1-s2.0-s0020768301000269-main

7/28/2019 1-s2.0-S0020768301000269-main

1/28

Analytical modeling of viscoelastic dampers for structural and

vibration control q

S.W. Park *

Georgia Institute of Technology, Atlanta, GA 30332, USA

Received 28 March 2000

Abstract

Dierent approaches to the mathematical modeling of viscoelastic dampers are addressed and their theoretical basis

and performance are compared. The standard mechanical model (SMM) comprising linear springs and dashpots is

shown to accurately describe the broad-band rheological behavior of common viscoelastic dampers and be more ef-

cient than other models such as the fractional derivative model and the modied power law. The SMM renders a

Prony series expression for the modulus and compliance functions in the time domain, and the remarkable mathe-

matical eciency associated with the exponential basis functions of a Prony series greatly facilitates model calibration

and interconversion. While cumbersome, nonlinear regression is usually required for other models, a simple collocation

or least-squares method can be used to t the SMM to available experimental data. The model allows viscoelastic

material functions to be readily determined either directly from the experimental data or through interconversion from

a function established in another domain. Numerical examples on two common viscoelastic dampers demonstrate theadvantages of the SMM over fractional derivative and power-law models. Detailed computational procedures for tting

and interconversion are discussed and illustrated. Published experimental data from a viscoelastic liquid damper and a

viscoelastic solid damper are used in the examples. 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Viscoelastic damper; Rheological model; Standard mechanical model; Prony series; Fitting; Interconversion

1. Introduction

Protection of constructed facilities from damaging natural hazards has become an increasingly impor-

tant issue. Recurring destructive seismic events and hurricanes in the United States and elsewhere point to acompelling need for the development of eective protective systems against such hazards. Various means

have been developed and implemented over the years to control excessive structural response to environ-

mental forces induced by earthquakes or winds. For example, in passive structural control, energy dissi-

pation devices are added to a structure so that a large portion of the input energy can be dissipated through

International Journal of Solids and Structures 38 (2001) 80658092

www.elsevier.com/locate/ijsolstr

qDisclaimer: the views expressed in this article are those of the author and do not necessarily represent the views of the FHWA.*Now at U.S. Federal Highway Administration, Turner-Fairbank Highway Research Center, Oce of Infrastructure R&D, HRDI/

PSI, 6300 Georgetown Pike, McLean, VA 22101, USA.

E-mail address: [email protected] (S.W. Park).

0020-7683/01/$ - see front matter

2001 Elsevier Science Ltd. All rights reserved.PII: S0 0 2 0 -7 6 8 3 (0 1 )0 0 0 2 6 -9

7/28/2019 1-s2.0-S0020768301000269-main

2/28

these devices, thereby reducing energy dissipation demand on the original structure. Such devices in-

clude metallic yield dampers, friction dampers, viscous or viscoelastic dampers, and tuned mass dampers

(Housner et al., 1997; Soong and Dargush, 1977).

Viscoelastic dampers have long been used in the control of vibration and noise in aerospace structures

and industrial machines (Kerwin, 1959; Ross et al., 1959; Jones, 1980; Torvik, 1980; Morgenthaler, 1987;

Ader et al., 1995). Similar applications have been undertaken for civil engineering structures. A pioneering

example is the 10,000 viscoelastic dampers installed in the twin towers of the World Trade Center in New

York in 1969 to mitigate the eects of wind loads (Mahmoodi et al., 1987). This was followed by a number

of other similar applications in the United States and abroad. The implementation of viscoelastic dampers

for seismic mitigation has been realized more recently (Zhang et al., 1989; Zhang and Soong, 1992; Chang

et al., 1993, 1995; Hanson, 1993; Bergman and Hanson, 1993; Tsai, 1993, 1994; Tsai and Lee, 1993a,b; Li

and Tsai, 1994; Samali and Kwok, 1995; Aprile et al., 1997; Hayes et al., 1999; Shukla and Datta, 1999;

Zou and Ou, 2000). Lately, electrorheological and magnetorheological uid dampers whose rheological

properties vary with applied electric or magnetic eld have received keen attention for their potential

applications in semi-active structural and vibration control (Gavin et al., 1996a,b; Dyke et al., 1996, 1998;

Makris, 1997; Sunakoda et al., 2000; Xu et al., 2000; Yang et al., 2000).Analysis of a structure that incorporates viscoelastic dampers normally requires an analytical charac-

terization of the rheological behavior of the dampers. Dierent approaches to the analytical modeling of

the rheological behavior of a linear viscoelastic system are available in the literature. A classical approach

uses a mechanical model comprising a combination of linear springs and dashpots (Bland, 1960; Findley

et al., 1976; Ferry, 1980; Christensen, 1982; Tschoegl, 1989). The stressstrain relation for a linear viscoelastic

system represented by a springdashpot mechanical model is commonly expressed in a dierential operator

form, and the time-domain material functions derived from such a model is expressed by a series of de-

caying exponentials, often referred to as a Prony series. The model has been proved to be consistent with

the molecular theory (Rouse, 1953; Ferry et al., 1955) and the thermodynamic theory (Biot, 1954; Schapery,

1964). The mechanical model analogs and corresponding Prony series representations have long been used

to express the material functions of linear viscoelastic media, and the related topics including tting andinterconversion are well established; see Fung (1965), Findley et al. (1976), Ferry (1980), and Tschoegl

(1989) for a comprehensive treatment of mechanical model theories.

A modeling approach based on fractional calculus has also received considerable attention and been

used in characterizing the rheological behavior of linear viscoelastic systems by a number of authors (e.g.,

Gemant, 1938; Smit and de Vries, 1970; Bagley and Torvik, 1983a,b; Rogers, 1983; Koeller, 1984). This

approach uses the framework of a standard springdashpot mechanical model except that the regular

dierential operators are replaced by fractional-order dierential operators. The primary motivation for the

use of fractional derivatives comes from their ability to describe the broad-band behavior of many vi-

scoelastic materials with a small number of parameters. Other widely used phenomenological models for

linear viscoelasticity include dierent forms of power laws (Schapery, 1974). In particular, the so-called

modied power law (MPL) (Williams, 1964), derived from the phenomenology of polymers, often provides

an excellent representation of the broad-band relaxation or creep behavior of amorphous polymers above

their glass transition temperature. Numerous other mathematical models for linear viscoelasticity are also

available (e.g., Tschoegl, 1989).

A review of the literature indicates that the fractional derivative model (FDM) has predominantly been

used for viscoelastic dampers (Koh and Kelly, 1990; Makris, 1991; Makris and Constantinou, 1991, 1992,

1993; Tsai and Lee, 1993a,b; Makris et al., 1993a,b, 1995; Aprile et al., 1997). For example, Koh and Kelly

(1990) modeled elastomeric bearings using a fractional-order Kelvin model and observed the superiority

of its performance to that of the standard Kelvin model. Makris and Constantinou (1991) modeled a

viscoelastic uid damper using a fractional-order Maxwell model and reported its advantage over the

standard Maxwell model in describing the viscoelastic material functions over a broad range of frequency.

8066 S.W. Park / International Journal of Solids and Structures 38 (2001) 80658092

7/28/2019 1-s2.0-S0020768301000269-main

3/28

A MPL was also used to characterize the relaxation function of a viscoelastic solid damper (Shen and

Soong, 1995). This MPL, with a small number of parameters, was found to describe the broad-band be-

havior of the damper quite well both in the time domain and frequency domain. No particular report has

been identied that addresses the use of the standard mechanical model (SMM) for viscoelastic dampers.

Although FDM is capable of characterizing a broad-band viscoelastic behavior with a small number of

model constants, the complex mathematical expressions of material functions and associated cumbersome

numerical operations make the model less ecient for a routine implementation. Determination of model

constants from experimental data normally requires a dicult, nonlinear regression procedure. In addition,

interconversion between material functions is not always feasible with FDM. The use of a small number of

constants in FDM signies a limited degree of freedom associated with the model and consequently the

model often results in a crude representation of the given data. It is known that a power-law model is

intrinsically linked to FDM (Bagley, 1989), and, therefore, a power-law model has limitations similar to

those of FDM.

In contrast, SMM, due to its sound physical basis and remarkable computational eciency, lends itself

to a better alternative to other prevailing models in the characterization of viscoelastic dampers. Although

each term in a Prony representation can depict only a narrow-band behavior, the series as a whole candescribe a broad-band behavior very accurately. The ndings reported by some investigators (e.g., Koh and

Kelly, 1990; Makris and Constantinou, 1991) that the standard Voigt or Maxwell model does not ade-

quately describe the rheological behavior of viscoelastic dampers can be explained by the narrow-band

representation capabilities of these simple models. Although useful for a conceptual illustration of a vi-

scoelastic phenomenon, these simple models are not adequate for characterization of the broad-band be-

havior of real viscoelastic media (Tschoegl, 1989). Instead, a model composed of multiple Voigt or Maxwell

elements, i.e., the generalized Voigt or generalized Maxwell model, can be used to characterize the broad-

band rheological behavior of linear viscoelastic media.

