Research ArticleAnalytical and Experimental Vibration of Sandwich BeamsHaving Various Boundary Conditions

Eshagh F Joubaneh 1 Oumar R Barry 12 and Hesham E Tanbour1

1College of Engineering and Science Central Michigan University 1200 S Franklin St Mount Pleasant MI 48859 USA2Department of Mechanical Engineering Virginia Tech 635 Prices Fork Road Blacksburg VA 2406 USA

Correspondence should be addressed to Oumar R Barry barry1ocmichedu

Received 8 February 2018 Revised 18 April 2018 Accepted 30 April 2018 Published 25 June 2018

Academic Editor Matteo Strozzi

Copyright copy 2018 Eshagh F Joubaneh et al 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

Generalized differential quadrature (GDQ) method is used to analyze the vibration of sandwich beams with different boundaryconditionsThe equations of motion of the sandwich beam are derived using higher-order sandwich panel theory (HSAPT) Sevenpartial differential equations of motions are obtained through the use of Hamiltonrsquos principle The GDQmethod is utilized to solvethe equations of motion Experiments are conducted to validate the proposed theoryThe results from the analytical model are alsocompared to those from the literature and finite element method (FEM) Parametric studies are conducted to investigate the effectsof different parameters on the natural frequency and response of the sandwich beam under various boundary conditions

1 Introduction

Many engineering applications in aerospace aeronauticsand robotics use multibody systems that consist of flexibleappendages containing sandwich panels These appendagesexperience large deflections due to vibration excitationsSandwich panel structures with soft cores made of foam orlow strength honeycomb like aramid or nomex are used invarious industrial applicationsThey consist of two compositeormetallic layers that are separated by a thick and lightweightcoreThe types of cores in sandwich beam can vary fromhon-eycomb web core and balsa wood to foamed polymer Suchconfigurations in sandwich panels result in high stiffnesslight weight characteristics and high energy absorption capa-bility These kinds of properties make sandwich structuresvery useful in aerospace application Herrmann et al gave abrief overview on sandwich beam explaining the importanceof sandwich panels in the aerospace industry [1] Vinson alsoreviewed the advantages of sandwich beam construction andits role in many applications [2]

The behavior of sandwich panels can be described usingvarious theories based on the type of core In the analysis of

sandwich panels with very rigid cores it is a commonpracticeto neglect the transverse deformation of the core Severalauthors have investigated these types of sandwich panelsAn early theory of sandwich structures known as the first-order shear deformation theory (FSDT) employs the platetheory by taking the shear rigidity of the core into accountbut assumes that the longitudinal deformation is linear in thethickness coordinate and that the core is infinitely rigid inthe transverse direction Although this model is simple theassumptions made are validated by the fact that the sandwichcore is statically loaded and stiff in the thickness coordi-nate

In reality the cores of modern sandwich panels areflexible in all directions Hohe et al acknowledged thisphenomenon by investigating the effect of the transversecompressibility of the core of sandwich beams on the tran-sient dynamic response of structural sandwich panels underrapid loading conditions [3] Many modified theories takeinto consideration different assumptions to more effectivelymodel the stress strain and displacement distributions alongthe thickness coordinate Among the modified theories aretwo different higher-order sandwich panel theory (HSAPT)

HindawiShock and VibrationVolume 2018 Article ID 3682370 15 pageshttpsdoiorg10115520183682370

2 Shock and Vibration

models for sandwich structures The first model is HSAPT(mixed) in which the displacements of the face sheetsalong with the shear stresses in the cores are considered tobe unknown Using various computational models Frostigand Baruch investigated the free vibration of unidirectionalsandwich panels of two core types compressible and incom-pressible using the HSAPT (mixed) model [4] Frostig et alalso studied the behavior of sandwich panels using HSAPT(mixed) for compressible cores while neglecting in-planestresses [5]TheHSAPT approach has also been used in otherinvestigations [6ndash8]

The second model is HSAPT (displacement) whichassumes cubic longitudinal displacement and quadratic ver-tical displacement through the thickness of the sandwichbeam The formulation of this model considers shear straininstead of shear stress Regardless of axial stress in the coreand according to the static equilibrium equation the secondmodel will consider a constant shear distribution withinthe core thickness This type of approximation for sand-wich beam construction with a soft core is appropriate forstatic problems Experimental investigations have shown thatHSAPT accurately predicts displacements and axial strain atthe surface points of support and regions of concentratedloading In certain regions of the core using HSAPT resultsare acceptable approximations for shear stress and axialstrain values through thickness distributions that are awayfrom points of support and areas of concentrated loadingIn contrast obtaining the same values in regions of thecore adjacent to concentrated loading and points of supportusing HSAPT yields inaccurate approximations of the samevalues mentioned previously Since axial stress through thelength of the core in HSAPT is neglected this theory canbe used for the study of composite beams and compositeplates with soft cores possessing only lateral stress Frostiget al studied the free vibration of a unidirectional sandwichpanel which consists of compressible and incompressiblecore by using various computational models [10] His studyalso included both displacement and mix HSAPT formula-tion

Different methods can be used to solve equations ofmotion of HSAPT models The most common methods aredynamic stiffnessmethod (DSM) andGDQmethodDaman-pack and Khalili investigated the higher-order free vibrationof sandwich beams with flexible cores using DSM [11]Tornabene andViola examined general higher-order theoriesof doubly curved laminated composite shells and panels usingthe local GDQ method [15] Hong et al examined thermalinduced vibration of a thermal sleeve using GDQ methodwhich was used to obtain numerical results of two-layercross-ply laminated tubes under thermal vibration [16] Theclassical finite element method has also been used in numer-ousworks to examine the vibration of sandwich beam [17ndash21]Similar to the FEMmethod the quadrature element method(QEM) has also been examined by numerous investigators tostudy sandwich structures [13 14 22ndash25] It should be notedthat the GDQmethod is much accurate and computationallyless intensive than the classical finite element method [26]This makes the GDQ method more appealing than othermethods

Figure 1 Geometry of sandwich beam

The GDQ method is adopted in this paper The objectiveis to (1) show that HSAPT displacement based model issufficient for predicting the dynamic behavior of a sandwichbeam with soft core under various boundary conditions (2)conduct experiments to validate the proposed formulation(3) compare the performance of the GDQ to that of QEMHQEM and FEM ABAQUS for a sandwich beam under var-ious boundary conditions and (4) carry out forced vibrationanalyses to determine the effect of key design variables onthe frequency response curves To this end the governingpartial differential equations of motion are derived usingHamiltonrsquos principle and then the GDQmethod is employedto reduce the partial differential equations to ordinary differ-ential equations Natural frequencies and vibration responsesobtained from the analytical model are compared to resultsfrom the literature In addition finite element analyses areconducted using SolidWorks software to assess the validityof the obtained analytical solutions Experiments were alsoperformed on sandwich beams with different thicknesses andthe results were compared to those obtained using the GDQmethod Parametric studies are carried out to study the effectsof various parameters on the natural frequency and responseof different sandwich beams

2 Derivation of the Governing Equations

Figure 1 depicts a schematic of a cantilever sandwich beamwith length 119871 and width 119887 This beam is composed of threelayers two thin stiff face sheets with thickness ℎ119891 and a thicksoft core with thickness ℎ119888

The general assumptions for the derivation of the govern-ing equation of motion are as follows [11]

(i) All deformations and strains are very small

(ii) The face sheets and the core of the beam are made ofisotropic and homogeneous materials

(iii) The sandwich beam is assumed to be symmetric

(iv) Transverse normal strains are negligible in the facesheets

(v) There is no slippage between layers

(vi) The face sheets are modeled by Euler-Bernoulli beamtheory and the core is modeled using 2D elasticitytheory

Shock and Vibration 3

It should be noted that the core is more soft throughthickness in comparison with the face sheets and the normalstress of the core is negligible The axial and transversedisplacements of a point on the middle of the top face sheetare 119906119905(119909 119905) and119908119905(119909 119905) respectivelyThe transverse and axialdisplacements of a point on the middle of the bottom facesheets are 119906119887(119909 119905) and 119908119887(119909 119905) respectively The axial andtransverse displacements of a point on the middle of the corelayer are 119906119888(119909 119905) and 119908119888(119909 119905) respectively 1205931 is the slope atthe centroid of the core 1205932 and 1205933 are unknown in-planerotations of the core 1205951 and 1205952 are unknown transverserotation of the core The displacement field for the sandwichbeam can be expressed as

119883

119906119905 minus 1199101199081015840119905 minusℎ1198912 lt 119910 lt ℎ1198912119906119888 + 1205931119910 + 12059321199102 + 12059331199103 minusℎ1198882 lt 119910 lt ℎ1198882119906119887 minus 1199101199081015840119887 minusℎ1198912 lt 119910 lt ℎ1198912

119884

119908119905 minusℎ1198912 lt 119910 lt ℎ1198912119908119888 + 1205951119910 + 12059521199102 minusℎ1198882 lt 119910 lt ℎ1198882119908119887 minusℎ1198912 lt 119910 lt ℎ1198912

(1)

HereX andY are displacement fields in x and y directionsBy using the compatibility conditions given as

119883

119883119862119900119903119890 (119910 = ℎ1198882 ) = 119883119879119900119901119891119886119888119890 (119910 = minusℎ1198912 )

119883119862119900119903119890 (119910 = minusℎ1198882 ) = 119883119861119900119905119905119900119898119891119886119888119890 (119910 =ℎ1198912 )

119884

119884119862119900119903119890 (119910 = ℎ1198882 ) = 119884119879119900119901119891119886119888119890 (119910 = minusℎ1198912 )

119884119862119900119903119890 (119910 = minusℎ1198882 ) = 119884119861119900119905119905119900119898119891119886119888119890 (119910 =ℎ1198912 )

(2)

the displacement fields becomes

119883

119906119905 minus 1199101199081015840119905 minusℎ1198912 lt 119910 lt ℎ1198912119906119888 + 1205931119910 + (119906119905 + 119906119887 + ℎ1198912 1199081015840119905 minus

ℎ1198912 1199081015840119887 minus 2119906119888) 21199102

ℎ1198882 + (119906119905 minus 119906119887 +ℎ1198912 1199081015840119905 +

ℎ1198912 1199081015840119887 minus ℎ1198881205931) 41199103

ℎ3119888 minusℎ1198882 lt 119910 lt ℎ1198882119906119887 minus 1199101199081015840119887 minusℎ1198912 lt 119910 lt ℎ1198912

119884

119908119905 minusℎ1198912 lt 119910 lt ℎ1198912119908119888 + (119908119905 minus 119908119887) 119910ℎ119888 + (119908119905 + 119908119887 minus 2119908119888) 21199102ℎ1198882 minusℎ1198882 lt 119910 lt ℎ1198882119908119887 minusℎ1198912 lt 119910 lt ℎ1198912

119885 (119891119900119903 119886119897119897 119897119886119910119890119903119904) 0

(3)

Here 1199081015840119905 and 1199081015840119887 are the rotational displacements of topand bottom face sheets Z defines the displacement field in zdirection which is assumed to be zero Potential energy forsandwich beam 119880 is given as

119880 = 12 (int119881119905 1198641198911205762119909119889119881119905 + int

119881119888

(1198641198881205762119910 + 1198661198881205742119909119910) 119889119881119888+ int119881119887

1198641198911205762119909119889119881119887)(4)

Here 119864119891 and 119864119888 are modulus of elasticity of face sheetsand the core respectively and 119866119888 is the shear modulus of thecore 120576119909 and 120576119910 are normal strains in x and y directions and 120574119909119910is the shear strain of the core119881119905119881119887 and119881119888 are volume of thetop bottom and core layers respectively 119860 119905 119860119887 and 119860119888 arethe area of top bottom and core layers By substituting thestrain displacement equation (3) in (4) the potential energybecomes [11]

119880 = 12 int119871

0int119860119905

119864119891 (1199061015840119905 minus 11991011990810158401015840119905 )2 119889119860 119905 + int119860119887

119864119891 (1199061015840119887 minus 11991011990810158401015840119887 )2 119889119860119887 + int119860119888

119864119888 [(119908119905 minus 119908119887) 1ℎ119888 + (119908119905 + 119908119887 minus 2119908119888)4119910ℎ2119888 ]2 119889119860119888

+ int119860119888

119866119888 [(1205931 + 1199081015840119888) + (119906119905 + 119906119887 + 2ℎ119891 + ℎ1198884 1199081015840119905 minus 2ℎ119891 + ℎ1198884 1199081015840119887 minus 2119906119888) 4119910ℎ2119888 119889119860119888 + int119860119888 (119906119905 minus 119906119887 +3ℎ119891 + ℎ1198886 1199081015840119905 + 3ℎ119891 + ℎ1198886 1199081015840119887 minus ℎ1198881205931 minus ℎ1198883 1199081015840119888) 12119910

2

ℎ3119888 ]2 119889119860119888119889119909

(5)

4 Shock and Vibration

The kinetic energy is given as

119879 = 12 (int119881119905 120588119891 ( 1199091199052 + 1199101199052) 119889119881119905

+ int119881119887

120588119891 ( 1199091198872 + 1199101198872) 119889119881119887 + int119881119888

120588119888 ( 1199091198882 + 1199101198882) 119889119881119888)(6)

where 120588119891 is the face sheets density and 120588119888 is the core den-sity 119909119905 119909119887 and 119909119888 are velocities of top and bottom face sheetsand core in x direction 119910119905 119910119887 and 119910119888 are velocities of top andbottom face sheets and core in y direction By substituting thedisplacement equations into (5) and (6) the governing partialdifferential equations ofmotion and boundary conditions canbe obtained from Hamiltonrsquos principle as

120575int11990521199051

(119879 minus 119880) 119889119905 = 0 (7)

where 120575 is the first variation operation and 1199051 and 1199052are defined as the time period The governing equations ofmotion are obtained by dividing and factoring the coefficientsof 120575119906119905 120575119906119887 120575119908119905 120575119908119887 120575119906119888 120575120593 120575119908119888 in (7) The followingequation is one of the seven obtained equations of motionThe rest of the equations are presented in the Appendix

119898119891 (119905) + 119898119888 [ 120 (119905 + 119887) + 128 (119905 minus 119887) + 115 119888+ 170ℎ119888 1205931 + 140ℎ119891 (1015840119905 minus 1015840119887) + 156ℎ119891 (1015840119905 + 1015840119887)]minus 11986411989111986011989111990610158401015840119905 + 119866119888119860119888ℎ2119888 [43 (119906119905 + 119906119887) + 95 (119906119905 minus 119906119887)minus 83119906119888 minus 45ℎ1198881205931 + (23ℎ119891 + 13ℎ119888) (1199081015840119905 minus 1199081015840119887)+ ( 910ℎ119891 + 310ℎ119888) (1199081015840119905 + 1199081015840119887) + 25ℎ1198881199081015840119888] = 0

(8)

where 119898119888 is the corersquos mass 119898119891 is the surface layersrsquo mass119868119891 is the surface moment of inertia 119860119888 is the corersquos crosssection and 119860119891 is the surface layersrsquo cross section 119905 and119887 are also the accelerations at the middle of the top andbottom face sheets and 119888 is the acceleration at the middle ofthe core 11990610158401015840119905 defines the variation of normal strain related tomiddle of the top face sheet in x direction

3 GDQ Method

TheGDQmethod is applied in order to discrete the equationof motion and boundary condition along the length of the

beam Applying GDQmethod to the equation of motion (see(8)) yields

[119898119891 + 119898119888 ( 120 + 128)]Nsum119895=1

119862(0)119894119895 119905 (119909119895)

+ 119898119888 ( 120 minus 128)119873sum119895=1

119862(0)119894119895 119887 (119909119895) + 119898119888 ( 115)

sdot 119873sum119895=1

119862(0)119894119895 119888 (119909119895) + 119898119888ℎ119888 170119873sum119895=1

119862(0)119894119895 1205931 (119909119895)

+ 119898119888 ( 140 + 156) ℎ119891119873sum119895=1

119862(1)119894119895 119905 (119909119895)

+ 119898119888 (minus 140 + 156) ℎ119891119873sum119895=1

119862(1)119894119895 119887 (119909119895)

minus 119864119891119860119891 119873sum119895=1

119862(2)119894119895 119906119905 (119909119895) + 119866119888119860119888ℎ2119888 (43 + 95)

sdot 119873sum119895=1

119862(0)119894119895 119906119905 (119909119895) + 119866119888119860119888ℎ2119888 (43 minus 95)119873sum119895=1

119862(0)119894119895 119906119887 (119909119895)

minus 119866119888119860119888ℎ2119888 (83)119873sum119895=1

119862(0)119894119895 119906119888 (119909119895)+ 119866119888119860119888ℎ2119888 [(23ℎ119891 + 13ℎ119888) + ( 910ℎ119891 + 310ℎ119888)]

sdot 119873sum119895=1

119862(1)119894119895 119908119905 (119909119895)+ 119866119888119860119888ℎ2119888 [minus (23ℎ119891 + 13ℎ119888) + ( 910ℎ119891 + 310ℎ119888)]

sdot 119873sum119895=1

119862(1)119894119895 119908119887 (119909119895) + 119866119888119860119888ℎ2119888 (25ℎ119888)119873sum119895=1

119862(1)119894119895 119908119888 (119909119895)

minus 119866119888119860119888ℎ2119888 (45ℎ119888)119873sum119895=1

119862(0)119894119895 1205931 (119909119895)

(9)

