ANewMethodtoCalculateAdditionalDamping … · 2020. 8. 24. · the aseismic design of structures conﬁgured with VFDs, in which the mechanical characteristics of the VFDs are easily

Research ArticleA New Method to Calculate Additional DampingRatio considering the Effect of Excitation Frequency

Weizhi Xu Dongsheng Du Shuguang Wang and Weiwei Li

College of Civil Engineering Nanjing Tech University Nanjing 211816 China

Correspondence should be addressed to Dongsheng Du ddshynjtecheducn

Received 27 December 2019 Revised 10 July 2020 Accepted 29 July 2020 Published 24 August 2020

Academic Editor John Mander

Copyright copy 2020 Weizhi Xu et al )is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

)e additional damping ratio (ADR) is an important indicator for evaluating the damping effect of structures with energy-dissipation devices Most existing methods for determining the ADR require an analysis of the structural dynamic response andcomplex iterative calculations An innovative simplified calculationmethod for determining the ADR of a structure supplementedby nonlinear viscous dampers is proposed )is method does not require the dynamic response of the structure to be calculatedand only requires the structural characteristics excitation frequency and damper parameters In this study several typicalcalculation methods for the ADR were analysed )en a calculation formula for the ADR was derived with consideration ofharmonic excitation under the condition where the excitation frequency is equal to the structural natural frequency withoutcalculation of the structural dynamic response or an iterative process)e effect of the excitation frequency on the calculated valueof the ADR with different damping exponents was studied Accordingly the response spectrum average period (RSAP) wasconsidered as the excitation period of groundmotion to evaluate the excitation frequency and a simplified calculation method forthe ADR considering the effect of the excitation frequency characterised by the RSAP of the ground motion was establishedFinally the accuracy and effectiveness of the proposed method were verified by comparison with ADRs calculated usingother methods

1 Introduction

In the past two decades many major earthquakes have oc-curred in the world which have caused many casualties andconsiderable property damage [1ndash3] Both structural andnonstructural failures induced by seismic excitation are re-sponsible for the losses )erefore it is vital to improve theseismic performance of building structures )e introductionof passive energy-dissipation technology to building struc-tures provides an effective solution to increase the structuraldamping and mitigate the effect of seismic action and theimportance of damping has been widely recognised in thefield of structural dynamics Effective damping has three maincomponents [4] inherent damping supplemental dampingprovided by the dampers the hysteretic damping which isrelated to the nonlinear behaviour of the structure

Microscopic internal friction local plastic deformationand plastic flow in a range of stresses within the apparent

elastic limit account for a major proportion of the energydissipation due to the inherent damping [5] In numericalanalysis it is important to select an appropriate inherentdamping model A few damping models have been proposedand widely used in numerical analysis eg viscous damping(VD) Rayleigh damping (RD) mass-proportional damping(MD) and stiffness-proportional damping (SD) )e RDmodel which consists of MD and SD is widely used incommercial finite-element platforms because of its mathe-matical simplicity Huang [5] reported that the desiredenergy-dissipation rate was achieved at only two targetfrequencies and that the mass-proportional term appliesdamping to rigid-body modes which cannot dissipate en-ergy in reality To solve this problem a versatile frequency-insensitive damping modelling method was developedwhich can achieve approximately uniform energy dissipa-tion in the specified frequency range and make the dampingorthogonal to the rigid-body motion In a few studies [6ndash8]

HindawiAdvances in Civil EngineeringVolume 2020 Article ID 3172982 15 pageshttpsdoiorg10115520203172982

the damping leakage phenomenon in structural systems wasinvestigated revealing that the damping leakage can result ina significant underestimation of the first-mode responseandor higher-mode responses To mitigate the problemsassociated with the improper damping models in the base-isolated structure Anajafi et al [6] proposed a dampingmodelling approach that involves removing the mass-pro-portional component of superstructure damping from theglobal damping matrix

In addition to the inherent damping the additionaldamping ratio (ADR) which consists of contributions fromvarious energy-dissipation devices (EDDs) has been utilisedin different types of structures to mitigate the structuralresponses induced by seismic excitation [9ndash16] )e viscousfluid damper (VFD)mdasha rate-dependent EDDmdashis favouredby engineers and increasingly used in engineering structuresto reduce excessive vibrations because of the absence ofsupplemental stiffness and a high energy-dissipation rate indifferent deformation stages Wu et al [17] evaluated theseismic performance of a nine-story steel frame supple-mented with five typical dampers under near-fault earth-quakes )e results indicated that the VFD functioned betterthan other four types of dampers even under high pulsevelocity amplitude (PVA) To quantitatively evaluate thedamping effects for structures supplemented with VFDs theADR was proposed Additionally in China the ADR is a keystructural parameter for engineers in the design processFirstly the target ADR is determined then the aseismicdesign is conducted via themode analysis response spectrummethod considering the total damping ratio the sum of theinherent damping ratio and the ADR Subsequently thedamper parameters and the layout are designed to achievethe target ADR )e design process determines the im-portance of estimating the ADR of the structures equippedwith EDDs

Recently scholars have proposed various calculationmethods for the ADR Lee et al [18] proposed an evaluationmethod for the equivalent damping ratio for systems con-figured with dampers In that study the Lyapunov functionwas first defined in the form of the modal energy then theRiccati matrix and damping ratio parameters were derivedFinally a formula for the equivalent damping ratio wasobtained by analogising the viscous modal damping ratioCharney et al [19] used the modal strain energymethod freevibration logarithmic decay method and eigenvectormethod for complex eigenvalues to analyse the equivalentVD ratio of a single-story single-bay frame )e resultsindicated that higher flexibility of the nodes corresponded toa larger phase difference between the deformation velocityand the relative horizontal velocity of the interlayer devicesHowever this conclusion was only applicable to the single-layer single-span structure Occhiuzzi et al [20] proposed amethod for calculating the modal damping ratio using anequation for the dynamic system state An analysis revealedthat the first vibration mode significantly affected thedamping ratio whereas the effects of the higher-order vi-bration modes were negligible )e aforementioned calcu-lation methods which involve complex theoretical formulasand large amounts of calculations are difficult for engineers

to apply )erefore a simpler direct method needs to bedeveloped

Silvestri et al [21ndash23] proposed a practical method forthe aseismic design of structures configured with VFDs inwhich the mechanical characteristics of the VFDs are easilydetermined according to the performance objectives of thestructure However the calculation method adopted for theADR still cannot be widely used in practical engineeringDiotallevi et al [24] proposed a method for evaluating theADR according to the damping index )is method has theadvantage of realising direct evaluation of ADR withoutiteration but is limited to the evaluation of linear-elasticsystems Weng et al [25] proposed a method for calculatingthe equivalent ADR of the viscous damper according to thecode response spectrum Landi et al [26] proposed a sim-plified method for calculating the ADR according to thecapability and demand spectra Under the condition ofconfirmed yield acceleration and target displacement of theknown structure the corresponding ductility factorequivalent total damping ratio and significant period couldbe confirmed directly Kudu et al [27] compared the modaldamping ratios determined with consideration of themeasurement time frequency scope and sample rate Whenthe modal damping ratio changed significantly the naturalfrequency of the structure did not Ishimaru et al [28]presented an approach for calculating the optimal VD ratioand accumulated ductility of a structure configured with abilinear hysteretic damper and a dynamic quality damperwhich has a very high accuracy for calculating the accu-mulated ductility coefficient A method for calculating theADR based on the concept of effective modal dampingenergy dissipation was proposed by Weng et al [29] whichhas a clear physical significance and fully considers the time-varying characteristics of the effective damping ratio Itsadvantages include simple calculations and a high accuracyLandi et al [30] performed probabilistic seismic assessmentof RC structures with and without a damper using a sim-plified SAC Federal Emergency Management Agency (SAC-FEMA) program )e results showed that the method waseffective and could be used as a simplified alternative tononlinear dynamic analyses for probabilistic assessmentpurposes Love and Tait [31] proposed a simple method forpredicting the effective damping of a linear structure withnonlinear EDDs)is method exploits the characteristic thatthe output mean energy of the structure remains unchangedunder a wind load to make predictions However the errorbetween the prediction results of this method and the cal-culation results based on time-history analysis was signifi-cant He et al [32] proposed an effective finite-elementanalysis method for calculating the modal damping ratio ofcomplex materials and proved the effectiveness and accuracyof the method through a theoretical analysis and a com-parison with experimental results

Most of the aforementioned methods for calculating theADR require the calculation of the structural dynamic re-sponse and iteration making them cumbersome and in-convenient to use directly in practical engineeringAdditionally the solutions of the ADR for supplementalnonlinear VFDs based on the energy principle rarely

2 Advances in Civil Engineering

consider the effect of the excitation frequency on the ADRHence it is necessary to establish a calculation method forthe ADR that reflects the effect of the excitation frequency onthe dynamic response of the structure with VFDs

)e remainder of this paper is organised as followsSection 2 introduces two typical calculation methods for theADR the energy-ratio method (ERM) and the dampingindex method (DIM) Section 3 presents the proposedmethod for calculating the ADR under harmonic excitationconsidering the impact of the excitation frequency which istheoretically derived )en the response spectrum averageperiod (RSAP) is used to distinguish the frequency prop-erties of different ground motions Subsequently a simpli-fied calculationmethod for the ADR under groundmotion ispresented In Section 4 the accuracy and effectiveness of theproposed calculation method for the ADR are confirmedthrough a case study Conclusions and directions for im-provement of the proposed method are presented in Section5

2 Existing Methods for Calculating ADR

21 ERM Chopra [33] proposed a classical calculationequation for the ADR based on the principle of hysteresisenergy equivalence

ζsd wD

4πwSo

ωΩ

(1)

where wSo represents the maximum strain energy of thedamping structure with a viscous damper under the ex-pected displacement wD represents the energy consumed bythe viscous damper in one cycle with the expected defor-mation and ω and Ω represent the structural natural fre-quency and the excitation frequency respectively

)e damping force is given as follows

FD cαsgn( _u)| _u|α (2)

where cα is the damping coefficient corresponding to dif-ferent velocity exponent (α) values and sgn (x) is a signfunction

For a single-degree-of-freedom (SDOF) system sub-jected to a harmonic displacement excitation with thefunction of u(t) u0 sin Ωt the energy consumption of thenonlinear viscous damper is given as follows

wD 1113946 FDdu 1113946

2πΩ

0

cα| _u|1+αdt (3)

Integration of equation (3) yields

wD λcαu1+α0 Ω

α (4)

where λ is a constant related to αWhen α 1 λ π and the following equation can be

obtained

wD πc1u20Ω (5)

Linearisation of the nonlinear viscous damper is per-formed according to the equal dissipating energy of a linearviscous damper and the equivalent linear damping coeffi-cient c1 can be determined using the following equation

πc1u20Ω λcαu

1+α0 Ω

α (6)

By dividing both sides of equation (6) by 2mω the ADRof nonlinear viscous dampers can be calculated as follows

ζsd λπ

cα

2mω1Ωuo( 1113857

1minus α (7)

where m represents the mass of the structure)is classical method is easy to understand However

the structural dynamic response must be calculated to de-termine the maximum displacement and the calculationprocess is complex which is not convenient for practicalengineering

22DIM To overcome the shortcomings of the ERM Landi[25] proposed a method for evaluating the ADR directlyaccording to the damping index For an SDOF system underharmonic excitation the damping index ε is defined by thefollowing relationship

ε λπ

cα

2m

ω1minus 2α

Ω1minus α aαminus 10 (8)

where a0 represents the peak acceleration of harmonicexcitation

Under the resonance condition ieΩω equations (7)and (8) can be rewritten as follows

ζsd λπ

cα

2mω2minus α1

uo( 11138571minus α (9)

