Research Article Dynamic Analysis of Three-Layer Sandwich...

Research ArticleDynamic Analysis of Three-Layer Sandwich Beams with ThickViscoelastic Damping Core for Finite Element Applications

Fernando Corteacutes1 and Imanol Sarriacutea2

1Deusto Institute of Technology (DeustoTech) Faculty of Engineering University of DeustoAvenida de las Universidades 24 48007 Bilbao Spain2Faculty of Engineering University of Deusto Avenida de las Universidades 24 48007 Bilbao Spain

Correspondence should be addressed to Fernando Cortes fernandocortesdeustoes

Received 12 October 2014 Revised 15 January 2015 Accepted 23 February 2015

Academic Editor Ahmet S Yigit

Copyright copy 2015 F Cortes and I Sarrıa This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper presents an analysis of the dynamic behaviour of constrained layer damping (CLD) beams with thick viscoelastic layerAhomogenisedmodel for the flexural stiffness is formulated usingReddy-Bickfordrsquos quadratic shear in each layer and it is comparedwith Ross-Kerwin-Ungar (RKU) classical model which considers a uniform shear deformation for the viscoelastic core In order toanalyse the efficiency of both models a numerical application is accomplished and the provided results are compared with those ofa 2Dmodel using finite elements which considers extensional and shear stress and longitudinal transverse and rotational inertiasThe intermediate viscoelastic material is characterised by a fractional derivative model with a frequency dependent complexmodulus Eigenvalues and eigenvectors are obtained from an iterative method avoiding the computational problems derived fromthe frequency dependence of the stiffness matrices Also frequency response functions are calculated The results show that thenew model provides better accuracy than the RKU one as the thickness of the core layer increases In conclusion a new model hasbeen developed being able to reproduce the mechanical behaviour of thick CLD beams reducing storage needs and computationaltime compared with a 2D model and improving the results from the RKU model

1 Introduction

In the last yearsmany studies have been presented concerningthe structural vibration reduction making use of passivedamping control techniques by means of surface treatmentswith viscoelastic materials A survey of different subjectson viscoelastic treatments can be found in [1] This kindof vibration control technique is largely used nowadaysfor several industrial applications such as aeronautical andautomotive components The free layer damping (FLD) andconstrained layer damping (CLD) technologies are two ofthese viscoelastic surface treatments consisting of adding adamping viscoelastic layer to the structural system Specif-ically FLD consists of adding that viscoelastic layer on avibratingmetallic base and the configuration can be analyzedas the flexural behavior of a two-layer beam In this contextOberst and Frankenfeldrsquos model [2] is traditionally usedThis model assumes that the layers are thin and the shear

effect is negligible In one of the authorsrsquo works [3] a modelwas presented in which shear effects are taken into accountimproving the results given by Oberst and Frankenfeldrsquosmodel for thick layers On the other hand CLD technolo-gies are based on a viscoelastic core working under sheardeformation In this configuration this damping viscoelasticmaterial is in the core of a three-layer sandwich structureThe other two layers are vibrating structural elements usuallymade of metallic materials subjected to flexural moment andaxial and shear loads The effect of this shear load is moreimportant as the thickness of the layers increases Comparedwith other damping techniques this viscoelastic treatmentpresents the disadvantage of adding mass into the originalsystem but offers a better damping-weight ratio than the freelayer damping configuration [4 5]

In Figure 1 the cross section of a three-layer composedbeam is represented the thickness of the base viscoelasticand constraining layers being 119867

1 1198672 and 119867

3 respectively

Hindawi Publishing CorporationShock and VibrationVolume 2015 Article ID 736256 9 pageshttpdxdoiorg1011552015736256

2 Shock and Vibration

H1

H2

H3

hn

b

Damping layer

Constraining layer

Neutral axis

Base beam

Figure 1 Cross section of a three-layer sandwich beam with aviscoelastic damping layer

and 119887 being the width of the beam Although in CLD beams1198672

≪ 1198671 in this work three-layer sandwich beams with

any value for 1198672are considered The properties for these

elastic materials are Youngrsquos moduli 1198641and 119864

3 Poissonrsquos

coefficients ]1and ]3 and densities 120588

1and 1205883 whereas those

of the viscoelastic core are 119864lowast

2 ]2 and 120588

2 respectively If the

materials are considered isotropic the shear moduli 1198661 119866lowast2

and 1198663are related to the extensional ones by 119866

1= 11986412(1 +

]1) 119866lowast2= 119864lowast

22(1 + ]

2) and 119866

3= 11986432(1 + ]

3) for the three

layersThe complex nature of themodulus of viscoelasticmateri-

als represented by the asterisk as (sdot)lowast is due to their ability todissipatemechanical energy A complex relationship betweenstress and strain allows representing the hysteretic behaviourof this kind of materials [6] The dissipative property may bedescribed by the loss factor 120578

2

1205782=

11986410158401015840

2

1198641015840

2

(1)

where 1198641015840

2and 119864

10158401015840

2are the real and imaginary components

of the complex modulus 119864lowast

2= 119864

1015840

2+ 11989411986410158401015840

2 which are

known as the storage and loss modulus respectively theformer being also frequently represented by119864

2 For polymers

both storage and loss moduli and consequently loss factorvary with temperature and frequency in terms of threedifferent behaviours rubbery vitreous and transition (seeeg [7]) In this sense Jones [8] reviews experimental datafor complex modulus of some typical viscoelastic materialsincluding elastomers adhesives and specially compoundedmaterials The experimental characterisation of low stiffnessdampingmaterials which are not appropriate to prepare self-supporting test specimens may be achieved by means of theASTM E 756-05 standard [9] wherefrom the nomenclaturehas been taken

The classical RKU [10] model is one of the first modelsdeveloped in order to calculate the flexural stiffness forsandwich beams with viscoelastic core and is one of the mostused models nowadays although it may lose accuracy inseveral cases for example when rotational or extensionalinertias are not negligible and specifically when a thickviscoelastic core in a CLD beam is studied This is due to theassumptions which this model takes

(i) The shear deformation in the elastic base and con-straining layers is considered to be negligible

(ii) The longitudinal direct stress in the viscoelastic layeris negligible

(iii) The shear stress in the viscoelastic layer is assumed tobe uniform

(iv) The plane cross section in each layer remains planeafter deformation

There are some other models beyond these restrictions suchas [11ndash18] or more recently [19 20] but they are not socomputationally efficient for finite elements applications Infact RKU is used in most engineering applications basicallydue to its very easy computational implementation AlsoRKU strictly applies to simply supported beams although itis much more generally used for other boundary conditionssuch as the fixed supportWe will refer to the results obtainedas RKU results although RKU theory for simply supportedbeams has been used in this work for a cantilever beam in thecontext of a finite element analysis with displacement fieldsrestricted to those of the RKU theory

In short following a similar approach as for the study ofthe FLD configuration for thick beams previously mentioned[3] this work is aimed at developing a new model improvingthe accuracy of the RKU model for three-layer sandwichbeams with a thick viscoelastic core but maintaining itscomputational benefit Thus an equivalent complex flexuralstiffness is derived considering quadratic shear stress basedon Reddy-Bickfordrsquos theory In order to prove the improve-ment achieved by the new model a numerical applicationfor a cantilever beam is presented comparing the solutionsprovided by three different finite elementmodels a 2Dmodel(whose results are considered to be the reference ones) andtwo 1D models based on the RKU theory and the newone respectively The damping material is characterised bymeans of a fractional derivative model involving the fre-quency dependence of the complex modulus which impliesimportant disadvantages for the dynamic analysis Thus theextraction of the eigenvalues and eigenvectors is carried outby a simple and effective iterative algorithm [21] Finally thedynamic response of the three models is compared in termsof the frequency response function

As a result the new model is able to reproduce themechanical behaviour of three-layer sandwich beams reduc-ing storage needs and computational time compared with a2Dmodel and improving the results provided by the classicalRKU model