The solution of the system is achieved by employing alocal version of the well-known GDQ method With respectto the global form this local approach considers localizedinterpolating basis function The numerical technique inGDQ is able to evaluate the 119899th derivative at a generic pointof a sufficiently smooth function 119891(119909) as a weighted linearsum of the function values at some chosen grid points in xdirection These grid points can be distributed through thethickness of the sandwich beam too [14]

119889119899119891 (119909)119889119909119899 (119909119894) cong119895=119873sum119895=1

119862(119899)119894119895 119891 (119909119895) 119891119900119903 119894 = 1 2 119873 (10)

Shock and Vibration 5

where 119862(119899)119894119895 is the weighting coefficient associated with thenth-order derivative and 119873 is the number of grid points inx direction It should be noted that the GDQ is quite differentfrom QEM and HQEM GDQ method uses Chebyshev gridpoints distribution and employs Lagrange function for alldisplacement fields namely 119906119905 119906119887 119906119888 1205931 119908119905 119908119887 119908119888 and1205931 The QEM on the other hand uses expanded Chebyshevgrids and employs Hermit functions for 119908119905 119908119887 and 119908119888 and

Lagrange function for displacement fields 119906119905 119906119887 119906119888 and 1205931HQEM which is an extended version of the QEM employsnodes on both longitudinal and through the thickness (trans-verse) directions whereas the GDQ method and QEM onlyuse nodes in the longitudinal direction The GDQ method isadopted in this paper as such using Lagrange interpolatedpolynomials for all displacements the function 119891(119909) can bewritten as

119891 (119909) = 119895=119873sum119895=1

(119909 minus 1199091) (119909 minus 1199092) (119909 minus 119909119895minus1) (119909 minus 119909119895+1) (119909 minus 119909119873)(119909119895 minus 1199091) (119909119895 minus 1199092) (119909119895 minus 119909119895minus1) (119909119895 minus 119909119873) 119891 (119909119895) (11)

Taking the derivative of (11) with respect to x yields

1198911015840 (119909) = 119895=119873sum119895=1

(119909 minus 1199091) (119909 minus 1199092) (119909 minus 119909119895minus1) (119909 minus 119909119895+1) (119909 minus 119909119873)(119909119895 minus 1199091) (119909119895 minus 1199092) (119909119895 minus 119909119895minus1) (119909119895 minus 119909119873) 1015840

119891 (119909119895) (12)

The coefficient of 119891(119909119895) calls119862(1)119894119895 which can be written as119862(1)119894119895 = 119872(1) (119909119894)(119909119894 minus 119909119895)119872(1) (119909119895)119862(1)119894119894 = minus 119873sum

119895=1119895 =119894

119862(1)119894119895(13)

where

119872(1) (119909119895) = 119873prod119896=1119896 =119895

(119909119895 minus 119909119896) (14)

The higher-order derivatives are obtained as

119862(119898)119894119895 = 119898[[119862(1)119894119895 119862(119898minus1)119894119894 minus 119862(119898minus1)119894119895119909119894 minus 119909119895]]

119894 119895 = 1 2 119873 119898 = 2 3 119873 minus 1 119894 = 119895119862(119898)119894119894 = minus 119873sum

119895=1119895 =119894

119862(119898)119894119895(15)

There are also different arrangements of grid pointswhich are used in GDQ method It has been shown in theliterature that using equal spacing sample of grid points givesinaccurate results Therefore the Chebyshev-Gauss-Lobattodistribution is employed to discretize the spatial domain asfollows [27]

119909119894 = 1198712 [1 minus cos( 119894 minus 1119873 minus 1120587)] 119894 = 1 119873 (16)

Other grid distributions such as expandedChebyshev [1314] can also be used

4 Numerical Results and Discussion

Theproperties of the sandwich beam used for conducting thenumerical simulation are listed in Table 1 Numerical resultsare presented for the free and forced vibration analyses ofsandwich beams with various boundary conditions includingcantilever (C-F) clamped-clamped (C-C) simply-simply (S-S) free-free (F-F) and simply-clamped (S-C)

41 Free Vibration To examine the validity of the presentformulation Table 2 compares the results of theGDQmethodfor the simply supported case to those of [11 12] The resultsshow excellent agreement with the closed-form solutionemployed in [11] Table 2 also indicates that the GDQmethodperforms better than the method used in [12] The validationof the rest of the boundary conditions namely C-F C-C S-SF-F and S-C is shown in Table 3 The natural frequency ofthe sandwich beam for GDQmethod is compared to those ofthe QEM harmonic quadrature element method (HQEM)and finite element method of ABAQUS The results showvery good agreement for all five boundary conditions Ingeneral it can be observed from Table 3 that the results of theGDQmethod and the HQEM are closer to those of ABAQUSthan the QEM This is an indication that the GDQ is moreaccurate than the QEMThe first twomode shapes for C-F S-S and C-C boundary conditions are shown in Figure 2 Theresults show that the plottedmode shapes are similar to thoseobtained in [14]

42 Forced Vibration This section discusses the time andfrequency responses of the sandwich beam with variousboundary conditions All the numerical simulations in thissection are based on a harmonic applied load 119865 at the tip ofthe sandwich beam defined as 119865 = 04 times 103 sin(120596119905) and the

6 Shock and Vibration

Table 1 Parameters of sandwich beam

References 119864119891 (119866119875119886) 119866119888 (119872119875119886) 120588119891 (1198961198921198983) 120588119888 (1198961198921198983) ℎ119891 (119898119898) ℎ119888 (119898119898) 119887 (119898119898) 119871 (119898119898)[11] 210 221 7900 60 19 348 599 260[12] 36 200 4400 521 05 200 200 300[13] 136 128 1800 585 5 19 20 152Experiment 210 221 7900 60 19 15 and 25 599 260

Table 2 The five low natural frequencies for a simply supported sandwich beam (rads)

Mode Present method (GDQ) Exact [11] Model (C) [12]1 204846 204841 2048192 518973 518967 5183373 825024 825019 8224064 1122532 1122527 11159635 1413922 1413919 1400912

material property from [11] listed in Table 1 Figure 3 showsthe time response of the cantilever sandwich beam for threeexcitation frequencies namely 120596 = 011205961 120596 = 0871205961 and120596 ≃ 11205961 Here 1205961 is the first natural frequency of sandwichbeam Beating phenomenon is observed when the excitationfrequency is 120596 = 0871205961 and resonance phenomenon isdemonstrated when the excitation frequency matches thefundamental frequency of the sandwich beam that is 120596 ≃11205961 Figure 3 also shows much smaller vibration amplitudewhen the fundamental frequency is far from the excitationfrequency 120596 = 011205961

The validation of the forced vibration analysis is demon-strated in Figure 4 by comparing the results of the GDQmethod (Case (a) (c) (e)) to those of SolidWorks simulation(Case (b) (d) (f)) It should be noted that this figure showsthe time response measured at the tip of the cantileversandwich beam with varying thickness and width For eachcase the cantilever sandwich beam is excited with its cor-responding fundamental frequency since natural frequencychanges with beam thicknesswidthThe results in this figureshow very good agreement between the GDQ method andSolidWorks simulations

It is also observed in Figure 4 that the vibration amplitudedecreases with increasing core thickness andor increasingface sheet thickness This is expected as increasing thicknessincreases bending stiffness thus resulting in lower deflectionFigure 4 also shows the role of the beam width on thetime response The results show that the vibration amplitudedecreases with increasing beam width This is also expectedas the width increases with bending stiffness Although boththickness and width affect the vibration amplitude only thethickness affects the natural frequency of the beam Thisobservation is in agreement with literatures [28 29]

Figures 5 6 and 7 show the role of geometry parameterson the frequency response curves for cantilever simply-simply supported and clamped-clamped boundary condi-tions respectively Similar trends are observed for all threeboundary conditions The frequency response curves aresignificantly affected by the core and face sheet thicknessesas well as the width of the sandwich beam Increasing

any of these geometry parameters obviously decreases thevibration amplitude The peaks corresponding to the exci-tation frequency change with varying thickness but remainconstant with varying width The natural frequency in gen-eral increases with increasing core thickness for all studiedboundary conditions As for the face sheet thickness thenatural frequency can increase or decrease with varying facesheet thickness The reason for this can be attributed tothe core thickness having more effect on the stiffness thanthe mass whereas the sheet thickness has similar effect onboth mass and stiffness The beam width shows no effecton the natural frequency for all boundary conditions Thisobservation is also in agreement with earlier discussion aboutthe role of the beam width on the natural frequency

5 Experimental Analysis and Discussion

An experiment is conducted to determine the frequencyresponse curve of two cantilever sandwich beams with dif-ferent core thicknesses The material properties of the testedsandwich beams are listed in Table 1 A schematic of theexperimental setup is depicted in Figure 8 The cantileverbeam is installed on a VDL shaker (B ampKV830-335-SPA16K)using a head plate and a fixture both made of aluminum

To reduce experimental error the fixture is designed inSolidWorks such that its fundamental frequency is well abovethe maximum excitation frequency A signal generated fromthe controller (Laser USB 14V) is fed to the power amplifier(LSD SPA 16K) and then to the vibration shaker The signalis fed back to the controller to ensure precise measurementTwo accelerometers (B amp K 8325) are employed for input andoutput measurements One is placed at the top of the fixtureto measure the input acceleration and the other is placedat the tip of the beam to measure the output response Thetest is carried out with a constant velocity of 2 mms and asine sweep is performed for a frequency range of 10 to 2000Hz Measurements are sent to the data acquisition systemthrough a signal analyzer The frequency response plots areobtained and the frequencies corresponding to the peaks ofthese plots are the natural frequencies of the tested sandwich

Shock and Vibration 7

(a) (b)

(c) (d)

(e) (f)

Figure 2 The first two mode shapes of sandwich beam with cantilever ((a) and (b)) simply-simply ((c) and (d)) and clamped-clamped ((e)and (f)) supported edges

beams The experimental results are shown in Figure 9which also illustrates the role of the core thickness on thefrequency response curve The results clearly indicate thatnatural frequency increases with increasing thickness while

vibration response decreases with increasing thickness Thisobservation is in agreement with the results obtained usingthe GDQ method Table 4 compares natural frequenciesobtained experimentally and numerically (GDQmethod) for

8 Shock and Vibration

Figure 3 Tip deflection of free edge of cantilever sandwich beam (120596119905) for applied frequency 1205961205961 = 01 1205961205961 = 087 and 1205961205961 ≃ 1 [9]

Table 3 The five low natural frequencies for different boundary conditions (Hz)

BCs Mode 1 2 3 4 5

C-F

ABAQUS [14] 32740 10791 21380 21753 22164GDQ 32918 108322 215254 220158 225815

QEM [13] 32966 108843 216459 218378 222572HQEM (full [M]) [14] 32782 10824 21539 21759 22179

C-C

ABAQUS [14] 83992 19567 22556 27026 34778GDQ 84417 197658 227456 270982 354127

QEM [13] 84871 198813 226266 272513 356396HQEM (full [M]) [14] 84403 19781 22571 27175 35475

S-S

ABAQUS [14] 61856 13491 15033 21704 23584GDQ 61905 134825 150881 221264 241309

QEM [13] 62222 135286 151545 217686 236977HQEM (full [M]) [14] 61891 13493 15082 21712 23623

F-F

ABAQUS [14] 12902 13669 21537 21692 22146GDQ 129210 136687 219787 220386 224586

QEM [13] 129919 137239 216280 217640 222519HQEM (full [M] [14] 12918 13676 21542 21701 22152

S-CABAQUS [14] 71545 17144 21964 25144 31171

GDQ 71725 172564 22339 25413 315728HQEM (full [M]) [14] 71718 17258 21975 25212 31607

Table 4 The natural frequencies (Hz) for cantilever beam with two different thicknesses

Modes ℎ119888 = 15 mm ℎ119888 = 25 mmGDQmethod Experiment GDQ method Experiment

1 11800 11119 14392 143022 38100 36460 45227 475213 71984 68080 82186 824334 114194 106350 126156 1323815 167551 165207 180017 177260

Shock and Vibration 9

(a) The effect of core by GDQmethod (b) The effect of core by SolidWorks

(c) The effect of face sheets by GDQmethod (d) The effect of face sheets by SolidWorks

(e) The effect of width by GDQmethod (f) The effect of width by SolidWorks

Figure 4 The effect of geometric parameters on the time response at the tip of the cantilever sandwich beam

10 Shock and Vibration

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 5 Vibration amplitude at the tip of the cantilever beam with respect to applied frequency

a core thickness of 15 and 25 mm The results in this tablealso show very good agreementThis is also an indication thatthe present formulation for both free and forced vibrationanalyses of sandwich beams is accurate

6 Conclusions

This paper presents the vibration analysis of sandwich beamswith various boundary conditions The governing equationsof motion are derived using Hamiltonrsquos principle The GDQmethod is utilized for solving the problem SolidWorkssimulation and experimental analyses are conducted to assessthe validity of the GDQ method and the results showvery good agreement Parametric studies are conducted toexamine the role of geometric properties on the time andfrequency response curvesThe results indicate that vibrationamplitude decreases with increasing both core and facesheet thicknesses whereas natural frequency increases withincreasing core sheet thickness but can increase or decreasewith varying face sheet thickness The results also show thatvibration amplitude decreases with increasing beam widthwhereas no effect is observed on the natural frequency for

varying the width of the beam Numerical examples alsodemonstrate that all studied boundary conditions exhibitsimilar effect with varying the aforementioned geometricproperties The findings in this paper are anticipated to beappealing to research communities because of the experi-mental and FEM validation of the GDQ method for thefree and forced vibration analyses of sandwich beams undervarious boundary conditions

Appendix

The rest of the six equations of motion are given as follows

119898119891 (119887) + 119898119888 [ 120 (119905 + 119887) minus 128 (119905 minus 119887) + 115 119888minus 170ℎ119888 1205931 + 140ℎ119891 (1015840119905 minus 1015840119887) minus 156ℎ119891 (1015840119905 + 1015840119887)]minus 11986411989111986011989111990610158401015840119887 + 119866119888119860119888ℎ2119888 [43 (119906119905 + 119906119887) minus 95 (119906119905 minus 119906119887)minus 83119906119888 + 45ℎ1198881205931 + (23ℎ119891 + 13ℎ119888) (1199081015840119905 minus 1199081015840119887)

Shock and Vibration 11

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 6 Vibration amplitude at the middle of the simply-simply supported edges sandwich beam with respect to applied frequency

minus ( 910ℎ119891 + 310ℎ119888) (1199081015840119905 + 1199081015840119887) minus 25ℎ1198881199081015840119888] = 0(A1)

119898119891 (119905 minus 112ℎ211989110158401015840119905 ) + 119898119888 [minus 140ℎ119891 (1015840119905 + 1015840119887)minus 156ℎ119891 (1015840119905 minus 1015840119887) minus 130ℎ1198911015840119888 minus 1140ℎ119888ℎ119891 12059311015840minus 180ℎ2119891 (10158401015840119905 minus 10158401015840119887 ) + 112 (119905 minus 119887)minus 1112ℎ2119891 (10158401015840119905 + 10158401015840119887 ) + 120 (119905 + 119887) + 115119888]+ 1198641198911198681198911199081015840101584010158401015840119905 + 119866119888119860119888ℎ2119888 [minus (23ℎ119891 + 13ℎ119888) (1199061015840119905 + 1199061015840119887)minus ( 910ℎ119891 + 310ℎ119888) (1199061015840119905 minus 1199061015840119887) + (43ℎ119891 + 23ℎ119888) 1199061015840119888+ (25ℎ119891ℎ119888 + 215ℎ2119888) (12059310158401)

minus ( 920ℎ2119891 + 310ℎ119888ℎ119891 + 120ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 )minus (13ℎ119891ℎ119888 + 13ℎ2119891 + 112ℎ2119888) (11990810158401015840119905 minus 11990810158401015840119887 )minus (15ℎ119891ℎ119888 + 115ℎ2119888)11990810158401015840119888 ] + 119864119888119860119888ℎ2119888 [43 (119908119905 + 119908119887)+ (119908119905 minus 119908119887) minus 83119908119888] = 0

(A2)

119898119891 (119887 minus 112ℎ211989110158401015840119887 ) + 119898119888 [ 140ℎ119891 (1015840119905 + 1015840119887)minus 156ℎ119891 (1015840119905 minus 1015840119887) + 130ℎ1198911015840119888 minus 1140ℎ119888ℎ119891 12059311015840+ 180ℎ2119891 (10158401015840119905 minus 10158401015840119887 ) minus 112 (119905 minus 119887)minus 1112ℎ2119891 (10158401015840119905 + 10158401015840119887 ) + 120 (119905 + 119887) + 115119888]

12 Shock and Vibration

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 7 Vibration amplitude at the middle of the clamped-clamped edges sandwich beam with respect to applied frequency

+ 1198641198911198681198911199081015840101584010158401015840119887 + 119866119888119860119888ℎ2119888 [(23ℎ119891 + 13ℎ119888) (1199061015840119905 + 1199061015840119887)minus ( 910ℎ119891 + 310ℎ119888) (1199061015840119905 minus 1199061015840119887) minus (43ℎ119891 + 23ℎ119888) 1199061015840119888+ (25ℎ119891ℎ119888 + 215ℎ2119888) (12059310158401)minus ( 920ℎ2119891 + 310ℎ119888ℎ119891 + 120ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 )+ (13ℎ119891ℎ119888 + 13ℎ2119891 + 112ℎ2119888) (11990810158401015840119905 minus 11990810158401015840119887 )minus (15ℎ119891ℎ119888 + 115ℎ2119888)11990810158401015840119888 ] + 119864119888119860119888ℎ2119888 [43 (119908119905 + 119908119887)minus (119908119905 minus 119908119887) minus 83119908119888] = 0

