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Marcelo A. TrindadeDepartment of Mechanical Engineering,

So Carlos School of Engineering,

University of So Paulo,

Av. Trabalhador So-Carlense, 400,

So Carlos-SP, 13566-590, Brazil

e-mail: [email protected]

Reduced-Order Finite ElementModels of ViscoelasticallyDamped Beams Through InternalVariables ProjectionFor a growing number of applications, the well-known passive viscoelastic constrainedlayer damping treatments need to be augmented by some active control technique. How-ever, active controllers generally require time-domain modeling and are very sensitive tosystem changes while viscoelastic materials properties are highly frequency dependent.

Hence, effective methods for time-domain modeling of viscoelastic damping are needed.This can be achieved through internal variables methods, such as the anelastic displace-ments fields and the Golla-Hughes-McTavish. Unfortunately, they increase considerablythe order of the model as they add dissipative degrees of freedom to the system. There-

fore, the dimension of the resulting augmented model must be reduced. Several research-ers have presented successful methods to reduce the state space coupled system, resulting

from a finite element structural model combined with an internal variables viscoelasticmodel. The present work presents an alternative two-step reduction method for such

problems. The first reduction is applied to the second-order model, through a projectionof the dissipative modes onto the structural modes. It is then followed by a second

reduction applied to the resulting coupled state space model. The reduced-order modelsare compared in terms of performance and computational efficiency for a cantileverbeam with a passive constrained layer damping treatment. Results show a reduction of upto 67% of added dissipative degrees of freedom at the first reduction step leading to much

faster computations at the second reduction step. DOI: 10.1115/1.2202155

1 Introduction

It is well known that viscoelastic constrained layer dampingtreatments can reduce resonant structural vibrations. For a grow-ing number of applications, however, and due to material and/orgeometrical limitations, this passive damping needs to be aug-mented by some active control technique. Indeed, several hybrid

active-passive damping mechanisms were proposed in the last de-cade through the combination of piezoelectric-actuated active vi-bration control and viscoelastic damping treatments 1,2. Themain difficulty when associating active control and viscoelastictreatments is that active controllers are generally very sensitive tosystem changes while viscoelastic materials properties are highlyfrequency dependent. In addition, most modern control techniquesrequire a time-domain model representation. Hence, effectivemethods for time-domain modeling of viscoelastic damping areneeded. This can be achieved through internal variables methods,such as the anelastic displacements fields ADF, proposed byLesieutre and Bianchini 3, and the Golla-Hughes-McTavishGHM, proposed by Golla and Hughes 4 and McTavish andHughes5, that allow effective time-domain modeling of the fre-quency dependence of stiffness and damping properties of vis-coelastically damped structures. It was shown in 6 that bothADF and GHM are effective for time-domain analyses of highlydamped structures. Unfortunately, they increase considerably thedimension of the resulting augmented model as they add dissipa-tive degrees of freedom to the system. Therefore, the resultingmodel must be reduced.

There are mainly three strategies published in the literature forthe reduction of viscoelastic finite element FE models with in-

ternal variables. The first one consists in applying some reductionmethod to the undamped structural FE model before adding theinternal variables, which has the advantage of handling smallermatrices but generally leads to erroneous or less precise estima-tions of viscoelastic damping. This was shown by Friswell andInman7who considered reducing the order of the physical FEmodel, through a system equivalent reduction expansion process,before and after assembly, previous to introducing the extra dissi-pative coordinates for the GHM method. The second reductionstrategy consists in applying a reduction method to the second-order augmented FE model, after inclusion of internal variables.Park, Inman and Lam8have studied Guyan reduction method toreduce the order of a FE model augmented by GHM viscoelasticdissipative coordinates, but they showed that the method performspoorly. The third strategy is based on the application of a reduc-tion method to the augmented state space system, that is afterinclusion of the internal variables and transformation to the statespace form. This has the advantage of allowing the application ofmodern reduction methods, developed for state space systems,such as internal balancing method. Friswell and Inman 7 andPark, Inman and Lam8applied an internal balancing method toa state space system augmented by GHM dissipative coordinates.Friswell and Inman 7 also applied eigensystem truncation forcomparison. Trindade, Benjeddou and Ohayon 9 have also ap-plied eigensystem truncation, followed by a real representation ofthe complex reduced system, to a state space system augmentedby ADF dissipative coordinates. Both balanced realization andeigensystem truncation methods were shown to be effective inreducing the order of the state space system while retaining theviscoelastic damping behavior. However, they require a high com-putational effort due to a generally large dimension of augmentedstate space matrices.