2 Homogenised Model for a CLD Beam

Next the theoretical study of a three-layer sandwich beamis presented in which quadratic shear stress is taken intoaccount An equivalent flexural stiffness will be deduced forpinned-pinned beams In order to simplify the notation itis assumed that the behaviour of all materials is linear andelastic (ie all the magnitudes are real) and the complexcharacter of themodulus in the viscoelastic core will be takeninto account in the numerical examples of the subsequentsections This substitution of a complex modulus into anelastic solution is usually known as the ldquocorrespondenceprinciplerdquo (see any standard text on viscoelasticity eg [22])

Shock and Vibration 3

The equivalent flexural stiffness 119861eq may be obtained bythe addition of the individual contribution of each layer

119861eq = 1198611+ 1198612+ 1198613 (2)

where 1198611= 11986411198681 1198612= 11986421198682 and 119861

3= 11986431198683are the complex

flexural stiffness of the three layers and 1198681 1198682 and 119868

3are the

complex cross-sectional second order moments computedwith respect to the neutral axis given by

1198681=

1

121198871198673

1+ 1198871198671(ℎ119899minus

1198671

2)

2

(3)

1198682=

1

121198871198673

2+ 1198871198672(1198671+

1198672

2minus ℎ119899)

2

(4)

1198683=

1

121198871198673

3+ 1198871198673(1198671+ 1198672+

1198673

2minus ℎ119899)

2

(5)

respectively in which the complex position of the neutral axisis represented by

ℎ119899=

11986411198672

12+11986421198672(1198671+11986722)+119864

31198673(1198671+ 1198672+ 11986732)

11986411198671+ 11986421198672+ 11986431198673

(6)

Following the same methodology as in [3] and decouplingthe transverse displacement of any cross section in a termdue to the flexural moment and in another term derived bythe shearing force (see any book of strength of materials fordetails eg [23]) the equivalent shear stiffness of the crosssection can be found to be decomposed as

1

119870eq=

1

1198701

+1

1198702

+1

1198703

(7)

For the geometry represented in Figure 1 the stiffness of 1198701

1198702 and119870

3of the individual layers satisfies

1

1198701

=1

11986611198612eq119887

int

minusℎ119899+1198671

minusℎ119899

Ω2

1(119910) d119910 (8)

1

1198702

=1

11986621198612eq119887

int

minusℎ119899+1198671+1198672

minusℎ119899+1198671

Ω2

2(119910) d119910 (9)

1

1198703

=1

11986631198612eq119887

int

minusℎ119899+1198671+1198672+1198673

minusℎ119899+1198671+1198672

Ω2

3(119910) d119910 (10)

respectively where

Ω2

1(119910) =

11988721198642

1(ℎ2

119899minus 1199102)2

4

(11)

Ω2

2(119910) = 119887

2

11986411198671(ℎ119899minus

1198671

2) +

1198642[(ℎ119899minus 1198671)2

minus 1199102]

2

2

(12)

Ω2

3(119910) =

11988721198642

3[(1198671+ 1198672+ 1198673minus ℎ119899)2

minus 1199102]2

4

(13)

respectively In these equations 119861eq is the flexural stiffnessgiven by (2)

By solving these integrals it yields

1

1198701

=6

511986611198871198671

(101199032

119899minus 15119903119899+ 6)

(1 + 11990321198722+ 11990331198723)2

(1 + 3 (119903119899minus 1)2

)2

(14)

1

1198702

=1

11986621198871198671

sdot361198792(119903119899minus 1)2

minus 1211987221198792

2(119903119899minus 1) (2119879

2minus 3119903119899+ 6)

(1 + 11990321198722+ 11990331198723)2

(1 + 3 (119903119899minus 1)2

)2

+6

511986621198871198672

sdot

1198722

21198794

2(101199032

119899minus 151199031198991198792minus 40119903119899+ 61198792

2+ 30119879

2+ 40)

(1 + 11990321198722+ 11990331198723)2

(1 + 3 (119903119899minus 1)2

)2

(15)

1

1198703

=61198722

31198794

3

511986631198871198673

times (101199032

119899minus 251199031198991198793minus 401199031198991198792minus 40119903119899+ 16119879

2

3

+5011987931198792+ 50119879

3+ 40119879

2

2+ 80119879

2+ 40)

sdot ((1 + 11990321198722+ 11990331198723)2

(1 + 3 (119903119899minus 1)2

)2

)

minus1

(16)

respectively where 1198722

= 11986421198641 1198723

= 11986431198641 1198792

= 1198672

1198671 1198793

= 11986731198671119903119899

= 2ℎ1198991198671 1199032

= 11986821198681 and 119903

3= 1198683

1198681 For base and constraining layers made of metallic mate-

rials which are much thinner and more rigid than theviscoelastic layer the terms 1119870

1and 1119870

3may be neglected

with respect to the coefficient of the polymeric layer 11198702

It can be pointed out that if the three layers were composedof the same material (14) provides the well-known result forhomogeneous rectangular sections

1

119870eq=

6

5119866119878 (17)

where 119878 is the total cross-sectional areaBy considering the shear coefficient119870eq the flexural field

equation in free vibration is given by Timoshenkorsquos formula[24]

119861eq

120588ℓ

1205974V (119909 119905)1205971199094

minus

119861eq

119870eq

1205974V (119909 119905)12059711990921205971199052

+1205972V (119909 119905)1205971199052

= 0 (18)

where 120588ℓis the mass per unit length In (18) the rotational

inertia has been neglected When the shear stiffness tends toinfinity (18) degenerates on the well-known Euler-Bernoullifield equation Following the samemethodology as in [3] theequivalent flexural stiffness 119861

119870 considering shear effects and


satisfying the homogenised Euler-Bernoulli field equationcan be found to be

119861119870

=

119861eq

(radic1 + 1205932(120596) + 120593(120596))

2 (19)

The function 120593(120596) takes into account the shear effects and isgiven by

120593 (120596) =

120596radic120588ℓ119861eq

2119870eq (20)

In order to compare RKU and the new model it can benoted that for the static values 120596 rarr 0 both models givethe same result the equivalent 119861eq However as a differencewhen frequency tends to infinity 120596 rarr infin the stiffness ofthe present model tends to zero whereas that of the RKU onetends to a finite value

These two models have another common property whenshear stiffness 119870eq tends to infinity that is when sheardeformations are negligible In this case the function 120593(120596)

tends to zero and the equivalent flexural stiffness 119861119870tends to

the classic 119861eq

3 Dynamic Analysis ofa Three-Layer Sandwich Beam UsingFinite Element Procedures

31 Problem Definition In this section the harmonic analysisof a three-layer sandwich beam in aCLDconfigurationwill becompleted using finite element procedure techniques Threedifferent thicknesses of the viscoelastic core layer will be stud-ied so as to evaluate the accuracy of the homogenizedmodelscompared to a 2Dmodel whose solutionwill be considered asexact considering the nonexistence of experimental results

The length of the beam is ℓ = 200mm the width is 119887 =

20mm the thickness of the base and constraining metalliclayers is119867

1= 1198673= 1mm and for the viscoelastic layer119867

2=

1 5 and 10mm is chosenThe properties of the materials are taken from the

experimental characterisation effectuated by Cortes and Ele-jabarrieta [25] on AISI T 316 L stainless steel laminated sheetand on Soundown Vibration Damping Tile material [26]Indeed Youngrsquos modulus and density of the elastic materialare 1198641= 1762 times 10

9 Pa and 1205881= 7782 kgm3 respectively

and density of the damping material is 1205882

= 1423 kgm3Poissonrsquos coefficient ]

1= ]3

= 03 is chosen for the elasticmaterials and ]

2= 045 is chosen for the viscoelastic

material The experimental data of the storage modulus andloss factor for the viscoelastic core layer were fitted to a four-parameter fractional model [27 28] given by

119864lowast

2(120596) =

119864119903+ 119864119906(119894120591120596)120572

1 + (119894120591120596)120572

(21)

where 119864119903and 119864

119906represent the relaxed and unrelaxed modu-

lus respectively 120591 is the relaxation time and120572 is the fractionalparameter The parameter values are summarised in Table 1