ε λπ

cα

2m

1ωα euroug01113872 1113873

αminus 1 (10)

respectively where euroug0 represents the peak acceleration ofthe seismic excitation

)e relationship between ADR ζsd and the dampingindex ε can be expressed as follows

ζsd εRαminus 1d (11)

where Rd (ω2u0 euroug0) reflecting the amplification of thedynamic response

)e acceleration amplification factor Ra can be com-puted via numerical analysis and the damping index ε can bedetermined according to the structural and input charac-teristics )en ADR ζsd is calculated using equation (11)Figure 1 shows the ADR-frequency ratio curves with dif-ferent damping indices (ε) and α 050 under harmonicexcitation and Figure 2 shows the ADR spectra for differentdamping indices (ε) and α 050 under the El Centro groundmotion recorded during the Imperial Valley Earthquake onMay 18 1940 According to the curves in Figures 1 and 2ADR ζsd is directly obtained when the damping index ε andfrequency ratio (Ωω) or natural period (T) of the structure

Advances in Civil Engineering 3

are determined It appears that the ADR can be determineddirectly without calculation of the structural responseFigures 1 and 2 were obtained via numerical analysis of thestructure with additional dampers

3 New Simplified Calculation Method for ADR

To calculate the ADR of a structure equipped with nonlinearviscous dampers more conveniently a simplified method is

proposed on the basis of the foregoing methods In theproposed method the structural response need not becalculated rather only the spectral characteristics of groundmotions must be considered

31 ADR of Damping System under Harmonic Excitation)e dynamic equation of the SDOF system with a lineardamper under harmonic excitation is

meurou + c1 _u + ku a0m sinΩt (12)

where c1 is the damping coefficient of the linear viscousdamper when the structural damping is 0 and a0 representsthe peak acceleration of harmonic excitation

)e complete solution of equation (12) [33] is

u(t) eminus ξ1ωt

A cosωDt + BsinωDt( 1113857 +(C sinΩt + D cosΩt)

(13)

where ωD ω

1 minus ζ211113969

represents the frequency of thestructure considering damping ω represents the naturalfrequency of the structure and ζ1 represents the dampingratio of the linear viscous damper

Considering the initial conditions u (0) 0 and _u(0) 0the following can be obtained

C ζ1Ωω

1113874 1113875 a0

ω21 minus (Ωω)2

1 minus (Ωω)21113960 11139612

+ 2ζ1(Ωω)21113960 11139612

D ζ1Ωω

1113874 1113875a0

ω2minus 2ζ(Ωω)

1 minus (Ωω)21113960 11139612

+ 2ζ1(Ωω)1113858 11138592

(14)

According to equation (13) the steady-state dynamicresponse us is calculated as follows

us us0 sin(Ωt + φ) (15)

where us0 C2 + D2

radicand tan(φ) DC

)e steady-state response is taken as the approximateresponse of the structure ie

u0 us0 (16)

Considering the equal energy consumption of linear andnonlinear dampers and the identical peak dynamic re-sponses of the SDOF for linear and nonlinear dampers theADRs of the two damping systems should be equal

ζ1 ζs d (17)

According to equations (7) (16) and (17) the ADR canbe determined as follows

ζsd λπ

cα

2mω1

ΩC2 + D2

radic( 1113857

1minus α (18)

)e ADR can be easily determined using equation (18)without calculating the response of the damping structurebut with iterative calculations Comparisons of the ADRscalculated using equation (18) and the ERM are presented inFigures 3 and 4 As shown when the frequency ratio Ωω

ndash El Centro ndash

ε = 20

ε = 5ε = 10ε = 15

ε = 25ε = 30ε = 35

ε = 40ε = 45ε = 50

00

100

80

90

60

70

40

50

20

30

10

005 10 15

T (s)

ζ sd (

)

20 25 30 35 40 45 50

Figure 2 ADR spectra for different values of ε under the excitationof the north-south component of El Centro ground motion(α 050)

ndash α = 050 ndash

ε = 5

ε = 10

ε = 15

ε = 20

ε = 25

ε = 30

00

100

80

60

40

20

005 10 15

Ωω

ζ sd (

)

20 25 30

Figure 1 Relationship between the ADR and the frequency ratiofor different damping indices (ε) under harmonic excitation


was le1 the calculation results of equation (18) agreed wellwith the results obtained using the ERM when the frequencyratio was gt1 the calculation results of equation (18) werelarger than those of the ERM owing to the impact of thetransient-state solution on the structural response indi-cating that the frequency ratio affected the difference be-tween the ADRs calculated using equation (18) and theERM

Using equation (18) the ADR can be determined withoutcalculating the dynamic response but iterative calculationsare required Considering that the results obtained via the twocalculation methods are similar particularly when Ωωfurther analysis is conducted for the resonance condition

When Ωω equation (15) is simplified as follows

us0 a0

2ω2ζ1 (19)

Substituting equation (19) into equation (6) and settingζ1 c12ωm yields

c1 λπ

1a0m

1113888 1113889

1minus α

cα⎡⎣ ⎤⎦

1α

(20)

)en by dividing both sides of equation (6) by 2mω theADR under the resonance condition is obtained

ζsdres 1

2mωλπ

1a0m

1113888 1113889

1minus α

cα⎡⎣ ⎤⎦

1α

(21)

Figure 5 presents the relationship curves of ζsdres and a0for different values of α and cα As shown the calculationresults of the two methods agreed well and the differencebetween them decreased with the increasing damping ex-ponent Additionally the ADR increased with the dampingcoefficient owing to the considerable energy dissipation

32 Simplified Calculation of ADR considering ExcitationSpectral Characteristics Because of the difference betweenthe ADRs calculated using equation (18) and the ERMdepended on the frequency ratio it was important to studythe variation of the ADR with different frequency ratiosFigure 6 shows the curves for the relationship between ζsdand Ωω obtained via the ERM

In Figure 6 the ADRs on the left and right sides of theresonance point exhibit different trends and the curve isapproximately parabolic when Ωωlt 1 and approximatelylinear when Ωωgt 1 )erefore the ζsd vs Ωω curve can bedefined as a piecewise function containing the structuralresonance point (1 ζsdres) as follows

ζsd

a1Ω

ω minus 11113874 1113875

2+ ζsdres

Ωωlt 1

ζsdresΩω

1

a2Ω

ω minus 11113874 1113875 + ζsdres

Ωωgt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)

where a1 and a2 are the coefficients that must be determinedvia fitting

Figures 7 and 8 present the curves of a1 and a2 underdifferent damping exponents (α) Numerical fitting revealedthat

a1 β1ec1α (23)

where β1 1096 and c1 ndash5489 and

a2 β2ec2α (24)

where β21002 and c2 ndash4649


00 05 10 15

Ωω

ζ sd (

)

(1 ζsd res)

20 25 30

20

15

10

5

0

ndash5

ERMEquation (18)

Figure 3 Comparison of the ADRs calculated using the twomethods with different frequency ratios (α 050)

ndash Ωω = 150 ndash

00 02 04

a0 (ms2)

ζ sd (

)

06 08 10

10

8

6

4

2

0

ERMEquation (18)

Figure 4 Comparison of the ADRs calculated using the twomethods with different excitation amplitudes (Ωω 150)


Serialno123456

100020003000400050006000

Dampingcoefficient

12

3

45

6

16

14

12

10

8

6

4

2

0

ndash α = 030 ndashζ sd

res (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(a)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

56

16

14

12

10

8

6

4

2

0


ζ sdre

s (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(b)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

5

6

8

7

6

5

4

3

2

1

0


ζ sdre

s (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(c)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

56

8

7

6

5

4

3

2

1

0


res (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(d)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

56

6

5

4

2

3

1

0


ζ sdre

s (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(e)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

56

5

4

3

2

1

0


ζ sdre

s (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(f )

Figure 5 ADRs calculated using the two methods for cα 1000 2000 3000 4000 5000 and 6000 kN sm (a) α 030 (b) α 040 (c)α 050 (d) α 060 (e) α 070 (f ) α 090


Equation (22) was used to determine the ADR afterconsidering the excitation spectral characteristic withoutcalculating the structural dynamic response Figure 9presents the relationship curve between ζsd and Ωω un-der different damping exponents (α) )e calculation resultsof the ERM equation (18) and equation (22) are comparedAs shown the calculation results of equation (18) were thelargest among the three calculation methods when Ωωgt 1however the calculation results of equation (22) and theERM agreed well with sufficient precision for practicalengineering

33 Calculation of ADR of Damping Structure under GroundMotion Excitation )e spectral characteristics of groundmotions affect the structural dynamic response as well as theADR However the foregoing simplified methods do notconsider the impact of the ground vibration frequency onthe ADR )us a calculation method for the ADR that doesnot require calculation of the structural dynamic responsebut considers the spectral characteristics of the groundmotions is derived

Considering the condition of structural resonance underharmonic excitation u0 is similar to a02ω2ζ If

uw euroug0

2ω2ζ1 (25)

where uw represents the assumed value of the peakdisplacement response which is related to the peakground acceleration (PGA) )ere is a relationship be-tween u0 and uw )e ratio between them is defined asf1 = u0uw and reflects the displacement amplificationbetween the nonresonant and resonant responses of thestructure

)en u0 is determined as follows

u0 f1 middot euroug0

2ω2ζ1 (26)

where euroug0 represents the seismic-excitation peak accelera-tion and ζ1 represents the ADR of the linear viscous damper

For a linear structural system subjected to a seismicexcitation the coefficient f1 is unrelated to the PGA but is


00 05 10 15

Ωω

ζ sd (

)

(1 ζsdres)

20

ERM

25 30

30

25

20

15

10

5

0

ndash5

(a)


(1 ζsdres)

ERM

00 05 10 15

Ωω

20 25 30

ζ sd (

)

20

15

10

5

0

ndash5

(b)


(1 ζsdres)

ERM

00 05 10 15

Ωω

20 25 30

ζ sd (

)

8

6

4

2

0

(c)


(1 ζsdres)

ERM

00 05 10 15

Ωω

20 25 30

ζ sd (

)

4

3

2

1

0

(d)

Figure 6 Relationship between ADR ζsd and the frequency ratio Ωω (a) α 030 (b) α 050 (c) α 070 (d) α 090


related to ADR (ζ1) )erefore f1 is designated as a unarypolynomial of ζ1

f1 ζ1( 1113857 b1ζ31 + b2ζ

21 + b3ζ1 + b4 (27)

where b1 b2 b3 and b4 are the coefficients of the polynomialwhich can be determined via fitting )e ADR of thestructure is typically lt05 thus considering a cubic poly-nomial can ensure an appropriate accuracy

Considering the different natural cycle frequencies (ω)of the structure a series of coordinate points (ζ1 f1) can beobtained through time-history analyses of the SDOF systemwith different ADRs (ζ1) Taking the El Centro seismicrecord of the Imperial Valley earthquake as an exampleconsidering the different natural cycle frequencies of thestructure ζ1 and corresponding f1 values can be obtainedas shown in Figure 10 Table 1 presents the fitting

coefficients for the unary polynomial of ζ1 )e fittingcoefficients with ω 539 are used in the case study inSection 4

By substituting equation (26) into equation (9) the ADRwith Ωω is obtained

ζsdres λπ

cα

2mω2minus α1

f1 euroug02ω2ζ11113872 11138731minus α (28)

According to equation (17) equation (28) can betransformed into

ζsdres λπ

cα

2mω2minus α1

f1 euroug02ω2ζsdres1113872 11138731minus α (29)