The present work presents an alternative two-step reductionmethod for internal variables-based viscoelastic FE models. The

Contributed by the Technical Committee on Vibration and Sound of ASME forpublication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript receivedSeptember 15, 2004; final manuscript received January 24, 2006. Assoc. Editor:William W. Clark.

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first reduction is applied to the second-order model, through aprojection of the dissipative modes onto the structural modes. Thisapproach also provides an effective measure of the coupling be-tween the dissipative modes and structural modes, and enablesphysical interpretation of the former. The second-order model re-duction is then followed by another reduction applied to the re-sulting coupled state space model. The objective is to allow areduction of the extra dissipative coordinates before transforma-tion to the state space, so as to provide faster computations on thereduction of the state space system.

2 Finite Element ModelLet us consider the following equations of motion for a finite

element structural model

Mq+ Dq+Ke+Kvq=F 1

where q is the degrees of freedom dofvector and, q andq are,respectively, the velocity and acceleration vectors. M is the massmatrix,D is a viscous damping matrix introduced a posteriori, andFis a mechanical perturbation input. K

vis the part of the stiffness

matrix corresponding to the contribution of the viscoelastic mate-rial and Keis the stiffness matrix corresponding to the remainingstiffness contributions in the structure.

The frequency dependence of the viscoelastic material proper-ties is modeled through the ADF model 3. The ADF model isbased on a separation of the viscoelastic material strains in an

elastic part, instantaneously proportional to the stress, and an an-elasticor dissipativepart, representing material relaxation. Thiscould be applied to Eq. 1 by replacing the dof vector q by qe

= q iqid in the viscoelastic strain energy; where qe and qi

d rep-resent the dof vectors associated with the elastic and anelasticstrains, respectively. Adding a system of equations describing thetime-domain evolution of the dissipative dofqi

d to Eq.1, we get

Mq+ Dq+Ke+KvqK

v

i

qid=F 2

Ci

i

Kv

qid+CiKv

qidK

v

q=0 3

whereKv

=GKv, for G=G01+ iiand Ci =1+ ii/i3.

ADF parameters G0, i and i are evaluated by curve fitting ofthe measurements ofG*, represented as a series of functions inthe frequency domain

G*=G0+G0i

i

2 + ji2 +i

2 4

The form of the series of functions used to construct G* iswell adapted to fit the behavior of complex modulus frequencydependence for generic viscoelastic materials, which presentstrong frequency dependence. Nevertheless, modern viscoelasticmaterials tend to be less frequency dependent so as to maintain ahigh loss factor over a wide frequency range of interest, and con-sequently being more effective in damping vibrations. For suchmaterials, a larger number of series terms must be used to providea satisfactory curve fit of complex modulus frequency depen-dence. A more detailed analysis of curve fitting will be presentedlater, but it is worthwhile advancing that, for modern viscoelasticmaterials used for vibration damping, more than three ADF seriesterms are generally required and the larger the number of ADFseries terms considered the better fitting of materials properties isobtained. Notice, however, that there is one system of equationsEq. 3 for each ADF series term considered. Thus, there mustbe a compromise between the quality of material properties curvefitting and the number of extra systems of equations included intothe final augmented system. Since the extra dissipative dof qi

d

included for each ADF series term has the same dimension ofq ,the dimension of the final augmented system will be at least four

times that of the original FE systemconsidering three ADF seriesterms to represent frequency dependence behavior.

