Research Article Dynamic Analysis of Three-Layer Sandwich...
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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
4 Shock and Vibration
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
Shock and Vibration 5
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 (4 minus ]2) 3120574 (1 minus 3]) minus31205742
(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)
6 Shock and Vibration
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
Table 3 Modal properties of the sandwich cantilever beam with1198672= 5mm
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
Table 4 Modal properties of the sandwich cantilever beam with1198672= 10mm
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
Shock and Vibration 7
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
2D modelPresent modelRKU model
(b)
Disp
lace
men
t (m
)
10minus2
10minus4
10minus6
10minus8
Frequency (kHz)0 05 1 15 2 25 3 35 4
2D modelPresent modelRKU model
(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
8 Shock and Vibration
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
Table 6 Logarithm of the resonance peak amplitude (m) of the freeedge with core layer thickness119867
2= 5mm
1198601
1198602
1198603
2D model minus7552 minus1144 minus1310
Present model minus7285 minus1116 minus1304
RKU model minus7058 minus1087 minus1268
Table 7 Logarithm of the resonance peak amplitude (m) of the freeedge with core layer thickness119867
2= 10mm
1198601
1198602
1198603
2D model minus9120 minus1271 minus1415
Present model minus8812 minus1248 minus1413
RKU model minus8451 minus1206 minus1372
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
Shock and Vibration 9
[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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International Journal of
![Page 2: Research Article Dynamic Analysis of Three-Layer Sandwich …downloads.hindawi.com/journals/sv/2015/736256.pdf · 2019-07-31 · Shock and Vibration H 1 H 2 H 3 hn b Damping layer](https://reader034.fdocuments.us/reader034/viewer/2022042310/5ed82ff90fa3e705ec0dffaa/html5/thumbnails/2.jpg)
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
4 Shock and Vibration
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
Shock and Vibration 5
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 (4 minus ]2) 3120574 (1 minus 3]) minus31205742
(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)
6 Shock and Vibration
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
Table 3 Modal properties of the sandwich cantilever beam with1198672= 5mm
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
Table 4 Modal properties of the sandwich cantilever beam with1198672= 10mm
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
Shock and Vibration 7
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
2D modelPresent modelRKU model
(b)
Disp
lace
men
t (m
)
10minus2
10minus4
10minus6
10minus8
Frequency (kHz)0 05 1 15 2 25 3 35 4
2D modelPresent modelRKU model
(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
8 Shock and Vibration
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
Table 6 Logarithm of the resonance peak amplitude (m) of the freeedge with core layer thickness119867
2= 5mm
1198601
1198602
1198603
2D model minus7552 minus1144 minus1310
Present model minus7285 minus1116 minus1304
RKU model minus7058 minus1087 minus1268
Table 7 Logarithm of the resonance peak amplitude (m) of the freeedge with core layer thickness119867
2= 10mm
1198601
1198602
1198603
2D model minus9120 minus1271 minus1415
Present model minus8812 minus1248 minus1413
RKU model minus8451 minus1206 minus1372
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
Shock and Vibration 9
[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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DistributedSensor Networks
International Journal of
![Page 3: Research Article Dynamic Analysis of Three-Layer Sandwich …downloads.hindawi.com/journals/sv/2015/736256.pdf · 2019-07-31 · Shock and Vibration H 1 H 2 H 3 hn b Damping layer](https://reader034.fdocuments.us/reader034/viewer/2022042310/5ed82ff90fa3e705ec0dffaa/html5/thumbnails/3.jpg)
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
4 Shock and Vibration
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
Shock and Vibration 5
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 (4 minus ]2) 3120574 (1 minus 3]) minus31205742
(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)
6 Shock and Vibration
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
Table 3 Modal properties of the sandwich cantilever beam with1198672= 5mm
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
Table 4 Modal properties of the sandwich cantilever beam with1198672= 10mm
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
Shock and Vibration 7
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
