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Viscoelastic Damping TechnologiesPart I:Modeling and Finite Element

Implementation

C.M.A. Vasques , R.A.S. Moreira and J. Dias Rodrigues

Departamento de Engenharia Mec anica, Faculdade de Engenharia,Universidade do Porto, Rua Dr. Roberto Frias s/n,

4200-465 Porto, Portugal e-mail: [email protected] and [email protected]

Departamento de Engenharia Mec anica, Universidade de Aveiro,Campus Santiago, 3810-193 Aveiro, Portugal

e-mail: [email protected]

Submitted: 05/03/2010

Accepted: 18/04/2010 Appeared: 29/04/2010

c HyperSciences.Publisher

Abstract: This is the rst of two companion articles addressing an integrated study onthe mathematical modeling and assessment of the efficiency of surface mounted or embeddedviscoelastic damping treatments, typically used to reduce structural vibration and/or noise radi-ation from structures, incorporating the adequate use and development of viscoelastic (arbitraryfrequency dependent) damping models, along with their nite element (FE) implementation,and the experimental identication of the constitutive behavior of viscoelastic materials. Thisrst article (Part I) is devoted to the development of mathematical descriptions of materialdamping to represent the linear viscoelastic constitutive behavior, their implementation intoFE formulations and the use of the underlying different solution methods. To this end, internal

variables models, such as the Golla-Hughes-McTavish (GHM) and anelastic displacement elds (ADF) models, and other methods such as the direct frequency response (DFR), based onthe complex modulus approach (CMA), iterative modal strain energy (IMSE) and an approachbased on an iterative complex eigensolution (ICE) are described and implemented at the globalFE model level. The experimental identication of viscoelastic materials properties and theaforementioned viscoelastically damped FE modeling approaches are assessed and validatedin the companion article [Vasques, C.M.A. et al., Viscoelastic damping technologiesPartII: Experimental identication procedure and validation, Journal of Advanced Research in Mechanical Engineering 1(2): 96-110 (2010)].

Keywords: Finite element, damping, viscoelastic, complex modulus, internal variables.

1. INTRODUCTION

Since the seminal developments in the 1950s, viscoelasticdamping technologies are nowadays well established andconstitute widespread means of controlling the dynamicsof structures, reducing and controlling structural vibra-tions and/or noise radiation. These technologies have beenwidely used in several technological areas (e.g. automotive,aeronautics, aerospace, acoustics). Several reasons couldexplain its development and widespread use. However, themost intuitive one relies on the evident growing interest inthe use of composite materials that have been intensivelyapplied to produce lighter and stiffer composite structures. The joint funding scheme provided by the European Social Fund

and Portuguese funds from MCTES through POPH/QREN/Tipo-logia 4.2 and project PTDC/EME-PME/66741/2006 are gratefullyacknowledged by the authors.

However, detrimentally, along with new and improvedassembly techniques, these new materials have also beenresponsible for an important reduction of the inherentmaterial and joint damping characteristics of exible com-posite structures.

In general, viscoelastic damping technologies have beenused mainly as distributed surface mounted or embed-ded damping treatments, utilizing passive viscoelastic ma-terials alone, the so-called passive damping treatments [see, for example, Kerwin (1959), Mead and Markus(1969), Nashif et al. (1985), Mace (1994)] or, more re-cently, in an unied way making use also of active materi-als such as piezoelectric polymers or ceramics, the so-calledhybrid damping treatments [see, for example, Plump andHubbard (1986), Baz (1993), Benjeddou (2001), Vasquesand Rodrigues (2008)]. When adequately designed, the

Journal of Advanced Research in Mechanical Engineering (Vol.1-2010/Iss.2)Vasques et al. / Viscoelastic Damping TechnologiesPart I : Modeling and / pp. 76-95

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Viscoelastic Damping TechnologiesPart I: Modeling and Finite Element Implementation 2

use of viscoelastic materials in passive or hybrid dampingtreatments furnishes structures with articial dampingmechanisms and might provide high damping capabilityover wide temperature and frequency ranges. However,their design and analysis (numerical and/or analytical) isquite difficult and cumbersome. One of these difficulties is

related with the development, application and implemen-tation of efficient constitutive mathematical models ableto represent the frequency- and temperature-dependentconstitutive properties of general viscoelastic materials.Another difficulty is related with the reduced efforts thathave been made in the publication and dissemination of databases with material data of general viscoelastic mate-rials, making it sometimes difficult to nd accurate datafor the material under consideration.

Since a generalized and widespread used viscoelastic con-stitutive model has not yet been recognized as such andtherefore is not available, the necessary viscoelastic mate-rial information is usually identied from low resolutionand outdated nomograms published by the material man-ufacturers in the form of frequency dependent complexmoduli data, which has been the most current and univer-sal means of providing material data used by the manu-facturers, or eventually obtained by adverse published datarelated with fortuitous heuristic theoretical models and/orby non-standardized experimental characterization proce-dures. In fact, quite often, when dealing with new ornon-standard viscoelastic materials (e.g. cork compounds),experimental identication tests for these materials needto be carried out in order to furnish the designer withthe most important information at the start of the designstage: the frequency- and temperature-dependent consti-tutive law of the material.

In virtue of the aforementioned issues, the present workpresents an integrated study on the mathematical model-ing and assessment of the efficiency of surface mountedor embedded viscoelastic damping treatments, incorpo-rating the adequate use and development of viscoelastic(arbitrary frequency dependent) damping models, alongwith their nite element (FE) implementation, and theexperimental identication of the constitutive behavior of viscoelastic materials. The work is presented in two com-panion papers; the rst, Part I, addresses mathematicalmodeling and implementation issues and the second, PartII, experimental identication and modeling validationones.

The present article (Part I) is devoted to the developmentof mathematical descriptions of material damping to repre-sent the linear viscoelastic constitutive behavior, their im-plementation into FE formulations and the use of the un-derlying different solution methods. To this end, the maindifficulties of the different approaches to modeling, FEimplementation and solution methods proposed in the lastdecades are shortly reviewed and the new trends in bothtime and frequency domain viscoelastic damping modelingand their FE implementations are hereby presented anddiscussed. Both time and frequency domain damping mod-els, typically necessary in FE analysis of structural systemswith embedded or surface mounted hybrid or purely pas-

sive damping treatments, are presented. Internal variablesmodels, such as the Golla-Hughes-McTavish (GHM) andanelastic displacement elds (ADF) models, and other

methods such as the direct frequency response (DFR),based on the complex modulus approach (CMA), iterative modal strain energy (IMSE) and an approach based on aniterative complex eigensolution (ICE) are described andimplemented at the global FE model level. Additionally,a state space representation of the integrated damped FE

models and model reduction procedures are discussed.Regarding the experimental identication of viscoelasticmaterials and the assessment and validation of the afore-mentioned integrated FE damping modeling approaches,they are presented in the companion article (Part II) byVasques et al. (2010) considering the well known viscoelas-tic material 3M ISD112 and a sandwiched viscoelasticplate.

2. VISCOELASTIC CONSTITUTIVE MODELING

2.1 Historical Developments of Viscoelasticity

In general, the constitutive behavior of viscoelastic mate-rials might be said to depend upon the frequency, work-ing temperature, amplitude and type of excitation [Ferry(1980), Nashif et al. (1985), Riande et al. (2000), Jones(2001)]. A mathematical model considering all these ef-fects simultaneously has not yet been developed, is verydifficult to conceive and in practice has somewhat lim-ited interest and applicability. Thus, for simplicity, sincethe amplitude and type of excitation effects have beenreported to be of reduced importance, these parametersare often overlooked. However, the temperature and fre-quency dependent mechanical properties of the viscoelasticmaterials still introduce serious difficulties in the deni-

tion of an accurate mathematical model able to simulateproperly the dynamic behavior of the damped structure.Therefore, for practical reasons, isothermal conditions areusually assumed in the simulation conditions and merelythe frequency dependent constitutive behavior is directlytaken into account upon the constitutive mathematicalmodel. Following this assumption, the design of passiveviscoelastic damping treatments, for broad temperaturerange applications, is usually conduced at several con-stant temperature levels, selected within the temperaturerange, considering isothermal conditions. A few excep-tions, though, where the isothermal simplication is notconsidered and where both the temperature and frequencyeffects are captured in a coupled model able to accountfor the the self-heating effects of the viscoelastic material,were reported by Schapery (1964), who extended Biotstheory [Biot (1954, 1955)] in the application of the ther-modynamics of irreversible processes to viscoelastic mate-rials, yielding a unied theory of the thermomechanicalbehavior of viscoelastic materials with an explicit tem-perature dependence and a consistent thermodynamicallyinclusion of the time-temperature superposition principle,and by Lesieutre and his co-workers [Lesieutre and Govin-dswamy (1996), Brackbill et al. (1996)], that extended theADF method to include thermal effects by means of theuse of the linear time-temperature superposition principle ,valid for thermorheologically simple materials , yielding a

nonlinear amplitude-dependent FE model which requiresspecic nonlinear solution strategies. Other studies wherethe temperature effects are also considered can be found


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Viscoelastic Damping TechnologiesPart I: Modeling and Finite Element Implementation 3

in [Taylor et al. (1970), Zocher et al. (1997), Hammerandand Kapania (1999), Demirdzic et al. (2005), Cliffordet al. (2006)]. The effect of the operating temperatureupon the performance of hybrid damping treatments andviscoelastic damping efficiency were analyzed, for example,in [Baz (1998), Friswell and Inman (1998), Trindade et al.

