves is distinct When the mass ratio is 05the minimum damping rate is 1047 and the

0 02 04 06 08 10

10

20

30

40

50

60

micro

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(a)

0 02 04 06 08 1micro

0

10

20

30

40

50

60

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(b)

0 02 04 06 08 10

10

20

30

40

50

60

micro

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(c)

0 02 04 06 08 1micro

0

10

20

30

40

50

60Re

(RM

S)

EI CentroHachinoheKobe

NorthridgeMean

(d)

Figure 13e damping eect of dierent mass ratios (TMD) on the mean square root of structural displacement response (a) T 05 s (b)T10 s (c) T 20 s (d) T 30 s

12 Shock and Vibration

Table 3 Maximum peak value of relative displacement between a TMD with optimal parameters and the main structure (cm)

T (s) μ El Centro Hachinohe Northridge Kobe Mean

05

005 436 296 381 312 356010 268 214 298 22 25050 214 254 335 191 249075 183 26 275 279 249100 224 274 224 337 265

10

005 818 883 1039 698 859010 353 483 512 598 487050 341 535 376 588 46075 391 59 355 546 47100 356 494 358 502 428

20

005 1477 209 866 1834 1567010 794 1197 554 1042 897050 53 863 417 858 667075 394 795 345 705 56100 389 846 332 629 549

30

005 1509 3529 666 1993 1925010 682 13 378 889 812050 491 1128 329 654 651075 463 1037 293 553 586100 509 1125 262 536 608

minus002

0

002

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(a)

minus002

0

002

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

micro = 005micro = 075

wo

(b)

Figure 14 Continued

Shock and Vibration 13

maximum is 4622 which is usually mainly due tothe dierence in the frequency relationship betweenthe structure and the seismic waves

Table 3 shows the statistical results of the relative dis-placement between the TMD and main structure underdierent time history analysis conditions It can be seen thatthe relative displacement between the TMD and the mainstructure is inversely proportional to the mass ratio of theTMD that is when a large mass ratio TMD is used thedisplacement response of the structure is eectively con-trolled and the displacement stroke of the TMD is clearlyreduced which reduces the requirements for the elasticcomponents and the damping components used to constructthe TMD

e displacement time histories of the structures withperiods of 10 and 20 s and the relative displacement timehistories between the TMD and the structures are shown inFigures 14 and 15 respectively e schemes of mass ratiosof 005 and 075 are compared It can be clearly seen that theeect of a TMD in controlling the structural response andthe TMD displacement stroke is more obvious for the TMDwith a mass ratio of 075 than for the one with a mass ratio of005

In summary the large mass ratio TMD has a moresignicant eect in seismic control of the main structurethan the small mass ratio TMD

6 Conclusions

In order to control the dynamic response and improve theaseismic performance of a structure a large mass ratio TMDdamping system is formed by using the equipment in thebuilding structure or relying on new structural forms e

existing optimal parameter tting formula is not applicableto large mass ratio TMDs so it is revised by numericaloptimisation and curve tting and the dynamic time historyanalysis method is used to study the eect of vibrationdamping control of large mass ratio TMDs e followingconclusions are obtained

Compared with the traditional small mass ratio (lt005)TMD the large mass ratio (gt05) TMD has obvious ad-vantages in controlling the displacement response of themain structure e control eect is about 15sim325 timeshigher the damping eect of the structural displacementpeak can reach about 30 and the damping ratio of the rootmean square displacement can reach about 436 At thesame time the relative stroke between the TMD and themain structure can be reduced with up to 30sim65 which ishighly benecial to the practical engineering design of TMDstructures

When the mass ratio of a TMD is relatively large (gt02)the results calculated by the existing tting formula diersignicantly from the actual optimal value and the calcu-lated values of the revised formula proposed in this paper areshown to be in good agreement with the actual optimalvalue In general the revised formula can be applied to bothtraditional small mass ratio and large mass ratio (le1) TMDsWhen the mass ratio is greater than 1 the optimal pa-rameters of TMD can also be obtained by the methodpresented in this paper

When the mass ratio is greater than 02 the optimalparameters of the stationary white noise random load andthat of the ltered white noise random load are more similarbut the dierence between the two is gradually increasedwith the increase of the TMD mass ratio For the optimalparameters of large mass ratio TMDs (gt02) the error is lessthan 005 when the ratio of the predominant frequency of the

0 5 10 15 20 25 30

minus002minus001

0001002

t (s)

0 5 10 15 20 25 30t (s)

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(c)

0 5 10 15 20 25 30t (s)

0 5 10 15 20 25 30t (s)

minus002

0

002

004

Disp

lace

men

t (m

)

minus01

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(d)

Figure 14 e damping eect of dierent mass ratio TMDs on the displacement-time history of the main structure (T10 s) and therelative displacement-time history between the TMD and the structure (wo denotes the displacement of a structure without a TMD) (a) ElCentro (b) Hachinohe (c) Northridge (d) Kobe

14 Shock and Vibration

base soil and the vibration frequency of the structure isgreater than 4 and the optimal parameters of the TMD canbe calculated by the tting formula proposed in this paperUnder other conditions it is suggested to use an optimi-sation method to determine the optimal value of TMDparameters

At present the actual engineering projects with largemass ratio TMD damping systems are less prominent buttheir aseismic advantages will bring a broad range of benetsfor research and practice

Data Availability

e data used to support the ndings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no concurrenicts of interestregarding the publication of this article

Acknowledgments

is research was supported by Grant no 51478023 from theNational Natural Science Foundation of China

References

[1] M Gutierrez Soto and H Adeli ldquoTuned mass dampersrdquoArchives of Computational Methods in Engineering vol 20no 4 pp 419ndash431 2013

ndash005

0

005

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

ndash01ndash005

0005

01

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(a)

ndash02

ndash01

0

01

02

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

ndash005

005

0

micro = 005micro = 075

wo

(b)

0 5 10 15 20 25 30

minus005

0

005

Disp

lace

men

t (m

)

t (s)

0 5 10 15 20 25 30t (s)

minus01

0

01

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(c)

minus002

0

002

004D

ispla

cem

ent (

m)

0 5 10 15 20 25 30t (s)

0

minus005

005

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30t (s)

micro = 005micro = 075

wo

(d)

Figure 15 e damping eect of dierent mass ratio TMDs on the displacement-time history of the main structure (T 20 s) and therelative displacement-time history between the TMD and the structure (wo denotes the displacement of a structure without a TMD) (a) ElCentro (b) Hachinohe (c) Northridge (d) Kobe

Shock and Vibration 15

[2] S Elias and VMatsagar ldquoResearch developments in vibrationcontrol of structures using passive tuned mass dampersrdquoAnnual Reviews in Control vol 44 pp 129ndash156 2017

[3] D Wang T K T Tse Y Zhou and Q Li ldquoStructural per-formance and cost analysis of wind-induced vibration controlschemes for a real super-tall buildingrdquo Structure and In-frastructure Engineering vol 11 no 8 pp 990ndash1011 2014

[4] N Longarini and M Zucca ldquoA chimneyrsquos seismic assessmentby a tuned mass damperrdquo Engineering Structures vol 79pp 290ndash296 2014

[5] L Tian and Y Zeng ldquoParametric study of tunedmass dampersfor long span transmission tower-line system under windloadsrdquo Shock and Vibration vol 2016 Article ID 496505611 pages 2016

[6] N Hoang Y Fujino and PWarnitchai ldquoOptimal tuned massdamper for seismic applications and practical design for-mulasrdquo Engineering Structures vol 30 no 3 pp 707ndash7152008

[7] J P Den Hartog Mechanical vibrations McGraw-Hill NewYork NY USA 1956

[8] G B Warburton ldquoOptimum absorber parameters for variouscombinations of response and excitation parametersrdquoEarthquake Engineering amp Structural Dynamics vol 10 no 3pp 381ndash401 1982

[9] H-C Tsai and G-C Lin ldquoExplicit formulae for optimumabsorber parameters for force-excited and viscously dampedsystemsrdquo Journal of Sound and Vibration vol 176 no 5pp 585ndash596 1994

[10] H-C Tsai and G-C Lin ldquoOptimum tuned-mass dampers forminimizing steady-state response of support-excited anddamped systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 22 no 11 pp 957ndash973 1993

[11] S V Bakre and R S Jangid ldquoOptimum parameters of tunedmass damper for damped main systemrdquo Structural Controland Health Monitoring vol 14 no 3 pp 448ndash470 2007

[12] C C Lin C M Hu J F Wang and R Y Hu ldquoVibrationcontrol effectiveness of passive tuned mass dampersrdquo Journalof the Chinese Institute of Engineers vol 17 pp 367ndash376 1994

[13] A Y T Leung and H Zhang ldquoParticle swarm optimization oftuned mass dampersrdquo Engineering Structures vol 31 no 3pp 715ndash728 2009

[14] M Q Feng and A Mita ldquoVibration control of tall buildingsusing mega SubConfigurationrdquo Journal of Engineering Me-chanics vol 121 no 10 pp 1082ndash1088 1995

[15] X X Li P Tan X J Li and A W Liu ldquoMechanism analysisand parameter optimisation of mega-sub-isolation systemrdquoShock and Vibration vol 2016 p 12 2016

[16] A Reggio and M D Angelis ldquoOptimal energy-based seismicdesign of non-conventional tuned mass damper (TMD)implemented via inter-story isolationrdquo Earthquake Engi-neering amp Structural Dynamics vol 44 no 10 pp 1623ndash16422015

[17] S J Wang B H Lee W C Chuang and K C ChangldquoOptimal dynamic characteristic control approach forbuilding mass damper designrdquo Earthquake Engineering ampStructural Dynamics vol 47 no 3 2017

[18] H Anajafi and R A Medina ldquoPartial mass isolation systemfor seismic vibration control of buildingsrdquo Structural Controlamp Health Monitoring vol 25 no 2 article e2088 2017

[19] K Yuan M S He and Y M Li ldquoShaking table tests forenergy-dissipation steel frame structures with infilled wallMTMDrdquo Journal of Vibration and Shock vol 33 no 11pp 200ndash207 2014

[20] R Ding M X Tao M Zhou and J G Nie ldquoSeismic behaviorof RC structures with absence of floor slab constraints andlarge mass turbine as a non-conventional TMD a case studyrdquoBulletin of Earthquake Engineering vol 13 no 11 pp 3401ndash3422 2015

[21] L Y Peng Y J Kang Z R Lai and Y K Deng ldquoOptimisationand damping performance of a coal-fired power plantbuilding equipped with multiple coal bucket dampersrdquo Ad-vances in Civil Engineering vol 2018 p 19 2018

[22] Z Shu S Li J Zhang and M He ldquoOptimum seismic designof a power plant building with pendulum tuned mass dampersystem by its heavy suspended bucketsrdquo Engineering Struc-tures vol 136 pp 114ndash132 2017

[23] K Dai B Li J Wang et al ldquoOptimal probability-based partialmass isolation of elevated coal scuttle in thermal power plantbuildingrdquo Structural Design of Tall and Special Buildingsvol 27 no 11 article e1477 2018

[24] M De Angelis S Perno and A Reggio ldquoDynamic responseand optimal design of structures with large mass ratio TMDrdquoEarthquake Engineering amp Structural Dynamics vol 41 no 1pp 41ndash60 2015

[25] K Kanai ldquoSemi-empirical formula for the seismic charac-teristics of the groundrdquo Bulletin of the Earthquake ResearchInstitute gte University of Tokyo vol 35 pp 309ndash325 1957

[26] Code for seismic design of buildings GB50011-2010 BeijingChina 2010

[27] B F Spencer R E Christenson and S J Dyke ldquoNextgeneration benchmark control problem for seismically excitedbuildingsrdquo in Proceedings of the 2nd World Conference onStructural Control pp 1351ndash1360 Kyoto Japan June 1998

16 Shock and Vibration

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Substituting S(ω) into equation (5) the mean square ofthe structural displacement response is obtained as

σ21 int+infin

minusinfinh1(ω)∣∣∣∣

∣∣∣∣2ω4g + 4ζ2gω

2gω

2

ω2g minusω2( )

2 + 4ζ2gω2gω2

S0 dω (17)

Because solving equation (17) analytically is very tediousa numerical integration method can be used e results byBakre [11] show that the TMD optimal parameters arerelatively close when the structure is excited by white noiserandom load and ltered white noise random load but onlywhen a mass ratio below 01 is examined e analysis resultobtained by Hoang et al [6] shows that there is a certaindierence between the optimal parameters for the largermass ratio TMD

When solving the optimal parameters the eect of themain structure damping ratio should be taken into accountthe predominant frequency and damping ratio of the ground

soil that is ωg and ζg are also taken into considerationHowever it is not practical to construct a correspondingtting formula erefore the optimisation process of TMDoptimal parameters is described in detail and the corre-sponding tting formula is no longer determined e de-tailed optimisation currenow chart is shown in Figure 9 Firstly

Table 1 Optimal design of large mass ratio TMD according toreference [16] and this study

Structuralparameters Method fopt ζopt σ21(2πS0ω3

1)

μ 0431 ζ1 002Reference

[16] 06871 02729 23830

is study 05970 02939 22429

μ 05 ζ1 002Reference

[16] 06543 02900 23862

is study 05549 03135 22246

μ 05 ζ1 003Reference

[16] 06492 02861 22947

is study 05432 03144 21320

Main structure parametersvibration frequency and damping ratio

The initial iteration value of the optimal parametercalculated by fitting formula (15)

Objective function the root mean square of the displacement response of the main structure

MATLAB optimization toolboxfunction Fmincon

Mass ratio of TMD

Parameters of ground motion power spectrum modelpredominant frequency and damping ratio

Optimal parameters and the correspondingobjective function values

Figure 9 e numerical optimisation process of TMD parametersunder ltered white noise excitations

0 02 04 06μ

08 1ndash04

ndash02

0

02

04

06

08

1

12Er

ror (

)

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004

ζ1 = 005ζ1 = 075ζ1 = 010

(a)

μ

ζ1 = 0ζ1 = 002ζ1 = 003ζ1 = 004

ζ1 = 005ζ1 = 075ζ1 = 010

0 02 04 06 08 1ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

Erro

r (

)(b)

Figure 8 e error rate of the revised tting formula value with the TMD mass ratio and the damping ratio of the main structure understationary white noise excitations (a) Error rate of the frequency ratio (b) Error rate of the damping ratio

8 Shock and Vibration

the dynamic characteristics of the main structure includingnatural frequency and damping ratio are determined enthe mass ratio of TMD and site condition parameters in-cluding the predominant frequency and damping ratio areselected After that the objective of optimisation is de-termined that is the mean square of the structural dis-placement Finally taking the calculated value of equation (15)as the initial value the optimal parameters can be obtained byoptimum analysis using MATLAB optimisation toolbox

Taking class II sites [26] as an example the parameters ofthe Kanai-Tajimi model ωg 15708 and ζg 072 aredetermined S0 is always taken as a constant and it has noeect on TMD parameter optimisation At the same time

λ ωgω1 is dened as the ratio between the predominantfrequency of ltered white noise and the natural frequency ofthe structure In order to study the TMD optimal parametersand the relative relationship between the excitation of sta-tionary white noise random load and ltered white noiserandom load the following denitions are given

ηf fopt_Num

fopt_Form

ηζ2 ζ2opt_Numζ2opt_Form

(18)

0 02 04 06 08 102

03

04

05

06

07

08

09

1

μ

f opt

λ = 1λ = 2λ = 3λ = 4

λ = 5λ = 75Equation (15)

(a)

μ0 02 04 06 08 1

098

1

102

104

106

108

11

112

114

λ = 1λ = 2λ = 3

λ = 4λ = 5λ = 75

η f(b)

λ = 1λ = 2λ = 3λ = 4

λ = 5λ = 75Equation (15)

0 02 04 06 08 101

015

02

025

03

035

04

045

05

μ

ζ 2op

t

(c)

λ = 1λ = 2λ = 3

λ = 4λ = 5λ = 75

μ0 02 04 06 08 1

084

086

088

09

092

094

096

098

1

102

104η ζ

2

(d)

Figure 10 e relation between optimal parameters with λ and μ and its deviation rate with equation (15) under ltered white noiseexcitations (a) Optimal frequency ratio (b) ηf (c) Optimal damping ratio (d) ηζ2opt