2D modelPresent modelRKU model
(b)
Disp
lace
men
t (m
)
10minus2
10minus4
10minus6
10minus8
Frequency (kHz)0 05 1 15 2 25 3 35 4
2D modelPresent modelRKU model
(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
8 Shock and Vibration
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
Table 6 Logarithm of the resonance peak amplitude (m) of the freeedge with core layer thickness119867
2= 5mm
1198601
1198602
1198603
2D model minus7552 minus1144 minus1310
Present model minus7285 minus1116 minus1304
RKU model minus7058 minus1087 minus1268
Table 7 Logarithm of the resonance peak amplitude (m) of the freeedge with core layer thickness119867
2= 10mm
1198601
1198602
1198603
2D model minus9120 minus1271 minus1415
Present model minus8812 minus1248 minus1413
RKU model minus8451 minus1206 minus1372
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
Shock and Vibration 9
[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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DistributedSensor Networks
International Journal of
![Page 4: Research Article Dynamic Analysis of Three-Layer Sandwich …downloads.hindawi.com/journals/sv/2015/736256.pdf · 2019-07-31 · Shock and Vibration H 1 H 2 H 3 hn b Damping layer](https://reader034.fdocuments.us/reader034/viewer/2022042310/5ed82ff90fa3e705ec0dffaa/html5/thumbnails/4.jpg)
4 Shock and Vibration
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
Shock and Vibration 5
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 (4 minus ]2) 3120574 (1 minus 3]) minus31205742
(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)
6 Shock and Vibration
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
Table 3 Modal properties of the sandwich cantilever beam with1198672= 5mm
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
Table 4 Modal properties of the sandwich cantilever beam with1198672= 10mm
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
Shock and Vibration 7
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
2D modelPresent modelRKU model
(b)
Disp
lace
men
t (m
)
10minus2
10minus4
10minus6
10minus8
Frequency (kHz)0 05 1 15 2 25 3 35 4
2D modelPresent modelRKU model
(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
8 Shock and Vibration
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
Table 6 Logarithm of the resonance peak amplitude (m) of the freeedge with core layer thickness119867
2= 5mm
1198601
1198602
1198603
2D model minus7552 minus1144 minus1310
Present model minus7285 minus1116 minus1304
RKU model minus7058 minus1087 minus1268
Table 7 Logarithm of the resonance peak amplitude (m) of the freeedge with core layer thickness119867
2= 10mm
1198601
1198602
1198603
2D model minus9120 minus1271 minus1415
Present model minus8812 minus1248 minus1413
RKU model minus8451 minus1206 minus1372
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
Shock and Vibration 9
[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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DistributedSensor Networks
International Journal of
![Page 5: Research Article Dynamic Analysis of Three-Layer Sandwich …downloads.hindawi.com/journals/sv/2015/736256.pdf · 2019-07-31 · Shock and Vibration H 1 H 2 H 3 hn b Damping layer](https://reader034.fdocuments.us/reader034/viewer/2022042310/5ed82ff90fa3e705ec0dffaa/html5/thumbnails/5.jpg)
Shock and Vibration 5
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 (4 minus ]2) 3120574 (1 minus 3]) minus31205742
(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)
6 Shock and Vibration
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
Table 3 Modal properties of the sandwich cantilever beam with1198672= 5mm
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
Table 4 Modal properties of the sandwich cantilever beam with1198672= 10mm
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
Shock and Vibration 7
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
2D modelPresent modelRKU model
(b)
Disp
lace
men
t (m
)
10minus2
10minus4
10minus6
10minus8
Frequency (kHz)0 05 1 15 2 25 3 35 4
2D modelPresent modelRKU model
(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
8 Shock and Vibration
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
Table 6 Logarithm of the resonance peak amplitude (m) of the freeedge with core layer thickness119867
2= 5mm
1198601
1198602
1198603
2D model minus7552 minus1144 minus1310
Present model minus7285 minus1116 minus1304
RKU model minus7058 minus1087 minus1268
Table 7 Logarithm of the resonance peak amplitude (m) of the freeedge with core layer thickness119867
2= 10mm
1198601
1198602
1198603
2D model minus9120 minus1271 minus1415
Present model minus8812 minus1248 minus1413
RKU model minus8451 minus1206 minus1372
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
Shock and Vibration 9
[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
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
