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Viscoelastic Damping TechnologiesPart II:

Experimental Identification Procedure and

Validation

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

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

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

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

e-mail: [email protected]

Submitted: 05/03/2010

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

cHyperSciences.Publisher

Abstract: This is the second 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 finite element (FE) implementation,and the experimental identification of the constitutive behavior of viscoelastic materials. In thefirst article [Vasques, C.M.A. et al., Viscoelastic damping technologiesPart I: Modeling andfinite element implementation, Journal of Advanced Research in Mechanical Engineering 1(2):76-95 (2010)] viscoelastic damping has been tackled from a mathematical point of view and the

implementation, at the global FE model level, of time and frequency domain methods, namelythe internal variables models,Golla-Hughes-McTavish(GHM) andanelastic displacement fields(ADF), and the complex modulus approach based ones, direct frequency response (DFR),iterative modal strain energy(IMSE) and aniterative complex eigensolution(ICE), respectively,were described and formulated. This second article is a natural extension of the first one. Itpresents a generic methodology to identify the complex shear modulus of viscoelastic materials.In this case, the complex shear modulus of the well-known viscoelastic material 3M ISD112 isidentified and up-to-date values for this material are used and curve-fitted in order to obtainthe modeling parameters of the GHM and ADF models. Afterward, a viscoelastic sandwich(three-layered) plate specimen and the correspondent FE model are considered numerically andexperimentally. Measured and predicted frequency response functions (FRFs) are compared withthe purpose of assessing the performance of the damping models presented in the companionarticle. The analysis allows to assess the validity of the methodology to determine the frequencydependent complex modulus, the GHM and ADF parameters identification procedure andthe outcomes and drawbacks of the DFR, IMSE, ICE, GHM and ADF viscoelastic dampingmodeling strategies and their FE implementations, with the aim of assisting structural designersin the selection of the most appropriate viscoelastic damping modeling approach for their specificneeds.

Keywords: Damping, viscoelastic, experimental, complex modulus, internal variables, ISD112.

Journal of Advanced Research in Mechanical Engineering (Vol.1-2010/Iss.2)Vasques et al. / Viscoelastic Damping TechnologiesPart II : Experimental / pp. 96-110

96Copyright 2010 HyperSciences_Publisher. All rights reserved www.hypersciences.org

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1. INTRODUCTION

This is the second of two companion articles addressingan integrated study on the mathematical modeling andassessment of the efficiency of viscoelastic damping treat-ments; it incorporates the adequate use and development

of viscoelastic (arbitrary frequency dependent) dampingmodels, along with their finite element (FE) implementa-tion, and the experimental identification of the materialproperties of viscoelastic materials, which allows the useof more realistic material properties data. In the firstarticle Vasques et al. (2010) alternative approaches tothe generation of representative viscoelastically dampedstructural FE models, e.g. beam, plate and shell structureswith surface mounted or embedded viscoelastic dampinglayers, are discussed. Both time and frequency domain-based viscoelastic damping models are tackled from amathematical point of view and their implementation atthe global FE level is described. This second companionarticle is a natural extension of the first one, addressing the

experimental identification of viscoelastic material prop-erties, the identification of the Golla-Hughes-McTavish(GHM) and anelastic displacement fields (ADF) modelparameters, and the validation and assessment of thedifferent viscoelastic damping models/approaches underconsideration.

The extensive use of passive or hybrid damping treatmentsusing viscoelastic materials to reduce vibration and soundradiation from structures [Nashif et al. (1985), Mace(1994), Sun and Lu (1995), Benjeddou (2001), Vasqueset al. (2006), Vasques and Rodrigues (2008), Vasquesand Dias Rodrigues (2008)] has motivated the integra-tion of the viscoelastic damping models discussed in the

companion article (Part I) [Vasques et al. (2010)] intocommercial or in-house FE codes. As far as commercialFE codes are concerned, they often offer only solutionmethods based upon the use of direct frequency responseand direct integration methods in the frequency and timedomains, respectively, while the remainder models (inter-nal variables models and the iterative complex modulusbased approaches) proposed in the companion article areusually developed and implemented by the users and struc-tural designers into in-house codes or adaptations of thecommercial ones.

The direct frequency response (DFR) approach is a verycommon strategy used by the major part of the structural

designers for studying complex structures with viscoelasticdamping treatments. This approach is based on the directapplication of the complex modulus and allows directlyobtaining frequency response functions (FRFs) with theshortcoming of, in principle, demanding a high compu-tational cost. On the contrary, in general, time domainmodels represent better alternatives to frequency, or com-plex modulus approach (CMA), based models, allowingthe reduction of the computational burden due to the re-calculation of the stiffness matrix during the frequencysweep and the study of the transient response in a morestraightforward (direct) manner, even for highly damped

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.

structural systems. This latter approach may lead to a nu-merical analysis with lower computational cost. Therefore,in principle, time domain models would represent betteralternatives than frequency domain ones, but this state-ment cannot be generalized since it strongly depends onparameters of the analysis such as the problem dimension,

frequency range of the analysis, constitutive model detaillevel and damping properties of the materials.

Among the time domain models, internal variables modelsare more interesting from the computational point of viewand easiness of implementation into FE codes. Thus, theGHM and ADF models are two attractive alternativemethods, used to model the damping behavior of vis-coelastic materials in FE analysis, which yield a standardFE formulation (however with the addition of some non-physical dissipative variables). In order to use them, oneneeds the GHM and ADF characteristic parameters whichallow characterizing the complex (frequency dependent)constitutive behavior of the viscoelastic material beingused. To this end, experimental procedures to measure theisotropic constitutive behavior (usually the shear modu-lus) may become necessary to be devised/performed andnumerical identification procedures of the measured dataneed to be developed.

In order to choose the most appropriate material for aspecific application, the designer needs some informationregarding the damping capabilities of these materials. Thelatter is usually obtained through the analysis of thecomplex (frequency dependent) constitutive behavior ofviscoelastic materials at different temperatures. To thatend, normalized and proprietary experimental proceduresto the measurement and analysis of the dynamic constitu-tive behavior (usually isotropic) and numerical proceduresfor the identification and graphical representation of thecomplex moduli of viscoelastic materials have been pro-posed.