Equation (22) is used to determine the ADR whenΩneω

ζsd

a1Ωm

ω minus 11113874 1113875

2+ ζsdres

Ωm

ωlt 1

ζsdresΩm

ω 1

a2Ωm

ω minus 11113874 1113875 + ζsdres

Ωm

ωgt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(30)

where Ωm represents the average frequency of the seismicexcitation

)e average period Tm of a seismic wave can be calcu-lated as follows [29]

Tm 1113936iC

2i 1Fi( 1113857

1113936iC2i

(31)

where Ci is the Fourier peak coefficient and Fi represents thediscrete frequency of the fast Fourier transform

)en average frequency Ωm of the seismic input groundmotion acceleration is calculated as follows

Ωm 2πTm

(32)

Figures 11 and 12 present the curves of ADR (ζsd) vs thenatural vibration period of the structure (T) and the PGA(euroug0) with different damping coefficients (ca in kNmiddotsm) forthe El Centro ground motion obtained using the ERM andequations (29) and (30) Here the damping exponent wasα 050

As shown in the figures forΩω the calculation resultsof equation (29) and the ERM agreed well and forΩne ω thecalculation results of equation (30) were larger than thecalculation results of the ERM )e calculation results ofequation (30) were more accurate than those of the equation(29) because this equation considered the impact of theground motion frequency and the change law before andafter the resonance point was consistent with that underharmonic excitation

01 02 03 04 05 06 07α

a 1

08 09 10

70

60

40

30

20

50

10

0

Calculated valuesEquation (23)

Figure 7 Fitting curve of a1

a 2

01 02 03 04 05 06 07α

08 09 10

10

6

4

8

2

ndash2

0




4 Case Study

In 1997 Takewaki [34] presented an optimal vibration-re-duction design for a six-story shear frame (Figure 13) )emass was identical for all the floors of the frame iem1m2 m6 080times105 kg and the lateral stiffnesswas evenly distributed ie k1 k2 k6 400times107Nm )e initial damping coefficient of each VFD wasc1 150times106 (N sm) and the inherent structural dampingratio was taken as ζ0 005

According to the foregoing conditions it is easy todetermine the natural period T1 11656 s and the first vi-bration mode of the structure ϕT 02411 04681 0668008290 09419 10000 According to the equivalent SDOFmethod [33] the equivalent massMr equivalent stiffness Krand equivalent damping Cr can be obtained as follows

Mr ϕ1113864 1113865T[M] ϕ1113864 1113865

ϕ1113864 1113865T[M] 1

ϕ1113864 1113865T[M] ϕ1113864 1113865

Cr ϕ1113864 1113865T[C] ϕ1113864 1113865

ϕ1113864 1113865T[M] 1

ϕ1113864 1113865T

[M] ϕ1113864 1113865

Kr ϕ1113864 1113865T[K] ϕ1113864 1113865

ϕ1113864 1113865T[M] 1

ϕ1113864 1113865T[M] ϕ1113864 1113865

(33)

whereM represents the mass matrix K represents the lateralstiffness matrix C represents the damping matrix corre-sponding to the linear VFDs and ϕ represents the firstvibration mode of the structure


00 05 10 15

Ωω

ζ sd (

)

(1 ζsdres)

20 25 30

30

25

20

15

10

5

0

ndash5

ERMEquation (18)Equation (22)

(a)


(1 ζsdres)

00 05 10 15

Ωω

20 25 30

ζ sd (

)

20

15

10

5

0

ndash5


(b)


(1 ζsdres)

00 05 10 15

Ωω

20 25 30

ζ sd (

)

8

6

4

2

0


(c)


(1 ζsdres)

00 05 10 15

Ωω

20 25 30

ζ sd (

)4

3

2

1

0


(d)

Figure 9 ADR (ζsd) vs frequency ratio (Ωω) (a) α 030 (b) α 050 (c) α 070 (d) α 090


According to equation (33) the equivalent mass isMr 332times105 kg the equivalent rigidity isKr 964times106Nm and the equivalent damping isCr 608times105 (N sm) Additionally the equivalent periodof the structure is T11656 s which is consistent with thatof the original frame

)e information of 10 ground motions selected from thePacific Earthquake Engineering Research Center (PEER) aswell as the corresponding average frequency Ωm is pre-sented in Table 2)e PGA of each groundmotion was takenas 500 Gal which is similar to the intensity of a rare 8-degreeearthquake for the time-history analyses Figure 14 showsthe interstory displacement of the shear frame equipped withlinear VFDs subjected to the seismic excitation Clearly thedisplacement response of the frame structure tended to belarger under seismic motions with a lower average frequencyΩm

When α= 1 the ADR of the linear viscous dampers iscalculated as follows using equation (21) ζ1 =Cr(2Mrω)=CrT4πMr= 1700 )e calculation results of the time-history analyses are presented in Table 2 which are close to1700 indicating that the calculation result of equation(28) can be used to quickly evaluate the ADR with con-sideration of the resonance condition despite the error

When αne 1 Ωω is set then equation (6) is trans-formed into

c1 λπ

cα

ωuo( 11138571minus α (34)

)enonlinear damping exponent is set as α 05 and thepeak deformation is set as u0 005m which is approxi-mately 160 of the floor height (3m) )is is close to theinterstory displacement limit for a structure under rare

0 20 40 60


ζ1

f 1

80

ω = 20 rads

ω = 14 rads

ω = 10 rads

ω = 539 rads

ω = 2 rads

100

14

12

10

00

02

04

06

08

Numerical pointFitting curve

Figure 10 Relationship between f1 and ζ1 under El Centro seismicexcitation

Table 1 Fitting coefficients of the polynomial for different ωvalues

ω (rads) Fitting coefficientsb 1 b 2 b 3 b 4

2 023 ndash042 030 0014 034 ndash086 080 001539 040 ndash089 084 0048 111 ndash239 181 00410 176 ndash370 266 00712 184 ndash399 304 00814 119 ndash313 302 00216 050 ndash200 273 00018 014 ndash128 244 00120 ndash005 ndash086 227 003

50

0 1 2

T (s)

ζ sd (

)

3 4 5

40

30

20

10

0

cα = 284

cα = 400

cα = 100


ERM

Equation (29)

Equation (30)

Figure 11 ADR spectra with different values of ca (kNmiddotsm) for theEl Centro ground motion (α 050)

50

05 1510 20 3025

uumlg0 (ms2)

cα = 100

cα = 284

cα = 400


ζ sd (

)

35 4540 50

40

30

20

10

0


Figure 12 Relationship between the ADR and the seismic-exci-tation peak acceleration with different values of ca (kNmiddotsm) for theEl Centro ground motion (α 050)


earthquakes based on the Chinese seismic code [35] )eother parameters are identical to those of the originalstructure According to equation (34) the equivalentdamping coefficient is ca 284times105 (N sm)

Taking the El Centro seismic record of the ImperialValley earthquake as an example according to equations(31) and (32) the average period is Tm 056 s andΩm 1120 rads According to the fitting results in Table 1f1 in equation (29) can be expressed as follows

f1 04ζ3sdres minus 09ζ2sdres + 084ζsdres + 004 (35)

By solving equation (29) ADR ζsdres of the shear frameunder the structural resonance condition is calculated as974 By substituting this value of ζsdres into equation(30) the ADR under the El Centro seismic excitationconsidering the spectral characteristics is determined to be10 Additionally the ADR of the nonlinear viscousdamper can be determined using Figures 11 and 12

according to the damping coefficient ca natural vibra-tion period of the structure T or PGA euroug0 of the El Centromotion

Taking the 10 input ground motions listed in Table 2 asseismic excitations the ADRs were calculated using fivemethods the ERM two nonlinear time-history analysismethods (NMA [28] and Code [35] methods) and equations(29) and (30) )e results are presented in Table 3 and acomparison of the ADRs calculated using three of the methodswith different frequency ratios is presented in Figure 15 Asindicated in Table 3 the difference between the results obtainedusing the ERM and equation (29) under the resonance con-dition was very small )e results obtained using the NMAmethod were larger than those obtained using the Codemethod indicating that the Code method is relatively con-servative with regard to structural safety)e calculation resultsof equation (30) were larger than those of equation (29) whichis consistent with the change law of the ADR under harmonicexcitation with a frequency ratio of gt1 (see Figure 15) Because

k5

k6

m5

m6

c5

k4

k3

k2

k1

m4

m3

m2

m1

c4

c3

c2

c1

c6

Figure 13 Six-story shearing frame model with VFDs

Table 2 Information on the considered earthquake records

Name Location Year M s Station PGA (g) Ωm ζ sd (α1)KEAT Kern County 1952 74 Athenaeum 0048 681 1406SATI Northridge 1966 67 Saticoy 0341 797 1415KEHO Kern County 1952 74 Hollywood 0042 834 1552NEWH Northridge 1994 64 Newhall 0 590 900 1593ELCE Imperial Valley 1940 70 El Centro 0348 1120 1697TAFT Kern County 1952 74 Taft 0159 1122 1826TABA Tabas 1978 74 Tabas 0854 1216 1795IRAR Irpinia 1980 69 Arienzo 0027 1299 1808IRAU Irpinia 1980 69 Auletta 0055 1479 1901MACO Mammoth 1980 57 Convict Creek 0178 1490 1898


the spectral characteristics of the seismic excitation significantlyaffected the ADR of the damping structure the calculationresults of equation (30) which considered the excitation fre-quency were more accurate than those of equation (29)

Figure 16 presents a comparison of the roof-displace-ment time histories of the shear frame with ADRs calculated

using different methods As shown the displacement re-sponses calculated using the ERM and equation (30) weresimilar and the displacement responses obtained using theCode method were smaller than those for the other twomethods indicating that the proposed calculation method isrelatively conservative

6

NEWH

KEAT

SATI

KEHO

ELCE

TAFT

TABA

IRAR

IRAU

MACO

5

4

3

2

1

1 10 20 30 40

Stor

y

Interstory displacement (mm)

50 60 70

Figure 14 Interstory displacement of the six-story shear frame equipped with linear VFDs

Table 3 ADRs calculated using different methods

ADR ()

Seismic waveΩm ω Ωmneω

ERM Equation (29)Nonlinear time-history analysis method

Equation (30)NMA [29] Code method [35]

KEAT 1219 1223 1661 1392 1343SATI 800 813 957 805 879KEHO 1227 1292 1559 1416 1397NEWH 932 978 1352 1159 1032ELCE 968 974 1353 1259 1000TAFT 1039 1068 1525 1377 1174TABA 966 1005 1395 1230 1052IRAR 1113 1116 1392 1182 1244IRAU 1368 1374 1898 1535 1545MACO 1364 1407 1695 1476 1580Note ADR of NMA (energy dissipated by VFDsenergy dissipated by modal damping)times inherent structural damping ratio


5 Conclusions

A new simplified calculation method for the ADR of adamping structure equipped with a nonlinear viscousdamper was developed on the basis of previous studies )eaccuracy of this method was verified by analysing a six-storyRC frame structure According to the calculation results thefollowing conclusions are drawn

(1) )e average frequency Ωm was introduced to char-acterise the seismic excitation and the ADR calcu-lated using the proposed method considering theeffect of the average frequency was larger than that

calculated using the ERM Moreover as the earth-quake intensity increased the ADR decreased

(2) Studying the effect of the excitation frequency on thecalculated value of the ADR with different velocityexponents revealed that the ADR of the dampingstructure was the smallest for Ωmω 1 and it de-creased with the increasing frequency ratio for Ωmωlt 1 Additionally the ADR of the dampingstructure decreased with an increase in the seismic-excitation peak acceleration

(3) According to the case study of a shear frame thedisplacement response of the frame structure tends

22

20

18

16

14

12

10

8

608 10 12 14 16 18 20 22 24 26 28 30 32

Ωmω

ζ sd (

)