(2000a), Silva et al. (2005), Pradeep and Ganesan (2006)].Literature covering the three primary mechanisms of damping which are important in the study of mechanicalsystems, namely internal damping (or material ), struc-tural damping (at joints and interfaces) and uid damping (through uid-structure interaction), is very vast and it isnot the purpose here to survey it. The interested readersare referred to some textbooks and articles on generaldamping theory [Bishop (1955), Crandall (1970), Ewins(2000), Lesieutre (2001), Inman (2001), Cremer et al.(2005), de Silva (2005)] in order to become acquaintedwith the developments on damping theories. Worth tomention is a recent historical and contemporary criticaldiscussion on damping theory, also pointing out the per-spective on how the treatment of damping is likely toevolve in the future, performed by Peters (2005).

Directing now our attention to material damping, sincethe rst experimental observation of material (or inter-nal ) damping performed by Coulomb in the 1780s withhis memoire Sur la force de torsion et sur lelasticitedes ls de metal [Coulomb (1784)], where he not onlyhypothesized regarding the microstructural mechanismsof damping but also undertook experiments which provedthat the damping of torsional oscillations is not caused byair friction but by internal losses in the material, over thefollowing centuries different methods to characterize fre-quency dependent damping, in general, and the viscoelas-tic materials constitutive behavior (viscoelastic damping),in particular, were proposed.

As discussed by Snowdon (1968), the mechanical prop-erties of rubber-like materials may be considered at twodifferent damping levels: (i) low-damping materials , whichhave the dynamic modulus and damping factor (typicallywith a low value of the order 0.1) varying slowly with fre-quency, which enables them to be considered as constantsthrough the range of frequencies normally of concern in vi-bration problems; and (ii) high-damping materials , whereboth the dynamic modulus and damping factor vary sig-nicantly with frequency. For the latter case, during thetransition zone, where the material undergoes a changefrom rubber to glassy dominated behavior, the dynamicmodulus increases very rapidly with frequency and thedamping factor is large (typically with values of order 1)and might vary slowly or more strongly with frequency.

Under the foregoing assumptions, Snowdon has consideredfour different models of damping, which he called dampingof the solid type I and II, of the viscous type and of athree-element spring and dashpot combination . The rstdamping type is nowadays referred to in the literatureas hysteretic damping [Banks and Pinter (2001), Inman(2001)], where both the storage modulus and loss factor donot depend on frequency and is typically applied to modelthe small energy dissipation effects due to the internalfriction which occurs in metals in general. The second andthird damping types alleviate, respectively, the assumption

that the storage modulus or the loss factor do not dependon frequency. The interest and applicability of them forviscoelastic modeling is somewhat reduced since usuallymodels of rubber-like materials need to have explicit fre-quency dependent mechanical properties in order to ac-curately capture their real physical behavior. Regarding

the third damping type, it is considered by an equivalentconstitutive model comprising an Hookean spring in par-allel with a Newtonian dashpot, which justies the termviscous used for its designation, where the loss factor isconsidered frequency-dependent (the well known viscous damping [Gandhi (2001), Inman (2001)]). It should bementioned, though, that the viscous equivalent consti-tutive model, also known as Kelvin-Voigt model , poorlyrepresents the dynamic mechanical properties of rubber-like materials justifying the need for other type of models.In view of that, the fourth model, known as standard linear solid , considering a three-element combination of a springand dashpot in series (Maxwell model) placed in parallelwith another spring, allows both the dynamic modulus and

damping factor to depend on frequency. In fact, all the rstthree damping types can be seen as particular cases of thismore general last model.

The aforementioned four different types of damping illus-trate well the real physical damping behavior observed inthe different kinds of materials that we have at our disposaland it is up to the designer to choose the model thatbest suits the real physical behavior of the material underconsideration. Usually, in the authors opinion, the desig-nation of viscoelastic damping refers to the fourth casementioned above where both the storage modulus and lossfactor are physically observed to depend signicantly withfrequency. However, depending on the degree of frequency

dependence, some viscoelastic models might be preferredover the others depending on how well the observed creepand relaxation, in the time domain, and broadband, inthe frequency domain, behaviors are approximated by theconsidered viscoelastic damping model.

The four models mentioned before can be regarded asclassical viscoelastic damping models. These modelswere gradually developed since the 18th century in an at-tempt to mathematically represent the real physically ob-served behavior of a class of materials somewhat evidenc-ing a mechanical behavior composed by a blend of pureviscous uids and elastic solid materials constitutive char-acteristics. These mechanical model analogies of viscoelas-

tic materials consider selected combinations of discreteelastic and viscous damping elements, ranging from basicdiscrete systems such as the Maxwell model , established byMaxwell (1868), or Kelvin-Voigt model , rst established byKelvin (1875) and later by Voigt (1892) (sometimes alsoreferred to in the literature as only Kelvin or Voigt model),to more complex models combining these basic elementsin parallel or series, namely, the linear standard solid model , also known as Kelvin or Zener model [Zener (1948)],and the Burgers model [Burgers (1935)], which representmore realistic viscoelastic models with good complexity toaccuracy trade-off, or even generalized versions combiningthese elements (e.g. the generalized Maxwell model andgeneralized Kelvin model ). The latter generalized mod-els yield frequency- and time-domain mathematical seriesrepresentations of the viscoelastic constitutive behavior in


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terms of the so-called Dirichlet or, alternatively, Pronysseries [Prony (1795), Williams (1964)]; see also some worksusing Pronys series in [Park and Schapery (1999), Slaniket al. (2000)]. Lastly, all these models, though, can alsobe seen as particular cases of a more general formulation,namely, the so-called differential equation form of the

linear viscoelastic constitutive behavior under isothermalconditions. The interested reader can nd further detailson classical models and differential equation forms usingmechanical analogies in [Alfrey and Doty (1945), Williams(1964), Ferry (1980), Findley et al. (1989), Johnson(1999), Fung and Tong (2001), Ottosen and Ristinmaa(2005)] and in the references therein.

Certainly, a very comprehensive and accurate model maybe constructed by combining generalized models in anarbitrary way. Irrespective of how we combine springs anddashpots, it comes as no surprise that it is always possibleto write the constitutive relation in a differential equationform. An advantage of the differential equation approachis that each model is easy to physically understand andinterpret and models can be constructed in an intuitivefashion. The drawback, however, is that more advancedmodels soon become cumbersome since the initial con-ditions become difficult to deal with when using higher-order differential equation in the time domain [Ottosenand Ristinmaa (2005)].

Since to have good accuracy representing complex rate-dependent materials, such as viscoelastic materials, morecomplex networks of springs and dashpots are usually re-quired, an alternative approach is to change the differentialequation for the evolution of the dissipative stresses bymeans of the tools of fractional calculus ; see, for example,Gemant (1936), Stiassnie (1979), Bagley and Torvik(1983), Koeller (1984), Bagley and Torvik (1985), Koeller(1986), Padovan (1987), Bagley (1989) and Pritz (1996).In that case, the viscoelastic constitutive behavior is de-ned by a fractional differential equation form where thetime derivatives order are not integer numbers. As sug-gested by Koeller (1984), the dashpots used in the classicalrheological models might be substituted by the so-calledspring-pot element, yielding a fractional differential modelwhere the spring-pots are elements that, depending on thenon-integer order (or memory parameter) of the derivative,allow a continuous transition of the constitutive behaviorfrom the solid state (full memory representing a springwith complete elastic recovery) to the uid state (no mem-

ory representing a dashpot with unrecoverable behavior)when the memory parameter varies from zero to one.

Using the fractional derivative model instead of the clas-sical differential approach, one advantage is the easinesshow with a reduced number of parameters, typically three[Bagley and Torvik (1985)], four [Pritz (1996), Galucioet al. (2004)] or ve [Bagley and Torvik (1983)], the vis-coelastic material properties are modeled in the frequencydomain. However, its representation and use in the timedomain is far more complicated than in the frequencydomain requiring specic solution procedures to integratethe governing equations; see, for example, Enelund andJosefson (1997).

As discussed previously, there were several early contri-butions to the theory of linear viscoelasticity stated in

terms of differential equations, such as the ones given byMaxwell, Kelvin and Voigt. However, integral equationscould be used as well, and in general terms, the integralapproach might be seen as the general approach and start-ing point from which the viscoelastic theory should startfrom. Boltzmann developed in 1874 the rst formulation

of a three-dimensional theory of linear isotropic viscoelas-tic stress-strain relations while Volterra (1909, 1913) ob-tained comparable forms for anisotropic solids in 1909.Boltzmann initiated a linear hereditary theory of materialdamping by formulating an integral representation of thestress-strain relation which allows greater freedom whenconstructing models than the differential approach thatrelies on the concepts of certain combinations of springsand dashpots. Integrals are summing operations, and thisview of viscoelasticity takes the response of the materialat a certain instant of time to be the sum of the responsesto excitations imposed at all previous instants of time.The ability to sum these individual responses requires thematerial to obey a more general statement of linearity than

we have invoked previously, specically that the responseto a number of individual excitations be the sum of the re-sponses that would have been generated by each excitationacting alone.

