Analytical modeling of viscoelastic dampers for structural and vibration control

International Journal of Solids and Structures 38 (2001) 8065±8092

www.elsevier.com/locate/ijsolstr

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 Di€erent 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 modi®ed power law. The SMM renders a Prony series expression for the modulus and compliance functions in the time domain, and the remarkable mathematical eciency 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 the advantages 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. Ó 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 important issue. Recurring destructive seismic events and hurricanes in the United States and elsewhere point to a compelling need for the development of e€ective protective systems against such hazards. Various means have been developed and implemented over the years to control excessive structural response to environmental forces induced by earthquakes or winds. For example, in passive structural control, energy dissipation devices are added to a structure so that a large portion of the input energy can be dissipated through

q

Disclaimer: 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, Oce 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 Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 6 8 3 ( 0 1 ) 0 0 0 2 6 - 9

8066

S.W. Park / International Journal of Solids and Structures 38 (2001) 8065±8092

these devices, thereby reducing energy dissipation demand on the original structure. Such devices include 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 e€ects 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 characterization of the rheological behavior of the dampers. Di€erent 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 stress±strain relation for a linear viscoelastic system represented by a spring±dashpot mechanical model is commonly expressed in a di€erential operator form, and the time-domain material functions derived from such a model is expressed by a series of decaying 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 and interconversion 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 spring±dashpot mechanical model except that the regular di€erential operators are replaced by fractional-order di€erential operators. The primary motivation for the use of fractional derivatives comes from their ability to describe the broad-band behavior of many viscoelastic materials with a small number of parameters. Other widely used phenomenological models for linear viscoelasticity include di€erent forms of power laws (Schapery, 1974). In particular, the so-called modi®ed 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.

S.W. Park / International Journal of Solids and Structures 38 (2001) 8065±8092

8067

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 behavior of the damper quite well both in the time domain and frequency domain. No particular report has been identi®ed 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 ecient for a routine implementation. Determination of model constants from experimental data normally requires a dicult, 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 signi®es 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 eciency, 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 can describe 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 adequately 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 viscoelastic phenomenon, these simple models are not adequate for characterization of the broad-band behavior 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 broadband rheological behavior of linear viscoelastic media. In Section 2, the linear viscoelastic stress±strain relationship and di€erent 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 in di€erent 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 discussion on the theoretical basis for di€erent 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 stress±strain equation for a nonaging, linear viscoelastic material can be represented by the following Boltzmann superposition integral: Z t de…s† r…t† ˆ ds …1† E…t s† ds 0 where E…t† is the relaxation modulus, and E…t† ˆ e…t† ˆ 0 for 1 < t < 0. Eq. (1) follows from the memory hypothesis, smoothness assumptions and mathematical representation theorem (Christensen, 1982). For a thermorheologically simple material (Morland and Lee, 1960), the stress±strain equation under transient temperatures (or nonisothermal condition) can also be expressed by Eq. (1) but with the time variable, s, Rt replaced with the so-called reduced time de®ned as n ˆ 0 ds=aT where aT is a function of temperature

8068

S.W. Park / International Journal of Solids and Structures 38 (2001) 8065±8092

called the time±temperature 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, E…t†. 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 stress±strain equation for such a model involves standard (or ordinary) di€erential operators. A general form of the stress±strain equation in di€erential operators is given by (Fung, 1965) M X mˆ0

am

N dm r X dn e ˆ bn n m dt dt nˆ0

…2†

where am and bn are constants, and dm … †=dtm denotes the mth-order time derivative of the function … †. Mechanical models with di€erent arrangements of springs and dashpots render di€erent mechanical interpretations of the constants am and bn in Eq. (2). The generalized Maxwell model, widely used to characterize the modulus functions of linear viscoelastic media, 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) E…t† ˆ Ee ‡

m X

Ei e

t=qi

…3†

iˆ1

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 de®ned by qi ˆ gi =Ei where gi is the 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 di€erential 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 stress±strain equation, Eq. (1)

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

S.W. Park / International Journal of Solids and Structures 38 (2001) 8065±8092

8069

and the stress±strain equation in a di€erential-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 di€erential operators in its mathematical representation) has long been accepted as an accurate, ecient tool for the characterization of the viscoelastic behavior of many polymers and polymeric composites. 2.2. The fractional derivative model In an e€ort to further generalize the stress±strain equation in a di€erential 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 M N X dpm r X dqn e am p m ˆ bn qn dt dt mˆ0 nˆ0

…4†

where pm and qn are real constants with 0 6 pm , qn 6 1. The ordinary time derivatives acting on the timedependent stress and strain ®elds in Eq. (2) are now replaced with corresponding fractional-order time derivatives. A fractional time-derivative of order a is de®ned, in an integral form, as (Bagley and Torvik, 1983a) Z t  da f …t† 1 d f …s†  …5† a ds dta C…1 a† dt 0 …t s† where 0 < a < 1 and C…
† denotes the gamma function. Although a summation of multiple terms with di€erent fractional orders are implied on both sides of Eq. (4), only a few terms are used in practice. For example, a model with M ˆ N ˆ 1, p0 ˆ q0 ˆ 0 and p1 ˆ q1 ˆ a in Eq. (4) is often used to describe the stress±strain behavior of a class of viscoelastic materials, i.e., r‡b

da r da e ˆ E0 e ‡ E1 a a dt dt

…6†

where b, E0 and E1 are constants and 0 6 a 6 1. An explicit form of the relaxation modulus E…t† can be derived from Eqs. (1) and (6) but its expression would be much more involved than Eq. (3) derived from SMM. 2.3. Power-law representations Di€erent 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: E…t† ˆ Ee ‡ E1 t

n

…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 of Ee 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 following form (Williams, 1964):