It is worthwhile to notice also that in the case of a structurepartially covered with the viscoelastic treatment, the viscoelasticstiffness matrix K

v

will possess a number of rigid body modes,corresponding to the FE dof of the nontreated parts of the struc-ture. Consequently, there will be a number of equations in Eq. 3that will be automatically satisfied. Hence, the increase in theaugmented system dimension will be also dependent on the per-cent of area covered with the viscoelastic material throughout thestructure surface.

The rigid body modes ofKv

can be eliminated through a modaldecompositionqid= Tdq i

d, such that d=TdTK

v

Tdand Eqs.2and3can be rewritten as

Mq+ Dq+Ke+KvqTdd

i

q id=F 5

Ci

i

dq id+Cidq i

d dTdTq=0 6

The equations that are automatically satisfied correspond to thenull eigenvalues in matrix d. Notice that these rigid body modesofK

v

do not contribute to the overall structural damping. Hence,the null eigenvalues are eliminated from dand so are the corre-sponding eigenvectors from T d. The combination of Eqs. 5and6leads to the following augmented system

Mq +Dq +Kq=F 7

with

M = M 00 0

; D= D 00 Ddd

; F= F0

K= Ke+Kv KedKed

T Kdd ; q= colq,q 1d, .. . ,q nd

where

Ddd= diag C11d

Cn

n

d ; Kdd= diagC1dCnd;

Ked=TddTdd

3 State Space Model Construction

In order to eliminate the apparent singularity of the mass matrixof system7and to provide a transformation to an elastic onlymodal reduced model, Eq. 7 is rewritten in a state space form.Therefore, a state vector x is formed by the augmented vector qand the time derivative of the mechanical dof vector q . The timederivatives of the dissipative dofqi

d are not included in the statevector since these variables are massless. This leads to

x=Ax +p ; y=Cx 8

where the perturbation vector p is the state distribution of themechanical loads F and the output vector y is, generally, com-

posed of the measured quantities, written in terms of the statevector x through the output matrix C. The system dynamics isdetermined by the square matrix A . These are

A= 0 0 0 I

1

C1Td

T 1I 0 0

0

n

CnTd

T 0 nI 0

M1Ke+Kv M1Tdd M

1Tdd M1D

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x= qq ; p= 0

M1F ; C=CqCq

where Cq and Cqare output matrices relative to augmented dofvector q and mechanical dof derivatives q , respectively.

4 Model Reduction

It is evident from Eq. 7 that inclusion of dissipative dofgreatly increases the dimension of the FE model, even for a partial

treatment, corresponding to a great increase also in the final statespace model Eq. 8. Since our final objective is to apply thestate space model for control design and optimization, leading toCPU-demanding computations for a large number of candidateconfigurations, some model reduction is required. Hence, in thissection some techniques are presented to provide an reduced-orderstate space model, which dimension is small enough to allowapplication to control design and optimization and that is still ableto well represent the viscoelastic damping of the structure.

4.1 State Space Model Reduction.In principle, all reductiontechniques for state space systems may be applied to Eq. 8. Themost standard ones are the reduction to modal coordinates and thereduction via internal balancing methods. While the latter leads tomore precise results for a given input and output configuration,the former is independent of input and output configurations andalso allow faster computations. Details on reduction via internalbalancing methods can be found in 8. Details on modal reduc-tion can be found in 9and are briefly resumed in this section.

By neglecting the contributions of viscoelastic relaxationmodes and some elastic modes, related to eigenfrequencies out ofthe frequency-range considered, a complex-based modal reductioncan be applied to the state space system 8. The eigenvaluesmatrix and, leftT land rightTr, eigenvectors of Eq.8are firstevaluated from

ATr=Tr; ATTl=Tl 9

so that TlTTr= I, then decomposed as following

=

r 0 0

0 ne 0

0 0 nd

; Tl=TlrT lne T lnd;

10Tr=TrrT rne T rnd

The state vector is then approximated as x Trrxr, so that thecontribution of out-of-frequency-range elastic and viscoelastic re-laxation modes Tlne, Trne, T lndand T rnd is neglected. Hence, thesystem8may be reduced to

xr=rxr+TlrTp y=CTrrxr 11

The main disadvantage of the reduced state space system 11isthat its matrices are complex. Fortunately, since all overdampedrelaxationmodes were neglected, all elements of the system11are composed of complex conjugates, such that it is possible touse a state transformation x = Tcxr 10 and write the following

real state space system equivalent to Eq.11

x =A x+p y=C x 12

where

A =TcrTc1 =

0 I

j2 2Rj

p= TcTlrTp; C =CTrrTc

1

It is clear that the eigenvalues of the real matrix A are exactlythe elements of r, i.e. the retained elastic eigenvalues j and

their complex conjugates j. In the form of Eq.12, the new statevariablesx represent the modal displacements and velocities.