In Section 2, the linear viscoelastic stressstrain relationship and dierent approaches to the modeling of

linear viscoelasticity are reviewed. In Section 3, characterization of the rheological behavior of viscoelastic

dampers using SMM is presented. Analytical representation of linear viscoelastic material functions indierent domains and the methods of tting and interconversion are discussed. Two numerical examples,

one on a viscoelastic liquid damper and the other on a viscoelastic solid damper, are presented in Sections 4

and 5, respectively. The examples illustrate the detailed procedures for tting and interconversion, and the

performance of SMM is compared with that of other models including FDM and MPL. A further dis-

cussion on the theoretical basis for dierent models discussed in the text are provided in Section 6.

2. Linear viscoelasticity and mathematical models

2.1. The standard mechanical model

The uniaxial, isothermal stressstrain equation for a nonaging, linear viscoelastic material can be rep-

resented by the following Boltzmann superposition integral:

rt

Zt0

Et sdes

dsds 1

where Et is the relaxation modulus, and Et et 0 for I < t< 0. Eq. (1) follows from the memoryhypothesis, smoothness assumptions and mathematical representation theorem (Christensen, 1982). For a

thermorheologically simple material (Morland and Lee, 1960), the stressstrain equation under transient

temperatures (or nonisothermal condition) can also be expressed by Eq. (1) but with the time variable, s,

replaced with the so-called reduced time dened as n

Rt

0ds=aT where aT is a function of temperature

S.W. Park / International Journal of Solids and Structures 38 (2001) 80658092 8067

7/28/2019 1-s2.0-S0020768301000269-main

4/28

called the timetemperature shift factor. Since Eq. (1) is founded on general principles, the equation is valid

for any linear viscoelastic material irrespective of the model employed to express the material function, Et.A number of phenomenological models of the behavior of linear viscoelastic materials are available in the

literature (Ferry, 1980; Tschoegl, 1989).

A classical approach to the modeling of linear viscoelastic behavior employs a mechanical model

composed of linear springs and dashpots, and the stressstrain equation for such a model involves standard

(or ordinary) dierential operators. A general form of the stressstrain equation in dierential operators is

given by (Fung, 1965)

XMm0

amdmr

dtmXNn0

bndne

dtn2

where am and bn are constants, and dm =dtm denotes the mth-order time derivative of the function .

Mechanical models with dierent arrangements of springs and dashpots render dierent mechanical in-

terpretations of the constants am and bn in Eq. (2).

The generalized Maxwell model, widely used to characterize the modulus functions of linear viscoelasticmedia, consists of a spring and m Maxwell units connected in parallel as illustrated in Fig. 1(a). A series

combination of a spring and a dashpot constitutes a Maxwell unit. The relaxation modulus derived from

the generalized Maxwell model is given by (Tschoegl, 1989)

Et Ee Xmi1

Eiet=qi 3

where Ee, Ei and qi are all positive constants representing the equilibrium modulus, relaxation strengths and

relaxation times, respectively; the relaxation time of the ith Maxwell unit is dened by qi gi=Ei where gi isthe viscosity of the unit. A typical term under the summation symbol in Eq. (3) represents the relaxation

modulus of the ith Maxwell unit. The series expression in Eq. (3) is often referred to as a Prony or Dirichlet

series.The specialized forms of the dierential operator equation, Eq. (2), for some common mechanical

models including the generalized Maxwell model and the generalized Voigt model are given by Findley

et al. (1976). Note that Eq. (3) is derived from a relation between the general stressstrain equation, Eq. (1)

Fig. 1. SMMs; (a) generalized Maxwell model, (b) generalized Voigt model.


7/28/2019 1-s2.0-S0020768301000269-main

5/28

and the stressstrain equation in a dierential-operator form corresponding to the generalized Maxwell

model. The use of a mechanical model not only leads to an explicit form of the material function such as

Eq. (3) but also makes it possible to relate thermodynamic and molecular parameters to measured time- or

frequency-dependent mechanical properties (Ferry, 1980). The SMM (that invokes standard dierential

operators in its mathematical representation) has long been accepted as an accurate, ecient tool for the

characterization of the viscoelastic behavior of many polymers and polymeric composites.

2.2. The fractional derivative model

In an eort to further generalize the stressstrain equation in a dierential operator form, Eq. (2), a

number of authors (e.g., Gemant, 1938; Smit and de Vries, 1970; Bagley and Torvik, 1983a,b; Rogers, 1983;

Koeller, 1984) have applied a notion of the fractional derivative to the equation in such a way that

XM

m0

amdpmr

dtpm X

N

n0

bndqne

dtqn4

where pm and qn are real constants with 06pm, qn6 1. The ordinary time derivatives acting on the time-

dependent stress and strain elds in Eq. (2) are now replaced with corresponding fractional-order time

derivatives. A fractional time-derivative of order a is dened, in an integral form, as (Bagley and Torvik,

1983a)

daft

dta

1

C1 a

d

dt

Zt0

fs

t sa ds

5

where 0 < a < 1 and C denotes the gamma function.Although a summation of multiple terms with dierent fractional orders are implied on both sides of

Eq. (4), only a few terms are used in practice. For example, a model with M N 1, p0 q0 0 and

p1 q1 a in Eq. (4) is often used to describe the stressstrain behavior of a class of viscoelastic materials,i.e.,

r bdar

dta E0e E1

dae

dta6

where b, E0 and E1 are constants and 06 a6 1. An explicit form of the relaxation modulus Et can bederived from Eqs. (1) and (6) but its expression would be much more involved than Eq. (3) derived from

SMM.

2.3. Power-law representations

Dierent forms of power law have long been used to describe the relaxation and creep behaviors of linear

viscoelastic materials (e.g, Nutting, 1921). In particular, a power law of the following form is widely used

for its simplicity:

Et Ee E1tn 7

where Ee, E1 and n are positive constants; Ee here denotes the equilibrium (or rubbery) modulus and has the

same physical meaning as that ofEe in Eq. (3).

Although Eq. (7) describes the relaxation behavior in the rubbery and transition regions well, it does not

provide a good representation for the glassy behavior of the material because Eq. (7) renders unbounded

values at short times. This shortcoming has prompted an introduction of the so-called MPL of the fol-

lowing form (Williams, 1964):


7/28/2019 1-s2.0-S0020768301000269-main

6/28

Et Ee Eg Ee

1 t=qn 8

where Ee, Eg, q and n are all positive constants which represent, respectively, the equilibrium modulus,

glassy modulus, relaxation time and power. With its broad-band representation characteristic, Eq. (8) canreasonably depict the glassy and rubbery plateau behavior as well as the transition-region behavior of a

large class of polymers and polymer-based composites. Note that when t=q ) 1, Eq. (8) reduces to Eq. (7).It is informative to note that the FDM and power laws are closely interrelated (Bagley, 1989; Tschoegl,

1989). For example, Bagley (1989) matched Et derived from Eq. (6) with that of Eq. (8) and found thefollowing asymptotic relations between the two groups of model parameters:

E0 Ee;E1

b Eg; a n;

b

C1 a qn 9

However, the two models are not completely equivalent. The discrepancy lies primarily in their short-time

relaxation behaviors. Numerous other phenomenological models for the behavior of linear viscoelastic

materials are available in the literature (e.g., Tschoegl, 1989).

2.4. Linear viscoelastic material functions in dierent domains

It is well known that all linear viscoelastic material functions are mathematically equivalent for a given

mode of loading. Each function contains essentially the same rheological information of the material.

However, depending on the nature of the input excitation, a certain material function can be used ad-

vantageously over others in computing the response of a viscoelastic medium. For example, a modulus

function can be used more conveniently when strain is specied as the input, and a creep function for

problems with a stress input. Similarly, a time-dependent or frequency-dependent material function may be

used advantageously when the input is transient or steady-state harmonic, respectively. Further, in solving

viscoelastic boundary-value problems following the Laplace transform-based elastic-viscoelastic corre-spondence principle, one needs to deal with material functions dened in the Laplace-transform domain.

For later references, the linear viscoelastic material functions in dierent domains are briey reviewed in

Appendix A.

3. Characterization of viscoelastic damers using the standard mechanical model

Characterization of the rheological behavior of a viscoelastic damper requires the knowledge of the

geometry and material properties of each individual component constituting the damper unit (or system). A

constitutive relation for the damper is determined by relating the macroscopic response of the unit with the

applied excitation. A rigorous treatment entails the solution of a boundary value problem dealing with the

damper unit as a whole (e.g., Makris et al., 1995). However, more practically, a macroscopic (or eective)

material function of the damper system can be obtained from a physical experiment in which the system

input and output values are measured and related (e.g., Makris and Constantinou, 1991). A direct ex-

perimental characterization is simple and straightforward and does not require an explicit account of the

individual components of the damper system. The rheological behavior of a damper may be described

analytically using a mathematical model. The SMM is adopted in this paper as a prefered tool for char-

acterization of the rheological behavior of viscoelastic dampers. The details of the modeling procedure are

illustrated through the use of experimental data from common viscoelastic dampers. First, analytical ex-

pressions for the general linear viscoelastic material functions based on the SMM are discussed. Then, the

procedure for model tting and the issue on interconversion between material functions are addressed.