8070

S.W. Park / International Journal of Solids and Structures 38 (2001) 8065±8092

E…t† ˆ Ee ‡

E g Ee …1 ‡ t=q†n

…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) can reasonably 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 E…t† derived from Eq. (6) with that of Eq. (8) and found the following asymptotic relations between the two groups of model parameters: E 0 ˆ Ee ;

E1 ˆ Eg ; b

a ˆ n;

b C…1

a†

ˆ 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 di€erent 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 advantageously over others in computing the response of a viscoelastic medium. For example, a modulus function can be used more conveniently when strain is speci®ed 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 correspondence principle, one needs to deal with material functions de®ned in the Laplace-transform domain. For later references, the linear viscoelastic material functions in di€erent domains are brie¯y 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 e€ective) 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 experimental 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 characterization 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 expressions 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.

S.W. Park / International Journal of Solids and Structures 38 (2001) 8065±8092

8071

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), D…t† ˆ Dg ‡

n X t ‡ Dj …1 g0 jˆ1

e

t=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 de®ned by sj ˆ Dj gj where gj is the viscosity of the unit. For viscoelastic solids, g0 ! 1 and thus the second term in Eq. (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): E0 …x† ˆ Ee ‡

E00 …x† ˆ

m X x2 q2i Ei x2 q2i ‡ 1 iˆ1

m X xqi Ei 2 q2 ‡ 1 x i iˆ1

D0 …x† ˆ Dg ‡

D00 …x† ˆ

n X

Dj ‡1

x2 s2j jˆ1

n X 1 xsj Dj ‡ g0 x jˆ1 x2 s2j ‡ 1

~ ˆ Ee ‡ E…s†

m X sqi Ei sq i ‡1 iˆ1

n X Dj ~ ˆ Dg ‡ 1 ‡ D…s† g0 s jˆ1 ssj ‡ 1

…11†

…12†

…13†

…14†

…15†

…16†

where E0 , E00 , D0 and D00 are commonly referred to as the storage modulus, loss modulus, storage compliance ~  sD  are often called the operational modulus and and loss compliance, respectively, and E~  sE and D  denote the Laplace transforms of E…t† and D…t†, respectively. The symbols x and compliance where E and D s 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).

8072

S.W. Park / International Journal of Solids and Structures 38 (2001) 8065±8092

3.2. Fitting of material functions The constants in the series representation of a material function can be evaluated by ®tting the expression 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 di€erent times, and the m unknowns, Ei …i ˆ 1; . . . ; m†, are found by solving the resulting system of m linear algebraic equations. In the 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 ˆ 1; . . . ; m† in general, which would lead to a system of 2m nonlinear equations. However, the relaxation times qi are usually speci®ed 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 inspecting the long-time behavior of the relaxation modulus data. It is to be noted that the model parameters in 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-de®ned subsets of the experimental data to enhance the quality of a ®t, Kaschta and Schwarzl (1994a,b) proposed a method that ensures positive coecients through an interactive adjustment of relaxation or retardation times, and Baumgaertel and Winter (1989) employed a nonlinear regression technique in which the coecients, 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 Elster et al. (1991) applied the so-called Tikhonov regularization techniques, and Elster and Honerkamp (1991) used the modi®ed 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 di€erent 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 dicult to subject sti€ materials to a constant-strain, relaxation test because of the requirement of a robust testing device. However, 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

S.W. Park / International Journal of Solids and Structures 38 (2001) 8065±8092

8073

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 integral equations, and demonstrated the e€ectiveness of the method through the determination of the relaxation 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 representation, and tested its e€ectiveness 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 speci®c 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 di€erent 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 properties of the ¯uid and the geometric details of the damper unit. Fig. 2(b) displays typical force±displacement 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) force±displacement hysteresis loops for vertical motion at f ˆ 2 Hz and T  25°C (Makris, 1991).

8074

S.W. Park / International Journal of Solids and Structures 38 (2001) 8065±8092

piping networks and by Hu€mann (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 di€erent 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. K 0 (x) and K 00 (x) by the fractional derivative model The experimental data for the storage and loss sti€ness 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, sti€ness rather than modulus is used since a force±displacement (rather than stress±strain) relation is discussed. 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) and Makris and Constantinou (1991), using FDM, obtained the following force±displacement relationship for the longitudinal motion of the piston: P ‡k

dr P dq u ˆ C0 q r dt dt

…17†

where P and u are the axial force and displacement, respectively, and the damper parameters C0 , k, r and q represent the zero-frequency damping coecient, relaxation time, and orders of fractional derivative, respectively. When r ˆ q ˆ 1, the model reduces to the classical Maxwell model with k being the relaxation time and C0 the viscosity. Eq. (17) is slightly di€erent 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), P^…x† ˆ K  …ix†^ u…x†