4.2 Second-Order Model Reduction.The main difficulty inusing reduction methods for state space systems, either via modaltruncation or balanced realization, is that the dimension of thestate space matrix A may be very large due to the inclusion ofinternal variables in the FE model. Consequently, when there is aneed for repeating the reduced model evaluation for several treat-

ment configurations, which is often the case for control design andoptimization, it easily becomes an impractical task. Hence, anovel model reduction method is presented in this section. It con-sists in reducing the dissipative system Eq.6before construc-tion of the augmented state space system. Since Eq. 6is alreadyconstructed in terms of a modal decomposition of the viscoelasticstiffness matrix K

v

, that is, in terms of viscoelastic dissipativemodes, one could consider retaining only a few dissipative modesto reduce the augmented system dimension. Obviously, the diffi-culty would be to guess which dissipative modes to retain.

Let us suppose that the damped solution for the FE dof may bewritten as q = Teq . Replacing this expression in Eq. 6leads to

TeTMTeq

+TeTDTeq +Te

TKe+KvTeq Te

TTddi

q id=Te

TF

13

Ci

i

dq id+Cidq i

ddTdTTeq= 0 14

Since the null eigenvalues were eliminated from d, Eq. 14could also be written as

q id= iq i

d+ i

CiTd

TTeq 15

Notice from the last equation that the jkth element of the matrixTd

TTerepresents the contribution of the kth response mode to thejth dissipative mode of the viscoelastic substructure, that is, ameasure of how the kth response mode excites the j th mode of theviscoelastic substructure. Consequently, supposing that the energyof the overall response is concentrated in certain responsemodes, we might be able to identify the dissipative modes whichare the most excited by the response. Alternatively, from Eq.13,one may notice that the elements of matrix Te

TTddalso give ameasure of how each viscoelastic dissipative mode contributes tothe structural response. Hence, a technique was tested to selectsome viscoelastic dissipative modes based on their contribution tothe dynamics of the overall structure.

Let us define the matrix R as

R=dTdTTe 16

such that its elements Rjk represent the weighted residuals be-tween viscoelastic dissipative mode Td

j and response mode Tek.

Supposing that the majority of structural response energy is con-tained in the Nk modes in Te, the selection of the dissipativemodes that contribute the most to the structural response may beperformed through the sorting of the following residual vector r

rj=Rjk, for k Nk 17

Notice that each element ofrcorresponds to a column ofTd,that is a viscoelastic dissipative mode. Thus, it is proposed toeliminate the dissipative modes from Td corresponding to thesmallest residualsrj, which are thought to be those that contributethe least to the structural response. This is done through the fol-lowing decomposition

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d= dr 00 dn

; Td=TdrT dn 18where drcontains the eigenvalues of the Njdissipative modeswith the largest residualsrj.Tdrcontains the corresponding eigen-vectors, that are the dissipative modes to be retained in the model.The other dissipative modes Tdn and their corresponding eigen-values dn are then neglected. Therefore, the dissipative dof isapproximated byq i

d Tdrq idr. The reduced modal matrix T drcon-

tains thus only the retained dissipative modes and q idr are their

corresponding coordinates. Since the eigenvalues matrix is alsoreduced to dr, the residual matrix becomes R r=drTdrT Te.