7/28/2019 1-s2.0-S0020768301000269-main

7/28

3.1. Linear viscoelastic material functions

The Prony series representation of the relaxation modulus derived from the generalized Maxwell model

was given in Eq. (3). Similarly, the creep compliance can be conveniently characterized by the generalized

Voigt model that comprises a spring, a dashpot and n Voigt units connected in series, see Fig. 1(b),

Dt Dg t

g0Xnj1

Dj1 et=sj 10

where Dg, g0, Dj and sj are all positive constants denoting the glassy compliance, zero shear-rate viscosity,

retardation strengths and retardation times, respectively; the retardation time of the jth Voigt unit is dened

by sj Djgj where gj is the viscosity of the unit. For viscoelastic solids, g0 3 I and thus the second term inEq. (10) vanishes. For viscoelastic uids, g0 is nite. A typical term under the summation symbol in Eq. (10)

represents the creep compliance of the jth Voigt unit consisting of a parallel combination of a spring and a

dashpot.

Now the time-domain material functions, Eqs. (3) and (10), can be substituted into Eqs. (A.7)(A.10)and (A.16), (A.17) to obtain the corresponding material functions in the frequency and Laplace-transform

domains as follows (Tschoegl, 1989):

EHx Ee Xmi1

x2q2iEi

x2q2i 111

EHHx Xmi1

xqiEi

x2q2i 112

D

H

x Dg Xnj1

Dj

x2s2j 1 13

DHHx 1

g0xXnj1

xsjDj

x2s2j 114

~Es Ee Xmi1

sqiEi

sqi 115

~Ds Dg 1

g0s

Xn

j1

Dj

ssj 1

16

where EH, EHH, DH and DHH are commonly referred to as the storage modulus, loss modulus, storage compliance

and loss compliance, respectively, and ~E s E and ~D s D are often called the operational modulus andcompliance where Eand D denote the Laplace transforms ofEt and Dt, respectively. The symbols x ands denote the circular frequency and the Laplace transform parameter, respectively. Eqs. (3) and (10)(16)

indicate that once a material function (either modulus or compliance) is determined in a particular domain,

the corresponding material functions in other domains are automatically established in terms of the same

model constants. The compact, closed-form expressions for the material functions in the frequency and

Laplace-transform domains are due to the amenable operational properties associated with the exponential

basis functions in the series, Eqs. (3) and (10).


7/28/2019 1-s2.0-S0020768301000269-main

8/28

3.2. Fitting of material functions

The constants in the series representation of a material function can be evaluated by tting the ex-

pression to available experimental data. Various methods of tting have been introduced by others. For

example, in the collocation method, Eq. (3) is equated to the data measured at m dierent times, and the m

unknowns, Ei i 1; . . . ;m, are found by solving the resulting system of m linear algebraic equations. Inthe least squares method, the equation is equated to the data at more than m sampling points and the

resulting over-determined system is solved by minimizing the square errors. Schapery (1961) illustrated the

collocation method of tting using the relaxation modulus data from PMMA and shear storage compliance

data from polyisobutylene. Cost and Becker (1970) used the so-called multidata method (based in a least

squares scheme) to determine the model constants within the Laplace-transform domain. In both methods,

one faces 2m unknowns including Ei and qi i 1; . . . ;m in general, which would lead to a system of 2mnonlinear equations. However, the relaxation times qi are usually specied a priori from experience and

only Ei are determined by solving the resulting system of m linear equations. Selection of time constants

is discussed and illustrated in Sections 4 and 5. The equilibrium modulus, Ee, is usually estimated by in-

specting the long-time behavior of the relaxation modulus data. It is to be noted that the model parametersin a modulus function may equally be determined by tting Eq. (11) or Eq. (12) to the frequency-domain

experimental data, if available. Similarly, model parameters in a compliance function may be found by

tting Eqs. (10), (13) or (14) to available experimental data in the time or frequency domain.

The issue of tting the SMM to experimental data has been extensively discussed in the literature.

Recently, a number of researchers have proposed various techniques for improvement of traditional tting

methods. For instance, Emri and Tschoegl (1993, 1994, 1995) and Tschoegl and Emri (1992, 1993) used

only well-dened subsets of the experimental data to enhance the quality of a t, Kaschta and Schwarzl

(1994a,b) proposed a method that ensures positive coecients through an interactive adjustment of re-

laxation or retardation times, and Baumgaertel and Winter (1989) employed a nonlinear regression tech-

nique in which the coecients, time constants, and the number of terms in the series are all variable. Mead

(1994) used a constrained linear regression with regularization, Honerkamp and Weese (1989) and Elsteret al. (1991) applied the so-called Tikhonov regularization techniques, and Elster and Honerkamp (1991)

used the modied maximum entropy method to determine discrete viscoelastic spectra from rheological

measurements.

3.3. Interconversion between material functions

A linear viscoelastic material function can be converted into other equivalent material functions through

appropriate mathematical operations. Interconversion may be required for dierent reasons. The response

of a medium under a certain excitation condition inaccessible to direct experiment may be predicted from

measurements under other readily realizable conditions. For example, it is often dicult to subject sti

materials to a constant-strain, relaxation test because of the requirement of a robust testing device. How-

ever, a constant-stress, creep test is relatively easy to carry out on these materials. In this case, the relaxation

modulus can be determined from the measured creep compliance through an interconversion between the

relaxation modulus and creep compliance. Similarly, time-domain material functions can be obtained

through the conversion of corresponding frequency-domain material functions measured from steady-state

harmonic tests which usually yield more accurate information than quasistatic tests.

Numerous interconversion methods, either exact or approximate, have been proposed. Hopkins and

Hamming (1957) and Kno and Hopkins (1972) presented numerical interconversion techniques based

on the integral relationship between the relaxation modulus and creep compliance similar to Eq. (A.2),

Baumgaertel and Winter (1989) demonstrated an analytical conversion from the relaxation modulus to the

creep compliance using their interrelationship in the Laplace transform domain, and Mead (1994) presented


7/28/2019 1-s2.0-S0020768301000269-main

9/28

a technique based on a constrained linear regression and used the technique to determine the relaxation

modulus from a set of storage and loss modulus data. Ramkumar et al. (1997) proposed a regularization

method that employs the quadratic programming originally developed for solving ill-posed Fredholm in-

tegral equations, and demonstrated the eectiveness of the method through the determination of the re-

laxation spectrum from steady-state experimental data. Park and Schapery (1999) presented a numerical

method of interconversion between linear viscoelastic material functions based on a Prony series repre-

sentation, and tested its eectiveness using experimental data from selected polymeric materials. Their

method is applicable to interconversion between modulus and compliance functions in time, frequency, and

Laplace transform domains. An approximate, analytical method of interconversion was presented by

Schapery and Park (1999). For a comprehensive treatment of the subject, the reader is referred to the

treatises by Schwarzl and Struik (1967), Ferry (1980), and Tschoegl (1989).

4. Numerical example 1 a viscoelastic liquid damper

The methods of constitutive modeling of viscoelastic dampers discussed above will now be illustrated

through two specic examples. The rst example concerns a viscoelastic liquid damper and the second a

viscoelastic solid damper. In each example, the performance of the SMM and a comparable model such as

the FDM or MPL is discussed and contrasted.

Viscoelastic liquids possess excellent energy dissipation characteristics and are widely used in dierent

types of energy dissipation device (Harris and Crede, 1976). A dashpot is a classical example of this kind.

Viscoelastic liquid dampers consisting of a cylindrical piston immersed in a viscoelastic uid (Schwahn and

Delinic, 1988) and viscous damping walls constructed of a at-plate piston immersed in a viscoelastic uid

(Arima et al., 1988) are commonly used to reduce vibration or isolate structures from seismic or wind

disturbances. Fig. 2(a) shows a viscous damper manufactured by GERB vibration control (GERB, 1986)

and used by Makris and Constantinou (1991) among others. The cylindrical pot is lled with silicon gel, a

highly viscous substance. The rheological characteristic of the damper depends on the viscoelastic prop-erties of the uid and the geometric details of the damper unit. Fig. 2(b) displays typical forcedisplacement

hysteresis loops measured from vertical piston motion at the frequency of 2 Hz and at a room temperature

(Makris, 1991). Dampers of this type were used by Schwahn and Delinic (1988) for vibration control of

Fig. 2. A viscoelastic liquid damper; (a) geometry (Makris and Constantinou, 1991), (b) forcedisplacement hysteresis loops for vertical

motion at f 2 Hz and T % 25C (Makris, 1991).


7/28/2019 1-s2.0-S0020768301000269-main

10/28

piping networks and by Humann (1985) for seismic base isolation of structures. The experimental data

presented by Makris (1991) and Makris and Constantinou (1991) for a viscoelastic liquid damper (or

simply, a viscous damper) are used here to illustrate how the data can be represented by analytical models

discussed above, and how the system response (or kernel) functions can be interconverted between dierent

analysis domains and loading modes. The term, viscous damper, is frequently used to refer to a viscoelastic

liquid (or uid) damper in the literature; however, it is to be noted that a viscoelastic uid possesses

elasticity as well as viscosity, while a viscous uid possesses only viscosity. The terms, viscous damper and

viscoelastic liquid damper, will be used interchangeably in this paper for convenience.