(A3)

119898119888 [ 815 119888 + 115 (119887 + 119905) + 130ℎ119891 (1015840119905 minus 1015840119887)]+ 119866119888119860119888ℎ1198882 [minus83 (119906119905 + 119906119887) + 163 119906119888minus (43ℎ119891 + 23ℎ119888) (1199081015840119905 minus 1199081015840119887)] = 0

(A4)

119898119888 [ 2105ℎ2119888 1205931 + 170ℎ119888 (119905 minus 119887)+ 1140ℎ119891ℎ119888 (1015840119905 + 1015840119887)] + 119866119888119860119888ℎ1198882 [minus45ℎ119888 (119906119905 minus 119906119887)+ 45ℎ21198881205931 + 415ℎ21198881199081015840119888minus ( 215ℎ2119888 + 25ℎ119888ℎ119891) (1199081015840119905 + 1199081015840119887)] = 0

(A5)

Shock and Vibration 13

Figure 8 Schematic of experimental setup

Figure 9 Vibration amplitude of cantilever sandwich beam fordifferent core thicknesses by experiment

119898119888 [ 815119888 + 115ℎ119888 (119887 + 119905)]+ 119866119888119860119888ℎ1198882 [minus25ℎ119888 (1199061015840119905 minus 1199061015840119887) minus 415ℎ21198881205931minus ( 115ℎ119891ℎ119888 + 115ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 ) minus 815ℎ211988811990810158401015840119888 ]+ 119864119888119860119888ℎ2119888 [minus83 (119908119905 + 119908119887) + 163 119908119888] = 0

(A6)

The essential and natural boundary conditions related to theclamped-free case are given as

119906119905 (0 119905) = 0119875119905 (119897 119905) = 0119908119905 (0 119905) = 0119881119905 (119897 119905) = 01199081199051015840 (0 119905) = 0119872119905 (119897 119905) = 0119908119888 (0 119905) = 0119881119888 (119897 119905) = 0119906119887 (0 119905) = 0119875119887 (119897 119905) = 0119908119887 (0 119905) = 0119881119887 (119897 119905) = 01199081198871015840 (0 119905) = 0119872119887 (119897 119905) = 0

(A7)

where the left side of sandwich beam is clamped and the rightside is free

The axial forces 119875119905119887(119909 119905) the shear forces 119881119905119887119888(119909 119905) andthe bending moments119872119905119887(119909 119905) are obtained as

119875119905 = minus119864119891119860119891 120597119906119905120597119909 (A8)

119875119887 = minus119864119891119860119891 120597119906119887120597119909 (A9)

119881119905 = minus 112119898119891ℎ21198911015840119905 + 119898119888840 [minus36ℎ119891119905 minus 6ℎ119891119887minus 28ℎ119891119888 minus 6ℎ119891ℎ119888 1205931 + 3ℎ21198911015840119887 minus 18ℎ21198911015840119905)+ 1198641198911198681198911199081015840101584010158401015840119905 + 119866119888119860119888840ℎ2119888 [(196ℎ119891 minus 28ℎ119888) 119906119887minus (1316ℎ119891 + 532ℎ119888) 119906119905 + (560ℎ119888 + 1120ℎ119891) 119906119888+ (28ℎ2119888 minus 98ℎ2119891 + 28ℎ119891ℎ119888)11990810158401015840119887minus (112ℎ2119888 + 658ℎ2119891 + 532ℎ2119891 + 532ℎ119891ℎ119888)1199081015840119905+ (336ℎ119891ℎ119888 + 112ℎ2119888) 1205931minus (56ℎ2119888 + 168ℎ119891ℎ119888)1199081015840119888]

(A10)

119881119887 = minus 112119898119891ℎ21198911015840119887 + 119898119888840 (36ℎ119891119887 + 6ℎ119891119905+ 28ℎ119891119888 minus 6ℎ119891ℎ119888 1205931 + 3ℎ21198911015840119905 minus 18ℎ21198911015840119887)

14 Shock and Vibration

+ 1198641198911198681198911199081015840101584010158401015840119887 + 119866119888119860119888840ℎ2119888 [(minus196ℎ119891 + 28ℎ119888) 119906119905+ (1316ℎ119891 + 532ℎ119888) 119906119887 minus (560ℎ119888 + 1120ℎ119891) 119906119888+ (28ℎ2119888 minus 98ℎ2119891 + 28ℎ119891ℎ119888)11990810158401015840119905minus (112ℎ2119888 + 658ℎ2119891 + 532ℎ2119891 + 532ℎ119891ℎ119888)1199081015840119887+ (336ℎ119891ℎ119888 + 112ℎ2119888) 1205931minus (56ℎ2119888 + 168ℎ119891ℎ119888)1199081015840119888]

(A11)

119881119888 = 119866119888119860119888840ℎ2119888 [minus336ℎ119891 (119906119905 minus 119906119887) minus (224ℎ2119888) 1205931

minus (56ℎ2119888 + 168ℎ119891ℎ119888) (1199081015840119905 minus 1199081015840119887) minus 448ℎ21198881199081015840119888](A12)

119872119905 = minus11986411989111986811989111990810158401015840119905 (A13)

119872119887 = minus11986411989111986811989111990810158401015840119887 (A14)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] A S Herrmann P C Zahlen and I Zuardy ldquoSandwich Struc-tures Technology in Commercial Aviationrdquo in Proceedings of the7th International Conference on Sandwich Structures pp 13ndash262005

[2] J Vinson The behavior of sandwich structures of isotropic andcomposite materials Technomic Publishing Company Lan-caster Pennsylvania 1999

[3] J Hohe L Librescu and S Y Oh ldquoDynamic buckling of flat andcurved sandwich panels with transversely compressible corerdquoComposite Structures vol 74 no 1 pp 10ndash24 2006

[4] 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

[5] Y Frostig M Baruch O Vilnay and I Sheinman ldquoHigh-ordertheory for sandwich-beam behavior with transversely flexiblecorerdquo Journal of Engineering Mechanics vol 118 no 5 pp 1026ndash1043 1992

[6] O Rahmani SM R Khalili KMalekzadeh andHHadavinialdquoFree vibration analysis of sandwich structures with a flexiblefunctionally graded syntactic corerdquo Composite Structures vol91 no 2 pp 229ndash235 2009

[7] C Santiuste O TThomsen and Y Frostig ldquoThermo-mechani-cal load interactions in foam cored axi-symmetric sandwich cir-cular plates - High-order and FEmodelsrdquo Composite Structuresvol 93 no 2 pp 369ndash376 2011

[8] R K Fruehmann J M Dulieu-Barton O T Thomsen andS Zhang ldquoExperimental investigation of thermal effects infoam cored sandwich beamsrdquo in Thermomechanics and Infra-Red Imaging vol 7 of Conference Proceedings of the Society forExperimental Mechanics Series pp 83ndash90 Springer NY USA2011

[9] V S Sokolinsky and S R Nutt ldquoConsistent Higher-OrderDynamic Equations for Soft-Core Sandwich Beamsrdquo AIAAJournal vol 42 no 2 pp 374ndash382 2004

[10] Y Frostig C Phan and G Kardomateas ldquoFree vibration ofunidirectional sandwich panels Part I Compressible corerdquoJournal of Sandwich Structures and Materials vol 15 no 4 pp377ndash411 2013

[11] A R Damanpack and S M R Khalili ldquoHigh-order free vibra-tion analysis of sandwich beams with a flexible core usingdynamic stiffness methodrdquo Composite Structures vol 94 no 5pp 1503ndash1514 2012

[12] M Yang and P Qiao ldquoHigher-order impact modeling of sand-wich structures with flexible corerdquo International Journal ofSolids and Structures vol 42 no 20 pp 5460ndash5490 2005

[13] Y Wang and X Wang ldquoFree vibration analysis of soft-coresandwich beams by the novel weak form quadrature elementmethodrdquo Journal of Sandwich Structures and Materials vol 18no 3 pp 294ndash320 2015

[14] X Wang and X Liang ldquoFree vibration of soft-core sandwichpanels with general boundary conditions by harmonic quadra-ture element methodrdquoThin-Walled Structures vol 113 pp 253ndash261 2017

[15] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe local GDQmethod applied to general higher-order theories of doubly-curved laminated composite shells and panels the free vibrationanalysisrdquo Composite Structures vol 116 pp 637ndash660 2014

[16] C C Hong H W Liao L T Lee J B Ke and K C JaneldquoThermally induced vibration of a thermal sleeve with the GDQmethodrdquo International Journal of Mechanical Sciences vol 47no 12 pp 1789ndash1806 2005

[17] H D Chalak A Chakrabarti M A Iqbal and A H SheikhldquoVibration of laminated sandwich beams having soft corerdquoJournal of Vibration and Control vol 18 no 10 pp 1422ndash14352012

[18] K M Ahmed ldquoFree vibration of curved sandwich beams by themethod of finite elementsrdquo Journal of Sound and Vibration vol18 no 1 pp 61ndash74 1971

[19] N Nanda S Kapuria and S Gopalakrishnan ldquoSpectral finiteelement based on an efficient layerwise theory for wave prop-agation analysis of composite and sandwich beamsrdquo Journal ofSound and Vibration vol 333 no 31 pp 20ndash27 2014

[20] Y Pourvais P Asgari A R Moradi and O Rahmani ldquoExper-imental and finite element analysis of higher order behaviourof sandwich beams using digital projection moirerdquo PolymerTesting vol 38 pp 7ndash17 2014

[21] R Long O Barry and D C D Oguamanam ldquoFinite elementfree vibration analysis of soft-core sandwich beamsrdquo AIAAJournal vol 50 no 1 pp 235ndash238 2012

[22] W L Chen A G Striz and C W Bert ldquoHigh-accuracy planestress and plate elements in the quadrature element methodrdquoInternational Journal of Solids and Structures vol 37 no 4 pp627ndash647 2000

[23] Y Xing and B Liu ldquoHigh-accuracy differential quadraturefinite element method and its application to free vibrations ofthin plate with curvilinear domainrdquo International Journal for

Shock and Vibration 15

Numerical Methods in Engineering vol 80 no 13 pp 1718ndash17422009

[24] YWang and XWang ldquoStatic analysis of higher order sandwichbeams by weak form quadrature element methodrdquo CompositeStructures vol 116 no 1 pp 841ndash848 2014

[25] F Tornabene S Brischetto N Fantuzzi and M BacciocchildquoBoundary conditions in 2Dnumerical and 3D exactmodels forcylindrical bending analysis of functionally graded structuresrdquoShock and Vibration vol 2016 Article ID 2373862 2016

[26] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe GDQmethod for the free vibration analysis of arbitrarily shaped lami-nated composite shells using a NURBS-based isogeometricapproachrdquo Composite Structures vol 154 pp 190ndash218 2016

[27] Z Zong and Y Zhang Advanced differential quadrature meth-ods CRC Press NY USA 2009

[28] N A Rubayi and S Charoenree ldquoNatural frequencies of vibra-tion of cantilever sandwich beamsrdquoComputersampStructures vol7 no 6 pp 737ndash745 1977

[29] M Azimi M Shahravi and E F Joubaneh ldquoDynamic analysisof maneuvering flexible spacecraft appendage using higherorder sandwich panel theoryrdquo Latin American Journal of Solidsand Structures vol 13 no 2 pp 296ndash313 2016

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Shock and Vibration 3

It should be noted that the core is more soft throughthickness in comparison with the face sheets and the normalstress of the core is negligible The axial and transversedisplacements of a point on the middle of the top face sheetare 119906119905(119909 119905) and119908119905(119909 119905) respectivelyThe transverse and axialdisplacements of a point on the middle of the bottom facesheets are 119906119887(119909 119905) and 119908119887(119909 119905) respectively The axial andtransverse displacements of a point on the middle of the corelayer are 119906119888(119909 119905) and 119908119888(119909 119905) respectively 1205931 is the slope atthe centroid of the core 1205932 and 1205933 are unknown in-planerotations of the core 1205951 and 1205952 are unknown transverserotation of the core The displacement field for the sandwichbeam can be expressed as

119883

119906119905 minus 1199101199081015840119905 minusℎ1198912 lt 119910 lt ℎ1198912119906119888 + 1205931119910 + 12059321199102 + 12059331199103 minusℎ1198882 lt 119910 lt ℎ1198882119906119887 minus 1199101199081015840119887 minusℎ1198912 lt 119910 lt ℎ1198912

119884

119908119905 minusℎ1198912 lt 119910 lt ℎ1198912119908119888 + 1205951119910 + 12059521199102 minusℎ1198882 lt 119910 lt ℎ1198882119908119887 minusℎ1198912 lt 119910 lt ℎ1198912

(1)

HereX andY are displacement fields in x and y directionsBy using the compatibility conditions given as

119883

119883119862119900119903119890 (119910 = ℎ1198882 ) = 119883119879119900119901119891119886119888119890 (119910 = minusℎ1198912 )

119883119862119900119903119890 (119910 = minusℎ1198882 ) = 119883119861119900119905119905119900119898119891119886119888119890 (119910 =ℎ1198912 )

119884

119884119862119900119903119890 (119910 = ℎ1198882 ) = 119884119879119900119901119891119886119888119890 (119910 = minusℎ1198912 )

119884119862119900119903119890 (119910 = minusℎ1198882 ) = 119884119861119900119905119905119900119898119891119886119888119890 (119910 =ℎ1198912 )

(2)

the displacement fields becomes

119883

119906119905 minus 1199101199081015840119905 minusℎ1198912 lt 119910 lt ℎ1198912119906119888 + 1205931119910 + (119906119905 + 119906119887 + ℎ1198912 1199081015840119905 minus

ℎ1198912 1199081015840119887 minus 2119906119888) 21199102

ℎ1198882 + (119906119905 minus 119906119887 +ℎ1198912 1199081015840119905 +

ℎ1198912 1199081015840119887 minus ℎ1198881205931) 41199103

ℎ3119888 minusℎ1198882 lt 119910 lt ℎ1198882119906119887 minus 1199101199081015840119887 minusℎ1198912 lt 119910 lt ℎ1198912

119884

119908119905 minusℎ1198912 lt 119910 lt ℎ1198912119908119888 + (119908119905 minus 119908119887) 119910ℎ119888 + (119908119905 + 119908119887 minus 2119908119888) 21199102ℎ1198882 minusℎ1198882 lt 119910 lt ℎ1198882119908119887 minusℎ1198912 lt 119910 lt ℎ1198912

119885 (119891119900119903 119886119897119897 119897119886119910119890119903119904) 0

(3)

Here 1199081015840119905 and 1199081015840119887 are the rotational displacements of topand bottom face sheets Z defines the displacement field in zdirection which is assumed to be zero Potential energy forsandwich beam 119880 is given as

119880 = 12 (int119881119905 1198641198911205762119909119889119881119905 + int

119881119888

(1198641198881205762119910 + 1198661198881205742119909119910) 119889119881119888+ int119881119887

1198641198911205762119909119889119881119887)(4)

Here 119864119891 and 119864119888 are modulus of elasticity of face sheetsand the core respectively and 119866119888 is the shear modulus of thecore 120576119909 and 120576119910 are normal strains in x and y directions and 120574119909119910is the shear strain of the core119881119905119881119887 and119881119888 are volume of thetop bottom and core layers respectively 119860 119905 119860119887 and 119860119888 arethe area of top bottom and core layers By substituting thestrain displacement equation (3) in (4) the potential energybecomes [11]

119880 = 12 int119871

0int119860119905

119864119891 (1199061015840119905 minus 11991011990810158401015840119905 )2 119889119860 119905 + int119860119887

119864119891 (1199061015840119887 minus 11991011990810158401015840119887 )2 119889119860119887 + int119860119888

119864119888 [(119908119905 minus 119908119887) 1ℎ119888 + (119908119905 + 119908119887 minus 2119908119888)4119910ℎ2119888 ]2 119889119860119888

+ int119860119888

119866119888 [(1205931 + 1199081015840119888) + (119906119905 + 119906119887 + 2ℎ119891 + ℎ1198884 1199081015840119905 minus 2ℎ119891 + ℎ1198884 1199081015840119887 minus 2119906119888) 4119910ℎ2119888 119889119860119888 + int119860119888 (119906119905 minus 119906119887 +3ℎ119891 + ℎ1198886 1199081015840119905 + 3ℎ119891 + ℎ1198886 1199081015840119887 minus ℎ1198881205931 minus ℎ1198883 1199081015840119888) 12119910

2

ℎ3119888 ]2 119889119860119888119889119909

(5)

4 Shock and Vibration

The kinetic energy is given as

119879 = 12 (int119881119905 120588119891 ( 1199091199052 + 1199101199052) 119889119881119905

+ int119881119887

120588119891 ( 1199091198872 + 1199101198872) 119889119881119887 + int119881119888

120588119888 ( 1199091198882 + 1199101198882) 119889119881119888)(6)

where 120588119891 is the face sheets density and 120588119888 is the core den-sity 119909119905 119909119887 and 119909119888 are velocities of top and bottom face sheetsand core in x direction 119910119905 119910119887 and 119910119888 are velocities of top andbottom face sheets and core in y direction By substituting thedisplacement equations into (5) and (6) the governing partialdifferential equations ofmotion and boundary conditions canbe obtained from Hamiltonrsquos principle as