Equation (29)Equation (30)ERM

Groundmotion

KEATSATIKEHO

NEWHELCETAFTTABAIRARIRAUMACO 276

274

241

226

208

207

167

155

148

126

Ωmω

Figure 15 Comparison of the calculation results of equations (29) and (30) and the ERM

150

100

50

0

ndash50

ndash100

ndash1500 10 20 30 40 50 60

t (s)

Disp

(m

m)

ERMCode methodEquation (30)

Figure 16 Roof-displacement time histories of the shear frame with ADRs calculated using different methods


to be larger under seismic motions with a loweraverage frequency Ωm indicating the significantimpact of the frequency characteristics of the seismicexcitation on the structural response Comparedwith other methods tested the calculation results ofthe proposed simplified method were closer to thoseof the Code method confirming the effectiveness ofthe proposed method

When the ground motion characteristics structuralcharacteristics and damper parameters are known theproposed method can be used to calculate the ADR of thedamping structure conveniently However the method forcalculating the ground motion frequency using the RSAPrequires further improvement

Data Availability

Some or all data and models used during the study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare no conflicts of interest

Acknowledgments

)is research was funded by the National Natural ScienceFoundation of China (grant nos 51678301 and 51678302)and the National Key RampD Program of China (grant no2017YFC0703600)

References

[1] R D Bertero and Great ldquoAmerican earthquakes lessons forseismic design and constructionrdquo Journal of ConstructionEngineering amp Management vol 140 no 4 Article IDB4013003 2010

[2] D M Siringoringo and Y Fujino ldquoLong-term seismicmonitoring of base-isolated building with emphasis on ser-viceability assessmentrdquo Earthquake Engineering amp StructuralDynamics vol 44 no 4 pp 637ndash655 2015

[3] W Z Xu D S Du S G Wang et al ldquoShaking table test ofmulti-dimensional seismic response of long-span girdstructure with base-isolationrdquo Engineering Structuresvol 201 2019

[4] O M Ramirez M C Constantinou J D GomezA S Whittaker and C Z Chrysostomou ldquoEvaluation ofsimplified methods of analysis of yielding structures withdamping systemsrdquo Earthquake Spectra vol 18 no 3pp 501ndash530 2002

[5] Y Huang R Sturt and M Willford ldquoA damping model fornonlinear dynamic analysis providing uniform damping overa frequency rangerdquo Computers amp Structures vol 212pp 101ndash109 2019

[6] H Anajafi R A Medina RA and E S Bell ldquoEffects of theimproper modeling of viscous damping on the first-mode andhigher-mode dominated responses of base-isolated build-ingsrdquo Earthquake Engineering amp Structural Dynamics vol 49no 1 pp 151ndash173 2020

[7] A Sarlis and M C Constantinou Modeling Triple FrictionPendulum Isolators in Program SAP2000 Supplement toMCEER Report 05ndash009 Document Distributed to the

Engineering Community Together with Example Files Uni-versity at Buffalo Buffalo NY USA 2010

[8] A Bajric R Brincker and S )ons ldquoEvaluation of dampingestimates in the presence of closely spaced modes usingoperational modal analysis techniquesrdquo in Proceedings of the6th International Operational Modal Analysis ConferenceGijon Spain May 2015

[9] W-I Liao I Mualla and C-H Loh ldquoShaking-table test of afriction-damped frame structurerdquo Ce Structural Design ofTall and Special Buildings vol 13 no 1 pp 45ndash54 2004

[10] Y Liu C-L Wang and J Wu ldquoDevelopment of a newpartially restrained energy dissipater experimental and nu-merical analysesrdquo Journal of Constructional Steel Researchvol 147 pp 367ndash379 2018

[11] H AmiriA Ahmadie et al ldquoExperimental and analyticalstudy of block slit damperrdquo Journal of Constructional SteelResearch vol 141 pp 167ndash178 2018

[12] R H Zhang and T T Soong ldquoSeismic design of viscoelasticdampers for structural applicationsrdquo Journal of StructuralEngineering vol 118 no 5 pp 1375ndash1392 1992

[13] Y Q Ni J M Ko and Z G Ying ldquoRandom seismic responseanalysis of adjacent buildings coupled with non-linear hys-teretic dampersrdquo Journal of Sound and Vibration vol 246no 3 pp 403ndash417 2001

[14] F C Ponzo A Di Cesare D Nigro et al ldquoJet-pacs projectdynamic experimental tests and numerical results obtainedfor a steel frame equipped with hysteretic damped chevronbracesrdquo Journal of Earthquake Engineering vol 16 no 5pp 662ndash685 2012

[15] C-L Wang Y Liu X Zheng and J Wu ldquoExperimentalinvestigation of a precast concrete connection with all-steelbamboo-shaped energy dissipatersrdquo Engineering Structuresvol 178 pp 298ndash308 2019

[16] Y Ribakov and A N Dancygier ldquoHigh-efficiency amplifiersfor viscous damped structures subjected to strong earth-quakesrdquo Structural Design Tall Special Buildings vol 15 no 2pp 221ndash232 2016

[17] X L Wu W Guo P Hu et al ldquoSeismic performanceevaluation of building-damper system under near-faultearthquakerdquo Shock amp Vibration vol 2020 Article ID2763709 2020

[18] S-H Lee K-W Min J-S Hwang and J Kim ldquoEvaluation ofequivalent damping ratio of a structure with added dampersrdquoEngineering Structures vol 26 no 3 pp 335ndash346 2004

[19] F A Charney and R J McNamara ldquoComparison of methodsfor computing equivalent viscous damping ratios of structureswith added viscous dampingrdquo Journal of Structural Engi-neering vol 134 no 1 pp 32ndash44 2008

[20] A Occhiuzzi ldquoAdditional viscous dampers for civil struc-tures analysis of design methods based on effective evaluationof modal damping ratiosrdquo Engineering Structures vol 31no 5 pp 1093ndash1101 2009

[21] S Silvestri G Gasparini and T Trombetti ldquoA five-stepprocedure for the dimensioning of viscous dampers to beinserted in building structuresrdquo Journal of Earthquake En-gineering vol 14 no 3 pp 417ndash447 2010

[22] M Palermo S Silvestri G Gasparini and T Trombetti ldquoAdirect five-step procedure for the dimensioning of viscousdampers to Be inserted in frame structuresrdquo Applied Me-chanics and Materials vol 847 pp 233ndash239 2016

[23] M Palermo S Silvestri G Gasparini A Dib andT Trombetti ldquoA direct design procedure for frame structureswith added viscous dampers for the mitigation of earthquake-


induced vibrationsrdquo Procedia Engineering vol 199pp 1755ndash1760 2017

[24] P P Diotallevi L Landi and A Dellavalle ldquoA methodologyfor the direct assessment of the damping ratio of structuresequipped with nonlinear viscous dampersrdquo Journal ofEarthquake Engineering vol 16 no 3 pp 350ndash373 2012

[25] D G Weng C Zhang X L Lu S Zeng and S M Zhang ldquoAsimplified design procedure for seismic retrofit of earthquake-damaged RC frames with viscous dampersrdquo Structural En-gineering and Mechanics vol 44 no 5 pp 611ndash631 2012

[26] L Landi S Lucchi and P P Diotallevi ldquoA procedure for thedirect determination of the required supplemental dampingfor the seismic retrofit with viscous dampersrdquo EngineeringStructures vol 71 pp 137ndash149 2014

[27] F N Kudu S Uccedilak G Osmancikli T Turker andA Bayraktar ldquoEstimation of damping ratios of steel structuresby operational modal analysis methodrdquo Journal of Con-structional Steel Research vol 112 pp 61ndash68 2015

[28] S Ishimaru and C Kuo ldquoNew versionductility factor controlmethodrdquo Journal of Structural and Construction Engineering(Transactions of AIJ) vol 80 no 708 pp 241ndash251 2015

[29] D G Wen C Li X Y Hu et al ldquoCalculation of additionaleffective damping ratio of damping structure based on modaldamping energy consumptionrdquo Journal of Civil Engineeringvol 49 pp 19ndash24 2016

[30] L Landi C Vorabbi O Fabbri and P P Diotallevi ldquoSim-plified probabilistic seismic assessment of RC frames withadded viscous dampersrdquo Soil Dynamics and EarthquakeEngineering vol 97 pp 277ndash288 2017

[31] J S Love and M J Tait ldquoEstimating the added effectivedamping of SDOF systems incorporating multiple dynamicvibration absorbers with nonlinear dampingrdquo EngineeringStructures vol 130 pp 154ndash161 2016

[32] Y He Y Xiao Y Liu and Z Zhang ldquoAn efficient finiteelement method for computing modal damping of laminatedcomposites theory and experimentrdquo Composite Structuresvol 184 pp 728ndash741 2018

[33] A K Chopra Dynamics of Structures Ceory and Applica-tions to Earthquake Engineeringpp 66ndash103 Prentice-HallUpper Saddle River NJ USA 2nd ed edition 2001

[34] I Takewaki ldquoOptimal damper placement for minimumtransfer functionsrdquo Earthquake Engineering amp StructuralDynamics vol 26 no 11 pp 1113ndash1124 1997

[35] GB 50011-2010 Code for Seismic Design of Buildings ChinaArchitecture and Industry Press Beijing China 2010


the damping leakage phenomenon in structural systems wasinvestigated revealing that the damping leakage can result ina significant underestimation of the first-mode responseandor higher-mode responses To mitigate the problemsassociated with the improper damping models in the base-isolated structure Anajafi et al [6] proposed a dampingmodelling approach that involves removing the mass-pro-portional component of superstructure damping from theglobal damping matrix

In addition to the inherent damping the additionaldamping ratio (ADR) which consists of contributions fromvarious energy-dissipation devices (EDDs) has been utilisedin different types of structures to mitigate the structuralresponses induced by seismic excitation [9ndash16] )e viscousfluid damper (VFD)mdasha rate-dependent EDDmdashis favouredby engineers and increasingly used in engineering structuresto reduce excessive vibrations because of the absence ofsupplemental stiffness and a high energy-dissipation rate indifferent deformation stages Wu et al [17] evaluated theseismic performance of a nine-story steel frame supple-mented with five typical dampers under near-fault earth-quakes )e results indicated that the VFD functioned betterthan other four types of dampers even under high pulsevelocity amplitude (PVA) To quantitatively evaluate thedamping effects for structures supplemented with VFDs theADR was proposed Additionally in China the ADR is a keystructural parameter for engineers in the design processFirstly the target ADR is determined then the aseismicdesign is conducted via themode analysis response spectrummethod considering the total damping ratio the sum of theinherent damping ratio and the ADR Subsequently thedamper parameters and the layout are designed to achievethe target ADR )e design process determines the im-portance of estimating the ADR of the structures equippedwith EDDs