4.1. KH(x) and KHH(x) by the fractional derivative model

The experimental data for the storage and loss stiness of a viscous damper (Fig. 2) in the longitudinal

motion of the piston are shown in Fig. 3(a) and (b), respectively, together with their analytical ts. Here,

stiness rather than modulus is used since a forcedisplacement (rather than stressstrain) relation is dis-

cussed. The experimental data presented in a tabular form by Makris (1991) are used. The SMM and the

FDM are compared here in their analytical representation of the experimental data. Makris (1991) andMakris and Constantinou (1991), using FDM, obtained the following forcedisplacement relationship for

the longitudinal motion of the piston:

P kdrP

dtr C0

dqu

dtq17

where Pand u are the axial force and displacement, respectively, and the damper parameters C0, k, r and q

represent the zero-frequency damping coecient, relaxation time, and orders of fractional derivative, re-

spectively. When r q 1, the model reduces to the classical Maxwell model with k being the relaxationtime and C0 the viscosity. Eq. (17) is slightly dierent from Eq. (6) besides the fact that r and e are replaced

with P and u. Taking the Fourier transform of Eq. (17) and rearranging terms, one nds expressions

parallel to Eqs. (A.11) and (A.5),^Px Kixux 18

Kix KHx iKHHx 19

where ^P and u denote the Fourier transforms ofPand u, respectively, and K is the complex stiness whose

real and imaginary components, KH and KHH, are the storage and loss stiness given by (Makris and Con-

stantinou, 1991)

KHx C0x

q cos pq2

1 kxr cos pr2

C0kxqr sin pr

2 sin pq

2

1 k2x2r 2kxrcos pr2

20

KHHx C0xq

sin

pq

2 1 kxr

cos pr

2 C0kxqr

sin pr

2 cos

pq

2 1 k2x2r 2kxr cos pr

2

21

Values for the damper parameters, C0 15; 000 N s/m, k 0:3 s0:6, r 0:6 and q 1, were determined by

Makris and Constantinou (1991) by tting Eqs. (20) and (21) to the experimental data. The resulting FDM

analytical ts are shown in Fig. 3(a) and (b).

4.2. KH(x) and KHH(x) by the standard mechanical model

Using SMM (the generalized Maxwell model), the storage and loss stiness functions of a viscous

damper can be expressed as


7/28/2019 1-s2.0-S0020768301000269-main

11/28

KHx Xmi1

x2q2iKi

x2q2i 122

KHHx Xmi1

xqiKi

x2q2i 123

where the constants Ki, qi and m have the meanings similar to those of the parameters used in Eqs. (11) and

(12). Note that, for a viscoelastic liquid, the equilibrium stiness vanishes, i.e., Ke lim t3IKt limx30K

Hx 0 (Ferry, 1980). The coecients Ki i 1; . . . ;m can be found by tting Eq. (22) or

Fig. 3. Complex stiness for the viscous damper in the frequency domain; (a) storage stiness, (b) loss stiness.


7/28/2019 1-s2.0-S0020768301000269-main

12/28

Eq. (23) to available experimental data. A nine-term m 9 Prony series representation is employed in thecurrent example. Relaxation times with half-decade intervals, qi 10

i7=2i 1; . . . ; 9, are selected so thatthe entire frequency range of the data may be covered by the analytical representation. One-decade spacings

ofqi are usually adequate for many viscoelastic media (Schapery, 1974); however, for highly viscoelastic

media such as viscoelastic dampers, spacings closer than one decade are often required to accurately de-

scribe the data. If the spacing between qis becomes too small, oscillations occur in the tted curve and

deviations from the original data increase. Since the number of observations (or data points) is greater than

the number of unknowns (m 9 in our case), a least squares method can be employed. Although only oneset of data, either for KH or KHH, is sucient to determine Kis, both data sets are used here to enhance the

quality of the t. The best-t Kis are found by minimizing the following functional representing the

normalized square errors:

EXNk1

KHxk

KH

k

24 1!2 KHHxkK

HH

k

1

!235 24

where KH

k and KHH

k denote the experimental data at the circular frequency ofxk, and N is the total number

of data points available. The model constants thus found are presented in Table 1, and KH and KHH curves

represented by SMM are shown in Fig. 3(a) and (b). Both FDM and SMM represent the data very well.

4.3. K(t)

A signicant advantage of the use of SMM is that, once a linear viscoelastic material (or system)

function is established in one domain, the equivalent functions in other domains are automatically given.

For instance, the time-domain relaxation stiness, Kt, which relates the force history to the input dis-placement history in a manner similar to Eq. (1), can be obtained from Eq. (3) with Es replaced by Ks and

setting Ke 0. The values for the parameters, Ki and qi, presented in Table 1 are used here again. Fig. 4shows the graphical representation ofKt thus obtained. As long as the parameters Ki and qi in Table 1correctly represent the rheological properties of the viscous damper considered, the Kt curve shown inFig. 4 represents the exact relaxation stiness of the damper. Recently, Schapery and Park (1999) developed

an approximate method of analytical interconversion between linear viscoelastic material functions based

on a comprehensive study of the weighting functions involved in the mathematical interrelationships be-

tween the functions. For instance, Kt can be obtained from KHx by

Table 1

SMM constants for stiness functions of the viscoelastic liquid damper

i qi (s) Ki (kN/m)

1 1:00E 03 5:01E 022 3:16E 03 1:66E 023 1:00E 02 1:12E 024 3:16E 02 8:12E 015 1:00E 01 3:51E 016 3:16E 01 1:02E 017 1:00E 00 2:97E 008 3:16E 00 5:84E 019 1:00E 01 1:23E 01

Ke 0 kN/m


7/28/2019 1-s2.0-S0020768301000269-main

13/28

Kt 1

kKHx

x1=t

25

where k C1 n cos np=2; the symbol C denotes the Gamma function and n is the slope of thesource function on a loglog scale, i.e., n d logKH=d log x at x 1=t. No analytical expression ofKHx is required in Eq. (25) and the experimental data for KHx can directly be used to nd the Kt curve.The result of an approximate interconversion according to Eq. (25) is also shown in Fig. 4 and is seen to

practically coincide with the exact representation. A similar approximate interrelationship was given by

Schapery and Park (1999) for KHHx 3 Kt conversion. Note that, while SMM readily gives Kt once themodel is calibrated using the experimental data for KHx or KHHx, FDM does not provide an explicitanalytical expression for Kt for general orders of fractional derivative. Only for a special case ofq 1 and

r 0:5, an explicit expression for Kt in terms of the material parameters used in Eq. (17) is available(Makris, 1991).

4.4. Interconversion

A exibility function of the viscous damper, relating a displacement response to a force input, may now

be found from a stiness function that is already established, using an interrelationship between the two

functions. It is known that, when both the source and the target functions are expressed in Prony series, the

interconversion can be carried out very eciently (Park and Schapery, 1999). Specically, in view of Eq. (3)

and Eqs. (10)(16), if one set of constants, either fqi;Eii 1; . . . ;m and Eeg or fsj;Djj 1; . . . ; n;Dg and g0g, is known, the other set of unknown constants can be determined from a relationship, Eqs.(A.2), (A.13) or Eq. (A.18). In our case, we have already determined a set of constants,

fqi;Kii 1; . . . ;mg, for the stiness functions as shown in Table 1 and we seek to nd a set of constants,fsj;Ljj 1; . . . ; n; Lg and g0g, for the exibility functions. Again, here we deal with stiness K andexibility L functions that are parallel with modulus E and compliance D functions, respectively. Notethat Ke 0 for viscoelastic uid dampers. Following the procedure introduced by Park and Schapery(1999) and using the interrelationship (A.18) which, of the three alternate interrelationships, renders the

simplest system of equations, the problem of interconversion reduces to solving the following system of

linear algebraic equations for unknowns, Ljj 1; . . . ; n:

AfLg fBg or AkjLj Bk summed on j; j 1; . . . ; n; k 1; . . . ;p 26

Fig. 4. Relaxation stiness for the viscous damper in the time domain.


7/28/2019 1-s2.0-S0020768301000269-main

14/28

where

Akj Xmi1

skqiKi

1 skqi

!1

1 sksj

j 1; . . . ; n; k 1; . . . ;p 27

and

Bk 1 Xmi1

skqiKi

1 skqi

!1Pmi1 Ki

1

skPm

i1 qiKi

k 1; . . . ;p: 28

The symbol sk k 1; . . . ;p denote the discrete values of the Laplace-transform parameter at which theinterrelationship is established. Time constants sj j 1; . . . ; n can be either determined following a nu-merical procedure (Park and Schapery, 1999) or specied manually from experience to avoid dealing with a

system of nonlinear equations with 2n unknowns. The number of sampling points (or the number of

equations) should not be less than the number of unknowns (i.e., pP n). The collocation method is eectedwhen p n and the least squares method can be used when p> n. In the case of the least squares method,a minimization of the square error kfBg AfLgk

2with respect to Lj j 1; . . . ; n leads to the re-

placement ofAfLg fBg in Eq. (26) with ATAfLg ATfBg in which the product ATA is a squarematrix. In our example, a set of retardation time constants sj j 1; . . . ; 8 are numerically determinedfollowing the procedure given by Park and Schapery (1999) and is tabulated in Table 2 together with the

results for Lj j 1; . . . ; 8, Lg and g0. The least squares method is used to solve the system of equations. Itis to be noted that when a stiness function is represented by FDM, it cannot readily be converted into the

corresponding exibility function. For the details of the interconversion procedure used in the current and

next examples, the reader is referred to Park and Schapery (1999).