Shock and Vibration 9

Under stationary white noise excitations fopt_Form andζopt_Form are the TMD optimal frequency ratios and theoptimal damping ratios calculated by the tting equation(15) respectively Under ltered white noise excitationsfopt_Num and ζopt_Num are the TMD optimal frequency ratioand the optimal damping ratio respectively obtained by thenumerical optimisation method

When the main structure with a damping ratio of 005 isexcited by ltered white noise load the relationship betweenthe optimal parameters of the TMD and μ and λ is shown inFigures 10(a) and 10(c) In order to facilitate comparisonthe optimal parameter calculated by the tting formula inequation (15) is also plotted Figures 10(b) and 10(d) showrespectively the trends of ηf and ηζ2opt with μ and for dif-ferent values of λ

From the above results the following results areobtained

When the mass ratio μ is smaller than 02 the optimalparameters of the stationary white noise random load andthat of the ltered white noise random load are more similarbut the dierence between the two is increased graduallywith the increase of the mass ratio which can be clearly seenfrom the relationship curve of ηf to μ

e larger the λ the more ηf and ηζ2opt become closer to1 When λge 4 even if the TMD mass ratio reaches 1 theerror of the optimal parameters of the TMD under the

stationary white noise random load and the ltered whitenoise random load is not more than 5

In addition the damping ratio of the main structurehas a relatively small impact on the error therefore theincurrenuence of the damping ratio of the main structure onthe error of the tting formula is no longer discussed inthis paper In conclusion for the optimal design of a largemass ratio TMD when λge 4 the tting formula inequation (15) is also suitable for ltered white noise ex-citations while in the other cases it is suggested to de-termine the optimal TMD parameters using a numericaloptimisation method

5 Damping Effect Analysis of a Large MassRatio TMD

Four SDOF structures with 05 10 20 and 30 s periods (T)are selected and the damping ratio of the four structures is005 e TMD mass ratios are 005 025 050 075 and 1and the TMD parameters are calculated by the tting for-mula in equation (15) Two far-eld seismic waves (ElCentro Hachinohe) and two near-eld seismic waves(Northridge Kobe) are used for load input [27] Seismicwave information is provided in Table 2 e seismic waveacceleration-time history curve is shown in Figure 11 and

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

Acce

lera

tion

(ms

2 )Ac

cele

ratio

n (m

s2 )

El Centro

Hachinohe

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

(a)

Acce

lera

tion

(ms

2 )Ac

cele

ratio

n (m

s2 )

Northridge

Kobe

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

(b)

Figure 11 Time history curve of seismic records (a) Far-eld seismic waves (b) Near-eld seismic waves

Table 2 Seismic wave information used in the analyses

Type Name Earthquake Station Component Year

Far eld El Centro Imperial valley El Centro N-S 1940Hachinohe Tokachi-Oki Hachinohe city N-S 1968

Near eld Northridge Northridge SCH N-S 1994Kobe Hyogo-ken KJMA N-S 1995

10 Shock and Vibration

the seismic wave peak value is 1ms2 In order to investigatethe eect of the TMD mass ratio on the control of thestructural displacement response four sets of seismic timehistory analyses were carried out for structures with andwithout a TMD respectively

e reduction rate of peak displacement Re(Peak) andthe reduction rate of root mean square of structural dis-placement Re(RMS) are used as the evaluation index re-spectively e two formulas for calculating the dampingrate are shown as follows

Re(Peak) 1minusMax xTMD( )Max(x)

Re(RMS) 1minusRMS xTMD( )RMS(x)

(19)

where x and xTMD are the time history record of the dis-placement response of the structure without and with aTMD respectively e formula for calculating the rootmean square of the displacement is as follows

0 02 04 06 08 1micro

0

10

20

30

40

50Re

(Pea

k) (

)

EI CentroHachinoheKobe

NorthridgeMean

(a)

0 02 04 06 08 1micro

0

10

20

30

40

50

Re(P

eak)

()

EI CentroHachinoheKobe

NorthridgeMean

(b)

0 02 04 06 08 1micro

0

10

20

30

40

50

Re(P

eak)

()

EI CentroHachinoheKobe

NorthridgeMean

(c)

0 02 04 06 08 1micro

0

10

20

30

40

50Re

(Pea

k) (

)

EI CentroHachinoheKobe

NorthridgeMean

(d)

Figure 12 e damping eect of dierent mass ratio (TMD) on the peak value of structural displacement response (a) T 05 s (b)T10 s (c) T 20 s (d) T 30 s

Shock and Vibration 11

RMS(x) sqrt1NsumN

i1x2i (20)

where xi is the structural displacement response corre-sponding to the ith time andN is the total number of pointscollected

Re(Peak) and Re(RMS) of the displacement responsewith dierent mass ratio TMDs are shown in Figures 12 and13 e following conclusions can be obtained

(1) TMD can eectively control the displacement re-sponse of the structure and the large mass ratio

(gt025) TMD is more eective than the conventionalsmall mass ratio (lt005) TMD But it can also befound that when the mass ratio of the TMD is greaterthan 05 the gain eect will diminish with increasingmass ratio

(2) e TMD with the same mass ratio shows certaindiscreteness for the structures with dierent naturalvibration periods and dierent seismic waves Forexample as shown in Figure 12(b) when thestructurersquos period is 10 s the damping rate of fourseismic waves is distinct When the mass ratio is 05the minimum damping rate is 1047 and the

0 02 04 06 08 10

10

20

30

40

50

60

micro

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(a)

0 02 04 06 08 1micro

0

10

20

30

40

50

60

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(b)

0 02 04 06 08 10

10

20

30

40

50

60

micro

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(c)

0 02 04 06 08 1micro

0

10

20

30

40

50

60Re

(RM

S)

EI CentroHachinoheKobe

NorthridgeMean

(d)

Figure 13e damping eect of dierent mass ratios (TMD) on the mean square root of structural displacement response (a) T 05 s (b)T10 s (c) T 20 s (d) T 30 s

12 Shock and Vibration

Table 3 Maximum peak value of relative displacement between a TMD with optimal parameters and the main structure (cm)

T (s) μ El Centro Hachinohe Northridge Kobe Mean

05

005 436 296 381 312 356010 268 214 298 22 25050 214 254 335 191 249075 183 26 275 279 249100 224 274 224 337 265

10

005 818 883 1039 698 859010 353 483 512 598 487050 341 535 376 588 46075 391 59 355 546 47100 356 494 358 502 428

20

005 1477 209 866 1834 1567010 794 1197 554 1042 897050 53 863 417 858 667075 394 795 345 705 56100 389 846 332 629 549

30

005 1509 3529 666 1993 1925010 682 13 378 889 812050 491 1128 329 654 651075 463 1037 293 553 586100 509 1125 262 536 608

minus002

0

002

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(a)

minus002

0

002

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

micro = 005micro = 075

wo

(b)

Figure 14 Continued

Shock and Vibration 13

maximum is 4622 which is usually mainly due tothe dierence in the frequency relationship betweenthe structure and the seismic waves

Table 3 shows the statistical results of the relative dis-placement between the TMD and main structure underdierent time history analysis conditions It can be seen thatthe relative displacement between the TMD and the mainstructure is inversely proportional to the mass ratio of theTMD that is when a large mass ratio TMD is used thedisplacement response of the structure is eectively con-trolled and the displacement stroke of the TMD is clearlyreduced which reduces the requirements for the elasticcomponents and the damping components used to constructthe TMD

e displacement time histories of the structures withperiods of 10 and 20 s and the relative displacement timehistories between the TMD and the structures are shown inFigures 14 and 15 respectively e schemes of mass ratiosof 005 and 075 are compared It can be clearly seen that theeect of a TMD in controlling the structural response andthe TMD displacement stroke is more obvious for the TMDwith a mass ratio of 075 than for the one with a mass ratio of005

In summary the large mass ratio TMD has a moresignicant eect in seismic control of the main structurethan the small mass ratio TMD

6 Conclusions

In order to control the dynamic response and improve theaseismic performance of a structure a large mass ratio TMDdamping system is formed by using the equipment in thebuilding structure or relying on new structural forms e

existing optimal parameter tting formula is not applicableto large mass ratio TMDs so it is revised by numericaloptimisation and curve tting and the dynamic time historyanalysis method is used to study the eect of vibrationdamping control of large mass ratio TMDs e followingconclusions are obtained

Compared with the traditional small mass ratio (lt005)TMD the large mass ratio (gt05) TMD has obvious ad-vantages in controlling the displacement response of themain structure e control eect is about 15sim325 timeshigher the damping eect of the structural displacementpeak can reach about 30 and the damping ratio of the rootmean square displacement can reach about 436 At thesame time the relative stroke between the TMD and themain structure can be reduced with up to 30sim65 which ishighly benecial to the practical engineering design of TMDstructures

When the mass ratio of a TMD is relatively large (gt02)the results calculated by the existing tting formula diersignicantly from the actual optimal value and the calcu-lated values of the revised formula proposed in this paper areshown to be in good agreement with the actual optimalvalue In general the revised formula can be applied to bothtraditional small mass ratio and large mass ratio (le1) TMDsWhen the mass ratio is greater than 1 the optimal pa-rameters of TMD can also be obtained by the methodpresented in this paper

When the mass ratio is greater than 02 the optimalparameters of the stationary white noise random load andthat of the ltered white noise random load are more similarbut the dierence between the two is gradually increasedwith the increase of the TMD mass ratio For the optimalparameters of large mass ratio TMDs (gt02) the error is lessthan 005 when the ratio of the predominant frequency of the

0 5 10 15 20 25 30

minus002minus001

0001002

t (s)

0 5 10 15 20 25 30t (s)

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(c)

0 5 10 15 20 25 30t (s)

0 5 10 15 20 25 30t (s)

minus002

0

002

004

Disp

lace

men

t (m

)

minus01

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(d)

Figure 14 e damping eect of dierent mass ratio TMDs on the displacement-time history of the main structure (T10 s) and therelative displacement-time history between the TMD and the structure (wo denotes the displacement of a structure without a TMD) (a) ElCentro (b) Hachinohe (c) Northridge (d) Kobe

14 Shock and Vibration

base soil and the vibration frequency of the structure isgreater than 4 and the optimal parameters of the TMD canbe calculated by the tting formula proposed in this paperUnder other conditions it is suggested to use an optimi-sation method to determine the optimal value of TMDparameters

At present the actual engineering projects with largemass ratio TMD damping systems are less prominent buttheir aseismic advantages will bring a broad range of benetsfor research and practice

Data Availability

e data used to support the ndings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no concurrenicts of interestregarding the publication of this article

Acknowledgments

is research was supported by Grant no 51478023 from theNational Natural Science Foundation of China

References

[1] M Gutierrez Soto and H Adeli ldquoTuned mass dampersrdquoArchives of Computational Methods in Engineering vol 20no 4 pp 419ndash431 2013

ndash005

0

005

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

ndash01ndash005

0005

01

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(a)

ndash02

ndash01

0

01

02

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

ndash005

005

0

micro = 005micro = 075

wo

(b)

0 5 10 15 20 25 30

minus005

0

005

Disp

lace

men

t (m

)

t (s)

0 5 10 15 20 25 30t (s)

minus01

0

01

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(c)

minus002

0

002

004D

ispla

cem

ent (

m)

0 5 10 15 20 25 30t (s)

0

minus005

005

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30t (s)

micro = 005micro = 075

wo

(d)

Figure 15 e damping eect of dierent mass ratio TMDs on the displacement-time history of the main structure (T 20 s) and therelative displacement-time history between the TMD and the structure (wo denotes the displacement of a structure without a TMD) (a) ElCentro (b) Hachinohe (c) Northridge (d) Kobe

Shock and Vibration 15

[2] S Elias and VMatsagar ldquoResearch developments in vibrationcontrol of structures using passive tuned mass dampersrdquoAnnual Reviews in Control vol 44 pp 129ndash156 2017

[3] D Wang T K T Tse Y Zhou and Q Li ldquoStructural per-formance and cost analysis of wind-induced vibration controlschemes for a real super-tall buildingrdquo Structure and In-frastructure Engineering vol 11 no 8 pp 990ndash1011 2014

[4] N Longarini and M Zucca ldquoA chimneyrsquos seismic assessmentby a tuned mass damperrdquo Engineering Structures vol 79pp 290ndash296 2014

[5] L Tian and Y Zeng ldquoParametric study of tunedmass dampersfor long span transmission tower-line system under windloadsrdquo Shock and Vibration vol 2016 Article ID 496505611 pages 2016

[6] N Hoang Y Fujino and PWarnitchai ldquoOptimal tuned massdamper for seismic applications and practical design for-mulasrdquo Engineering Structures vol 30 no 3 pp 707ndash7152008

[7] J P Den Hartog Mechanical vibrations McGraw-Hill NewYork NY USA 1956

[8] G B Warburton ldquoOptimum absorber parameters for variouscombinations of response and excitation parametersrdquoEarthquake Engineering amp Structural Dynamics vol 10 no 3pp 381ndash401 1982

[9] H-C Tsai and G-C Lin ldquoExplicit formulae for optimumabsorber parameters for force-excited and viscously dampedsystemsrdquo Journal of Sound and Vibration vol 176 no 5pp 585ndash596 1994

[10] H-C Tsai and G-C Lin ldquoOptimum tuned-mass dampers forminimizing steady-state response of support-excited anddamped systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 22 no 11 pp 957ndash973 1993

[11] S V Bakre and R S Jangid ldquoOptimum parameters of tunedmass damper for damped main systemrdquo Structural Controland Health Monitoring vol 14 no 3 pp 448ndash470 2007

[12] C C Lin C M Hu J F Wang and R Y Hu ldquoVibrationcontrol effectiveness of passive tuned mass dampersrdquo Journalof the Chinese Institute of Engineers vol 17 pp 367ndash376 1994

[13] A Y T Leung and H Zhang ldquoParticle swarm optimization oftuned mass dampersrdquo Engineering Structures vol 31 no 3pp 715ndash728 2009

[14] M Q Feng and A Mita ldquoVibration control of tall buildingsusing mega SubConfigurationrdquo Journal of Engineering Me-chanics vol 121 no 10 pp 1082ndash1088 1995

[15] X X Li P Tan X J Li and A W Liu ldquoMechanism analysisand parameter optimisation of mega-sub-isolation systemrdquoShock and Vibration vol 2016 p 12 2016

[16] A Reggio and M D Angelis ldquoOptimal energy-based seismicdesign of non-conventional tuned mass damper (TMD)implemented via inter-story isolationrdquo Earthquake Engi-neering amp Structural Dynamics vol 44 no 10 pp 1623ndash16422015

[17] S J Wang B H Lee W C Chuang and K C ChangldquoOptimal dynamic characteristic control approach forbuilding mass damper designrdquo Earthquake Engineering ampStructural Dynamics vol 47 no 3 2017

[18] H Anajafi and R A Medina ldquoPartial mass isolation systemfor seismic vibration control of buildingsrdquo Structural Controlamp Health Monitoring vol 25 no 2 article e2088 2017

[19] K Yuan M S He and Y M Li ldquoShaking table tests forenergy-dissipation steel frame structures with infilled wallMTMDrdquo Journal of Vibration and Shock vol 33 no 11pp 200ndash207 2014

[20] R Ding M X Tao M Zhou and J G Nie ldquoSeismic behaviorof RC structures with absence of floor slab constraints andlarge mass turbine as a non-conventional TMD a case studyrdquoBulletin of Earthquake Engineering vol 13 no 11 pp 3401ndash3422 2015

[21] L Y Peng Y J Kang Z R Lai and Y K Deng ldquoOptimisationand damping performance of a coal-fired power plantbuilding equipped with multiple coal bucket dampersrdquo Ad-vances in Civil Engineering vol 2018 p 19 2018

[22] Z Shu S Li J Zhang and M He ldquoOptimum seismic designof a power plant building with pendulum tuned mass dampersystem by its heavy suspended bucketsrdquo Engineering Struc-tures vol 136 pp 114ndash132 2017

[23] K Dai B Li J Wang et al ldquoOptimal probability-based partialmass isolation of elevated coal scuttle in thermal power plantbuildingrdquo Structural Design of Tall and Special Buildingsvol 27 no 11 article e1477 2018