120575int11990521199051

(119879 minus 119880) 119889119905 = 0 (7)

where 120575 is the first variation operation and 1199051 and 1199052are defined as the time period The governing equations ofmotion are obtained by dividing and factoring the coefficientsof 120575119906119905 120575119906119887 120575119908119905 120575119908119887 120575119906119888 120575120593 120575119908119888 in (7) The followingequation is one of the seven obtained equations of motionThe rest of the equations are presented in the Appendix

119898119891 (119905) + 119898119888 [ 120 (119905 + 119887) + 128 (119905 minus 119887) + 115 119888+ 170ℎ119888 1205931 + 140ℎ119891 (1015840119905 minus 1015840119887) + 156ℎ119891 (1015840119905 + 1015840119887)]minus 11986411989111986011989111990610158401015840119905 + 119866119888119860119888ℎ2119888 [43 (119906119905 + 119906119887) + 95 (119906119905 minus 119906119887)minus 83119906119888 minus 45ℎ1198881205931 + (23ℎ119891 + 13ℎ119888) (1199081015840119905 minus 1199081015840119887)+ ( 910ℎ119891 + 310ℎ119888) (1199081015840119905 + 1199081015840119887) + 25ℎ1198881199081015840119888] = 0

(8)

where 119898119888 is the corersquos mass 119898119891 is the surface layersrsquo mass119868119891 is the surface moment of inertia 119860119888 is the corersquos crosssection and 119860119891 is the surface layersrsquo cross section 119905 and119887 are also the accelerations at the middle of the top andbottom face sheets and 119888 is the acceleration at the middle ofthe core 11990610158401015840119905 defines the variation of normal strain related tomiddle of the top face sheet in x direction

3 GDQ Method

TheGDQmethod is applied in order to discrete the equationof motion and boundary condition along the length of the

beam Applying GDQmethod to the equation of motion (see(8)) yields

[119898119891 + 119898119888 ( 120 + 128)]Nsum119895=1

119862(0)119894119895 119905 (119909119895)

+ 119898119888 ( 120 minus 128)119873sum119895=1

119862(0)119894119895 119887 (119909119895) + 119898119888 ( 115)

sdot 119873sum119895=1

119862(0)119894119895 119888 (119909119895) + 119898119888ℎ119888 170119873sum119895=1

119862(0)119894119895 1205931 (119909119895)

+ 119898119888 ( 140 + 156) ℎ119891119873sum119895=1

119862(1)119894119895 119905 (119909119895)

+ 119898119888 (minus 140 + 156) ℎ119891119873sum119895=1

119862(1)119894119895 119887 (119909119895)

minus 119864119891119860119891 119873sum119895=1

119862(2)119894119895 119906119905 (119909119895) + 119866119888119860119888ℎ2119888 (43 + 95)

sdot 119873sum119895=1

119862(0)119894119895 119906119905 (119909119895) + 119866119888119860119888ℎ2119888 (43 minus 95)119873sum119895=1

119862(0)119894119895 119906119887 (119909119895)

minus 119866119888119860119888ℎ2119888 (83)119873sum119895=1

119862(0)119894119895 119906119888 (119909119895)+ 119866119888119860119888ℎ2119888 [(23ℎ119891 + 13ℎ119888) + ( 910ℎ119891 + 310ℎ119888)]

sdot 119873sum119895=1

119862(1)119894119895 119908119905 (119909119895)+ 119866119888119860119888ℎ2119888 [minus (23ℎ119891 + 13ℎ119888) + ( 910ℎ119891 + 310ℎ119888)]

sdot 119873sum119895=1

119862(1)119894119895 119908119887 (119909119895) + 119866119888119860119888ℎ2119888 (25ℎ119888)119873sum119895=1

119862(1)119894119895 119908119888 (119909119895)

minus 119866119888119860119888ℎ2119888 (45ℎ119888)119873sum119895=1

119862(0)119894119895 1205931 (119909119895)

(9)

The solution of the system is achieved by employing alocal version of the well-known GDQ method With respectto the global form this local approach considers localizedinterpolating basis function The numerical technique inGDQ is able to evaluate the 119899th derivative at a generic pointof a sufficiently smooth function 119891(119909) as a weighted linearsum of the function values at some chosen grid points in xdirection These grid points can be distributed through thethickness of the sandwich beam too [14]

119889119899119891 (119909)119889119909119899 (119909119894) cong119895=119873sum119895=1

119862(119899)119894119895 119891 (119909119895) 119891119900119903 119894 = 1 2 119873 (10)

Shock and Vibration 5

where 119862(119899)119894119895 is the weighting coefficient associated with thenth-order derivative and 119873 is the number of grid points inx direction It should be noted that the GDQ is quite differentfrom QEM and HQEM GDQ method uses Chebyshev gridpoints distribution and employs Lagrange function for alldisplacement fields namely 119906119905 119906119887 119906119888 1205931 119908119905 119908119887 119908119888 and1205931 The QEM on the other hand uses expanded Chebyshevgrids and employs Hermit functions for 119908119905 119908119887 and 119908119888 and

Lagrange function for displacement fields 119906119905 119906119887 119906119888 and 1205931HQEM which is an extended version of the QEM employsnodes on both longitudinal and through the thickness (trans-verse) directions whereas the GDQ method and QEM onlyuse nodes in the longitudinal direction The GDQ method isadopted in this paper as such using Lagrange interpolatedpolynomials for all displacements the function 119891(119909) can bewritten as

119891 (119909) = 119895=119873sum119895=1

(119909 minus 1199091) (119909 minus 1199092) (119909 minus 119909119895minus1) (119909 minus 119909119895+1) (119909 minus 119909119873)(119909119895 minus 1199091) (119909119895 minus 1199092) (119909119895 minus 119909119895minus1) (119909119895 minus 119909119873) 119891 (119909119895) (11)

Taking the derivative of (11) with respect to x yields

1198911015840 (119909) = 119895=119873sum119895=1

(119909 minus 1199091) (119909 minus 1199092) (119909 minus 119909119895minus1) (119909 minus 119909119895+1) (119909 minus 119909119873)(119909119895 minus 1199091) (119909119895 minus 1199092) (119909119895 minus 119909119895minus1) (119909119895 minus 119909119873) 1015840

119891 (119909119895) (12)

The coefficient of 119891(119909119895) calls119862(1)119894119895 which can be written as119862(1)119894119895 = 119872(1) (119909119894)(119909119894 minus 119909119895)119872(1) (119909119895)119862(1)119894119894 = minus 119873sum

119895=1119895 =119894

119862(1)119894119895(13)

where

119872(1) (119909119895) = 119873prod119896=1119896 =119895

(119909119895 minus 119909119896) (14)

The higher-order derivatives are obtained as

119862(119898)119894119895 = 119898[[119862(1)119894119895 119862(119898minus1)119894119894 minus 119862(119898minus1)119894119895119909119894 minus 119909119895]]

119894 119895 = 1 2 119873 119898 = 2 3 119873 minus 1 119894 = 119895119862(119898)119894119894 = minus 119873sum

119895=1119895 =119894

119862(119898)119894119895(15)

There are also different arrangements of grid pointswhich are used in GDQ method It has been shown in theliterature that using equal spacing sample of grid points givesinaccurate results Therefore the Chebyshev-Gauss-Lobattodistribution is employed to discretize the spatial domain asfollows [27]

119909119894 = 1198712 [1 minus cos( 119894 minus 1119873 minus 1120587)] 119894 = 1 119873 (16)

Other grid distributions such as expandedChebyshev [1314] can also be used

4 Numerical Results and Discussion

Theproperties of the sandwich beam used for conducting thenumerical simulation are listed in Table 1 Numerical resultsare presented for the free and forced vibration analyses ofsandwich beams with various boundary conditions includingcantilever (C-F) clamped-clamped (C-C) simply-simply (S-S) free-free (F-F) and simply-clamped (S-C)

41 Free Vibration To examine the validity of the presentformulation Table 2 compares the results of theGDQmethodfor the simply supported case to those of [11 12] The resultsshow excellent agreement with the closed-form solutionemployed in [11] Table 2 also indicates that the GDQmethodperforms better than the method used in [12] The validationof the rest of the boundary conditions namely C-F C-C S-SF-F and S-C is shown in Table 3 The natural frequency ofthe sandwich beam for GDQmethod is compared to those ofthe QEM harmonic quadrature element method (HQEM)and finite element method of ABAQUS The results showvery good agreement for all five boundary conditions Ingeneral it can be observed from Table 3 that the results of theGDQmethod and the HQEM are closer to those of ABAQUSthan the QEM This is an indication that the GDQ is moreaccurate than the QEMThe first twomode shapes for C-F S-S and C-C boundary conditions are shown in Figure 2 Theresults show that the plottedmode shapes are similar to thoseobtained in [14]

42 Forced Vibration This section discusses the time andfrequency responses of the sandwich beam with variousboundary conditions All the numerical simulations in thissection are based on a harmonic applied load 119865 at the tip ofthe sandwich beam defined as 119865 = 04 times 103 sin(120596119905) and the

6 Shock and Vibration

Table 1 Parameters of sandwich beam

References 119864119891 (119866119875119886) 119866119888 (119872119875119886) 120588119891 (1198961198921198983) 120588119888 (1198961198921198983) ℎ119891 (119898119898) ℎ119888 (119898119898) 119887 (119898119898) 119871 (119898119898)[11] 210 221 7900 60 19 348 599 260[12] 36 200 4400 521 05 200 200 300[13] 136 128 1800 585 5 19 20 152Experiment 210 221 7900 60 19 15 and 25 599 260

Table 2 The five low natural frequencies for a simply supported sandwich beam (rads)

Mode Present method (GDQ) Exact [11] Model (C) [12]1 204846 204841 2048192 518973 518967 5183373 825024 825019 8224064 1122532 1122527 11159635 1413922 1413919 1400912

material property from [11] listed in Table 1 Figure 3 showsthe time response of the cantilever sandwich beam for threeexcitation frequencies namely 120596 = 011205961 120596 = 0871205961 and120596 ≃ 11205961 Here 1205961 is the first natural frequency of sandwichbeam Beating phenomenon is observed when the excitationfrequency is 120596 = 0871205961 and resonance phenomenon isdemonstrated when the excitation frequency matches thefundamental frequency of the sandwich beam that is 120596 ≃11205961 Figure 3 also shows much smaller vibration amplitudewhen the fundamental frequency is far from the excitationfrequency 120596 = 011205961

The validation of the forced vibration analysis is demon-strated in Figure 4 by comparing the results of the GDQmethod (Case (a) (c) (e)) to those of SolidWorks simulation(Case (b) (d) (f)) It should be noted that this figure showsthe time response measured at the tip of the cantileversandwich beam with varying thickness and width For eachcase the cantilever sandwich beam is excited with its cor-responding fundamental frequency since natural frequencychanges with beam thicknesswidthThe results in this figureshow very good agreement between the GDQ method andSolidWorks simulations

It is also observed in Figure 4 that the vibration amplitudedecreases with increasing core thickness andor increasingface sheet thickness This is expected as increasing thicknessincreases bending stiffness thus resulting in lower deflectionFigure 4 also shows the role of the beam width on thetime response The results show that the vibration amplitudedecreases with increasing beam width This is also expectedas the width increases with bending stiffness Although boththickness and width affect the vibration amplitude only thethickness affects the natural frequency of the beam Thisobservation is in agreement with literatures [28 29]

Figures 5 6 and 7 show the role of geometry parameterson the frequency response curves for cantilever simply-simply supported and clamped-clamped boundary condi-tions respectively Similar trends are observed for all threeboundary conditions The frequency response curves aresignificantly affected by the core and face sheet thicknessesas well as the width of the sandwich beam Increasing

any of these geometry parameters obviously decreases thevibration amplitude The peaks corresponding to the exci-tation frequency change with varying thickness but remainconstant with varying width The natural frequency in gen-eral increases with increasing core thickness for all studiedboundary conditions As for the face sheet thickness thenatural frequency can increase or decrease with varying facesheet thickness The reason for this can be attributed tothe core thickness having more effect on the stiffness thanthe mass whereas the sheet thickness has similar effect onboth mass and stiffness The beam width shows no effecton the natural frequency for all boundary conditions Thisobservation is also in agreement with earlier discussion aboutthe role of the beam width on the natural frequency

5 Experimental Analysis and Discussion

An experiment is conducted to determine the frequencyresponse curve of two cantilever sandwich beams with dif-ferent core thicknesses The material properties of the testedsandwich beams are listed in Table 1 A schematic of theexperimental setup is depicted in Figure 8 The cantileverbeam is installed on a VDL shaker (B ampKV830-335-SPA16K)using a head plate and a fixture both made of aluminum

To reduce experimental error the fixture is designed inSolidWorks such that its fundamental frequency is well abovethe maximum excitation frequency A signal generated fromthe controller (Laser USB 14V) is fed to the power amplifier(LSD SPA 16K) and then to the vibration shaker The signalis fed back to the controller to ensure precise measurementTwo accelerometers (B amp K 8325) are employed for input andoutput measurements One is placed at the top of the fixtureto measure the input acceleration and the other is placedat the tip of the beam to measure the output response Thetest is carried out with a constant velocity of 2 mms and asine sweep is performed for a frequency range of 10 to 2000Hz Measurements are sent to the data acquisition systemthrough a signal analyzer The frequency response plots areobtained and the frequencies corresponding to the peaks ofthese plots are the natural frequencies of the tested sandwich

Shock and Vibration 7

(a) (b)

(c) (d)

(e) (f)

Figure 2 The first two mode shapes of sandwich beam with cantilever ((a) and (b)) simply-simply ((c) and (d)) and clamped-clamped ((e)and (f)) supported edges

beams The experimental results are shown in Figure 9which also illustrates the role of the core thickness on thefrequency response curve The results clearly indicate thatnatural frequency increases with increasing thickness while

vibration response decreases with increasing thickness Thisobservation is in agreement with the results obtained usingthe GDQ method Table 4 compares natural frequenciesobtained experimentally and numerically (GDQmethod) for

8 Shock and Vibration

Figure 3 Tip deflection of free edge of cantilever sandwich beam (120596119905) for applied frequency 1205961205961 = 01 1205961205961 = 087 and 1205961205961 ≃ 1 [9]

Table 3 The five low natural frequencies for different boundary conditions (Hz)

BCs Mode 1 2 3 4 5

C-F

ABAQUS [14] 32740 10791 21380 21753 22164GDQ 32918 108322 215254 220158 225815

QEM [13] 32966 108843 216459 218378 222572HQEM (full [M]) [14] 32782 10824 21539 21759 22179

C-C

ABAQUS [14] 83992 19567 22556 27026 34778GDQ 84417 197658 227456 270982 354127

QEM [13] 84871 198813 226266 272513 356396HQEM (full [M]) [14] 84403 19781 22571 27175 35475

S-S

ABAQUS [14] 61856 13491 15033 21704 23584GDQ 61905 134825 150881 221264 241309

QEM [13] 62222 135286 151545 217686 236977HQEM (full [M]) [14] 61891 13493 15082 21712 23623

F-F

ABAQUS [14] 12902 13669 21537 21692 22146GDQ 129210 136687 219787 220386 224586

QEM [13] 129919 137239 216280 217640 222519HQEM (full [M] [14] 12918 13676 21542 21701 22152

S-CABAQUS [14] 71545 17144 21964 25144 31171

GDQ 71725 172564 22339 25413 315728HQEM (full [M]) [14] 71718 17258 21975 25212 31607

Table 4 The natural frequencies (Hz) for cantilever beam with two different thicknesses

Modes ℎ119888 = 15 mm ℎ119888 = 25 mmGDQmethod Experiment GDQ method Experiment

1 11800 11119 14392 143022 38100 36460 45227 475213 71984 68080 82186 824334 114194 106350 126156 1323815 167551 165207 180017 177260

Shock and Vibration 9

(a) The effect of core by GDQmethod (b) The effect of core by SolidWorks

(c) The effect of face sheets by GDQmethod (d) The effect of face sheets by SolidWorks

(e) The effect of width by GDQmethod (f) The effect of width by SolidWorks

Figure 4 The effect of geometric parameters on the time response at the tip of the cantilever sandwich beam

10 Shock and Vibration

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 5 Vibration amplitude at the tip of the cantilever beam with respect to applied frequency

a core thickness of 15 and 25 mm The results in this tablealso show very good agreementThis is also an indication thatthe present formulation for both free and forced vibrationanalyses of sandwich beams is accurate

6 Conclusions

This paper presents the vibration analysis of sandwich beamswith various boundary conditions The governing equationsof motion are derived using Hamiltonrsquos principle The GDQmethod is utilized for solving the problem SolidWorkssimulation and experimental analyses are conducted to assessthe validity of the GDQ method and the results showvery good agreement Parametric studies are conducted toexamine the role of geometric properties on the time andfrequency response curvesThe results indicate that vibrationamplitude decreases with increasing both core and facesheet thicknesses whereas natural frequency increases withincreasing core sheet thickness but can increase or decreasewith varying face sheet thickness The results also show thatvibration amplitude decreases with increasing beam widthwhereas no effect is observed on the natural frequency for

varying the width of the beam Numerical examples alsodemonstrate that all studied boundary conditions exhibitsimilar effect with varying the aforementioned geometricproperties The findings in this paper are anticipated to beappealing to research communities because of the experi-mental and FEM validation of the GDQ method for thefree and forced vibration analyses of sandwich beams undervarious boundary conditions