Table 1 Parameters of the fractional derivative model

119864119903(GPa) 119864

119906(GPa) 120591 (10minus6 s) 120572

0353 3462 3149 0873

F

u120001

Figure 2 Finite element model for the 2D dynamical analysis of asandwich beam

The dynamic behaviour of the three-layer sandwich beamin a CLD configuration is studied on the basis of threedifferent finite element models The first of them is a 2Dmodel discretised in bilinear quadrilateral elements with fournodes under plane-stress assumption (see eg [29ndash31] fordetails about finite element formulations) All the three layersare modelled with 4 elements along thickness to assure thecontinuous evolution of the shear stress and with 60 elementsalong the length (see Figure 2) in order to obtain the firstthree eigenvalues accurately enough it has been checked thatthis number of finite elements is enough to get convergencefor any of the results shown in the tables

The consistent mass matrix and the stiffness matrixobtained using reduced integration with Kosloff and Frazier[32] hourglass control are summarised as follows

MassM and Stiffness KMatrices for the 2D Model Quadrilat-eral Finite Elements See Figure 3

Consistent Mass MatrixM Consider

120588119886119887119905

36

[[[[[[[[[[[[[[[[[[[[[[[

[

4 0 2 0 1 0 2 0

4 0 2 0 1 0 2

4 0 2 0 1 0

4 0 2 0 1

4 0 2 0

4 0 2

4 0

4

]]]]]]]]]]]]]]]]]]]]]]]

]

(22)

Stiffness Matrix K Obtained by Reduced Integration withHourglass Control Consider


a

b

1 2

34

Thickness tAspect ratio 120574 = ba

Material properties E 120588 and

Figure 3 Rectangular Finite Element (Plane-Stress Assumption)

119864119905

24120574 (1 minus ]2)

times

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

3 (1 minus ]) + 21205742 (4 minus ]2) 3120574 (1 + ]) 3 (1 minus ]) minus 21205742 (4 minus ]2) minus3120574 (1 minus 3]) minus3 (1 minus ]) minus 21205742 (2 + ]2) minus3120574 (1 + ]) minus3 (1 minus ]) + 21205742 (2 + ]2) 3120574 (1 minus 3])

31205742

(1 minus ]) + 2 (4 minus ]2) 3120574 (1 minus 3]) minus31205742

(1 minus ]) + 2 (2 + ]2) minus3120574 (1 + ]) minus31205742

(1 minus ]) minus 2 (2 + ]2) minus3119887 (1 minus 3]) 31205742

(1 minus ]) minus 2 (4 minus ]2)

3 (1 minus ]) + 21205742 (4 minus ]2) minus3120574 (1 + ]) minus3 (1 minus ]) + 21205742 (2 + ]2) minus3120574 (1 minus 3]) minus3 (1 minus ]) minus 21205742 (2 + ]2) 3120574 (1 + ])

31205742

(1 minus ]) + 2 (4 minus ]2) 3120574 (1 minus 3]) 31205742

(1 minus ]) minus 2 (4 minus ]2) 3120574 (1 + ]) minus31205742

(1 minus ]) minus 2 (2 + ]2)

3 (1 minus ]) + 21205742 (4 minus ]2) 3120574 (1 + ]) 3 (1 minus ]) minus 21205742 (4 minus ]2) minus3120574 (1 minus 3])

31205742


(1 minus ]) + 2 (2 + ]2)

3 (1 minus ]) + 21205742 (4 minus ]2) minus3120574 (1 + ])

31205742

(1 minus ]) + 2 (4 minus ]2)

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(23)

Extensional and shear strain and transverse extensionaland rotational inertias are considered in this 2D finiteelementmodel which is then able to reproduce the dynamicalbehaviour of the three-layer sandwich beam with any thick-ness of the viscoelastic core

The two othermodels are 1Dbeammodelswhose 119894thmassM119894and stiffness K

119894matrices are

M119894=

(12058811198671+ 12058821198672+ 12058831198673) 119887119886119894

420

sdot

[[[[[

[

156 22119886119894

54 minus13119886119894

22119886119894

41198862

11989413119886119894

minus31198862

119894

54 13119886119894

156 minus22119886119894

minus13119886119894

minus31198862

119894minus22119886119894

41198862

119894

]]]]]

]

(24)

K119894=

119861

1198863

119894

[[[[[

[

12 6119886119894

minus12 6119886119894

6119886119894

41198862

119894minus6119886119894

21198862

119894

minus12 minus6119886119894

12 minus6119886119894

6119886119894

21198862

119894minus6119886119894

41198862

119894

]]]]]

]

(25)

respectively where 119886119894is the length of the 119894th finite element

The complex flexural stiffness 119861 of (25) is given by (2) 119861 = 119861lowast

eqfor the RKUmodel and by (19)119861 = 119861

lowast

119870for the new thick beam

modelThe discretisation is alsomade with 60 finite elementsalong span

32 Extraction of Eigenvalues The equation from whichthe complex eigenvalues of the system under study can beobtained is

(minus120582lowast

119903M + Klowast (120596

119903))120601lowast

119903= 0 (26)

where 120582lowast119903and 120601lowast

119903are the complex eigenvalue and eigenvector

of the 119903th mode respectively M is the mass matrix andKlowast is the complex stiffness matrix which is dependent onfrequency This 120596

119903is the real part of the square root of the

complex eigenvalue 120582lowast119903

120596119903= Re(radic120582lowast

119903) (27)

which induces a nonlinearity into the eigenproblem Thereare several methods such as those of Lanczos [33] or Arnoldi[34] which use iterative procedures involving importantcomputational time In order to decrease this computationaleffort Cortes and Elejabarrieta [21] developed an iterativeprocedure that approximates in a simple and accurate waythe complex eigenpairThismethod begins by considering thestatic stiffness matrix Klowast(0) in (26) yielding

(minus1205821199030M + Klowast (0))120601

1199030= 0 (28)


Table 2 Modal properties of the sandwich cantilever beam with1198672= 1mm

1205961(rads) 120578

11205962(rads) 120578

21205963(rads) 120578

3

2D model 4111 0024 2523 0049 6851 0047

Present model 4122 0019 2547 0037 6912 0043

RKU model 4124 0018 2552 0034 6952 0037


1205961(rads) 120578

11205962(rads) 120578

21205963(rads) 120578

3

2D model 1003 0080 5659 0113 13987 0091

Present model 1013 0066 5909 0094 14504 0103

RKU model 1019 0052 6051 0068 15461 0063


1205961(rads) 120578

11205962(rads) 120578

21205963(rads) 120578

3

2D model 1561 0118 8106 0138 18828 0099

Present model 1589 0098 8714 0127 19779 0131

RKU model 1613 0069 9291 0076 22875 0064

and then the undamped eigensolutions 1205821199030

and 1206011199030

can beobtained Then Nelsonrsquos method [35] is used in order tocalculate the eigenvector derivative 1206011015840

119903 From this eigenvector

derivative and taking into account the variation of thecomplex stiffness for the obtained eigenfrequency

ΔKlowast (1205961199030

) = Klowast (1205961199030

) minus Klowast (0) (29)

and by means of Taylorrsquos series approach a complex finiteincrement Δ120601lowast

119903of the eigenvector is obtained The complex

eigenvector can be approximated with

120601lowast

119903= 1206011199030

+ Δ120601lowast

119903 (30)

with which the complex eigenvalue 120582lowast119903is estimated according

to

120582lowast

119903=120601lowast119867119903Κlowast (120596

119903)120601lowast119903

120601lowast119867119903

M120601lowast119903

(31)

where (sdot)119867 denotes the Hermitian transpose operator that

is the complex conjugate transposition Equations (29)ndash(31)can be iterated making use of the new eigenfrequency 120596

119903

given by (27) in order to obtain the desired convergencetolerance As a main difference with other iterative methodsthis one presents the advantage of solving only once theundamped eigenproblem and the iterations are carried outon the derivatives reducing computational resources

If damping in the system is very large the accuracy ofthe method can be improved by means of the incrementalapproach of the method as seen in [21]

Making use of this new method with the correspondingincremental approach the first three modal natural frequen-cies 120596

119903derived from (27) and loss factor derived from

120582lowast

119903= 1205962

119903(1 + 119894120578

119903) (32)

can be computed The corresponding results for the threethicknesses and for the three models under study are shownin Tables 2ndash4