Thus far we have devoted our attention to time-dependentformulations by means of differential or integral equationsforms where the typical creep and relaxation behaviorof viscoelastic materials were more directly observableand identied from experimental measurements. However,since one of the simplest ways of experimentally determinethe viscoelastic properties is to subject the material toperiodic dynamic oscillations, the dynamic representationof the material properties by means of a complex modulus

became a current practice and is perhaps the most com-monly used approach when modeling the material in thefrequency domain.

As far as material damping is concerned, and in par-ticular the viscoelastic damping, the interested read-ers in further details are referred to a report witha unique comprehensive bibliography survey until theearly-mid of the 20th century of the material dampingeld compiled by Demer in the 1950s [Demer (1956)]and to some articles [Alfrey and Doty (1945), Lazan(1959), Ungar and Kerwin (1962), Williams (1964), Bert(1973), Tschoegl (1997), Johnson (1999), Mead (2002)]and textbooks [Zener (1948), Snowdon (1968), Chris-

tensen (1982), Nashif et al. (1985), Tschoegl (1989), Fung(1993), Sun and Lu (1995), Mead (1998), Riande et al.(2000), Fung and Tong (2001), Jones (2001), Goodman(2002), Ottosen and Ristinmaa (2005)] which illustratewell the later developments in the eld.

Within the scope of linear theory, time and frequencydomain constitutive models are presented in what follows.

2.2 Time Domain Representation: Hereditary Approach

Viscoelastic materials are sometimes called materials withinnite memory, in the sense that their actual mechan-ical response is modulated by the past history. There-

fore, the viscoelastic constitutive behavior relies on theassumption that the current value of the stress tensordepends upon the complete past history of the strain


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tensor components. This latter assumption therefore im-plies that the behavior of any viscoelastic material maybe represented by a hereditary approach. Furthermore, if the net strain response of the material to an arbitrarysequence of stimulus can be determined through the sumof the responses which would have been obtained if all

stimuli had act independently, time dependence linearity ,and if an increase in the strain stimulus by an arbitraryfactor increases the stress response by the same factor,stress-strain linearity , the material is said to have a linear viscoelastic behavior. These are the type of viscoelasticmaterials discussed in what follows.

Considering an isotropic viscoelastic material under isother-mal conditions and under small (innitesimal) deformationconditions, the theory of linear viscoelasticity [Christensen(1982), Fung and Tong (2001)] states that the constitutiverelationship for a generic one-dimensional isotropic stress-strain system (e.g. in shear or extension) can be given bya Riemann convolution integral,

(t) = t

Grel (t )

( )

d , (1)

where (t) and (t) are the time dependent stress andarbitrary strain component histories and Grel (t) is calledthe constitutive time varying (shear) characteristic relax-ation function of the material (the stress response to aunit-step strain input) which is also utilized in the litera-ture under many different names, such as damping kernel ,retardation , hereditary or after-effect function. Equation(1) expresses an essential feature of linear behavior of viscoelastic materials known as Boltzmanns superposition principle [Boltzmann (1874)]. Since the lower limit of inte-gration is taken as , it is to mean that the integrationis to be taken before the very beginning of the motion.Thus, if the motion starts at time t = 0, meaning thatthe stress and strain are equal to zero up until time zerowhere the loading begins, i.e., (t) = (t) = 0 for t < 0,and discontinuous strain histories with a step discontinuityat t = 0 are to be considered, Equation (1) reduces to

(t) = Grel (t)(0) + t

0Grel (t )

( )

d , (2)

where (0) is the limiting value of (t) when t 0 fromthe positive side. The rst term in Equation (2) gives theeffect of the initial disturbance and it arises from allowinga jump of (t) at t = 0. Furthermore, when Equation (1)was written it was tacitly assumed that (t) is continuousand differentiable [see Fung and Tong (2001) for furtherdetails].

Considering nil initial conditions, i.e., (0) = 0, the La-place transform of Equation (2) yields

(s) = G(s)(s) s Grel (s)(s), (3)where G(s) s Grel (s) is a characteristic material func-tion , which should be experimentally identied somehow.Among all the possible identication tests [see Tschoegl

(1989) for further details], the relaxation test , where astep relaxation strain stimulus with height rel is applied,i.e. (s) = rel /s , is considered here. Substituting it into

Equation (3), the relaxation stress is given by

rel (s) = rel Grel (s). (4)

Transforming the previous equation back to the time do-main, the constitutive time varying characteristic relax-

ation function of the material is given by

Grel (t) = rel (t)

rel. (5)

The previous equation allows us to directly determine itsvalue from the measured time varying relaxation stresshistory to a step strain stimulus which is obtained from arelaxation test applied to a viscoelastic material. In fact,the typical relaxation behavior is well known and is usuallydescribed by a time domain curve composed by the sum of a constant step part and a time decaying counterpart withthe simplest relaxation function having the fading memorycharacteristics being that of a single decaying exponential

[Christensen (1982)]. Therefore, in general, the relaxationbehavior might be expressed as

Grel (t) = rel (t)/ rel G + r (t), (6)which in the Laplace domain becomes

Grel (s) = G

s + r (s), (7)

where G is the so-called relaxed (also known as static )modulus, which for an isotropic material might be theshear or extension modulus, obtained after the materialrelaxation, i.e., G is the limiting value of Grel (t) when

t . The resultant relaxation modulus in Equation (7)therefore has two components; since G is constant, therst component, G /s , represents the recoverable (elastic)counterpart of the material stress, while the second com-ponent, r (s), represents the non-recoverable (dissipated)one. For that reason, r (s) is usually termed as dissipa-tion , or more appropriately, relaxation function. Thesetwo parameters, G and r (s), are unique characteristicproperties of the viscoelastic material which are obtainedby curve tting experimental data, obtained from a relax-ation identication test where a step function deformationstimulus is utilized, according to the mathematical vis-coelastic damping representation chosen by the analyst.In fact, it is worthy to emphasize here that in order tocharacterize the viscoelastic material behavior to any typeof strain stimulus, Equation (3) should be used; the nec-essary characteristic material function, G(s) s Grel (s),can readily be obtained after the relaxation stress hasbeen determined in a relaxation test. Depending uponthe nature of the viscoelastic material being considered,as discussed for example by Brinson and Brinson (2008),Equation (6) is pretty general being able to represent boththe behavior of thermosetting (crosslinked) polymers if G = 0 or thermoplastic (linear) polymers if G = 0assuming in both cases that r = r (t).Many authors have proposed different mathematical rep-resentations of the relaxation behavior given by r (s) in thelast decades, which in turn leads to different mathematicalmodels of the viscoelastic damping behavior and different


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Table 1. Mathematical representations of eligible relaxation functions in Laplace and time domains.

Function r (s) Function r (t), t 0 Author and yearn

s =1

D s p + s

+ D n

s =1D s e s t + (t)D Biot (1954)

0

D() () p +

d + D

0D() ()e t d + (t)D Biot (1954)Buhariwala and Hansen (1988)

E 1 s E 0 bss(1 + bs )

E 1 bE 0b

E [(t/ b) ] Bagley and Torvik (1983)

n

i=1 is + 2 i i

s2 + 2 i i s + 2i

n

i=1 ib2 i e b1 i t

b1 i e b2 i t

b2 i b1 i Golla and Hughes (1985)

McTavish and Hughes (1993)

n

i=1

is + i

n

i=1 i e i t Lesieutre (1992)

n

i=1

ci i s + 1

+ c0n

i=1(ci / i )e t/ i + (t)c0 Yiu (1994)

gx1 e st 1

sgx , t < t 10, t > t 1 Adhikari (1998)

gx

s

1 e st 3 + 2( st 3 / )21 + 2( st 3 / )2

gx2

1 + costt

3

, t < t 30, t > t 3

Adhikari (1998)

es

2

4 2 erfc s

2 2 2 2 e 2 t 2 Adhikari and Woodhouse (2001a) In this expression a one-dimensional constitutive behavior was considered and the original notation used by Biot (1954)

was retained. Here p = d / dt is a time operator which in the Laplace domain corresponds to considering p = s and theequality G D was considered for the denition the relaxation function. Is is worthy to mention that, strictly speaking,the second term of the relaxation function has no inverse in the time domain and therefore will not inuence the time domaindenition of the relaxation function, which is only applicable for a relaxation test considering a step strain stimulus. However,for completeness, as the delta function is often used in mathematical physics as a generalized function, it can be formallyadded to the time domain relaxation function, which will allow to represent the relaxation modulus, including already a pureelastic term given by G , also by an extra pure viscous term when the correspondent relaxation modulus is substituted intoEquation 2.