There are numerous methods for evaluating the per-formance of damping materials. These methods can beroughly divided into two categories: those whose purposeis to rank the performance of damping materials on adefined structure [e.g. the SAE J1637 (2007) test of theSociety of Automotive Engineers, which is based on theso-calledOberst Bar Test Methodand is used to rank ordermaterials used in passenger vehicle applications] and thosewhose purpose is to measure the properties of the dampingmaterial alone so that mathematical models can be usedto predict its damping performance when applied to manydifferent types of structures.

As far as measuring the complex moduli of viscoelasticdamping materials is concerned, there are various testingmethodologies that have been devised through time andthey can be divided in two distinct and somewhat comple-mentary categories: (i) direct methods, which are basedon dynamic measurements of bare viscoelastic materialsamples, and (ii) indirect methods, where the propertiesare inferred from dynamic measurements made on barswith surface mounted or sandwiched viscoelastic dampinglayers and from assumed underlying mathematical modelswhich relate the composite bar damping behavior to raw

viscoelastic material one. Both methods, though, rely onthe time-temperature superposition principle to construct


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master curves of the storage modulus and loss factor fromdata sets measured at several distinct temperatures, in-herently assuming that the material is thermorheologicallysimple[Ferry (1980)].

Regarding the direct methods, they usually consider rawmaterial samples in a single degree of freedom (DoF)

system configuration, which is conceptually the simplestmeans for obtaining complex moduli data, where thedamping material sample is deformed in a specific mode,such as shear or extension. These tests can be dynamic orpseudo-static and involve the measurement and acquisitionof force and displacement time histories, which are laterprocessed to obtain the complex modulus data. In theformer case, the material is dynamically (usually harmon-ically) deformed and the frequency dependent complexmodulus is directly obtained through transmissibility- orimpedance-based approaches. Regarding the latter ap-proach, usually a step deformation (relaxation test) orforce (creep test), or other types of stimuli, is used todeform the material and it yields the complex moduli in-directly through time-frequency equivalences, based uponthe identification of a representative time domain modelby fitting the measured response to the model.

On the other hand, indirect methods consider test spec-imens where the damping material is surface mountedor embedded in beam structures, with the most popularconfiguration of the test specimen being the Oberst beamwhere the damping material is surface mounted only inone side of the beam being subjected mainly to extensionaldeformations, which was alleged first proposed and studiedby Lienard (1951) and Oberst (1952) in the early 1950s.Other configurations have also been used considering asymmetric Oberst beam with damping material on bothsides (so-called Van Hoort beam) or sandwiched beams[Nashif et al. (1985), Jones (2001), ASTM E756-05(2005)], or even beams with constrained damping layers[e.g. Hambric et al. (2007)]. These test specimens areused to infer complex elastic moduli by resonance- orimpedance-based methods, which use analytic expressions[e.g. Ross et al. (1959), Kerwin (1959), Liao and Wells(2008)] or approximated numerical models [e.g. Wojtow-icki et al. (2004), Hambric et al. (2007), Castello et al.(2008)] of distributed parameters systems, that relate thecomplex modulus of the viscoelastic material to the reso-nances and loss factors or to the FRFs of the test specimen(beam and the damping material), respectively.

The different experimental apparatus for measurementof the viscoelastic constitutive behavior by the afore-mentioned approaches (e.g. resonant beam tests, SDoFresonant and dynamic modulus testing, creep and re-laxation tests), along with measurement and determina-tion procedures and representation of viscoelastic ma-terials properties are described and discussed, for ex-ample, in monographs and books [Drake and Soovere(1984), Nashif et al. (1985), Tschoegl (1989), Findleyet al. (1989), Sun and Lu (1995), Jones (2001), Cre-mer et al. (2005)], articles [Paxson (1975), Ferguson(1984), Allen and Pinson (1991), Allen (1996), Williset al. (2001), Etchessahar et al. (2005), Sorvari and

Malinen (2007), Jaouen et al. (2008)] and standards[ISO 10112:1991 (1991), ASTM D5026-95a (1995), ANSIS2.21-1998 (R2007) (2007), ANSI S2.22-1998 (R2007)

(2007), ANSI S2.23-1998 (R2007) (2007), ANSI S2.24-2001 (R2006) (2006), ASTM D5023-01 (2001), ASTME756-05 (2005), ASTM D5418-07 (2007), ASTM D5024-07 (2007)] and the references therein. Furthermore, someinformation of proprietary apparatus and test methodssuch as the Dynamic Mechanical Analyzer (DMA), Dy-

namic Mechanical Thermal Analyzer (DMTA), Rheovi-bron, Autovibron, Viscoanalyzer, RSA II, to name afew, from companies and laboratories such as CSA Engi-neering (Mountain View, CA, US), Polymer Laboratories(Amherst, MA, US) and Roush Technologies (Livonia, MI,US), can be found in [Allen (1996), Jones (2001), ASTME756-05 (2005), Melo and Radford (2005), Price et al.(2008)] and the references therein.

In the previous references, viscoelastic material mastercurves, tables and empirical and fractional derivative ana-lytical expressions of the complex modulus and shift factorare presented for several types of viscoelastic materials.However, as referred by Jones (2001), that information is

provided from various sources and is fairly accurate, ingeneral, but it is by no means the best data which can beobtained. Better, trustworthy and up-to-date data may beobtained by additional testing or purchase of data from in-house and proprietary data bases [e.g Drake and Terborg(1980), Drake (1988)].