It can be pointed out that the results for the naturalfrequency 120596

119903are practically the same for the thinnest beam

(see Table 2) and the differences between the present modeland the RKU one are more important as the thickness of theviscoelastic layer increases which is an expected behaviourthe reason is that the shear contribution is more importantfor larger thickness and the present model considers a morerealistic shear stress distribution Also the most importantdifferences take place at higher order modes This is becauseat higher frequencies the shear effects acquire more impor-tance and as previously mentioned the present model takesinto account shear effects in a more effective way

Specifically for the third mode of the beam with aviscoelastic layer thickness equal to 5mm (see Table 3) thepresentmodel improves the RKU result in a 68 (from 105down to 37)This improvement is evenmore important forthe beamwith 10mmof viscoelastic layer (see Table 4) wherethe difference between the present model and the RKU onewith respect to the 2Dmodel goes up to 165 (from 215 to50)

As for the results of the modal loss factor 120578119903 an erratic

behaviour in both RKU and thick beammodels can be notedIt should be highlighted that this parameter cannot be directlycompared because the damping of the viscoelastic materialrepresented by the loss factor 120578

119903depends on frequency

according to (21) and the natural frequencies for the modelsare not the same Instead of modal loss factor 120578

119903 the

amplitudes of the resonance peaks will be compared in thenext section


Disp

lace

men

t (m

)10minus2

10minus4

10minus6

10minus8

Frequency (kHz)0 05 1 15 2 25 3 35 4

2D modelPresent modelRKU model

(a)

Disp

lace

men

t (m

) 10minus4

10minus6

10minus8

Frequency (kHz)0 05 1 15 2 25 3 35 4


(b)

Disp

lace

men

t (m

)

10minus2

10minus4

10minus6

10minus8

Frequency (kHz)0 05 1 15 2 25 3 35 4


(c)

Figure 4 Frequency response function for the free-end for the cantilever beam (a)1198672= 1mm (b)119867

2= 5mm and (c)119867

2= 10mm

33 Analysis of the Frequency Response Function Thematrixsystem for a cantilever beam is given by

(minus1205962

119896M + Klowast (120596

119896))Ulowast119896= Flowast (33)

where Flowast and Ulowast are the nodal vectors of the amplitude ofthe force and the displacement respectively The frequencydependence of the complex stiffness matrix Klowast involves thefact that the modal superposition cannot be applied andtherefore (33)must be solved forUlowast

119896at each desired frequency

120596119896 Figure 4 represents the frequency response up to 3 kHz of

the free-end of the three-layer sandwich beam in all the threemodels when a harmonic unitary force 119865 = 1 times exp(119894120596119905) 119873

is applied on the right-hand side of our systemFigure 4(a) indicates that the responses provided by the

three models are practically the same as expected for sucha thin beam in which the shear is not so important On thecontrary Figures 4(b) and 4(c) illustrate a displacement of thecurves to the right with respect to that given by the 2DmodelThis is due to the overestimation of their natural frequenciesas it was mentioned in the previous section The differencesaremore patent as the frequency is larger and as the thicknessis bigger mostly for the RKU model

Regarding the amplitudes of the resonance peaks Tables5ndash7 showhow themodel presented in this paper improves theresults of the RKU model the improvement being better asthe thickness of the viscoelastic layer increases For instancefor the firstmode of the thickest beam the difference betweenthe errors in RKU and present models goes down from 73to 34 which is a significant difference taking into accountthe logarithmic scale

Since a quadratic distribution of the shear stress isrepresentative of the state of stress in beams the differencesbetween the new model and the 2D model are mainly due tothe inertias

4 Conclusions

In this paper a formulation for three-layer sandwich beamshas been presented which takes into account the deflectiondue to shearing forces For this quadratic distribution ofshear stress was considered along thickness In order totest the performance of the presented model a dynamicalanalysis has been carried out on a CLD beam in whichthe viscoelastic material has been modelled by means ofa fractional derivative model Consequently the natural


Table 5 Logarithm of the resonance peak amplitude (m) of the freeedge with core layer thickness119867

2= 1mm

1198601

1198602

1198603

2D model minus6357 minus8608 minus1058

Present model minus6299 minus8304 minus1046

RKU model minus6203 minus8211 minus1031


2= 5mm

1198601

1198602

1198603





2= 10mm

1198601

1198602

1198603




frequencies and resonance peak amplitudes provided by thenew model and by the classical RKU model have beencompared with those given by a 2D finite element modelAs a conclusion both models present similar results for athin core layer but the results are significantly improved asthe thickness of the core layer increases This formulationis very simple to be implemented in finite element analysispresenting the same complexity level as the RKU one Thisrepresents an important advantage with respect to otherformulations

In short a new model for finite element calculationsin three-layer sandwich beams with viscoelastic core layerhas been developed improving the results of the RKUone because the former considers shearing force deflectionwithout enlarging computational efforts

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] F Cortes M Martinez and M J Elejabarrieta ViscoelasticSurface Treatments for Passive Control of Structural VibrationNova Publishers New York NY USA 2012

[2] H Oberst and K Frankenfeld ldquoUber die dampfung derbiegeschwingungen duner bleche durch fest haftende belagerdquoAcustica vol 2 pp 181ndash194 1952

[3] F Cortes andM J Elejabarrieta ldquoStructural vibration of flexuralbeams with thick unconstrained layer dampingrdquo InternationalJournal of Solids and Structures vol 45 no 22-23 pp 5805ndash5813 2008

[4] A D Nashif D I G Jones and J P Henderson VibrationDamping John Wiley amp Sons New York NY USA 1995

[5] C T Sun andY P LuVibrationDamping of Structural ElementsPrentice Hall Upper Saddle River NJ USA 1995

[6] N O Myklestad ldquoThe concept of complex dampingrdquo Journal ofApplied Mechanics vol 19 pp 284ndash286 1952

[7] I MWard and DW HadleyAn Introduction to the MechanicalProperties of Polymers JohnWileyampSons ChichesterUK 1993

[8] D I G JonesHandbook of Viscoelastic VibrationDamping JohnWiley amp Sons Chichester UK 2001

[9] American Society for Testing and Material ldquoStandard testmethod for measuring vibration-damping properties of mate-rialsrdquo ASTM E 756-05 2005

[10] D Ross E M Kerwin and E E Ungar ldquoDamping of plate flex-ural vibration by means of viscoelastic laminaerdquo in StructuralDamping Section II ASME New York NY USA 1959

[11] R A DiTaranto ldquoTheory of vibratory bending for elastic andviscoelastic layered finite-length beamsrdquo Journal of AppliedMechanics vol 32 no 4 p 881 1965

[12] D J Mead and SMarkus ldquoThe forced vibration of a three-layerdamped sandwich beam with arbitrary boundary conditionsrdquoJournal of Sound and Vibration vol 10 no 2 pp 163ndash175 1969

[13] M J Yan and E H Dowell ldquoGoverning equations for vibratingconstrainedndashlayer damping of sandwich beams and platesrdquoJournal of Applied Mechanics vol 39 no 4 pp 1041ndash1046 1972

[14] YVK SadasivaRao andBCNakra ldquoVibrations of unsymmet-rical sandwich beams and plates with viscoelastic coresrdquo Journalof Sound and Vibration vol 34 no 3 pp 309ndash326 1974

[15] M Mace ldquoDamping of beam vibrations by means of a thinconstrained viscoelastic layer evaluation of a new theoryrdquoJournal of Sound and Vibration vol 172 no 5 pp 577ndash591 1994

[16] Z Liu S A Trogdon and J Yong ldquoModeling and analysis ofa laminated beamrdquoMathematical and Computer Modelling vol30 no 1-2 pp 149ndash167 1999

[17] J M Bai and C T Sun ldquoEffect of viscoelastic adhesive layers onstructural damping of sandwich beamsrdquoMechanics of Structuresand Machines vol 23 no 1 pp 1ndash16 1995

[18] Y Frostig and M Baruch ldquoFree vibrations of sandwich beamswith a transversely flexible core a high order approachrdquo Journalof Sound and Vibration vol 176 no 2 pp 195ndash208 1994

[19] A C Galucio J-F Deui and R Ohayon ldquoFinite elementformulation of viscoelastic sandwich beams using fractionalderivative operatorsrdquo Computational Mechanics vol 33 no 4pp 282ndash291 2004