The ve parameter constitutive behavior considering the fractional derivative operator D [ ], so that 0 < < 1 and0 < < 1, with the property in the Laplace domain L{D [x(t )]} = s L{ x(t)}, is given by (t) + bD [(t)] = E 0 (t ) +E 1 D [(t )]; the original notation in Bagley and Torvik (1983) was used; however, it is worthy to mention that E 0 in theoriginal notation represents the relaxed (static) modulus denoted in the present work as G . Since the fractional derivative identication of most viscoelastic materials has usually resulted in and that, in addition

to this experimental nding, it has been proved theoretically that for the model to be consistent with thermodynamic principles = [Bagley and Torvik (1986)], yielding a four parameter model, the latter model is considered enough to describe thedynamic behavior of real viscoelastic materials in a wide frequency range; the time domain relaxation function presented istherefore dened under those assumptions, i.e., that = , and is expressed in terms of the one parameter Mittag-Lefflerfunction E (z) =

i =1 z

i / (1 + i ) [see Enelund and Olsson (1999), Welch et al. (1999) and Adolfsson et al. (2005) forfurther details].

The constants b1 i and b2 i are given by b1 i , b2 i = i i i ( i 2 1)1/2 .


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Viscoelastic Damping TechnologiesPart I: Modeling and Finite Element Implementation

solution methods of the governing equations in the timeand frequency domains. As a result of a compilation fromseveral sources, some different relaxation functions weresummarized by Golla and Hughes (1985), Park et al.(1999), Trindade and Benjeddou (2002) and Adhikariand Woodhouse (2003), where the different models and

solution methods of the resultant mathematical modelsare also thoroughly discussed. Some of the most wellknown relaxation functions, frequently referred in the openliterature, are the ones used by the internal variables GHMand ADF models, which will be considered in this workand described in the following sections. The correspondentrelaxation functions along with some other mathematicalrepresentations that have been used and presented in theliterature are summarized in Table 1.

Following the thermodynamics of irreversible phenomenaand applying it to viscoelasticity, the engineer, physicistand applied mathematician Maurice Biot was a pioneerrst giving the subject in the 1950s an original treat-ment 1 based upon relaxation modes and hidden variablesand establishing a linear viscoelastic constitutive theory[Biot (1954)]. It was shown that the proposed model wasvery general in nature and that it could represent thebehavior of a wide variety of phenomena in the processof their response to external actions which may involve,for example, the application of external stresses, chemicalreactions, heat transfer, etc. In fact, Biots work comprisedan unusually broad range of scientic and technologicalareas including applied mechanics, acoustics, heat transfer,thermodynamics, aeronautics, geophysics and electromag-netism, to which his contributions on the thermodynamicsof irreversible phenomena were extremely important. Hisapproach is presented in the rst two lines of Table 1.

As reported by Biot (1954), the summation presented inthe rst line of Table 1 is extended to all internal relaxationconstants, s . In order to fully represent the effects of allthe internal variables of a solid, since it has too manydegrees of freedom, there may be in some cases an almostcontinuous distribution of relaxation modes. That willcorrespond to a spectrum or spectral density distributionof the relaxation constant D() with a density distributionfunction (), as presented in the second line of Table1. That constitutive model was later utilized to modelviscoelastic beams and plates as reported by Buhariwalaand Hansen (1988) and Biot (1972), respectively.

The exponential function introduced by Biot is proba-bly the simplest physically realistic non-viscous dampingmodel [Adhikari and Woodhouse (2001b)] and has beenused extensively in the context of viscoelastic systems.In fact, Biot has shown that, not only in the contextof mechanical models but also involving other coupledphenomena such as chemical, thermodynamic, etc., anyrelaxation phenomena may be represented by a spring, adashpot and a sum of a great many elements made up of aMaxwell type material. As particular cases of Biots modelwe have the double exponential relaxation function, known1 The rst treatment given by Biot has been shown over thefollowing decades to include other theories and relaxation models,as the ones presented in Table 1 which, with the exception of the

fractional calculus approach proposed by Bagley and Torvik (1983)and presented in the third line of Table 1, correspond to particularcases of Biots treatment.

as the GHM model, or the single exponential relaxationfunction of the ADF model. Regarding Yius model, itwas proposed in the context of nite element solutions of structures with viscoelastic treatments and is equal to themodel proposed by Biot, however with a slight change inthe constants of the coefficients.

2.3 Frequency Domain Representation: Complex Modulus

There are practical situations in which structures withviscoelastic materials may be subjected to steady-stateoscillatory forcing conditions. Under these conditions, thecharacteristic (shear) material function previously denedin the stress-strain relationship in Equation (3) is denedby assuming s as a pure imaginary variable (or similarlythrough its Fourier transformation), so that

G(j) = G(s) for s = j, (8)

where is the frequency, yielding the so-called complex (shear ) modulus in the form

G(j) = G () + j G (), (9)

where G () is the (shear) storage modulus , which accountsfor the recoverable energy, and G () is the (shear) loss modulus , which represents the energy dissipation effects[Christensen (1982), Nashif et al. (1985), Jones (2001)].The loss factor of the viscoelastic materials is dened as

() = G ()G ()

, (10)

which alternatively allows writing Equation (9) as

G(j) = G () [1 + j()] . (11)

For a linear, homogeneous and isotropic viscoelastic mate-rial, equivalent representations of the previous equationshold for the complex extensional modulus E (j) and therelationship

G(j) = E (j)

2 [1 + (j)], (12)

where (j) is the Poissons ratio, establishes a relation-ship between the two. However, for simplicity, a real fre-quency independent Poissons ratio (j) = is usuallyassumed, leading to identical loss factors of the shear andextensional complex moduli, i.e., E () = G () = ().Further details on the dynamic properties of the Poissonsratio in linearly viscoelastic solids can be found in Chenand Lakes (1993), Pritz (2000), Tschoegl et al. (2002)and Lakes and Wineman (2006).

Before concluding this section on the constitutive mod-eling of viscoelastic materials, it is worthy to empha-size that in fact elastic materials are particular cases of viscoelastic ones. Therefore, imposing the restrictions of time-independent relaxation behavior, i.e. r(t) = 0 ,or alternatively r(t) = const ., whether in the time ortransformed (Laplace or Fourier) domains, a pure elasticbehavior is obtained where for an elastic isotropic materialwe have G(t) = G and E (t) = E in the time domain.As a consequence, in the frequency domain, the complex


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modulus representation yields nil loss modulus and lossfactor without any time or phase lag between the stressand the corresponding strain. In general, the materialproperties may still be frequency dependent but withoutany (or a weak) damping behavior, which corresponds toa very restricted class of materials which have somewhat

reduced interest and applicability.

3. FINITE ELEMENT IMPLEMENTATION OFVISCOELASTIC DAMPING

In the analysis of viscoelastically damped structures thechoice of the most adequate solution method is stronglydictated by the applied temperature- and time-dependentviscoelastic constitutive model and by the desired type andaccuracy of the system responses. This relation is strik-ingly noticeable and both issues, viscoelastic model andsolution method, are often associated and no distinctionbetween them is usually reported in the open literature.

However, a clear distinction between the two must bemade, keeping still in mind that a close relation betweenthem exists. These methods are reviewed and presented inwhat follows.

3.1 Damping Modeling and Solution Approaches

As previously referred, in general, the elastic and dissipa-tive properties of viscoelastic materials may depend uponthe frequency, operating temperature, amplitude and typeof excitation [Ferry (1980), Nashif et al. (1985), Riandeet al. (2000), Jones (2001)]. These dependencies of theviscoelastic material properties make a mathematical de-scription of the viscoelastic constitutive behavior and ma-terial damping more difficult and complicated to obtain,and might turn the underlying FE implementations andsolution methods more troublesome and difficult endeav-ors. Thus, for simplicity, the amplitude and type of exci-tation effects are often overlooked, isothermal conditionsare usually assumed and only the frequency dependency of the viscoelastic constitutive behavior is usually taken intoaccount.

The direct frequency response (DFR) method, which isbased upon the complex modulus approach (CMA), earlypresented in the 1950s by Myklestad (1952) and subse-quently further discussed and utilized by Snowdon (1968)and Bert (1973), is a frequency domain method thatutilizes a time-domain based model which is limited tosteady state harmonic vibrations. This corresponds toa simple way of modeling viscoelastic damping effectswhere the material properties are continuously updatedfor each discrete frequency value [Moreira and Rodrigues(2004)]. However, the structural model can alternativelybe formulated in the frequency domain, yielding the so-called wave models , which are based on the denitionof a dynamic stiffness matrix. These wave methods canalso be extended to discontinuous structures and damp-ing treatments, for example by the use of the so-calledspectral nite element method (SFEM) proposed by Doyle(1997) which utilizes dynamic interpolation functions.

These wave-based methods, which also allow viscoelasticdamping effects to be accurately considered by deningthe material constitutive behavior for the current waves

specic frequency, have been used, for example, by Wangand Wereley (1998), Baz (2000) and Wang and Wereley(2002). This solution method is usually denoted as a wave propagation method (WPM) and, according to Wang andWereley (2002), Douglas was the rst to explore wavesolutions in order to implicitly account for the frequency

dependent complex modulus of viscoelastic components inthe solution method [Douglas (1977), Douglas and Yang(1978)].