[24] M De Angelis S Perno and A Reggio ldquoDynamic responseand optimal design of structures with large mass ratio TMDrdquoEarthquake Engineering amp Structural Dynamics vol 41 no 1pp 41ndash60 2015

[25] K Kanai ldquoSemi-empirical formula for the seismic charac-teristics of the groundrdquo Bulletin of the Earthquake ResearchInstitute gte University of Tokyo vol 35 pp 309ndash325 1957

[26] Code for seismic design of buildings GB50011-2010 BeijingChina 2010

[27] B F Spencer R E Christenson and S J Dyke ldquoNextgeneration benchmark control problem for seismically excitedbuildingsrdquo in Proceedings of the 2nd World Conference onStructural Control pp 1351ndash1360 Kyoto Japan June 1998

16 Shock and Vibration

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the dynamic characteristics of the main structure includingnatural frequency and damping ratio are determined enthe mass ratio of TMD and site condition parameters in-cluding the predominant frequency and damping ratio areselected After that the objective of optimisation is de-termined that is the mean square of the structural dis-placement Finally taking the calculated value of equation (15)as the initial value the optimal parameters can be obtained byoptimum analysis using MATLAB optimisation toolbox

Taking class II sites [26] as an example the parameters ofthe Kanai-Tajimi model ωg 15708 and ζg 072 aredetermined S0 is always taken as a constant and it has noeect on TMD parameter optimisation At the same time

λ ωgω1 is dened as the ratio between the predominantfrequency of ltered white noise and the natural frequency ofthe structure In order to study the TMD optimal parametersand the relative relationship between the excitation of sta-tionary white noise random load and ltered white noiserandom load the following denitions are given

ηf fopt_Num

fopt_Form

ηζ2 ζ2opt_Numζ2opt_Form

(18)

0 02 04 06 08 102

03

04

05

06

07

08

09

1

μ

f opt

λ = 1λ = 2λ = 3λ = 4

λ = 5λ = 75Equation (15)

(a)

μ0 02 04 06 08 1

098

1

102

104

106

108

11

112

114

λ = 1λ = 2λ = 3

λ = 4λ = 5λ = 75

η f(b)

λ = 1λ = 2λ = 3λ = 4

λ = 5λ = 75Equation (15)

0 02 04 06 08 101

015

02

025

03

035

04

045

05

μ

ζ 2op

t

(c)

λ = 1λ = 2λ = 3

λ = 4λ = 5λ = 75

μ0 02 04 06 08 1

084

086

088

09

092

094

096

098

1

102

104η ζ

2

(d)

Figure 10 e relation between optimal parameters with λ and μ and its deviation rate with equation (15) under ltered white noiseexcitations (a) Optimal frequency ratio (b) ηf (c) Optimal damping ratio (d) ηζ2opt

Shock and Vibration 9

Under stationary white noise excitations fopt_Form andζopt_Form are the TMD optimal frequency ratios and theoptimal damping ratios calculated by the tting equation(15) respectively Under ltered white noise excitationsfopt_Num and ζopt_Num are the TMD optimal frequency ratioand the optimal damping ratio respectively obtained by thenumerical optimisation method

When the main structure with a damping ratio of 005 isexcited by ltered white noise load the relationship betweenthe optimal parameters of the TMD and μ and λ is shown inFigures 10(a) and 10(c) In order to facilitate comparisonthe optimal parameter calculated by the tting formula inequation (15) is also plotted Figures 10(b) and 10(d) showrespectively the trends of ηf and ηζ2opt with μ and for dif-ferent values of λ

From the above results the following results areobtained

When the mass ratio μ is smaller than 02 the optimalparameters of the stationary white noise random load andthat of the ltered white noise random load are more similarbut the dierence between the two is increased graduallywith the increase of the mass ratio which can be clearly seenfrom the relationship curve of ηf to μ

e larger the λ the more ηf and ηζ2opt become closer to1 When λge 4 even if the TMD mass ratio reaches 1 theerror of the optimal parameters of the TMD under the

stationary white noise random load and the ltered whitenoise random load is not more than 5

In addition the damping ratio of the main structurehas a relatively small impact on the error therefore theincurrenuence of the damping ratio of the main structure onthe error of the tting formula is no longer discussed inthis paper In conclusion for the optimal design of a largemass ratio TMD when λge 4 the tting formula inequation (15) is also suitable for ltered white noise ex-citations while in the other cases it is suggested to de-termine the optimal TMD parameters using a numericaloptimisation method

5 Damping Effect Analysis of a Large MassRatio TMD

Four SDOF structures with 05 10 20 and 30 s periods (T)are selected and the damping ratio of the four structures is005 e TMD mass ratios are 005 025 050 075 and 1and the TMD parameters are calculated by the tting for-mula in equation (15) Two far-eld seismic waves (ElCentro Hachinohe) and two near-eld seismic waves(Northridge Kobe) are used for load input [27] Seismicwave information is provided in Table 2 e seismic waveacceleration-time history curve is shown in Figure 11 and

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

Acce

lera

tion

(ms

2 )Ac

cele

ratio

n (m

s2 )

El Centro

Hachinohe

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

(a)

Acce

lera

tion

(ms

2 )Ac

cele

ratio

n (m

s2 )

Northridge

Kobe

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

(b)

Figure 11 Time history curve of seismic records (a) Far-eld seismic waves (b) Near-eld seismic waves

Table 2 Seismic wave information used in the analyses

Type Name Earthquake Station Component Year

Far eld El Centro Imperial valley El Centro N-S 1940Hachinohe Tokachi-Oki Hachinohe city N-S 1968

Near eld Northridge Northridge SCH N-S 1994Kobe Hyogo-ken KJMA N-S 1995

10 Shock and Vibration

the seismic wave peak value is 1ms2 In order to investigatethe eect of the TMD mass ratio on the control of thestructural displacement response four sets of seismic timehistory analyses were carried out for structures with andwithout a TMD respectively

e reduction rate of peak displacement Re(Peak) andthe reduction rate of root mean square of structural dis-placement Re(RMS) are used as the evaluation index re-spectively e two formulas for calculating the dampingrate are shown as follows

Re(Peak) 1minusMax xTMD( )Max(x)

Re(RMS) 1minusRMS xTMD( )RMS(x)

(19)

where x and xTMD are the time history record of the dis-placement response of the structure without and with aTMD respectively e formula for calculating the rootmean square of the displacement is as follows

0 02 04 06 08 1micro

0

10

20

30

40

50Re

(Pea

k) (

)

EI CentroHachinoheKobe

NorthridgeMean

(a)

0 02 04 06 08 1micro

0

10

20

30

40

50

Re(P

eak)

()

EI CentroHachinoheKobe

NorthridgeMean

(b)

0 02 04 06 08 1micro

0

10

20

30

40

50

Re(P

eak)

()

EI CentroHachinoheKobe

NorthridgeMean

(c)

0 02 04 06 08 1micro

0

10

20

30

40

50Re

(Pea

k) (

)

EI CentroHachinoheKobe

NorthridgeMean

(d)

Figure 12 e damping eect of dierent mass ratio (TMD) on the peak value of structural displacement response (a) T 05 s (b)T10 s (c) T 20 s (d) T 30 s

Shock and Vibration 11

RMS(x) sqrt1NsumN

i1x2i (20)

where xi is the structural displacement response corre-sponding to the ith time andN is the total number of pointscollected

Re(Peak) and Re(RMS) of the displacement responsewith dierent mass ratio TMDs are shown in Figures 12 and13 e following conclusions can be obtained

(1) TMD can eectively control the displacement re-sponse of the structure and the large mass ratio

(gt025) TMD is more eective than the conventionalsmall mass ratio (lt005) TMD But it can also befound that when the mass ratio of the TMD is greaterthan 05 the gain eect will diminish with increasingmass ratio

(2) e TMD with the same mass ratio shows certaindiscreteness for the structures with dierent naturalvibration periods and dierent seismic waves Forexample as shown in Figure 12(b) when thestructurersquos period is 10 s the damping rate of fourseismic waves is distinct When the mass ratio is 05the minimum damping rate is 1047 and the

0 02 04 06 08 10

10

20

30

40

50

60

micro

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(a)

0 02 04 06 08 1micro

0

10

20

30

40

50

60

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(b)

0 02 04 06 08 10

10

20

30

40

50

60

micro

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(c)

0 02 04 06 08 1micro

0

10

20

30

40

50

60Re

(RM

S)

EI CentroHachinoheKobe

NorthridgeMean

(d)

Figure 13e damping eect of dierent mass ratios (TMD) on the mean square root of structural displacement response (a) T 05 s (b)T10 s (c) T 20 s (d) T 30 s

12 Shock and Vibration

Table 3 Maximum peak value of relative displacement between a TMD with optimal parameters and the main structure (cm)

T (s) μ El Centro Hachinohe Northridge Kobe Mean

05

005 436 296 381 312 356010 268 214 298 22 25050 214 254 335 191 249075 183 26 275 279 249100 224 274 224 337 265

10

005 818 883 1039 698 859010 353 483 512 598 487050 341 535 376 588 46075 391 59 355 546 47100 356 494 358 502 428

20

005 1477 209 866 1834 1567010 794 1197 554 1042 897050 53 863 417 858 667075 394 795 345 705 56100 389 846 332 629 549

30

005 1509 3529 666 1993 1925010 682 13 378 889 812050 491 1128 329 654 651075 463 1037 293 553 586100 509 1125 262 536 608

minus002

0

002

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(a)

minus002

0

002

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

micro = 005micro = 075

wo

(b)

Figure 14 Continued

Shock and Vibration 13

maximum is 4622 which is usually mainly due tothe dierence in the frequency relationship betweenthe structure and the seismic waves

Table 3 shows the statistical results of the relative dis-placement between the TMD and main structure underdierent time history analysis conditions It can be seen thatthe relative displacement between the TMD and the mainstructure is inversely proportional to the mass ratio of theTMD that is when a large mass ratio TMD is used thedisplacement response of the structure is eectively con-trolled and the displacement stroke of the TMD is clearlyreduced which reduces the requirements for the elasticcomponents and the damping components used to constructthe TMD

e displacement time histories of the structures withperiods of 10 and 20 s and the relative displacement timehistories between the TMD and the structures are shown inFigures 14 and 15 respectively e schemes of mass ratiosof 005 and 075 are compared It can be clearly seen that theeect of a TMD in controlling the structural response andthe TMD displacement stroke is more obvious for the TMDwith a mass ratio of 075 than for the one with a mass ratio of005

In summary the large mass ratio TMD has a moresignicant eect in seismic control of the main structurethan the small mass ratio TMD

6 Conclusions

In order to control the dynamic response and improve theaseismic performance of a structure a large mass ratio TMDdamping system is formed by using the equipment in thebuilding structure or relying on new structural forms e

existing optimal parameter tting formula is not applicableto large mass ratio TMDs so it is revised by numericaloptimisation and curve tting and the dynamic time historyanalysis method is used to study the eect of vibrationdamping control of large mass ratio TMDs e followingconclusions are obtained

Compared with the traditional small mass ratio (lt005)TMD the large mass ratio (gt05) TMD has obvious ad-vantages in controlling the displacement response of themain structure e control eect is about 15sim325 timeshigher the damping eect of the structural displacementpeak can reach about 30 and the damping ratio of the rootmean square displacement can reach about 436 At thesame time the relative stroke between the TMD and themain structure can be reduced with up to 30sim65 which ishighly benecial to the practical engineering design of TMDstructures

When the mass ratio of a TMD is relatively large (gt02)the results calculated by the existing tting formula diersignicantly from the actual optimal value and the calcu-lated values of the revised formula proposed in this paper areshown to be in good agreement with the actual optimalvalue In general the revised formula can be applied to bothtraditional small mass ratio and large mass ratio (le1) TMDsWhen the mass ratio is greater than 1 the optimal pa-rameters of TMD can also be obtained by the methodpresented in this paper

When the mass ratio is greater than 02 the optimalparameters of the stationary white noise random load andthat of the ltered white noise random load are more similarbut the dierence between the two is gradually increasedwith the increase of the TMD mass ratio For the optimalparameters of large mass ratio TMDs (gt02) the error is lessthan 005 when the ratio of the predominant frequency of the

0 5 10 15 20 25 30

minus002minus001

0001002

t (s)

0 5 10 15 20 25 30t (s)

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(c)

0 5 10 15 20 25 30t (s)

0 5 10 15 20 25 30t (s)

minus002

0

002

004

Disp

lace

men

t (m

)

minus01

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(d)

Figure 14 e damping eect of dierent mass ratio TMDs on the displacement-time history of the main structure (T10 s) and therelative displacement-time history between the TMD and the structure (wo denotes the displacement of a structure without a TMD) (a) ElCentro (b) Hachinohe (c) Northridge (d) Kobe

14 Shock and Vibration

base soil and the vibration frequency of the structure isgreater than 4 and the optimal parameters of the TMD canbe calculated by the tting formula proposed in this paperUnder other conditions it is suggested to use an optimi-sation method to determine the optimal value of TMDparameters

At present the actual engineering projects with largemass ratio TMD damping systems are less prominent buttheir aseismic advantages will bring a broad range of benetsfor research and practice

Data Availability

e data used to support the ndings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no concurrenicts of interestregarding the publication of this article

Acknowledgments

is research was supported by Grant no 51478023 from theNational Natural Science Foundation of China

References

[1] M Gutierrez Soto and H Adeli ldquoTuned mass dampersrdquoArchives of Computational Methods in Engineering vol 20no 4 pp 419ndash431 2013

ndash005

0

005

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

ndash01ndash005

0005

01

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(a)

ndash02

ndash01

0

01

02

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

ndash005

005

0

micro = 005micro = 075

wo

(b)

0 5 10 15 20 25 30

minus005

0

005

Disp

lace

men

t (m

)

t (s)

0 5 10 15 20 25 30t (s)

minus01

0

01

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(c)

minus002

0

002

004D

ispla

cem

ent (

m)

0 5 10 15 20 25 30t (s)

0

minus005

005

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30t (s)

micro = 005micro = 075

wo

(d)

Figure 15 e damping eect of dierent mass ratio TMDs on the displacement-time history of the main structure (T 20 s) and therelative displacement-time history between the TMD and the structure (wo denotes the displacement of a structure without a TMD) (a) ElCentro (b) Hachinohe (c) Northridge (d) Kobe

Shock and Vibration 15

[2] S Elias and VMatsagar ldquoResearch developments in vibrationcontrol of structures using passive tuned mass dampersrdquoAnnual Reviews in Control vol 44 pp 129ndash156 2017

[3] D Wang T K T Tse Y Zhou and Q Li ldquoStructural per-formance and cost analysis of wind-induced vibration controlschemes for a real super-tall buildingrdquo Structure and In-frastructure Engineering vol 11 no 8 pp 990ndash1011 2014

[4] N Longarini and M Zucca ldquoA chimneyrsquos seismic assessmentby a tuned mass damperrdquo Engineering Structures vol 79pp 290ndash296 2014

[5] L Tian and Y Zeng ldquoParametric study of tunedmass dampersfor long span transmission tower-line system under windloadsrdquo Shock and Vibration vol 2016 Article ID 496505611 pages 2016

[6] N Hoang Y Fujino and PWarnitchai ldquoOptimal tuned massdamper for seismic applications and practical design for-mulasrdquo Engineering Structures vol 30 no 3 pp 707ndash7152008

[7] J P Den Hartog Mechanical vibrations McGraw-Hill NewYork NY USA 1956

[8] G B Warburton ldquoOptimum absorber parameters for variouscombinations of response and excitation parametersrdquoEarthquake Engineering amp Structural Dynamics vol 10 no 3pp 381ndash401 1982

[9] H-C Tsai and G-C Lin ldquoExplicit formulae for optimumabsorber parameters for force-excited and viscously dampedsystemsrdquo Journal of Sound and Vibration vol 176 no 5pp 585ndash596 1994

[10] H-C Tsai and G-C Lin ldquoOptimum tuned-mass dampers forminimizing steady-state response of support-excited anddamped systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 22 no 11 pp 957ndash973 1993

[11] S V Bakre and R S Jangid ldquoOptimum parameters of tunedmass damper for damped main systemrdquo Structural Controland Health Monitoring vol 14 no 3 pp 448ndash470 2007