[20] Y Frostig and O T Thomsen ldquoHigh-order free vibration ofsandwich panels with a flexible corerdquo International Journal ofSolids and Structures vol 41 no 5-6 pp 1697ndash1724 2004

[21] F Cortes and M J Elejabarrieta ldquoAn approximate numericalmethod for the complex eigenproblem in systems characterisedby a structural dampingmatrixrdquo Journal of Sound andVibrationvol 296 no 1-2 pp 166ndash182 2006

[22] R M Christensen Theory of Viscoelasticity Academic PressLondon UK 1982

[23] T H G Megson Structural and Stress Analysis Butterworth-Heinemann Oxford UK 2005

[24] S P Timoshenko ldquoLXVI On the correction for shear of thedifferential equation for transverse vibrations of prismatic barsrdquoPhilosophical Magazine Series 6 vol 41 no 245 pp 744ndash7461921


[25] F Cortes andM J Elejabarrieta ldquoViscoelastic materials charac-terisation using the seismic responserdquo Materials amp Design vol28 no 7 pp 2054ndash2062 2007

[26] Soundown Corporation November 2012 httpwwwsound-owncom

[27] R L Bagley and P J Torvik ldquoOn the fractional calculus modelof viscoelastic behaviourrdquo Journal of Rheology vol 30 no 1 pp133ndash155 1986

[28] T Pritz ldquoAnalysis of four-parameter fractional derivativemodelof real solid materialsrdquo Journal of Sound and Vibration vol 195no 1 pp 103ndash115 1996

[29] O C Zienkiewicz and R L Taylor Finite Element MethodButterworth-Heinemann Oxford UK 2000

[30] T J R HughesThe Finite Element Method Dover PublicationsNew York NY USA 2000

[31] K J Bathe Finite Element Procedures Prentice Hall UpperSaddle River NJ USA 1996

[32] D Kosloff and G A Frazier ldquoTreatment of hourglass patternsin low order finite element codesrdquo Numerical and AnalyticalMethods in Geomechanics vol 2 no 1 pp 57ndash72 1978

[33] C Lanczos ldquoAn iteration method for the solution of theeigenvalue problemof linear differential and integral operatorsrdquoJournal of Research of the National Bureau of Standards vol 45pp 255ndash282 1950

[34] W E Arnoldi ldquoThe principle of minimized iteration in thesolution of thematrix eigenvalue problemrdquoQuarterly of AppliedMathematics vol 9 pp 17ndash29 1951

[35] R B Nelson ldquoSimplified calculation of eigenvector derivativesrdquoAIAA Journal vol 14 no 9 pp 1201ndash1205 1976

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Control Scienceand Engineering

Journal of



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Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

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Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



H1

H2

H3

hn

b

Damping layer

Constraining layer

Neutral axis

Base beam

Figure 1 Cross section of a three-layer sandwich beam with aviscoelastic damping layer

and 119887 being the width of the beam Although in CLD beams1198672

≪ 1198671 in this work three-layer sandwich beams with

any value for 1198672are considered The properties for these

elastic materials are Youngrsquos moduli 1198641and 119864

3 Poissonrsquos

coefficients ]1and ]3 and densities 120588

1and 1205883 whereas those

of the viscoelastic core are 119864lowast

2 ]2 and 120588

2 respectively If the

materials are considered isotropic the shear moduli 1198661 119866lowast2

and 1198663are related to the extensional ones by 119866

1= 11986412(1 +

]1) 119866lowast2= 119864lowast

22(1 + ]

2) and 119866

3= 11986432(1 + ]

3) for the three

layersThe complex nature of themodulus of viscoelasticmateri-

als represented by the asterisk as (sdot)lowast is due to their ability todissipatemechanical energy A complex relationship betweenstress and strain allows representing the hysteretic behaviourof this kind of materials [6] The dissipative property may bedescribed by the loss factor 120578

2

1205782=

11986410158401015840

2

1198641015840

2

(1)

where 1198641015840

2and 119864

10158401015840

2are the real and imaginary components

of the complex modulus 119864lowast

2= 119864

1015840

2+ 11989411986410158401015840

2 which are

known as the storage and loss modulus respectively theformer being also frequently represented by119864

2 For polymers

both storage and loss moduli and consequently loss factorvary with temperature and frequency in terms of threedifferent behaviours rubbery vitreous and transition (seeeg [7]) In this sense Jones [8] reviews experimental datafor complex modulus of some typical viscoelastic materialsincluding elastomers adhesives and specially compoundedmaterials The experimental characterisation of low stiffnessdampingmaterials which are not appropriate to prepare self-supporting test specimens may be achieved by means of theASTM E 756-05 standard [9] wherefrom the nomenclaturehas been taken

The classical RKU [10] model is one of the first modelsdeveloped in order to calculate the flexural stiffness forsandwich beams with viscoelastic core and is one of the mostused models nowadays although it may lose accuracy inseveral cases for example when rotational or extensionalinertias are not negligible and specifically when a thickviscoelastic core in a CLD beam is studied This is due to theassumptions which this model takes

(i) The shear deformation in the elastic base and con-straining layers is considered to be negligible

(ii) The longitudinal direct stress in the viscoelastic layeris negligible

(iii) The shear stress in the viscoelastic layer is assumed tobe uniform

(iv) The plane cross section in each layer remains planeafter deformation

There are some other models beyond these restrictions suchas [11ndash18] or more recently [19 20] but they are not socomputationally efficient for finite elements applications Infact RKU is used in most engineering applications basicallydue to its very easy computational implementation AlsoRKU strictly applies to simply supported beams although itis much more generally used for other boundary conditionssuch as the fixed supportWe will refer to the results obtainedas RKU results although RKU theory for simply supportedbeams has been used in this work for a cantilever beam in thecontext of a finite element analysis with displacement fieldsrestricted to those of the RKU theory

In short following a similar approach as for the study ofthe FLD configuration for thick beams previously mentioned[3] this work is aimed at developing a new model improvingthe accuracy of the RKU model for three-layer sandwichbeams with a thick viscoelastic core but maintaining itscomputational benefit Thus an equivalent complex flexuralstiffness is derived considering quadratic shear stress basedon Reddy-Bickfordrsquos theory In order to prove the improve-ment achieved by the new model a numerical applicationfor a cantilever beam is presented comparing the solutionsprovided by three different finite elementmodels a 2Dmodel(whose results are considered to be the reference ones) andtwo 1D models based on the RKU theory and the newone respectively The damping material is characterised bymeans of a fractional derivative model involving the fre-quency dependence of the complex modulus which impliesimportant disadvantages for the dynamic analysis Thus theextraction of the eigenvalues and eigenvectors is carried outby a simple and effective iterative algorithm [21] Finally thedynamic response of the three models is compared in termsof the frequency response function

As a result the new model is able to reproduce themechanical behaviour of three-layer sandwich beams reduc-ing storage needs and computational time compared with a2Dmodel and improving the results provided by the classicalRKU model

2 Homogenised Model for a CLD Beam

Next the theoretical study of a three-layer sandwich beamis presented in which quadratic shear stress is taken intoaccount An equivalent flexural stiffness will be deduced forpinned-pinned beams In order to simplify the notation itis assumed that the behaviour of all materials is linear andelastic (ie all the magnitudes are real) and the complexcharacter of themodulus in the viscoelastic core will be takeninto account in the numerical examples of the subsequentsections This substitution of a complex modulus into anelastic solution is usually known as the ldquocorrespondenceprinciplerdquo (see any standard text on viscoelasticity eg [22])



119861eq = 1198611+ 1198612+ 1198613 (2)

where 1198611= 11986411198681 1198612= 11986421198682 and 119861



3are the


1198681=

1

121198871198673

1+ 1198871198671(ℎ119899minus

1198671

2)

2

(3)

1198682=

1

121198871198673

2+ 1198871198672(1198671+

1198672

2minus ℎ119899)

2

(4)

1198683=

1

121198871198673

3+ 1198871198673(1198671+ 1198672+

1198673

2minus ℎ119899)

2

(5)


ℎ119899=

11986411198672

12+11986421198672(1198671+11986722)+119864

31198673(1198671+ 1198672+ 11986732)

11986411198671+ 11986421198672+ 11986431198673

(6)