The CMA is also the basis of the so-called modal strain energy (MSE) method, rst derived and utilized by Mead(1960) and later popularized by Johnson et al. (1980),where the loss factor of each individual mode is determinedfrom the ratio between the dissipated modal strain energyof the viscoelastic counterpart and the storage modalstrain energy of the whole structural system. However,MSE based methods are known to lead to poor viscoelasticdamping estimation of highly damped structural systems;iterative versions of the MSE have been successfully used,though, only for moderate damping values [Trindade et al.(2000b)].

In more recent years and in opposition to frequency do-main based, i.e. CMA based, approaches, time domainmodels, such as the Golla-Hughes-McTavish (GHM), afterHughes and his colleagues [Golla and Hughes (1985), Mc-Tavish and Hughes (1993)], anelastic displacement elds (ADF), after Lesieutre and co-workers [Lesieutre andBianchini (1995), Lesieutre et al. (1996)], and otherspresented by Yiu (1993) and more recently by Silva(2003), utilizing additional internal (or dissipation ) vari-ables [Johnson (1999)], have been successfully utilized andwere shown to yield good damping estimates.

Alternatively, the use of fractional calculus (FC) models,initially proposed by Bagley and Torvik (1983, 1985) andrecently revisited, among others, by Enelund and Lesieutre(1999), Jones (2001), Schmidt and Gaul (2002) and Galu-cio et al. (2004, 2005, 2006), provides a simpler and moreeconomic descriptor of the complex constitutive behaviorof viscoelastic materials, being able to represent the stor-age and loss modulus of frequency dependent viscoelasticmaterials using a low order parameter set formed by oneor two model parameter series and based upon the useof fractional derivatives. It has the drawback, though, of generating a non-standard FE time domain formulation,with a more burdensome characteristic solution proceduremaking use of some special solution artices, but whichstill allows to obtain good damping estimates of the sys-tem.

Other relevant contributions for damping, in general, andviscoelastic damping modeling, in particular, were given,among others, by Adhikari and Woodhouse (2003) andthe references therein on non-viscous damping in dis-crete linear systems, Balmes and his co-workers [Balmes(1997), Plouin and Balmes (1998, 1999)], who have useddifferent variations of modal subspace models speciallydened and assumed to be representative of the dampedsystem response, and Kelly and Stevens (1989) and Linand Lim (1996), that have used perturbation method ap-proaches. Since they fall out of the scope of the dampingstrategies utilized in this work, it is not intended to reviewthose works here and the interested readers are referred to


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the aforementioned references for further details on themethods used.

To sum sup, the solution and analysis methods canbe divided into frequency and time domain based solu-tion strategies, which clearly can admit distinct specicconstitutive models. Comparative analysis and surveysof modern methods for modeling frequency dependentdamping in FE models were performed by Slater et al.(1993), Trindade et al. (2000b), Benjeddou (2001) andTrindade and Benjeddou (2002). The next sections addressthe two alternative solution methods and describe differentalternatives as far as the FE implementation is concerned.

3.2 Frequency Domain Based Approaches

Direct Frequency Response (DFR) Analysis Model Con-sidering the frequency dependent constitutive behavior of viscoelastic materials discussed previously and adopting acomplex modulus representation, the general time depen-

dent FE equations of motion of a viscoelastically dampedgeneral structural system can be written as

Mu (t) + D u (t) + [ K E + K V (j)]u (t) = f (t), (13)

where M and D are the global mass and viscous dampingmatrices, K E and K V (j) are the elastic and complexfrequency dependent viscoelastic stiffness matrices, andu (t) and f (t) are the displacement degrees of freedom(DoFs) and applied loads vectors. It is worthy to mentionthat two types of damping models are considered in theprevious equation: (i) an arbitrary frequency dependenthysteretic (or viscoelastic) damping type, represented byIm[K V (j)], whose terms are frequency dependent, repre-

senting the viscoelastic dissipation (relaxation) behavior;and (ii) a viscous damping type, which is described by D ,representing other general sources of damping (e.g. air-based damping, energy dissipation in the supports, etc.),which are assumed to be proportional to the velocity.

Using Equation (13) as it is, the frequency dependentmatrix denition implies that its use and analysis canonly be performed in the frequency domain, based onthe CMA, where the material properties of the stiffnessmatrix of the viscoelastic parts are dened for each discretefrequency value [Vasques et al. (2004), Moreira andRodrigues (2004), Vasques et al. (2006)]. Thus, theDFR is a frequency domain method where the frequency

response model [Ewins (2000)] can be generated in astraightforward manner from the results of many discretefrequency calculations of the equations of motion, in whichthe complex stiffness matrix of the viscoelastic parts is re-calculated at each frequency value of the desired discretefrequency range.

Considering simple harmonic excitation, with

f (t) = F e jt , (14)

where F is the amplitude vector of the applied mechanicalforces, the steady state harmonic response of the systemcan be written as

u (t) = U (j)e jt , (15)

where U (j) is the complex response vector (displace-ments phasor). Substituting Equations (14) and (15) intoEquation (13), yields

[K (j) + j D 2 M ]U (j) = F , (16)

where K (j) = KE

+ KV

(j), from which the complexsolution vector U (j) can be obtained.

For a force applied in the ith DoF and a displacementmeasured in the oth DoF, the FRF (frequency responsefunction) can be obtained by solving Equation (16) fordifferent values of frequency,

[K (jl ) + j l D 2l M ]U (jl ) = F i , (17)where F i denotes a force vector with a non-zero componentin the ith DoF and all other elements equal to zero, andU (jl ) is the resulting complex response vector (displace-ments) solution at frequency l . Thus, the receptance FRFat frequency l is given by

H oi (jl ) =U o(jl )

F i, (18)

where F i is the amplitude of the force input and U o(jl )is the displacement response amplitude extracted fromthe oth DoF of the vector U (jl ). Lastly, the frequencyresponse model (FRFs) can be generated from the resultsof many discrete frequency calculations of Equations (17),in which the complex stiffness matrix of the viscoelasticlayers is re-calculated at each frequency value comprisedin the discrete frequency range = 0 , . . . , l , . . . , f , asdepicted in Figure 1.

E V(j ) (j ) K K K

2[ (j ) j ] (j ) i

K D M U F

l

(j )oi H

0, , , ,l f

Fig. 1. Generation diagram of the direct frequency respon-se (DFR) analysis model.

Modal Strain Energy (MSE) Approach Assuming thatthe normal modes obtained from the undamped system are

representative of the damped system, which in principleis true only for lightly damped structures, and that afrequency independent stiffness matrix might be assumed,


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an approximation to the modal loss factors can thereforebe obtained from the ratio between the loss and thestorage parts of the modal strain energy. Based on thisassumption, Johnson and Kienholz (1982) proposed adirect method for the FE prediction of the modal dampingratio in structures with viscoelastic damping layers. While

the storage energy is calculated from the modal strainenergy of the entire structure for each individual mode,the dissipated energy is calculated by multiplying themodal strain energy corresponding to the viscoelasticmaterial FEs by the respective material loss factor atsome arbitrarily chosen value of frequency, which ideallywould be the value of the natural frequency (not yetknown though) of the current mode. Thus, establishinga relationship between the loss and storage modal strainenergies of a generic mode r, the modal loss factor of ther th mode might be expressed as

r = r

T Im[K (j)] r r T Re[K (j)] r

, (19)

where r is the r th mode shape vector. In physical terms,the previous relationship represents the ratio between thedissipated energy, proportional to the imaginary compo-nent of the complex stiffness matrix, and the stored energy,proportional to the real counterpart.

Conceptually, the MSE approach was initially introducedby Mead (1960) and Ungar and Kerwin (1962) in the early1960s and later popularized and applied to FE analysisby Johnson and Kienholz (1982) and Soni and Bogner(1982). This approximated method is simple to implementin a FE simulation and is computationally cost-effective,providing a good damping estimator in comparative design

analysis or optimization procedures [Hwang and Gibson(1992), Moreira and Rodrigues (2006)]. However, itsusage is restricted to lightly damped structures whereinthe undamped real modal model can effectively representthe damped structure. For highly damped structures,where the modal coupling is noteworthy, and especiallyfor those where the added viscoelastic material induces aconsiderable mass and stiffness modication, the originallyproposed MSE approach usage may lead to considerablerepresentativeness errors. In view of this, modied versionshave been recently proposed in an attempt to nd moreaccurate approaches. Among others, the one proposed byHu et al. (1995) is one of these modied MSE versionsand is based on the application of a modal vectors derivedfrom a modied eigenvalue problem which considers alsothe imaginary counterpart of the stiffness matrix.

Iterative Modal Strain Energy (IMSE) Approach As pre-viously referred, the MSE approach is based on the princi-ple that the undamped natural modes of vibration of theviscoelastically damped structure are representative of thedamped modal model and that a frequency independentstiffness matrix might be assumed. However, and despitethe fact that this approximation might be valid for low tomoderate additions of damping materials, the applicationof single or multi-layer viscoelastic sandwiched dampingtreatments to structures might modify the stiffness sub-stantially due to the decoupling effects derived from theviscoelastic core/layers. In view of this, the mechanicalproperties may alternatively be assumed constant in the

neighborhood of each mode and the eigensolution of eachindividual mode can be determined independently, whichcorresponds to a more realistic approach. This method-ology can, however, be also applied under an iterativeapproach where the stiffness matrix is updated iterativelywith the new corrected complex stiffness of the calculated

mode. In general, this procedure allows to obtain morerealistic values of the modal loss factors and natural fre-quencies with the drawback of requiring more processingtime.