4.5. LH

(x) and LHH

(x)

The storage and loss exibility functions for a viscous damper can be represented by SMM (the gen-

eralized Voigt model) as

LHx Lg Xnj1

Lj

x2s2j 129

LHHx 1

g0xXnj1

xsjLj

x2s2j 1: 30

Table 2

SMM constants for exibility functions of the viscoelastic liquid damper

i si (s) Li (m/kN)

1 1:78E 03 4:77E 042 5:13E 03 9:90E 043 1:70E 02 1:35E 034 6:46E 02 2:78E 035 2:40E 01 4:92E 036 8:13E 01 1:04E 027 2:82E 00 1:76E 028 9:33E 00 3:40E 02

Lg 1:10E 03 m/kN g0 1:75E 01 kNs/m


7/28/2019 1-s2.0-S0020768301000269-main

15/28

Eqs. (29) and (30) follow from Eqs. (13) and (14) with Ds replaced with Ls. The functions LH and LHH are the

real and imaginary components of the complex exibility function L that relates the displacement to the

force in the frequency domain so that

ux Lix ^Px 31

Lix LHx iLHHx: 32

where u and ^P denote the Fourier transforms of u and P, respectively. The curves for LH and LHH represented

by Eqs. (29) and (30) are shown in Fig. 5. The constants presented in Table 2 are used in the evaluation of

these functions.

4.6. L(t)

The time-domain exibility function, Lt, relating an applied force history to the resulting displacementhistory in a manner similar to that of Eq. (A.1), can be obtained from Eq. (10) with Ds replaced with Ls.

The resulting Lt is shown graphically in Fig. 6. Again, the constants shown in Table 2 are used. Also

Fig. 6. Creep exibility for the viscous damper in the time domain.

Fig. 5. Complex exibility for the viscous damper in the frequency domain.


7/28/2019 1-s2.0-S0020768301000269-main

16/28

shown in Fig. 6, for a comparison, is an approximate Lt curve obtained simply by taking the reciprocal ofKt shown in Fig. 4. It can be clearly seen that the quasi-elastic relationship, KtLt 1, renders sig-nicant errors especially in the long-time behavior.

4.7. ~K(s) and ~L(s)

Finally, the operational stiness and exibility functions, ~Ks and ~Ls, dened by Eqs. (15) and (16)with Es and Ds replaced respectively with Ks and Ls, are shown in Fig. 7. Again, the model constants

presented in Tables 1 and 2 are used in generating these curves. One may easily check the reciprocal re-

lationship of ~Kand ~L. These functions are useful in solving viscoelastic boundary value problems following

the Laplace-transform-based elastic-viscoelastic correspondence principle.

5. Numerical example 2 a viscoelastic solid damper

Viscoelastic solid dampers have been used in the control of vibration and noise in aircrafts and machines,

and more recently in the mitigation of wind or earthquake-induced vibration of structures. Fig. 8(a) shows

a viscoelastic solid damper (or simply, a viscoelastic damper) composed of viscoelastic layers bonded with

Fig. 7. Operational stiness and exibility for the viscous damper in the Laplace-transform domain.

Fig. 8. A viscoelastic solid damper; (a) geometry (Mahmoodi, 1969), (b) forcedisplacement hysteresis loops at f 3 Hz and T 21C(Shen and Soong, 1995).


7/28/2019 1-s2.0-S0020768301000269-main

17/28

three parallel steel plates (Mahmoodi, 1969). When incorporated into a structure subjected to dynamic

loading that induces a relative motion between the outer steel anges and the center plate, a portion of the

input energy is dissipated through shear deformation of the viscoelastic layers. The concept and theory for

damping devices that use constrained viscoelastic layers were originally developed by Kerwin (1959) and

Ross et al. (1959) and further expanded by Torvik (1980) and others. A viscoelastic solid damper of the type

shown in Fig. 8(a) was experimented by Shen and Soong (1995), and they used the MPL for analytical

modeling of the damper and calibrated the model using the experimental data from stress relaxation and

steady-state harmonic tests. Fig. 8(b) displays forcedisplacement hysteresis loops measured in a sinusoidal

test conducted at the frequency of 3 Hz and the temperature of 21C. The data presented by Shen and

Soong (1995) are used here to illustrate the modeling of the damper using SMM, then the performance of

MPL and SMM are discussed and compared.

5.1. G(t)

Fig. 9 shows the time-domain shear relaxation modulus, Gt, of the viscoelastic solid damper. Becausethe experimental data were not available in a tabular form, data reconstructed from the MPL represen-

tation given by Shen and Soong (1995) are used here to calibrate the generalized Maxwell model expression

of Gt, Eq. (3), with Es replaced by Gs. Shen and Soong (1995) gave Ge 0:05861 MPa, Gg 1039:1MPa, q 1:26E 6 s, and n 0:586 for an MPL representation of Eq. (8), with Es replaced by Gs. Notethat for a viscoelastic solid, Ge is nonzero. An 18-term m 18 Prony series is used and the model con-stants are determined in a manner similar to that described in the above example of a viscoelastic liquid

damper. The resulting numerical values for the Prony series constants are tabulated in Table 3. Except in

the glassy and rubbery plateau regions, relaxation times with half-decade intervals are used. Compared to

the earlier viscoelastic liquid damper, the current viscoelastic solid damper requires more series terms be-

cause the latter is dened over a broader range of time (or frequency).

5.2. GH(x) and GHH(x)

The real and imaginary components of the complex shear modulus are presented in Fig. 10(a) and (b).

The expressions for GH and GHH, corresponding to the MPL representation ofGt as in Eq. (8), are given by(Shen and Soong, 1995)

Fig. 9. Shear relaxation modulus for the viscoelastic damper in the time domain.


7/28/2019 1-s2.0-S0020768301000269-main

18/28

GHx Ge Gg GexqnC1

" n cos

np

2

xq

xq1n

1 ncos

p

2

xq

#33

GHH

x Gg Gexqn

C1"

n sin

np

2

xq

xq1n

1 n sin

p

2

xq#

34

The SMM representations for GH and GHH in the form of Eqs. (11) and (12) are readily available using the

constants presented in Table 3. Fig. 10(a) and (b) show that MPL and SMM representations for GH and GHH

match very well except in the high frequency region. It is found that Eqs. (33) and (34) do not properly

represent GH and GHH in the high frequency region where xqP 1. Additional corrective terms are required for

Eqs. (33) and (34) to accurately depict GH and GHH in that region as pointed out in the Appendix of Shen and

Soong (1995). For comparison purposes, GH and GHH are also obtained directly from Gt curve using theapproximate interconversion method of Schapery and Park (1999) and are plotted in Fig. 10. The results of

the approximate interconversion match very well with the SMM representation for GH. The approximate

curve for GHH obtained directly from Gt shows some departures from the SMM representation, although

the overall trend of the curve agrees with that of the SMM curve. Such departures are due to some intrinsicmathematical incompatibilities between the functions Gt and GHHx (Schapery and Park, 1999). Note thedecreasing GHH with frequency in the low and high frequency regions, which is consistent with the following

general constraints on GHH (Christensen, 1982):

limx30

GHHx 0 and limx3I

GHHx 0 35

Shen and Soong (1995) also addressed the use of a simplied power law, basically in the form of Eq. (7).

Although the simplied power law, Eq. (7), oers an excellent description of the rubber and transition

behavior, it does not adequately describe the short-time (or high-frequency), glassy behavior of most vi-

scoelastic materials.

Table 3

SMM constants for modulus functions of the viscoelastic solid damper

i qi (s) Gi (MPa)

1 1:00E 07 1:33E 012 1:00E 06 2:86E 023 3:16E 06 2:91E 024 1:00E 05 2:12E 025 3:16E 05 1:12E 026 1:00E 04 6:16E 017 3:16E 04 2:98E 018 1:00E 03 1:61E 019 3:16E 03 7:83E 0010 1:00E 02 4:15E 0011 3:16E 02 2:03E 0012 1:00E 01 1:11E 0013 3:16E 01 4:91E 0114 1:00E 00 3:26E 01

15 3:16E 00 8:25E 0216 1:00E 01 1:26E 0117 1:00E 02 3:73E 0218 1:00E 03 1:18E 02

Ge 5:86E 02 MPa


7/28/2019 1-s2.0-S0020768301000269-main

19/28

5.3. J(t), JH(x) and JHH(x)

The corresponding shear compliance functions, J(t), JHx and JHHx, of the viscoelastic solid dampermay easily be determined from an available modulus function through interconversion when both the

source and target functions are represented by SMM. The method (Park and Schapery, 1999) used above

for the viscoelastic liquid damper may equally be used for the viscoelastic solid damper in the current

example. However, slight modications to Eqs. (26)(28) are required. The unknown constants, Jj, for the

compliance functions may be found by solving the following set of equations:

AfJg fBg or AkjJj Bk summed on j; j 1; . . . ; n; k 1; . . . ;p 36

where

Akj Ge

Xmi1

skqiGi

1 skqi

!1

1 sksj

j 1; . . . ; n; k 1; . . . ;p 37

and

Bk 1 Ge

Xmi1

skqiGi

1 skqi

!,Ge

Xmi1

Gi

!k 1; . . . ;p: 38

Note that nonzero Ge is present in Eqs. (37) and (38). The model constants thus obtained are presented in

Table 4. Equation solution techniques similar to those used earlier are employed here. The glassy com-

pliance and zero shear-rate viscosity are determined from the following relations (Park and Schapery,1999):

Jg 1

Ge Pm

i1 Giand g0

Xg03Ii1

qiGi: 39

The resulting shear creep compliance, Jt, and the storage and loss compliance, JHx and JHHx, areshown in Figs. 11 and 12, respectively. Their analytical forms are the same as those of Eqs. (10), (13) and

(14) with simple notational substitutions of Js for Ds. Function Jt obtained from the quasi-elastic ap-proximation is also presented in Fig. 11 for comparison. At t 1 s, for example, Jt obtained from thequasi-elastic relationship is 74% greater than the exact value. The quasi-elastic approximation, although

Fig. 10. Complex modulus for the viscoelastic damper in the frequency domain; (a) storage modulus, (b) loss modulus.