Recently scholars have proposed various calculationmethods for the ADR Lee et al [18] proposed an evaluationmethod for the equivalent damping ratio for systems con-figured with dampers In that study the Lyapunov functionwas first defined in the form of the modal energy then theRiccati matrix and damping ratio parameters were derivedFinally a formula for the equivalent damping ratio wasobtained by analogising the viscous modal damping ratioCharney et al [19] used the modal strain energymethod freevibration logarithmic decay method and eigenvectormethod for complex eigenvalues to analyse the equivalentVD ratio of a single-story single-bay frame )e resultsindicated that higher flexibility of the nodes corresponded toa larger phase difference between the deformation velocityand the relative horizontal velocity of the interlayer devicesHowever this conclusion was only applicable to the single-layer single-span structure Occhiuzzi et al [20] proposed amethod for calculating the modal damping ratio using anequation for the dynamic system state An analysis revealedthat the first vibration mode significantly affected thedamping ratio whereas the effects of the higher-order vi-bration modes were negligible )e aforementioned calcu-lation methods which involve complex theoretical formulasand large amounts of calculations are difficult for engineers

to apply )erefore a simpler direct method needs to bedeveloped

Silvestri et al [21ndash23] proposed a practical method forthe aseismic design of structures configured with VFDs inwhich the mechanical characteristics of the VFDs are easilydetermined according to the performance objectives of thestructure However the calculation method adopted for theADR still cannot be widely used in practical engineeringDiotallevi et al [24] proposed a method for evaluating theADR according to the damping index )is method has theadvantage of realising direct evaluation of ADR withoutiteration but is limited to the evaluation of linear-elasticsystems Weng et al [25] proposed a method for calculatingthe equivalent ADR of the viscous damper according to thecode response spectrum Landi et al [26] proposed a sim-plified method for calculating the ADR according to thecapability and demand spectra Under the condition ofconfirmed yield acceleration and target displacement of theknown structure the corresponding ductility factorequivalent total damping ratio and significant period couldbe confirmed directly Kudu et al [27] compared the modaldamping ratios determined with consideration of themeasurement time frequency scope and sample rate Whenthe modal damping ratio changed significantly the naturalfrequency of the structure did not Ishimaru et al [28]presented an approach for calculating the optimal VD ratioand accumulated ductility of a structure configured with abilinear hysteretic damper and a dynamic quality damperwhich has a very high accuracy for calculating the accu-mulated ductility coefficient A method for calculating theADR based on the concept of effective modal dampingenergy dissipation was proposed by Weng et al [29] whichhas a clear physical significance and fully considers the time-varying characteristics of the effective damping ratio Itsadvantages include simple calculations and a high accuracyLandi et al [30] performed probabilistic seismic assessmentof RC structures with and without a damper using a sim-plified SAC Federal Emergency Management Agency (SAC-FEMA) program )e results showed that the method waseffective and could be used as a simplified alternative tononlinear dynamic analyses for probabilistic assessmentpurposes Love and Tait [31] proposed a simple method forpredicting the effective damping of a linear structure withnonlinear EDDs)is method exploits the characteristic thatthe output mean energy of the structure remains unchangedunder a wind load to make predictions However the errorbetween the prediction results of this method and the cal-culation results based on time-history analysis was signifi-cant He et al [32] proposed an effective finite-elementanalysis method for calculating the modal damping ratio ofcomplex materials and proved the effectiveness and accuracyof the method through a theoretical analysis and a com-parison with experimental results

Most of the aforementioned methods for calculating theADR require the calculation of the structural dynamic re-sponse and iteration making them cumbersome and in-convenient to use directly in practical engineeringAdditionally the solutions of the ADR for supplementalnonlinear VFDs based on the energy principle rarely






ζsd wD

4πwSo

ωΩ

(1)






wD 1113946 FDdu 1113946

2πΩ

0

cα| _u|1+αdt (3)


wD λcαu1+α0 Ω

α (4)


obtained

wD πc1u20Ω (5)


πc1u20Ω λcαu

1+α0 Ω

α (6)


ζsd λπ

cα

2mω1Ωuo( 1113857

1minus α (7)




ε λπ

cα

2m

ω1minus 2α




ζsd λπ

cα

2mω2minus α1

uo( 11138571minus α (9)

ε λπ

cα

2m

1ωα euroug01113872 1113873

αminus 1 (10)















u(t) eminus ξ1ωt


(13)

where ωD ω

1 minus ζ211113969



C ζ1Ωω

1113874 1113875 a0

ω21 minus (Ωω)2

1 minus (Ωω)21113960 11139612

+ 2ζ1(Ωω)21113960 11139612

D ζ1Ωω

1113874 1113875a0

ω2minus 2ζ(Ωω)

1 minus (Ωω)21113960 11139612

+ 2ζ1(Ωω)1113858 11138592

(14)



where us0 C2 + D2

radicand tan(φ) DC


u0 us0 (16)


ζ1 ζs d (17)


ζsd λπ

cα

2mω1

ΩC2 + D2

radic( 1113857

1minus α (18)



ε = 20

ε = 5ε = 10ε = 15

ε = 25ε = 30ε = 35

ε = 40ε = 45ε = 50

00

100

80

90

60

70

40

50

20

30

10

005 10 15

T (s)

ζ sd (

)

20 25 30 35 40 45 50



ε = 5

ε = 10

ε = 15

ε = 20

ε = 25

ε = 30

00

100

80

60

40

20

005 10 15

Ωω

ζ sd (

)

20 25 30






us0 a0

2ω2ζ1 (19)


c1 λπ

1a0m

1113888 1113889

1minus α

cα⎡⎣ ⎤⎦

1α

(20)


ζsdres 1

2mωλπ

1a0m

1113888 1113889

1minus α

cα⎡⎣ ⎤⎦

1α

(21)




ζsd

a1Ω

ω minus 11113874 1113875

2+ ζsdres

Ωωlt 1

ζsdresΩω

1

a2Ω

ω minus 11113874 1113875 + ζsdres

Ωωgt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)



a1 β1ec1α (23)


a2 β2ec2α (24)



00 05 10 15

Ωω

ζ sd (

)

(1 ζsd res)

20 25 30

20

15

10

5

0

ndash5

ERMEquation (18)



00 02 04

a0 (ms2)

ζ sd (

)

06 08 10

10

8

6

4

2

0

ERMEquation (18)



Serialno123456

100020003000400050006000

Dampingcoefficient

12

3

45

6

16

14

12

10

8

6

4

2

0


res (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(a)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

56

16

14

12

10

8

6

4

2

0


ζ sdre

s (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(b)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

5

6

8

7

6

5

4

3

2

1

0


ζ sdre

s (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(c)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

56

8

7

6

5

4

3

2

1

0


res (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(d)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

56

6

5

4

2

3

1

0


ζ sdre

s (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(e)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

56

5

4

3

2

1

0


ζ sdre

s (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(f )






uw euroug0

2ω2ζ1 (25)




2ω2ζ1 (26)




00 05 10 15

Ωω

ζ sd (

)

(1 ζsdres)

20

ERM

25 30

30

25

20

15

10

5

0

ndash5

(a)


(1 ζsdres)

ERM

00 05 10 15

Ωω

20 25 30

ζ sd (

)

20

15

10

5

0

ndash5

(b)


(1 ζsdres)

ERM

00 05 10 15

Ωω

20 25 30

ζ sd (

)

8

6

4

2

0

(c)


(1 ζsdres)

ERM

00 05 10 15

Ωω

20 25 30

ζ sd (

)

4

3

2

1

0

(d)




f1 ζ1( 1113857 b1ζ31 + b2ζ

21 + b3ζ1 + b4 (27)





ζsdres λπ

cα

2mω2minus α1



ζsdres λπ

cα

2mω2minus α1



ζsd

a1Ωm

ω minus 11113874 1113875

2+ ζsdres

Ωm

ωlt 1

ζsdresΩm

ω 1

a2Ωm

ω minus 11113874 1113875 + ζsdres

Ωm

ωgt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(30)



Tm 1113936iC

2i 1Fi( 1113857

1113936iC2i

(31)



Ωm 2πTm

(32)



01 02 03 04 05 06 07α

a 1

08 09 10

70

60

40

30

20

50

10

0



a 2

01 02 03 04 05 06 07α

08 09 10

10

6

4

8

2

ndash2

0




4 Case Study



Mr ϕ1113864 1113865T[M] ϕ1113864 1113865

ϕ1113864 1113865T[M] 1

ϕ1113864 1113865T[M] ϕ1113864 1113865

Cr ϕ1113864 1113865T[C] ϕ1113864 1113865

ϕ1113864 1113865T[M] 1

ϕ1113864 1113865T

[M] ϕ1113864 1113865

Kr ϕ1113864 1113865T[K] ϕ1113864 1113865

ϕ1113864 1113865T[M] 1

ϕ1113864 1113865T[M] ϕ1113864 1113865

(33)



00 05 10 15

Ωω

ζ sd (

)

(1 ζsdres)

20 25 30

30

25

20

15

10

5

0

ndash5


(a)


(1 ζsdres)

00 05 10 15

Ωω

20 25 30

ζ sd (

)

20

15

10

5

0

ndash5


(b)


(1 ζsdres)

00 05 10 15

Ωω

20 25 30

ζ sd (

)

8

6

4

2

0


(c)


(1 ζsdres)

00 05 10 15

Ωω

20 25 30

ζ sd (

)4

3

2

1

0


(d)







c1 λπ

cα

ωuo( 11138571minus α (34)


0 20 40 60


ζ1

f 1

80

ω = 20 rads

ω = 14 rads

ω = 10 rads

ω = 539 rads

ω = 2 rads

100

14

12

10

00

02

04

06

08






50

0 1 2

T (s)

ζ sd (

)

3 4 5

40

30

20

10

0

cα = 284

cα = 400

cα = 100


ERM

Equation (29)

Equation (30)


50

05 1510 20 3025

uumlg0 (ms2)

cα = 100

cα = 284

cα = 400


ζ sd (

)

35 4540 50

40

30

20

10

0










k5

k6

m5

m6

c5

k4

k3

k2

k1

m4

m3

m2

m1

c4

c3

c2

c1

c6








6

NEWH

KEAT

SATI

KEHO

ELCE

TAFT

TABA

IRAR

IRAU

MACO

5

4

3

2

1

1 10 20 30 40

Stor

y


50 60 70



ADR ()






5 Conclusions






22

20

18

16

14

12

10

8

608 10 12 14 16 18 20 22 24 26 28 30 32

Ωmω

ζ sd (

)


Groundmotion

KEATSATIKEHO


274

241

226

208

207

167

155

148

126

Ωmω


150

100

50

0

ndash50

ndash100

ndash1500 10 20 30 40 50 60

t (s)

Disp

(m

m)






Data Availability




Acknowledgments


References












































ζsd wD

4πwSo

ωΩ

(1)






wD 1113946 FDdu 1113946

2πΩ

0

cα| _u|1+αdt (3)


wD λcαu1+α0 Ω

α (4)


obtained

wD πc1u20Ω (5)


πc1u20Ω λcαu

1+α0 Ω

α (6)


ζsd λπ

cα

2mω1Ωuo( 1113857

1minus α (7)




ε λπ

cα

2m

ω1minus 2α




ζsd λπ

cα

2mω2minus α1

uo( 11138571minus α (9)

ε λπ

cα

2m

1ωα euroug01113872 1113873

αminus 1 (10)















u(t) eminus ξ1ωt


(13)

where ωD ω

1 minus ζ211113969



C ζ1Ωω

1113874 1113875 a0

ω21 minus (Ωω)2

1 minus (Ωω)21113960 11139612

+ 2ζ1(Ωω)21113960 11139612

D ζ1Ωω

1113874 1113875a0

ω2minus 2ζ(Ωω)

1 minus (Ωω)21113960 11139612

+ 2ζ1(Ωω)1113858 11138592

(14)



where us0 C2 + D2

radicand tan(φ) DC


u0 us0 (16)


ζ1 ζs d (17)


ζsd λπ

cα

2mω1

ΩC2 + D2

radic( 1113857

1minus α (18)



ε = 20

ε = 5ε = 10ε = 15

ε = 25ε = 30ε = 35

ε = 40ε = 45ε = 50

00

100

80

90

60

70

40

50

20

30

10

005 10 15

T (s)

ζ sd (

)

20 25 30 35 40 45 50



ε = 5

ε = 10

ε = 15

ε = 20

ε = 25

ε = 30

00

100

80

60

40

20

005 10 15

Ωω

ζ sd (

)

20 25 30






us0 a0

2ω2ζ1 (19)


c1 λπ

1a0m

1113888 1113889

1minus α

cα⎡⎣ ⎤⎦

1α

(20)