Appendix

The rest of the six equations of motion are given as follows

119898119891 (119887) + 119898119888 [ 120 (119905 + 119887) minus 128 (119905 minus 119887) + 115 119888minus 170ℎ119888 1205931 + 140ℎ119891 (1015840119905 minus 1015840119887) minus 156ℎ119891 (1015840119905 + 1015840119887)]minus 11986411989111986011989111990610158401015840119887 + 119866119888119860119888ℎ2119888 [43 (119906119905 + 119906119887) minus 95 (119906119905 minus 119906119887)minus 83119906119888 + 45ℎ1198881205931 + (23ℎ119891 + 13ℎ119888) (1199081015840119905 minus 1199081015840119887)

Shock and Vibration 11

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 6 Vibration amplitude at the middle of the simply-simply supported edges sandwich beam with respect to applied frequency

minus ( 910ℎ119891 + 310ℎ119888) (1199081015840119905 + 1199081015840119887) minus 25ℎ1198881199081015840119888] = 0(A1)

119898119891 (119905 minus 112ℎ211989110158401015840119905 ) + 119898119888 [minus 140ℎ119891 (1015840119905 + 1015840119887)minus 156ℎ119891 (1015840119905 minus 1015840119887) minus 130ℎ1198911015840119888 minus 1140ℎ119888ℎ119891 12059311015840minus 180ℎ2119891 (10158401015840119905 minus 10158401015840119887 ) + 112 (119905 minus 119887)minus 1112ℎ2119891 (10158401015840119905 + 10158401015840119887 ) + 120 (119905 + 119887) + 115119888]+ 1198641198911198681198911199081015840101584010158401015840119905 + 119866119888119860119888ℎ2119888 [minus (23ℎ119891 + 13ℎ119888) (1199061015840119905 + 1199061015840119887)minus ( 910ℎ119891 + 310ℎ119888) (1199061015840119905 minus 1199061015840119887) + (43ℎ119891 + 23ℎ119888) 1199061015840119888+ (25ℎ119891ℎ119888 + 215ℎ2119888) (12059310158401)

minus ( 920ℎ2119891 + 310ℎ119888ℎ119891 + 120ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 )minus (13ℎ119891ℎ119888 + 13ℎ2119891 + 112ℎ2119888) (11990810158401015840119905 minus 11990810158401015840119887 )minus (15ℎ119891ℎ119888 + 115ℎ2119888)11990810158401015840119888 ] + 119864119888119860119888ℎ2119888 [43 (119908119905 + 119908119887)+ (119908119905 minus 119908119887) minus 83119908119888] = 0

(A2)

119898119891 (119887 minus 112ℎ211989110158401015840119887 ) + 119898119888 [ 140ℎ119891 (1015840119905 + 1015840119887)minus 156ℎ119891 (1015840119905 minus 1015840119887) + 130ℎ1198911015840119888 minus 1140ℎ119888ℎ119891 12059311015840+ 180ℎ2119891 (10158401015840119905 minus 10158401015840119887 ) minus 112 (119905 minus 119887)minus 1112ℎ2119891 (10158401015840119905 + 10158401015840119887 ) + 120 (119905 + 119887) + 115119888]

12 Shock and Vibration

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 7 Vibration amplitude at the middle of the clamped-clamped edges sandwich beam with respect to applied frequency

+ 1198641198911198681198911199081015840101584010158401015840119887 + 119866119888119860119888ℎ2119888 [(23ℎ119891 + 13ℎ119888) (1199061015840119905 + 1199061015840119887)minus ( 910ℎ119891 + 310ℎ119888) (1199061015840119905 minus 1199061015840119887) minus (43ℎ119891 + 23ℎ119888) 1199061015840119888+ (25ℎ119891ℎ119888 + 215ℎ2119888) (12059310158401)minus ( 920ℎ2119891 + 310ℎ119888ℎ119891 + 120ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 )+ (13ℎ119891ℎ119888 + 13ℎ2119891 + 112ℎ2119888) (11990810158401015840119905 minus 11990810158401015840119887 )minus (15ℎ119891ℎ119888 + 115ℎ2119888)11990810158401015840119888 ] + 119864119888119860119888ℎ2119888 [43 (119908119905 + 119908119887)minus (119908119905 minus 119908119887) minus 83119908119888] = 0

(A3)

119898119888 [ 815 119888 + 115 (119887 + 119905) + 130ℎ119891 (1015840119905 minus 1015840119887)]+ 119866119888119860119888ℎ1198882 [minus83 (119906119905 + 119906119887) + 163 119906119888minus (43ℎ119891 + 23ℎ119888) (1199081015840119905 minus 1199081015840119887)] = 0

(A4)

119898119888 [ 2105ℎ2119888 1205931 + 170ℎ119888 (119905 minus 119887)+ 1140ℎ119891ℎ119888 (1015840119905 + 1015840119887)] + 119866119888119860119888ℎ1198882 [minus45ℎ119888 (119906119905 minus 119906119887)+ 45ℎ21198881205931 + 415ℎ21198881199081015840119888minus ( 215ℎ2119888 + 25ℎ119888ℎ119891) (1199081015840119905 + 1199081015840119887)] = 0

(A5)

Shock and Vibration 13

Figure 8 Schematic of experimental setup

Figure 9 Vibration amplitude of cantilever sandwich beam fordifferent core thicknesses by experiment

119898119888 [ 815119888 + 115ℎ119888 (119887 + 119905)]+ 119866119888119860119888ℎ1198882 [minus25ℎ119888 (1199061015840119905 minus 1199061015840119887) minus 415ℎ21198881205931minus ( 115ℎ119891ℎ119888 + 115ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 ) minus 815ℎ211988811990810158401015840119888 ]+ 119864119888119860119888ℎ2119888 [minus83 (119908119905 + 119908119887) + 163 119908119888] = 0

(A6)

The essential and natural boundary conditions related to theclamped-free case are given as

119906119905 (0 119905) = 0119875119905 (119897 119905) = 0119908119905 (0 119905) = 0119881119905 (119897 119905) = 01199081199051015840 (0 119905) = 0119872119905 (119897 119905) = 0119908119888 (0 119905) = 0119881119888 (119897 119905) = 0119906119887 (0 119905) = 0119875119887 (119897 119905) = 0119908119887 (0 119905) = 0119881119887 (119897 119905) = 01199081198871015840 (0 119905) = 0119872119887 (119897 119905) = 0

(A7)

where the left side of sandwich beam is clamped and the rightside is free

The axial forces 119875119905119887(119909 119905) the shear forces 119881119905119887119888(119909 119905) andthe bending moments119872119905119887(119909 119905) are obtained as

119875119905 = minus119864119891119860119891 120597119906119905120597119909 (A8)

119875119887 = minus119864119891119860119891 120597119906119887120597119909 (A9)

119881119905 = minus 112119898119891ℎ21198911015840119905 + 119898119888840 [minus36ℎ119891119905 minus 6ℎ119891119887minus 28ℎ119891119888 minus 6ℎ119891ℎ119888 1205931 + 3ℎ21198911015840119887 minus 18ℎ21198911015840119905)+ 1198641198911198681198911199081015840101584010158401015840119905 + 119866119888119860119888840ℎ2119888 [(196ℎ119891 minus 28ℎ119888) 119906119887minus (1316ℎ119891 + 532ℎ119888) 119906119905 + (560ℎ119888 + 1120ℎ119891) 119906119888+ (28ℎ2119888 minus 98ℎ2119891 + 28ℎ119891ℎ119888)11990810158401015840119887minus (112ℎ2119888 + 658ℎ2119891 + 532ℎ2119891 + 532ℎ119891ℎ119888)1199081015840119905+ (336ℎ119891ℎ119888 + 112ℎ2119888) 1205931minus (56ℎ2119888 + 168ℎ119891ℎ119888)1199081015840119888]

(A10)

119881119887 = minus 112119898119891ℎ21198911015840119887 + 119898119888840 (36ℎ119891119887 + 6ℎ119891119905+ 28ℎ119891119888 minus 6ℎ119891ℎ119888 1205931 + 3ℎ21198911015840119905 minus 18ℎ21198911015840119887)

14 Shock and Vibration

+ 1198641198911198681198911199081015840101584010158401015840119887 + 119866119888119860119888840ℎ2119888 [(minus196ℎ119891 + 28ℎ119888) 119906119905+ (1316ℎ119891 + 532ℎ119888) 119906119887 minus (560ℎ119888 + 1120ℎ119891) 119906119888+ (28ℎ2119888 minus 98ℎ2119891 + 28ℎ119891ℎ119888)11990810158401015840119905minus (112ℎ2119888 + 658ℎ2119891 + 532ℎ2119891 + 532ℎ119891ℎ119888)1199081015840119887+ (336ℎ119891ℎ119888 + 112ℎ2119888) 1205931minus (56ℎ2119888 + 168ℎ119891ℎ119888)1199081015840119888]

(A11)

119881119888 = 119866119888119860119888840ℎ2119888 [minus336ℎ119891 (119906119905 minus 119906119887) minus (224ℎ2119888) 1205931

minus (56ℎ2119888 + 168ℎ119891ℎ119888) (1199081015840119905 minus 1199081015840119887) minus 448ℎ21198881199081015840119888](A12)

119872119905 = minus11986411989111986811989111990810158401015840119905 (A13)

119872119887 = minus11986411989111986811989111990810158401015840119887 (A14)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] A S Herrmann P C Zahlen and I Zuardy ldquoSandwich Struc-tures Technology in Commercial Aviationrdquo in Proceedings of the7th International Conference on Sandwich Structures pp 13ndash262005

[2] J Vinson The behavior of sandwich structures of isotropic andcomposite materials Technomic Publishing Company Lan-caster Pennsylvania 1999

[3] J Hohe L Librescu and S Y Oh ldquoDynamic buckling of flat andcurved sandwich panels with transversely compressible corerdquoComposite Structures vol 74 no 1 pp 10ndash24 2006

[4] 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

[5] Y Frostig M Baruch O Vilnay and I Sheinman ldquoHigh-ordertheory for sandwich-beam behavior with transversely flexiblecorerdquo Journal of Engineering Mechanics vol 118 no 5 pp 1026ndash1043 1992

[6] O Rahmani SM R Khalili KMalekzadeh andHHadavinialdquoFree vibration analysis of sandwich structures with a flexiblefunctionally graded syntactic corerdquo Composite Structures vol91 no 2 pp 229ndash235 2009

[7] C Santiuste O TThomsen and Y Frostig ldquoThermo-mechani-cal load interactions in foam cored axi-symmetric sandwich cir-cular plates - High-order and FEmodelsrdquo Composite Structuresvol 93 no 2 pp 369ndash376 2011

[8] R K Fruehmann J M Dulieu-Barton O T Thomsen andS Zhang ldquoExperimental investigation of thermal effects infoam cored sandwich beamsrdquo in Thermomechanics and Infra-Red Imaging vol 7 of Conference Proceedings of the Society forExperimental Mechanics Series pp 83ndash90 Springer NY USA2011

[9] V S Sokolinsky and S R Nutt ldquoConsistent Higher-OrderDynamic Equations for Soft-Core Sandwich Beamsrdquo AIAAJournal vol 42 no 2 pp 374ndash382 2004

[10] Y Frostig C Phan and G Kardomateas ldquoFree vibration ofunidirectional sandwich panels Part I Compressible corerdquoJournal of Sandwich Structures and Materials vol 15 no 4 pp377ndash411 2013

[11] A R Damanpack and S M R Khalili ldquoHigh-order free vibra-tion analysis of sandwich beams with a flexible core usingdynamic stiffness methodrdquo Composite Structures vol 94 no 5pp 1503ndash1514 2012

[12] M Yang and P Qiao ldquoHigher-order impact modeling of sand-wich structures with flexible corerdquo International Journal ofSolids and Structures vol 42 no 20 pp 5460ndash5490 2005

[13] Y Wang and X Wang ldquoFree vibration analysis of soft-coresandwich beams by the novel weak form quadrature elementmethodrdquo Journal of Sandwich Structures and Materials vol 18no 3 pp 294ndash320 2015

[14] X Wang and X Liang ldquoFree vibration of soft-core sandwichpanels with general boundary conditions by harmonic quadra-ture element methodrdquoThin-Walled Structures vol 113 pp 253ndash261 2017

[15] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe local GDQmethod applied to general higher-order theories of doubly-curved laminated composite shells and panels the free vibrationanalysisrdquo Composite Structures vol 116 pp 637ndash660 2014

[16] C C Hong H W Liao L T Lee J B Ke and K C JaneldquoThermally induced vibration of a thermal sleeve with the GDQmethodrdquo International Journal of Mechanical Sciences vol 47no 12 pp 1789ndash1806 2005

[17] H D Chalak A Chakrabarti M A Iqbal and A H SheikhldquoVibration of laminated sandwich beams having soft corerdquoJournal of Vibration and Control vol 18 no 10 pp 1422ndash14352012

[18] K M Ahmed ldquoFree vibration of curved sandwich beams by themethod of finite elementsrdquo Journal of Sound and Vibration vol18 no 1 pp 61ndash74 1971

[19] N Nanda S Kapuria and S Gopalakrishnan ldquoSpectral finiteelement based on an efficient layerwise theory for wave prop-agation analysis of composite and sandwich beamsrdquo Journal ofSound and Vibration vol 333 no 31 pp 20ndash27 2014

[20] Y Pourvais P Asgari A R Moradi and O Rahmani ldquoExper-imental and finite element analysis of higher order behaviourof sandwich beams using digital projection moirerdquo PolymerTesting vol 38 pp 7ndash17 2014

[21] R Long O Barry and D C D Oguamanam ldquoFinite elementfree vibration analysis of soft-core sandwich beamsrdquo AIAAJournal vol 50 no 1 pp 235ndash238 2012

[22] W L Chen A G Striz and C W Bert ldquoHigh-accuracy planestress and plate elements in the quadrature element methodrdquoInternational Journal of Solids and Structures vol 37 no 4 pp627ndash647 2000

[23] Y Xing and B Liu ldquoHigh-accuracy differential quadraturefinite element method and its application to free vibrations ofthin plate with curvilinear domainrdquo International Journal for

Shock and Vibration 15

Numerical Methods in Engineering vol 80 no 13 pp 1718ndash17422009

[24] YWang and XWang ldquoStatic analysis of higher order sandwichbeams by weak form quadrature element methodrdquo CompositeStructures vol 116 no 1 pp 841ndash848 2014

[25] F Tornabene S Brischetto N Fantuzzi and M BacciocchildquoBoundary conditions in 2Dnumerical and 3D exactmodels forcylindrical bending analysis of functionally graded structuresrdquoShock and Vibration vol 2016 Article ID 2373862 2016

[26] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe GDQmethod for the free vibration analysis of arbitrarily shaped lami-nated composite shells using a NURBS-based isogeometricapproachrdquo Composite Structures vol 154 pp 190ndash218 2016

[27] Z Zong and Y Zhang Advanced differential quadrature meth-ods CRC Press NY USA 2009

[28] N A Rubayi and S Charoenree ldquoNatural frequencies of vibra-tion of cantilever sandwich beamsrdquoComputersampStructures vol7 no 6 pp 737ndash745 1977

[29] M Azimi M Shahravi and E F Joubaneh ldquoDynamic analysisof maneuvering flexible spacecraft appendage using higherorder sandwich panel theoryrdquo Latin American Journal of Solidsand Structures vol 13 no 2 pp 296ndash313 2016

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4 Shock and Vibration

The kinetic energy is given as

119879 = 12 (int119881119905 120588119891 ( 1199091199052 + 1199101199052) 119889119881119905

+ int119881119887

120588119891 ( 1199091198872 + 1199101198872) 119889119881119887 + int119881119888

120588119888 ( 1199091198882 + 1199101198882) 119889119881119888)(6)

where 120588119891 is the face sheets density and 120588119888 is the core den-sity 119909119905 119909119887 and 119909119888 are velocities of top and bottom face sheetsand core in x direction 119910119905 119910119887 and 119910119888 are velocities of top andbottom face sheets and core in y direction By substituting thedisplacement equations into (5) and (6) the governing partialdifferential equations ofmotion and boundary conditions canbe obtained from Hamiltonrsquos principle as

120575int11990521199051

(119879 minus 119880) 119889119905 = 0 (7)

where 120575 is the first variation operation and 1199051 and 1199052are defined as the time period The governing equations ofmotion are obtained by dividing and factoring the coefficientsof 120575119906119905 120575119906119887 120575119908119905 120575119908119887 120575119906119888 120575120593 120575119908119888 in (7) The followingequation is one of the seven obtained equations of motionThe rest of the equations are presented in the Appendix

119898119891 (119905) + 119898119888 [ 120 (119905 + 119887) + 128 (119905 minus 119887) + 115 119888+ 170ℎ119888 1205931 + 140ℎ119891 (1015840119905 minus 1015840119887) + 156ℎ119891 (1015840119905 + 1015840119887)]minus 11986411989111986011989111990610158401015840119905 + 119866119888119860119888ℎ2119888 [43 (119906119905 + 119906119887) + 95 (119906119905 minus 119906119887)minus 83119906119888 minus 45ℎ1198881205931 + (23ℎ119891 + 13ℎ119888) (1199081015840119905 minus 1199081015840119887)+ ( 910ℎ119891 + 310ℎ119888) (1199081015840119905 + 1199081015840119887) + 25ℎ1198881199081015840119888] = 0

(8)

where 119898119888 is the corersquos mass 119898119891 is the surface layersrsquo mass119868119891 is the surface moment of inertia 119860119888 is the corersquos crosssection and 119860119891 is the surface layersrsquo cross section 119905 and119887 are also the accelerations at the middle of the top andbottom face sheets and 119888 is the acceleration at the middle ofthe core 11990610158401015840119905 defines the variation of normal strain related tomiddle of the top face sheet in x direction