…18†

K  …ix† ˆ K 0 …x† ‡ iK 00 …x†

…19†

where P^ and u^ denote the Fourier transforms of P and u, respectively, and K is the complex sti€ness whose real and imaginary components, K 0 and K 00 , are the storage and loss sti€ness given by (Makris and Constantinou, 1991) 

K 0 …x† ˆ K 00 …x† ˆ

C0 xq cos …pq †‰1 ‡ kxr cos …pr2 †Š ‡ C0 kxq‡r sin …pr2 † sin …pq † 2 2 1 ‡ k2 x2r ‡ 2kxr cos…pr2 †

C0 xq sin …pq †‰1 ‡ kxr cos…pr2 †Š 2

C0 kxq‡r sin …pr2 † cos…pq † 2

1 ‡ k2 x2r ‡ 2kxr cos …pr2 †

…20†

…21†

Values for the damper parameters, C0 ˆ 15; 000 N s/m, k ˆ 0:3 s0:6 , r ˆ 0:6 and q ˆ 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. K 0 (x) and K 00 (x) by the standard mechanical model Using SMM (the generalized Maxwell model), the storage and loss sti€ness functions of a viscous damper can be expressed as

S.W. Park / International Journal of Solids and Structures 38 (2001) 8065±8092

8075

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

K 0 …x† ˆ

m X x2 q2i Ki x2 q2i ‡ 1 iˆ1

…22†

K 00 …x† ˆ

m X xqi Ki 2 q2 ‡ 1 x i iˆ1

…23†

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 sti€ness vanishes, i.e., Ke ˆ lim t!1 K…t† ˆ lim x!0 K 0 …x† ˆ 0 (Ferry, 1980). The coecients Ki …i ˆ 1; . . . ; m† can be found by ®tting Eq. (22) or

8076

S.W. Park / International Journal of Solids and Structures 38 (2001) 8065±8092

Table 1 SMM constants for sti€ness functions of the viscoelastic liquid damper i

qi (s)

Ki (kN/m)

1 2 3 4 5 6 7 8 9

1:00E 03 3:16E 03 1:00E 02 3:16E 02 1:00E 01 3:16E 01 1:00E ‡ 00 3:16E ‡ 00 1:00E ‡ 01

5:01E ‡ 02 1:66E ‡ 02 1:12E ‡ 02 8:12E ‡ 01 3:51E ‡ 01 1:02E ‡ 01 2:97E ‡ 00 5:84E 01 1:23E 01

Ke ˆ 0 kN/m

Eq. (23) to available experimental data. A nine-term …m ˆ 9† Prony series representation is employed in the current example. Relaxation times with half-decade intervals, qi ˆ 10…i 7†=2 …i ˆ 1; . . . ; 9†, are selected so that the entire frequency range of the data may be covered by the analytical representation. One-decade spacings of qi 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 describe the data. If the spacing between qi Õs 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 ˆ 9 in our case), a least squares method can be employed. Although only one set of data, either for K 0 or K 00 , is sucient to determine Ki Õs, both data sets are used here to enhance the quality of the ®t. The best-®t Ki Õs are found by minimizing the following functional representing the normalized square errors: 2 !2 !2 3 N 0 00 X K …x † K …x † k k 4 Eˆ 1 ‡ …24† 1 5 0 00 K K kˆ1 k k 0

00

where K k and K k denote the experimental data at the circular frequency of xk , and N is the total number of data points available. The model constants thus found are presented in Table 1, and K 0 and K 00 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 signi®cant 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 sti€ness, K…t†, which relates the force history to the input displacement history in a manner similar to Eq. (1), can be obtained from Eq. (3) with EÕs replaced by KÕs and setting Ke ˆ 0. The values for the parameters, Ki and qi , presented in Table 1 are used here again. Fig. 4 shows the graphical representation of K…t† thus obtained. As long as the parameters Ki and qi in Table 1 correctly represent the rheological properties of the viscous damper considered, the K…t† curve shown in Fig. 4 represents the exact relaxation sti€ness 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 between the functions. For instance, K…t† can be obtained from K 0 …x† by

S.W. Park / International Journal of Solids and Structures 38 (2001) 8065±8092

8077

Fig. 4. Relaxation sti€ness for the viscous damper in the time domain.

K…t† 

1 0 K …x† k xˆ1=t

…25†

where k ˆ C…1 n† cos …np=2†; the symbol C…
† denotes the Gamma function and n is the slope of the source function on a log±log scale, i.e., n ˆ d… log K 0 †=d… log x† at x ˆ 1=t. No analytical expression of K 0 …x† is required in Eq. (25) and the experimental data for K 0 …x† can directly be used to ®nd the K…t† 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 K 00 …x† ! K…t† conversion. Note that, while SMM readily gives K…t† once the model is calibrated using the experimental data for K 0 …x† or K 00 …x†, FDM does not provide an explicit analytical expression for K…t† for general orders of fractional derivative. Only for a special case of q ˆ 1 and r ˆ 0:5, an explicit expression for K…t† 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 sti€ness 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 eciently (Park and Schapery, 1999). Speci®cally, in view of Eq. (3) and Eqs. (10)±(16), if one set of constants, either fqi ; Ei …i ˆ 1; . . . ; m† and Ee g or fsj ; Dj …j ˆ 1; . . . ; n†; Dg and g0 g, 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 ; Ki …i ˆ 1; . . . ; m†g, for the sti€ness functions as shown in Table 1 and we seek to ®nd a set of constants, fsj ; Lj …j ˆ 1; . . . ; n†; Lg and g0 g, for the ¯exibility functions. Again, here we deal with sti€ness …K† and ¯exibility …L† functions that are parallel with modulus …E† and compliance …D† functions, respectively. Note that Ke ˆ 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, Lj …j ˆ 1; . . . ; n†: ‰AŠfLg ˆ fBg