Following this line of thought, a modication to the orig-inal MSE algorithm is hereby proposed in order to intro-duce the stiffness changes of the structural damped systemthrough an iterative procedure which more appropriatelyconsiders the effects of the variation of the storage andloss moduli with frequency. As a consequence, an itera-tive calculation of the real (undamped) eigensolution isperformed using the continuously iteratively updated realpart of the stiffness matrix, which is updated accordingto the viscoelastic material properties at the value of frequency of the current iteration within the vicinity of the natural mode being considered. Once the convergenceof the undamped natural frequency of the natural modeunder analysis is veried, the correspondent modal lossfactor is determined by means of the same methodologyused in the original MSE method but utilizing the cor-respondent imaginary part of the stiffness matrix denedat the converged value of frequency. When necessary, thismethod might be repeated according to the modal densityof the bandwidth of interest and employed to determinethe individual modal loss factors and natural frequenciesof the several modes within the considered bandwidth.Some examples of the use of the IMSE in the context

of FE models can be found, for example, in [Trindadeet al. (2000b), Zhang and Chen (2006)]. A schematic of the algorithm describing the IMSE approach is presentedin Figure 2 where it is assumed that the eigensolutionalgorithm calculates the smallest eigenvalues.

Iterative Complex Eigensolution (ICE) Approach Re-sembling the IMSE method, an alternative iterative com-plex eigensolution (ICE) algorithm is proposed in thissection. This algorithm is exact in the sense that it doesnot estimate the modal loss factor in the same way asthe MSE but, instead, it uses the determined complexeigenvalue to calculate the exact modal loss factor value.The resultant ICE method can therefore be seen as a moreaccurate and generally applicable method. However, it hasthe drawback of requiring an higher computational effortsince, for each mode, repeated complex eigensolutions aresought before the convergence is achieved.

As the previous method it starts by determining thereal eigensolution of the frequency dependent undampedFE model at a dened initial frequency. Based on thatpreliminary analysis to give a rst guess of the valueof the damped natural frequencies, the proposed methodsuccessively updates the complex stiffness matrix and thecorresponding complex eigenvalues are obtained from acomplex eigensolution until the convergence satisfying thenecessary accuracy condition is achieved. A schematic of the algorithm describing the ICE approach is presented inFigure 3 where, in a similar way, it is assumed that the


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IMSE algorithm

Step 1. Eigensolution with = 0

Re[K (j0

)] r = ( 0

r )2

M rStep 2. Loop for each eigenpair 0r , r with r = 1 , . . . , pi. Initial value

ir = 0r

ii. Iterative loop for each natural frequency and mode shape

Eigensolution 0s , s with s = 1, . . . , rRe[K (jir )] s = (

i+1s )

2 M s

Iterated natural frequencyi +1s : rejected , for s < r

i +1r = (i +1s )2 , for s = r

Convergence condition test

=i+1r ir

i+1r maxiii. Modal loss factor

r = Tr Im[K (ji+1r )] r Tr Re[K (j

i+1r )] r

Fig. 2. Iterative modal strain energy (IMSE) algorithm.

ICE algorithm

Step 1. Eigensolution with = 0

Re[K (j0 )] r = ( 0r )

2 M rStep 2. Loop for each eigenpair ( r , r ) with r = 1, . . . , p

i. Initial valueir =

0r

ii. Iterative loop for each complex eigenvalue and eigenvector

Complex eigensolution ( s , s ) with s = 1 , . . . , rK (jir ) s =

i+1s M s

Iterated damped natural frequency i +1s : rejected , for s < ri+1r = Re( i +1s ), for s = r Convergence condition test

=i +1r ir

i+1r maxiii. Modal loss factor

r = Im( i+1r )Re( i +1r )

Fig. 3. Iterative complex eigensolution (ICE) algorithm.


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eigensolution algorithm calculates the smallest magnitudeeigenvalues.

When compared with conventional methods, the ICEmethod improves considerably the accuracy while it some-what maintains the computational efficiency and simplic-ity required for practical applications. Referring to thetwo iterative approaches, namely the IMSE and ICE, inboth methods as the iteration continues, the estimatedi+1r , i+1r and

i+1r will converge to more accurate solu-

tions, r , r and r . However, the computational effortinvolved may increase with the modal density and/orthe bandwidth of interest, since, for each mode, severalreal or complex eigensolutions are required before theconvergence is achieved. In opposition to that we havethe too simplistic MSE approach with the advantage thatthese shortcomings are not an issue. However, the MSEis an approximate method which considers some unre-alistic simplications which make its accuracy stronglydependent upon the level of frequency dependent damping

and stiffness and also of the degree of isolation of themode to be analyzed. When compared with the DFR,where the modal parameters are indirectly approximatelydetermined by modal identication procedures [Maia andSilva (1997), Ewins (2000)], the IMSE and ICE allowthe modal parameters r , r , and r of the rth modeof a generic viscoelastically damped structural system tobe obtained directly. However, with the aforementionedadvantages and disadvantages and modeling strategies, inprinciple all the models can be used to build a modalmodel of the damped structural system in a more or lessstraightforward manner, which can be used to estimateboth frequency and time domain responses.

In conclusion, the proposed ICE method makes use of classical complex eigensolution algorithms and representsan extension and improvement to IMSE method, whereonly real eigensolution algorithms are utilized. To theauthors best knowledge, this method has been seldomused in the context of viscoelastically damped structuresand the discussion in the open literature regarding itsapplication and performance is very scarce [Lin and Lim(1996)].

Methods based in the denition of time domain viscoelas-tically damped spatial models which afterwards allow ob-taining representative truncated modal models will bediscussed in the following sections.

3.3 Time Domain Based Approaches

Golla-Hughes-McTavish (GHM) Damping Model Whengeneral transient responses are required, time domainmodels are more suitable and versatile and they might rep-resent better alternatives than the CMA-based frequencydomain methods, since they allow the reduction of thecomputational burden due to the re-calculation of thestiffness matrix for each discrete frequency value (DFR)and the use of iterative eigenproblem calculations (IMSEand ICE). One alternative is the GHM ( Golla-Hughes-McTavish ) model [Golla and Hughes (1985), McTavishand Hughes (1993)], which assumes that the characteristic

material function G(s) s Grel (s) in Equation (3) may berepresented in terms of a series of mini-oscillator terms(see Table 1),

s Grel (s) = G [1 + sr (s)]

= G 1 +n

i=1 i

s2 + 2 i i ss2 + 2 i i s +

2i

, (20)

where G is the relaxed modulus and each ith mini-oscillator term is a second order rational function involvingthree positive constant parameters, i , i and i . Theseparameters govern the shape of the characteristic materialfunction (usually for isotropic materials the shear modu-lus) over the complex plane (or over the frequency domain,assuming only the imaginary part of the complex plane),and depending upon the nature of the viscoelastic materialand the range of s (or frequency) over which it is to bemodeled, the number of mini-oscillator terms is denedaccording to the required accuracy of the representation.

Considering the FE equations of motion of a viscoelas-tically damped general structural system expressed inEquation (13) with the shear modulus factored out of the viscoelastic stiffness matrix and assuming that thestructural system possesses only one type of viscoelasticmaterial, following the hereditary stress-strain law givenin (2), yields

Mu (t) + D u (t) + K E u (t) + Grel (t)K V u (0)

+ t

0Grel (t )K V

u ( )

d = f (t), (21)

where K V is the remaining viscoelastic stiffness term afterfactoring out the shear modulus; it is worthy to mention atthis point that both shear and extensional stiffness termsmay be considered in the viscoelastic stiffness matrix byassuming a frequency independent Poissons coefficient inthe extensional and shear modulus relationship denedin Equation (12). Considering nil initial conditions andtransforming Equation (21) to the Laplace domain, wehave

s2 M + sD + K E u (s) + s Grel (s)K V u (s) = f (s). (22)

Substituting the characteristic material function represen-tation in Equation (20) into (22) yields

s2 M + sD + K E u (s) + G K V

1 +n

i=1 i

s2 + 2 i i s

s2

+ 2 i i s +

2

i

u (s) = f (s). (23)

Then, introducing a set of n series of dissipation (orinternal ) variables u Di (s), with i = 1 , . . . , n , for each series,we can establish the relationship

u (s) u Di (s) = s2 + 2 i i s

s2 + 2 i i s + 2i

u (s). (24)

Substituting Equation (24) into (23) yields

s2 M + sD + K E + K V u (s)

+ K V n

i=1 i u (s) u Di (s) = f (s), (25)

where K V = G K V is the relaxed (or static ) stiffnessmatrix of the viscoelastic components. Next, in order to


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describe the dissipative behavior of the internal DoFs,after some algebra, Equation (24) can be rewritten as

u Di (s) = 2i

s2 + 2 i i s + 2i

u (s). (26)