1

119870eq=

1

1198701

+1

1198702

+1

1198703

(7)


1198702 and119870


1

1198701

=1

11986611198612eq119887

int

minusℎ119899+1198671

minusℎ119899

Ω2

1(119910) d119910 (8)

1

1198702

=1

11986621198612eq119887

int

minusℎ119899+1198671+1198672

minusℎ119899+1198671

Ω2

2(119910) d119910 (9)

1

1198703

=1

11986631198612eq119887

int

minusℎ119899+1198671+1198672+1198673

minusℎ119899+1198671+1198672

Ω2

3(119910) d119910 (10)

respectively where

Ω2

1(119910) =

11988721198642

1(ℎ2

119899minus 1199102)2

4

(11)

Ω2

2(119910) = 119887

2

11986411198671(ℎ119899minus

1198671

2) +

1198642[(ℎ119899minus 1198671)2

minus 1199102]

2

2

(12)

Ω2

3(119910) =

11988721198642

3[(1198671+ 1198672+ 1198673minus ℎ119899)2

minus 1199102]2

4

(13)



1

1198701

=6

511986611198871198671

(101199032

119899minus 15119903119899+ 6)

(1 + 11990321198722+ 11990331198723)2

(1 + 3 (119903119899minus 1)2

)2

(14)

1

1198702

=1

11986621198871198671

sdot361198792(119903119899minus 1)2

minus 1211987221198792

2(119903119899minus 1) (2119879

2minus 3119903119899+ 6)

(1 + 11990321198722+ 11990331198723)2

(1 + 3 (119903119899minus 1)2

)2

+6

511986621198871198672

sdot

1198722

21198794

2(101199032

119899minus 151199031198991198792minus 40119903119899+ 61198792

2+ 30119879

2+ 40)

(1 + 11990321198722+ 11990331198723)2

(1 + 3 (119903119899minus 1)2

)2

(15)

1

1198703

=61198722

31198794

3

511986631198871198673

times (101199032


2

3

+5011987931198792+ 50119879

3+ 40119879

2

2+ 80119879

2+ 40)

sdot ((1 + 11990321198722+ 11990331198723)2

(1 + 3 (119903119899minus 1)2

)2

)

minus1

(16)


= 11986421198641 1198723

= 11986431198641 1198792

= 1198672

1198671 1198793

= 11986731198671119903119899

= 2ℎ1198991198671 1199032

= 11986821198681 and 119903

3= 1198683



1and 1119870

3may be neglected



1

119870eq=

6

5119866119878 (17)



119861eq

120588ℓ

1205974V (119909 119905)1205971199094

minus

119861eq

119870eq

1205974V (119909 119905)12059711990921205971199052

+1205972V (119909 119905)1205971199052

= 0 (18)






119861119870

=

119861eq

(radic1 + 1205932(120596) + 120593(120596))

2 (19)


120593 (120596) =

120596radic120588ℓ119861eq

2119870eq (20)










2=






1= ]3




119864lowast

2(120596) =

119864119903+ 119864119906(119894120591120596)120572

1 + (119894120591120596)120572

(21)

where 119864119903and 119864




119864119903(GPa) 119864

119906(GPa) 120591 (10minus6 s) 120572

0353 3462 3149 0873

F

u120001






120588119886119887119905

36

[[[[[[[[[[[[[[[[[[[[[[[

[

4 0 2 0 1 0 2 0

4 0 2 0 1 0 2

4 0 2 0 1 0

4 0 2 0 1

4 0 2 0

4 0 2

4 0

4

]]]]]]]]]]]]]]]]]]]]]]]

]

(22)



a

b

1 2

34




119864119905

24120574 (1 minus ]2)

times

[[[[[[[[[[[[[[[[[[[[[[[[[[

[


31205742






31205742



(1 minus ]) minus 2 (2 + ]2)


31205742


(1 minus ]) + 2 (2 + ]2)


31205742

(1 minus ]) + 2 (4 minus ]2)

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(23)



119894matrices are

M119894=

(12058811198671+ 12058821198672+ 12058831198673) 119887119886119894

420

sdot

[[[[[

[

156 22119886119894

54 minus13119886119894

22119886119894

41198862

11989413119886119894

minus31198862

119894

54 13119886119894

156 minus22119886119894

minus13119886119894

minus31198862

119894minus22119886119894

41198862

119894

]]]]]

]

(24)

K119894=

119861

1198863

119894

[[[[[

[

12 6119886119894

minus12 6119886119894

6119886119894

41198862

119894minus6119886119894

21198862

119894

minus12 minus6119886119894

12 minus6119886119894

6119886119894

21198862

119894minus6119886119894

41198862

119894

]]]]]

]

(25)




lowast




(minus120582lowast

119903M + Klowast (120596

119903))120601lowast

119903= 0 (26)






120596119903= Re(radic120582lowast

119903) (27)


(minus1205821199030M + Klowast (0))120601

1199030= 0 (28)



1205961(rads) 120578

11205962(rads) 120578

21205963(rads) 120578

3

2D model 4111 0024 2523 0049 6851 0047

Present model 4122 0019 2547 0037 6912 0043

RKU model 4124 0018 2552 0034 6952 0037


1205961(rads) 120578

11205962(rads) 120578

21205963(rads) 120578

3

2D model 1003 0080 5659 0113 13987 0091

Present model 1013 0066 5909 0094 14504 0103

RKU model 1019 0052 6051 0068 15461 0063


1205961(rads) 120578

11205962(rads) 120578

21205963(rads) 120578

3

2D model 1561 0118 8106 0138 18828 0099

Present model 1589 0098 8714 0127 19779 0131

RKU model 1613 0069 9291 0076 22875 0064


and 1206011199030




ΔKlowast (1205961199030

) = Klowast (1205961199030





120601lowast

119903= 1206011199030

+ Δ120601lowast

119903 (30)


to

120582lowast

119903=120601lowast119867119903Κlowast (120596

119903)120601lowast119903

120601lowast119867119903

M120601lowast119903

(31)



119903





120582lowast

119903= 1205962

119903(1 + 119894120578

119903) (32)










119903 the



Disp

lace

men

t (m

)10minus2

10minus4

10minus6

10minus8

Frequency (kHz)0 05 1 15 2 25 3 35 4


(a)

Disp

lace

men

t (m

) 10minus4

10minus6

10minus8

Frequency (kHz)0 05 1 15 2 25 3 35 4


(b)

Disp

lace

men

t (m

)

10minus2

10minus4

10minus6

10minus8

Frequency (kHz)0 05 1 15 2 25 3 35 4


(c)


2= 5mm and (c)119867

2= 10mm


(minus1205962

119896M + Klowast (120596










4 Conclusions




2= 1mm

1198601

1198602

1198603





2= 5mm

1198601

1198602

1198603





2= 10mm

1198601

1198602

1198603








References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119861eq = 1198611+ 1198612+ 1198613 (2)

where 1198611= 11986411198681 1198612= 11986421198682 and 119861



3are the


1198681=

1

121198871198673

1+ 1198871198671(ℎ119899minus

1198671

2)

2

(3)

1198682=

1

121198871198673

2+ 1198871198672(1198671+

1198672

2minus ℎ119899)

2

(4)

1198683=

1

121198871198673

3+ 1198871198673(1198671+ 1198672+

1198673

2minus ℎ119899)

2

(5)


ℎ119899=

11986411198672

12+11986421198672(1198671+11986722)+119864

31198673(1198671+ 1198672+ 11986732)

11986411198671+ 11986421198672+ 11986431198673

(6)


1

119870eq=

1

1198701

+1

1198702

+1

1198703

(7)


1198702 and119870


1

1198701

=1

11986611198612eq119887

int

minusℎ119899+1198671

minusℎ119899

Ω2

1(119910) d119910 (8)

1

1198702

=1

11986621198612eq119887

int

minusℎ119899+1198671+1198672

minusℎ119899+1198671

Ω2

2(119910) d119910 (9)

1

1198703

=1

11986631198612eq119887

int

minusℎ119899+1198671+1198672+1198673

minusℎ119899+1198671+1198672

Ω2

3(119910) d119910 (10)

respectively where

Ω2

1(119910) =

11988721198642

1(ℎ2

119899minus 1199102)2

4

(11)