7/28/2019 1-s2.0-S0020768301000269-main

20/28

useful for providing rough estimates, is not good enough for accurate characterization of material functions

for viscoelastic dampers.

5.4. ~G(s) and ~J(s)

Fig. 13 shows the variation of the operational modulus and compliance functions in the Laplace-

transform domain. It is informative to note that, in view of Figs. 913, the curve shapes for Gt, JHx and~Js are all similar to each other, and likewise, Jt, GHx and ~Gs share the same trends. These general

Table 4

SMM constants for compliance functions of the viscoelastic solid damper

i si (s) Ji (MPa1)

1 1:02E 07 5:50E 052 1:29E 06 1:55E 043 4:79E 06 4:42E 044 1:70E 05 8:90E 045 5:62E 05 1:69E 036 1:91E 04 3:29E 037 5:89E 04 5:96E 038 1:95E 03 1:22E 029 6:03E 03 2:50E 0210 1:95E 02 3:97E 0211 6:03E 02 1:03E 0112 2:00E 01 1:55E 0113 5:75E 01 3:58E 0114 2:09E 00 7:17E 01

15 4:79E 00 1:12E 0016 2:57E 01 5:15E 0017 1:62E 02 6:03E 0018 1:23E 03 3:34E 00

Jg 9:64E 04 MPa1 g0 3 I

Fig. 11. Creep compliance for the viscoelastic damper in the time domain.


7/28/2019 1-s2.0-S0020768301000269-main

21/28

trends provide a useful rule of thumb for the engineer to predict the response of the damper subjected to

dierent modes of excitation. Similar trends exist for viscoelastic liquid dampers.

6. Further discussions

The two examples presented above demonstrate that the SMM can accurately represent the material (or

system) functions of viscoelastic dampers and that the numerical procedure involved is simple, straight-

forward and ecient compared to other models. A Prony series is highly amenable to various mathematical

operations with its exponential basis functions. This amenable nature of a Prony series renders the pro-

cedures for calibration (or tting) of a model and interconversion between material functions computa-

tionally ecient. For example, in contrast to FDM or MPL which normally requires an involved, nonlinear

curve-tting technique, SMM can be calibrated by a standard, linear curve-tting procedure as demon-

strated above. Also, while FDM and MPL oer very limited closed-form expressions for material func-

tions in dierent domains, SMM readily provides the explicit expressions for these functions. In addition,

SMM are grounded in a well-dened physical basis. Some molecular theories indeed predict the SMM

Fig. 12. Storage and loss compliance for the viscoelastic damper in the frequency domain.

Fig. 13. Operational modulus and compliance for the viscoelastic damper in the Laplace-transform domain.


7/28/2019 1-s2.0-S0020768301000269-main

22/28

representation of linear viscoelastic material functions. Specically, the Rouses theory for dilute polymer

solutions (Rouse, 1953) and the modied Rouse theory for undiluted polymers (Ferry et al., 1955) specify

the storage and loss modulus in the form of Eqs. (11) and (12). Also, it should be noted that the constants in

the generalized Maxwell and generalized Voigt models can be chosen so that the two models are mathe-

matically equivalent, and thus a viscoelastic material depicted by one model may also be depicted by the

other. Further, Biot (1954), using the thermodynamics of linear irreversible process, showed that Eqs. (3)

and (10) are the most general representations possible for Et and Dt for the isothermal case, andSchapery (1964) similarly showed the same for certain important nonisothermal cases such as thermo-

rheologically simple behavior.

An exponential basis function for a Prony series have a very narrow range of transition behavior while a

single-term in MPL has a broad range of transition behavior extending over many decades of logarithmic

time, which explains why many terms are required for a Prony series to depict a broad-band behavior. The

function, et, when plotted against log t, is transient only within the approximate range of 2 < log t< 1,and is practically constant outside of this range (Cost and Becker, 1970). Fig. 14 shows the variations of

typical, narrow-band, exponential basis functions for a Prony series and a broadband, MPL function. For

brevity but without loss of generality, simple values are assigned to the constants involved. While an MPLfunction can cover a broad range with a single term, it often fails to accurately depict the entire range of

experimental data because of its limited degrees of freedom. Whereas, a multi-term Prony series can de-

scribe the same data much more accurately with expanded degrees of freedom. As pointed out above, there

is a close interrelationship between FDM and MPL. Therefore, FDM can also describe a broad-band

viscoelastic behavior with a small number of parameters. A comprehensive discussion of various basis

functions for dierent analytical representations of linear viscoelastic material behavior is given by

Tschoegl (1989).

Finally, it is to be noted that, from the theory of linear viscoelasticity, EH and EHH are not independent but

are related to each other (Tschoegl, 1989),

EHx Ee 2x2

p

ZI

0

EHHk 1kx2 k2

dk 40

or

Fig. 14. Behavior of sample Prony series basis functions in comparison with a MPL representation.


7/28/2019 1-s2.0-S0020768301000269-main

23/28

EHHx 2x

p

ZI0

EHk Ee1

k2 x2

dk 41

Eqs. (40) and (41) are known as KronigKramers relations and the integrals are to be interpreted as Cauchy

principal values. Schapery and Park (1999), based on Eqs. (40) and (41), developed the following ap-proximate analytical interrelationships between EH and EHH:

EHx Ee 1

k E

HHxxx

42

EHHx kEHx Eexx

43

where k tannp=2 and n d logEHH=d log x when EHH is the source function, or n d log EH Ee=d log x when EH is the source function. Relations (42) and (43) may be used by the engineer to check theadequacy of the experimental data obtained from a steady-state harmonic test.

7. Conclusions

Dierent approaches to the mathematical modeling of the rheological behavior of viscoelastic dampers

are discussed. Their theoretical basis and performance are reviewed and compared. The classical, SMM

consisting of a combination of linear springs and dashpots is shown to be more ecient than other widely

used models such as the FDM and the MPL in the mechanical characterization of viscoelastic dampers. It

is found that, despite the widespread notion of the inadequacy of spring-dashpot mechanical models

for viscoelastic dampers, the generalized Maxwell or generalized Voigt model, with their expanded degrees

of freedom, accurately describes the broad-band rheological behavior of common viscoelastic dampers.

Some reported inadequacies are attributed to the use of unduly simple mechanical models such as those

consisting of a single Maxwell or Voigt unit. In the time domain, the SMM renders a Prony series ex-

pression for the modulus or compliance function. The outstanding mathematical eciency associated withthe exponential basis functions of a Prony series greatly facilitates numerical operations that involve the

series. The methods of tting and interconversion associated with the SMM are straightforward and e-

cient, without requiring cumbersome nonlinear regression or equation-solving procedures. The material

functions in dierent domains for viscoelastic dampers can be readily determined from experimental data

(either from quasi-static or dynamic tests) or through interconversion from a function established in an-

other domain. Numerical examples on two commonly used viscoelastic dampers illustrating the detailed

procedure for tting and interconversion demonstrate the superiority of the SMM to other prevailing

models in accuracy and computational facility.

Appendix A. Interrelationships between linear viscoelastic material functions

A.1. In the time domain

The stressstrain relation (1) is used to nd the stress response of a linear viscoelastic medium to a strain

input. Conversely, the strain response to a given stress input is given by

et

Zt0

Dt sdrs

dsds A:1

where Dt is the creep compliance. From Eqs. (1) and (A.1), setting, e.g., rt Ht where Ht is theHeaviside step function with Ht 1 for t> 0 and Ht 0 for t< 0, one nds the following integralrelationship between the relaxation modulus and creep compliance:


7/28/2019 1-s2.0-S0020768301000269-main

24/28

Zt0

Et sdDs

dsds 1 t> 0: A:2

A.2. In the frequency domain

The complex material functions arise from a steady-state harmonic input. For instance, substituting

et eAeixt and rt rAe

ixt respectively into Eqs. (1) and (A.1) as input, one nds

rt Eixet A:3

et Dixrt A:4

where E and D are the complex modulus and complex compliance, respectively, and have their real and

imaginary parts such that

Eix EHx iEHHx A:5

Dix DHx iDHHx: A:6

The real and imaginary components of these complex material functions can be expressed in terms of

transient material functions (Tschoegl, 1989) as

EHx Ee x

ZI0

Et Ee sinxtdt A:7

EHHx x ZI0

Et Ee cosxtdt A:8

DHx De x

ZI0

Dt De sinxtdt A:9

DHHx x

ZI0

Dt De cosxtdt A:10

in which Ee and De denote the long-time equilibrium modulus and compliance, respectively; i.e., Ee limt3IEt and De limt3IDt. Note that a minus sign is used in Eq. (A.6) so that D

HH will be positive.