3 GDQ Method

TheGDQmethod is applied in order to discrete the equationof motion and boundary condition along the length of the

beam Applying GDQmethod to the equation of motion (see(8)) yields

[119898119891 + 119898119888 ( 120 + 128)]Nsum119895=1

119862(0)119894119895 119905 (119909119895)

+ 119898119888 ( 120 minus 128)119873sum119895=1

119862(0)119894119895 119887 (119909119895) + 119898119888 ( 115)

sdot 119873sum119895=1

119862(0)119894119895 119888 (119909119895) + 119898119888ℎ119888 170119873sum119895=1

119862(0)119894119895 1205931 (119909119895)

+ 119898119888 ( 140 + 156) ℎ119891119873sum119895=1

119862(1)119894119895 119905 (119909119895)

+ 119898119888 (minus 140 + 156) ℎ119891119873sum119895=1

119862(1)119894119895 119887 (119909119895)

minus 119864119891119860119891 119873sum119895=1

119862(2)119894119895 119906119905 (119909119895) + 119866119888119860119888ℎ2119888 (43 + 95)

sdot 119873sum119895=1

119862(0)119894119895 119906119905 (119909119895) + 119866119888119860119888ℎ2119888 (43 minus 95)119873sum119895=1

119862(0)119894119895 119906119887 (119909119895)

minus 119866119888119860119888ℎ2119888 (83)119873sum119895=1

119862(0)119894119895 119906119888 (119909119895)+ 119866119888119860119888ℎ2119888 [(23ℎ119891 + 13ℎ119888) + ( 910ℎ119891 + 310ℎ119888)]

sdot 119873sum119895=1

119862(1)119894119895 119908119905 (119909119895)+ 119866119888119860119888ℎ2119888 [minus (23ℎ119891 + 13ℎ119888) + ( 910ℎ119891 + 310ℎ119888)]

sdot 119873sum119895=1

119862(1)119894119895 119908119887 (119909119895) + 119866119888119860119888ℎ2119888 (25ℎ119888)119873sum119895=1

119862(1)119894119895 119908119888 (119909119895)

minus 119866119888119860119888ℎ2119888 (45ℎ119888)119873sum119895=1

119862(0)119894119895 1205931 (119909119895)

(9)

The solution of the system is achieved by employing alocal version of the well-known GDQ method With respectto the global form this local approach considers localizedinterpolating basis function The numerical technique inGDQ is able to evaluate the 119899th derivative at a generic pointof a sufficiently smooth function 119891(119909) as a weighted linearsum of the function values at some chosen grid points in xdirection These grid points can be distributed through thethickness of the sandwich beam too [14]

119889119899119891 (119909)119889119909119899 (119909119894) cong119895=119873sum119895=1

119862(119899)119894119895 119891 (119909119895) 119891119900119903 119894 = 1 2 119873 (10)

Shock and Vibration 5

where 119862(119899)119894119895 is the weighting coefficient associated with thenth-order derivative and 119873 is the number of grid points inx direction It should be noted that the GDQ is quite differentfrom QEM and HQEM GDQ method uses Chebyshev gridpoints distribution and employs Lagrange function for alldisplacement fields namely 119906119905 119906119887 119906119888 1205931 119908119905 119908119887 119908119888 and1205931 The QEM on the other hand uses expanded Chebyshevgrids and employs Hermit functions for 119908119905 119908119887 and 119908119888 and

Lagrange function for displacement fields 119906119905 119906119887 119906119888 and 1205931HQEM which is an extended version of the QEM employsnodes on both longitudinal and through the thickness (trans-verse) directions whereas the GDQ method and QEM onlyuse nodes in the longitudinal direction The GDQ method isadopted in this paper as such using Lagrange interpolatedpolynomials for all displacements the function 119891(119909) can bewritten as

119891 (119909) = 119895=119873sum119895=1

(119909 minus 1199091) (119909 minus 1199092) (119909 minus 119909119895minus1) (119909 minus 119909119895+1) (119909 minus 119909119873)(119909119895 minus 1199091) (119909119895 minus 1199092) (119909119895 minus 119909119895minus1) (119909119895 minus 119909119873) 119891 (119909119895) (11)

Taking the derivative of (11) with respect to x yields

1198911015840 (119909) = 119895=119873sum119895=1

(119909 minus 1199091) (119909 minus 1199092) (119909 minus 119909119895minus1) (119909 minus 119909119895+1) (119909 minus 119909119873)(119909119895 minus 1199091) (119909119895 minus 1199092) (119909119895 minus 119909119895minus1) (119909119895 minus 119909119873) 1015840

119891 (119909119895) (12)

The coefficient of 119891(119909119895) calls119862(1)119894119895 which can be written as119862(1)119894119895 = 119872(1) (119909119894)(119909119894 minus 119909119895)119872(1) (119909119895)119862(1)119894119894 = minus 119873sum

119895=1119895 =119894

119862(1)119894119895(13)

where

119872(1) (119909119895) = 119873prod119896=1119896 =119895

(119909119895 minus 119909119896) (14)

The higher-order derivatives are obtained as

119862(119898)119894119895 = 119898[[119862(1)119894119895 119862(119898minus1)119894119894 minus 119862(119898minus1)119894119895119909119894 minus 119909119895]]

119894 119895 = 1 2 119873 119898 = 2 3 119873 minus 1 119894 = 119895119862(119898)119894119894 = minus 119873sum

119895=1119895 =119894

119862(119898)119894119895(15)

There are also different arrangements of grid pointswhich are used in GDQ method It has been shown in theliterature that using equal spacing sample of grid points givesinaccurate results Therefore the Chebyshev-Gauss-Lobattodistribution is employed to discretize the spatial domain asfollows [27]

119909119894 = 1198712 [1 minus cos( 119894 minus 1119873 minus 1120587)] 119894 = 1 119873 (16)

Other grid distributions such as expandedChebyshev [1314] can also be used

4 Numerical Results and Discussion

Theproperties of the sandwich beam used for conducting thenumerical simulation are listed in Table 1 Numerical resultsare presented for the free and forced vibration analyses ofsandwich beams with various boundary conditions includingcantilever (C-F) clamped-clamped (C-C) simply-simply (S-S) free-free (F-F) and simply-clamped (S-C)

41 Free Vibration To examine the validity of the presentformulation Table 2 compares the results of theGDQmethodfor the simply supported case to those of [11 12] The resultsshow excellent agreement with the closed-form solutionemployed in [11] Table 2 also indicates that the GDQmethodperforms better than the method used in [12] The validationof the rest of the boundary conditions namely C-F C-C S-SF-F and S-C is shown in Table 3 The natural frequency ofthe sandwich beam for GDQmethod is compared to those ofthe QEM harmonic quadrature element method (HQEM)and finite element method of ABAQUS The results showvery good agreement for all five boundary conditions Ingeneral it can be observed from Table 3 that the results of theGDQmethod and the HQEM are closer to those of ABAQUSthan the QEM This is an indication that the GDQ is moreaccurate than the QEMThe first twomode shapes for C-F S-S and C-C boundary conditions are shown in Figure 2 Theresults show that the plottedmode shapes are similar to thoseobtained in [14]

42 Forced Vibration This section discusses the time andfrequency responses of the sandwich beam with variousboundary conditions All the numerical simulations in thissection are based on a harmonic applied load 119865 at the tip ofthe sandwich beam defined as 119865 = 04 times 103 sin(120596119905) and the

6 Shock and Vibration

Table 1 Parameters of sandwich beam

References 119864119891 (119866119875119886) 119866119888 (119872119875119886) 120588119891 (1198961198921198983) 120588119888 (1198961198921198983) ℎ119891 (119898119898) ℎ119888 (119898119898) 119887 (119898119898) 119871 (119898119898)[11] 210 221 7900 60 19 348 599 260[12] 36 200 4400 521 05 200 200 300[13] 136 128 1800 585 5 19 20 152Experiment 210 221 7900 60 19 15 and 25 599 260

Table 2 The five low natural frequencies for a simply supported sandwich beam (rads)

Mode Present method (GDQ) Exact [11] Model (C) [12]1 204846 204841 2048192 518973 518967 5183373 825024 825019 8224064 1122532 1122527 11159635 1413922 1413919 1400912

material property from [11] listed in Table 1 Figure 3 showsthe time response of the cantilever sandwich beam for threeexcitation frequencies namely 120596 = 011205961 120596 = 0871205961 and120596 ≃ 11205961 Here 1205961 is the first natural frequency of sandwichbeam Beating phenomenon is observed when the excitationfrequency is 120596 = 0871205961 and resonance phenomenon isdemonstrated when the excitation frequency matches thefundamental frequency of the sandwich beam that is 120596 ≃11205961 Figure 3 also shows much smaller vibration amplitudewhen the fundamental frequency is far from the excitationfrequency 120596 = 011205961

The validation of the forced vibration analysis is demon-strated in Figure 4 by comparing the results of the GDQmethod (Case (a) (c) (e)) to those of SolidWorks simulation(Case (b) (d) (f)) It should be noted that this figure showsthe time response measured at the tip of the cantileversandwich beam with varying thickness and width For eachcase the cantilever sandwich beam is excited with its cor-responding fundamental frequency since natural frequencychanges with beam thicknesswidthThe results in this figureshow very good agreement between the GDQ method andSolidWorks simulations

It is also observed in Figure 4 that the vibration amplitudedecreases with increasing core thickness andor increasingface sheet thickness This is expected as increasing thicknessincreases bending stiffness thus resulting in lower deflectionFigure 4 also shows the role of the beam width on thetime response The results show that the vibration amplitudedecreases with increasing beam width This is also expectedas the width increases with bending stiffness Although boththickness and width affect the vibration amplitude only thethickness affects the natural frequency of the beam Thisobservation is in agreement with literatures [28 29]

Figures 5 6 and 7 show the role of geometry parameterson the frequency response curves for cantilever simply-simply supported and clamped-clamped boundary condi-tions respectively Similar trends are observed for all threeboundary conditions The frequency response curves aresignificantly affected by the core and face sheet thicknessesas well as the width of the sandwich beam Increasing

any of these geometry parameters obviously decreases thevibration amplitude The peaks corresponding to the exci-tation frequency change with varying thickness but remainconstant with varying width The natural frequency in gen-eral increases with increasing core thickness for all studiedboundary conditions As for the face sheet thickness thenatural frequency can increase or decrease with varying facesheet thickness The reason for this can be attributed tothe core thickness having more effect on the stiffness thanthe mass whereas the sheet thickness has similar effect onboth mass and stiffness The beam width shows no effecton the natural frequency for all boundary conditions Thisobservation is also in agreement with earlier discussion aboutthe role of the beam width on the natural frequency

5 Experimental Analysis and Discussion

An experiment is conducted to determine the frequencyresponse curve of two cantilever sandwich beams with dif-ferent core thicknesses The material properties of the testedsandwich beams are listed in Table 1 A schematic of theexperimental setup is depicted in Figure 8 The cantileverbeam is installed on a VDL shaker (B ampKV830-335-SPA16K)using a head plate and a fixture both made of aluminum

To reduce experimental error the fixture is designed inSolidWorks such that its fundamental frequency is well abovethe maximum excitation frequency A signal generated fromthe controller (Laser USB 14V) is fed to the power amplifier(LSD SPA 16K) and then to the vibration shaker The signalis fed back to the controller to ensure precise measurementTwo accelerometers (B amp K 8325) are employed for input andoutput measurements One is placed at the top of the fixtureto measure the input acceleration and the other is placedat the tip of the beam to measure the output response Thetest is carried out with a constant velocity of 2 mms and asine sweep is performed for a frequency range of 10 to 2000Hz Measurements are sent to the data acquisition systemthrough a signal analyzer The frequency response plots areobtained and the frequencies corresponding to the peaks ofthese plots are the natural frequencies of the tested sandwich

Shock and Vibration 7

(a) (b)

(c) (d)

(e) (f)

Figure 2 The first two mode shapes of sandwich beam with cantilever ((a) and (b)) simply-simply ((c) and (d)) and clamped-clamped ((e)and (f)) supported edges

beams The experimental results are shown in Figure 9which also illustrates the role of the core thickness on thefrequency response curve The results clearly indicate thatnatural frequency increases with increasing thickness while

vibration response decreases with increasing thickness Thisobservation is in agreement with the results obtained usingthe GDQ method Table 4 compares natural frequenciesobtained experimentally and numerically (GDQmethod) for

8 Shock and Vibration

Figure 3 Tip deflection of free edge of cantilever sandwich beam (120596119905) for applied frequency 1205961205961 = 01 1205961205961 = 087 and 1205961205961 ≃ 1 [9]

Table 3 The five low natural frequencies for different boundary conditions (Hz)

BCs Mode 1 2 3 4 5

C-F

ABAQUS [14] 32740 10791 21380 21753 22164GDQ 32918 108322 215254 220158 225815

QEM [13] 32966 108843 216459 218378 222572HQEM (full [M]) [14] 32782 10824 21539 21759 22179

C-C

ABAQUS [14] 83992 19567 22556 27026 34778GDQ 84417 197658 227456 270982 354127

QEM [13] 84871 198813 226266 272513 356396HQEM (full [M]) [14] 84403 19781 22571 27175 35475

S-S

ABAQUS [14] 61856 13491 15033 21704 23584GDQ 61905 134825 150881 221264 241309

QEM [13] 62222 135286 151545 217686 236977HQEM (full [M]) [14] 61891 13493 15082 21712 23623

F-F

ABAQUS [14] 12902 13669 21537 21692 22146GDQ 129210 136687 219787 220386 224586

QEM [13] 129919 137239 216280 217640 222519HQEM (full [M] [14] 12918 13676 21542 21701 22152

S-CABAQUS [14] 71545 17144 21964 25144 31171

GDQ 71725 172564 22339 25413 315728HQEM (full [M]) [14] 71718 17258 21975 25212 31607

Table 4 The natural frequencies (Hz) for cantilever beam with two different thicknesses

Modes ℎ119888 = 15 mm ℎ119888 = 25 mmGDQmethod Experiment GDQ method Experiment

1 11800 11119 14392 143022 38100 36460 45227 475213 71984 68080 82186 824334 114194 106350 126156 1323815 167551 165207 180017 177260

Shock and Vibration 9

(a) The effect of core by GDQmethod (b) The effect of core by SolidWorks

(c) The effect of face sheets by GDQmethod (d) The effect of face sheets by SolidWorks

(e) The effect of width by GDQmethod (f) The effect of width by SolidWorks

Figure 4 The effect of geometric parameters on the time response at the tip of the cantilever sandwich beam

10 Shock and Vibration

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 5 Vibration amplitude at the tip of the cantilever beam with respect to applied frequency

a core thickness of 15 and 25 mm The results in this tablealso show very good agreementThis is also an indication thatthe present formulation for both free and forced vibrationanalyses of sandwich beams is accurate

6 Conclusions

This paper presents the vibration analysis of sandwich beamswith various boundary conditions The governing equationsof motion are derived using Hamiltonrsquos principle The GDQmethod is utilized for solving the problem SolidWorkssimulation and experimental analyses are conducted to assessthe validity of the GDQ method and the results showvery good agreement Parametric studies are conducted toexamine the role of geometric properties on the time andfrequency response curvesThe results indicate that vibrationamplitude decreases with increasing both core and facesheet thicknesses whereas natural frequency increases withincreasing core sheet thickness but can increase or decreasewith varying face sheet thickness The results also show thatvibration amplitude decreases with increasing beam widthwhereas no effect is observed on the natural frequency for

varying the width of the beam Numerical examples alsodemonstrate that all studied boundary conditions exhibitsimilar effect with varying the aforementioned geometricproperties The findings in this paper are anticipated to beappealing to research communities because of the experi-mental and FEM validation of the GDQ method for thefree and forced vibration analyses of sandwich beams undervarious boundary conditions

Appendix

The rest of the six equations of motion are given as follows

119898119891 (119887) + 119898119888 [ 120 (119905 + 119887) minus 128 (119905 minus 119887) + 115 119888minus 170ℎ119888 1205931 + 140ℎ119891 (1015840119905 minus 1015840119887) minus 156ℎ119891 (1015840119905 + 1015840119887)]minus 11986411989111986011989111990610158401015840119887 + 119866119888119860119888ℎ2119888 [43 (119906119905 + 119906119887) minus 95 (119906119905 minus 119906119887)minus 83119906119888 + 45ℎ1198881205931 + (23ℎ119891 + 13ℎ119888) (1199081015840119905 minus 1199081015840119887)

Shock and Vibration 11

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 6 Vibration amplitude at the middle of the simply-simply supported edges sandwich beam with respect to applied frequency

minus ( 910ℎ119891 + 310ℎ119888) (1199081015840119905 + 1199081015840119887) minus 25ℎ1198881199081015840119888] = 0(A1)