or Akj Lj ˆ Bk

…summed on j; j ˆ 1; . . . ; n; k ˆ 1; . . . ; p†

…26†

8078

S.W. Park / International Journal of Solids and Structures 38 (2001) 8065±8092

Table 2 SMM constants for ¯exibility functions of the viscoelastic liquid damper i

si (s)

Li (m/kN)

1 2 3 4 5 6 7 8

1:78E 03 5:13E 03 1:70E 02 6:46E 02 2:40E 01 8:13E 01 2:82E ‡ 00 9:33E ‡ 00

4:77E 9:90E 1:35E 2:78E 4:92E 1:04E 1:76E 3:40E

Lg ˆ 1:10E

03 m/kN g0 ˆ 1:75E ‡ 01 kN s/m

where Akj ˆ

04 04 03 03 03 02 02 02

m X sk qi Ki 1 ‡ sk qi iˆ1

and Bk ˆ 1

!

1 1 ‡ s k sj

 …j ˆ 1; . . . ; n; k ˆ 1; . . . ; p†

!  m X sk qi Ki 1 1 Pm ‡ Pm sk iˆ1 qi Ki 1 ‡ sk q i iˆ1 Ki iˆ1

…k ˆ 1; . . . ; p†:

…27†

…28†

The symbol sk …k ˆ 1; . . . ; p† denote the discrete values of the Laplace-transform parameter at which the interrelationship is established. Time constants sj …j ˆ 1; . . . ; n† can be either determined following a numerical procedure (Park and Schapery, 1999) or speci®ed 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., p P n). The collocation method is e€ected when p ˆ n and the least squares method can be used when p > n. In the case of the least squares method, 2 a minimization of the square error kfBg ‰AŠfLgk with respect to Lj …j ˆ 1; . . . ; n† leads to the reT placement of ‰AŠfLg ˆ fBg in Eq. (26) with ‰AŠ ‰AŠfLg ˆ ‰AŠT fBg in which the product ‰AŠT ‰AŠ is a square matrix. In our example, a set of retardation time constants sj …j ˆ 1; . . . ; 8† are numerically determined following the procedure given by Park and Schapery (1999) and is tabulated in Table 2 together with the results for Lj …j ˆ 1; . . . ; 8†, Lg and g0 . The least squares method is used to solve the system of equations. It is to be noted that when a sti€ness 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. L0 (x) and L00 (x) The storage and loss ¯exibility functions for a viscous damper can be represented by SMM (the generalized Voigt model) as n X Lj …29† L0 …x† ˆ Lg ‡ 2 x s2j ‡ 1 jˆ1 L00 …x† ˆ

n X 1 xsj Lj : ‡ g0 x jˆ1 x2 s2j ‡ 1

…30†

S.W. Park / International Journal of Solids and Structures 38 (2001) 8065±8092

8079

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

Eqs. (29) and (30) follow from Eqs. (13) and (14) with DÕs replaced with LÕs. The functions L0 and L00 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 u^…x† ˆ L …ix†P^…x†

…31†

L …ix† ˆ L0 …x† ‡ iL00 …x†:

…32†

where u^ and P^ denote the Fourier transforms of u and P, respectively. The curves for L0 and L00 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, L…t†, relating an applied force history to the resulting displacement history in a manner similar to that of Eq. (A.1), can be obtained from Eq. (10) with DÕs replaced with LÕs. The resulting L…t† 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.

8080

S.W. Park / International Journal of Solids and Structures 38 (2001) 8065±8092

Fig. 7. Operational sti€ness and ¯exibility for the viscous damper in the Laplace-transform domain.

shown in Fig. 6, for a comparison, is an approximate L…t† curve obtained simply by taking the reciprocal of K…t† shown in Fig. 4. It can be clearly seen that the quasi-elastic relationship, K…t†L…t† ˆ 1, renders signi®cant errors especially in the long-time behavior. ~ ~ 4.7. K(s) and L(s) ~ ~ Finally, the operational sti€ness and ¯exibility functions, K…s† and L…s†, de®ned by Eqs. (15) and (16) with EÕs and DÕs replaced respectively with KÕs and LÕs, 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~ These functions are useful in solving viscoelastic boundary value problems following lationship of K~ and L. 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. 8. A viscoelastic solid damper; (a) geometry (Mahmoodi, 1969), (b) force±displacement hysteresis loops at f ˆ 3 Hz and T ˆ 21°C (Shen and Soong, 1995).