Two main factors determine the performance of the proposedreduction technique: 1 the basis considered for the structuralresponse Te and 2the number of dissipative modes kept in themodel. As for the basis considered, let us suppose as a first ap-proximation that the damped modes are similar to the undampedmodes. Then, assuming that Te

TMTe =I and TeTKe +Kv

Te =e,Eqs.13and 14can be rewritten as

q +TeTDTeq +eq Rr

Ti

q idr =Te

TF 19

Ci

idrq i

dr

+Cidrq idr

Rrq=0 20

The combination of Eqs.19and 20leads then to a reduced-order augmented system

Mq +Dq +Kq=F 21

with

M = I 00 0

; D= TeTDTe 00 Ddd

; F= TeTF0

K

=

e K

ed

KedT Kdd ; q= colq ,q 1

dr, .. . ,q ndr

where

Ddd= diag C11dr

Cn

n

dr;

Kdd= diagC1drCndr ; Ked=RrT Rr

T

Notice that the structural model could be, but is not, reducedusing its undamped modes Te, although writing the equations interms ofq instead ofq has some advantages, such as to provide adiagonal structural model, specially if damping matrix D is a pro-

portional damping suchD

will be a diagonal matrix. However, thesame technique for reducing the dimension of the dissipative sys-tem could still be used with a nondiagonal structural model in Eq.21. Notice also, from Eq. 21, that the reduced dissipative co-ordinatesq i

dr contain now only those coordinates corresponding tothe selected dissipative modes according to their residual and,thus, matrices Rrand drhave a reduced dimension. This reduc-tion can be specially important since each eliminated dissipativemode leads to a reduction of n dof in Eq. 21, where n is thenumber of ADF series terms considered generally at least three.

The state space system matrices and vectors of Eq. 8can thenbe rewritten as

A= 0 0 0 I

1

C1Tdr

T Te 1I 0 0

0

n

CnTdr

T Te 0 nI 0

e RrT Rr

T TeTDTe

x=

q

q ; p= 0

TeTF ; C=C

qTe C qTe

These, now reduced, state space system matrices can then befurther reduced through the state space model reduction presentedpreviously, saving a large amount of computation effort. In thenext section, the technique presented here is validated for a can-tilever beam with viscoelastic treatment. Also an analysis of thenumber of dissipative modes that should be kept in the model ispresented.

5 Validation of Reduced-Order Models

Let us consider the aluminum cantilever beam partially coveredwith a constrained layer treatment as presented in Fig. 1. The

beam is of length 300 mm and thickness 1 mm and is made ofaluminum with Youngs modulus 70 GPa and mass density2700 kg/m3. No viscous damping is considered in this example,that isD =0. The constraining layer is also made of aluminum andhas thickness 0.5 mm and length 270 mm, that is the treatmentcovers 90% of the beam and is centered. The viscoelastic layerhas thickness 0.254 mm10 miland is made of 3M ISD112 vis-coelastic material, with a mass density of 1000 kg/m3. The vis-coelastic material shear modulus is frequency dependent. Thecurve fitting of ADF parameters to the measured shear modulusprovided by 3M is presented in the next section.

5.1 Curve Fitting of Viscoelastic Material Properties.TheADF parameters G, Ci and i needed to build system 8 arebased on ADF relaxation function parametersG0, iand i, from

Eq. 4, which must be curve fitted relative to the measurementsofG*. In the present work, a nonlinear least squares optimiza-tion method was used to evaluate the ADF parameters. Figure 2shows the measured and approximated storage modulus Gandloss factor for 3M ISD112 viscoelastic material at 20C,where

G*=G+ jG=G1 + j 22

As shown in Fig. 2, both storage modulus and loss factor are wellrepresented by five series terms of ADF parameters, whereas threeADF series terms provide only a first approximationwithin 15%error margin for the frequency dependence. Nevertheless, theseparameters are valid only in the frequency range considered, thatis, the frequency range for which material properties were fur-nished by 3M. Therefore, it is necessary to ensure a reasonable

behavior of estimated material properties outside the frequencyrange, since arbitrary external perturbations will generally excitemodes lying on this interval. Required asymptotical properties are

Fig. 1 Cantilever beam partially covered with a passive con-strained layer damping treatment


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lim0

G*=G0

limG*

=G

, whereGG0R

+ 23

Meaning that the shear modulus tends to its static relaxed andinstantaneous unrelaxed values at the boundaries 0 and , re-spectively. This also imposes that 0 ,=0, that is, dissipa-tion only occurs in the transition region. Curve-fitted ADF param-eters for viscoelastic material 3M ISD112 at 20C respectingasymptotical behavior are presented in Table 1.