119898119891 (119905 minus 112ℎ211989110158401015840119905 ) + 119898119888 [minus 140ℎ119891 (1015840119905 + 1015840119887)minus 156ℎ119891 (1015840119905 minus 1015840119887) minus 130ℎ1198911015840119888 minus 1140ℎ119888ℎ119891 12059311015840minus 180ℎ2119891 (10158401015840119905 minus 10158401015840119887 ) + 112 (119905 minus 119887)minus 1112ℎ2119891 (10158401015840119905 + 10158401015840119887 ) + 120 (119905 + 119887) + 115119888]+ 1198641198911198681198911199081015840101584010158401015840119905 + 119866119888119860119888ℎ2119888 [minus (23ℎ119891 + 13ℎ119888) (1199061015840119905 + 1199061015840119887)minus ( 910ℎ119891 + 310ℎ119888) (1199061015840119905 minus 1199061015840119887) + (43ℎ119891 + 23ℎ119888) 1199061015840119888+ (25ℎ119891ℎ119888 + 215ℎ2119888) (12059310158401)

minus ( 920ℎ2119891 + 310ℎ119888ℎ119891 + 120ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 )minus (13ℎ119891ℎ119888 + 13ℎ2119891 + 112ℎ2119888) (11990810158401015840119905 minus 11990810158401015840119887 )minus (15ℎ119891ℎ119888 + 115ℎ2119888)11990810158401015840119888 ] + 119864119888119860119888ℎ2119888 [43 (119908119905 + 119908119887)+ (119908119905 minus 119908119887) minus 83119908119888] = 0

(A2)

119898119891 (119887 minus 112ℎ211989110158401015840119887 ) + 119898119888 [ 140ℎ119891 (1015840119905 + 1015840119887)minus 156ℎ119891 (1015840119905 minus 1015840119887) + 130ℎ1198911015840119888 minus 1140ℎ119888ℎ119891 12059311015840+ 180ℎ2119891 (10158401015840119905 minus 10158401015840119887 ) minus 112 (119905 minus 119887)minus 1112ℎ2119891 (10158401015840119905 + 10158401015840119887 ) + 120 (119905 + 119887) + 115119888]

12 Shock and Vibration

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 7 Vibration amplitude at the middle of the clamped-clamped edges sandwich beam with respect to applied frequency

+ 1198641198911198681198911199081015840101584010158401015840119887 + 119866119888119860119888ℎ2119888 [(23ℎ119891 + 13ℎ119888) (1199061015840119905 + 1199061015840119887)minus ( 910ℎ119891 + 310ℎ119888) (1199061015840119905 minus 1199061015840119887) minus (43ℎ119891 + 23ℎ119888) 1199061015840119888+ (25ℎ119891ℎ119888 + 215ℎ2119888) (12059310158401)minus ( 920ℎ2119891 + 310ℎ119888ℎ119891 + 120ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 )+ (13ℎ119891ℎ119888 + 13ℎ2119891 + 112ℎ2119888) (11990810158401015840119905 minus 11990810158401015840119887 )minus (15ℎ119891ℎ119888 + 115ℎ2119888)11990810158401015840119888 ] + 119864119888119860119888ℎ2119888 [43 (119908119905 + 119908119887)minus (119908119905 minus 119908119887) minus 83119908119888] = 0

(A3)

119898119888 [ 815 119888 + 115 (119887 + 119905) + 130ℎ119891 (1015840119905 minus 1015840119887)]+ 119866119888119860119888ℎ1198882 [minus83 (119906119905 + 119906119887) + 163 119906119888minus (43ℎ119891 + 23ℎ119888) (1199081015840119905 minus 1199081015840119887)] = 0

(A4)

119898119888 [ 2105ℎ2119888 1205931 + 170ℎ119888 (119905 minus 119887)+ 1140ℎ119891ℎ119888 (1015840119905 + 1015840119887)] + 119866119888119860119888ℎ1198882 [minus45ℎ119888 (119906119905 minus 119906119887)+ 45ℎ21198881205931 + 415ℎ21198881199081015840119888minus ( 215ℎ2119888 + 25ℎ119888ℎ119891) (1199081015840119905 + 1199081015840119887)] = 0

(A5)

Shock and Vibration 13

Figure 8 Schematic of experimental setup

Figure 9 Vibration amplitude of cantilever sandwich beam fordifferent core thicknesses by experiment

119898119888 [ 815119888 + 115ℎ119888 (119887 + 119905)]+ 119866119888119860119888ℎ1198882 [minus25ℎ119888 (1199061015840119905 minus 1199061015840119887) minus 415ℎ21198881205931minus ( 115ℎ119891ℎ119888 + 115ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 ) minus 815ℎ211988811990810158401015840119888 ]+ 119864119888119860119888ℎ2119888 [minus83 (119908119905 + 119908119887) + 163 119908119888] = 0

(A6)

The essential and natural boundary conditions related to theclamped-free case are given as

119906119905 (0 119905) = 0119875119905 (119897 119905) = 0119908119905 (0 119905) = 0119881119905 (119897 119905) = 01199081199051015840 (0 119905) = 0119872119905 (119897 119905) = 0119908119888 (0 119905) = 0119881119888 (119897 119905) = 0119906119887 (0 119905) = 0119875119887 (119897 119905) = 0119908119887 (0 119905) = 0119881119887 (119897 119905) = 01199081198871015840 (0 119905) = 0119872119887 (119897 119905) = 0

(A7)

where the left side of sandwich beam is clamped and the rightside is free

The axial forces 119875119905119887(119909 119905) the shear forces 119881119905119887119888(119909 119905) andthe bending moments119872119905119887(119909 119905) are obtained as

119875119905 = minus119864119891119860119891 120597119906119905120597119909 (A8)

119875119887 = minus119864119891119860119891 120597119906119887120597119909 (A9)

119881119905 = minus 112119898119891ℎ21198911015840119905 + 119898119888840 [minus36ℎ119891119905 minus 6ℎ119891119887minus 28ℎ119891119888 minus 6ℎ119891ℎ119888 1205931 + 3ℎ21198911015840119887 minus 18ℎ21198911015840119905)+ 1198641198911198681198911199081015840101584010158401015840119905 + 119866119888119860119888840ℎ2119888 [(196ℎ119891 minus 28ℎ119888) 119906119887minus (1316ℎ119891 + 532ℎ119888) 119906119905 + (560ℎ119888 + 1120ℎ119891) 119906119888+ (28ℎ2119888 minus 98ℎ2119891 + 28ℎ119891ℎ119888)11990810158401015840119887minus (112ℎ2119888 + 658ℎ2119891 + 532ℎ2119891 + 532ℎ119891ℎ119888)1199081015840119905+ (336ℎ119891ℎ119888 + 112ℎ2119888) 1205931minus (56ℎ2119888 + 168ℎ119891ℎ119888)1199081015840119888]

(A10)

119881119887 = minus 112119898119891ℎ21198911015840119887 + 119898119888840 (36ℎ119891119887 + 6ℎ119891119905+ 28ℎ119891119888 minus 6ℎ119891ℎ119888 1205931 + 3ℎ21198911015840119905 minus 18ℎ21198911015840119887)

14 Shock and Vibration

+ 1198641198911198681198911199081015840101584010158401015840119887 + 119866119888119860119888840ℎ2119888 [(minus196ℎ119891 + 28ℎ119888) 119906119905+ (1316ℎ119891 + 532ℎ119888) 119906119887 minus (560ℎ119888 + 1120ℎ119891) 119906119888+ (28ℎ2119888 minus 98ℎ2119891 + 28ℎ119891ℎ119888)11990810158401015840119905minus (112ℎ2119888 + 658ℎ2119891 + 532ℎ2119891 + 532ℎ119891ℎ119888)1199081015840119887+ (336ℎ119891ℎ119888 + 112ℎ2119888) 1205931minus (56ℎ2119888 + 168ℎ119891ℎ119888)1199081015840119888]

(A11)

119881119888 = 119866119888119860119888840ℎ2119888 [minus336ℎ119891 (119906119905 minus 119906119887) minus (224ℎ2119888) 1205931

minus (56ℎ2119888 + 168ℎ119891ℎ119888) (1199081015840119905 minus 1199081015840119887) minus 448ℎ21198881199081015840119888](A12)

119872119905 = minus11986411989111986811989111990810158401015840119905 (A13)

119872119887 = minus11986411989111986811989111990810158401015840119887 (A14)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] A S Herrmann P C Zahlen and I Zuardy ldquoSandwich Struc-tures Technology in Commercial Aviationrdquo in Proceedings of the7th International Conference on Sandwich Structures pp 13ndash262005

[2] J Vinson The behavior of sandwich structures of isotropic andcomposite materials Technomic Publishing Company Lan-caster Pennsylvania 1999

[3] J Hohe L Librescu and S Y Oh ldquoDynamic buckling of flat andcurved sandwich panels with transversely compressible corerdquoComposite Structures vol 74 no 1 pp 10ndash24 2006

[4] 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

[5] Y Frostig M Baruch O Vilnay and I Sheinman ldquoHigh-ordertheory for sandwich-beam behavior with transversely flexiblecorerdquo Journal of Engineering Mechanics vol 118 no 5 pp 1026ndash1043 1992

[6] O Rahmani SM R Khalili KMalekzadeh andHHadavinialdquoFree vibration analysis of sandwich structures with a flexiblefunctionally graded syntactic corerdquo Composite Structures vol91 no 2 pp 229ndash235 2009

[7] C Santiuste O TThomsen and Y Frostig ldquoThermo-mechani-cal load interactions in foam cored axi-symmetric sandwich cir-cular plates - High-order and FEmodelsrdquo Composite Structuresvol 93 no 2 pp 369ndash376 2011

[8] R K Fruehmann J M Dulieu-Barton O T Thomsen andS Zhang ldquoExperimental investigation of thermal effects infoam cored sandwich beamsrdquo in Thermomechanics and Infra-Red Imaging vol 7 of Conference Proceedings of the Society forExperimental Mechanics Series pp 83ndash90 Springer NY USA2011

[9] V S Sokolinsky and S R Nutt ldquoConsistent Higher-OrderDynamic Equations for Soft-Core Sandwich Beamsrdquo AIAAJournal vol 42 no 2 pp 374ndash382 2004

[10] Y Frostig C Phan and G Kardomateas ldquoFree vibration ofunidirectional sandwich panels Part I Compressible corerdquoJournal of Sandwich Structures and Materials vol 15 no 4 pp377ndash411 2013

[11] A R Damanpack and S M R Khalili ldquoHigh-order free vibra-tion analysis of sandwich beams with a flexible core usingdynamic stiffness methodrdquo Composite Structures vol 94 no 5pp 1503ndash1514 2012

[12] M Yang and P Qiao ldquoHigher-order impact modeling of sand-wich structures with flexible corerdquo International Journal ofSolids and Structures vol 42 no 20 pp 5460ndash5490 2005

[13] Y Wang and X Wang ldquoFree vibration analysis of soft-coresandwich beams by the novel weak form quadrature elementmethodrdquo Journal of Sandwich Structures and Materials vol 18no 3 pp 294ndash320 2015

[14] X Wang and X Liang ldquoFree vibration of soft-core sandwichpanels with general boundary conditions by harmonic quadra-ture element methodrdquoThin-Walled Structures vol 113 pp 253ndash261 2017

[15] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe local GDQmethod applied to general higher-order theories of doubly-curved laminated composite shells and panels the free vibrationanalysisrdquo Composite Structures vol 116 pp 637ndash660 2014

[16] C C Hong H W Liao L T Lee J B Ke and K C JaneldquoThermally induced vibration of a thermal sleeve with the GDQmethodrdquo International Journal of Mechanical Sciences vol 47no 12 pp 1789ndash1806 2005

[17] H D Chalak A Chakrabarti M A Iqbal and A H SheikhldquoVibration of laminated sandwich beams having soft corerdquoJournal of Vibration and Control vol 18 no 10 pp 1422ndash14352012

[18] K M Ahmed ldquoFree vibration of curved sandwich beams by themethod of finite elementsrdquo Journal of Sound and Vibration vol18 no 1 pp 61ndash74 1971

[19] N Nanda S Kapuria and S Gopalakrishnan ldquoSpectral finiteelement based on an efficient layerwise theory for wave prop-agation analysis of composite and sandwich beamsrdquo Journal ofSound and Vibration vol 333 no 31 pp 20ndash27 2014

[20] Y Pourvais P Asgari A R Moradi and O Rahmani ldquoExper-imental and finite element analysis of higher order behaviourof sandwich beams using digital projection moirerdquo PolymerTesting vol 38 pp 7ndash17 2014

[21] R Long O Barry and D C D Oguamanam ldquoFinite elementfree vibration analysis of soft-core sandwich beamsrdquo AIAAJournal vol 50 no 1 pp 235ndash238 2012

[22] W L Chen A G Striz and C W Bert ldquoHigh-accuracy planestress and plate elements in the quadrature element methodrdquoInternational Journal of Solids and Structures vol 37 no 4 pp627ndash647 2000

[23] Y Xing and B Liu ldquoHigh-accuracy differential quadraturefinite element method and its application to free vibrations ofthin plate with curvilinear domainrdquo International Journal for

Shock and Vibration 15

Numerical Methods in Engineering vol 80 no 13 pp 1718ndash17422009

[24] YWang and XWang ldquoStatic analysis of higher order sandwichbeams by weak form quadrature element methodrdquo CompositeStructures vol 116 no 1 pp 841ndash848 2014

[25] F Tornabene S Brischetto N Fantuzzi and M BacciocchildquoBoundary conditions in 2Dnumerical and 3D exactmodels forcylindrical bending analysis of functionally graded structuresrdquoShock and Vibration vol 2016 Article ID 2373862 2016

[26] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe GDQmethod for the free vibration analysis of arbitrarily shaped lami-nated composite shells using a NURBS-based isogeometricapproachrdquo Composite Structures vol 154 pp 190ndash218 2016

[27] Z Zong and Y Zhang Advanced differential quadrature meth-ods CRC Press NY USA 2009

[28] N A Rubayi and S Charoenree ldquoNatural frequencies of vibra-tion of cantilever sandwich beamsrdquoComputersampStructures vol7 no 6 pp 737ndash745 1977

[29] M Azimi M Shahravi and E F Joubaneh ldquoDynamic analysisof maneuvering flexible spacecraft appendage using higherorder sandwich panel theoryrdquo Latin American Journal of Solidsand Structures vol 13 no 2 pp 296ndash313 2016

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Shock and Vibration 5

where 119862(119899)119894119895 is the weighting coefficient associated with thenth-order derivative and 119873 is the number of grid points inx direction It should be noted that the GDQ is quite differentfrom QEM and HQEM GDQ method uses Chebyshev gridpoints distribution and employs Lagrange function for alldisplacement fields namely 119906119905 119906119887 119906119888 1205931 119908119905 119908119887 119908119888 and1205931 The QEM on the other hand uses expanded Chebyshevgrids and employs Hermit functions for 119908119905 119908119887 and 119908119888 and

Lagrange function for displacement fields 119906119905 119906119887 119906119888 and 1205931HQEM which is an extended version of the QEM employsnodes on both longitudinal and through the thickness (trans-verse) directions whereas the GDQ method and QEM onlyuse nodes in the longitudinal direction The GDQ method isadopted in this paper as such using Lagrange interpolatedpolynomials for all displacements the function 119891(119909) can bewritten as

119891 (119909) = 119895=119873sum119895=1

(119909 minus 1199091) (119909 minus 1199092) (119909 minus 119909119895minus1) (119909 minus 119909119895+1) (119909 minus 119909119873)(119909119895 minus 1199091) (119909119895 minus 1199092) (119909119895 minus 119909119895minus1) (119909119895 minus 119909119873) 119891 (119909119895) (11)

Taking the derivative of (11) with respect to x yields

1198911015840 (119909) = 119895=119873sum119895=1

(119909 minus 1199091) (119909 minus 1199092) (119909 minus 119909119895minus1) (119909 minus 119909119895+1) (119909 minus 119909119873)(119909119895 minus 1199091) (119909119895 minus 1199092) (119909119895 minus 119909119895minus1) (119909119895 minus 119909119873) 1015840

119891 (119909119895) (12)

The coefficient of 119891(119909119895) calls119862(1)119894119895 which can be written as119862(1)119894119895 = 119872(1) (119909119894)(119909119894 minus 119909119895)119872(1) (119909119895)119862(1)119894119894 = minus 119873sum

119895=1119895 =119894

119862(1)119894119895(13)

where

119872(1) (119909119895) = 119873prod119896=1119896 =119895

(119909119895 minus 119909119896) (14)

The higher-order derivatives are obtained as

119862(119898)119894119895 = 119898[[119862(1)119894119895 119862(119898minus1)119894119894 minus 119862(119898minus1)119894119895119909119894 minus 119909119895]]

119894 119895 = 1 2 119873 119898 = 2 3 119873 minus 1 119894 = 119895119862(119898)119894119894 = minus 119873sum

119895=1119895 =119894

119862(119898)119894119895(15)

There are also different arrangements of grid pointswhich are used in GDQ method It has been shown in theliterature that using equal spacing sample of grid points givesinaccurate results Therefore the Chebyshev-Gauss-Lobattodistribution is employed to discretize the spatial domain asfollows [27]

119909119894 = 1198712 [1 minus cos( 119894 minus 1119873 minus 1120587)] 119894 = 1 119873 (16)

Other grid distributions such as expandedChebyshev [1314] can also be used

4 Numerical Results and Discussion

Theproperties of the sandwich beam used for conducting thenumerical simulation are listed in Table 1 Numerical resultsare presented for the free and forced vibration analyses ofsandwich beams with various boundary conditions includingcantilever (C-F) clamped-clamped (C-C) simply-simply (S-S) free-free (F-F) and simply-clamped (S-C)