S.W. Park / International Journal of Solids and Structures 38 (2001) 8065±8092

8081

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 force±displacement hysteresis loops measured in a sinusoidal test conducted at the frequency of 3 Hz and the temperature of 21°C. 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, G…t†, of the viscoelastic solid damper. Because the experimental data were not available in a tabular form, data reconstructed from the MPL representation given by Shen and Soong (1995) are used here to calibrate the generalized Maxwell model expression of G…t†, Eq. (3), with EÕs replaced by GÕs. Shen and Soong (1995) gave Ge ˆ 0:05861 MPa, Gg ˆ 1039:1 MPa, q ˆ 1:26E 6 s, and n ˆ 0:586 for an MPL representation of Eq. (8), with EÕs replaced by GÕs. Note that for a viscoelastic solid, Ge is nonzero. An 18-term …m ˆ 18† Prony series is used and the model constants 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 because the latter is de®ned over a broader range of time (or frequency). 5.2. G 0 (x) and G 00 (x) The real and imaginary components of the complex shear modulus are presented in Fig. 10(a) and (b). The expressions for G0 and G00 , corresponding to the MPL representation of G…t† as in Eq. (8), are given by (Shen and Soong, 1995)

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

8082

S.W. Park / International Journal of Solids and Structures 38 (2001) 8065±8092

Table 3 SMM constants for modulus functions of the viscoelastic solid damper i

qi (s)

Gi (MPa)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1:00E 07 1:00E 06 3:16E 06 1:00E 05 3:16E 05 1:00E 04 3:16E 04 1:00E 03 3:16E 03 1:00E 02 3:16E 02 1:00E 01 3:16E 01 1:00E ‡ 00 3:16E ‡ 00 1:00E ‡ 01 1:00E ‡ 02 1:00E ‡ 03

1:33E ‡ 01 2:86E ‡ 02 2:91E ‡ 02 2:12E ‡ 02 1:12E ‡ 02 6:16E ‡ 01 2:98E ‡ 01 1:61E ‡ 01 7:83E ‡ 00 4:15E ‡ 00 2:03E ‡ 00 1:11E ‡ 00 4:91E 01 3:26E 01 8:25E 02 1:26E 01 3:73E 02 1:18E 02

Ge ˆ 5:86E

02 MPa

" 0

G …x† ˆ Ge ‡ …Gg

n

Ge †…xq† C…1

n† cos

" 00

G …x† ˆ …Gg

n

Ge †…xq† C…1

n† sin

 np 2

 np 2

‡ xq

 ‡ xq



# p  …xq†1 n cos ‡ xq 2 1 n

# p  …xq†1 n sin ‡ xq 2 1 n

…33†

…34†

The SMM representations for G0 and G00 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 G0 and G00 match very well except in the high frequency region. It is found that Eqs. (33) and (34) do not properly represent G0 and G00 in the high frequency region where xq P 1. Additional corrective terms are required for Eqs. (33) and (34) to accurately depict G0 and G00 in that region as pointed out in the Appendix of Shen and Soong (1995). For comparison purposes, G0 and G00 are also obtained directly from G…t† curve using the approximate 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 G0 . The approximate curve for G00 obtained directly from G…t† 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 intrinsic mathematical incompatibilities between the functions G…t† and G00 …x† (Schapery and Park, 1999). Note the decreasing G00 with frequency in the low and high frequency regions, which is consistent with the following general constraints on G00 (Christensen, 1982): lim G00 …x† ˆ 0

x!0

and

lim G00 …x† ˆ 0

x!1

…35†

Shen and Soong (1995) also addressed the use of a simpli®ed power law, basically in the form of Eq. (7). Although the simpli®ed power law, Eq. (7), o€ers an excellent description of the rubber and transition behavior, it does not adequately describe the short-time (or high-frequency), glassy behavior of most viscoelastic materials.

S.W. Park / International Journal of Solids and Structures 38 (2001) 8065±8092

8083

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

5.3. J(t), J 0 (x) and J 00 (x) The corresponding shear compliance functions, J(t), J 0 …x† and J 00 …x†, of the viscoelastic solid damper may 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 modi®cations to Eqs. (26)±(28) are required. The unknown constants, Jj , for the compliance functions may be found by solving the following set of equations: ‰AŠfJg ˆ fBg or

Akj Jj ˆ Bk

where Akj ˆ

m X sk qi Gi Ge ‡ 1 ‡ sk qi iˆ1

!

and Bk ˆ 1

m X sk qi Gi Ge ‡ 1 ‡ sk qi iˆ1

…summed on j; j ˆ 1; . . . ; n; k ˆ 1; . . . ; p†

1 1 ‡ s k sj

!,

…36†

 …j ˆ 1; . . . ; n; k ˆ 1; . . . ; p†

m X Gi Ge ‡

…37†

! …k ˆ 1; . . . ; p†:

…38†

iˆ1

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

Ge ‡

1 Pm

iˆ1

Gi

and

g0 ˆ

gX 0 !1 iˆ1

qi Gi :

…39†

The resulting shear creep compliance, J …t†, and the storage and loss compliance, J 0 …x† and J 00 …x†, are shown 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 JÕs for DÕs. Function J …t† obtained from the quasi-elastic approximation is also presented in Fig. 11 for comparison. At t ˆ 1 s, for example, J …t† obtained from the quasi-elastic relationship is 74% greater than the exact value. The quasi-elastic approximation, although