[12] C C Lin C M Hu J F Wang and R Y Hu ldquoVibrationcontrol effectiveness of passive tuned mass dampersrdquo Journalof the Chinese Institute of Engineers vol 17 pp 367ndash376 1994

[13] A Y T Leung and H Zhang ldquoParticle swarm optimization oftuned mass dampersrdquo Engineering Structures vol 31 no 3pp 715ndash728 2009

[14] M Q Feng and A Mita ldquoVibration control of tall buildingsusing mega SubConfigurationrdquo Journal of Engineering Me-chanics vol 121 no 10 pp 1082ndash1088 1995

[15] X X Li P Tan X J Li and A W Liu ldquoMechanism analysisand parameter optimisation of mega-sub-isolation systemrdquoShock and Vibration vol 2016 p 12 2016

[16] A Reggio and M D Angelis ldquoOptimal energy-based seismicdesign of non-conventional tuned mass damper (TMD)implemented via inter-story isolationrdquo Earthquake Engi-neering amp Structural Dynamics vol 44 no 10 pp 1623ndash16422015

[17] S J Wang B H Lee W C Chuang and K C ChangldquoOptimal dynamic characteristic control approach forbuilding mass damper designrdquo Earthquake Engineering ampStructural Dynamics vol 47 no 3 2017

[18] H Anajafi and R A Medina ldquoPartial mass isolation systemfor seismic vibration control of buildingsrdquo Structural Controlamp Health Monitoring vol 25 no 2 article e2088 2017

[19] K Yuan M S He and Y M Li ldquoShaking table tests forenergy-dissipation steel frame structures with infilled wallMTMDrdquo Journal of Vibration and Shock vol 33 no 11pp 200ndash207 2014

[20] R Ding M X Tao M Zhou and J G Nie ldquoSeismic behaviorof RC structures with absence of floor slab constraints andlarge mass turbine as a non-conventional TMD a case studyrdquoBulletin of Earthquake Engineering vol 13 no 11 pp 3401ndash3422 2015

[21] L Y Peng Y J Kang Z R Lai and Y K Deng ldquoOptimisationand damping performance of a coal-fired power plantbuilding equipped with multiple coal bucket dampersrdquo Ad-vances in Civil Engineering vol 2018 p 19 2018

[22] Z Shu S Li J Zhang and M He ldquoOptimum seismic designof a power plant building with pendulum tuned mass dampersystem by its heavy suspended bucketsrdquo Engineering Struc-tures vol 136 pp 114ndash132 2017

[23] K Dai B Li J Wang et al ldquoOptimal probability-based partialmass isolation of elevated coal scuttle in thermal power plantbuildingrdquo Structural Design of Tall and Special Buildingsvol 27 no 11 article e1477 2018

[24] M De Angelis S Perno and A Reggio ldquoDynamic responseand optimal design of structures with large mass ratio TMDrdquoEarthquake Engineering amp Structural Dynamics vol 41 no 1pp 41ndash60 2015

[25] K Kanai ldquoSemi-empirical formula for the seismic charac-teristics of the groundrdquo Bulletin of the Earthquake ResearchInstitute gte University of Tokyo vol 35 pp 309ndash325 1957

[26] Code for seismic design of buildings GB50011-2010 BeijingChina 2010

[27] B F Spencer R E Christenson and S J Dyke ldquoNextgeneration benchmark control problem for seismically excitedbuildingsrdquo in Proceedings of the 2nd World Conference onStructural Control pp 1351ndash1360 Kyoto Japan June 1998

16 Shock and Vibration

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Under stationary white noise excitations fopt_Form andζopt_Form are the TMD optimal frequency ratios and theoptimal damping ratios calculated by the tting equation(15) respectively Under ltered white noise excitationsfopt_Num and ζopt_Num are the TMD optimal frequency ratioand the optimal damping ratio respectively obtained by thenumerical optimisation method

When the main structure with a damping ratio of 005 isexcited by ltered white noise load the relationship betweenthe optimal parameters of the TMD and μ and λ is shown inFigures 10(a) and 10(c) In order to facilitate comparisonthe optimal parameter calculated by the tting formula inequation (15) is also plotted Figures 10(b) and 10(d) showrespectively the trends of ηf and ηζ2opt with μ and for dif-ferent values of λ

From the above results the following results areobtained

When the mass ratio μ is smaller than 02 the optimalparameters of the stationary white noise random load andthat of the ltered white noise random load are more similarbut the dierence between the two is increased graduallywith the increase of the mass ratio which can be clearly seenfrom the relationship curve of ηf to μ

e larger the λ the more ηf and ηζ2opt become closer to1 When λge 4 even if the TMD mass ratio reaches 1 theerror of the optimal parameters of the TMD under the

stationary white noise random load and the ltered whitenoise random load is not more than 5

In addition the damping ratio of the main structurehas a relatively small impact on the error therefore theincurrenuence of the damping ratio of the main structure onthe error of the tting formula is no longer discussed inthis paper In conclusion for the optimal design of a largemass ratio TMD when λge 4 the tting formula inequation (15) is also suitable for ltered white noise ex-citations while in the other cases it is suggested to de-termine the optimal TMD parameters using a numericaloptimisation method

5 Damping Effect Analysis of a Large MassRatio TMD

Four SDOF structures with 05 10 20 and 30 s periods (T)are selected and the damping ratio of the four structures is005 e TMD mass ratios are 005 025 050 075 and 1and the TMD parameters are calculated by the tting for-mula in equation (15) Two far-eld seismic waves (ElCentro Hachinohe) and two near-eld seismic waves(Northridge Kobe) are used for load input [27] Seismicwave information is provided in Table 2 e seismic waveacceleration-time history curve is shown in Figure 11 and

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

Acce

lera

tion

(ms

2 )Ac

cele

ratio

n (m

s2 )

El Centro

Hachinohe

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

(a)

Acce

lera

tion

(ms

2 )Ac

cele

ratio

n (m

s2 )

Northridge

Kobe

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

0 5 10 15 20 25 30 35ndash1

ndash05

0

05

1

Time (s)

(b)

Figure 11 Time history curve of seismic records (a) Far-eld seismic waves (b) Near-eld seismic waves

Table 2 Seismic wave information used in the analyses

Type Name Earthquake Station Component Year

Far eld El Centro Imperial valley El Centro N-S 1940Hachinohe Tokachi-Oki Hachinohe city N-S 1968

Near eld Northridge Northridge SCH N-S 1994Kobe Hyogo-ken KJMA N-S 1995

10 Shock and Vibration

the seismic wave peak value is 1ms2 In order to investigatethe eect of the TMD mass ratio on the control of thestructural displacement response four sets of seismic timehistory analyses were carried out for structures with andwithout a TMD respectively

e reduction rate of peak displacement Re(Peak) andthe reduction rate of root mean square of structural dis-placement Re(RMS) are used as the evaluation index re-spectively e two formulas for calculating the dampingrate are shown as follows

Re(Peak) 1minusMax xTMD( )Max(x)

Re(RMS) 1minusRMS xTMD( )RMS(x)

(19)

where x and xTMD are the time history record of the dis-placement response of the structure without and with aTMD respectively e formula for calculating the rootmean square of the displacement is as follows

0 02 04 06 08 1micro

0

10

20

30

40

50Re

(Pea

k) (

)

EI CentroHachinoheKobe

NorthridgeMean

(a)

0 02 04 06 08 1micro

0

10

20

30

40

50

Re(P

eak)

()

EI CentroHachinoheKobe

NorthridgeMean

(b)

0 02 04 06 08 1micro

0

10

20

30

40

50

Re(P

eak)

()

EI CentroHachinoheKobe

NorthridgeMean

(c)

0 02 04 06 08 1micro

0

10

20

30

40

50Re

(Pea

k) (

)

EI CentroHachinoheKobe

NorthridgeMean

(d)

Figure 12 e damping eect of dierent mass ratio (TMD) on the peak value of structural displacement response (a) T 05 s (b)T10 s (c) T 20 s (d) T 30 s

Shock and Vibration 11

RMS(x) sqrt1NsumN

i1x2i (20)

where xi is the structural displacement response corre-sponding to the ith time andN is the total number of pointscollected

Re(Peak) and Re(RMS) of the displacement responsewith dierent mass ratio TMDs are shown in Figures 12 and13 e following conclusions can be obtained

(1) TMD can eectively control the displacement re-sponse of the structure and the large mass ratio

(gt025) TMD is more eective than the conventionalsmall mass ratio (lt005) TMD But it can also befound that when the mass ratio of the TMD is greaterthan 05 the gain eect will diminish with increasingmass ratio

(2) e TMD with the same mass ratio shows certaindiscreteness for the structures with dierent naturalvibration periods and dierent seismic waves Forexample as shown in Figure 12(b) when thestructurersquos period is 10 s the damping rate of fourseismic waves is distinct When the mass ratio is 05the minimum damping rate is 1047 and the

0 02 04 06 08 10

10

20

30

40

50

60

micro

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(a)

0 02 04 06 08 1micro

0

10

20

30

40

50

60

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(b)

0 02 04 06 08 10

10

20

30

40

50

60

micro

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(c)

0 02 04 06 08 1micro

0

10

20

30

40

50

60Re

(RM

S)

EI CentroHachinoheKobe

NorthridgeMean

(d)

Figure 13e damping eect of dierent mass ratios (TMD) on the mean square root of structural displacement response (a) T 05 s (b)T10 s (c) T 20 s (d) T 30 s

12 Shock and Vibration

Table 3 Maximum peak value of relative displacement between a TMD with optimal parameters and the main structure (cm)

T (s) μ El Centro Hachinohe Northridge Kobe Mean

05

005 436 296 381 312 356010 268 214 298 22 25050 214 254 335 191 249075 183 26 275 279 249100 224 274 224 337 265

10

005 818 883 1039 698 859010 353 483 512 598 487050 341 535 376 588 46075 391 59 355 546 47100 356 494 358 502 428

20

005 1477 209 866 1834 1567010 794 1197 554 1042 897050 53 863 417 858 667075 394 795 345 705 56100 389 846 332 629 549

30

005 1509 3529 666 1993 1925010 682 13 378 889 812050 491 1128 329 654 651075 463 1037 293 553 586100 509 1125 262 536 608

minus002

0

002

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(a)

minus002

0

002

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

micro = 005micro = 075

wo

(b)

Figure 14 Continued

Shock and Vibration 13

maximum is 4622 which is usually mainly due tothe dierence in the frequency relationship betweenthe structure and the seismic waves

Table 3 shows the statistical results of the relative dis-placement between the TMD and main structure underdierent time history analysis conditions It can be seen thatthe relative displacement between the TMD and the mainstructure is inversely proportional to the mass ratio of theTMD that is when a large mass ratio TMD is used thedisplacement response of the structure is eectively con-trolled and the displacement stroke of the TMD is clearlyreduced which reduces the requirements for the elasticcomponents and the damping components used to constructthe TMD

e displacement time histories of the structures withperiods of 10 and 20 s and the relative displacement timehistories between the TMD and the structures are shown inFigures 14 and 15 respectively e schemes of mass ratiosof 005 and 075 are compared It can be clearly seen that theeect of a TMD in controlling the structural response andthe TMD displacement stroke is more obvious for the TMDwith a mass ratio of 075 than for the one with a mass ratio of005

In summary the large mass ratio TMD has a moresignicant eect in seismic control of the main structurethan the small mass ratio TMD

6 Conclusions

In order to control the dynamic response and improve theaseismic performance of a structure a large mass ratio TMDdamping system is formed by using the equipment in thebuilding structure or relying on new structural forms e

existing optimal parameter tting formula is not applicableto large mass ratio TMDs so it is revised by numericaloptimisation and curve tting and the dynamic time historyanalysis method is used to study the eect of vibrationdamping control of large mass ratio TMDs e followingconclusions are obtained

Compared with the traditional small mass ratio (lt005)TMD the large mass ratio (gt05) TMD has obvious ad-vantages in controlling the displacement response of themain structure e control eect is about 15sim325 timeshigher the damping eect of the structural displacementpeak can reach about 30 and the damping ratio of the rootmean square displacement can reach about 436 At thesame time the relative stroke between the TMD and themain structure can be reduced with up to 30sim65 which ishighly benecial to the practical engineering design of TMDstructures

When the mass ratio of a TMD is relatively large (gt02)the results calculated by the existing tting formula diersignicantly from the actual optimal value and the calcu-lated values of the revised formula proposed in this paper areshown to be in good agreement with the actual optimalvalue In general the revised formula can be applied to bothtraditional small mass ratio and large mass ratio (le1) TMDsWhen the mass ratio is greater than 1 the optimal pa-rameters of TMD can also be obtained by the methodpresented in this paper

When the mass ratio is greater than 02 the optimalparameters of the stationary white noise random load andthat of the ltered white noise random load are more similarbut the dierence between the two is gradually increasedwith the increase of the TMD mass ratio For the optimalparameters of large mass ratio TMDs (gt02) the error is lessthan 005 when the ratio of the predominant frequency of the

0 5 10 15 20 25 30

minus002minus001

0001002

t (s)

0 5 10 15 20 25 30t (s)

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(c)

0 5 10 15 20 25 30t (s)

0 5 10 15 20 25 30t (s)

minus002

0

002

004

Disp

lace

men

t (m

)

minus01

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(d)

Figure 14 e damping eect of dierent mass ratio TMDs on the displacement-time history of the main structure (T10 s) and therelative displacement-time history between the TMD and the structure (wo denotes the displacement of a structure without a TMD) (a) ElCentro (b) Hachinohe (c) Northridge (d) Kobe

14 Shock and Vibration

base soil and the vibration frequency of the structure isgreater than 4 and the optimal parameters of the TMD canbe calculated by the tting formula proposed in this paperUnder other conditions it is suggested to use an optimi-sation method to determine the optimal value of TMDparameters

At present the actual engineering projects with largemass ratio TMD damping systems are less prominent buttheir aseismic advantages will bring a broad range of benetsfor research and practice

Data Availability

e data used to support the ndings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no concurrenicts of interestregarding the publication of this article

Acknowledgments

is research was supported by Grant no 51478023 from theNational Natural Science Foundation of China

References

[1] M Gutierrez Soto and H Adeli ldquoTuned mass dampersrdquoArchives of Computational Methods in Engineering vol 20no 4 pp 419ndash431 2013

ndash005

0

005

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

ndash01ndash005

0005

01

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(a)

ndash02

ndash01

0

01

02

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

ndash005

005

0

micro = 005micro = 075

wo

(b)

0 5 10 15 20 25 30

minus005

0

005

Disp

lace

men

t (m

)

t (s)

0 5 10 15 20 25 30t (s)

minus01

0

01

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(c)

minus002

0

002

004D

ispla

cem

ent (

m)

0 5 10 15 20 25 30t (s)

0

minus005

005

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30t (s)

micro = 005micro = 075

wo

(d)

Figure 15 e damping eect of dierent mass ratio TMDs on the displacement-time history of the main structure (T 20 s) and therelative displacement-time history between the TMD and the structure (wo denotes the displacement of a structure without a TMD) (a) ElCentro (b) Hachinohe (c) Northridge (d) Kobe

Shock and Vibration 15

[2] S Elias and VMatsagar ldquoResearch developments in vibrationcontrol of structures using passive tuned mass dampersrdquoAnnual Reviews in Control vol 44 pp 129ndash156 2017

[3] D Wang T K T Tse Y Zhou and Q Li ldquoStructural per-formance and cost analysis of wind-induced vibration controlschemes for a real super-tall buildingrdquo Structure and In-frastructure Engineering vol 11 no 8 pp 990ndash1011 2014

[4] N Longarini and M Zucca ldquoA chimneyrsquos seismic assessmentby a tuned mass damperrdquo Engineering Structures vol 79pp 290ndash296 2014

[5] L Tian and Y Zeng ldquoParametric study of tunedmass dampersfor long span transmission tower-line system under windloadsrdquo Shock and Vibration vol 2016 Article ID 496505611 pages 2016

[6] N Hoang Y Fujino and PWarnitchai ldquoOptimal tuned massdamper for seismic applications and practical design for-mulasrdquo Engineering Structures vol 30 no 3 pp 707ndash7152008