ζsdres 1

2mωλπ

1a0m

1113888 1113889

1minus α

cα⎡⎣ ⎤⎦

1α

(21)




ζsd

a1Ω

ω minus 11113874 1113875

2+ ζsdres

Ωωlt 1

ζsdresΩω

1

a2Ω

ω minus 11113874 1113875 + ζsdres

Ωωgt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)



a1 β1ec1α (23)


a2 β2ec2α (24)



00 05 10 15

Ωω

ζ sd (

)

(1 ζsd res)

20 25 30

20

15

10

5

0

ndash5

ERMEquation (18)



00 02 04

a0 (ms2)

ζ sd (

)

06 08 10

10

8

6

4

2

0

ERMEquation (18)



Serialno123456

100020003000400050006000

Dampingcoefficient

12

3

45

6

16

14

12

10

8

6

4

2

0


res (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(a)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

56

16

14

12

10

8

6

4

2

0


ζ sdre

s (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(b)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

5

6

8

7

6

5

4

3

2

1

0


ζ sdre

s (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(c)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

56

8

7

6

5

4

3

2

1

0


res (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(d)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

56

6

5

4

2

3

1

0


ζ sdre

s (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(e)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

56

5

4

3

2

1

0


ζ sdre

s (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(f )






uw euroug0

2ω2ζ1 (25)




2ω2ζ1 (26)




00 05 10 15

Ωω

ζ sd (

)

(1 ζsdres)

20

ERM

25 30

30

25

20

15

10

5

0

ndash5

(a)


(1 ζsdres)

ERM

00 05 10 15

Ωω

20 25 30

ζ sd (

)

20

15

10

5

0

ndash5

(b)


(1 ζsdres)

ERM

00 05 10 15

Ωω

20 25 30

ζ sd (

)

8

6

4

2

0

(c)


(1 ζsdres)

ERM

00 05 10 15

Ωω

20 25 30

ζ sd (

)

4

3

2

1

0

(d)




f1 ζ1( 1113857 b1ζ31 + b2ζ

21 + b3ζ1 + b4 (27)





ζsdres λπ

cα

2mω2minus α1



ζsdres λπ

cα

2mω2minus α1



ζsd

a1Ωm

ω minus 11113874 1113875

2+ ζsdres

Ωm

ωlt 1

ζsdresΩm

ω 1

a2Ωm

ω minus 11113874 1113875 + ζsdres

Ωm

ωgt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(30)



Tm 1113936iC

2i 1Fi( 1113857

1113936iC2i

(31)



Ωm 2πTm

(32)



01 02 03 04 05 06 07α

a 1

08 09 10

70

60

40

30

20

50

10

0



a 2

01 02 03 04 05 06 07α

08 09 10

10

6

4

8

2

ndash2

0




4 Case Study



Mr ϕ1113864 1113865T[M] ϕ1113864 1113865

ϕ1113864 1113865T[M] 1

ϕ1113864 1113865T[M] ϕ1113864 1113865

Cr ϕ1113864 1113865T[C] ϕ1113864 1113865

ϕ1113864 1113865T[M] 1

ϕ1113864 1113865T

[M] ϕ1113864 1113865

Kr ϕ1113864 1113865T[K] ϕ1113864 1113865

ϕ1113864 1113865T[M] 1

ϕ1113864 1113865T[M] ϕ1113864 1113865

(33)



00 05 10 15

Ωω

ζ sd (

)

(1 ζsdres)

20 25 30

30

25

20

15

10

5

0

ndash5


(a)


(1 ζsdres)

00 05 10 15

Ωω

20 25 30

ζ sd (

)

20

15

10

5

0

ndash5


(b)


(1 ζsdres)

00 05 10 15

Ωω

20 25 30

ζ sd (

)

8

6

4

2

0


(c)


(1 ζsdres)

00 05 10 15

Ωω

20 25 30

ζ sd (

)4

3

2

1

0


(d)







c1 λπ

cα

ωuo( 11138571minus α (34)


0 20 40 60


ζ1

f 1

80

ω = 20 rads

ω = 14 rads

ω = 10 rads

ω = 539 rads

ω = 2 rads

100

14

12

10

00

02

04

06

08






50

0 1 2

T (s)

ζ sd (

)

3 4 5

40

30

20

10

0

cα = 284

cα = 400

cα = 100


ERM

Equation (29)

Equation (30)


50

05 1510 20 3025

uumlg0 (ms2)

cα = 100

cα = 284

cα = 400


ζ sd (

)

35 4540 50

40

30

20

10

0










k5

k6

m5

m6

c5

k4

k3

k2

k1

m4

m3

m2

m1

c4

c3

c2

c1

c6








6

NEWH

KEAT

SATI

KEHO

ELCE

TAFT

TABA

IRAR

IRAU

MACO

5

4

3

2

1

1 10 20 30 40

Stor

y


50 60 70



ADR ()






5 Conclusions






22

20

18

16

14

12

10

8

608 10 12 14 16 18 20 22 24 26 28 30 32

Ωmω

ζ sd (

)


Groundmotion

KEATSATIKEHO


274

241

226

208

207

167

155

148

126

Ωmω


150

100

50

0

ndash50

ndash100

ndash1500 10 20 30 40 50 60

t (s)

Disp

(m

m)






Data Availability




Acknowledgments


References
















































u(t) eminus ξ1ωt


(13)

where ωD ω

1 minus ζ211113969



C ζ1Ωω

1113874 1113875 a0

ω21 minus (Ωω)2

1 minus (Ωω)21113960 11139612

+ 2ζ1(Ωω)21113960 11139612

D ζ1Ωω

1113874 1113875a0

ω2minus 2ζ(Ωω)

1 minus (Ωω)21113960 11139612

+ 2ζ1(Ωω)1113858 11138592

(14)



where us0 C2 + D2

radicand tan(φ) DC


u0 us0 (16)


ζ1 ζs d (17)


ζsd λπ

cα

2mω1

ΩC2 + D2

radic( 1113857

1minus α (18)



ε = 20

ε = 5ε = 10ε = 15

ε = 25ε = 30ε = 35

ε = 40ε = 45ε = 50

00

100

80

90

60

70

40

50

20

30

10

005 10 15

T (s)

ζ sd (

)

20 25 30 35 40 45 50



ε = 5

ε = 10

ε = 15

ε = 20

ε = 25

ε = 30

00

100

80

60

40

20

005 10 15

Ωω

ζ sd (

)

20 25 30






us0 a0

2ω2ζ1 (19)


c1 λπ

1a0m

1113888 1113889

1minus α

cα⎡⎣ ⎤⎦

1α

(20)


ζsdres 1

2mωλπ

1a0m

1113888 1113889

1minus α

cα⎡⎣ ⎤⎦

1α

(21)




ζsd

a1Ω

ω minus 11113874 1113875

2+ ζsdres

Ωωlt 1

ζsdresΩω

1

a2Ω

ω minus 11113874 1113875 + ζsdres

Ωωgt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)



a1 β1ec1α (23)


a2 β2ec2α (24)



00 05 10 15

Ωω

ζ sd (

)

(1 ζsd res)

20 25 30

20

15

10

5

0

ndash5

ERMEquation (18)



00 02 04

a0 (ms2)

ζ sd (

)

06 08 10

10

8

6

4

2

0

ERMEquation (18)



Serialno123456

100020003000400050006000

Dampingcoefficient

12

3

45

6

16

14

12

10

8

6

4

2

0


res (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(a)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

56

16

14

12

10

8

6

4

2

0


ζ sdre

s (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(b)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

5

6

8

7

6

5

4

3

2

1

0


ζ sdre

s (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(c)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

56

8

7

6

5

4

3

2

1

0


res (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(d)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

56

6

5

4

2

3

1

0


ζ sdre

s (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(e)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

56

5

4

3

2

1

0


ζ sdre

s (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(f )






uw euroug0

2ω2ζ1 (25)




2ω2ζ1 (26)




00 05 10 15

Ωω

ζ sd (

)

(1 ζsdres)

20

ERM

25 30

30

25

20

15

10

5

0

ndash5

(a)


(1 ζsdres)

ERM

00 05 10 15

Ωω

20 25 30

ζ sd (

)

20

15

10

5

0

ndash5

(b)


(1 ζsdres)

ERM

00 05 10 15

Ωω

20 25 30

ζ sd (

)

8

6

4

2

0

(c)


(1 ζsdres)

ERM

00 05 10 15

Ωω

20 25 30

ζ sd (

)

4

3

2

1

0

(d)




f1 ζ1( 1113857 b1ζ31 + b2ζ

21 + b3ζ1 + b4 (27)





ζsdres λπ

cα

2mω2minus α1



ζsdres λπ

cα

2mω2minus α1



ζsd

a1Ωm

ω minus 11113874 1113875

2+ ζsdres

Ωm

ωlt 1

ζsdresΩm

ω 1

a2Ωm

ω minus 11113874 1113875 + ζsdres

Ωm

ωgt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(30)



Tm 1113936iC

2i 1Fi( 1113857

1113936iC2i

(31)



Ωm 2πTm

(32)



01 02 03 04 05 06 07α

a 1

08 09 10

70

60

40

30

20

50

10

0



a 2

01 02 03 04 05 06 07α

08 09 10

10

6

4

8

2

ndash2

0




4 Case Study



Mr ϕ1113864 1113865T[M] ϕ1113864 1113865

ϕ1113864 1113865T[M] 1

ϕ1113864 1113865T[M] ϕ1113864 1113865

Cr ϕ1113864 1113865T[C] ϕ1113864 1113865

ϕ1113864 1113865T[M] 1

ϕ1113864 1113865T

[M] ϕ1113864 1113865

Kr ϕ1113864 1113865T[K] ϕ1113864 1113865

ϕ1113864 1113865T[M] 1

ϕ1113864 1113865T[M] ϕ1113864 1113865

(33)



00 05 10 15

Ωω

ζ sd (

)

(1 ζsdres)

20 25 30

30

25

20

15

10

5

0

ndash5


(a)


(1 ζsdres)

00 05 10 15

Ωω

20 25 30

ζ sd (

)

20

15

10

5

0

ndash5


(b)


(1 ζsdres)

00 05 10 15

Ωω

20 25 30

ζ sd (

)

8

6

4

2

0


(c)


(1 ζsdres)

00 05 10 15

Ωω

20 25 30

ζ sd (

)4

3

2

1

0


(d)







c1 λπ

cα

ωuo( 11138571minus α (34)


0 20 40 60


ζ1

f 1

80

ω = 20 rads

ω = 14 rads

ω = 10 rads

ω = 539 rads

ω = 2 rads

100

14

12

10

00

02

04

06

08






50

0 1 2

T (s)

ζ sd (

)

3 4 5

40

30

20

10

0

cα = 284

cα = 400

cα = 100


ERM

Equation (29)

Equation (30)


50

05 1510 20 3025

uumlg0 (ms2)

cα = 100

cα = 284

cα = 400


ζ sd (

)

35 4540 50

40

30

20

10

0










k5

k6

m5

m6

c5

k4

k3

k2

k1

m4

m3

m2

m1

c4

c3

c2

c1

c6








6

NEWH

KEAT

SATI

KEHO

ELCE

TAFT

TABA

IRAR

IRAU

MACO

5

4

3

2

1

1 10 20 30 40

Stor

y


50 60 70



ADR ()






5 Conclusions






22

20

18

16

14

12

10

8

608 10 12 14 16 18 20 22 24 26 28 30 32

Ωmω

ζ sd (

)


Groundmotion

KEATSATIKEHO


274

241

226

208

207

167

155

148

126

Ωmω


150

100

50

0

ndash50

ndash100

ndash1500 10 20 30 40 50 60

t (s)

Disp

(m

m)






Data Availability




Acknowledgments


References











































us0 a0

2ω2ζ1 (19)


c1 λπ

1a0m

1113888 1113889

1minus α

cα⎡⎣ ⎤⎦

1α

(20)


ζsdres 1

2mωλπ

1a0m

1113888 1113889

1minus α

cα⎡⎣ ⎤⎦

1α

(21)




ζsd

a1Ω

ω minus 11113874 1113875

2+ ζsdres

Ωωlt 1

ζsdresΩω

1

a2Ω

ω minus 11113874 1113875 + ζsdres

Ωωgt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)



a1 β1ec1α (23)


a2 β2ec2α (24)



00 05 10 15

Ωω

ζ sd (

)

(1 ζsd res)

20 25 30

20

15

10

5

0

ndash5

ERMEquation (18)



00 02 04

a0 (ms2)

ζ sd (

)

06 08 10

10

8

6

4

2

0

ERMEquation (18)



Serialno123456

100020003000400050006000

Dampingcoefficient

12

3

45

6

16

14

12

10

8

6

4

2

0


res (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(a)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

56

16

14

12

10

8

6

4

2

0


ζ sdre

s (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(b)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

5

6

8

7

6

5

4

3

2

1

0


ζ sdre

s (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(c)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

56

8

7

6

5

4

3

2

1

0


res (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(d)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

56

6

5

4

2

3

1

0


ζ sdre

s (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(e)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

56

5

4

3

2

1

0


ζ sdre

s (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(f )






uw euroug0

2ω2ζ1 (25)




2ω2ζ1 (26)




00 05 10 15

Ωω

ζ sd (

)

(1 ζsdres)

20

ERM

25 30

30

25

20

15

10

5

0

ndash5

(a)


(1 ζsdres)

ERM

00 05 10 15

Ωω

20 25 30

ζ sd (

)

20

15

10

5

0

ndash5

(b)


(1 ζsdres)

ERM

00 05 10 15

Ωω

20 25 30

ζ sd (

)

8

6

4

2

0

(c)


(1 ζsdres)

ERM

00 05 10 15

Ωω

20 25 30

ζ sd (

)

4

3

2

1

0

(d)




f1 ζ1( 1113857 b1ζ31 + b2ζ

21 + b3ζ1 + b4 (27)





ζsdres λπ

cα

2mω2minus α1



ζsdres λπ

cα

2mω2minus α1



ζsd

a1Ωm

ω minus 11113874 1113875

2+ ζsdres

Ωm

ωlt 1

ζsdresΩm

ω 1

a2Ωm

ω minus 11113874 1113875 + ζsdres

Ωm

ωgt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(30)



Tm 1113936iC

2i 1Fi( 1113857

1113936iC2i

(31)



Ωm 2πTm

(32)



01 02 03 04 05 06 07α

a 1

08 09 10

70

60

40

30

20

50

10

0



a 2

01 02 03 04 05 06 07α

08 09 10

10

6

4

8

2

ndash2

0




4 Case Study



Mr ϕ1113864 1113865T[M] ϕ1113864 1113865

ϕ1113864 1113865T[M] 1

ϕ1113864 1113865T[M] ϕ1113864 1113865

Cr ϕ1113864 1113865T[C] ϕ1113864 1113865

ϕ1113864 1113865T[M] 1

ϕ1113864 1113865T

[M] ϕ1113864 1113865

Kr ϕ1113864 1113865T[K] ϕ1113864 1113865

ϕ1113864 1113865T[M] 1

ϕ1113864 1113865T[M] ϕ1113864 1113865

(33)



00 05 10 15

Ωω

ζ sd (

)

(1 ζsdres)

20 25 30

30

25

20

15

10

5

0

ndash5


(a)


(1 ζsdres)

00 05 10 15

Ωω

20 25 30

ζ sd (

)

20

15

10

5

0

ndash5


(b)


(1 ζsdres)

00 05 10 15

Ωω

20 25 30

ζ sd (

)

8

6

4

2

0


(c)


(1 ζsdres)

00 05 10 15

Ωω

20 25 30

ζ sd (

)4

3

2

1

0


(d)







c1 λπ

cα

ωuo( 11138571minus α (34)


0 20 40 60


ζ1

f 1

80

ω = 20 rads

ω = 14 rads

ω = 10 rads

ω = 539 rads

ω = 2 rads

100

14

12

10

00

02

04

06

08






50

0 1 2

T (s)

ζ sd (

)

3 4 5

40

30

20

10

0

cα = 284

cα = 400

cα = 100


ERM

Equation (29)

Equation (30)


50

05 1510 20 3025

uumlg0 (ms2)

cα = 100

cα = 284

cα = 400


ζ sd (

)

35 4540 50

40

30

20

10

0










k5

k6

m5

m6

c5

k4

k3

k2

k1

m4

m3

m2

m1

c4

c3

c2

c1

c6








6

NEWH

KEAT

SATI

KEHO

ELCE

TAFT

TABA

IRAR

IRAU

MACO

5

4

3

2

1

1 10 20 30 40

Stor

y


50 60 70



ADR ()






5 Conclusions






22

20

18

16

14

12

10

8

608 10 12 14 16 18 20 22 24 26 28 30 32

Ωmω

ζ sd (

)


Groundmotion

KEATSATIKEHO


274

241

226

208

207

167

155

148

126

Ωmω


150

100

50

0

ndash50

ndash100

ndash1500 10 20 30 40 50 60

t (s)

Disp

(m

m)






Data Availability




Acknowledgments


References








































Serialno123456

100020003000400050006000

Dampingcoefficient

12

3

45

6

16

14

12

10

8

6

4

2

0


res (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(a)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

56

16

14

12

10

8

6

4

2

0


ζ sdre

s (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(b)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

5

6

8

7

6

5

4

3

2

1

0


ζ sdre

s (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(c)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

56

8

7

6

5

4

3

2

1

0


res (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(d)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

56

6

5

4

2

3

1

0


ζ sdre

s (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(e)

Serialno123456

100020003000400050006000

Dampingcoefficient

12

34

56

5

4

3

2

1

0


ζ sdre

s (

)

01 02 03 04 05 06 07 08 09 10

a0 (ms2)

ERMEquation (21)

(f )






uw euroug0

2ω2ζ1 (25)




2ω2ζ1 (26)




00 05 10 15

Ωω

ζ sd (

)

(1 ζsdres)

20

ERM

25 30

30

25

20

15

10

5

0

ndash5

(a)


(1 ζsdres)

ERM

00 05 10 15

Ωω

20 25 30

ζ sd (

)

20

15

10

5

0

ndash5

(b)


(1 ζsdres)

ERM

00 05 10 15

Ωω

20 25 30

ζ sd (

)

8

6

4

2

0

(c)


(1 ζsdres)

ERM

00 05 10 15

Ωω

20 25 30

ζ sd (

)

4

3

2

1

0

(d)




f1 ζ1( 1113857 b1ζ31 + b2ζ

21 + b3ζ1 + b4 (27)





ζsdres λπ

cα

2mω2minus α1



ζsdres λπ

cα

2mω2minus α1



ζsd

a1Ωm

ω minus 11113874 1113875

2+ ζsdres

Ωm

ωlt 1

ζsdresΩm

ω 1

a2Ωm

ω minus 11113874 1113875 + ζsdres

Ωm

ωgt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(30)



Tm 1113936iC

2i 1Fi( 1113857

1113936iC2i

(31)



Ωm 2πTm

(32)



01 02 03 04 05 06 07α

a 1

08 09 10

70

60

40

30

20

50

10

0



a 2

01 02 03 04 05 06 07α

08 09 10

10

6

4

8

2

ndash2

0




4 Case Study



Mr ϕ1113864 1113865T[M] ϕ1113864 1113865

ϕ1113864 1113865T[M] 1

ϕ1113864 1113865T[M] ϕ1113864 1113865

Cr ϕ1113864 1113865T[C] ϕ1113864 1113865

ϕ1113864 1113865T[M] 1

ϕ1113864 1113865T

[M] ϕ1113864 1113865

Kr ϕ1113864 1113865T[K] ϕ1113864 1113865

ϕ1113864 1113865T[M] 1

ϕ1113864 1113865T[M] ϕ1113864 1113865

(33)



00 05 10 15

Ωω

ζ sd (

)

(1 ζsdres)

20 25 30

30

25

20

15

10

5

0

ndash5


(a)


(1 ζsdres)

00 05 10 15

Ωω

20 25 30

ζ sd (

)

20

15

10

5

0

ndash5


(b)


(1 ζsdres)

00 05 10 15

Ωω

20 25 30

ζ sd (

)

8

6

4

2

0


(c)


(1 ζsdres)

00 05 10 15

Ωω

20 25 30

ζ sd (

)4

3

2

1

0


(d)







c1 λπ

cα

ωuo( 11138571minus α (34)


0 20 40 60


ζ1

f 1

80

ω = 20 rads

ω = 14 rads

ω = 10 rads

ω = 539 rads

ω = 2 rads

100

14

12

10

00

02

04

06

08






50

0 1 2

T (s)

ζ sd (

)

3 4 5

40

30

20

10

0

cα = 284

cα = 400

cα = 100


ERM

Equation (29)

Equation (30)


50

05 1510 20 3025

uumlg0 (ms2)

cα = 100

cα = 284

cα = 400


ζ sd (

)

35 4540 50

40

30

20

10

0










k5

k6

m5

m6

c5

k4

k3

k2

k1

m4

m3

m2

m1

c4

c3

c2

c1

c6








6

NEWH

KEAT

SATI

KEHO

ELCE

TAFT

TABA

IRAR

IRAU

MACO

5

4

3

2

1

1 10 20 30 40

Stor

y


50 60 70



ADR ()






5 Conclusions






22

20

18

16

14

12

10

8

608 10 12 14 16 18 20 22 24 26 28 30 32

Ωmω

ζ sd (

)


Groundmotion

KEATSATIKEHO


274

241

226

208

207

167

155

148

126

Ωmω


150

100

50

0

ndash50

ndash100

ndash1500 10 20 30 40 50 60

t (s)

Disp

(m

m)






Data Availability




Acknowledgments


References











































uw euroug0

2ω2ζ1 (25)




2ω2ζ1 (26)




00 05 10 15

Ωω

ζ sd (

)

(1 ζsdres)

20

ERM

25 30

30

25

20

15

10

5

0

ndash5

(a)


(1 ζsdres)

ERM

00 05 10 15

Ωω

20 25 30

ζ sd (

)

20

15

10

5

0

ndash5

(b)


(1 ζsdres)

ERM

00 05 10 15

Ωω

20 25 30

ζ sd (

)

8

6

4

2

0

(c)


(1 ζsdres)

ERM

00 05 10 15

Ωω

20 25 30

ζ sd (

)

4

3

2

1

0

(d)




f1 ζ1( 1113857 b1ζ31 + b2ζ

21 + b3ζ1 + b4 (27)





ζsdres λπ

cα

2mω2minus α1



ζsdres λπ

cα

2mω2minus α1



ζsd

a1Ωm

ω minus 11113874 1113875

2+ ζsdres

Ωm

ωlt 1

ζsdresΩm

ω 1

a2Ωm

ω minus 11113874 1113875 + ζsdres

Ωm

ωgt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(30)



Tm 1113936iC

2i 1Fi( 1113857

1113936iC2i

(31)



Ωm 2πTm

(32)



01 02 03 04 05 06 07α

a 1

08 09 10

70

60

40

30

20

50

10

0



a 2

01 02 03 04 05 06 07α

08 09 10

10

6

4

8

2

ndash2

0




4 Case Study



Mr ϕ1113864 1113865T[M] ϕ1113864 1113865

ϕ1113864 1113865T[M] 1

ϕ1113864 1113865T[M] ϕ1113864 1113865

Cr ϕ1113864 1113865T[C] ϕ1113864 1113865

ϕ1113864 1113865T[M] 1

ϕ1113864 1113865T

[M] ϕ1113864 1113865

Kr ϕ1113864 1113865T[K] ϕ1113864 1113865

ϕ1113864 1113865T[M] 1

ϕ1113864 1113865T[M] ϕ1113864 1113865

(33)



00 05 10 15

Ωω

ζ sd (

)

(1 ζsdres)

20 25 30

30

25

20

15

10

5

0

ndash5


(a)


(1 ζsdres)

00 05 10 15

Ωω

20 25 30

ζ sd (

)

20

15

10

5

0

ndash5


(b)


(1 ζsdres)

00 05 10 15

Ωω

20 25 30

ζ sd (

)

8

6

4

2

0


(c)


(1 ζsdres)

00 05 10 15

Ωω

20 25 30

ζ sd (

)4

3

2

1

0


(d)







c1 λπ

cα

ωuo( 11138571minus α (34)


0 20 40 60


ζ1

f 1

80

ω = 20 rads

ω = 14 rads

ω = 10 rads

ω = 539 rads

ω = 2 rads

100

14

12

10

00

02

04

06

08






50

0 1 2

T (s)

ζ sd (

)

3 4 5

40

30

20

10

0

cα = 284

cα = 400

cα = 100


ERM

Equation (29)

Equation (30)


50

05 1510 20 3025

uumlg0 (ms2)

cα = 100

cα = 284

cα = 400


ζ sd (

)

35 4540 50

40

30

20

10

0










k5

k6

m5

m6

c5

k4

k3

k2

k1

m4

m3

m2

m1

c4

c3

c2

c1

c6








6

NEWH

KEAT

SATI

KEHO

ELCE

TAFT

TABA

IRAR

IRAU

MACO

5

4

3

2

1

1 10 20 30 40

Stor

y


50 60 70



ADR ()






5 Conclusions






22

20

18

16

14

12

10

8

608 10 12 14 16 18 20 22 24 26 28 30 32

Ωmω

ζ sd (

)


Groundmotion

KEATSATIKEHO


274

241

226

208

207

167

155

148

126

Ωmω


150

100

50

0

ndash50

ndash100

ndash1500 10 20 30 40 50 60

t (s)

Disp

(m

m)






Data Availability




Acknowledgments


References









































f1 ζ1( 1113857 b1ζ31 + b2ζ

21 + b3ζ1 + b4 (27)





ζsdres λπ

cα

2mω2minus α1



ζsdres λπ

cα

2mω2minus α1



ζsd

a1Ωm

ω minus 11113874 1113875

2+ ζsdres

Ωm

ωlt 1

ζsdresΩm

ω 1

a2Ωm

ω minus 11113874 1113875 + ζsdres

Ωm

ωgt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(30)



Tm 1113936iC

2i 1Fi( 1113857

1113936iC2i

(31)



Ωm 2πTm

(32)



01 02 03 04 05 06 07α

a 1

08 09 10

70

60

40

30

20

50

10

0



a 2

01 02 03 04 05 06 07α

08 09 10

10

6

4

8

2

ndash2

0




4 Case Study



Mr ϕ1113864 1113865T[M] ϕ1113864 1113865

ϕ1113864 1113865T[M] 1

ϕ1113864 1113865T[M] ϕ1113864 1113865

Cr ϕ1113864 1113865T[C] ϕ1113864 1113865

ϕ1113864 1113865T[M] 1

ϕ1113864 1113865T

[M] ϕ1113864 1113865

Kr ϕ1113864 1113865T[K] ϕ1113864 1113865

ϕ1113864 1113865T[M] 1

ϕ1113864 1113865T[M] ϕ1113864 1113865

(33)



00 05 10 15

Ωω

ζ sd (

)

(1 ζsdres)

20 25 30

30

25

20

15

10

5

0

ndash5


(a)


(1 ζsdres)

00 05 10 15

Ωω

20 25 30

ζ sd (

)

20

15

10

5

0

ndash5


(b)


(1 ζsdres)

00 05 10 15

Ωω

20 25 30

ζ sd (

)

8

6

4

2

0


(c)


(1 ζsdres)

00 05 10 15

Ωω

20 25 30

ζ sd (

)4

3

2

1

0


(d)







c1 λπ

cα

ωuo( 11138571minus α (34)


0 20 40 60


ζ1

f 1

80

ω = 20 rads

ω = 14 rads

ω = 10 rads

ω = 539 rads

ω = 2 rads

100

14

12

10

00

02

04

06

08






50

0 1 2

T (s)

ζ sd (

)

3 4 5

40

30

20

10

0

cα = 284

cα = 400

cα = 100


ERM

Equation (29)

Equation (30)


50

05 1510 20 3025

uumlg0 (ms2)

cα = 100

cα = 284

cα = 400


ζ sd (

)

35 4540 50

40

30

20

10

0










k5

k6

m5

m6

c5

k4

k3

k2

k1

m4

m3

m2

m1

c4

c3

c2

c1

c6








6

NEWH

KEAT

SATI

KEHO

ELCE

TAFT

TABA

IRAR

IRAU

MACO

5

4

3

2

1

1 10 20 30 40

Stor

y


50 60 70



ADR ()






5 Conclusions






22

20

18

16

14

12

10

8

608 10 12 14 16 18 20 22 24 26 28 30 32

Ωmω

ζ sd (

)


Groundmotion

KEATSATIKEHO


274

241

226

208

207

167

155

148

126

Ωmω


150

100

50

0

ndash50

ndash100

ndash1500 10 20 30 40 50 60

t (s)

Disp

(m

m)






Data Availability




Acknowledgments


References








































4 Case Study



Mr ϕ1113864 1113865T[M] ϕ1113864 1113865

ϕ1113864 1113865T[M] 1

ϕ1113864 1113865T[M] ϕ1113864 1113865

Cr ϕ1113864 1113865T[C] ϕ1113864 1113865

ϕ1113864 1113865T[M] 1

ϕ1113864 1113865T

[M] ϕ1113864 1113865

Kr ϕ1113864 1113865T[K] ϕ1113864 1113865

ϕ1113864 1113865T[M] 1

ϕ1113864 1113865T[M] ϕ1113864 1113865

(33)



00 05 10 15

Ωω

ζ sd (

)

(1 ζsdres)

20 25 30

30

25

20

15

10

5

0

ndash5


(a)


(1 ζsdres)

00 05 10 15

Ωω

20 25 30

ζ sd (

)

20

15

10

5

0

ndash5


(b)


(1 ζsdres)

00 05 10 15

Ωω

20 25 30

ζ sd (

)

8

6

4

2

0


(c)


(1 ζsdres)

00 05 10 15

Ωω

20 25 30

ζ sd (

)4

3

2

1

0


(d)







c1 λπ

cα

ωuo( 11138571minus α (34)


0 20 40 60


ζ1

f 1

80

ω = 20 rads

ω = 14 rads

ω = 10 rads

ω = 539 rads

ω = 2 rads

100

14

12

10

00

02

04

06

08






50

0 1 2

T (s)

ζ sd (

)

3 4 5

40

30

20

10

0

cα = 284

cα = 400

cα = 100


ERM

Equation (29)

Equation (30)


50

05 1510 20 3025

uumlg0 (ms2)

cα = 100

cα = 284

cα = 400


ζ sd (

)

35 4540 50

40

30

20

10

0










k5

k6

m5

m6

c5

k4

k3

k2

k1

m4

m3

m2

m1

c4

c3

c2

c1

c6








6

NEWH

KEAT

SATI

KEHO

ELCE

TAFT

TABA

IRAR

IRAU

MACO

5

4

3

2

1

1 10 20 30 40

Stor

y


50 60 70



ADR ()






5 Conclusions






22

20

18

16

14

12

10

8

608 10 12 14 16 18 20 22 24 26 28 30 32

Ωmω

ζ sd (

)


Groundmotion

KEATSATIKEHO


274

241

226

208

207

167

155

148

126

Ωmω


150

100

50

0

ndash50

ndash100

ndash1500 10 20 30 40 50 60

t (s)

Disp

(m

m)






Data Availability




Acknowledgments


References












































c1 λπ

cα

ωuo( 11138571minus α (34)


0 20 40 60


ζ1

f 1

80

ω = 20 rads

ω = 14 rads

ω = 10 rads

ω = 539 rads

ω = 2 rads

100

14

12

10

00

02

04

06

08






50

0 1 2

T (s)

ζ sd (

)

3 4 5

40

30

20

10

0

cα = 284

cα = 400

cα = 100


ERM

Equation (29)

Equation (30)


50

05 1510 20 3025

uumlg0 (ms2)

cα = 100

cα = 284

cα = 400


ζ sd (

)

35 4540 50

40

30

20

10

0










k5

k6

m5

m6

c5

k4

k3

k2

k1

m4

m3

m2

m1

c4

c3

c2

c1

c6








6

NEWH

KEAT

SATI

KEHO

ELCE

TAFT

TABA

IRAR

IRAU

MACO

5

4

3

2

1

1 10 20 30 40

Stor

y


50 60 70



ADR ()






5 Conclusions






22

20

18

16

14

12

10

8

608 10 12 14 16 18 20 22 24 26 28 30 32

Ωmω

ζ sd (

)


Groundmotion

KEATSATIKEHO


274

241

226

208

207

167

155

148

126

Ωmω


150

100

50

0

ndash50

ndash100

ndash1500 10 20 30 40 50 60

t (s)

Disp

(m

m)






Data Availability




Acknowledgments


References














































k5

k6

m5

m6

c5

k4

k3

k2

k1

m4

m3

m2

m1

c4

c3

c2

c1

c6








6

NEWH

KEAT

SATI

KEHO

ELCE

TAFT

TABA

IRAR

IRAU

MACO

5

4

3

2

1

1 10 20 30 40

Stor

y


50 60 70



ADR ()






5 Conclusions






22

20

18

16

14

12

10

8

608 10 12 14 16 18 20 22 24 26 28 30 32

Ωmω

ζ sd (

)


Groundmotion

KEATSATIKEHO


274

241

226

208

207

167

155

148

126

Ωmω


150

100

50

0

ndash50

ndash100

ndash1500 10 20 30 40 50 60

t (s)

Disp

(m

m)






Data Availability




Acknowledgments


References











































6

NEWH

KEAT

SATI

KEHO

ELCE

TAFT

TABA

IRAR

IRAU

MACO

5

4

3

2

1

1 10 20 30 40

Stor

y


50 60 70



ADR ()






5 Conclusions






22

20

18

16

14

12

10

8

608 10 12 14 16 18 20 22 24 26 28 30 32

Ωmω

ζ sd (

)


Groundmotion

KEATSATIKEHO


274

241

226

208

207

167

155

148

126

Ωmω


150

100

50

0

ndash50

ndash100

ndash1500 10 20 30 40 50 60

t (s)

Disp

(m

m)






Data Availability




Acknowledgments


References








































5 Conclusions






22

20

18

16

14

12

10

8

608 10 12 14 16 18 20 22 24 26 28 30 32

Ωmω

ζ sd (

)


Groundmotion

KEATSATIKEHO


274

241

226

208

207

167

155

148

126

Ωmω


150

100

50

0

ndash50

ndash100

ndash1500 10 20 30 40 50 60

t (s)

Disp

(m

m)






Data Availability




Acknowledgments


References










































Data Availability




Acknowledgments


References






















































Documents

ANewMethodtoCalculateAdditionalDamping … · 2020. 8. 24. · the aseismic design of structures conﬁgured with VFDs, in which the mechanical characteristics of the VFDs are easily