Considering Equations (25) and (26), an augmented cou-pled system can be written as

s2 M + sD + K E + K V0 u (s)

K V n

i=1 i u Di (s) = f (s), (27a)

s2 12i

+ s2 ii

+ 1 u Di (s) u (s) = 0 , (27b)

whereK V0 = 1 +

n

i =1 i K V . (28)

In order to obtain the time-dependent behavior of the aug-mented system, multiplying Equation (27b) by i K V ,and since all matrices are independent of s, a linear timedomain model is readily recovered by the inverse Laplacetransform of Equations (27), yielding

Mu (t) + D u (t) + K E + K V0 u (t)

K V n

i=1 i u Di (t) = f (t), (29a)

i2i

K V u Di (t) + 2 i i

iK V u Di (t)

+ i K V u Di (t)

i K V u (t) = 0 . (29b)

The augmented coupled system in Equations (29) may stillbe expressed in compact matrix form as

Mz (t) + D z(t) + Kz (t) = f (t), (30)

where

M = M 00 M DD , D = D 00 D DD ,

K = K EE K EDK DE K DD ,

(31a-c)z(t) = col u (t), u D1 (t), . . . , u

Dn (t) ,

f (t) = col [ f (t), 0 , . . . , 0 ] , (32a,b)and

M DD = diag 121

K V , . . . , n2n

K V ,

D DD = diag2 1 1

1K V , . . . ,

2 n nn

K V ,

K DD = diag 1 K V , . . . , n K V ,

K ED = 1 K V , . . . , n K V ,K EE = K E + K V0 ,

K DE = K TED . (33a-f)

Equation (30) represents a time domain mathematicalrealization of a viscoelastically damped FE structuralsystem comprising viscoelastic and elastic materials. Itis important to emphasize that the derived model isexpressed in a second-order form. However, rst-orderstate space formulations and model reduction techniques,

adapted to the problem and matrix topology in hands,are often more convenient to solve the system equations.Further details on these issues will be given in the nextsections.

Anelastic Displacement Fields (ADF) Damping Model Atime domain model based on a variation, or Laplace trans-formed, formulation of the ADF ( anelastic displacement elds ) model, originally proposed by Lesieutre and his co-workers [Lesieutre and Bianchini (1995), Lesieutre et al.(1996)], is presented in this work. It takes a denitionof the complex modulus in the frequency (or Laplace)domain (see Table 1) and utilizes the so-called internal ,dissipation or anelastic (after Lesieutre) variables to sim-

plify and linearize the damped equations, although withthe drawback of increasing the size of the problem. Af-terward, through an inverse Laplace transform we obtainan amenable and computationally tractable augmentedsystem of linear ordinary differential equations that canbe solved by standard numerical methods applicable torst-order linear systems. With this procedure, the FEmodel implementation of the ADF model is more straight-forward when compared with the Lesieutres original di-rect time domain formulation, based upon the methodof irreversible thermodynamics and a decomposition of the total displacement eld in an elastic and anelastic counterpart. Thus, the process of deriving the augmentedcoupled elastic-anelastic system (using the original desig-nation of Lesieutre) is similar to what has been presentedfor the GHM method. In this case, a different denitionof the characteristic material function G(s) is utilized. Itis also worthy to mention that Lesieutres denition of the relaxation function is in fact a particular case of therelaxation function derived by Biot and presented in Table1, where the viscous term proportional to velocity is notconsidered in the relaxation function denition. That is infact expected since both are derived from thermodynamicprinciples. However, velocity proportional damping is stillconsidered here by means of a viscous damping model,through the introduction of the viscous damping matrixD , which is used here to model other general sources

of damping and not the viscoelastic material damping.Furthermore, when compared with the GHM model, whichcan also be seen as a particular case of Biots modelassuming a time-domain relaxation function as a seriesof double exponential terms, the ADF approach has beenshown to yield the same damping capabilities with lessparameters per each series.

As reported by Lesieutre, the characteristic material func-tion, which in this case corresponds to the frequency de-pendent viscoelastic shear modulus, described by the ADFmodel is represented by a series of functions in the Laplacedomain and is given by [Lesieutre (1992), Lesieutre andBianchini (1995); cf. Table 1]

s Grel (s) = G [1 + sr (s)] = G 1 +n

i=1

i ss + i

, (34)


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where, again, G is the relaxed (or static, low-frequency)shear modulus, and i is the inverse of the characteristicrelaxation time at constant strain and i the correspon-dent relaxation resistance . To take into consideration therelaxation behavior, the entire ADF model itself may becomprised of several individual elds, where n series of

ADF are used to describe the material behavior. Givena set of measured values of the shear modulus in theform of a frequency dependent complex modulus G(j),the relaxed shear modulus G and the series of materialparameters i and i can be determined through curve t-ting techniques. The number of series of ADF parametersdetermines the accuracy of the matching of the measuredmaterial data over the frequency range of interest.

Substituting Equation (34) into (22) yields

s2 M + sD + K E u (s)

+ G K V 1 +n

i=1

i ss + i

u (s) = f (s). (35)

Then, in a similar way to the GHM, introducing a setof n series of anelastic (or internal, dissipation) variablesu Ai (s), with i = 1 , . . . , n , for each series, one can assumethe relationship

u (s) u Ai (s) = s

s + iu (s). (36)

Substituting Equation (36) into (35) and considering thedissipative behavior of the anelastic DoFs given fromEquation (36) as

uAi (s) =

is + i u (s), (37)

after some algebra, we get the following augmented elastic -anelastic coupled system,

s2 M + sD + K E + K V0 u (s)

K V n

i=1 i u Ai (s) = f (s), (38a)

si

+ 1 u Ai (s) u (s) = 0 , (38b)where, in this case,

K V0 = 1 +n

i=1 i K V . (39)

Multiplying Equation (38b) by i K V , the time-depen-dent behavior of the augmented system is recovered by theinverse Laplace transform of Equations (38), yielding

Mu (t) + D u (t) + K E + K V0 u (t)

K V n

i=1 i u Ai (t) = f (t), (40a)

ii

K V u Ai (t) + i KV u Ai (t) i K V u (t) = 0 .

(40b)

The augmented coupled system in Equations (40) can alsobe expressed in compact matrix form, as in Equation (30),where, for the ADF model, we get the following differentdenitions,

M = M 00 0 , D = D 00 D

AA

, K = K EE K EAKAE

KAA

,

(41a-c)z(t) = col u (t), u A1 (t), . . . , u

An (t) , (42)

where

D AA = diag 11

K V , . . . , nn

K V ,

K AA = diag 1 K V , . . . , n K V ,

K EE = K E + K V0 ,

K EA =

1 K V , . . . ,

n K V ,

K AE = K TEA . (43a-e)

Regarding the augmented damped system in Equation(30), we may notice from matrix M that, when comparedwith the GHM model, the anelastic DoFs of the ADFmodel have no inertia and therefore the augmented massmatrix M is singular and hence is not positive-denite.However, the singularity of the augmented mass matrixcan be overcome if, instead of solving the second-ordersystem (30) of the ADF, a state-space rst-order repre-sentation with an adequate design of the state variables isconsidered. While demanding rst-order solution methods,an advantage that immediately results of the rst-orderrepresentation is that the size of the ADF augmentedmodel is smaller than the one obtained with a state-spacerst-order representation of the augmented GHM model,as will be presented in the following section. With the rst-order representation, the number of exible modes is keptthe same and the dissipative modes, which correspond tothe internal relaxations of the viscoelastic material, areoverdamped with low observability.

3.4 Model Reduction and State-Space Representation

Associated with the FE discretization and in order toaccount for the frequency dependence of the viscoelastic

material, as evidenced by Equations (29) and (40), themain disadvantage of the GHM and ADF models is thatthey require the use of additional DoFs which lead to thedenition of augmented systems with a higher number of DoFs. In view of this, the initial system size is increased bya number of additional DoFs equal to the initial number of DoFs times the number of series utilized by the GHM andADF models to t the viscoelastic material constitutivebehavior. However, model reduction techniques might beutilized in order to reduce the size of the augmentedproblem and in part to circumvent this shortcoming.

As suggested by Trindade et al. (2000b), which followsa similar treatment to the one given by Biot (1954) to

the unobservable variables of a generic irreversible physicalsystem, the matrices corresponding to the internal DoFsmight be reduced and diagonalized through a projection


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in a suitable reduced modal basis, to reduce the com-putational cost. Thus, dening I = (D , A) and using ithere to denote internal variables in general, which may beboth the dissipative or anelastic variables denitions of theGHM and ADF approaches, respectively, and consideringthe linear coordinate transformation

u Ii (t) = I u Ii (t), (44)

where I = TI K V I is a diagonal matrix composedby the non-zero eigenvalues of K V and I is the cor-respondent matrix of normalized eigenvectors, such that TI I = I , the vector z(t) in Equation (30) is alternativelymodied to

z(t) = col u (t), u I1 (t), . . . , uIn (t) (45)

and the matrices M DD , D II , K II and K IE are modied to

M DD = diag 1

21

D , . . . , n

2

n

D ,

D DD = diag2 1 1

1 D , . . . ,

2 n nn

D ,

K DD = diag [ 1 D , . . . , n D ] ,

K ED = 1 K V D , . . . , n K V D , (46a-d)for the GHM model, and for the ADF model to

D AA = diag 11

A , . . . , nn

A

K AA = diag 1 A , . . . ,n

A ,

K EA = 1 K V A , . . . , n K V A . (47a-c)The advantages of this alternative (transformed) repre-sentation are that in the case where only some part of thestructure has surface mounted or embedded viscoelasticmaterials, only some FEs have viscoelastic componentsand K V can have several rows and columns of zeros,which in turn leads to some nil eigenvalues. Thus, the sizeof u Ii (t) can be substantially smaller than that of u Ii (t).Therefore, through an adequate coordinate transformationbased on the eigenvalues and eigenvectors of K V andelimination of the nil (spurious) eigenvalues, the size of

the problem can be substantially reduced.As previously mentioned, in comparison with the GHMmodel, the ADF model due to the singularity of theaugmented mass matrix M dened in Equation (41a),demands a rst-order solution method of the ADF sys-tem and therefore a state-space representation of theaugmented system which, following the forementionedprocedure, might have also the matrices associated withthe internal DoFs reduced and diagonalized. Therefore,state-space model realizations are mandatory for the ADFmodel and optional to the GHM model, which can also besolved using classic second-order solution methods.

The state-space vector denition depends on the viscoelas-tic damping model being used. For the GHM model, a

suitable and adequate design of the state variables wouldyield a denition of the state vector x (t) given by

x D (t)=z (t)z (t) , (48)

where the chosen state variables are the reduced aug-mented vector z(t) given in Equation (45), which is com-posed by the mechanical elastic DoFs vector u (t) and a setof reduced dissipative (internal) variables vectors u Di (t),and its time derivative. However, for the ADF model, inorder to overcome the singularity of the augmented massmatrix, a suitable and convenient denition of x (t) wouldbe given by

x A (t)=z(t)u (t) , (49)

where the chosen state variables are the reduced aug-mented vector z(t), composed by the mechanical elastic DoFs vector u (t) and a set of reduced anelastic (internal)

variables vectors uAi (t), and the time derivative of theelastic DoFs vector, u (t). It is worthy to emphasize at this

point that the time derivatives of u Ai (t) are not consideredhere since these variables are massless, i.e. the matrix Mis singular.

Thus, taking the forementioned state-space vector deni-tions into account and considering also the reduced anddiagonalized matrix denitions in Equations (46)-(47), theaugmented coupled system given in Equations (29) and(40) for the GHM and ADF models, respectively, expressedby the generic second-order system equation in compactmatrix form given in Equation (30), can be written in arst-order state-space form in terms of the generic state

variables vector x I (t), yieldingx I (t) = Ax I (t) + Bf (t), (50)

where the state-space system and input matrices, A andB , for the GHM model are given by

A =

0 0 I 00 0 0 I

M 1 K EE M 1 K ED M 1 D 0M 1DD K DE M 1DD K DD 0 M 1DD D DD

,

B =

00

M 10

,

(51a,b)and for the ADF model are modied to

A =0 0 I

D 1AA K AE D 1AA K AA 0M 1 K EE M 1 K EA M 1 D

,

B =00

M 1. (52a,b)

The analysis of the previous matrix denitions, consistentwith the state-space vector denitions in Equations (48)and (49), demonstrates that the rst-order form of the


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GHM model has an higher number of DoFs than the ADFmodel. In general terms, the number of total DoFs of theGHM state-space model is given by 2 nE + 2 nn D and of the ADF model by 2 nE + nn A , where nE is the number of elastic DoFs, n is the number of series of model parametersused to t the viscoelastic constitutive behavior, and nD

and nA

are the number of dissipative and anelastic DoFsper series. In view of this, the difference between the GHMand ADF system sizes is equal to nn I with nI being thenumber of internal (dissipative, anelastic) DoFs per series.

It is worth to emphasize that the alternative transformed(reduced) denitions of the matrices presented in Equa-tions (46)-(47) should be used in order to reduce thesize of the system which, as previously discussed, canincrease signicantly, sometimes dramatically affecting therequired computational time. This procedure has two mainadvantages: (i) diagonalization of the matrices, reducingthe number of non-zero elements, and (ii) matrices reduc-tion, only if some rows and columns of the matrices arezeros. The latter advantage means that some of the elasticDoFs may not be coupled with the internal DoFs, as is thecase for example when we have a structure only partiallycovered by viscoelastic damping treatments. Therefore,for a fully covered structure, i.e. with all the elastic andinternal DoFs coupled, with this basis transformation wewould only benet from advantage (i) since the matricesdo not have any null eigenvalue and therefore the size of the matrices would be the kept the same. In those cases,the general GHM and ADF denitions of the total numberof DoFs can be particularized since the number of elasticDoFs is equal to the number of internal DoFs per series,i.e. n E = n I . In view of this, the total number of DoFs of the augmented fully coupled system is given by (2+2 n)nE

for the GHM and (2+ n)nE

for the ADF state-space model.Even with a modal reduction of the DoFs of the viscoelas-tic elements, the order of the system quickly increases asthe number of series of GHM or ADF parameters used inthe summation is increased. Larger order models makessimulation and design more difficult. It is therefore ad-vantageous to look at more model reduction techniques toreduce the systems size even further. Model reduction instructural dynamics, where the original state space systemis approximated by an equivalent system with a lowerdimension, can be achieved by a complex modal projectionof the original system and a subsequent truncation of thenumber of modes considered. A study addressing reduced-

order FE models of viscoelastically damped beams, wherethe selective signicance of the non-physical modes tothe net mechanical response is performed through internalvariables projection in order to signicantly reduce thecomputational cost, was recently presented by Trindade(2006), and the reader is also referred to Biot (1954), Yiu(1993, 1994), Friswell and Inman (1999), Park et al.(1999), Trindade et al. (2000b), Vasques and Dias Ro-drigues (2008) and Vasques (2008) for further details aboutstate space design and model reduction techniques.

4. CONCLUSION

This article presents FE-based mathematical strategies tomodel the damped constitutive behavior of viscoelasticmaterials used as surface mounted, constrained or embed-

ded damping treatments in structures, in order to reducevibrations and/or noise radiation.

Both time and frequency domain based techniques wereconsidered to model the constitutive behavior and theimplementation of these approaches into FE solution pro-cedures was presented and discussed. Time domain tech-niques regard the use of the GHM and ADF internalvariables models. When a state-space rst-order represen-tation with an adequate design of the state variables is con-sidered, the ADF model is known to lead to an augmentedmodel of the damped structural system with a smaller sizethan the GHM model. In the authors opinion the ADFmodel represents the best internal variables alternativeto accurately model the damping behavior since it yieldsgood trade-off between accuracy and complexity. One ma- jor disadvantage in using internal variables models, such asthe GHM or ADF, is the creation of additional dissipationvariables increasing the size of the coupled damped FEmodel. Frequency domain techniques comprise the use of CMA-based methods, namely the DFR, IMSE and ICE(which is recalled in this work), where the FE spatialmodel is used by re-calculating the complex viscoelasticstiffness matrix for each discrete frequency value, in thecase of the DFR, or during the iterative eigensolution pro-cess, in the case of the IMSE and ICE. These approachesare more straightforward to use and implement at theglobal FE level. That is the reason why the CMA-basedmethod, DFR, is the most common approach implementedin commercial FE codes incorporating viscoelastic damp-ing modeling capabilities.

I d e n t i f i c a t i o n

S y n t h e t i z a t i o n

E i g e n s o l u t i o n

REQUENCY ESPONSE IME ESPONSE

Fig. 4. FE-based viscoelastic time and frequency domainsolution alternatives.

The ultimate aim of all these viscoelastic damping modelsis to be able to simulate the time and frequency responseof viscoelastically damped structural systems. While thefrequency response is straightforward to obtain, whetherdirectly obtained (as is the case for the DFR approach) orobtained through modal models derived from the spatialmodel either by iterative frequency dependent eigenso-lutions (IMSE and ICE) or from the augmented spatialmodel eigensolution (GHM and ADF), the time domainresponse can be obtained from the spatial model eitherby direct integration methods or by the modal modelsusing the superposition principle (see Figure 4). With the


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advantages and disadvantages mentioned thus far, all theapproaches can be used to build a truncated modal modelof the damped structural system, whether from the spatialmodel or from a FRF model generated with the DFRmethod, which can be used to estimate both frequencyand time domain responses. However, it is important to

emphasize that when considering structures with a highmodal density, possessing modes not well separated andstrongly damped, the modal identication methods maynot be accurate and efficient, rendering the identicationprocedure more troublesome, if not impossible.

The experimental identication of viscoelastic materials,with the 3M ISD112 being considered in this work, andthe aforementioned FE integrated damped modeling ap-proaches are assessed and validated in the companionarticle [Vasques et al. (2010)].

REFERENCES

Adhikari, S. (1998). Energy Dissipation in Vibrating Struc-tures , PhD (First Year Report), Engineering Depart-ment, University of Cambridge, Cambridge, UK.

Adhikari, S. and Woodhouse, J. (2001a). Identicationof dam

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