[7] J P Den Hartog Mechanical vibrations McGraw-Hill NewYork NY USA 1956

[8] G B Warburton ldquoOptimum absorber parameters for variouscombinations of response and excitation parametersrdquoEarthquake Engineering amp Structural Dynamics vol 10 no 3pp 381ndash401 1982

[9] H-C Tsai and G-C Lin ldquoExplicit formulae for optimumabsorber parameters for force-excited and viscously dampedsystemsrdquo Journal of Sound and Vibration vol 176 no 5pp 585ndash596 1994

[10] H-C Tsai and G-C Lin ldquoOptimum tuned-mass dampers forminimizing steady-state response of support-excited anddamped systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 22 no 11 pp 957ndash973 1993

[11] S V Bakre and R S Jangid ldquoOptimum parameters of tunedmass damper for damped main systemrdquo Structural Controland Health Monitoring vol 14 no 3 pp 448ndash470 2007

[12] C C Lin C M Hu J F Wang and R Y Hu ldquoVibrationcontrol effectiveness of passive tuned mass dampersrdquo Journalof the Chinese Institute of Engineers vol 17 pp 367ndash376 1994

[13] A Y T Leung and H Zhang ldquoParticle swarm optimization oftuned mass dampersrdquo Engineering Structures vol 31 no 3pp 715ndash728 2009

[14] M Q Feng and A Mita ldquoVibration control of tall buildingsusing mega SubConfigurationrdquo Journal of Engineering Me-chanics vol 121 no 10 pp 1082ndash1088 1995

[15] X X Li P Tan X J Li and A W Liu ldquoMechanism analysisand parameter optimisation of mega-sub-isolation systemrdquoShock and Vibration vol 2016 p 12 2016

[16] A Reggio and M D Angelis ldquoOptimal energy-based seismicdesign of non-conventional tuned mass damper (TMD)implemented via inter-story isolationrdquo Earthquake Engi-neering amp Structural Dynamics vol 44 no 10 pp 1623ndash16422015

[17] S J Wang B H Lee W C Chuang and K C ChangldquoOptimal dynamic characteristic control approach forbuilding mass damper designrdquo Earthquake Engineering ampStructural Dynamics vol 47 no 3 2017

[18] H Anajafi and R A Medina ldquoPartial mass isolation systemfor seismic vibration control of buildingsrdquo Structural Controlamp Health Monitoring vol 25 no 2 article e2088 2017

[19] K Yuan M S He and Y M Li ldquoShaking table tests forenergy-dissipation steel frame structures with infilled wallMTMDrdquo Journal of Vibration and Shock vol 33 no 11pp 200ndash207 2014

[20] R Ding M X Tao M Zhou and J G Nie ldquoSeismic behaviorof RC structures with absence of floor slab constraints andlarge mass turbine as a non-conventional TMD a case studyrdquoBulletin of Earthquake Engineering vol 13 no 11 pp 3401ndash3422 2015

[21] L Y Peng Y J Kang Z R Lai and Y K Deng ldquoOptimisationand damping performance of a coal-fired power plantbuilding equipped with multiple coal bucket dampersrdquo Ad-vances in Civil Engineering vol 2018 p 19 2018

[22] Z Shu S Li J Zhang and M He ldquoOptimum seismic designof a power plant building with pendulum tuned mass dampersystem by its heavy suspended bucketsrdquo Engineering Struc-tures vol 136 pp 114ndash132 2017

[23] K Dai B Li J Wang et al ldquoOptimal probability-based partialmass isolation of elevated coal scuttle in thermal power plantbuildingrdquo Structural Design of Tall and Special Buildingsvol 27 no 11 article e1477 2018

[24] M De Angelis S Perno and A Reggio ldquoDynamic responseand optimal design of structures with large mass ratio TMDrdquoEarthquake Engineering amp Structural Dynamics vol 41 no 1pp 41ndash60 2015

[25] K Kanai ldquoSemi-empirical formula for the seismic charac-teristics of the groundrdquo Bulletin of the Earthquake ResearchInstitute gte University of Tokyo vol 35 pp 309ndash325 1957

[26] Code for seismic design of buildings GB50011-2010 BeijingChina 2010

[27] B F Spencer R E Christenson and S J Dyke ldquoNextgeneration benchmark control problem for seismically excitedbuildingsrdquo in Proceedings of the 2nd World Conference onStructural Control pp 1351ndash1360 Kyoto Japan June 1998

16 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

the seismic wave peak value is 1ms2 In order to investigatethe eect of the TMD mass ratio on the control of thestructural displacement response four sets of seismic timehistory analyses were carried out for structures with andwithout a TMD respectively

e reduction rate of peak displacement Re(Peak) andthe reduction rate of root mean square of structural dis-placement Re(RMS) are used as the evaluation index re-spectively e two formulas for calculating the dampingrate are shown as follows

Re(Peak) 1minusMax xTMD( )Max(x)

Re(RMS) 1minusRMS xTMD( )RMS(x)

(19)

where x and xTMD are the time history record of the dis-placement response of the structure without and with aTMD respectively e formula for calculating the rootmean square of the displacement is as follows

0 02 04 06 08 1micro

0

10

20

30

40

50Re

(Pea

k) (

)

EI CentroHachinoheKobe

NorthridgeMean

(a)

0 02 04 06 08 1micro

0

10

20

30

40

50

Re(P

eak)

()

EI CentroHachinoheKobe

NorthridgeMean

(b)

0 02 04 06 08 1micro

0

10

20

30

40

50

Re(P

eak)

()

EI CentroHachinoheKobe

NorthridgeMean

(c)

0 02 04 06 08 1micro

0

10

20

30

40

50Re

(Pea

k) (

)

EI CentroHachinoheKobe

NorthridgeMean

(d)

Figure 12 e damping eect of dierent mass ratio (TMD) on the peak value of structural displacement response (a) T 05 s (b)T10 s (c) T 20 s (d) T 30 s

Shock and Vibration 11

RMS(x) sqrt1NsumN

i1x2i (20)

where xi is the structural displacement response corre-sponding to the ith time andN is the total number of pointscollected

Re(Peak) and Re(RMS) of the displacement responsewith dierent mass ratio TMDs are shown in Figures 12 and13 e following conclusions can be obtained

(1) TMD can eectively control the displacement re-sponse of the structure and the large mass ratio

(gt025) TMD is more eective than the conventionalsmall mass ratio (lt005) TMD But it can also befound that when the mass ratio of the TMD is greaterthan 05 the gain eect will diminish with increasingmass ratio

(2) e TMD with the same mass ratio shows certaindiscreteness for the structures with dierent naturalvibration periods and dierent seismic waves Forexample as shown in Figure 12(b) when thestructurersquos period is 10 s the damping rate of fourseismic waves is distinct When the mass ratio is 05the minimum damping rate is 1047 and the

0 02 04 06 08 10

10

20

30

40

50

60

micro

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(a)

0 02 04 06 08 1micro

0

10

20

30

40

50

60

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(b)

0 02 04 06 08 10

10

20

30

40

50

60

micro

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(c)

0 02 04 06 08 1micro

0

10

20

30

40

50

60Re

(RM

S)

EI CentroHachinoheKobe

NorthridgeMean

(d)

Figure 13e damping eect of dierent mass ratios (TMD) on the mean square root of structural displacement response (a) T 05 s (b)T10 s (c) T 20 s (d) T 30 s

12 Shock and Vibration

Table 3 Maximum peak value of relative displacement between a TMD with optimal parameters and the main structure (cm)

T (s) μ El Centro Hachinohe Northridge Kobe Mean

05

005 436 296 381 312 356010 268 214 298 22 25050 214 254 335 191 249075 183 26 275 279 249100 224 274 224 337 265

10

005 818 883 1039 698 859010 353 483 512 598 487050 341 535 376 588 46075 391 59 355 546 47100 356 494 358 502 428

20

005 1477 209 866 1834 1567010 794 1197 554 1042 897050 53 863 417 858 667075 394 795 345 705 56100 389 846 332 629 549

30

005 1509 3529 666 1993 1925010 682 13 378 889 812050 491 1128 329 654 651075 463 1037 293 553 586100 509 1125 262 536 608

minus002

0

002

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(a)

minus002

0

002

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

micro = 005micro = 075

wo

(b)

Figure 14 Continued

Shock and Vibration 13

maximum is 4622 which is usually mainly due tothe dierence in the frequency relationship betweenthe structure and the seismic waves

Table 3 shows the statistical results of the relative dis-placement between the TMD and main structure underdierent time history analysis conditions It can be seen thatthe relative displacement between the TMD and the mainstructure is inversely proportional to the mass ratio of theTMD that is when a large mass ratio TMD is used thedisplacement response of the structure is eectively con-trolled and the displacement stroke of the TMD is clearlyreduced which reduces the requirements for the elasticcomponents and the damping components used to constructthe TMD

e displacement time histories of the structures withperiods of 10 and 20 s and the relative displacement timehistories between the TMD and the structures are shown inFigures 14 and 15 respectively e schemes of mass ratiosof 005 and 075 are compared It can be clearly seen that theeect of a TMD in controlling the structural response andthe TMD displacement stroke is more obvious for the TMDwith a mass ratio of 075 than for the one with a mass ratio of005

In summary the large mass ratio TMD has a moresignicant eect in seismic control of the main structurethan the small mass ratio TMD

6 Conclusions

In order to control the dynamic response and improve theaseismic performance of a structure a large mass ratio TMDdamping system is formed by using the equipment in thebuilding structure or relying on new structural forms e

existing optimal parameter tting formula is not applicableto large mass ratio TMDs so it is revised by numericaloptimisation and curve tting and the dynamic time historyanalysis method is used to study the eect of vibrationdamping control of large mass ratio TMDs e followingconclusions are obtained

Compared with the traditional small mass ratio (lt005)TMD the large mass ratio (gt05) TMD has obvious ad-vantages in controlling the displacement response of themain structure e control eect is about 15sim325 timeshigher the damping eect of the structural displacementpeak can reach about 30 and the damping ratio of the rootmean square displacement can reach about 436 At thesame time the relative stroke between the TMD and themain structure can be reduced with up to 30sim65 which ishighly benecial to the practical engineering design of TMDstructures

When the mass ratio of a TMD is relatively large (gt02)the results calculated by the existing tting formula diersignicantly from the actual optimal value and the calcu-lated values of the revised formula proposed in this paper areshown to be in good agreement with the actual optimalvalue In general the revised formula can be applied to bothtraditional small mass ratio and large mass ratio (le1) TMDsWhen the mass ratio is greater than 1 the optimal pa-rameters of TMD can also be obtained by the methodpresented in this paper

When the mass ratio is greater than 02 the optimalparameters of the stationary white noise random load andthat of the ltered white noise random load are more similarbut the dierence between the two is gradually increasedwith the increase of the TMD mass ratio For the optimalparameters of large mass ratio TMDs (gt02) the error is lessthan 005 when the ratio of the predominant frequency of the

0 5 10 15 20 25 30

minus002minus001

0001002

t (s)

0 5 10 15 20 25 30t (s)

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(c)

0 5 10 15 20 25 30t (s)

0 5 10 15 20 25 30t (s)

minus002

0

002

004

Disp

lace

men

t (m

)

minus01

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(d)

Figure 14 e damping eect of dierent mass ratio TMDs on the displacement-time history of the main structure (T10 s) and therelative displacement-time history between the TMD and the structure (wo denotes the displacement of a structure without a TMD) (a) ElCentro (b) Hachinohe (c) Northridge (d) Kobe

14 Shock and Vibration

base soil and the vibration frequency of the structure isgreater than 4 and the optimal parameters of the TMD canbe calculated by the tting formula proposed in this paperUnder other conditions it is suggested to use an optimi-sation method to determine the optimal value of TMDparameters

At present the actual engineering projects with largemass ratio TMD damping systems are less prominent buttheir aseismic advantages will bring a broad range of benetsfor research and practice

Data Availability

e data used to support the ndings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no concurrenicts of interestregarding the publication of this article

Acknowledgments

is research was supported by Grant no 51478023 from theNational Natural Science Foundation of China

References

[1] M Gutierrez Soto and H Adeli ldquoTuned mass dampersrdquoArchives of Computational Methods in Engineering vol 20no 4 pp 419ndash431 2013

ndash005

0

005

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

ndash01ndash005

0005

01

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(a)

ndash02

ndash01

0

01

02

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

ndash005

005

0

micro = 005micro = 075

wo

(b)

0 5 10 15 20 25 30

minus005

0

005

Disp

lace

men

t (m

)

t (s)

0 5 10 15 20 25 30t (s)

minus01

0

01

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(c)

minus002

0

002

004D

ispla

cem

ent (

m)

0 5 10 15 20 25 30t (s)

0

minus005

005

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30t (s)

micro = 005micro = 075

wo

(d)

Figure 15 e damping eect of dierent mass ratio TMDs on the displacement-time history of the main structure (T 20 s) and therelative displacement-time history between the TMD and the structure (wo denotes the displacement of a structure without a TMD) (a) ElCentro (b) Hachinohe (c) Northridge (d) Kobe

Shock and Vibration 15

[2] S Elias and VMatsagar ldquoResearch developments in vibrationcontrol of structures using passive tuned mass dampersrdquoAnnual Reviews in Control vol 44 pp 129ndash156 2017

[3] D Wang T K T Tse Y Zhou and Q Li ldquoStructural per-formance and cost analysis of wind-induced vibration controlschemes for a real super-tall buildingrdquo Structure and In-frastructure Engineering vol 11 no 8 pp 990ndash1011 2014

[4] N Longarini and M Zucca ldquoA chimneyrsquos seismic assessmentby a tuned mass damperrdquo Engineering Structures vol 79pp 290ndash296 2014

[5] L Tian and Y Zeng ldquoParametric study of tunedmass dampersfor long span transmission tower-line system under windloadsrdquo Shock and Vibration vol 2016 Article ID 496505611 pages 2016

[6] N Hoang Y Fujino and PWarnitchai ldquoOptimal tuned massdamper for seismic applications and practical design for-mulasrdquo Engineering Structures vol 30 no 3 pp 707ndash7152008

[7] J P Den Hartog Mechanical vibrations McGraw-Hill NewYork NY USA 1956

[8] G B Warburton ldquoOptimum absorber parameters for variouscombinations of response and excitation parametersrdquoEarthquake Engineering amp Structural Dynamics vol 10 no 3pp 381ndash401 1982

[9] H-C Tsai and G-C Lin ldquoExplicit formulae for optimumabsorber parameters for force-excited and viscously dampedsystemsrdquo Journal of Sound and Vibration vol 176 no 5pp 585ndash596 1994

[10] H-C Tsai and G-C Lin ldquoOptimum tuned-mass dampers forminimizing steady-state response of support-excited anddamped systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 22 no 11 pp 957ndash973 1993

[11] S V Bakre and R S Jangid ldquoOptimum parameters of tunedmass damper for damped main systemrdquo Structural Controland Health Monitoring vol 14 no 3 pp 448ndash470 2007

[12] C C Lin C M Hu J F Wang and R Y Hu ldquoVibrationcontrol effectiveness of passive tuned mass dampersrdquo Journalof the Chinese Institute of Engineers vol 17 pp 367ndash376 1994

[13] A Y T Leung and H Zhang ldquoParticle swarm optimization oftuned mass dampersrdquo Engineering Structures vol 31 no 3pp 715ndash728 2009

[14] M Q Feng and A Mita ldquoVibration control of tall buildingsusing mega SubConfigurationrdquo Journal of Engineering Me-chanics vol 121 no 10 pp 1082ndash1088 1995

[15] X X Li P Tan X J Li and A W Liu ldquoMechanism analysisand parameter optimisation of mega-sub-isolation systemrdquoShock and Vibration vol 2016 p 12 2016

[16] A Reggio and M D Angelis ldquoOptimal energy-based seismicdesign of non-conventional tuned mass damper (TMD)implemented via inter-story isolationrdquo Earthquake Engi-neering amp Structural Dynamics vol 44 no 10 pp 1623ndash16422015

[17] S J Wang B H Lee W C Chuang and K C ChangldquoOptimal dynamic characteristic control approach forbuilding mass damper designrdquo Earthquake Engineering ampStructural Dynamics vol 47 no 3 2017