The real parts, EH and DH, are commonly referred to as the storage modulus and compliance, and the

imaginary parts, EHH and DHH, the loss modulus and compliance, respectively.

It should be noted that the elastic-like Eqs. (A.3) and (A.4) apply only to steady-state harmonic motions.

However, the same form of equations apply to transient motions when Eqs. (1) and (A.1) are Fourier

transformed, i.e.,

rx Eixex A:11

ex Dixrx A:12

where r and e denote the Fourier transforms of r and e, respectively; rx RI

I rteixtdt. From Eqs.

(A.3) and (A.4), or Eqs. (A.11) and (A.12), the following relationship between the complex modulus and

compliance can be seen:


7/28/2019 1-s2.0-S0020768301000269-main

25/28

EixDix 1: A:13

A.3. In the Laplace-transform domain

Taking Laplace transforms of Eqs. (1) and (A.1), one nds

rs ~Eses A:14

es ~Dsrs A:15

where r and e denote the Laplace transforms ofr and e, respectively; rs RI

0rtestdt. The functions ~E

and ~D in Eqs. (A.14) and (A.15) are often referred to as the operational modulus and operational com-

pliance, respectively, and are dened by

~Es s Es A:16

~Ds s Ds A:17

where Eand D are the Laplace transforms ofEt and Dt, respectively. From Eqs. (A.14) and (A.15), orby taking the Laplace transform of Eq. (A.2), one readily nds

~Es ~Ds 1: A:18

Eqs. (A.14), (A.15), and (A.18) have forms identical to those of the corresponding elastic equations,

providing the basis for the so-called elasticviscoelastic correspondence principle. The operational modulus

and compliance, Eqs. (A.16) and (A.17), are commonly involved in solving linear viscoelastic boundary

value problems via the correspondence principle.

Further, from Eqs. (A.16), (A.17) and (A.5)(A.10), the following useful relationships between the

operational and complex material functions result (Pipkin, 1972; Tschoegl, 1989):

Eix ~Esjs3ix A:19

Dix ~Dsjs3ix A:20

References

Ader, C., Jendrzejczyk, J., Mangra, D., 1995. The application of viscoelastic damping materials to control the vibration of magnets in a

synchrotron radiation facility. In: Karim-Panahi, K. (Ed.), Advances in Vibration Issues, Active and Passive Vibration Mitigation,

Damping and Seismic Isolation, PVP-vol. 309. ASME, New York, pp. 18.

Aprile, A., Inaudi, J.A., Kelly, J.M., 1997. Evolutionary model of viscoelastic dampers for structural applications. J. Engng. Mech. 123

(6), 551560.

Arima, F., Miyazaki, M., Tanaka, H., Yamazaki, Y., 1988. A study on buildings with large damping using viscous damping walls.

Ninth World Conference on Earthquake Engineering, Tokyo, V, pp. 821826.

Bagley, R.L., 1989. Power law and fractional calculus model of viscoelasticity. AIAA J. 27 (10), 14121417.

Bagley, R.L., Torvik, P.J., 1983a. Fractional calculus a dierent approach to the analysis of viscoelastically damped structures.

AIAA J. 21 (5), 741748.

Bagley, R.L., Torvik, P.J., 1983b. A theoretical basis for the application of fractional calculus. J. Rheol. 27, 201210.

Baumgaertel, M., Winter, H.H., 1989. Determination of discrete relaxation and retardation time spectra from dynamic mechanical

data. Rheol. Acta 28, 511519.

Bergman, D.M., Hanson, R.D., 1993. Viscoelastic mechanical damping devices tested at real earthquake displacements. Earthquake

Spectra 9 (3), 389417.


7/28/2019 1-s2.0-S0020768301000269-main

26/28

Biot, M.A., 1954. Theory of stress-strain relations in anisotropic viscoelasticity and relaxation phenomena. J. Appl. Phys. 25, 1385

1391.

Bland, D.R., 1960. The Theory of Linear Viscoelasticity. Pergamon Press, New York.

Chang, K.C., Soong, T.T., Lai, M.L., Nielsen, E.J., 1993. Viscoelastic dampers as energy dissipation devices for seismic applications.

Earthquake Spectra 9 (3), 371387.Chang, K.C., Soong, T.T., Oh, S.T., Lai, M.L., 1995. Seismic behavior of steel frame with added viscoelastic dampers. J. Struct.

Engng. 121 (10), 14181426.

Christensen, R.M., 1982. Theory of Viscoelasticity, second ed. Academic Press, New York.

Cost, T.L., Becker, E.B., 1970. A multidata method of approximate Laplace transform inversion. Int. J. Numer. Meth. Engng. 2, 207

219.

Dyke, S.J., Spencer, B.F., Sain, M.K., Carlson, J.D., 1996. Modeling and control of magnetorheological dampers for seismic response

reduction. Smart Mater. Struct. 5 (5), 565575.

Dyke, S.J., Spencer Jr., B.F., Sain, M.K., Carlson, J.D., 1998. Experimental study of MR dampers for seismic protection. Smart

Mater. Struct. 7 (5), 693703.

Elster, C., Honerkamp, J., 1991. Modied maximum entropy method and its application to creep data. Macromolecules 24, 310314.

Elster, C., Honerkamp, J., Weese, J., 1991. Using regularization methods for the determination of relaxation and retardation spectra of

polymeric liquids. Rheol. Acta 30, 161174.

Emri, I., Tschoegl, N.W., 1993. Generating line spectra from experimental responses. Part I: Relaxation modulus and creep

compliance. Rheol. Acta 32, 311321.Emri, I., Tschoegl, N.W., 1994. Generating line spectra from experimental responses. Part IV: Application to experimental data.

Rheol. Acta 33, 6070.

Emri, I., Tschoegl, N.W., 1995. Determination of mechanical spectra from experimental responses. Int. J. Solids Struct. 32, 817826.

Ferry, J.D., 1980. Viscoelastic properties of polymers, third ed. Wiley, New York.

Ferry, J.D., Landel, R.F., Williams, M.L., 1955. Extensions of the Rouse theory of viscoelastic properties to undiluted linear polymers.

J. Appl. Phys. 26 (4), 359362.

Findley, W.N., Lai, J.S., Onaran, K., 1976. Creep and Relaxation of Nonlinear Viscoelastic Materials. Dover, New York.

Fung, Y.C., 1965. Foundations of Solid Mechanics. Prentice-Hall, Englewood Clis, New Jersey.

Gavin, H.P., Hanson, R.D., Filisko, F.E., 1996a. Electrorheological dampers. 1. Analysis and design. J. Appl. Mech. 63 (3), 669675.

Gavin, H.P., Hanson, R.D., Filisko, F.E., 1996b. Electrorheological dampers. 2. Testing and modeling. J. Appl. Mech. 63 (3), 676682.

Gemant, A., 1938. On fractional dierentials. Philos. Mag. 25, 540549.

GERB, 1986. Pipework Dampers A Technical Report, GERB Vibration Control, Westmont, IL.

Hanson, R.D., 1993. Supplemental damping for improved seismic performance. Earthquake Spectra 9 (3), 319334.

Harris, C.M., Crede, C.E., 1976. Shock and Vibration Handbook, second ed. McGraw-Hill, New York.

Hayes Jr., J.R., Foutch, D.A., Wood, S.L., 1999. Inuence of viscoelastic dampers on the seismic response of a lightly reinforced

concrete at slab structure. Earthquake Spectra 15 (4), 681710.

Honerkamp, J., Weese, J., 1989. Determination of the relaxation spectrum by a regularization method. Macromolecules 22, 43724377.

Hopkins, I.L., Hamming, R.W., 1957. On creep and relaxation. J. Appl. Phys. 28 (8), 906909.

Housner, G.W., Bergman, L.A., Caughey, T.K., Chassiakos, A.G., Claus, R.O., Masri, S.F., Skelton, R.E., Soong, T.T., Spencer,

B.F., Yao, J.T.P., 1997. Structural control: past, present, and future. J. Engng. Mech. 123 (9), 897971.

Humann, G.K., 1985. Full base isolation for earthquake protection by helical springs and viscodampers. Nucl. Engng. Des. 84 (2),

331338.

Jones, D.I.J., 1980. Viscoelastic materials for damping applications. Damping Applications for Vibration Control, AMD-vol. 38.

ASME, New York, pp. 2751.

Kaschta, J., Schwarzl, F.R., 1994a. Calculation of discrete retardation spectra from creep data: I. method. Rheol. Acta 33, 517529.

Kaschta, J., Schwarzl, F.R., 1994b. Calculation of discrete retardation spectra from creep data: II. Analysis of measured creep curves.