8084

S.W. Park / International Journal of Solids and Structures 38 (2001) 8065±8092

Table 4 SMM constants for compliance functions of the viscoelastic solid damper i

si (s)

Ji (MPa 1 )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1:02E 07 1:29E 06 4:79E 06 1:70E 05 5:62E 05 1:91E 04 5:89E 04 1:95E 03 6:03E 03 1:95E 02 6:03E 02 2:00E 01 5:75E 01 2:09E ‡ 00 4:79E ‡ 00 2:57E ‡ 01 1:62E ‡ 02 1:23E ‡ 03

5:50E 05 1:55E 04 4:42E 04 8:90E 04 1:69E 03 3:29E 03 5:96E 03 1:22E 02 2:50E 02 3:97E 02 1:03E 01 1:55E 01 3:58E 01 7:17E 01 1:12E ‡ 00 5:15E ‡ 00 6:03E ‡ 00 3:34E ‡ 00

Jg ˆ 9:64E

04 MPa

1

g0 ! 1

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

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 Laplacetransform domain. It is informative to note that, in view of Figs. 9±13, the curve shapes for G…t†, J 0 …x† and ~ share the same trends. These general J~…s† are all similar to each other, and likewise, J …t†, G0 …x† and G…s†

S.W. Park / International Journal of Solids and Structures 38 (2001) 8065±8092

8085

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.

trends provide a useful rule of thumb for the engineer to predict the response of the damper subjected to di€erent 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, straightforward and ecient 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 procedures for calibration (or ®tting) of a model and interconversion between material functions computationally ecient. 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 demonstrated above. Also, while FDM and MPL o€er very limited closed-form expressions for material functions in di€erent domains, SMM readily provides the explicit expressions for these functions. In addition, SMM are grounded in a well-de®ned physical basis. Some molecular theories indeed predict the SMM

8086

S.W. Park / International Journal of Solids and Structures 38 (2001) 8065±8092

representation of linear viscoelastic material functions. Speci®cally, the RouseÕs theory for dilute polymer solutions (Rouse, 1953) and the modi®ed 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 mathematically 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 E…t† and D…t† for the isothermal case, and Schapery (1964) similarly showed the same for certain important nonisothermal cases such as thermorheologically 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, e t , 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 MPL function 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 describe 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 di€erent 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, E0 and E00 are not independent but are related to each other (Tschoegl, 1989), 2x2 E …x† ˆ Ee ‡ p 0

Z 0

1

E00 …k†

1 k…x2

k2 †

dk

or

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

…40†

S.W. Park / International Journal of Solids and Structures 38 (2001) 8065±8092

8087

Z

2x 1 0 1 ‰E …k† Ee Š 2 dk …41† p 0 k x2 Eqs. (40) and (41) are known as Kronig±Kramers 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 approximate analytical interrelationships between E0 and E00 : 1 ^ …42† E0 …x†  Ee ‡  E00 …x† ^ xˆx k ^ E00 …x†  k ‰E0 …x† Ee Š …43† E00 …x† ˆ

^ xˆx

where k ˆ tan …np=2† and n ˆ d… log E00 †=d… log x† when E00 is the source function, or n ˆ d… log …E0 Ee ††= d… log x† when E0 is the source function. Relations (42) and (43) may be used by the engineer to check the adequacy of the experimental data obtained from a steady-state harmonic test. 7. Conclusions Di€erent 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 ecient 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 expression for the modulus or compliance function. The outstanding mathematical eciency associated with the 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 di€erent 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 another 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 stress±strain 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 Z t dr…s† e…t† ˆ ds …A:1† D…t s† ds 0 where D…t† is the creep compliance. From Eqs. (1) and (A.1), setting, e.g., r…t† ˆ H …t† where H …t† is the Heaviside step function with H …t† ˆ 1 for t > 0 and H …t† ˆ 0 for t < 0, one ®nds the following integral relationship between the relaxation modulus and creep compliance:

8088

S.W. Park / International Journal of Solids and Structures 38 (2001) 8065±8092

Z

t 0

E…t

s†

dD…s† ds ˆ 1 ds

…t > 0†:

…A:2†

A.2. In the frequency domain The complex material functions arise from a steady-state harmonic input. For instance, substituting e…t† ˆ eA eixt and r…t† ˆ rA eixt respectively into Eqs. (1) and (A.1) as input, one ®nds r…t† ˆ E …ix†e…t†

…A:3†

e…t† ˆ D …ix†r…t†

…A:4†

where E and D are the complex modulus and complex compliance, respectively, and have their real and imaginary parts such that E …ix† ˆ E0 …x† ‡ iE00 …x†

…A:5†

D …ix† ˆ D0 …x†

…A:6†

iD00 …x†:

The real and imaginary components of these complex material functions can be expressed in terms of transient material functions (Tschoegl, 1989) as Z 1 0 E …x† ˆ Ee ‡ x ‰E…t† Ee Š sin xt dt …A:7† 0