Notice also that these properties are valid for a temperature of20C and it is well known that temperature decrease will movethese master curves to the left and vice versa. On the other hand,some viscoelastic materials present optimal loss factor at lowerfrequencies. So that it is normally possible to select a materialaccording to frequency range of interest and operation tempera-ture.

5.2 Comparison of Reduced- and Full-Order ViscoelasticModels.The reduced state space systems, with and without pre-vious reduction of the dissipative system, are now compared forthe cantilever beam introduced previously. This is done using a FEmodel considering 35 sandwich beam elements, with 6 dof perelement, thus leading to a total of 105 mechanical doffor detailson the FE model, please refer to11. Five ADF series terms wereconsidered in both cases, leading to an inclusion of 445 dissipa-tive dof 89 dof 5 ADF series terms in addition to the 105mechanical dof from the FE model.

Figures 3 and 4 present the eigenfrequency and modal dampingfactor errors, respectively, when using different numbers of dissi-pative modes prior to state space modal reduction compared tousing all but rigid body dissipative modes. These selected dissi-pative modes correspond to the ones with largest residuals. Com-

parison of Figs. 3 and 4 shows that eigenfrequency errors are

much smaller than damping factors errors. This is probably due tothe fact that the dissipative coordinates are solely responsible forthe damping in the structure and neglecting all dissipative modesleads to the absence of damping. Moreover, although the dampingfactor error decreases quite rapidly for modes 510, it isstill larger than 20% for modes 1 and 3 when using less than20 dissipative modes. Nevertheless, when using 30 dissipa-tive modes the damping factor errors decrease to0.06,0.08,0.09,0.11,0.12,0.14,0.16,0.19,0.24,0.31 %, respec-tively, while the maximum eigenfrequency error is as low as0.01%.

From Figs. 3 and 4, one may notice that both eigenfrequencyand damping factor errors decay is achieved through a series oflarge steps. This can be explained by the fact that the dampingfactor of a specific vibration mode will be well represented onlywhen a set of dissipative modes with strong coupling with thisspecific vibration mode is included. Since dissipative modes are

sorted in terms of their weighted residuals, the inclusion of a fewfirst dissipative modes with weak coupling with one or more vi-bration modes will not affect their damping factors. For example,

Fig. 2 Frequency dependence of 3M ISD112 viscoelastic ma-terial properties at 20C solid line and curve fit using threedashed lineand five dashed-dotted lineADF series terms

Table 1 Curve-fitted ADF parameters for viscoelastic material3M ISD112 at 20C

Fig. 3 Eigenfrequency error when using different numbers ofdissipative modes in second-order system compared to usingall but rigid body modes

Fig. 4 Damping factor error when using different numbers ofdissipative modes in second-order system compared to usingall but rigid body modes


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it seems for this example that the coupling between the 26 firstsorted dissipative modes and the first vibration mode is not verystrong but it is for the other vibration modes. That is why the

damping factor error for the first vibration mode is reduced from91% to 1% when the 27th dissipative mode is included Fig. 4.On the other hand, the damping factor error for the tenth vibrationmode is decreased gradually by the inclusion of a set of dissipa-tive modes.

Table 2 shows the eigenfrequencies and damping factors fordifferent levels of reduction of dissipative modes. It is clear thatthe reduction to 29 dissipative modesof the 89 availableleads toan accurate representation of the eigenfrequencies and dampingfactors. Indeed, from Fig. 5, one can observe that the 31st largestresidual is only 0.6% of the first one so that most of the couplingbetween elastic and dissipative coordinates is provided by the first30 dissipative modes. Alternatively, one may also observe, fromFig. 5, that the cumulative sum of the normalized residuals ismore than 99% for 30 dissipative modes.