Rheol. Acta 33, 530541.Kerwin Jr., E.M., 1959. Damping of exural waves by a constrained viscoelastic layer. J. Acous. Soc. Am. 31 (7), 952962.

Kno, W.F., Hopkins, I.L., 1972. An improved numerical interconversion for creep compliance and relaxation modulus. J. Appl.

Polym. Sci. 16, 29632972.

Koeller, R.C., 1984. Applications of fractional calculus to the theory of viscoelasticity. J. Appl. Mech. 51, 299307.

Koh, C.G., Kelly, J.M., 1990. Application of fractional derivatives to seismic analysis of base-isolated models. Earthquake Engng.

Struct. Dyn. 19, 229241.

Li, W.Q., Tsai, C.S., 1994. Seismic mitigation of structures by using viscoelastic dampers. Nucl. Engng. Des. 147 (3), 263274.

Mahmoodi, P., 1969. Structural dampers. ASCE J. Struct. Div. 95 (8), 16611672.

Mahmoodi, P., Robertson, L.E., Yontar, M., Moy, C., Feld, L., 1987. Performance of viscoelastic dampers in World Trade Center

towers. Dynamics of Structures. In: Roesset, J.M. (Ed.), Proc. Structures Congress '87, Orlando, Florida, ASCE, New York,

pp. 632644.


7/28/2019 1-s2.0-S0020768301000269-main

27/28

Makris, N., 1991. Theoretical and experimental investigation of viscous dampers in applications of seismic and vibration isolation.

Ph.D. Thesis, State University of New York at Baalo, Bualo, New York.

Makris, N., 1997. Rigidity-plasticity-viscosity: Can electrorheological dampers protect base-isolated structures from near-source

ground motions? Earthquake Engng. Struct. Dyn. 26 (5), 571591.

Makris, N., Constantinou, M.C., 1991. Fractional-derivative Maxwell model for viscous dampers. J. Struct. Engng. 117 (9), 27082724.

Makris, N., Constantinou, M.C., 1992. Spring-viscous damper systems for combined seismic and vibration isolation. Earthquake

Engng. Struct. Dyn. 21, 649664.

Makris, N., Constantinou, M.C., 1993. Models of viscoelasticity with complex-order derivatives. J. Engng. Mech. 119 (7), 14531464.

Makris, N., Constantinou, M.C., Dargush, G.F., 1993a. Analytical model of viscoelastic uid dampers. J. Struct. Engng. 119 (11),

33103325.

Makris, N., Dargush, G.F., Constantinou, M.C., 1993b. Dynamic analysis of generalized viscoelastic uids. J. Engng. Mech. 119 (8),

16631679.

Makris, N., Dargush, G.F., Constantinou, M.C., 1995. Dynamic analysis of viscoelastic-uid dampers. J. Engng. Mech. 121 (10),

11141121.

Mead, D.W., 1994. Numerical interconversion of linear viscoelastic material functions. J. Rheol. 38, 17691795.

Morgenthaler, D.R., 1987. Design and analysis of passive damped large space structures. The Role of Damping in Vibration and Noise

Control. ASCE, New York, pp. 18.

Morland, L.W., Lee, E.H., 1960. Stress analysis for linear viscoelastic materials with temperature variation. Trans. Soc. Rheol. 4, 233.Nutting, P.G., 1921. A new generalized law of deformation. J. Franklin Institute 191, 679685.

Park, S.W., Schapery, R.A., 1999. Methods of interconversion between linear viscoelastic material functions. Part 1-A numerical

method based on Prony series. Int. J. Solids Struct. 36 (11), 16531675.

Pipkin, A.C., 1972. Lectures on Viscoelasticity Theory. Springer, Berlin.

Ramkumar, D.H.S., Caruthers, J.M., Mavridis, H., Shro, R., 1997. Computation of the linear viscoelastic relaxation spectrum from

experimental data. J. Appl. Polym. Sci. 64, 21772189.

Rogers, L., 1983. Operators and fractional derivatives for viscoelastic constitutive equations. J. Rheol. 27 (4), 351372.

Ross, D., Ungar, E.E., Kerwin Jr., E.M., 1959. Damping of plate exural vibrations by means of viscoelastic laminae. In: Ruzicka,

J.E., (Ed.), Structural Damping. ASME, New York, pp. 4987.

Rouse Jr., P.E., 1953. The theory of the linear viscoelastic properties of dilute solutions of coiling polymers. J. Chem. Phys. 21 (7),

12721280.

Samali, B., Kwok, K.C.S., 1995. Use of viscoelastic dampers in reducing wind and earthquake-induced motion of building structures.

Engng. Struct. 17 (9), 639654.

Schapery, R.A., 1961. A simple collocation method for tting viscoelastic models to experimental data. Report GALCIT SM 61-23A,

California Institute of Technology, Pasadena, CA. November.

Schapery, R.A., 1964. Application of thermodynamics to thermomechanical, fracture, and birefringent phenomena in viscoelastic

media. J. Appl. Phys. 35, 14511465.

Schapery, R.A., 1974. Viscoelastic behavior and analysis of composite materials. In: Sendeckyj, G.P. (Ed.), In Mechanics of Composite

Materials, vol. 2, Academic Press, New York, pp. 85168.

Schapery, R.A., Park, S.W., 1999. Methods of interconversion between linear viscoelastic material functions. Part 2 An approximate

analytical method. Int. J. Solids Struct. 36 (11), 16771699.

Schwahn, K.J., Delinic, K., 1988. Verication of the reduction of structural vibrations by means of viscous dampers, ASME Pressure

Vessels and Piping Conference, PVP vol. 144, June1923, Pittsburgh, PA, pp. 8795.

Schwarzl, F.R., Struik, L.C.E., 1967. Analysis of relaxation measurements. Adv. Mol. Relaxation Processes 1, 201255.

Shen, K.L., Soong, T.T., 1995. Modeling of viscoelastic dampers for structural applications. J. Engng. Mech. 121 (6), 694701.

Shukla, A.K., Datta, T.K., 1999. Optimal use of viscoelastic dampers in building frames for seismic force. ASCE J. Struct. Engng. 125

(4), 401409.Smit, W., deVries, H., 1970. Rheological models containing fractional derivatives. Rheol. Acta 9, 525534.

Soong, T.T., Dargush, G.F., 1997. Passive Energy Dissipation Systems in Structural Engineering. Wiley, Chichester, UK.

Sunakoda, K., Sodeyama, H., Iwata, N., Fujitani, H., Soda, S., 2000. Dynamic characteristics of magneto-rheological uid damper.

Proceedings of the seventh SPIE Smart Structures and Materials Conference, March 59, Newport Beach, CA.

Torvik, P.J., 1980. The analysis and design of constrained layer damping treatments. Damping Applications for Vibration Control,

AMD-vol. 38. ASME, New York, pp. 85112.

Tsai, C.S., 1993. Innovative design of viscoelastic dampers for seismic mitigation. Nucl. Engng. Des. 139, 165182.

Tsai, C.S., 1994. Temperature eect of viscoelastic dampers during earthquakes. J. Struct. Engng. 120 (2), 394409.

Tsai, C.S., Lee, H.H., 1993a. Applications of viscoelastic dampers to high-rise buildings. J. Struct. Engng. 119 (4), 12221233.

Tsai, C.S., Lee, H.H., 1993b. Seismic mitigation of bridges by using viscoelastic dampers. Comput. Struct. 48 (4), 719727.

Tschoegl, N.W., 1989. The Phenomenological Theory of Linear Viscoelastic Behavior. Springer, Berlin.


7/28/2019 1-s2.0-S0020768301000269-main

28/28

Tschoegl, N.W., Emri, I., 1992. Generating line spectra from experimental responses. Part III: interconversion between relaxation and

retardation behavior. Int. J. Polym. Mater. 18, 117127.

Tschoegl, N.W., Emri, I., 1993. Generating line spectra from experimental responses. Part II: storage and loss functions. Rheol. Acta

32, 322327.

Williams, M.L., 1964. Structural analysis of viscoelastic materials. AIAA J. 2 (5), 785808.Xu, Y.L., Qu, W.L., Ko, J.M., 2000. Seismic response control of frame structures using magnetorheological/electrorheological

dampers. Earthquake Engng. Struct. Dyn. 29 (5), 557575.

Yang, G., Ramallo, J.C., Spencer Jr., B.F., Carlson, J.D., Sain, M.K., 2000. Dynamic performance of large-scale MR uid dampers.

Proceedings of the 14th ASCE Engineering Mechanics Conference, May 2124, Austin, TX.

Zhang, R.H., Soong, T.T., 1992. Seismic design of viscoelastic dampers for structural applications. J. Struct. Engng. 118 (5), 1375

1392.

Zhang, R.H., Soong, T.T., Mahmoodi, P., 1989. Seismic response of steel frame structures with added viscoelastic dampers.

Earthquake Engng. Struct. Dyn. 18, 389396.

Zou, X., Ou, J., 2000. Experimental study on properties of viscoelastic dampers and seismic vibration-suppression eects of viscoelastic

damped structures. Shock Vib. Dig. 32 (1), 38.


1-s2.0-s0020768301000269-main

Documents