[18] H Anajafi and R A Medina ldquoPartial mass isolation systemfor seismic vibration control of buildingsrdquo Structural Controlamp Health Monitoring vol 25 no 2 article e2088 2017

[19] K Yuan M S He and Y M Li ldquoShaking table tests forenergy-dissipation steel frame structures with infilled wallMTMDrdquo Journal of Vibration and Shock vol 33 no 11pp 200ndash207 2014

[20] R Ding M X Tao M Zhou and J G Nie ldquoSeismic behaviorof RC structures with absence of floor slab constraints andlarge mass turbine as a non-conventional TMD a case studyrdquoBulletin of Earthquake Engineering vol 13 no 11 pp 3401ndash3422 2015

[21] L Y Peng Y J Kang Z R Lai and Y K Deng ldquoOptimisationand damping performance of a coal-fired power plantbuilding equipped with multiple coal bucket dampersrdquo Ad-vances in Civil Engineering vol 2018 p 19 2018

[22] Z Shu S Li J Zhang and M He ldquoOptimum seismic designof a power plant building with pendulum tuned mass dampersystem by its heavy suspended bucketsrdquo Engineering Struc-tures vol 136 pp 114ndash132 2017

[23] K Dai B Li J Wang et al ldquoOptimal probability-based partialmass isolation of elevated coal scuttle in thermal power plantbuildingrdquo Structural Design of Tall and Special Buildingsvol 27 no 11 article e1477 2018

[24] M De Angelis S Perno and A Reggio ldquoDynamic responseand optimal design of structures with large mass ratio TMDrdquoEarthquake Engineering amp Structural Dynamics vol 41 no 1pp 41ndash60 2015

[25] K Kanai ldquoSemi-empirical formula for the seismic charac-teristics of the groundrdquo Bulletin of the Earthquake ResearchInstitute gte University of Tokyo vol 35 pp 309ndash325 1957

[26] Code for seismic design of buildings GB50011-2010 BeijingChina 2010

[27] B F Spencer R E Christenson and S J Dyke ldquoNextgeneration benchmark control problem for seismically excitedbuildingsrdquo in Proceedings of the 2nd World Conference onStructural Control pp 1351ndash1360 Kyoto Japan June 1998

16 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

RMS(x) sqrt1NsumN

i1x2i (20)

where xi is the structural displacement response corre-sponding to the ith time andN is the total number of pointscollected

Re(Peak) and Re(RMS) of the displacement responsewith dierent mass ratio TMDs are shown in Figures 12 and13 e following conclusions can be obtained

(1) TMD can eectively control the displacement re-sponse of the structure and the large mass ratio

(gt025) TMD is more eective than the conventionalsmall mass ratio (lt005) TMD But it can also befound that when the mass ratio of the TMD is greaterthan 05 the gain eect will diminish with increasingmass ratio

(2) e TMD with the same mass ratio shows certaindiscreteness for the structures with dierent naturalvibration periods and dierent seismic waves Forexample as shown in Figure 12(b) when thestructurersquos period is 10 s the damping rate of fourseismic waves is distinct When the mass ratio is 05the minimum damping rate is 1047 and the

0 02 04 06 08 10

10

20

30

40

50

60

micro

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(a)

0 02 04 06 08 1micro

0

10

20

30

40

50

60

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(b)

0 02 04 06 08 10

10

20

30

40

50

60

micro

Re(R

MS)

EI CentroHachinoheKobe

NorthridgeMean

(c)

0 02 04 06 08 1micro

0

10

20

30

40

50

60Re

(RM

S)

EI CentroHachinoheKobe

NorthridgeMean

(d)

Figure 13e damping eect of dierent mass ratios (TMD) on the mean square root of structural displacement response (a) T 05 s (b)T10 s (c) T 20 s (d) T 30 s

12 Shock and Vibration

Table 3 Maximum peak value of relative displacement between a TMD with optimal parameters and the main structure (cm)

T (s) μ El Centro Hachinohe Northridge Kobe Mean

05

005 436 296 381 312 356010 268 214 298 22 25050 214 254 335 191 249075 183 26 275 279 249100 224 274 224 337 265

10

005 818 883 1039 698 859010 353 483 512 598 487050 341 535 376 588 46075 391 59 355 546 47100 356 494 358 502 428

20

005 1477 209 866 1834 1567010 794 1197 554 1042 897050 53 863 417 858 667075 394 795 345 705 56100 389 846 332 629 549

30

005 1509 3529 666 1993 1925010 682 13 378 889 812050 491 1128 329 654 651075 463 1037 293 553 586100 509 1125 262 536 608

minus002

0

002

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(a)

minus002

0

002

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

micro = 005micro = 075

wo

(b)

Figure 14 Continued

Shock and Vibration 13

maximum is 4622 which is usually mainly due tothe dierence in the frequency relationship betweenthe structure and the seismic waves

Table 3 shows the statistical results of the relative dis-placement between the TMD and main structure underdierent time history analysis conditions It can be seen thatthe relative displacement between the TMD and the mainstructure is inversely proportional to the mass ratio of theTMD that is when a large mass ratio TMD is used thedisplacement response of the structure is eectively con-trolled and the displacement stroke of the TMD is clearlyreduced which reduces the requirements for the elasticcomponents and the damping components used to constructthe TMD

e displacement time histories of the structures withperiods of 10 and 20 s and the relative displacement timehistories between the TMD and the structures are shown inFigures 14 and 15 respectively e schemes of mass ratiosof 005 and 075 are compared It can be clearly seen that theeect of a TMD in controlling the structural response andthe TMD displacement stroke is more obvious for the TMDwith a mass ratio of 075 than for the one with a mass ratio of005

In summary the large mass ratio TMD has a moresignicant eect in seismic control of the main structurethan the small mass ratio TMD

6 Conclusions

In order to control the dynamic response and improve theaseismic performance of a structure a large mass ratio TMDdamping system is formed by using the equipment in thebuilding structure or relying on new structural forms e

existing optimal parameter tting formula is not applicableto large mass ratio TMDs so it is revised by numericaloptimisation and curve tting and the dynamic time historyanalysis method is used to study the eect of vibrationdamping control of large mass ratio TMDs e followingconclusions are obtained

Compared with the traditional small mass ratio (lt005)TMD the large mass ratio (gt05) TMD has obvious ad-vantages in controlling the displacement response of themain structure e control eect is about 15sim325 timeshigher the damping eect of the structural displacementpeak can reach about 30 and the damping ratio of the rootmean square displacement can reach about 436 At thesame time the relative stroke between the TMD and themain structure can be reduced with up to 30sim65 which ishighly benecial to the practical engineering design of TMDstructures

When the mass ratio of a TMD is relatively large (gt02)the results calculated by the existing tting formula diersignicantly from the actual optimal value and the calcu-lated values of the revised formula proposed in this paper areshown to be in good agreement with the actual optimalvalue In general the revised formula can be applied to bothtraditional small mass ratio and large mass ratio (le1) TMDsWhen the mass ratio is greater than 1 the optimal pa-rameters of TMD can also be obtained by the methodpresented in this paper

When the mass ratio is greater than 02 the optimalparameters of the stationary white noise random load andthat of the ltered white noise random load are more similarbut the dierence between the two is gradually increasedwith the increase of the TMD mass ratio For the optimalparameters of large mass ratio TMDs (gt02) the error is lessthan 005 when the ratio of the predominant frequency of the

0 5 10 15 20 25 30

minus002minus001

0001002

t (s)

0 5 10 15 20 25 30t (s)

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(c)

0 5 10 15 20 25 30t (s)

0 5 10 15 20 25 30t (s)

minus002

0

002

004

Disp

lace

men

t (m

)

minus01

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(d)

Figure 14 e damping eect of dierent mass ratio TMDs on the displacement-time history of the main structure (T10 s) and therelative displacement-time history between the TMD and the structure (wo denotes the displacement of a structure without a TMD) (a) ElCentro (b) Hachinohe (c) Northridge (d) Kobe

14 Shock and Vibration

base soil and the vibration frequency of the structure isgreater than 4 and the optimal parameters of the TMD canbe calculated by the tting formula proposed in this paperUnder other conditions it is suggested to use an optimi-sation method to determine the optimal value of TMDparameters

At present the actual engineering projects with largemass ratio TMD damping systems are less prominent buttheir aseismic advantages will bring a broad range of benetsfor research and practice

Data Availability

e data used to support the ndings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no concurrenicts of interestregarding the publication of this article

Acknowledgments

is research was supported by Grant no 51478023 from theNational Natural Science Foundation of China

References

[1] M Gutierrez Soto and H Adeli ldquoTuned mass dampersrdquoArchives of Computational Methods in Engineering vol 20no 4 pp 419ndash431 2013

ndash005

0

005

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

ndash01ndash005

0005

01

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(a)

ndash02

ndash01

0

01

02

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

ndash005

005

0

micro = 005micro = 075

wo

(b)

0 5 10 15 20 25 30

minus005

0

005

Disp

lace

men

t (m

)

t (s)

0 5 10 15 20 25 30t (s)

minus01

0

01

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(c)

minus002

0

002

004D

ispla

cem

ent (

m)

0 5 10 15 20 25 30t (s)

0

minus005

005

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30t (s)

micro = 005micro = 075

wo

(d)

Figure 15 e damping eect of dierent mass ratio TMDs on the displacement-time history of the main structure (T 20 s) and therelative displacement-time history between the TMD and the structure (wo denotes the displacement of a structure without a TMD) (a) ElCentro (b) Hachinohe (c) Northridge (d) Kobe

Shock and Vibration 15

[2] S Elias and VMatsagar ldquoResearch developments in vibrationcontrol of structures using passive tuned mass dampersrdquoAnnual Reviews in Control vol 44 pp 129ndash156 2017

[3] D Wang T K T Tse Y Zhou and Q Li ldquoStructural per-formance and cost analysis of wind-induced vibration controlschemes for a real super-tall buildingrdquo Structure and In-frastructure Engineering vol 11 no 8 pp 990ndash1011 2014

[4] N Longarini and M Zucca ldquoA chimneyrsquos seismic assessmentby a tuned mass damperrdquo Engineering Structures vol 79pp 290ndash296 2014

[5] L Tian and Y Zeng ldquoParametric study of tunedmass dampersfor long span transmission tower-line system under windloadsrdquo Shock and Vibration vol 2016 Article ID 496505611 pages 2016

[6] N Hoang Y Fujino and PWarnitchai ldquoOptimal tuned massdamper for seismic applications and practical design for-mulasrdquo Engineering Structures vol 30 no 3 pp 707ndash7152008

[7] J P Den Hartog Mechanical vibrations McGraw-Hill NewYork NY USA 1956

[8] G B Warburton ldquoOptimum absorber parameters for variouscombinations of response and excitation parametersrdquoEarthquake Engineering amp Structural Dynamics vol 10 no 3pp 381ndash401 1982

[9] H-C Tsai and G-C Lin ldquoExplicit formulae for optimumabsorber parameters for force-excited and viscously dampedsystemsrdquo Journal of Sound and Vibration vol 176 no 5pp 585ndash596 1994

[10] H-C Tsai and G-C Lin ldquoOptimum tuned-mass dampers forminimizing steady-state response of support-excited anddamped systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 22 no 11 pp 957ndash973 1993

[11] S V Bakre and R S Jangid ldquoOptimum parameters of tunedmass damper for damped main systemrdquo Structural Controland Health Monitoring vol 14 no 3 pp 448ndash470 2007

[12] C C Lin C M Hu J F Wang and R Y Hu ldquoVibrationcontrol effectiveness of passive tuned mass dampersrdquo Journalof the Chinese Institute of Engineers vol 17 pp 367ndash376 1994

[13] A Y T Leung and H Zhang ldquoParticle swarm optimization oftuned mass dampersrdquo Engineering Structures vol 31 no 3pp 715ndash728 2009

[14] M Q Feng and A Mita ldquoVibration control of tall buildingsusing mega SubConfigurationrdquo Journal of Engineering Me-chanics vol 121 no 10 pp 1082ndash1088 1995

[15] X X Li P Tan X J Li and A W Liu ldquoMechanism analysisand parameter optimisation of mega-sub-isolation systemrdquoShock and Vibration vol 2016 p 12 2016

[16] A Reggio and M D Angelis ldquoOptimal energy-based seismicdesign of non-conventional tuned mass damper (TMD)implemented via inter-story isolationrdquo Earthquake Engi-neering amp Structural Dynamics vol 44 no 10 pp 1623ndash16422015

[17] S J Wang B H Lee W C Chuang and K C ChangldquoOptimal dynamic characteristic control approach forbuilding mass damper designrdquo Earthquake Engineering ampStructural Dynamics vol 47 no 3 2017

[18] H Anajafi and R A Medina ldquoPartial mass isolation systemfor seismic vibration control of buildingsrdquo Structural Controlamp Health Monitoring vol 25 no 2 article e2088 2017

[19] K Yuan M S He and Y M Li ldquoShaking table tests forenergy-dissipation steel frame structures with infilled wallMTMDrdquo Journal of Vibration and Shock vol 33 no 11pp 200ndash207 2014

[20] R Ding M X Tao M Zhou and J G Nie ldquoSeismic behaviorof RC structures with absence of floor slab constraints andlarge mass turbine as a non-conventional TMD a case studyrdquoBulletin of Earthquake Engineering vol 13 no 11 pp 3401ndash3422 2015

[21] L Y Peng Y J Kang Z R Lai and Y K Deng ldquoOptimisationand damping performance of a coal-fired power plantbuilding equipped with multiple coal bucket dampersrdquo Ad-vances in Civil Engineering vol 2018 p 19 2018

[22] Z Shu S Li J Zhang and M He ldquoOptimum seismic designof a power plant building with pendulum tuned mass dampersystem by its heavy suspended bucketsrdquo Engineering Struc-tures vol 136 pp 114ndash132 2017

[23] K Dai B Li J Wang et al ldquoOptimal probability-based partialmass isolation of elevated coal scuttle in thermal power plantbuildingrdquo Structural Design of Tall and Special Buildingsvol 27 no 11 article e1477 2018

[24] M De Angelis S Perno and A Reggio ldquoDynamic responseand optimal design of structures with large mass ratio TMDrdquoEarthquake Engineering amp Structural Dynamics vol 41 no 1pp 41ndash60 2015

[25] K Kanai ldquoSemi-empirical formula for the seismic charac-teristics of the groundrdquo Bulletin of the Earthquake ResearchInstitute gte University of Tokyo vol 35 pp 309ndash325 1957

[26] Code for seismic design of buildings GB50011-2010 BeijingChina 2010

[27] B F Spencer R E Christenson and S J Dyke ldquoNextgeneration benchmark control problem for seismically excitedbuildingsrdquo in Proceedings of the 2nd World Conference onStructural Control pp 1351ndash1360 Kyoto Japan June 1998

16 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Table 3 Maximum peak value of relative displacement between a TMD with optimal parameters and the main structure (cm)

T (s) μ El Centro Hachinohe Northridge Kobe Mean

05

005 436 296 381 312 356010 268 214 298 22 25050 214 254 335 191 249075 183 26 275 279 249100 224 274 224 337 265

10

005 818 883 1039 698 859010 353 483 512 598 487050 341 535 376 588 46075 391 59 355 546 47100 356 494 358 502 428

20

005 1477 209 866 1834 1567010 794 1197 554 1042 897050 53 863 417 858 667075 394 795 345 705 56100 389 846 332 629 549

30

005 1509 3529 666 1993 1925010 682 13 378 889 812050 491 1128 329 654 651075 463 1037 293 553 586100 509 1125 262 536 608

minus002

0

002

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(a)

minus002

0

002

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

micro = 005micro = 075

wo

(b)

Figure 14 Continued

Shock and Vibration 13

maximum is 4622 which is usually mainly due tothe dierence in the frequency relationship betweenthe structure and the seismic waves