Figure 6 shows the frequency response function between theimpact force input and displacement output, both colocated at10 mm from the clamped end, using all but rigid body dissipativemodes, as a reference, and reduced-order models using only 9, 19and 29 dissipative modes of the 89 available. It is possible to

observe that higher-frequency modes are better represented bylow-order models as previously shown in frequency and dampingerrors analyses. It can be seen, however, that, when using only

nine dissipative modes, the frequency response around the first,second, and fourth eigenfrequencies is not correctly represented.However, when including 19 dissipative modes, the differencebetween the reduced-order model and the full-dissipative model isalmost only perceptible around the first eigenfrequency. The fre-quency response for the reduced-order model with 29 dissipativemodes matches almost exactly the full-dissipative model.

As it is guessed that the importance of the dissipative coordi-nates in the representation of damping may be dependent on theoverall damping level induced in the structure by the viscoelasticdamping treatment, a similar analysis was performed for a canti-lever beam with only 50% of area covered with the viscoelastictreatment. This is done by changing the length of the viscoelasticand constraining layers to 150 mm in Fig. 1, whereas still cen-tered in the beam surface. This leads to a much less damped

structure, such that the ten first modal damping factors are50%= 5.6,9.2,5.3,5.2,6.7,5.3,4.7,4.7,3.8,2.8% comparedto 90%= 5.8, 11.2,11.7, 11.8,11.6, 10.8,9.4,7.8,6.3,5.1%

Table 2 Eigenfrequencies and damping factors for different levels of reduction

Fig. 5 Normalized residual and cumulative sum of residualsfor the viscoelastic dissipative modes

Fig. 6 Frequency response function using different numbersof dissipative modes in second-order system. : all but rigidbody modes 89, : 9 modes, - - -: 19 modes, .: 29 modes


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of the previous configuration.Similarly to the previous case, Figs. 7 and 8 present the eigen-

frequency and modal damping factor errors, respectively, whenusing different numbers of dissipative modes compared to usingall but rigid body dissipative modes. For this case, the eigenfre-quency errors are also much smaller than that for the dampingfactors. Here, both the eigenfrequency and damping factor errorsdecrease more rapidly than in the previous case. It may be guessedthat this is due to the fact that a less damped structure requires lessdissipative coordinates. Indeed, in this case, only 18 dissipativemodes are required to reduce the damping error to less than1% for all ten first vibration modes. Indeed, when using 18dissipative modes, the damping factor errors are0.75,0.87,0.66,0.01,0.14,0.28,0.20,0.10,0.17,0.36 %, respec-tively, and the maximum eigenfrequency error is 0.07%. Also, asin the previous case, the cumulative sum of the normalized residu-als is more than 99%.

For the case of 50% coverage, the frequency response functionwas also analyzed, and it is shown in Fig. 9, using all but rigidbody dissipative modes, as a reference, and reduced-order models

using only 5, 9, and 18 dissipative modes of the 52 available. Forthis case, the higher-frequency modes are also better represented

by low-order models and when using a reduced-order model with18 dissipative modes the frequency response function matchesalmost exactly the full-dissipative model.

Although, for a less damped structure, less dissipative modeswere necessary to represent correctly its viscoelastic damping, it isworthwhile noticing that, for the structure with smaller coverage,there are less dissipative coordinates to be reduced. This is due tothe fact that there are more rigid body dissipative modes, corre-sponding to the mechanical dof of the beam uncovered areas.Notice, however, that in both cases it is possible to reduce thenumber of dissipative modes to approximately one third of allnon-rigid body ones. Since five ADF series terms were necessaryto correctly represent the frequency dependence of the viscoelasticmaterial, that reduction represents a gain of 300 dof reductionfrom 655 to 355 dof in the state space system for the 90% cov-erage case. Since the calculation of the eigenvalues of the statespace matrix requires a number of operations approximately equalto N3, where Nis the matrix size, this reduction would lead to areduction in 84% of computational effort.

6 Conclusions

The present work has presented an alternative reduction methodfor internal variables-based viscoelastic finite element models. Apreviously developed sandwich/multilayer beam finite elementmodel combined with the internal variables-based anelastic dis-placement fields viscoelastic model was used. Through a physicalinterpretation of the dissipative modes, due to the added internalvariables, and their coupling with structural vibration modes in thesecond-order model, a technique for the reduction of the extradissipative coordinates before transformation to the state space

system was proposed. This method has led to much faster com-putations on the reduction of the state space system. Comparisonbetween the reduced-order and full-dissipative models for aclamped beam with passive constrained layer damping treatmenthas shown satisfactory results. In particular, a reduction of 67% ofdissipative dof has led to errors smaller than 0.5% for dampingfactors of the coupled structure. Similar results were obtained forthe two examples considered with different treatment lengths and,thus, different levels of modal damping factors. A similar reduc-tion can be expected for more complicated structures, although itis worthwhile to notice that the full state space system dimensiondepends on the number of both mechanical and dissipative dof.Hence, for structures with a large number of mechanical dof, the

Fig. 7 Eigenfrequency error when using different numbers ofdissipative modes in second-order system compared to usingall but rigid body modes

Fig. 8 Damping factor error when using different numbers ofdissipative modes in second-order system compared to usingall but rigid body modes

Fig. 9 Frequency response function using different numbersof dissipative modes in second-order system. : all but rigidbody modes 52, : 5 modes, - - -: 9 modes, .: 18 modes


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second-order structural model should also be reduced prior totransformation to a state space model. The effect of also reducingthe structural model, using the undamped vibration modes, on thecorrect representation of viscoelastic damping will be studied in afuture work.

References1 Garg, D. P., and Anderson, G. L., 2003, Structural Damping and Vibration

Control Via Smart Sensors and Actuators, J. Vib. Control, 9, pp. 14211452.2 Trindade, M. A., and Benjeddou, A., 2002, Hybrid Active-Passive Damping

Treatments Using Viscoelastic and Piezoelectric Materials: Review and As-sessment, J. Vib. Control, 86, pp. 699746.

3 Lesieutre, G. A., and Bianchini, E., 1995, Time Domain Modeling of LinearViscoelasticity Using Anelastic Displacement Fields, ASME J. Vibr. Acoust.,1174, pp. 424430.

4 Golla, D. F., and Hughes, P. C., 1985, Dynamics of ViscoelasticStructuresA Time-Domain, Finite Element Formulation, ASME J. Appl.Mech., 524, pp. 897906.

5 McTavish, D. J., and Hughes, P. C., 1993, Modeling of Linear ViscoelasticSpace Structures, ASME J. Vibr. Acoust., 115, pp. 103110.

6 Trindade, M. A., Benjeddou, A., and Ohayon, R., 2000, Modeling ofFrequency-Dependent Viscoelastic Materials for Active-Passive VibrationDamping, ASME J. Vibr. Acoust., 1222, pp. 169174.

7 Friswell, M. I., and Inman, D. J., 1999, Reduced-Order Models of StructuresWith Viscoelastic Components, AIAA J., 3710, pp. 13181325.

8 Park, C. H., Inman, D. J., and Lam, M. J., 1999, Model Reduction of Vis-coelastic Finite Element Models, J. Sound Vib., 2194, pp. 619637.

9 Trindade, M. A., Benjeddou, A., and Ohayon, R., 2001, Piezoelectric ActiveVibration Control of Sandwich Damped Beams, J. Sound Vib., 2464, pp.653677.

10 Friot, E., and Bouc, R., 1996, Contrle Optimal par Rtroaction du Rayon-nement dune Plaque Munie de Capteurs et dActionneurs Pizo-lectriques

Non Colocaliss, 2eme Colloque GDR Vibroacoustique, LMA, Marseille pp.229248.

11 Trindade, M. A., Benjeddou, A., and Ohayon, R., 2001, Finite Element Mod-eling of Hybrid Active-Passive Vibration Damping of Multilayer PiezoelectricSandwich Beams. Part 1: Formulation and Part 2: System Analysis, Int. J.Numer. Methods Eng., 517, pp. 835864.


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