E00 …x† ˆ x

Z

1 0

‰E…t† Z

0

D …x† ˆ De ‡ x D00 …x† ˆ

Z x

0

1

1 0

Ee Š cosxt dt ‰D…t†

‰D…t†

…A:8†

De Š sin xt dt

…A:9†

De Š cosxt dt

…A:10†

in which Ee and De denote the long-time equilibrium modulus and compliance, respectively; i.e., Ee  limt!1 E…t† and De  limt!1 D…t†. Note that a minus sign is used in Eq. (A.6) so that D00 will be positive. The real parts, E0 and D0 , are commonly referred to as the storage modulus and compliance, and the imaginary parts, E00 and D00 , 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., r^…x† ˆ E …ix†^e…x† ^e…x† ˆ D …ix†^ r…x†

…A:11† R1

…A:12†

where r^ and ^e denote the Fourier transforms of r and e, respectively; r^…x†  1 r…t†e ixt dt. 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:

S.W. Park / International Journal of Solids and Structures 38 (2001) 8065±8092

E …ix†D …ix† ˆ 1:

8089

…A:13†

A.3. In the Laplace-transform domain Taking Laplace transforms of Eqs. (1) and (A.1), one ®nds ~ e…s† r…s† ˆ E…s† ~ r…s† e…s† ˆ D…s†

…A:14† R1

…A:15†

where r and e denote the Laplace transforms of r and e, respectively; r…s† ˆ 0 r…t†e st dt. The functions E~ ~ in Eqs. (A.14) and (A.15) are often referred to as the operational modulus and operational comand D pliance, respectively, and are de®ned by ~  sE…s†  E…s†

…A:16†

~  sD…s†  D…s†

…A:17†

 are the Laplace transforms of E…t† and D…t†, respectively. From Eqs. (A.14) and (A.15), or where E and D by taking the Laplace transform of Eq. (A.2), one readily ®nds ~ D…s† ~ ˆ 1: E…s†

…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 elastic±viscoelastic 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): ~ s!ix E …ix† ˆ E…s†j

…A:19†

~ s!ix D …ix† ˆ D…s†j

…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. 1±8. Aprile, A., Inaudi, J.A., Kelly, J.M., 1997. Evolutionary model of viscoelastic dampers for structural applications. J. Engng. Mech. 123 (6), 551±560. 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. 821±826. Bagley, R.L., 1989. Power law and fractional calculus model of viscoelasticity. AIAA J. 27 (10), 1412±1417. Bagley, R.L., Torvik, P.J., 1983a. Fractional calculus ±± a di€erent approach to the analysis of viscoelastically damped structures. AIAA J. 21 (5), 741±748. Bagley, R.L., Torvik, P.J., 1983b. A theoretical basis for the application of fractional calculus. J. Rheol. 27, 201±210. Baumgaertel, M., Winter, H.H., 1989. Determination of discrete relaxation and retardation time spectra from dynamic mechanical data. Rheol. Acta 28, 511±519. Bergman, D.M., Hanson, R.D., 1993. Viscoelastic mechanical damping devices tested at real earthquake displacements. Earthquake Spectra 9 (3), 389±417.

8090

S.W. Park / International Journal of Solids and Structures 38 (2001) 8065±8092

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), 371±387. 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), 1418±1426. 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), 565±575. 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), 693±703. Elster, C., Honerkamp, J., 1991. Modi®ed maximum entropy method and its application to creep data. Macromolecules 24, 310±314. Elster, C., Honerkamp, J., Weese, J., 1991. Using regularization methods for the determination of relaxation and retardation spectra of polymeric liquids. Rheol. Acta 30, 161±174. Emri, I., Tschoegl, N.W., 1993. Generating line spectra from experimental responses. Part I: Relaxation modulus and creep compliance. Rheol. Acta 32, 311±321. Emri, I., Tschoegl, N.W., 1994. Generating line spectra from experimental responses. Part IV: Application to experimental data. Rheol. Acta 33, 60±70. Emri, I., Tschoegl, N.W., 1995. Determination of mechanical spectra from experimental responses. Int. J. Solids Struct. 32, 817±826. 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), 359±362. 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 Cli€s, New Jersey. Gavin, H.P., Hanson, R.D., Filisko, F.E., 1996a. Electrorheological dampers. 1. Analysis and design. J. Appl. Mech. 63 (3), 669±675. Gavin, H.P., Hanson, R.D., Filisko, F.E., 1996b. Electrorheological dampers. 2. Testing and modeling. J. Appl. Mech. 63 (3), 676±682. Gemant, A., 1938. On fractional di€erentials. Philos. Mag. 25, 540±549. 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), 319±334. 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. In¯uence of viscoelastic dampers on the seismic response of a lightly reinforced concrete ¯at slab structure. Earthquake Spectra 15 (4), 681±710. Honerkamp, J., Weese, J., 1989. Determination of the relaxation spectrum by a regularization method. Macromolecules 22, 4372±4377. Hopkins, I.L., Hamming, R.W., 1957. On creep and relaxation. J. Appl. Phys. 28 (8), 906±909. 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), 897±971. Hu€mann, G.K., 1985. Full base isolation for earthquake protection by helical springs and viscodampers. Nucl. Engng. Des. 84 (2), 331±338. Jones, D.I.J., 1980. Viscoelastic materials for damping applications. Damping Applications for Vibration Control, AMD-vol. 38. ASME, New York, pp. 27±51. Kaschta, J., Schwarzl, F.R., 1994a. Calculation of discrete retardation spectra from creep data: I. method. Rheol. Acta 33, 517±529. Kaschta, J., Schwarzl, F.R., 1994b. Calculation of discrete retardation spectra from creep data: II. Analysis of measured creep curves. Rheol. Acta 33, 530±541. Kerwin Jr., E.M., 1959. Damping of ¯exural waves by a constrained viscoelastic layer. J. Acous. Soc. Am. 31 (7), 952±962. Kno€, W.F., Hopkins, I.L., 1972. An improved numerical interconversion for creep compliance and relaxation modulus. J. Appl. Polym. Sci. 16, 2963±2972. Koeller, R.C., 1984. Applications of fractional calculus to the theory of viscoelasticity. J. Appl. Mech. 51, 299±307. Koh, C.G., Kelly, J.M., 1990. Application of fractional derivatives to seismic analysis of base-isolated models. Earthquake Engng. Struct. Dyn. 19, 229±241. Li, W.Q., Tsai, C.S., 1994. Seismic mitigation of structures by using viscoelastic dampers. Nucl. Engng. Des. 147 (3), 263±274. Mahmoodi, P., 1969. Structural dampers. ASCE J. Struct. Div. 95 (8), 1661±1672. 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. 632±644.