Table 3 shows the statistical results of the relative dis-placement between the TMD and main structure underdierent time history analysis conditions It can be seen thatthe relative displacement between the TMD and the mainstructure is inversely proportional to the mass ratio of theTMD that is when a large mass ratio TMD is used thedisplacement response of the structure is eectively con-trolled and the displacement stroke of the TMD is clearlyreduced which reduces the requirements for the elasticcomponents and the damping components used to constructthe TMD

e displacement time histories of the structures withperiods of 10 and 20 s and the relative displacement timehistories between the TMD and the structures are shown inFigures 14 and 15 respectively e schemes of mass ratiosof 005 and 075 are compared It can be clearly seen that theeect of a TMD in controlling the structural response andthe TMD displacement stroke is more obvious for the TMDwith a mass ratio of 075 than for the one with a mass ratio of005

In summary the large mass ratio TMD has a moresignicant eect in seismic control of the main structurethan the small mass ratio TMD

6 Conclusions

In order to control the dynamic response and improve theaseismic performance of a structure a large mass ratio TMDdamping system is formed by using the equipment in thebuilding structure or relying on new structural forms e

existing optimal parameter tting formula is not applicableto large mass ratio TMDs so it is revised by numericaloptimisation and curve tting and the dynamic time historyanalysis method is used to study the eect of vibrationdamping control of large mass ratio TMDs e followingconclusions are obtained

Compared with the traditional small mass ratio (lt005)TMD the large mass ratio (gt05) TMD has obvious ad-vantages in controlling the displacement response of themain structure e control eect is about 15sim325 timeshigher the damping eect of the structural displacementpeak can reach about 30 and the damping ratio of the rootmean square displacement can reach about 436 At thesame time the relative stroke between the TMD and themain structure can be reduced with up to 30sim65 which ishighly benecial to the practical engineering design of TMDstructures

When the mass ratio of a TMD is relatively large (gt02)the results calculated by the existing tting formula diersignicantly from the actual optimal value and the calcu-lated values of the revised formula proposed in this paper areshown to be in good agreement with the actual optimalvalue In general the revised formula can be applied to bothtraditional small mass ratio and large mass ratio (le1) TMDsWhen the mass ratio is greater than 1 the optimal pa-rameters of TMD can also be obtained by the methodpresented in this paper

When the mass ratio is greater than 02 the optimalparameters of the stationary white noise random load andthat of the ltered white noise random load are more similarbut the dierence between the two is gradually increasedwith the increase of the TMD mass ratio For the optimalparameters of large mass ratio TMDs (gt02) the error is lessthan 005 when the ratio of the predominant frequency of the

0 5 10 15 20 25 30

minus002minus001

0001002

t (s)

0 5 10 15 20 25 30t (s)

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(c)

0 5 10 15 20 25 30t (s)

0 5 10 15 20 25 30t (s)

minus002

0

002

004

Disp

lace

men

t (m

)

minus01

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(d)

Figure 14 e damping eect of dierent mass ratio TMDs on the displacement-time history of the main structure (T10 s) and therelative displacement-time history between the TMD and the structure (wo denotes the displacement of a structure without a TMD) (a) ElCentro (b) Hachinohe (c) Northridge (d) Kobe

14 Shock and Vibration

base soil and the vibration frequency of the structure isgreater than 4 and the optimal parameters of the TMD canbe calculated by the tting formula proposed in this paperUnder other conditions it is suggested to use an optimi-sation method to determine the optimal value of TMDparameters

At present the actual engineering projects with largemass ratio TMD damping systems are less prominent buttheir aseismic advantages will bring a broad range of benetsfor research and practice

Data Availability

e data used to support the ndings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no concurrenicts of interestregarding the publication of this article

Acknowledgments

is research was supported by Grant no 51478023 from theNational Natural Science Foundation of China

References

[1] M Gutierrez Soto and H Adeli ldquoTuned mass dampersrdquoArchives of Computational Methods in Engineering vol 20no 4 pp 419ndash431 2013

ndash005

0

005

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

ndash01ndash005

0005

01

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(a)

ndash02

ndash01

0

01

02

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

ndash005

005

0

micro = 005micro = 075

wo

(b)

0 5 10 15 20 25 30

minus005

0

005

Disp

lace

men

t (m

)

t (s)

0 5 10 15 20 25 30t (s)

minus01

0

01

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(c)

minus002

0

002

004D

ispla

cem

ent (

m)

0 5 10 15 20 25 30t (s)

0

minus005

005

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30t (s)

micro = 005micro = 075

wo

(d)

Figure 15 e damping eect of dierent mass ratio TMDs on the displacement-time history of the main structure (T 20 s) and therelative displacement-time history between the TMD and the structure (wo denotes the displacement of a structure without a TMD) (a) ElCentro (b) Hachinohe (c) Northridge (d) Kobe

Shock and Vibration 15

[2] S Elias and VMatsagar ldquoResearch developments in vibrationcontrol of structures using passive tuned mass dampersrdquoAnnual Reviews in Control vol 44 pp 129ndash156 2017

[3] D Wang T K T Tse Y Zhou and Q Li ldquoStructural per-formance and cost analysis of wind-induced vibration controlschemes for a real super-tall buildingrdquo Structure and In-frastructure Engineering vol 11 no 8 pp 990ndash1011 2014

[4] N Longarini and M Zucca ldquoA chimneyrsquos seismic assessmentby a tuned mass damperrdquo Engineering Structures vol 79pp 290ndash296 2014

[5] L Tian and Y Zeng ldquoParametric study of tunedmass dampersfor long span transmission tower-line system under windloadsrdquo Shock and Vibration vol 2016 Article ID 496505611 pages 2016

[6] N Hoang Y Fujino and PWarnitchai ldquoOptimal tuned massdamper for seismic applications and practical design for-mulasrdquo Engineering Structures vol 30 no 3 pp 707ndash7152008

[7] J P Den Hartog Mechanical vibrations McGraw-Hill NewYork NY USA 1956

[8] G B Warburton ldquoOptimum absorber parameters for variouscombinations of response and excitation parametersrdquoEarthquake Engineering amp Structural Dynamics vol 10 no 3pp 381ndash401 1982

[9] H-C Tsai and G-C Lin ldquoExplicit formulae for optimumabsorber parameters for force-excited and viscously dampedsystemsrdquo Journal of Sound and Vibration vol 176 no 5pp 585ndash596 1994

[10] H-C Tsai and G-C Lin ldquoOptimum tuned-mass dampers forminimizing steady-state response of support-excited anddamped systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 22 no 11 pp 957ndash973 1993

[11] S V Bakre and R S Jangid ldquoOptimum parameters of tunedmass damper for damped main systemrdquo Structural Controland Health Monitoring vol 14 no 3 pp 448ndash470 2007

[12] C C Lin C M Hu J F Wang and R Y Hu ldquoVibrationcontrol effectiveness of passive tuned mass dampersrdquo Journalof the Chinese Institute of Engineers vol 17 pp 367ndash376 1994

[13] A Y T Leung and H Zhang ldquoParticle swarm optimization oftuned mass dampersrdquo Engineering Structures vol 31 no 3pp 715ndash728 2009

[14] M Q Feng and A Mita ldquoVibration control of tall buildingsusing mega SubConfigurationrdquo Journal of Engineering Me-chanics vol 121 no 10 pp 1082ndash1088 1995

[15] X X Li P Tan X J Li and A W Liu ldquoMechanism analysisand parameter optimisation of mega-sub-isolation systemrdquoShock and Vibration vol 2016 p 12 2016

[16] A Reggio and M D Angelis ldquoOptimal energy-based seismicdesign of non-conventional tuned mass damper (TMD)implemented via inter-story isolationrdquo Earthquake Engi-neering amp Structural Dynamics vol 44 no 10 pp 1623ndash16422015

[17] S J Wang B H Lee W C Chuang and K C ChangldquoOptimal dynamic characteristic control approach forbuilding mass damper designrdquo Earthquake Engineering ampStructural Dynamics vol 47 no 3 2017

[18] H Anajafi and R A Medina ldquoPartial mass isolation systemfor seismic vibration control of buildingsrdquo Structural Controlamp Health Monitoring vol 25 no 2 article e2088 2017

[19] K Yuan M S He and Y M Li ldquoShaking table tests forenergy-dissipation steel frame structures with infilled wallMTMDrdquo Journal of Vibration and Shock vol 33 no 11pp 200ndash207 2014

[20] R Ding M X Tao M Zhou and J G Nie ldquoSeismic behaviorof RC structures with absence of floor slab constraints andlarge mass turbine as a non-conventional TMD a case studyrdquoBulletin of Earthquake Engineering vol 13 no 11 pp 3401ndash3422 2015

[21] L Y Peng Y J Kang Z R Lai and Y K Deng ldquoOptimisationand damping performance of a coal-fired power plantbuilding equipped with multiple coal bucket dampersrdquo Ad-vances in Civil Engineering vol 2018 p 19 2018

[22] Z Shu S Li J Zhang and M He ldquoOptimum seismic designof a power plant building with pendulum tuned mass dampersystem by its heavy suspended bucketsrdquo Engineering Struc-tures vol 136 pp 114ndash132 2017

[23] K Dai B Li J Wang et al ldquoOptimal probability-based partialmass isolation of elevated coal scuttle in thermal power plantbuildingrdquo Structural Design of Tall and Special Buildingsvol 27 no 11 article e1477 2018

[24] M De Angelis S Perno and A Reggio ldquoDynamic responseand optimal design of structures with large mass ratio TMDrdquoEarthquake Engineering amp Structural Dynamics vol 41 no 1pp 41ndash60 2015

[25] K Kanai ldquoSemi-empirical formula for the seismic charac-teristics of the groundrdquo Bulletin of the Earthquake ResearchInstitute gte University of Tokyo vol 35 pp 309ndash325 1957

[26] Code for seismic design of buildings GB50011-2010 BeijingChina 2010

[27] B F Spencer R E Christenson and S J Dyke ldquoNextgeneration benchmark control problem for seismically excitedbuildingsrdquo in Proceedings of the 2nd World Conference onStructural Control pp 1351ndash1360 Kyoto Japan June 1998

16 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

maximum is 4622 which is usually mainly due tothe dierence in the frequency relationship betweenthe structure and the seismic waves

Table 3 shows the statistical results of the relative dis-placement between the TMD and main structure underdierent time history analysis conditions It can be seen thatthe relative displacement between the TMD and the mainstructure is inversely proportional to the mass ratio of theTMD that is when a large mass ratio TMD is used thedisplacement response of the structure is eectively con-trolled and the displacement stroke of the TMD is clearlyreduced which reduces the requirements for the elasticcomponents and the damping components used to constructthe TMD

e displacement time histories of the structures withperiods of 10 and 20 s and the relative displacement timehistories between the TMD and the structures are shown inFigures 14 and 15 respectively e schemes of mass ratiosof 005 and 075 are compared It can be clearly seen that theeect of a TMD in controlling the structural response andthe TMD displacement stroke is more obvious for the TMDwith a mass ratio of 075 than for the one with a mass ratio of005

In summary the large mass ratio TMD has a moresignicant eect in seismic control of the main structurethan the small mass ratio TMD

6 Conclusions

In order to control the dynamic response and improve theaseismic performance of a structure a large mass ratio TMDdamping system is formed by using the equipment in thebuilding structure or relying on new structural forms e

existing optimal parameter tting formula is not applicableto large mass ratio TMDs so it is revised by numericaloptimisation and curve tting and the dynamic time historyanalysis method is used to study the eect of vibrationdamping control of large mass ratio TMDs e followingconclusions are obtained

Compared with the traditional small mass ratio (lt005)TMD the large mass ratio (gt05) TMD has obvious ad-vantages in controlling the displacement response of themain structure e control eect is about 15sim325 timeshigher the damping eect of the structural displacementpeak can reach about 30 and the damping ratio of the rootmean square displacement can reach about 436 At thesame time the relative stroke between the TMD and themain structure can be reduced with up to 30sim65 which ishighly benecial to the practical engineering design of TMDstructures

When the mass ratio of a TMD is relatively large (gt02)the results calculated by the existing tting formula diersignicantly from the actual optimal value and the calcu-lated values of the revised formula proposed in this paper areshown to be in good agreement with the actual optimalvalue In general the revised formula can be applied to bothtraditional small mass ratio and large mass ratio (le1) TMDsWhen the mass ratio is greater than 1 the optimal pa-rameters of TMD can also be obtained by the methodpresented in this paper

When the mass ratio is greater than 02 the optimalparameters of the stationary white noise random load andthat of the ltered white noise random load are more similarbut the dierence between the two is gradually increasedwith the increase of the TMD mass ratio For the optimalparameters of large mass ratio TMDs (gt02) the error is lessthan 005 when the ratio of the predominant frequency of the

0 5 10 15 20 25 30

minus002minus001

0001002

t (s)

0 5 10 15 20 25 30t (s)

Disp

lace

men

t (m

)

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(c)

0 5 10 15 20 25 30t (s)

0 5 10 15 20 25 30t (s)

minus002

0

002

004

Disp

lace

men

t (m

)

minus01

minus005

0

005

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(d)

Figure 14 e damping eect of dierent mass ratio TMDs on the displacement-time history of the main structure (T10 s) and therelative displacement-time history between the TMD and the structure (wo denotes the displacement of a structure without a TMD) (a) ElCentro (b) Hachinohe (c) Northridge (d) Kobe

14 Shock and Vibration

base soil and the vibration frequency of the structure isgreater than 4 and the optimal parameters of the TMD canbe calculated by the tting formula proposed in this paperUnder other conditions it is suggested to use an optimi-sation method to determine the optimal value of TMDparameters

At present the actual engineering projects with largemass ratio TMD damping systems are less prominent buttheir aseismic advantages will bring a broad range of benetsfor research and practice

Data Availability

e data used to support the ndings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no concurrenicts of interestregarding the publication of this article

Acknowledgments

is research was supported by Grant no 51478023 from theNational Natural Science Foundation of China

References

[1] M Gutierrez Soto and H Adeli ldquoTuned mass dampersrdquoArchives of Computational Methods in Engineering vol 20no 4 pp 419ndash431 2013

ndash005

0

005

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

ndash01ndash005

0005

01

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(a)

ndash02

ndash01

0

01

02

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

ndash005

005

0

micro = 005micro = 075

wo

(b)

0 5 10 15 20 25 30

minus005

0

005

Disp

lace

men

t (m

)

t (s)

0 5 10 15 20 25 30t (s)

minus01

0

01

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(c)

minus002

0

002

004D

ispla

cem

ent (

m)

0 5 10 15 20 25 30t (s)

0

minus005

005

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30t (s)

micro = 005micro = 075

wo

(d)

Figure 15 e damping eect of dierent mass ratio TMDs on the displacement-time history of the main structure (T 20 s) and therelative displacement-time history between the TMD and the structure (wo denotes the displacement of a structure without a TMD) (a) ElCentro (b) Hachinohe (c) Northridge (d) Kobe

Shock and Vibration 15

[2] S Elias and VMatsagar ldquoResearch developments in vibrationcontrol of structures using passive tuned mass dampersrdquoAnnual Reviews in Control vol 44 pp 129ndash156 2017

[3] D Wang T K T Tse Y Zhou and Q Li ldquoStructural per-formance and cost analysis of wind-induced vibration controlschemes for a real super-tall buildingrdquo Structure and In-frastructure Engineering vol 11 no 8 pp 990ndash1011 2014

[4] N Longarini and M Zucca ldquoA chimneyrsquos seismic assessmentby a tuned mass damperrdquo Engineering Structures vol 79pp 290ndash296 2014

[5] L Tian and Y Zeng ldquoParametric study of tunedmass dampersfor long span transmission tower-line system under windloadsrdquo Shock and Vibration vol 2016 Article ID 496505611 pages 2016

[6] N Hoang Y Fujino and PWarnitchai ldquoOptimal tuned massdamper for seismic applications and practical design for-mulasrdquo Engineering Structures vol 30 no 3 pp 707ndash7152008

[7] J P Den Hartog Mechanical vibrations McGraw-Hill NewYork NY USA 1956

[8] G B Warburton ldquoOptimum absorber parameters for variouscombinations of response and excitation parametersrdquoEarthquake Engineering amp Structural Dynamics vol 10 no 3pp 381ndash401 1982

[9] H-C Tsai and G-C Lin ldquoExplicit formulae for optimumabsorber parameters for force-excited and viscously dampedsystemsrdquo Journal of Sound and Vibration vol 176 no 5pp 585ndash596 1994