The viscoelastic material considered in this work is theISD112 from 3M (1993), which was chosen because of itswidespread use, its commercial availability and becausemost of the related published work on the open litera-ture is related to the application or characterization ofthis specific material. Regarding its constitutive behavior,an empirical explicit analytical definition of the complexmodulus and shift factor obtained by curve fitting mas-ter curves of 3M ISD112 is given by Drake and Soovere(1984). However, the accuracy of these expressions is notclear and, with time, manufacturers have changed theproduction process, with implications on the properties ofthe viscoelastic materials, making old data inaccurate. Allthese aspects strongly justify the development and usageof an experimental methodology to alternatively test andmeasure the viscoelastic material complex modulus, whichis hereby proposed and validated for the 3M ISD112. How-ever, the proposed methodology to identify the materialproperties can obviously be adapted and applied to anyviscoelastic material and in fact the same experimentalprocedure and apparatus was also used to identify the fre-

quency dependent material properties of cork compoundsas described by Lopes et al. (2006).

Regarding the structure of the article it is divided intothree main sections: viscoelastic material characterization,identification of constitutive model parameters and exper-imental assessment and validation of the damping models.Therefore, the experimental test rig and methodology usedto identify the complex shear modulus of the viscoelasticmaterial 3M ISD112 are described first. Next, the curve-fitted viscoelastic material data is compared with themeasured one and the parameters of the GHM and ADFmodels are presented and used to represent the frequencydependent viscoelastic stiffness and to model the viscoelas-

tic damping behavior of the structure. The identified pa-rameters are then utilized in a FE model and the analysisof a sandwich plate with a viscoelastic core and elastic


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skins is performed. The measured and predicted FRFs arecompared with the purpose of assessing the performanceof all the damping models. The analysis allows to assessthe validity of the methodology to determine the frequencydependent complex modulus, the GHM and ADF parame-ters identification and the outcomes and drawbacks of the

direct frequency response(DFR),iterative modal strain en-ergy(IMSE),iterative complex eigensolution(ICE), GHMand ADF viscoelastic damping modeling strategies andtheir FE implementations presented in the companionarticle [Vasques et al. (2010)].

The main contributions and novelties of this work arerelated to the fact that both measured and predictedresults are utilized to validate the methodologies used toinclude viscoelastic damping into FE models. Therefore,the developed experimental procedure to determine theviscoelastic material properties is presented and discussed,and the characteristic material parameters of the GHMand ADF models of the 3M ISD112 are obtained by curve-fitting the measured shear storage modulus and loss factorof the actual 3M ISD112 material. It is important topoint out, though, that the experimental data is usuallytaken by a manual graphical procedure from the plot of thenomogram given by the manufacturer of the viscoelasticmaterial. A similar study is reported by Trindade et al.(2000) where the GHM, ADF and iterative MSE modelsare compared. However, such study does not compriseneither an effective assessment with measured experimen-tal results nor a performance comparison with predictedresults obtained by a DFR approach, which, being insome circumstances the most time consuming techniqueis, however, for sure, the most precise in the frequency do-main. Additionally, the proposed ICE approach is assessed

and compared with other damping modeling strategies.Therefore, to conclude, it is important to emphasize thatthe damping methodologies proposed here are assessed andcompared in a more rigorous way.

2. EXPERIMENTAL IDENTIFICATION OFVISCOELASTIC MATERIALS

2.1 Underlying Analytical Model

The proposed experimental methodology for the dynamiccharacterization of the complex modulus of viscoelasticmaterials is based on the direct identification of the com-

plex equivalent stiffness K(j) of a discrete single degreeof freedom (SDoF) system [Allen (1996)]. The complexstiffness is physically materialized by a thin viscoelasticlayer which is subjected to cyclic shear deformation im-posed by a dynamic exciter. The dynamic response of theconsidered SDoF system is then used to evaluate the shearstorage modulus and loss factor variation with frequencyand temperature.

ThereceptanceFRF of a SDoF system, assuming station-ary harmonic motion, is given by

H(j) =X(j)

F =

1

K(j) 2M, (1)

where F and X(j) are the amplitudes of the dynamicforce and displacement response and Mis the active mass.

The complex valued stiffness K(j) of the viscoelasticsample can be directly determined through the inverse ofthe receptance, the dynamic stiffness function Z(j) =H1(j), as

K(j) =2M+ Z(j). (2)

The viscoelastic material storage modulus G() and lossfactor () functions can then be evaluated from

G() = h

AS

2M+ Re [Z(j)]

, (3a)

() = Im [Z(j)]

2M+ Re [Z(j)], (3b)

whereh is the thickness of the viscoelastic sample and ASits shear area.

2.2 Experimental Apparatus

The test rig representing the dynamic SDoF system usedto identify the complex shear modulus of viscoelastic ma-terials is illustrated in Figure 1. The underlying analyticalmodel of the SDoF system previously described comprisesa moving mass M, represented by the upper bar (2),and a complex stiffness K(j), represented by the thinviscoelastic layer sample (4) introduced between two rigidblocks, as shown in Figure 2. The upper bar is guided bytwo thin lamina springs (3), which provide the restraintof the spurious DoFs, allowing the viscoelastic specimento deform mainly in shear due to the relative horizontalmotion between the moving bar (2) and the fixed bar (1).

Fig. 1. Test rig used to identify the complex shear modulus.

Fig. 2. Detailed view of the viscoelastic specimen.

The dynamic excitation of the moving bar is provided by

an electrodynamic shaker (Ling Dynamic Systems - model201), using a thin stinger to minimize the rotation andlateral excitation, driven by a random signal generated by


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the signal generator of a dynamic signal analyzer (Bruel& Kjr - model 2035) and amplified by a power amplifier(Ling Dynamic Systems - PA25E). The applied excitationforce is measured using a piezoelectric force transducer(Bruel & Kjr - model 8200). The acceleration responseof the moving mass is measured with a piezoelectric ac-

celerometer (Bruel & Kjr - model 4371) and the relativedisplacement response with a proximity probe (PhilipsPR6423). The signal conditioning and the frequency re-sponse functions are determined with the aforementioneddynamic signal analyzer.