![Page 6: Research Article Dynamic Analysis of Three-Layer Sandwich …downloads.hindawi.com/journals/sv/2015/736256.pdf · 2019-07-31 · Shock and Vibration H 1 H 2 H 3 hn b Damping layer](https://reader034.fdocuments.us/reader034/viewer/2022042310/5ed82ff90fa3e705ec0dffaa/html5/thumbnails/6.jpg)
6 Shock and Vibration
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
Table 3 Modal properties of the sandwich cantilever beam with1198672= 5mm
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
Table 4 Modal properties of the sandwich cantilever beam with1198672= 10mm
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
Shock and Vibration 7
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
2D modelPresent modelRKU model
(b)
Disp
lace
men
t (m
)
10minus2
10minus4
10minus6
10minus8
Frequency (kHz)0 05 1 15 2 25 3 35 4
2D modelPresent modelRKU model
(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
8 Shock and Vibration
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
Table 6 Logarithm of the resonance peak amplitude (m) of the freeedge with core layer thickness119867
2= 5mm
1198601
1198602
1198603
2D model minus7552 minus1144 minus1310
Present model minus7285 minus1116 minus1304
RKU model minus7058 minus1087 minus1268
Table 7 Logarithm of the resonance peak amplitude (m) of the freeedge with core layer thickness119867
2= 10mm
1198601
1198602
1198603
2D model minus9120 minus1271 minus1415
Present model minus8812 minus1248 minus1413
RKU model minus8451 minus1206 minus1372
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
Shock and Vibration 9
[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
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
![Page 7: Research Article Dynamic Analysis of Three-Layer Sandwich …downloads.hindawi.com/journals/sv/2015/736256.pdf · 2019-07-31 · Shock and Vibration H 1 H 2 H 3 hn b Damping layer](https://reader034.fdocuments.us/reader034/viewer/2022042310/5ed82ff90fa3e705ec0dffaa/html5/thumbnails/7.jpg)
Shock and Vibration 7
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
2D modelPresent modelRKU model
(b)
Disp
lace
men
t (m
)
10minus2
10minus4
10minus6
10minus8
Frequency (kHz)0 05 1 15 2 25 3 35 4
2D modelPresent modelRKU model
(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
8 Shock and Vibration
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
Table 6 Logarithm of the resonance peak amplitude (m) of the freeedge with core layer thickness119867
2= 5mm
1198601
1198602
1198603
2D model minus7552 minus1144 minus1310
Present model minus7285 minus1116 minus1304
RKU model minus7058 minus1087 minus1268
Table 7 Logarithm of the resonance peak amplitude (m) of the freeedge with core layer thickness119867
2= 10mm
1198601
1198602
1198603
2D model minus9120 minus1271 minus1415
Present model minus8812 minus1248 minus1413
RKU model minus8451 minus1206 minus1372
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
Shock and Vibration 9
[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
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
![Page 8: Research Article Dynamic Analysis of Three-Layer Sandwich …downloads.hindawi.com/journals/sv/2015/736256.pdf · 2019-07-31 · Shock and Vibration H 1 H 2 H 3 hn b Damping layer](https://reader034.fdocuments.us/reader034/viewer/2022042310/5ed82ff90fa3e705ec0dffaa/html5/thumbnails/8.jpg)
8 Shock and Vibration
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
Table 6 Logarithm of the resonance peak amplitude (m) of the freeedge with core layer thickness119867
2= 5mm
1198601
1198602
1198603
2D model minus7552 minus1144 minus1310
Present model minus7285 minus1116 minus1304
RKU model minus7058 minus1087 minus1268
Table 7 Logarithm of the resonance peak amplitude (m) of the freeedge with core layer thickness119867
2= 10mm
1198601
1198602
1198603
2D model minus9120 minus1271 minus1415
Present model minus8812 minus1248 minus1413
RKU model minus8451 minus1206 minus1372
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
Shock and Vibration 9
[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
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
![Page 9: Research Article Dynamic Analysis of Three-Layer Sandwich …downloads.hindawi.com/journals/sv/2015/736256.pdf · 2019-07-31 · Shock and Vibration H 1 H 2 H 3 hn b Damping layer](https://reader034.fdocuments.us/reader034/viewer/2022042310/5ed82ff90fa3e705ec0dffaa/html5/thumbnails/9.jpg)
Shock and Vibration 9
[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
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
![Page 10: Research Article Dynamic Analysis of Three-Layer Sandwich …downloads.hindawi.com/journals/sv/2015/736256.pdf · 2019-07-31 · Shock and Vibration H 1 H 2 H 3 hn b Damping layer](https://reader034.fdocuments.us/reader034/viewer/2022042310/5ed82ff90fa3e705ec0dffaa/html5/thumbnails/10.jpg)
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of