S.W. Park / International Journal of Solids and Structures 38 (2001) 8065±8092

8091

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 Ba€alo, Bu€alo, 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), 571±591. Makris, N., Constantinou, M.C., 1991. Fractional-derivative Maxwell model for viscous dampers. J. Struct. Engng. 117 (9), 2708± 2724. Makris, N., Constantinou, M.C., 1992. Spring-viscous damper systems for combined seismic and vibration isolation. Earthquake Engng. Struct. Dyn. 21, 649±664. Makris, N., Constantinou, M.C., 1993. Models of viscoelasticity with complex-order derivatives. J. Engng. Mech. 119 (7), 1453±1464. Makris, N., Constantinou, M.C., Dargush, G.F., 1993a. Analytical model of viscoelastic ¯uid dampers. J. Struct. Engng. 119 (11), 3310±3325. Makris, N., Dargush, G.F., Constantinou, M.C., 1993b. Dynamic analysis of generalized viscoelastic ¯uids. J. Engng. Mech. 119 (8), 1663±1679. Makris, N., Dargush, G.F., Constantinou, M.C., 1995. Dynamic analysis of viscoelastic-¯uid dampers. J. Engng. Mech. 121 (10), 1114±1121. Mead, D.W., 1994. Numerical interconversion of linear viscoelastic material functions. J. Rheol. 38, 1769±1795. 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. 1±8. 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, 679±685. 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), 1653±1675. 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, 2177±2189. Rogers, L., 1983. Operators and fractional derivatives for viscoelastic constitutive equations. J. Rheol. 27 (4), 351±372. 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. 49±87. Rouse Jr., P.E., 1953. The theory of the linear viscoelastic properties of dilute solutions of coiling polymers. J. Chem. Phys. 21 (7), 1272±1280. 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), 639±654. 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, 1451±1465. 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. 85±168. 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), 1677±1699. Schwahn, K.J., Delinic, K., 1988. Veri®cation of the reduction of structural vibrations by means of viscous dampers, ASME Pressure Vessels and Piping Conference, PVP vol. 144, June19±23, Pittsburgh, PA, pp. 87±95. Schwarzl, F.R., Struik, L.C.E., 1967. Analysis of relaxation measurements. Adv. Mol. Relaxation Processes 1, 201±255. Shen, K.L., Soong, T.T., 1995. Modeling of viscoelastic dampers for structural applications. J. Engng. Mech. 121 (6), 694±701. Shukla, A.K., Datta, T.K., 1999. Optimal use of viscoelastic dampers in building frames for seismic force. ASCE J. Struct. Engng. 125 (4), 401±409. Smit, W., deVries, H., 1970. Rheological models containing fractional derivatives. Rheol. Acta 9, 525±534. 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 5±9, 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. 85±112. Tsai, C.S., 1993. Innovative design of viscoelastic dampers for seismic mitigation. Nucl. Engng. Des. 139, 165±182. Tsai, C.S., 1994. Temperature e€ect of viscoelastic dampers during earthquakes. J. Struct. Engng. 120 (2), 394±409. Tsai, C.S., Lee, H.H., 1993a. Applications of viscoelastic dampers to high-rise buildings. J. Struct. Engng. 119 (4), 1222±1233. Tsai, C.S., Lee, H.H., 1993b. Seismic mitigation of bridges by using viscoelastic dampers. Comput. Struct. 48 (4), 719±727. Tschoegl, N.W., 1989. The Phenomenological Theory of Linear Viscoelastic Behavior. Springer, Berlin.

8092

S.W. Park / International Journal of Solids and Structures 38 (2001) 8065±8092

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, 117±127. Tschoegl, N.W., Emri, I., 1993. Generating line spectra from experimental responses. Part II: storage and loss functions. Rheol. Acta 32, 322±327. Williams, M.L., 1964. Structural analysis of viscoelastic materials. AIAA J. 2 (5), 785±808. 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), 557±575. 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 21±24, 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, 389±396. Zou, X., Ou, J., 2000. Experimental study on properties of viscoelastic dampers and seismic vibration-suppression e€ects of viscoelastic damped structures. Shock Vib. Dig. 32 (1), 38.