The test rig and the electrodynamic shaker are rigidlyassembled onto an inertial block (Figure 3), which issupported by rubber pads in order to reduce the influ-ence of the rigid body modes of the assembly on theutile bandwidth. Furthermore, the experimental apparatuswas introduced into a thermal chamber, providing nearlyisothermal conditions between 0 and 35 C; the tempera-ture range is limited by the shaker admissible range of op-

erating temperature. The analysis at various temperaturesallows the use of the temperature-frequency equivalenceprinciple [Jones (2001)], extending the frequency range ofthe characterization and allowing data correlation. Thetemperature of the viscoelastic material is evaluated usingtwo thermocouple probes located near the specimen, asdepicted in Figure 2.

Fig. 3. Experimental apparatus into the thermal chamber.

2.3 Analysis of Measured Complex Modulus Data

The receptance and accelerance FRFs of the SDoF dy-namic system described in the previous sections are mea-sured using the transducer response signals (proximityprobe, accelerometer and force transducer) for differenttemperatures. Since two different FRFs were measuredusing two different transducer responses (displacement andacceleration), the corresponding FRFs can be correlatedin order to identify high frequency noise generated bythe thin springs, the stinger and shaker trunnion (highlyevidenced by the receptance results), and the rigid bodymodes effects at low frequencies (evidenced in the accel-erance results). Therefore, the simultaneous use of bothtransducers allows to correlate the two response measure-ments in order to conveniently combine information fromboth measurements and to enlarge the utile frequencyrange.

Viscoelastic damping is a property exhibited by a widevariety of materials such as polymeric materials, rangingfrom natural or synthetic rubber to industrial plastics, oreven cork and cork compounds [Lopes et al. (2006)]. Theclass of polymer materials is extremely wide and manypolymeric compounds, displaying somewhat different com-

plex modulus properties, available from commercial man-ufacturers and other sources [see Nashif et al. (1985) andJones (2001) for manufacturers] can be found. Several vis-coelastic materials, specifically tailored for passive damp-ing treatments, are commercially available in the market,both as raw materials for OEM application or available aspart of a damping solution service/product. The ISD seriesfrom 3MTMcompany (ISD110, 112 and 113), the DYADseries from SoundcoatR company (DYAD601, 606 and609) and the HIP2 from the Heathcote Industrial Plasticscompany are just a few examples of such materials.

In this work, a special attention is given to the 3MISD112 material, which is experimentally characterizedand applied as a damping treatment in the experimentalspecimen used to assess the different viscoelastic dampingmodels and approaches. As stated before, the main reasonfor its choice is related to the fact that it is the viscoelasticmaterial more often used in most of the numerical andexperimental studies published in the open literature. Nev-ertheless, the experimental and numerical methodologieshereby implemented and applied may be straightforwardlyapplied and adapted to characterize other materials witha somewhat different viscoelastic constitutive behavior.

The receptance and accelerance FRFs of a 0.57mm 480mm2 3M ISD112 viscoelastic material specimen, mea-sured in a bandwidth of [0 400] Hz at nine differenttemperatures between 2.7 and 33.5 C are represented inFigure 4. The obtained FRFs, whether using directly thereceptance or the accelerance FRFs, are the inputs forthe complex modulus description calculated according toEquations (3).

In order to identify and filter possible random errors due tothe high frequency noise, rigid body modes or other errorsources, a Wicket plot representation, which is proved tobe a useful graphical representation to highlight erroneousdata values [see Jones (2001) for further details], is appliedto the identified complex modulus data. In this graph-ical representation the storage modulus calculated fromthe data at different temperatures is plotted against thecorresponding loss modulus values (or loss factor values).Figure 5 (a) represents the Wicket plot of the identifiedcomplex modulus data of the 3M ISD112 specimen withoutany filtering of scattered data. From the analysis of thisplot, the master curve of the complex modulus can bedetermined, making it possible to identify erroneous datasets and thus to eliminate the data not following the maintendency of the data distribution. After filtering the data,i.e. removing all scattered points, a reliable data set ofcomplex modulus is obtained, as illustrated in Figure 5 (b),which can be used to characterize the viscoelastic materialunder analysis.

The different sets of complex modulus functions, identifiedat different temperatures, should finally be correlated to

identify the shift factorT(T) distribution of the materialby using the frequency-temperature equivalence principle.


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0 100 200 300 400 0

180

Phase[]

Frequency [Hz]

107

106

105

Magnitude

[m/N]

Temperature

Fig. 4. Measured receptance and accelerance FRFs at 2 .7,

5.8, 7.1, 11.2, 14.4, 14.5, 18.8, 21.5 and 33.5C.

The shift factor relationship might be described by theArrhenius equation [Jones (2001)] as

log[T(T)] =TA

1

T

1

T0

, (4)

where T is the absolute temperature of each data set, T0is the reference temperature andTA is identified by fittingthe experimental data to the model. Using the identifiedshift factor distribution, the identified complex modulusfunctions can be represented in the unified and univer-sal representation, the so-called reduced-frequency nomo-gram [ISO 10112:1991 (1991), ANSI S2.24-2001 (R2006)(2006)]. This representation is very useful and broadlyapplied since it represents simultaneously the frequencyand the temperature dependence of the complex modulusof a viscoelastic material by a single pair of curves: onefor the storage modulus and the other representing theloss factor distribution. Figure 6 overlaps the originalnomogram of the 3M ISD112 viscoelastic material pro-vided by the manufacturer 3M (1993) with the identifiednomogram of the 3M ISD112 specimen under analysis. Theidentified nomogram agrees well with the one publishedby the viscoelastic material manufacturer and is consistentwith other nomograms obtained with other specimens with

different dimensions using the same methodology. Thereader is referred to the well-known books of Nashif et al.

105

106

107

105

106

107

108

Lossmodulu

sG"[Pa]

Storage modulus G [Pa]

(a) Unfiltered data

105

106

107

105

106

107

108

Storage modulus G [Pa]

StoragemodulusG"

[Pa]

(b) Filtered data

Fig. 5. Wicket plot of the identified complex modulus

functions (spectral filtering).

(1985) and Jones (2001), and standard ANSI S2.24-2001(R2006) (2006), for further details about the temperature-frequency equivalence principle and the nomogram (alsoknown as nomograph) elaboration.

3. IDENTIFICATION OF THE CONSTITUTIVEPARAMETERS OF TIME DOMAIN MODELS

As previously mentioned, time domain models may repre-sent a better alternative to CMA-based models, with thelatter not requiring any curve fitting procedure, since theydirectly allow a transient analysis to be performed, evenfor highly damped structural systems where CMA-basedmodels experience some difficulties. Additionally, time do-main models may also reduce the computational cost thatis usually significant when using a direct frequency domainresponse method such as the DFR approach, with thedifference being more significant as the required frequencyresolution and bandwidth increase.

Among the time domain models, internal variables modelsare more interesting from the computational cost andeasiness of implementation into FE codes viewpoints. TheGHM and ADF models are two of such methods, usedto model the damping behavior of viscoelastic materialsin FE analysis, which yield a standard FE formulation(however with the addition of some non-physical and non-


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Fig. 6. Nomogram of the viscoelastic material 3M ISD112: manufacturers (black) and identified (color).

observable dissipative variables). In order to use them, oneneeds the GHM and ADF characteristic parameters whichallow characterizing the complex (frequency dependent)constitutive behavior of the viscoelastic material beingused. To this end, experimental testing to measure theisotropic constitutive behavior (usually the shear modulus)may be necessary and numerical identification proceduresneed to be applied to determine the unknown modelparameters from the measured data.

As reported by Trindade and Benjeddou (2002), the pa-rameters of internal variables models, in general, are ad-justed by curve-fitting the viscoelastic material mastercurves provided by the manufacturer, in order to minimizethe difference between the measured and estimated data.In fact, such procedure was carried out in the 1990s byLesieutre and Bianchini (1995) that presented the resultsof the curve-fitting of the 3M ISD112 material data at27 C in the frequency range [8 8000] Hz. They con-cluded that five ADF series (with two ADF parametersper each series) would accurately represent the behaviorof the material shear modulus and loss factor. Friswellet al. (1997) performed the same analysis for the GHMmodel, with three or four parameters per series. They usedthe 3M ISD112 material at 20 C in the frequency range[10 4800] Hz and the DYAD601 material at 24C inthe frequency range [2 4800] Hz. Their results evidencedthat a more accurate constitutive description is obtained

with a model with four parameters than that with onlythree parameters. In general, GHM and ADF models fitwell the master curves of materials whose properties have

strong frequency dependence. Nevertheless, the number ofparameters needed is somewhat related with the degree offrequency dependence of the material properties and usu-ally increases with that dependence. These reasons moti-vated Enelund and Lesieutre (1999) to propose a combina-tion of the ADF model with a fractional derivative model,where the relaxation equation for the anelastic strain istaken as an in time differential equation of fractional-order,combining the best features of the ADF model and thefractional calculus(FC). It was shown that this fractional-order ADF model can predict the instantaneous transientresponse of the material over a wide frequency range usinga single anelastic strain field (only one ADF series) gov-erned by a fractional-order equation with the drawback of

yielding a non-standard FE formulation, which requiresa more complex characteristic solution procedure typicallyused for FC.

From the measured material properties (identified nomo-gram) of the 3M ISD112, and using the shift factor rela-tionship previously determined, the frequency dependentcomplex modulus for different temperature values can beobtained. In order to use the internal variables viscoelasticdamping models GHM and ADF defined in the companionarticle [Vasques et al. (2010)], the characteristic modelparameters (three and two per each GHM and ADF se-ries, respectively) were determined using a curve fittingprocedure based upon the minimization of the least mean

square error between the measured and predicted data;the latter data was obtained using the GHM and ADF


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Table 1. Identified GHM and ADF parameters for the 3M ISD112 at 27C using three series (n= 3).

GHM ADF

i G[M Pa] i i i G[M Pa] i i

1 0.1633 4.8278 28045 22.013 0.1789 3.5286 504.20

2 14.548 41494 3.1275 8.7533 4286.53 40.043 41601 0.6165 60.324 39313

definitions of the complex shear modulus, for a specifiedfrequency range of interest.

The number of series used in the GHM and ADF modelsdetermines how well the models are capable of matchingthe data over the frequency range of interest. For eachviscoelastic model, one and three series of parameterswere used to curve fit the identified complex modulusof the 3M ISD112 material at 27 C over the frequencyrange [10 3000] Hz. The fitted curves of both modelsare presented in Figures 7 and 8 and compared with themeasured data.

As can be verified, the quality of the fit strongly dependsupon the number of terms (series) retained in the models.Using more series improves the accuracy of the materialmodels, however increasing the size of the problem. Thevalues of the parameters determined in the fitting processfor both GHM and ADF models are presented in Table 1.It is worthy to mention that the values of the parametersdetermined are guaranteed to define the material proper-ties with accuracy only over the frequency range specifiedin the fitting process, which in this case is [10 3000] Hz.From the analysis of Figures 7 and 8, it may be concluded

that both models fit the measured data with a satisfactoryaccuracy with three series of parameters and that only oneseries would yield a bad fit.

4. ASSESSMENT AND VALIDATION OF THEDAMPING MODELS

The aim of this section is to assess and validate the timedomain based GHM and ADF viscoelastic damping mod-els, and their correspondent FE implementation and solu-tion method, and the frequency domain based ones, IMSEand ICE, presented and described in the companion article[Vasques et al. (2010)]. With that in mind, a comparisonbetween measured FRFs of a viscoelastic sandwich plateand predicted ones utilizing the aforementioned modelsand the DFR approach is performed.

4.1 Experimental specimen

To perform the experimental analysis, a sandwich platespecimen with an embedded viscoelastic damping treat-ment was utilized. The viscoelastic material consideredis the ISD112 from 3M (1993) which was used for theviscoelastic layer applied in the core of the sandwich plate.The specimen was manufactured using two aluminumplates (aluminum alloy 1050A H24) with dimensions 200

100 1 mm, to produce the sandwich skins, and a vis-coelastic core with thickness 0.127 mm (Figure 9). It wasproduced following the manufacturer instructions and it

was found that the 3M ISD112 can be easily bonded to themetallic substrate at room temperature. The propertiesof the viscoelastic material and aluminium applied in thespecimen are presented in Table 2.

Table 2. Material properties of the viscoelasticsandwich plate specimen.

Material E[ Pa] G [ Pa] [ k g m3]

AA 1050A H24 70

10

9

0.32 27083M ISD112 Figure 6 0.49 1140

4.2 Experimental setup

The aim of the experimental study was the determinationof a representative set of FRFs providing a reliable basisfor comparison and validation of the numerical models.The experimental specimen was suitably suspended in theair through thin lightweight nylon wires attached to arigid frame to get approximate free boundary conditions,which minimizes the boundary error effects. A grid with 15points, as depicted in Figure 9, was defined for measure-

ment. An electrodynamic shaker (Ling Dynamic Systems -model 201), suspended from an independent rigid frame,was utilized to generate a random [0 800] Hz excitationapplied in point 5 of the specimen. A thin and flexiblestinger was used to link the shaker to the miniature forcetransducer (Bruel & Kjr - model 8203) attached to theplate surface, which provided the measurement of theapplied dynamic force (Figure 10). The specimen responsewas evaluated by using a laser vibrometer (Polytec - modelOFV303) to measure the velocity of each point of themeasuring mesh (Figure 11). The FRFs determinationwas performed by a dynamic signal analyzer (Bruel &Kjr - model 2035). The temperature was continuouslymonitored by a thermocouple located near the specimen.

Fifteen FRFs (mobility functions) were determined for thespecimen. However, here, for comparison of the dampingmodels, only the driving point mobility FRF measured atpoint 5 is considered.

4.3 Comparison and assessment of the results

For comparison of results, the driving point mobility FRFmeasured at point 5 of the mesh depicted in Figure 9is utilized. The predicted FRFs were determined usinga 4-node facet type quadrangular shell FE based on alayerwise theory, developed for the dynamic modeling oflaminated structures [Moreira et al. (2006)]. A three-layered layerwise FE with 9 DoFs per node is consideredand the sandwich plate was modeled with a FE meshdiscretization of 32 16 elements with 5049 DoFs. The


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101 102 103 10410

1

100

101

Frequency [Hz]

Storagemodulus(G)[MPa]andlossfactor()

G(1 series)

G(3 series)

G(measured)

(1 series)

(3 series)(measured)

Fig. 7. Curve fitted GHM curves with 1 and 3 series of parameters at 27C.

101

102

103

104

101

100

101

Frequency [Hz]

Storagemodulus(G)[MPa]andlossfactor()

G(1 series)

G(3 series)

G(measured)

(1 series)

(3 series)

(measured)

Fig. 8. Curve fitted ADF curves with 1 and 3 series of parameters at 27C.

damping behavior of the viscoelastic sandwiched layer wasincorporated by the five methods described in Part I of thisarticle [Vasques et al. (2010)]: (i) DFR, where the stiffnessmatrix is re-calculated at each frequency value; (ii) IMSE,where an iterative procedure is used to estimate the modalparameters with the modal loss factor determined througha MSE approach; (iii) ICE, where, similarly to the IMSEmethod, the modal parameters are determined through aniterative solution usually requiring only a few iterations

(typically less than 10) where, in this case, a complexeigenvalue problem is considered instead of a real one,

which allows a direct estimation of the loss factor anddamped natural frequency from the complex eigenvalue;(iv and v) and GHM and ADF models, based in the use ofadditional internal (or dissipative) variables, utilizing thethree series of parameters previously identified (Table 1).

The measured and predicted FRFs are compared in Figure12. Results of the natural frequencies and modal loss fac-tors for the first 4 modes are presented in Table 3. In orderto quantify the correlation level between the measured andpredicted FRFs, the global amplitude criterion (GAC),proposed by Zang et al. (2001), is utilized here and the


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12

3

4

5

67

8

910

11

12

13

1415

100mm

200mm

Fig. 9. Measuring mesh and viscoelastic sandwich platespecimen.

Fig. 10. Experimental setup (specimen excitation).

Fig. 11. Experimental setup (response measurement).

frequency distributions of the GAC indicator for the fivedifferent methods are presented in Figure 13. It is worthyto mention that this correlation criterion will only be uni-tary if the compared FRFs have identical phase and mag-nitude, and since it is correlating mainly the amplitude,it allows more clearly to identify the damping deviation

between the measured and each of the five predicted FRFs[Moreira and Rodrigues (2002)].

From the results it can be seen that the predicted FRFsare well correlated with the measured one. Both naturalfrequencies and modal loss factors are well estimated withthe internal variables models, GHM and ADF, and withthe DFR and ICE approaches. Discrepant results, both interms of natural frequency and damping estimation, areobtained with the IMSE, which is shown to be less accurate(at least for highly damped systems, as demonstrated inthe present case study). It is worthy to mention thatthe DFR approach may be considerably time-consumingand that it does not directly provide the modal param-eters of the viscoelastically damped structure. Although,it directly provides FRFs that may be embodied into afrequency response model which can be directly correlatedwith experimental measurements. In opposition, both theinternal variables models and the two iterative approachesallow the definition of a mathematical model of the struc-ture allowing, as in this specific case, directly and itera-tively determining the natural frequencies and modal lossfactors from the extracted eigenvalues, identifying all themodes within the frequency range under analysis, whichsometimes might be difficult to identify from the post-processing of highly damped FRFs. Furthermore, thesemathematical modal models can be used both in time andfrequency domain analysis. Moreover, it is also shown that

the GHM and ADF models conduct to similar results, withthe size of the ADF model being smaller than the GHMone, and it is concluded that the fitting process is easierto perform with the ADF model. Lastly, the proposed ICEapproach is shown to be very accurate in terms of damp-ing estimation and less accurate in the natural frequencyestimation, with the drawback of requiring a complexeigenvalue solution but without the need to increase thesize of the problem, as is the case in the internal variablesmodels. A summary and comparison of the general featuresof the five damping methods is presented in Table 4.

5. CONCLUSION

This companion article has assessed FE-based analyticalstrategies to model the damping behavior of viscoelasticmaterials, which might be used as surface mounted, con-strained or embedded damping treatments in structures tocontrol noise and vibration levels. An experimental proce-dure to identify the complex shear modulus of viscoelasticmaterials was presented and the obtained data was usedto fit the internal variables models, GHM and ADF, andidentify the characteristic parameters for the 3M ISD112viscoelastic material. Measured and FE-based predictedFRFs based on a DFR, internal variables models (GHMand ADF), and two iterative approaches (IMSE and ICE),

were compared in order to assess the damping modelsand validate the experimental procedure for the materialproperties identification and the curve fitting process.


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Table 3. Measured and predicted natural frequencies [ Hz] and modal loss factors [%] of the first 4modes of the viscoelastic sandwich plate specimen.

1st mode 2nd mode 3rd mode 4th mode

Measured 235.1 (28.2) 521.0 (24.2) Predicted (DFR) 233.9 (26.0) 518.6 (30.2) Predicted (IMSE) 212.7 (39.0) 223.1 (38.0) 476.7 (38.2) 526.6 (41.4)Predicted (ICE) 230.3 (26.4) 237.1 (29.8) 504.6 (31.0) 558.9 (34.0)Predicted (GHM) 235.7 (27.4) 268.2 (25.4) 524.3 (33.6) 556.8 (38.6)Predicted (ADF) 233.8 (28.2) 269.4 (26.6) 524.2 (33.0) 554.5 (37.4)

Table 4. General features and comparison of the five damping methods.

DFR IMSE ICE GHM ADF

DoFsNumber of series (n) 3 32nd order form

Elastic (nE) 5049 5049 5049 5049 5049Internal: dissipative (nnD) or anelastic (nnA) 15147 15147Total (nE + nnD; nE + nnA) 5049 5049 5049 20196 20196

1st order formElastic (2nE) 10098 10098 10098 10098 10098Internal: dissipative (2nnD) or anelastic (nnA) 30294 15147Total (2nE + 2nnD; 2nE + nnA) 10098 10098 10098 40392 25245

Model Generation

Spatial model

Frequency response directly obtained N/A N/A Time response directly obtained N/A N/A N/A

Modal modelDirectly obtained N/A Identification procedure N/A N/A

Frequency response modelDirectly obtained N/A N/A Synthesized (generated)

Solution Methods

EigenanalysisReal N/A N/A N/A N/AComplex N/A

System of linear equationsComplex N/A N/A

Viscoelastic Constitutive Data

Creep/relaxation dataCurve fitting N/A N/A N/A Discrete data N/A N/A N/A N/A N/A

Frequency dependent dataCurve fitting N/A N/A N/A Discrete data N/A N/A

Notes: , necessary/appropriate; , not necessary/appropriate; N/A, not applicable/available. Singular augmented mass matrix. Computational effort depends upon the frequency bandwidth and required resolution of the analysis.

Iterative procedure applied to each mode.


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0 100 200 300 400 500 600 700 80090

0

90

Phase[]

Frequency [Hz]

102

101

100

Magnitude[ms

1/N]

Measured

Predicted (DFR)

Predicted (ADF)Predicted (GHM)

Predicted (IMSE)

Predicted (ICE)

Fig. 12. Direct mobility FRF of the viscoelastic sandwich plate at 27 C.

0.9

1

GAC

0.9

1

GAC

0.9

1

GAC

0.9

1

GAC

0 100 200 300 400 500 600 700 800

0.9

1

GAC

Frequency [Hz]

Predicted (ADF)

Predicted (GHM)

Predicted (IMSE)

Predicted (ICE)

Predicted (DFR)

Fig. 13. GAC correlation indicators for the five damping models.


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The application of a discrete SDoF dynamic system andthe underlying analytical model proved to be a reliableidentification methodology, since it is based on the directcharacterization of the complex stiffness of a viscoelasticmaterial sample in shear deformation (in opposition toindirect measuring approaches like the resonant beam

technique). The obtained results are consistent with thosepublished by the material manufacturer.

Regarding the internal variables models under analysishere, which were implemented at the global FE modellevel, the ADF model is shown to lead to an augmentedmodel of the damped structural system with a lower sizethan the GHM model when cast in a state space form.In first-order form and considering the present analysis,the ADF model represents the most interesting alternativeto accurately model the general damping behavior, sinceit yields good trade-off between accuracy and complexityproviding also a better description of the constitutivebehavior with only two parameters per series. One majordisadvantage in using internal variables models such as theGHM or ADF is the introduction of additional dissipation(or anelastic) variables increasing the size of the coupleddamped FE model.

Providing an alternative procedure, the DFR approachrelies on the use of the FE spatial model and in a frequencydomain step by step (discrete) solution, which sees itscomplex viscoelastic stiffness matrix being continuouslyupdated with the current value of the complex modulusfor the current value of frequency, as the current frequencyvalue is changed across the discrete range of frequenciesof the analysis. This procedure, while being simpler toimplement and not increasing the problem size, can leadto a time-consuming analysis with an high computationalcost, which mainly depends upon the number of discretefrequency values which in turn is related to the frequencyrange and required frequency resolution. Furthermore, thistype of analysis also does not allow the direct determina-tion of the modal model of the damped structural system,but enables a more straightforward comparison with ex-perimental FRFs as is often necessary in validation taskswhen experimental data is a set of measured FRFs. Themodal parameters can still be indirectly determined byusing a modal identification procedure, which may becomemore troublesome, if not impossible, for highly dampedstructures with an high modal density and with the modesnot well separated.

Alternatively, the ICE approach was successfully utilizedand it was shown that this strategy has some advantagesover the ADF model: it does not need to increase thesize of the problem and the eigensolution can consideronly a small number of modes necessary to build a reli-able truncated modal basis able to represent the dampedsystem response over a limited frequency range. However,it has the drawback of requiring complex eigensolutioncalculations (of a smaller size than the ADF approach,though) during an iterative process until convergence isachieved.

The IMSE method proved to be the less accurate ap-proach, specially for highly damped systems, though it is

an attractive approach to give an estimate of the FRFswhen low cost and fast analysis are required.

In general, it can be stated that all the models but theIMSE have similar accuracy and yield representative re-sults of viscoelastically damped structural systems. How-ever, they present significant differences in terms of easi-ness of implementation, which makes the decision on whichmethod is most adequate strongly problem-dependent,

i.e. it depends, among other aspects, upon the materialand geometric characteristics of the structural system andupon the type of analysis required. With this reflectionon state-of-the-art viscoelastic damping modeling method-ologies and the results of this research, presented in twocompanion articles, it is expected to somewhat contributein the selection of the best (most appropriate) methodmeeting the needs of structural designers.

REFERENCES

3M (1993). Scotchdamp Vibration Control Systems: Prod-uct Information and Performance Data, 3M IndustrialTape and Specialties Division, St. Paul, Minnesota, US.

Allen, B. R. (1996). A direct complex stiffness test systemfor viscoelastic material properties, Proceedings of 3rdSmart Structures and Materials (SPIE), Vol. 2720, SanDiego, CA, US, pp. 338345.

Allen, B. R. and Pinson, E. D. (1991). Complex stiffnesstest data for three viscoelastic materials by the directcomplex stiffness method, Proceedings of Damping 91,Vol. 2, San Diego, CA, US, pp. EAE 114.

ANSI S2.21-1998 (R2007) (2007). American NationalStandard: Method for Preparation of a Standard Mate-rial for Dynamic Mechanical Measurements, AmericanNational Standards Institute, New York, US.

ANSI S2.22-1998 (R2007) (2007). American National

Standard: Resonance Method for Measuring the Dy-namic Mechanical Properties of Viscoelastic Materials,American National Standards Institute, New York, US.

ANSI S2.23-1998 (R2007) (2007). American NationalStandard: Single Cantilever Beam Method for Measur-ing the Dynamic Mechanical Properties of ViscoelasticMaterials, American National Standards Institute, NewYork, US.

ANSI S2.24-2001 (R2006) (2006). American NationalStandard: Graphical Representation of the ComplexModulus of Viscoelastic Materials, American NationalStandards Institute, New York, US.

ASTM D5023-01 (2001). Standard Test Method for Mea-suring the Dynamic Mechanical Properties in Flexure

(Three-Point Bending), American Society for Testingand Materials, West Conshohocken, Pennsylvania, US.

ASTM D5024-07 (2007). Standard Test Method for Plas-tics: Dynamic Mechanical Properties in Compression,American Society for Testing and Materials, West Con-shohocken, Pennsylvania, US.

ASTM D5026-95a (1995). Standard Test Method forPlastics: Dynamic Mechanical Properties in Tension,American Society for Testing and Materials, West Con-shohocken, Pennsylvania, US.

ASTM D5418-07 (2007). Standard Test Method for Plas-tics: Dynamic Mechanical Properties in Flexure (DualCantilever Beam), American Society for Testing andMaterials, West Conshohocken, Pennsylvania, US.

ASTM E756-05 (2005).Standard Test Method for Measur-ing Vibration-Damping Properties of Materials, Amer-


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8/14/2019 JARME-2-2-2010

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Sun, C. T. and Lu, Y. P. (1995). Vibration Dampingof Structural Elements, Prentice Hall, Englewood Cliffs,NJ, US.

Trindade, M. A. and Benjeddou, A. (2002). Hybridactive-passive damping treatments using viscoelasticand piezoelectric materials: Review and assessment,

Journal of Vibration and Control8(6): 699745.Trindade, M. A., Benjeddou, A. and Ohayon, R. (2000).Modeling of frequency-dependent viscoelastic materialsfor active-passive vibration damping, Journal of Vibra-tion and Acoustics122(2): 169174.

Tschoegl, N. W. (1989). The Phenomenological The-ory of Linear Viscoelastic Behaviour: An Introduction,Springer-Verlag, Berlin, DE.

Vasques, C. M. A. and Dias Rodrigues, J. (2008). Com-bined feedback/feedforward active control of vibrationof beams with ACLD treatments: Numerical simulation,Computers and Structures86(3-5): 292306.

Vasques, C. M. A., Mace, B. R., Gardonio, P. and Ro-drigues, J. D. (2006). Arbitrary active constrained layer

damping treatments on beams: Finite element modellingand experimental validation,Computers and Structures84(22-23): 13841401.

Vasques, C. M. A., Moreira, R. A. S. and Dias Rodrigues,J. (2010). Viscoelastic damping technologiesPart I:Modeling and finite element implementation, Journal ofAdvanced Research in Mechanical Engineering1(2): 7695.

Vasques, C. M. A. and Rodrigues, J. D. (2008). Numericaland experimental comparison of the adaptive feedfor-ward control of vibration of a beam with hybrid active-passive damping treatments,Journal of Intelligent Ma-terial Systems and Structures19(7): 805813.

Willis, R. L., Wu, L. and Berthelot, Y. H. (2001). De-termination of the complex young and shear dynamicmoduli of viscoelastic materials, Journal of the Acousti-cal Society of America109(2): 611621.

Wojtowicki, J.-L., Jaouen, L. and Panneton, R. (2004).New approach for the measurement of damping prop-erties of materials using the Oberst beam, Review ofScientific Instruments75(8): 25692574.

Zang, C., Grafe, H. and Imregun, M. (2001). Frequency-domain criteria for correlating and updating dynamicfinite element models, Mechanical Systems and SignalProcessing15(1): 139155.


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