Ω2

2(119910) = 119887

2

11986411198671(ℎ119899minus

1198671

2) +

1198642[(ℎ119899minus 1198671)2

minus 1199102]

2

2

(12)

Ω2

3(119910) =

11988721198642

3[(1198671+ 1198672+ 1198673minus ℎ119899)2

minus 1199102]2

4

(13)



1

1198701

=6

511986611198871198671

(101199032

119899minus 15119903119899+ 6)

(1 + 11990321198722+ 11990331198723)2

(1 + 3 (119903119899minus 1)2

)2

(14)

1

1198702

=1

11986621198871198671

sdot361198792(119903119899minus 1)2

minus 1211987221198792

2(119903119899minus 1) (2119879

2minus 3119903119899+ 6)

(1 + 11990321198722+ 11990331198723)2

(1 + 3 (119903119899minus 1)2

)2

+6

511986621198871198672

sdot

1198722

21198794

2(101199032

119899minus 151199031198991198792minus 40119903119899+ 61198792

2+ 30119879

2+ 40)

(1 + 11990321198722+ 11990331198723)2

(1 + 3 (119903119899minus 1)2

)2

(15)

1

1198703

=61198722

31198794

3

511986631198871198673

times (101199032


2

3

+5011987931198792+ 50119879

3+ 40119879

2

2+ 80119879

2+ 40)

sdot ((1 + 11990321198722+ 11990331198723)2

(1 + 3 (119903119899minus 1)2

)2

)

minus1

(16)


= 11986421198641 1198723

= 11986431198641 1198792

= 1198672

1198671 1198793

= 11986731198671119903119899

= 2ℎ1198991198671 1199032

= 11986821198681 and 119903

3= 1198683



1and 1119870

3may be neglected



1

119870eq=

6

5119866119878 (17)



119861eq

120588ℓ

1205974V (119909 119905)1205971199094

minus

119861eq

119870eq

1205974V (119909 119905)12059711990921205971199052

+1205972V (119909 119905)1205971199052

= 0 (18)






119861119870

=

119861eq

(radic1 + 1205932(120596) + 120593(120596))

2 (19)


120593 (120596) =

120596radic120588ℓ119861eq

2119870eq (20)










2=






1= ]3




119864lowast

2(120596) =

119864119903+ 119864119906(119894120591120596)120572

1 + (119894120591120596)120572

(21)

where 119864119903and 119864




119864119903(GPa) 119864

119906(GPa) 120591 (10minus6 s) 120572

0353 3462 3149 0873

F

u120001






120588119886119887119905

36

[[[[[[[[[[[[[[[[[[[[[[[

[

4 0 2 0 1 0 2 0

4 0 2 0 1 0 2

4 0 2 0 1 0

4 0 2 0 1

4 0 2 0

4 0 2

4 0

4

]]]]]]]]]]]]]]]]]]]]]]]

]

(22)



a

b

1 2

34




119864119905

24120574 (1 minus ]2)

times

[[[[[[[[[[[[[[[[[[[[[[[[[[

[


31205742






31205742



(1 minus ]) minus 2 (2 + ]2)


31205742


(1 minus ]) + 2 (2 + ]2)


31205742

(1 minus ]) + 2 (4 minus ]2)

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(23)



119894matrices are

M119894=

(12058811198671+ 12058821198672+ 12058831198673) 119887119886119894

420

sdot

[[[[[

[

156 22119886119894

54 minus13119886119894

22119886119894

41198862

11989413119886119894

minus31198862

119894

54 13119886119894

156 minus22119886119894

minus13119886119894

minus31198862

119894minus22119886119894

41198862

119894

]]]]]

]

(24)

K119894=

119861

1198863

119894

[[[[[

[

12 6119886119894

minus12 6119886119894

6119886119894

41198862

119894minus6119886119894

21198862

119894

minus12 minus6119886119894

12 minus6119886119894

6119886119894

21198862

119894minus6119886119894

41198862

119894

]]]]]

]

(25)




lowast




(minus120582lowast

119903M + Klowast (120596

119903))120601lowast

119903= 0 (26)






120596119903= Re(radic120582lowast

119903) (27)


(minus1205821199030M + Klowast (0))120601

1199030= 0 (28)



1205961(rads) 120578

11205962(rads) 120578

21205963(rads) 120578

3

2D model 4111 0024 2523 0049 6851 0047

Present model 4122 0019 2547 0037 6912 0043

RKU model 4124 0018 2552 0034 6952 0037


1205961(rads) 120578

11205962(rads) 120578

21205963(rads) 120578

3

2D model 1003 0080 5659 0113 13987 0091

Present model 1013 0066 5909 0094 14504 0103

RKU model 1019 0052 6051 0068 15461 0063


1205961(rads) 120578

11205962(rads) 120578

21205963(rads) 120578

3

2D model 1561 0118 8106 0138 18828 0099

Present model 1589 0098 8714 0127 19779 0131

RKU model 1613 0069 9291 0076 22875 0064


and 1206011199030




ΔKlowast (1205961199030

) = Klowast (1205961199030





120601lowast

119903= 1206011199030

+ Δ120601lowast

119903 (30)


to

120582lowast

119903=120601lowast119867119903Κlowast (120596

119903)120601lowast119903

120601lowast119867119903

M120601lowast119903

(31)



119903





120582lowast

119903= 1205962

119903(1 + 119894120578

119903) (32)










119903 the



Disp

lace

men

t (m

)10minus2

10minus4

10minus6

10minus8

Frequency (kHz)0 05 1 15 2 25 3 35 4


(a)

Disp

lace

men

t (m

) 10minus4

10minus6

10minus8

Frequency (kHz)0 05 1 15 2 25 3 35 4


(b)

Disp

lace

men

t (m

)

10minus2

10minus4

10minus6

10minus8

Frequency (kHz)0 05 1 15 2 25 3 35 4


(c)


2= 5mm and (c)119867

2= 10mm


(minus1205962

119896M + Klowast (120596










4 Conclusions




2= 1mm

1198601

1198602

1198603





2= 5mm

1198601

1198602

1198603





2= 10mm

1198601

1198602

1198603








References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119861119870

=

119861eq

(radic1 + 1205932(120596) + 120593(120596))

2 (19)


120593 (120596) =

120596radic120588ℓ119861eq

2119870eq (20)










2=






1= ]3




119864lowast

2(120596) =

119864119903+ 119864119906(119894120591120596)120572

1 + (119894120591120596)120572

(21)

where 119864119903and 119864




119864119903(GPa) 119864

119906(GPa) 120591 (10minus6 s) 120572

0353 3462 3149 0873

F

u120001






120588119886119887119905

36

[[[[[[[[[[[[[[[[[[[[[[[

[

4 0 2 0 1 0 2 0

4 0 2 0 1 0 2

4 0 2 0 1 0

4 0 2 0 1

4 0 2 0

4 0 2

4 0

4

]]]]]]]]]]]]]]]]]]]]]]]

]

(22)



a

b

1 2

34




119864119905

24120574 (1 minus ]2)

times

[[[[[[[[[[[[[[[[[[[[[[[[[[

[


31205742






31205742



(1 minus ]) minus 2 (2 + ]2)


31205742


(1 minus ]) + 2 (2 + ]2)


31205742

(1 minus ]) + 2 (4 minus ]2)

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(23)



119894matrices are

M119894=

(12058811198671+ 12058821198672+ 12058831198673) 119887119886119894

420

sdot

[[[[[

[

156 22119886119894

54 minus13119886119894

22119886119894

41198862

11989413119886119894

minus31198862

119894

54 13119886119894

156 minus22119886119894

minus13119886119894

minus31198862

119894minus22119886119894

41198862

119894

]]]]]

]

(24)

K119894=

119861

1198863

119894

[[[[[

[

12 6119886119894

minus12 6119886119894

6119886119894

41198862

119894minus6119886119894

21198862

119894

minus12 minus6119886119894

12 minus6119886119894

6119886119894

21198862

119894minus6119886119894

41198862

119894

]]]]]

]

(25)




lowast




(minus120582lowast

119903M + Klowast (120596

119903))120601lowast

119903= 0 (26)






120596119903= Re(radic120582lowast

119903) (27)


(minus1205821199030M + Klowast (0))120601

1199030= 0 (28)



1205961(rads) 120578

11205962(rads) 120578

21205963(rads) 120578

3

2D model 4111 0024 2523 0049 6851 0047

Present model 4122 0019 2547 0037 6912 0043

RKU model 4124 0018 2552 0034 6952 0037


1205961(rads) 120578

11205962(rads) 120578

21205963(rads) 120578

3

2D model 1003 0080 5659 0113 13987 0091

Present model 1013 0066 5909 0094 14504 0103

RKU model 1019 0052 6051 0068 15461 0063


1205961(rads) 120578

11205962(rads) 120578

21205963(rads) 120578

3

2D model 1561 0118 8106 0138 18828 0099

Present model 1589 0098 8714 0127 19779 0131

RKU model 1613 0069 9291 0076 22875 0064


and 1206011199030




ΔKlowast (1205961199030

) = Klowast (1205961199030





120601lowast

119903= 1206011199030

+ Δ120601lowast

119903 (30)


to

120582lowast

119903=120601lowast119867119903Κlowast (120596

119903)120601lowast119903

120601lowast119867119903

M120601lowast119903

(31)



119903





120582lowast

119903= 1205962

119903(1 + 119894120578

119903) (32)










119903 the



Disp

lace

men

t (m

)10minus2

10minus4

10minus6

10minus8

Frequency (kHz)0 05 1 15 2 25 3 35 4


(a)

Disp

lace

men

t (m

) 10minus4

10minus6

10minus8

Frequency (kHz)0 05 1 15 2 25 3 35 4


(b)

Disp

lace

men

t (m

)

10minus2

10minus4

10minus6

10minus8

Frequency (kHz)0 05 1 15 2 25 3 35 4


(c)


2= 5mm and (c)119867

2= 10mm


(minus1205962

119896M + Klowast (120596










4 Conclusions




2= 1mm

1198601

1198602

1198603





2= 5mm

1198601

1198602

1198603





2= 10mm

1198601

1198602

1198603








References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










a

b

1 2

34




119864119905

24120574 (1 minus ]2)

times

[[[[[[[[[[[[[[[[[[[[[[[[[[

[


31205742






31205742



(1 minus ]) minus 2 (2 + ]2)


31205742


(1 minus ]) + 2 (2 + ]2)


31205742

(1 minus ]) + 2 (4 minus ]2)

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(23)



119894matrices are

M119894=

(12058811198671+ 12058821198672+ 12058831198673) 119887119886119894

420

sdot

[[[[[

[

156 22119886119894

54 minus13119886119894

22119886119894

41198862

11989413119886119894

minus31198862

119894

54 13119886119894

156 minus22119886119894

minus13119886119894

minus31198862

119894minus22119886119894

41198862

119894

]]]]]

]

(24)

K119894=

119861

1198863

119894

[[[[[

[

12 6119886119894

minus12 6119886119894

6119886119894

41198862

119894minus6119886119894

21198862

119894

minus12 minus6119886119894

12 minus6119886119894

6119886119894

21198862

119894minus6119886119894

41198862

119894

]]]]]

]

(25)




lowast




(minus120582lowast

119903M + Klowast (120596

119903))120601lowast

119903= 0 (26)






120596119903= Re(radic120582lowast

119903) (27)


(minus1205821199030M + Klowast (0))120601

1199030= 0 (28)



1205961(rads) 120578

11205962(rads) 120578

21205963(rads) 120578

3

2D model 4111 0024 2523 0049 6851 0047

Present model 4122 0019 2547 0037 6912 0043

RKU model 4124 0018 2552 0034 6952 0037


1205961(rads) 120578

11205962(rads) 120578

21205963(rads) 120578

3

2D model 1003 0080 5659 0113 13987 0091

Present model 1013 0066 5909 0094 14504 0103

RKU model 1019 0052 6051 0068 15461 0063


1205961(rads) 120578

11205962(rads) 120578

21205963(rads) 120578

3

2D model 1561 0118 8106 0138 18828 0099

Present model 1589 0098 8714 0127 19779 0131

RKU model 1613 0069 9291 0076 22875 0064


and 1206011199030




ΔKlowast (1205961199030

) = Klowast (1205961199030





120601lowast

119903= 1206011199030

+ Δ120601lowast

119903 (30)


to

120582lowast

119903=120601lowast119867119903Κlowast (120596

119903)120601lowast119903

120601lowast119867119903

M120601lowast119903

(31)



119903





120582lowast

119903= 1205962

119903(1 + 119894120578

119903) (32)










119903 the



Disp

lace

men

t (m

)10minus2

10minus4

10minus6

10minus8

Frequency (kHz)0 05 1 15 2 25 3 35 4


(a)

Disp

lace

men

t (m

) 10minus4

10minus6

10minus8

Frequency (kHz)0 05 1 15 2 25 3 35 4


(b)

Disp

lace

men

t (m

)

10minus2

10minus4

10minus6

10minus8

Frequency (kHz)0 05 1 15 2 25 3 35 4


(c)


2= 5mm and (c)119867

2= 10mm


(minus1205962

119896M + Klowast (120596










4 Conclusions




2= 1mm

1198601

1198602

1198603





2= 5mm

1198601

1198602

1198603





2= 10mm

1198601

1198602

1198603








References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1205961(rads) 120578

11205962(rads) 120578

21205963(rads) 120578

3

2D model 4111 0024 2523 0049 6851 0047

Present model 4122 0019 2547 0037 6912 0043

RKU model 4124 0018 2552 0034 6952 0037


1205961(rads) 120578

11205962(rads) 120578

21205963(rads) 120578

3

2D model 1003 0080 5659 0113 13987 0091

Present model 1013 0066 5909 0094 14504 0103

RKU model 1019 0052 6051 0068 15461 0063


1205961(rads) 120578

11205962(rads) 120578

21205963(rads) 120578

3

2D model 1561 0118 8106 0138 18828 0099

Present model 1589 0098 8714 0127 19779 0131

RKU model 1613 0069 9291 0076 22875 0064


and 1206011199030




ΔKlowast (1205961199030

) = Klowast (1205961199030





120601lowast

119903= 1206011199030

+ Δ120601lowast

119903 (30)


to

120582lowast

119903=120601lowast119867119903Κlowast (120596

119903)120601lowast119903

120601lowast119867119903

M120601lowast119903

(31)



119903





120582lowast

119903= 1205962

119903(1 + 119894120578

119903) (32)










119903 the



Disp

lace

men

t (m

)10minus2

10minus4

10minus6

10minus8

Frequency (kHz)0 05 1 15 2 25 3 35 4


(a)

Disp

lace

men

t (m

) 10minus4

10minus6

10minus8

Frequency (kHz)0 05 1 15 2 25 3 35 4


(b)

Disp

lace

men

t (m

)

10minus2

10minus4

10minus6

10minus8

Frequency (kHz)0 05 1 15 2 25 3 35 4


(c)


2= 5mm and (c)119867

2= 10mm


(minus1205962

119896M + Klowast (120596










4 Conclusions




2= 1mm

1198601

1198602

1198603





2= 5mm

1198601

1198602

1198603





2= 10mm

1198601

1198602

1198603








References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Disp

lace

men

t (m

)10minus2

10minus4

10minus6

10minus8

Frequency (kHz)0 05 1 15 2 25 3 35 4


(a)

Disp

lace

men

t (m

) 10minus4

10minus6

10minus8

Frequency (kHz)0 05 1 15 2 25 3 35 4


(b)

Disp

lace

men

t (m

)

10minus2

10minus4

10minus6

10minus8

Frequency (kHz)0 05 1 15 2 25 3 35 4


(c)


2= 5mm and (c)119867

2= 10mm


(minus1205962

119896M + Klowast (120596










4 Conclusions




2= 1mm

1198601

1198602

1198603





2= 5mm

1198601

1198602

1198603





2= 10mm

1198601

1198602

1198603








References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











2= 1mm

1198601

1198602

1198603





2= 5mm

1198601

1198602

1198603





2= 10mm

1198601

1198602

1198603








References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Dynamic Analysis of Three-Layer Sandwich...

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