[10] H-C Tsai and G-C Lin ldquoOptimum tuned-mass dampers forminimizing steady-state response of support-excited anddamped systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 22 no 11 pp 957ndash973 1993

[11] S V Bakre and R S Jangid ldquoOptimum parameters of tunedmass damper for damped main systemrdquo Structural Controland Health Monitoring vol 14 no 3 pp 448ndash470 2007

[12] C C Lin C M Hu J F Wang and R Y Hu ldquoVibrationcontrol effectiveness of passive tuned mass dampersrdquo Journalof the Chinese Institute of Engineers vol 17 pp 367ndash376 1994

[13] A Y T Leung and H Zhang ldquoParticle swarm optimization oftuned mass dampersrdquo Engineering Structures vol 31 no 3pp 715ndash728 2009

[14] M Q Feng and A Mita ldquoVibration control of tall buildingsusing mega SubConfigurationrdquo Journal of Engineering Me-chanics vol 121 no 10 pp 1082ndash1088 1995

[15] X X Li P Tan X J Li and A W Liu ldquoMechanism analysisand parameter optimisation of mega-sub-isolation systemrdquoShock and Vibration vol 2016 p 12 2016

[16] A Reggio and M D Angelis ldquoOptimal energy-based seismicdesign of non-conventional tuned mass damper (TMD)implemented via inter-story isolationrdquo Earthquake Engi-neering amp Structural Dynamics vol 44 no 10 pp 1623ndash16422015

[17] S J Wang B H Lee W C Chuang and K C ChangldquoOptimal dynamic characteristic control approach forbuilding mass damper designrdquo Earthquake Engineering ampStructural Dynamics vol 47 no 3 2017

[18] H Anajafi and R A Medina ldquoPartial mass isolation systemfor seismic vibration control of buildingsrdquo Structural Controlamp Health Monitoring vol 25 no 2 article e2088 2017

[19] K Yuan M S He and Y M Li ldquoShaking table tests forenergy-dissipation steel frame structures with infilled wallMTMDrdquo Journal of Vibration and Shock vol 33 no 11pp 200ndash207 2014

[20] R Ding M X Tao M Zhou and J G Nie ldquoSeismic behaviorof RC structures with absence of floor slab constraints andlarge mass turbine as a non-conventional TMD a case studyrdquoBulletin of Earthquake Engineering vol 13 no 11 pp 3401ndash3422 2015

[21] L Y Peng Y J Kang Z R Lai and Y K Deng ldquoOptimisationand damping performance of a coal-fired power plantbuilding equipped with multiple coal bucket dampersrdquo Ad-vances in Civil Engineering vol 2018 p 19 2018

[22] Z Shu S Li J Zhang and M He ldquoOptimum seismic designof a power plant building with pendulum tuned mass dampersystem by its heavy suspended bucketsrdquo Engineering Struc-tures vol 136 pp 114ndash132 2017

[23] K Dai B Li J Wang et al ldquoOptimal probability-based partialmass isolation of elevated coal scuttle in thermal power plantbuildingrdquo Structural Design of Tall and Special Buildingsvol 27 no 11 article e1477 2018

[24] M De Angelis S Perno and A Reggio ldquoDynamic responseand optimal design of structures with large mass ratio TMDrdquoEarthquake Engineering amp Structural Dynamics vol 41 no 1pp 41ndash60 2015

[25] K Kanai ldquoSemi-empirical formula for the seismic charac-teristics of the groundrdquo Bulletin of the Earthquake ResearchInstitute gte University of Tokyo vol 35 pp 309ndash325 1957

[26] Code for seismic design of buildings GB50011-2010 BeijingChina 2010

[27] B F Spencer R E Christenson and S J Dyke ldquoNextgeneration benchmark control problem for seismically excitedbuildingsrdquo in Proceedings of the 2nd World Conference onStructural Control pp 1351ndash1360 Kyoto Japan June 1998

16 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

base soil and the vibration frequency of the structure isgreater than 4 and the optimal parameters of the TMD canbe calculated by the tting formula proposed in this paperUnder other conditions it is suggested to use an optimi-sation method to determine the optimal value of TMDparameters

At present the actual engineering projects with largemass ratio TMD damping systems are less prominent buttheir aseismic advantages will bring a broad range of benetsfor research and practice

Data Availability

e data used to support the ndings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no concurrenicts of interestregarding the publication of this article

Acknowledgments

is research was supported by Grant no 51478023 from theNational Natural Science Foundation of China

References

[1] M Gutierrez Soto and H Adeli ldquoTuned mass dampersrdquoArchives of Computational Methods in Engineering vol 20no 4 pp 419ndash431 2013

ndash005

0

005

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

ndash01ndash005

0005

01

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(a)

ndash02

ndash01

0

01

02

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30 35t (s)

0 5 10 15 20 25 30 35t (s)

Disp

lace

men

t (m

)

ndash005

005

0

micro = 005micro = 075

wo

(b)

0 5 10 15 20 25 30

minus005

0

005

Disp

lace

men

t (m

)

t (s)

0 5 10 15 20 25 30t (s)

minus01

0

01

TMD

-disp

lace

men

t (m

)

micro = 005micro = 075

wo

(c)

minus002

0

002

004D

ispla

cem

ent (

m)

0 5 10 15 20 25 30t (s)

0

minus005

005

TMD

-disp

lace

men

t (m

)

0 5 10 15 20 25 30t (s)

micro = 005micro = 075

wo

(d)

Figure 15 e damping eect of dierent mass ratio TMDs on the displacement-time history of the main structure (T 20 s) and therelative displacement-time history between the TMD and the structure (wo denotes the displacement of a structure without a TMD) (a) ElCentro (b) Hachinohe (c) Northridge (d) Kobe

Shock and Vibration 15

[2] S Elias and VMatsagar ldquoResearch developments in vibrationcontrol of structures using passive tuned mass dampersrdquoAnnual Reviews in Control vol 44 pp 129ndash156 2017

[3] D Wang T K T Tse Y Zhou and Q Li ldquoStructural per-formance and cost analysis of wind-induced vibration controlschemes for a real super-tall buildingrdquo Structure and In-frastructure Engineering vol 11 no 8 pp 990ndash1011 2014

[4] N Longarini and M Zucca ldquoA chimneyrsquos seismic assessmentby a tuned mass damperrdquo Engineering Structures vol 79pp 290ndash296 2014

[5] L Tian and Y Zeng ldquoParametric study of tunedmass dampersfor long span transmission tower-line system under windloadsrdquo Shock and Vibration vol 2016 Article ID 496505611 pages 2016

[6] N Hoang Y Fujino and PWarnitchai ldquoOptimal tuned massdamper for seismic applications and practical design for-mulasrdquo Engineering Structures vol 30 no 3 pp 707ndash7152008

[7] J P Den Hartog Mechanical vibrations McGraw-Hill NewYork NY USA 1956

[8] G B Warburton ldquoOptimum absorber parameters for variouscombinations of response and excitation parametersrdquoEarthquake Engineering amp Structural Dynamics vol 10 no 3pp 381ndash401 1982

[9] H-C Tsai and G-C Lin ldquoExplicit formulae for optimumabsorber parameters for force-excited and viscously dampedsystemsrdquo Journal of Sound and Vibration vol 176 no 5pp 585ndash596 1994

[10] H-C Tsai and G-C Lin ldquoOptimum tuned-mass dampers forminimizing steady-state response of support-excited anddamped systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 22 no 11 pp 957ndash973 1993

[11] S V Bakre and R S Jangid ldquoOptimum parameters of tunedmass damper for damped main systemrdquo Structural Controland Health Monitoring vol 14 no 3 pp 448ndash470 2007

[12] C C Lin C M Hu J F Wang and R Y Hu ldquoVibrationcontrol effectiveness of passive tuned mass dampersrdquo Journalof the Chinese Institute of Engineers vol 17 pp 367ndash376 1994

[13] A Y T Leung and H Zhang ldquoParticle swarm optimization oftuned mass dampersrdquo Engineering Structures vol 31 no 3pp 715ndash728 2009

[14] M Q Feng and A Mita ldquoVibration control of tall buildingsusing mega SubConfigurationrdquo Journal of Engineering Me-chanics vol 121 no 10 pp 1082ndash1088 1995

[15] X X Li P Tan X J Li and A W Liu ldquoMechanism analysisand parameter optimisation of mega-sub-isolation systemrdquoShock and Vibration vol 2016 p 12 2016

[16] A Reggio and M D Angelis ldquoOptimal energy-based seismicdesign of non-conventional tuned mass damper (TMD)implemented via inter-story isolationrdquo Earthquake Engi-neering amp Structural Dynamics vol 44 no 10 pp 1623ndash16422015

[17] S J Wang B H Lee W C Chuang and K C ChangldquoOptimal dynamic characteristic control approach forbuilding mass damper designrdquo Earthquake Engineering ampStructural Dynamics vol 47 no 3 2017

[18] H Anajafi and R A Medina ldquoPartial mass isolation systemfor seismic vibration control of buildingsrdquo Structural Controlamp Health Monitoring vol 25 no 2 article e2088 2017

[19] K Yuan M S He and Y M Li ldquoShaking table tests forenergy-dissipation steel frame structures with infilled wallMTMDrdquo Journal of Vibration and Shock vol 33 no 11pp 200ndash207 2014

[20] R Ding M X Tao M Zhou and J G Nie ldquoSeismic behaviorof RC structures with absence of floor slab constraints andlarge mass turbine as a non-conventional TMD a case studyrdquoBulletin of Earthquake Engineering vol 13 no 11 pp 3401ndash3422 2015

[21] L Y Peng Y J Kang Z R Lai and Y K Deng ldquoOptimisationand damping performance of a coal-fired power plantbuilding equipped with multiple coal bucket dampersrdquo Ad-vances in Civil Engineering vol 2018 p 19 2018

[22] Z Shu S Li J Zhang and M He ldquoOptimum seismic designof a power plant building with pendulum tuned mass dampersystem by its heavy suspended bucketsrdquo Engineering Struc-tures vol 136 pp 114ndash132 2017

[23] K Dai B Li J Wang et al ldquoOptimal probability-based partialmass isolation of elevated coal scuttle in thermal power plantbuildingrdquo Structural Design of Tall and Special Buildingsvol 27 no 11 article e1477 2018

[24] M De Angelis S Perno and A Reggio ldquoDynamic responseand optimal design of structures with large mass ratio TMDrdquoEarthquake Engineering amp Structural Dynamics vol 41 no 1pp 41ndash60 2015

[25] K Kanai ldquoSemi-empirical formula for the seismic charac-teristics of the groundrdquo Bulletin of the Earthquake ResearchInstitute gte University of Tokyo vol 35 pp 309ndash325 1957

[26] Code for seismic design of buildings GB50011-2010 BeijingChina 2010

[27] B F Spencer R E Christenson and S J Dyke ldquoNextgeneration benchmark control problem for seismically excitedbuildingsrdquo in Proceedings of the 2nd World Conference onStructural Control pp 1351ndash1360 Kyoto Japan June 1998

16 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

[2] S Elias and VMatsagar ldquoResearch developments in vibrationcontrol of structures using passive tuned mass dampersrdquoAnnual Reviews in Control vol 44 pp 129ndash156 2017

[3] D Wang T K T Tse Y Zhou and Q Li ldquoStructural per-formance and cost analysis of wind-induced vibration controlschemes for a real super-tall buildingrdquo Structure and In-frastructure Engineering vol 11 no 8 pp 990ndash1011 2014

[4] N Longarini and M Zucca ldquoA chimneyrsquos seismic assessmentby a tuned mass damperrdquo Engineering Structures vol 79pp 290ndash296 2014

[5] L Tian and Y Zeng ldquoParametric study of tunedmass dampersfor long span transmission tower-line system under windloadsrdquo Shock and Vibration vol 2016 Article ID 496505611 pages 2016

[6] N Hoang Y Fujino and PWarnitchai ldquoOptimal tuned massdamper for seismic applications and practical design for-mulasrdquo Engineering Structures vol 30 no 3 pp 707ndash7152008

[7] J P Den Hartog Mechanical vibrations McGraw-Hill NewYork NY USA 1956

[8] G B Warburton ldquoOptimum absorber parameters for variouscombinations of response and excitation parametersrdquoEarthquake Engineering amp Structural Dynamics vol 10 no 3pp 381ndash401 1982

[9] H-C Tsai and G-C Lin ldquoExplicit formulae for optimumabsorber parameters for force-excited and viscously dampedsystemsrdquo Journal of Sound and Vibration vol 176 no 5pp 585ndash596 1994

[10] H-C Tsai and G-C Lin ldquoOptimum tuned-mass dampers forminimizing steady-state response of support-excited anddamped systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 22 no 11 pp 957ndash973 1993

[11] S V Bakre and R S Jangid ldquoOptimum parameters of tunedmass damper for damped main systemrdquo Structural Controland Health Monitoring vol 14 no 3 pp 448ndash470 2007

[12] C C Lin C M Hu J F Wang and R Y Hu ldquoVibrationcontrol effectiveness of passive tuned mass dampersrdquo Journalof the Chinese Institute of Engineers vol 17 pp 367ndash376 1994

[13] A Y T Leung and H Zhang ldquoParticle swarm optimization oftuned mass dampersrdquo Engineering Structures vol 31 no 3pp 715ndash728 2009

[14] M Q Feng and A Mita ldquoVibration control of tall buildingsusing mega SubConfigurationrdquo Journal of Engineering Me-chanics vol 121 no 10 pp 1082ndash1088 1995

[15] X X Li P Tan X J Li and A W Liu ldquoMechanism analysisand parameter optimisation of mega-sub-isolation systemrdquoShock and Vibration vol 2016 p 12 2016

[16] A Reggio and M D Angelis ldquoOptimal energy-based seismicdesign of non-conventional tuned mass damper (TMD)implemented via inter-story isolationrdquo Earthquake Engi-neering amp Structural Dynamics vol 44 no 10 pp 1623ndash16422015

[17] S J Wang B H Lee W C Chuang and K C ChangldquoOptimal dynamic characteristic control approach forbuilding mass damper designrdquo Earthquake Engineering ampStructural Dynamics vol 47 no 3 2017

[18] H Anajafi and R A Medina ldquoPartial mass isolation systemfor seismic vibration control of buildingsrdquo Structural Controlamp Health Monitoring vol 25 no 2 article e2088 2017

[19] K Yuan M S He and Y M Li ldquoShaking table tests forenergy-dissipation steel frame structures with infilled wallMTMDrdquo Journal of Vibration and Shock vol 33 no 11pp 200ndash207 2014

[20] R Ding M X Tao M Zhou and J G Nie ldquoSeismic behaviorof RC structures with absence of floor slab constraints andlarge mass turbine as a non-conventional TMD a case studyrdquoBulletin of Earthquake Engineering vol 13 no 11 pp 3401ndash3422 2015

[21] L Y Peng Y J Kang Z R Lai and Y K Deng ldquoOptimisationand damping performance of a coal-fired power plantbuilding equipped with multiple coal bucket dampersrdquo Ad-vances in Civil Engineering vol 2018 p 19 2018

[22] Z Shu S Li J Zhang and M He ldquoOptimum seismic designof a power plant building with pendulum tuned mass dampersystem by its heavy suspended bucketsrdquo Engineering Struc-tures vol 136 pp 114ndash132 2017

[23] K Dai B Li J Wang et al ldquoOptimal probability-based partialmass isolation of elevated coal scuttle in thermal power plantbuildingrdquo Structural Design of Tall and Special Buildingsvol 27 no 11 article e1477 2018

[24] M De Angelis S Perno and A Reggio ldquoDynamic responseand optimal design of structures with large mass ratio TMDrdquoEarthquake Engineering amp Structural Dynamics vol 41 no 1pp 41ndash60 2015

[25] K Kanai ldquoSemi-empirical formula for the seismic charac-teristics of the groundrdquo Bulletin of the Earthquake ResearchInstitute gte University of Tokyo vol 35 pp 309ndash325 1957

[26] Code for seismic design of buildings GB50011-2010 BeijingChina 2010

[27] B F Spencer R E Christenson and S J Dyke ldquoNextgeneration benchmark control problem for seismically excitedbuildingsrdquo in Proceedings of the 2nd World Conference onStructural Control pp 1351ndash1360 Kyoto Japan June 1998

16 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom