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An experimental study on a generalized Maxwell model for nonlinear

viscoelastic dampers used in seismic isolation

Lyan-Ywan Lu a,, Ging-Long Lin b, Ming-Hsiang Shih c

a Department of Construction Engineering, National Kaohsiung First University of Science and Technology, 1 University Road, Yenchao, Kaohsiung 824, Taiwanb Department of Civil Engineering, National Chung Hsing University, Taichung, Taiwanc Department of Civil Engineering, National Chi-Nan University, Puli, Nantou County, Taiwan

a r t i c l e i n f o

Article history:

Received 16 May 2011

Revised 7 September 2011

Accepted 7 September 2011

Available online 4 November 2011

Keywords:

Maxwell model

Fluid damper

Viscoelastic damper

Sliding isolator

Seismic isolation

Energy dissipation

Shaking table test

a b s t r a c t

Long-stroke fluid dampers may be installed under seismic isolation systems to provide supplementary

damping. Due to the larger vibration amplitude and velocity, highly nonlinear viscoelastic behavior

may exist in a long-stroke fluid damper. In order to accurately simulate the hysteretic behavior of such

a damper, this paper presents and experimentally verifies a mathematical model called the generalized

Maxwell model (GMM). Similar to the classic Maxwell model, the GMM is composed of a stiffness and a

viscous elements connected in series. However, nonlinearity is incorporated into both elements of the

GMM by assuming that their resistant forces are exponential functions of the relative velocity and defor-

mation of the damper. By adjusting the two exponential coefficients, the GMM is able to simulate the

more complicated viscoelastic behavior of fluid dampers. The GMM is reduced to the Maxwell model

when both exponential coefficients are set to one. To verify the GMM, both an element test with har-

monic excitations and a shaking table test with seismic excitations were conducted for a long-stroke fluid

damper with highly nonlinear viscoelastic behavior. The result of the element test confirms that the

GMM model is very accurate in simulating thehysteretic property of thefluid damper under a wide range

of excitation frequencies, while the classic Maxwell and theviscous models may only be accurate under a

certain excitation frequency. Moreover, the shaking table test, in which the fluid damper is used to pro-vide supplementary damping for a sliding isolation system, demonstrates that the GMM is able to more

accurately predict the amount of energy dissipation by the damper and also the peak isolator drift of the

isolation system, especially for an earthquake with a long-period pulse.

2011 Published by Elsevier Ltd.

1. Introduction

Viscoelastic (VE) dampers generally represent a wide class of

energy dissipation devices whose forcedisplacement relationship

has viscous or viscoelastic mechanical properties [13]. In recent

decades, VE dampers have been widely applied to mitigate the

effects of vibration in civil engineering structures caused by various

excitations, including traffic load, wind load and seismic load. Forinstance, they have been used to reduce the level of deck vertical

vibration of high-speed rail bridges [4], mitigate the wind response

of buildings [5], control the torsional seismic response of structures

[6], reduce structural motion due to near-fault earthquakes [7,8],

retrofit heritage buildings or high-tech factories under seismic load

[9,10], suppress the large seismic displacement for rocking struc-

tures [11], protect bridges by installing dampersat expansion joints

[12], suppress the seismic motions of two parallel structures [13],

and suppress the excessive isolator drift in an isolation system

induced by near-fault earthquakes [14,15]. The effectiveness of VE

dampers has been verified by experimental means, such as shaking

table tests [16] or mass exciters [17]. Moreover, for practical pur-

poses, some studies regarding design issues related to VE dampers,

such as spectra design methods [7,18], optimal design methods

[1921], a unified theory for both VE and Viscous dampers [1],

and a step by step design procedure [2], etc., have also been pro-

posed by researchers.A VE damper is usually connected to a structure through braces

(diagonal, chevron or toggled), and is activated by the relative

motion of the structure to which it is connected. The mechanical

property of a VE damper may behave linearly or nonlinearly,

depending on the constituent material (i.e., what the damper is

made of) and the fabrication technique (how the damper is made).

Generally speaking, a VE damper can be fabricated in either a fluid

or solid form. A fluid-type VE damper usually consists of an orifice

piston moving in a hollow cylinder filled with highly viscous fluid,

whilea solid-typeVE damperconsists of a solid viscoelasticmaterial

bonded to steel plates [3]. As compared to solid VE dampers, fluid

dampers are able to provide longer strokes and their mechanical

0141-0296/$ - see front matter 2011 Published by Elsevier Ltd.doi:10.1016/j.engstruct.2011.09.012

Corresponding author. Tel.: +886 7 6011000x2127.

E-mail address: [email protected] (L.-Y. Lu).

Engineering Structures 34 (2012) 111123

Contents lists available at SciVerse ScienceDirect

Engineering Structures

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e n g s t r u c t
http://dx.doi.org/10.1016/j.engstruct.2011.09.012mailto:[email protected]://dx.doi.org/10.1016/j.engstruct.2011.09.012http://www.sciencedirect.com/science/journal/01410296http://www.elsevier.com/locate/engstructhttp://www.elsevier.com/locate/engstructhttp://www.sciencedirect.com/science/journal/01410296http://dx.doi.org/10.1016/j.engstruct.2011.09.012mailto:[email protected]://dx.doi.org/10.1016/j.engstruct.2011.09.012

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propertiesare lesssensitiveto thevariationof ambienttemperature.

Therefore, fluid VE dampers have wider applications, especially in

the field of earthquake engineering, in which the vibration ampli-

tudes of protected structures are usuallyrelatively large when com-

pared to other engineering applications. Additionally, some

researchers have developed viscoplastic dampers which use visco-

elastic or elastomeric solid materials in series connected to a yield-

ing or friction device to control the peak damper force [22,23].

The accurate analysis of structures with VE dampers usually re-

lies on a mathematical model that is able to precisely capture the

mechanical properties of the installed VE dampers. In the afore-

mentioned studies about VE dampers, four types of classic VE mod-

els are often employed: the linear viscous, nonlinear viscous,

Maxwell and Kelvin models. The linear viscous model assumes that

the damper force is directly proportional to the relative velocity

across the damper [6,10], and the only parameter in this model

is the damping coefficient. In the nonlinear viscous model, the

damper force is assumed to be proportional to the power of the rel-

ative velocity [8,18,24]; therefore, this model has two parameters,

i.e., the damping coefficient and the power of the velocity. The

Maxwell model is composed of stiffness (spring) and viscous

(dashpot) elements connected in series [2527], soit has two mod-

el parameters, i.e., the stiffness and the damping coefficients. The

Kelvin model is similar to the Maxwell model, except that the stiff-

ness and viscous elements are placed in parallel, and thus it also

has two model parameters [12,19,28]. The hysteretic diagram

resulting from these four classic VE models are an eclipse, an in-

clined eclipse or a loop symmetric about the vertical and horizon-

tal axes of the coordinates.

The four classic models mentioned above, which have at most

two model parameters, are simple to use and easily applied to

numerically simulate the dynamic responses of structures with

VE dampers; however, they may be accurate only for dampers

operating at a relatively small amplitude or low frequency

[29,30]. Some studies have found experimental evidence that with

a longer stroke, higher velocity or more extreme conditions, the

hysteresis loop of a fluid damper may become a twisted ellipticshape that is anti-symmetric about the origin of the hysteretic

diagram [15,31,32]. This hysteretic behavior cannot be explained

by the simple classic models, and thus more sophisticated models

that consider more factors are needed. Some of the more advanced

VE models that may involve more computational complexity are

reviewed below.

Black and Makris [29] conducted an experimental study on the

viscous heating effect of fluid dampers. They concluded that under

long-stroke motions (more than two times the piston diameter),

only a few cycles are needed to raise the internal fluid temperature

significantly and cause a drop in the damper force. Their study also

proposed a cooling law to estimate the temperature change in a

fluid damper. Based on test data of full-scale fluid dampers, Wolfe

et al. [30] suggested that the inertia effect of a moving fluid massshould be included in the model, especially for large-size dampers

under high-frequency or high-velocity excitations. Miyamoto et al.

[33] developed and experimentally verified a mathematical model

for fluid dampers under limit-state conditions. Their model was

established in the OpenSees computer programwith the gap, hook,

dashpot, spring and undercut elements connected in parallel and in

series. By separating the measured damper forces, Hou et al. [32]

found that the force associated with the stiffness element in a

Maxwell model actually may not be linear, so they proposed a sim-

ple forcedisplacement relationship to account for this non-linear

stiffness effect. Based on experimental data, Mazza and Vulcano

[5] proposed a six-element generalized model, which can be con-

sidered as an in-parallel-combination of two Maxwell models

and one Kelvin model, for VE dampers used in the seismic andwind response controls of steel frames. Moreover, to accurately

capture the frequency dependence behavior of fluid dampers,

Makris et al. [25] proposed a mathematical model using fractional

derivatives, which has the ability to capture the viscoelastic behav-

ior using fewer model parameters [34,35].

As shown previously, although both simple and advanced mod-

els for VE dampers have been proposed, it may not be possible to

accurately represent all types of such dampers with one single

model, because the actual mechanical properties of an installed

VE damper may depend on several factors, including the constitu-

ent material, fabrication details (e.g., geometric parameters), and

operating and field conditions. To this end, the current study will

focus on a particular problem, i.e., how to accurately model a

fluid-type VE damper used for the protection of a seismic isolation

system. A seismic isolation system may suffer from excessive iso-

lator drift when subjected to a near-fault earthquake which con-

tains strong long-period components [36,37]. Nevertheless,

recent studies have also demonstrated that this excessive isolator

drift can be effectively suppressed by using a damper with visco-

elastic properties, without significantly interfering with the isola-

tion performance [14,15,38]. Moreover, since the stroke demand

for a damper installed under a seismic isolation system may be

ten times that of the dampers installed inside a structural frame,

this study will focus on fluid-type VE dampers, which are capable

of providing a longer stroke than solid-type dampers.

The objective of the current study is to propose and verify an

accurate mathematical model for a long-stroke fluid VE damper

used in conjunction with a seismic isolation system. The proposed

model, called the generalized Maxwell model (GMM), is a modifi-

cation of the familiar Maxwell model. The major difference be-

tween the two models is that the GMM incorporates nonlinearity

in both the spring and the viscous element. The resistant forces

of both elements have fractional exponential coefficients, and by

adjusting these coefficients GMM is able to simulate the more

complicated viscous-elastic behavior of a fluid damper. In order

to investigate its ability to capture the hysteretic behavior of a fluid

damper under excitations of different frequencies, the proposed

model will be verified experimentally by both an element testand a shaking table test. In the element test, harmonic excitations

with different frequencies will be imposed on the specimen fluid

damper, while in the shaking table test seismic ground motions

with strong near-fault characteristics will be imposed on a seismi-

cally isolated structure protected by the specimen damper.

2. Generalized Maxwell model (GMM)

Experimental studies have demonstrated that the resistance

forces of some viscous dampers are not only related to the velocity

of the damper, but also to the damper deformation. Such mechan-

ical properties may be mathematically modeled by the generalized

Maxwell model (GMM), as shown schematically in Fig. 1. Therestoring and viscous force components of this model are simu-

lated by the nonlinear elastic (stiffness) element and the nonlinear

viscous element, respectively, with these two elements connected

in series. Mechanically, this model must satisfy the following kine-

matic conditions

d(t) = de(t) + dv(t)

Elastic part Viscous part

de(t) dv(t)

fd(t)fd(t)

Fig. 1. A schematic of generalized Maxwell model (GMM) for a VE damper.

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dt det dvt 1:a

_dt _det _dvt 1:b

and also the following force condition

fdt sgndet ke j det jne sgn _dvt cv j

_dvt jnv 2

where the sign function sgn is defined as

sgnx

1 if x > 0

0 if x 0

1 if x < 0

8>: 3

In Eqs. (1) and (2), fd(t) represents the damper (axial) force; d(t)

denotes the total damper deformation; de(t) and dv(t) denote the

deformations of the stiffness and viscous components, respec-

tively; _dt, _det and_dvt represent the deformation rates (the

relative velocities) corresponding to d(t), de(t) and dv(t); ke denotes

the stiffness value of the stiffness element; cv is the damping

coefficient of the viscous element; ne and nv are the exponential

coefficients of the deformation de(t) and the velocity_dvt, respec-

tively. If ne and ne are taken to be unity, i.e., ne nv 1, Eq. (2)

reduces to the force equation of a classic Maxwell model, in which

both the forces of the elastic and viscous components behave

linearly, i.e.,

fdt ke det cv_dvt 4

In this study, the GMM model shown in Eq. (1)-(3) will be

employed to simulate a nonlinear fluid VE damper. As shown

above, there are four characteristic parameters to be determined

in the GMM, i.e., ke, cv, nv and ne.

3. Numerical method for the generalized maxwell model

The damper hysteretic response resulting from the GMM model

will be simulated and compared to test data in a latter section. Thissection will first explain the numerical method used in the simula-

tion of the GMM. Since the hysteresis loop of a damper represents

the relation between the damper force fd(t) and the total damper

deformation d(t), fd(t) has to be solved as a function of d(t) first.

To this end, Eq. (2) is rewritten as

kecv

sgndet jdetj

nenv sgn _dvt j

_dvtj 5

The last equation can be further expressed as

kecv

jdetj

nenv

1det _dvt 6

Substituting _dvt from Eq. (1.b) in Eq. (6) yields

_det At det _dt 7

where A(t) is a time-variant coefficient of the following form

At kecv

jdetj

nenv

1 8

If the damper total deformation _dt is prescribed and treated as

an external excitation, Eq. (7) represents the dynamic equation of

the GMM damper excited by the prescribed disturbance _dt.

Mathematically, this equation is also a first-order ordinary differ-

ential equation with de(t) being the variable and A(t) being a

time-variant coefficient, and it can be solved by many numerical

methods. Once de(t) is solved from Eq. (7), it is substituted back

to Eq. (2) to compute the damper force fd(t). In this study, Eq. (7)will be solved numerically by using the discrete-time state-space

technique [39]. To do so, let us first assume that A(t) is constant

within each computational time step, while _dt varies linearly in

each time step, i.e.,

At Ak; for k Dt6 t< k 1 Dt 9

_dt _dk _dk 1 _dk

Dtt k Dt;

for k Dt6 t< k 1 Dt 10

where k represents the number of the time step; Dt is the time

interval; A[k] denotes the value of the quantity A evaluated at the

k-th time step, i.e., at t= kDt. With the assumption of Eq. (9), within

each time step Eq. (7) is no different from a time-invariant state-

space equation with de(t) being the state variable. The discrete-time

solution of this state-spaceequation can be readilyobtained and ex-

pressed as [39]

dek 1 Adk dek B1k_dk B2k

_dk 1 11

where

Adk eAkDt 12

B1k Ak1

Adk 1

DtAk

21 Adk

B2k Ak1

1

DtAk

2Adk 1 13

After de[k + 1] is computed from Eq. (11), it is substituted back

to Eq. (2) to obtain the damper force at the (k + 1)-th time step, i.e.,

fdk 1 sgndek 1 ke jdek 1jne 14

Eqs. (11) and (14) together state that the damper force fd[k] of

the GMM damper can be evaluated step by step in time, provided

that the time history of the total deformation rate _dk across the

two ends of the damper is specified. Therefore, Eqs. (11) and (14)

can be treated as the numerical solution of the GMM and will be

employed in the later numerical simulation.Notably, since the classic Maxwell model is a special case of the

GMM with nv ne 1, the above numerical solution can also be

applied to simulate the Maxwell damper. In such a case, the

time-variant term A[k] in Eqs. (8) and (9) is reduced to a constant,

i.e.,Ak ke=cv, and consequently the termsAd[k], B1[k] and B2[k]

in Eqs. (12) and (13) are also reduced to constants. Therefore, the

numerical simulation of the Maxwell model is computationally

simpler than that of the GMM.

4. Effect of GMM parameters on damper hysteretic behavior

As mentioned previously, the GMM model has four characteris-

tic parameters, and this section examines how these affect the hys-

teretic behavior by employing the numerical method from theprevious section. This investigation can benefit the design of a fluid

damper with the GMM property, since the designer will have a

qualitative concept about the influence of each parameter. Fig. 2

illustrates the changes in the damper hysteresis loop when a cer-

tain parameter of the GMM is varied and the other three parame-

ters are fixed. The numerical values for the four parameters used in

the simulation are listed in Table 1. In Fig. 2, the damper force is

simulated by using the numerical method explained in the last sec-

tion. A harmonic displacement excitation d(t) of the following form

is considered as applied at the two ends of the damper

dt d0 sin2p f t 15

where d0 and f represent the displacement amplitude and the fre-

quency of the harmonic excitation, respectively. In the simulation,the values of d0 and fare taken to be 50 mm and 1 Hz, respectively.

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(1) Effect of parameter ke: Fig. 2(a) shows that the variation of the

damper hysteresis loop with different values of the stiffness

coefficient ke, when the other parameters are nv ne 1

and cv 100 (N-s/mm). In this case, the hysteretic behavior

of the GMM model is reduced to that of a classic Maxwell

model whose hysteresis loop is an inclined ellipse. As the stiff-ness coefficient ke decreases, the inclination of the loop

becomes more obvious and the area of the loop declines. In

contrast, as the stiffness increases, the loop area increasesand the Maxwell model approaches a linear viscous model

without the stiffness effect.

(2) Effect of parameter cv: Fig. 2(b) compares the hysteresis loops

for various values of the viscous coefficient cv, when

nv ne 1 and ke 1000 (N/mm). The figure shows that

the area of the hysteresis loop is enlarged as the viscous coef-

ficient increases. At the same time, due to the increase of the

damper force, the loop is more inclined and the stiffness effect

becomes more obvious.

(3) Effect of parameter ne: Fig. 2(c) illustrates the variation of the

hysteresis loop as the stiffness exponent ne of the GMM is

changing, while the other parameters are fixed at nv 1 (lin-

ear viscous force), cv 100 (N-s/mm) and ke 1000 (N/mm)

(see Table 1). It is evident that the hysteretic behavior of thedamper is insensitive to ne, when ne is greater than one. When

Fig. 2. Parametric study of the GMM model.

Table 1

Numerical values used in the parametric study of the generalized Maxwell model.Parameter name Value

Stiffness coefficient, ke 1000 (kN=mne ) or as shown in Fig. 2(a)

Damping coefficient, cv 100 (kN=m=snv ) or as shown in Fig. 2(b)

Stiffness exponent, ne 1.0 or as shown in Fig. 2(c)

Damping exponent, nv 1.0 or as shown in Fig. 2(d)


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ne is reduced to less than one, the hysteresis loop becomes a

distorted ellipse with two inclined line segments at the two

ends, and the loop area is significantly reduced.

(4) Effect of parameter nv: Fig. 2(d) depicts the hysteresis loop of

the GMM with the variation of the viscous exponent nv, while

the other three parameters are fixed at ne 1 (linear restoring

force), cv 100 (N-s/mm) and ke 1000 (N/mm) (see Table

1). As shown in the figure, the loop is more sensitive to the

value ofnv, when nv is less than one. The enclosed area quickly

diminishes when nv is reduced to less than 0.5. Moreover,

when nv is larger than unity, the hysteresis loop becomes more

like a parallelogram with four round corners. Two inclined line

segments which represent the stiffness effect can be clearly

observed at the two end of the loop.

(5) Effect of varying ne and nv: Different from Fig. 2(c) and (d), in

which one of the exponents nv and ne remains linear,

Fig. 2(e) illustrates the change of the hysteresis loop when

both coefficients nv and ne are identical and varied simulta-

neously. In this case, both the restoring and viscous forces

may become nonlinear. The figure shows that the enclosed

area of the hysteresis loop grows when a higher value is

adopted for nv and ne. When both nv and ne become greater

than one, the hysteresis loop becomes neither an inclined

ellipse nor a parallelogram. The hysteresis loop is transformed

into a mango-shaped loop that has slightly-crooked tips at

both ends. This shape does not have a symmetric axis, but

rather is anti-symmetric about the origin, and it cannot be

described by the classic Maxwell model (nv ne 1) or other

viscous models.

Furthermore, Fig. 2(d) actually represents the hysteresis loops

of a nonlinear viscoelastic model extensively implemented in some

well-known commercial structural analysis programs, such as the

SAP2000 software. This model, which will be called the SAP2000

model hereafter for convenience, generally consists of a linear

spring (ne 1) and a nonlinear viscous element (nv1) connected

in series, and is actually a special case of the GMM model withne 1. On the other hand, Fig. 2(e) represents the hysteresis loops

of a more general GMM model in which both the spring and vis-

cous elements are nonlinear (nv and ne not equal to one). The com-

parison between Fig. 2(d) and (e) demonstrates that without the

nonlinear spring effect, the SAP2000 model with the nonlinear vis-

cous element alone cannot achieve a similar hysteresis loop char-

acterized by the general GMM model.

5. Cyclic element test with harmonic excitations

5.1. Test setup

A long-stroke viscous fluid damper was fabricated to verify the

GMM model. The mechanical properties of this damper will betested by using an element cyclic test and a shaking table test.

The element test aims to investigate the nonlinear hysteretic

property of the damper, while the shaking table test is to demon-

strate the dynamic response of the damper to a seismic load. This

section reports the results of the element test, while the shaking

table test will be discussed in the next section.

Fig. 3 illustrates the setup of the element test. The fluid damperwas fabricated by using standard manufacture techniques and was

filled with regular automobile oil. Fig. 4 shows a photo of the dam-

per piston head with orifices. The number of orifices, which can be

reduced by screw bolts, determines the resulting damper coeffi-

cient. The maximum stroke of the damper was designed to be

250 mm, which is much larger than that of a common viscous

damper installed inside a seismic structure. In the test, the damper

was excited by a hydraulic actuator (see Fig. 3) with a prescribed

harmonic displacement in the form of Eq. (15). The magnitude of

the harmonic displacement was set to d0 = 50 mm, while the exci-

tation frequency was increased gradually from f= 0.6 to 1.5 Hz at

an interval of 0.1 Hz. In other words, there were a total of 10 exci-

tation frequencies tested, covering the primary working frequency

range of the damper. The axial resistance force and the relative dis-placement of the damper were all recorded in the test, so the

experimental hysteresis loop can be depicted after the test.

5.2. Identification of the parametric values for the GMM model

Using the experimental data of the damper hysteresis loop, this

subsection identifies the optimal parametric values that best fit the

test data for the GMM model, and this is achieved by using an opti-

mization searching scheme called the generalized pattern search

(GPS) algorithm [40]. Different from the traditional optimization

methods that usually require information about the gradient or

derivatives of the objective function to be minimized, the GPS algo-

rithm solves an optimization problem by directly searching a set of

points, called a mesh, around the current point, and then lookingfor the one whose value of the objective function is lower than

the value of the current point. Once the point with a lower value

Fig. 3. Setup of the damper element test.

Fig. 4. Piston head with orifices of the tested fluid damper.


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is found in the mesh, the GPS replaces the current point with this

new point at the next step of the algorithm. Since the GPS does not

require gradient and derivatives of the objective function, it can be

applied to solve complicated optimization problems whose objec-

tive function is not differentiable or even continuous.

The optimal parametric values of the GMM model are searched

for using the GPS algorithm to minimize the following objective

function,J, which is defined by the root mean square of the error

between the theoretical and experimental data,

Minimize JX10i1

Ji X10i1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

Ni

XNik1

fi

d; thek f

id; exp

k2

vuut 16

where Ji denotes the part of the objective function associated with

the i-th excitation frequency; fi

d; thek and fi

d; expk denote the k-th

data point of the damper force obtained from the experiment and

the theoretical model for the i-th excitation frequency, respectively;

Ni represents the total number of data points in the test cycles of the

i-th excitation frequency. Note that in the last equation i = 110,

since there are a total of 10 excitation frequencies in the test: 0.6,

0.7, . . ., 1.5 Hz. Eq. (16) means that if a certain set of the GMM

parameter values leads to the minimum value of J, it will also havethe least error between the experimental and theoretical data for

the damper force. In other words, the optimal values of the GMM

parameters are the ones that best describe the experimental dam-

per force. Notably, the theoretical value fd; thek in Eq. (16) is com-

puted by the numerical solution given in Eqs. (11) and (14).

Moreover, since only the damper relative displacement d[k] is mea-

sured in the element test, the relative velocity terms in Eq. (11) are

approximated by the following difference equation

_dk 1 1

Dtdk 1 dk 17

where the sampling period Dt is taken to be 0.01 s in the test.

The second row of Table 2 lists the results of searching for theoptimal GMM parameter values and the minimized function value

J. In addition, for the latter comparative study, the aforementioned

GPS algorithm is also employed to determine the optimal parame-

ter values for the other three models: the Maxwell model, the lin-

ear viscous model and the SAP2000 model mentioned previously,

when they are used to fit the test data. The optimal parameter

values and the minimized function value J are also summarized

in the third, fourth and fifth rows ofTable 2, for the Maxwell, linear

viscous, SAP2000 models, respectively. Fig. 5 compares the value of

Jas a function of the number of the searching steps for all models.

As shown by Table 2 and Fig. 5, the searching algorithm efficiently

converges on J= 109 kN for the GMM model, which is the lowest

objective function value among the four models. This implies that

the GMM model predicts the nonlinear damper force more pre-

cisely than the other models.

In addition, the last row ofTable 2 and Fig. 5 show that the min-imum value of the objective functionJand the optimal parameters

for the SAP2000 model are converged to those of the Maxwell

model, in which nv 1. This implies that, without the nonlinear

stiffness effect, the optimal SAP2000 model that minimizes the

error between the experimental and analytical data is reduced to

the optimal Maxwell model. Furthermore, since the Maxwell and

SAP2000 models lead to the same optimal parameters for the given

test data, only the Maxwell model was considered in the later com-

parative study.

5.3. Comparison of hysteresis loops predicted by various models

Fig. 6 compares the experimental hysteresis loops of the dam-

per with those simulated by the GMM, Maxwell and the linear-vis-cous models, for the excitation frequencies 0.6, 0.9, 1.2 and 1.5 Hz.

The dotted lines in Fig. 6 represent the experimental loops ob-

tained from the element test, while the solid grey lines depict

the theoretical loops simulated by using the parameter values

listed in Table 2 for each model. Notably, in Fig. 6 the damper rel-

ative displacement d[k] and velocity _dk taken from the test data

are used in simulating the theoretical loops. Therefore, the exper-

imental and theoretical loops have exactly the same relative

displacement and velocity, whereas the damper force of the theo-

retical loop is computed according to each of the models. The

experiment data in Fig. 6 show that when under harmonic excita-

tion the hysteresis loop of the damper specimen is olive-shaped

under a lower excitation frequency, but this is then distorted under

a higher frequency and transformed into more like a mango shape,

which is anti-symmetric about the origin. Fig. 6(a) demonstrates

Table 2

Optimal parametric values for various models obtained by GPS searching scheme.

Model name Equation Parameter Optimal value Index value

Generalized Maxwell fdt sgndet ke j det jne

sgn _dvt cv j _dvt jnv

Stiffness coef. ke 711103kN=mne J= 109 kN

Viscous coef. cv 242 kN=m=snv

Stiffness exponent ne 1.92

Damping exponent nv 2.03

Maxwell fdt ke det cv_dvt Stiffness coef. ke 2432 kN/m J= 437 kN

Viscous coef. cv 65.3 kN-s/m

Linear viscous fdt cv_dt Viscous coef. cv 64.7 kN-s/m J= 492 kN

SAP2000 model fdt ke det cv_dnvv t Stiffness coef. ke 2432 kN/m J= 437 kN

Viscous coef. cv 65.3kN=m=snv

Damping exponent nv 1.0

Fig. 5. Value of the objective function J vs. number of searching steps.


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that the simulated results obtained by the GMM model match very

well with the experimental loops under all the excitation frequen-

cies. On the other hand, Fig. 6(b) and (c) show that both the

Maxwell and linear-viscous models lead to elliptic hysteresis loops

that cannot match well with the experimental loops for all the fre-

quencies, even though they have less simulation error around the

frequency 0.9 Hz. Notably, under a higher frequency, the Maxwell

model leads to a rotated ellipse. Fig. 6(b) and (c) also illustrate that

both the Maxwell and the linear-viscous model overestimate the

damper force under a lower frequency, but underestimate the force

under a higher one. Fig. 6 thus indicates that the mechanical prop-

erties of the tested long-stroke damper can be well characterized

by the GMM model.

Moreover, since the area of a hysteresis loop is equivalent to the

amount of the energy dissipated by the damper per cycle, Table 3

compares the area of the experimental loop per cycle to those of

Fig. 6. Comparison of experimental and theoretical hysteresis loops for the element test.


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the theoretical loops predicted by the three models under each

excitation frequency, 0.6, 0.7, . . ., 1.5 Hz. Note that in the table

the numbers in parentheses denote the error percentages of the

dissipated energy predicted by each damper model. The error per-

centage is defined as

e Athe Aexp

Aexp

100% 18

where Athe and Aexp denote the areas of the theoretical and experi-

mental hysteresis loops, respectively. The areas of the experimental

hysteresis loops were integrated numerically by using the trapezoi-

dal rule that can be written as

Aexp XNkk1

1

2fd;expk fd;expk 1dk 1 dk 19

where Nk denotes the number of data points per cycle. Similarly, the

areas of the theoretical loops Athe were also calculated by Eq. (19).

Table 3 shows that the GMM is the most accurate model in predict-

ing the energy dissipation for the whole frequency range consid-ered, with an average error less than 1% (e 0:7%). On the other

hand, the energy dissipation predicted by the Maxwell and linear-

viscous models have an average error percentage around 19%, and

are only accurate around the mean excitation frequency 1.1 Hz. This

is consistent with the findings of Fig. 6, in which both the Maxwell

and the linear models overestimate and underestimate the energy

dissipation capacity of the damper, under a lower and a higher exci-

tation frequency, respectively. Therefore, Table 3 concludes that

with regard to the issue of energy dissipation, the GMM model is

still the most accurate one for fluid damper testing.

6. Shaking table test with seismic excitations

The previous section demonstrated that the GMM model is able

to accurately capture the hysteretic behavior of a highly-nonlinear

fluid damper under harmonic excitation. To further investigate the

ability of the GMM model to simulate the dynamic response of a

fluid damper subjected to a seismic load, a shaking table test was

conducted. In the test, the fluid damper was installed in an FPS

(friction pendulum system) sliding isolation system to provide

supplementary damping to the system. The setup and the results

of the shaking table test will be discussed in this section.

6.1. Test setup

(1) The isolated structure: Fig. 7(a) shows the overall setup of the

shaking table test, which was conducted in the earthquakesimulation laboratory of the National Center for Research on

Earthquake Engineering (NCREE, Taiwan). As shown, the

long-stroke fluid damper was placed between the shaking

table and the bottom of the base isolated structure. The iso-

lated structure has a height of 1.62 m and a square plane view

whose length and width are both 2.2 m (see Fig. 7(a)). The

fluid damper was placed along the center line of the structure

and parallel to the excitation direction. The isolated structure,

whose structural properties are listed in Table 4, is an assem-

bly of three mass layers. The mass layers are filled with lead

blocks and are connected to each other by vertical tension rods

through the four corners of the layers. The structure has very

high rigidity, with an identified natural frequency of 17.5 Hz,

so it may be considered as a rigid block. The reason for using

such a rigid structure is to focus on the response of the dam-

per, rather than the dynamic behavior of the super-structure.

As shown in Fig. 7, two displacement sensors (LVDT) were

deployed to measure the damper relative displacement and

the isolator displacement. A load-cell to measure the damper

force was also installed at one end of the damper. Additionally,

accelerometers were placed on the top of the rigid block andalso on the shaking table, to measure the structural and

ground accelerations.

(2) The FPS isolators: Four FPS-type sliding isolators were used in

the shaking table test. Each of the four columns of the isolated

structure was mounted on a sliding isolator (see Fig. 7), whose

properties are also summarized in Table 4. The radius of the

FPS sliding spherical surface was designed to be 560 mm,

which gives an isolation period of 1.5 s. The maximum isolator

drift is 150 mm. The slider of each isolator is composed of

polymer material. To identify the friction coefficient of the

sliding isolators, an isolator element test was conducted inde-

pendently before the shaking table test [36], and the results

show that the isolators have a stable and typical FPS hysteresis

loop with a friction coefficient of about 0.09.

Additionally, for later numerical simulation, Fig. 7(b) shows the

mathematical model of the tested system. As shown, the FPS isola-

tion system is modeled by a linear spring and a friction element

connected in parallel. The spring is used to simulate the resorting

force produced by the concave sliding surfaces of the FPS isolators,

while the friction element is used to simulate the friction force ex-

erted on the sliding surfaces. In the simulation, the FPS stiffness is

determined by ki M g=R, where M denotes the total mass of the

isolated structure and R represents the radius of the sliding sur-

faces (see Table 4) The friction coefficient li is determined by the

constituent materials of the slider and sliding surface. Notably, kiand li are the only two parameters in modeling the FPS system,

therefore, the accuracy of their values will affect the simulated re-sponse of the isolated system.

Table 3

Comparison of the areas of damper hysteresis-loops per cycle in the element test.

Excitation frequency (Hz) Experimental value (m-kN) Theoretical value (m-kN)

GMM Maxwell Linear viscous

0.6 0.999 0.999 (0.0%) 1.582 (58.4%) 1.586 (58.8%)

0.7 1.435 1.440 (0.3%) 2.022 (40.9%) 2.025 (41.1%)

0.8 1.943 1.914 (1.5%) 2.422 (25.7%) 2.426 (25.9%)

0.9 2.402 2.391 (0.5%) 2.772 (15.4%) 2.781 (15.8%)

1.0 2.863 2.874 (0.4%) 3.089 (7.9%) 3.106 (8.5%)

1.1 3.346 3.351 (0.2%) 3.374 (0.8%) 3.402 (1.7%)

1.2 3.797 3.781 (0.4%) 3.599 (5.2%) 3.647 (4.0%)

1.3 4.288 4.314 (0.6%) 3.899 (9.2%) 3.959 (7.7%)

1.4 4.775 4.730 (0.9%) 4.091 (14.3%) 4.173 (12.6%)

1.5 5.239 5.152 (1.7%) 4.286 (18.2%) 4.387 (16.3%)

Average error (0.7%) (19.6%) (19.2%)

Note: Numbers in parentheses denote the error percentages e of the theoretical values (see Eq. (18)).


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(3) Input ground accelerations: To be more representative, the fol-

lowing three ground excitations with very different character-

istics were used in the shaking table test: (a) the first one,

representing a typical far-field earthquake, is the famous

1940 El Centro earthquake (see Fig. 8(a)). (b) The second

one, representing a typical near-fault earthquake, is the

ground acceleration recorded by the El Centro station (Array

No. 6) in the 1979 Imperial Valley earthquake, California.

Fig. 8(b) depicts the waveform of this near-fault earthquake,

and it has an obvious pulse-like waveform between 5 and

15 s. (c) The third one is an artificially generated pulse-like

earthquake (see Fig. 8(c)), whose acceleration time-history is

generated by the following function:

ugt xp vp sinxpt; 0 6 t6 Tp 20

where ugt denotes the ground acceleration, xp is called the pulse

frequency, Tp ( 2p=xp) is the pulse period, vp is the velocityamplitude of the pulse in velocity domain. This pulse waveform,

which was proposed by Makris and Chang [38], is used to simulate

the typical pulse-component that exists in a near-fault earthquake.

Fig. 8(c) depicts the generated pulse earthquake with Tp = 2s.

6.2. Comparison of damper hysteresis loops

Under the El Centro Earthquake, Fig. 9 compares the damper

hysteresis loop obtained from the shaking table test to those sim-

ulated by the GMM, Maxwell and linear-viscous models. Similarly,

Figs. 10 and 11 compare the hysteresis loops for the Imperial Val-

ley and the artificial-pulse earthquakes, respectively. To compare

the energy dissipated by the damper, Table 5 calculates the areas

of the damper experimental and theoretical hysteresis loops in-

duced by all three types of earthquakes. Note that in the simulation

of Figs. 911 and Table 5, the following numerical measures are

adopted: (1) in order to compute the nonlinear time-history re-

sponse of the isolated system that involves the super-structure,the FPS-isolators and the fluid damper, the nonlinear numerical

method proposed by Lu et al. [41] was employed. This method,

developed on the basis of discrete-time state-space formulation,

is able to simulate the dynamic response of a seismic structure

with various nonlinear passive control devices. The GMM dis-

crete-time solution described in Section 3, which is also developed

based on state-space formulation, was then incorporated into this

numerical approach. (2) For each damper model, the parametric

values listed in Table 2 were used. (3) For the isolators and the iso-

lated structure, the parameters listed in Table 4 were used. (4) The

actual, measured accelerations of the shaking table were used as

the input ground accelerations in the simulation.

Because the modeling and measurement uncertainties in the

shaking table test are far more complicated than in the elementtest, Figs. 9(a), 10(a) and 11(a) show that for the GMM model the

Fig. 7. The shaking table test.

Table 4

Properties of isolated structure and isolators for the shaking table test.

Item Parameter Value

Isolated structure

(Rigid mass block)

Natural frequency 17.5 Hz

Total mass M 16.38 ton

Height 1.62 m

Height of mass center 1.08 m

FPS isolators Isolation period Ti 2pffiffiffiffiffiffiffiffiR=g

p1.5 s

Radius of sliding curvature R 0.56m

Friction coefficient li 0.09

Maximum isolator displacement 0.15 m


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0 10 20 30 40-4

-3

-2

-1

0

1

2

3

4

Time (s)

Acceleration(m/s2)

0 10 20 30 40-6

-4

-2

0

2

4

6

Time (s)

Acceleration(m/s2)

0 2 4 6 8-3

-2

-1

0

1

2

3

Time (s)

Acceleration(m/s2)

(a) El Centro (b) Imperial Valley (c) Artificial pulse

Fig. 8. Ground accelerations used in the shaking table test.

Fig. 9. Comparison of damper hysteresis loops for the El Centro Earthquake (PGA = 0.4 g).

Fig. 10. Comparison of damper hysteresis loops for the Imperial Valley Earthquake (PGA = 0.4 g).

Fig. 11. Comparison of damper hysteresis loops for the artificial-pulse earthquake (Pulse period = 1.75s, PGA = 0.2 g).


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consistency between the experimental and theoretical loops is not

as good as that in the element test (see Fig. 6(a)). Nevertheless,

when comparing the three damper models, Figs. 911 show that

the GMM model leads to a hysteresis loop whose shape is most

consistent with the experimental one under all types of the seismic

excitations. As for predicting the amount of energy dissipated, Ta-

ble 5 shows that the GMM model gives the most accurate areas of

the hysteresis loops with an average error of 7.2%, which is about

one-third of the Maxwell model error (21.6%) and one-fourth of

the linear-viscous model error (28.2%). The Maxwell model is thus

slightly more accurate than the linear-viscous model. Table 5 alsoshows that the GMM model is especially accurate for a seismic load

with a long-period pulse-like waveform, such as the Imperial Val-

ley and artificial-pulse earthquakes. The results ofTable 5 and Figs.

911 together demonstrate that under seismic excitations the

GMM is the model most capable of capturing the hysteretic behav-

ior of a nonlinear fluid damper.

Notably, due to the test arrangement, the damper elongation in

Figs. 911 is actually equal to the isolator displacement in the test.

Therefore, the relatively larger deviation observed in the damper

hysteresis loops shown in Figs. 10 and 11 for all models are primar-

ily due to the deviation of the simulated isolator displacement in

the time-history analysis. The deviation of the isolator displace-

ment can be due to the modeling or parametric errors contributed

by both the FPS isolators and the fluid damper.

6.3. Comparison of the isolation system responses

The previous section focused on the capability of the three

mathematical models with regard to simulating the hysteretic

property of the fluid damper under seismic loads. Nevertheless,

for engineering applications, it would be also desirable to investi-

gate the influences of using different models on the overall

response of the isolation system. Since the responses of the most

engineering importance for an isolation system are the maximum

isolator displacement (isolator drift) and the maximum super-

structural acceleration, the experimental data of these two peak

values will be investigated in this subsection.

For the different ground motions, Table 6 compares the exper-

imental and theoretical peak values of the isolator displacements

predicted by the three models. Similarly, Table 7 compares the

experimental and theoretical peak values of the structural acceler-

ations. For the artificial-pulse ground motion, Figs. 12 and 13

further compare the time-history responses of the isolator dis-

placement and the structural acceleration, respectively. Notably,

in Tables 6 and 7, a numbers in parentheses denote the error per-centage of a certain peak response predicted by the mathematical

model, and the error percentage is defined similar to Eq. (18). For

the prediction of the peak isolator displacement, Table 6 and

Fig. 12 show that the GMM model has the least average error of

about 10% among the three models. The artificial-pulse response

in Table 6 shows the GMM is especially accurate for the isolator

displacement induced by a long-period pulse-like earthquake, in

which the other two models result in a fairly large error (more

than 30%) for the isolator displacement. On the other hand, for

the prediction of the peak structural acceleration, the last row of

Table 7 shows that the three models have an equal level of the per-

centage error. This implies that the selection among the three

models has very little influence on the peak acceleration response

of the isolation system. Moreover, the last row of Table 7 showsthat for all three models the averages of the acceleration errors

are about 30%, which is higher than the displacement errors shown

in Table 6. However, Fig. 13 illustrates that the time-history re-

sponses simulated by all the three models are very consistent with

the experimental one, even though the peak values of the theoret-

ical responses have an error of about 10%. The relatively larger er-

rors in the acceleration responses of the three models listed in

Table 7 may be due to the measurement noise of the acceleration

signal, which is more easily contaminated by ambient electrical

noise.

Table 5

Comparison of the areas of the damper hysteresis-loops for the shaking table test.

Earthquake name (PGA) Experimentalvalue (m-kN) Theoretical value (103 J)


El Centro(0.4 g) 0.693 0.773 (11.5%) 0.858 (23.8%) 0.901 (30.0%)

Imperial Valley (0.4 g) 0.210 0.191 (9.0%) 0.270 (28.6%) 0.288 (37.1%)

Artificial pulse (0.2 g) 0.343 0.347 (1.2%) 0.300 (12.5%) 0.283 (17.5%)

Average error (7.2%) (21.6%) (28.2%)

Note: Numbers in parentheses denote the error percentages e of the theoretical values (see Eq. (18)).

Fig. 12. Comparison of isolator displacements for the artificial-pulse earthquake (Pulse period = 1.75s, PGA = 0.2 g).


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7. Conclusions

To provide a more accurate analytical tool for fluid dampers

with a highly nonlinear viscoelastic property, a mathematical mod-

el called the generalized Maxwell model (GMM) is proposed and

experimentally verified in this paper. Developed based on the Max-

well model, the GMM is composed of a nonlinear spring and non-

linear viscous elements, which are connected in series. The

resistant forces of both nonlinear elements in the GMM are propor-

tional to the powers of the relative displacement and velocity of

the damper. As a result, the GMM has four parameters, i.e., thestiffness coefficient, viscous coefficient, stiffness exponent and

damping exponent. The GMM will reduce to a Maxwell model

when both the stiffness and damping exponents are equal to one.

By adjusting these four parameters, the GMM model is able to sim-

ulate fluid dampers with more complicated hysteretic behavior.

To experimentally verify the proposed GMM model, both an

element test and a shaking table test were conducted on a long-

stroke fluid damper that was fabricated by using the standard

manufacturing technique. In the element test, harmonic displace-

ment excitations with different frequencies ranging from 0.6 to

1.5 Hz were imposed on the damper. The results show that the

specimen damper exhibits highly nonlinear viscoelastic behavior.

Under a lower excitation frequency, the olive-shaped hysteresis

loop is symmetrical about the coordinates axes; however, whenthe excitation frequency increases, its hysteresis loop gradually

transforms into a mango shape that is anti-symmetrical about

the origin. By using a numerical searching scheme, the optimal

parametric values that best fit the data of the element test were

sought for the GMM model, as well as for the classic Maxwell

model and the linear viscous model for comparison purposes.

The comparison between the experimental and simulated data

demonstrates that the GMM model is able to more accurately cap-

ture the aforementioned hysteretic behavior of the fluid damper

within the whole range of excitation frequencies, while the other

two models are only accurate under the mean excitation fre-

quency. Over the tested frequency range, the amount of energy dis-sipated per cycle predicted by the GMM has an average error of less

than 1%, while the average errors for the other two models are

about 19%.

Moreover, in the shakingtable test, the fluid damper was placed

under an FPS (friction pendulum system) isolated structure system

to collect the damper response data for seismic loads. Ground mo-

tions with far-field and near-fault characteristics were all consid-

ered in the shaking table test, so the exerted damper seismic

responses can be more representative. The responses of the dam-

per and the isolated system under the same ground motions were

then simulated by using the GMM, Maxwell and linear-viscous

models, separately. The comparison between experimental and

simulated responses shows that the GMM provides the most accu-

rate prediction with regard to the amount of the energy dissipatedby the damper under an earthquake with either far-field or near-

Fig. 13. Comparison of structural acceleration responses for the artificial-pulse earthquake (Pulse period = 1.75s, PGA = 0.2 g).

Table 6

Peak isolator displacements for the shaking table test.

Earthquake name (PGA) Experimental value (m) Theoretical value (m)


El Centro (0.4 g) 0.026 0.029 (11.5%) 0.025 (3.8%) 0.023 (11.5%)Imperial Valley (0.4 g) 0.016 0.018 (12.5%) 0.017 (6.3%) 0.014 (12.5%)


Average error (10.1%) (15.9%) (20.5%)

Note: Numbers in parentheses denote the error percentages of the theoretical values.

Table 7

Peak acceleration responses for the shaking table test.

Earthquake name (PGA) Experimental value (m=s2) Theoretical value (m=s2)


El Centro(0.4 g) 2.132 2.294 (7.6%) 2.386 (11.9%) 2.350 (10.2%)

Imperial Valley (0.4 g) 1.767 2.952 (67.1%) 3.004 (70.0%) 2.777 (57.2%)


Average error (28.9%) (31.3%) (26.6%)

Note: Numbers in parentheses denote the error percentages of the theoretical value.


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fault characteristics. The GMM model is also able to predict the

peak isolator displacement more accurately, especially for an

earthquake with a long-period pulse waveform. Nevertheless, the

comparative results also demonstrate that the three models have

the same accuracy in predicting the peak acceleration of the isola-

tion system.

Acknowledgements

This research was sponsored in part by the National Science

Council of ROC. (Taiwan), through Grant No. 91-2211-E-327-005.

The authors are grateful to the National Center for Research on

Earthquake Engineering (NCREE, Taipei) for their technical support

on the shaking table test, and also to Ms. Shiu-Wen Tzeng for pre-

paring the test data and plots.

References

[1] Fu Y, Kasai K. Comparative study of frames using viscoelastic and viscous

dampers. J Struct Eng (ASCE) 1998;124(5):51322.

[2] Lee KS, Fan CP, Sause R, Ricles J. Simplified design procedure for frame

buildings with viscoelastic or elastomeric structural dampers. Earthquake EngStruct Dynam 2005;34(10):127184.

[3] Symans MD, Charney FA, Whittaker AS, Constantinou MC, Kircher CA, Johnson

MW, McNamara RJ. Energy dissipation systems for seismic applications:

current practice and recent developments. J Struct Eng (ASCE)

2008;134(1):321.

[4] Martnez Rodrigo MD, Lavado J, Museros P. Dynamic performance of existing

high-speed railway bridges under resonant conditions retrofitted with fluid

viscous dampers. Eng Struct 2010;32(3):80828.

[5] Mazza F, Vulcano A. Control of the earthquake and wind dynamic response of

steel-framed buildings by using additional braces and/or viscoelastic dampers.

Earthquake Eng Struct Dynam 2011;40(2):15574.

[6] Petti L, De Iuliis M. Torsional seismic response control of asymmetric-plan

systems by using viscous dampers. Eng Struct 2008;30(11):337788.

[7] Hubbard DT, Mavroeidis GP. Damping coefficients for near-fault ground

motion response spectra. Soil Dynam Earthquake Eng 2011;31(3):40117.

[8] Xu Z, Agrawal AK, He WL, Tan P. Performance of passive energy dissipation

systems during near-field ground motion type pulses. Eng Struct

2007;29(2):22436.

[9] Chrysostomou CZ, Demetriou T, Pittas M, Stassis A. Retrofit of a church withlinear viscous dampers. Struct Control Health Monit 2005;12(2):197212.

[10] Hwang JH, Wang SJ, Huang YN, Chen JF. A seismic retrofit method by

connecting viscous dampers for microelectronics factories. Earthquake Eng

Struct Dynam 2007;36(11):146180.

[11] Roh H, Reinhorn AM. Modeling and seismic response of structures with

concrete rocking columns and viscous dampers. Eng Struct

2010;32(8):2096107.

[12] Feng MQ, Kim JM, Shinozuka M, Purasinghe R. Viscoelastic dampers at

expansion joints for seismic protection of bridges. J Bridge Eng (ASCE)

2000;5(1):6774.

[13] Zhu HP, Ge DD, Huang X. Optimum connecting dampers to reduce the seismic

responses of parallel structures. J Sound Vib 2011;330(9):193149.

[14] Providakis CP. Effect of supplemental damping on LRB and FPS seismic

isolators under near-fault ground motions. Soil Dynam Earthquake Eng

2009;29(1):8090.

[15] Lu LY, Shih MH, Yeh SW, Lin GL, Tzeng SW. Experimental verification of using

viscous damper for response mitigation of isolation systems subjected to near-

fault earthquakes. In: Proceedings of the 11th East Asian-pacific conference on

structural engineering and construction 2008, Taipei, Taiwan, November 1921, Paper No. 254.

[16] Chang KC, Lin YY, Chen CY. Shaking table study on displacement-based design

for seismic retrofit of existing buildings using nonlinear viscous dampers. J

Struct Eng (ASCE) 2008;134(4):67181.

[17] MinKW, Kim J, Lee SH.Vibration tests of 5-storey steel frame with viscoelastic

dampers. Eng. Struct. 2004;26(6):8319.

[18] Pekcan G, Mander JB, Chen SS. Fundamental considerations for the design of

non-linear viscous dampers. Earthquake Eng Struct Dynam

1999;28(11):140525.

[19] Gluck N, Reinhorn AM, Gluck J, Levy R. Design of supplemental dampers for

control of structures. J Struct Eng (ASCE) 1996;122(12):13949.

[20] Uetani K, Tsuji M, Takewaki I. Application of an optimum design method topractical building frames with viscous dampers and hysteretic dampers. Eng

Struct 2003;25(5):57992.

[21] Singh MP, Verma NP, Moreschi LM. Seismic analysis and design with Maxwell

dampers. J Eng Mech (ASCE) 2003;129(3):27382.

[22] Ibrahim YE, Marshall J, Charney FA. A visco-plastic device for seismic

protection of structures. J Constr Steel Res 2007;63(11):151528.

[23] Karavasilis TL, Ricles JM, Sause R, Chen C. Experimental evaluation of the

seismic performance of steel MRFs with compressed elastomer dampers using

large-scale real-time hybrid simulation. Eng Struct 2011;33(6):185969.

[24] Lin WH, Chopra AK. Earthquake response of elastic SDF systems with non-

linear fluid viscous dampers. Earthquake Eng Struct Dynam

2002;31(9):162342.

[25] Makris N, Constantinou MC, Dargush GF. Analytical model of viscoelastic fluid

dampers. J Struct Eng (ASCE) 1993;119(11):331025.

[26] Hatada T, Kobori T, Ishida M, Niwa N. Dynamic analysis of structures with

Maxwell model. Earthquake Eng Struct Dynam 2000;29(2):15976.

[27] Chen YT, Chai YH. Effects of brace stiffness on performance of structures with

supplemental Maxwell model-based brace-damper systems. Earthquake Eng

Struct Dynam 2011;40(1):7592.

[28] SinghMP, Chang TS. Seismic analysis of structures withviscoelastic dampers. J

Eng Mech (ASCE) 2009;135(6):57180.

[29] Black CJ, Makris N. Viscous heating of fluid dampers under small and large

amplitude motions: experimental studies and parametric modeling. J Eng

Mech (ASCE) 2007;133(5):56677.

[30] Wolfe RW, Yun HB, Masri S, Tasbihgoo F, Benzoni G. Fidelity of reduced-order

models for large-scale nonlinear orifice viscous dampers. Struc Control Health

Monit 2008;15(8):114363.

[31] Contantinou MC, Syman MD. Experimental study of seismic response of

buildings with supplemental fluid dampers. Struct Design Tall Build

1993;2:93132.

[32] Hou CY, Hsu DS, Chen HY. Experimental verification of the restoring-stiffness

concept used in fluid dampers for seismic energy dissipation. Struct Des Tall

Special Build 2005;14(1):113.

[33] Miyamoto HK, Gilani ASJ, Wada A, Ariyaratana C. Limit states and failure

mechanisms of viscous dampers and the implications for large earthquakes.

Earthquake Eng Struct Dynam 2010;39(11):127997.

[34] Lewandowski R, Chorazyczewski B. Identification of the parameters of theKelvinVoigt and the Maxwell fractional models, used to modeling of

viscoelastic dampers. Comput Struct 2010;88(12):117.

[35] Hwang JS, Tsai CH, Wang SJ, Huang YN. Experimental study of RC building

structures with supplemental viscous dampers and lightly reinforced walls.

Eng Struct 2006;28(13):181624.

[36] Lu LY, Shih MH, Tzeng SW, Chang Chien CS. Experiment of a sliding isolated

structure subjected to near-fault ground motion. In: Proceedings of the 7th

Pacific conference on earthquake engineering 2003, February 1315,

Christchurch, New Zealand.

[37] Jangid RS. Optimum lead-rubber isolation bearings for near-fault motions. Eng

Struct 2007;29:250313.

[38] Makris N, Chang SP. Effect of viscous, viscoplastic and friction damping on the

response of seismic isolated structures. Earthquake Eng Struct Dynam

2000;29(1):85107.

[39] Meirovitch L. Dynamics and control of structures. New York: John Wiley &

Sons; 1990.

[40] MATLAB. The Math Works, Inc., Natick, Massachusetts; 2000.

[41] Lu LY, Lin GL, Lin CH. A unified analysis model for energy dissipation devices

used in seismic structures. Comp-Aided Civil Infrastruct Eng 2009;24(1):4161.


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