41 Free Vibration To examine the validity of the presentformulation Table 2 compares the results of theGDQmethodfor the simply supported case to those of [11 12] The resultsshow excellent agreement with the closed-form solutionemployed in [11] Table 2 also indicates that the GDQmethodperforms better than the method used in [12] The validationof the rest of the boundary conditions namely C-F C-C S-SF-F and S-C is shown in Table 3 The natural frequency ofthe sandwich beam for GDQmethod is compared to those ofthe QEM harmonic quadrature element method (HQEM)and finite element method of ABAQUS The results showvery good agreement for all five boundary conditions Ingeneral it can be observed from Table 3 that the results of theGDQmethod and the HQEM are closer to those of ABAQUSthan the QEM This is an indication that the GDQ is moreaccurate than the QEMThe first twomode shapes for C-F S-S and C-C boundary conditions are shown in Figure 2 Theresults show that the plottedmode shapes are similar to thoseobtained in [14]

42 Forced Vibration This section discusses the time andfrequency responses of the sandwich beam with variousboundary conditions All the numerical simulations in thissection are based on a harmonic applied load 119865 at the tip ofthe sandwich beam defined as 119865 = 04 times 103 sin(120596119905) and the

6 Shock and Vibration

Table 1 Parameters of sandwich beam

References 119864119891 (119866119875119886) 119866119888 (119872119875119886) 120588119891 (1198961198921198983) 120588119888 (1198961198921198983) ℎ119891 (119898119898) ℎ119888 (119898119898) 119887 (119898119898) 119871 (119898119898)[11] 210 221 7900 60 19 348 599 260[12] 36 200 4400 521 05 200 200 300[13] 136 128 1800 585 5 19 20 152Experiment 210 221 7900 60 19 15 and 25 599 260

Table 2 The five low natural frequencies for a simply supported sandwich beam (rads)

Mode Present method (GDQ) Exact [11] Model (C) [12]1 204846 204841 2048192 518973 518967 5183373 825024 825019 8224064 1122532 1122527 11159635 1413922 1413919 1400912

material property from [11] listed in Table 1 Figure 3 showsthe time response of the cantilever sandwich beam for threeexcitation frequencies namely 120596 = 011205961 120596 = 0871205961 and120596 ≃ 11205961 Here 1205961 is the first natural frequency of sandwichbeam Beating phenomenon is observed when the excitationfrequency is 120596 = 0871205961 and resonance phenomenon isdemonstrated when the excitation frequency matches thefundamental frequency of the sandwich beam that is 120596 ≃11205961 Figure 3 also shows much smaller vibration amplitudewhen the fundamental frequency is far from the excitationfrequency 120596 = 011205961

The validation of the forced vibration analysis is demon-strated in Figure 4 by comparing the results of the GDQmethod (Case (a) (c) (e)) to those of SolidWorks simulation(Case (b) (d) (f)) It should be noted that this figure showsthe time response measured at the tip of the cantileversandwich beam with varying thickness and width For eachcase the cantilever sandwich beam is excited with its cor-responding fundamental frequency since natural frequencychanges with beam thicknesswidthThe results in this figureshow very good agreement between the GDQ method andSolidWorks simulations

It is also observed in Figure 4 that the vibration amplitudedecreases with increasing core thickness andor increasingface sheet thickness This is expected as increasing thicknessincreases bending stiffness thus resulting in lower deflectionFigure 4 also shows the role of the beam width on thetime response The results show that the vibration amplitudedecreases with increasing beam width This is also expectedas the width increases with bending stiffness Although boththickness and width affect the vibration amplitude only thethickness affects the natural frequency of the beam Thisobservation is in agreement with literatures [28 29]

Figures 5 6 and 7 show the role of geometry parameterson the frequency response curves for cantilever simply-simply supported and clamped-clamped boundary condi-tions respectively Similar trends are observed for all threeboundary conditions The frequency response curves aresignificantly affected by the core and face sheet thicknessesas well as the width of the sandwich beam Increasing

any of these geometry parameters obviously decreases thevibration amplitude The peaks corresponding to the exci-tation frequency change with varying thickness but remainconstant with varying width The natural frequency in gen-eral increases with increasing core thickness for all studiedboundary conditions As for the face sheet thickness thenatural frequency can increase or decrease with varying facesheet thickness The reason for this can be attributed tothe core thickness having more effect on the stiffness thanthe mass whereas the sheet thickness has similar effect onboth mass and stiffness The beam width shows no effecton the natural frequency for all boundary conditions Thisobservation is also in agreement with earlier discussion aboutthe role of the beam width on the natural frequency

5 Experimental Analysis and Discussion

An experiment is conducted to determine the frequencyresponse curve of two cantilever sandwich beams with dif-ferent core thicknesses The material properties of the testedsandwich beams are listed in Table 1 A schematic of theexperimental setup is depicted in Figure 8 The cantileverbeam is installed on a VDL shaker (B ampKV830-335-SPA16K)using a head plate and a fixture both made of aluminum

To reduce experimental error the fixture is designed inSolidWorks such that its fundamental frequency is well abovethe maximum excitation frequency A signal generated fromthe controller (Laser USB 14V) is fed to the power amplifier(LSD SPA 16K) and then to the vibration shaker The signalis fed back to the controller to ensure precise measurementTwo accelerometers (B amp K 8325) are employed for input andoutput measurements One is placed at the top of the fixtureto measure the input acceleration and the other is placedat the tip of the beam to measure the output response Thetest is carried out with a constant velocity of 2 mms and asine sweep is performed for a frequency range of 10 to 2000Hz Measurements are sent to the data acquisition systemthrough a signal analyzer The frequency response plots areobtained and the frequencies corresponding to the peaks ofthese plots are the natural frequencies of the tested sandwich

Shock and Vibration 7

(a) (b)

(c) (d)

(e) (f)

Figure 2 The first two mode shapes of sandwich beam with cantilever ((a) and (b)) simply-simply ((c) and (d)) and clamped-clamped ((e)and (f)) supported edges

beams The experimental results are shown in Figure 9which also illustrates the role of the core thickness on thefrequency response curve The results clearly indicate thatnatural frequency increases with increasing thickness while

vibration response decreases with increasing thickness Thisobservation is in agreement with the results obtained usingthe GDQ method Table 4 compares natural frequenciesobtained experimentally and numerically (GDQmethod) for

8 Shock and Vibration

Figure 3 Tip deflection of free edge of cantilever sandwich beam (120596119905) for applied frequency 1205961205961 = 01 1205961205961 = 087 and 1205961205961 ≃ 1 [9]

Table 3 The five low natural frequencies for different boundary conditions (Hz)

BCs Mode 1 2 3 4 5

C-F

ABAQUS [14] 32740 10791 21380 21753 22164GDQ 32918 108322 215254 220158 225815

QEM [13] 32966 108843 216459 218378 222572HQEM (full [M]) [14] 32782 10824 21539 21759 22179

C-C

ABAQUS [14] 83992 19567 22556 27026 34778GDQ 84417 197658 227456 270982 354127

QEM [13] 84871 198813 226266 272513 356396HQEM (full [M]) [14] 84403 19781 22571 27175 35475

S-S

ABAQUS [14] 61856 13491 15033 21704 23584GDQ 61905 134825 150881 221264 241309

QEM [13] 62222 135286 151545 217686 236977HQEM (full [M]) [14] 61891 13493 15082 21712 23623

F-F

ABAQUS [14] 12902 13669 21537 21692 22146GDQ 129210 136687 219787 220386 224586

QEM [13] 129919 137239 216280 217640 222519HQEM (full [M] [14] 12918 13676 21542 21701 22152

S-CABAQUS [14] 71545 17144 21964 25144 31171

GDQ 71725 172564 22339 25413 315728HQEM (full [M]) [14] 71718 17258 21975 25212 31607

Table 4 The natural frequencies (Hz) for cantilever beam with two different thicknesses

Modes ℎ119888 = 15 mm ℎ119888 = 25 mmGDQmethod Experiment GDQ method Experiment

1 11800 11119 14392 143022 38100 36460 45227 475213 71984 68080 82186 824334 114194 106350 126156 1323815 167551 165207 180017 177260

Shock and Vibration 9

(a) The effect of core by GDQmethod (b) The effect of core by SolidWorks

(c) The effect of face sheets by GDQmethod (d) The effect of face sheets by SolidWorks

(e) The effect of width by GDQmethod (f) The effect of width by SolidWorks

Figure 4 The effect of geometric parameters on the time response at the tip of the cantilever sandwich beam

10 Shock and Vibration

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 5 Vibration amplitude at the tip of the cantilever beam with respect to applied frequency

a core thickness of 15 and 25 mm The results in this tablealso show very good agreementThis is also an indication thatthe present formulation for both free and forced vibrationanalyses of sandwich beams is accurate

6 Conclusions

This paper presents the vibration analysis of sandwich beamswith various boundary conditions The governing equationsof motion are derived using Hamiltonrsquos principle The GDQmethod is utilized for solving the problem SolidWorkssimulation and experimental analyses are conducted to assessthe validity of the GDQ method and the results showvery good agreement Parametric studies are conducted toexamine the role of geometric properties on the time andfrequency response curvesThe results indicate that vibrationamplitude decreases with increasing both core and facesheet thicknesses whereas natural frequency increases withincreasing core sheet thickness but can increase or decreasewith varying face sheet thickness The results also show thatvibration amplitude decreases with increasing beam widthwhereas no effect is observed on the natural frequency for

varying the width of the beam Numerical examples alsodemonstrate that all studied boundary conditions exhibitsimilar effect with varying the aforementioned geometricproperties The findings in this paper are anticipated to beappealing to research communities because of the experi-mental and FEM validation of the GDQ method for thefree and forced vibration analyses of sandwich beams undervarious boundary conditions

Appendix

The rest of the six equations of motion are given as follows

119898119891 (119887) + 119898119888 [ 120 (119905 + 119887) minus 128 (119905 minus 119887) + 115 119888minus 170ℎ119888 1205931 + 140ℎ119891 (1015840119905 minus 1015840119887) minus 156ℎ119891 (1015840119905 + 1015840119887)]minus 11986411989111986011989111990610158401015840119887 + 119866119888119860119888ℎ2119888 [43 (119906119905 + 119906119887) minus 95 (119906119905 minus 119906119887)minus 83119906119888 + 45ℎ1198881205931 + (23ℎ119891 + 13ℎ119888) (1199081015840119905 minus 1199081015840119887)

Shock and Vibration 11

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 6 Vibration amplitude at the middle of the simply-simply supported edges sandwich beam with respect to applied frequency

minus ( 910ℎ119891 + 310ℎ119888) (1199081015840119905 + 1199081015840119887) minus 25ℎ1198881199081015840119888] = 0(A1)

119898119891 (119905 minus 112ℎ211989110158401015840119905 ) + 119898119888 [minus 140ℎ119891 (1015840119905 + 1015840119887)minus 156ℎ119891 (1015840119905 minus 1015840119887) minus 130ℎ1198911015840119888 minus 1140ℎ119888ℎ119891 12059311015840minus 180ℎ2119891 (10158401015840119905 minus 10158401015840119887 ) + 112 (119905 minus 119887)minus 1112ℎ2119891 (10158401015840119905 + 10158401015840119887 ) + 120 (119905 + 119887) + 115119888]+ 1198641198911198681198911199081015840101584010158401015840119905 + 119866119888119860119888ℎ2119888 [minus (23ℎ119891 + 13ℎ119888) (1199061015840119905 + 1199061015840119887)minus ( 910ℎ119891 + 310ℎ119888) (1199061015840119905 minus 1199061015840119887) + (43ℎ119891 + 23ℎ119888) 1199061015840119888+ (25ℎ119891ℎ119888 + 215ℎ2119888) (12059310158401)

minus ( 920ℎ2119891 + 310ℎ119888ℎ119891 + 120ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 )minus (13ℎ119891ℎ119888 + 13ℎ2119891 + 112ℎ2119888) (11990810158401015840119905 minus 11990810158401015840119887 )minus (15ℎ119891ℎ119888 + 115ℎ2119888)11990810158401015840119888 ] + 119864119888119860119888ℎ2119888 [43 (119908119905 + 119908119887)+ (119908119905 minus 119908119887) minus 83119908119888] = 0

(A2)

119898119891 (119887 minus 112ℎ211989110158401015840119887 ) + 119898119888 [ 140ℎ119891 (1015840119905 + 1015840119887)minus 156ℎ119891 (1015840119905 minus 1015840119887) + 130ℎ1198911015840119888 minus 1140ℎ119888ℎ119891 12059311015840+ 180ℎ2119891 (10158401015840119905 minus 10158401015840119887 ) minus 112 (119905 minus 119887)minus 1112ℎ2119891 (10158401015840119905 + 10158401015840119887 ) + 120 (119905 + 119887) + 115119888]

12 Shock and Vibration

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 7 Vibration amplitude at the middle of the clamped-clamped edges sandwich beam with respect to applied frequency

+ 1198641198911198681198911199081015840101584010158401015840119887 + 119866119888119860119888ℎ2119888 [(23ℎ119891 + 13ℎ119888) (1199061015840119905 + 1199061015840119887)minus ( 910ℎ119891 + 310ℎ119888) (1199061015840119905 minus 1199061015840119887) minus (43ℎ119891 + 23ℎ119888) 1199061015840119888+ (25ℎ119891ℎ119888 + 215ℎ2119888) (12059310158401)minus ( 920ℎ2119891 + 310ℎ119888ℎ119891 + 120ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 )+ (13ℎ119891ℎ119888 + 13ℎ2119891 + 112ℎ2119888) (11990810158401015840119905 minus 11990810158401015840119887 )minus (15ℎ119891ℎ119888 + 115ℎ2119888)11990810158401015840119888 ] + 119864119888119860119888ℎ2119888 [43 (119908119905 + 119908119887)minus (119908119905 minus 119908119887) minus 83119908119888] = 0

(A3)

119898119888 [ 815 119888 + 115 (119887 + 119905) + 130ℎ119891 (1015840119905 minus 1015840119887)]+ 119866119888119860119888ℎ1198882 [minus83 (119906119905 + 119906119887) + 163 119906119888minus (43ℎ119891 + 23ℎ119888) (1199081015840119905 minus 1199081015840119887)] = 0

(A4)

119898119888 [ 2105ℎ2119888 1205931 + 170ℎ119888 (119905 minus 119887)+ 1140ℎ119891ℎ119888 (1015840119905 + 1015840119887)] + 119866119888119860119888ℎ1198882 [minus45ℎ119888 (119906119905 minus 119906119887)+ 45ℎ21198881205931 + 415ℎ21198881199081015840119888minus ( 215ℎ2119888 + 25ℎ119888ℎ119891) (1199081015840119905 + 1199081015840119887)] = 0

(A5)

Shock and Vibration 13

Figure 8 Schematic of experimental setup

Figure 9 Vibration amplitude of cantilever sandwich beam fordifferent core thicknesses by experiment

119898119888 [ 815119888 + 115ℎ119888 (119887 + 119905)]+ 119866119888119860119888ℎ1198882 [minus25ℎ119888 (1199061015840119905 minus 1199061015840119887) minus 415ℎ21198881205931minus ( 115ℎ119891ℎ119888 + 115ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 ) minus 815ℎ211988811990810158401015840119888 ]+ 119864119888119860119888ℎ2119888 [minus83 (119908119905 + 119908119887) + 163 119908119888] = 0

(A6)

The essential and natural boundary conditions related to theclamped-free case are given as

119906119905 (0 119905) = 0119875119905 (119897 119905) = 0119908119905 (0 119905) = 0119881119905 (119897 119905) = 01199081199051015840 (0 119905) = 0119872119905 (119897 119905) = 0119908119888 (0 119905) = 0119881119888 (119897 119905) = 0119906119887 (0 119905) = 0119875119887 (119897 119905) = 0119908119887 (0 119905) = 0119881119887 (119897 119905) = 01199081198871015840 (0 119905) = 0119872119887 (119897 119905) = 0

(A7)

where the left side of sandwich beam is clamped and the rightside is free

The axial forces 119875119905119887(119909 119905) the shear forces 119881119905119887119888(119909 119905) andthe bending moments119872119905119887(119909 119905) are obtained as

119875119905 = minus119864119891119860119891 120597119906119905120597119909 (A8)

119875119887 = minus119864119891119860119891 120597119906119887120597119909 (A9)

119881119905 = minus 112119898119891ℎ21198911015840119905 + 119898119888840 [minus36ℎ119891119905 minus 6ℎ119891119887minus 28ℎ119891119888 minus 6ℎ119891ℎ119888 1205931 + 3ℎ21198911015840119887 minus 18ℎ21198911015840119905)+ 1198641198911198681198911199081015840101584010158401015840119905 + 119866119888119860119888840ℎ2119888 [(196ℎ119891 minus 28ℎ119888) 119906119887minus (1316ℎ119891 + 532ℎ119888) 119906119905 + (560ℎ119888 + 1120ℎ119891) 119906119888+ (28ℎ2119888 minus 98ℎ2119891 + 28ℎ119891ℎ119888)11990810158401015840119887minus (112ℎ2119888 + 658ℎ2119891 + 532ℎ2119891 + 532ℎ119891ℎ119888)1199081015840119905+ (336ℎ119891ℎ119888 + 112ℎ2119888) 1205931minus (56ℎ2119888 + 168ℎ119891ℎ119888)1199081015840119888]

(A10)

119881119887 = minus 112119898119891ℎ21198911015840119887 + 119898119888840 (36ℎ119891119887 + 6ℎ119891119905+ 28ℎ119891119888 minus 6ℎ119891ℎ119888 1205931 + 3ℎ21198911015840119905 minus 18ℎ21198911015840119887)

14 Shock and Vibration

+ 1198641198911198681198911199081015840101584010158401015840119887 + 119866119888119860119888840ℎ2119888 [(minus196ℎ119891 + 28ℎ119888) 119906119905+ (1316ℎ119891 + 532ℎ119888) 119906119887 minus (560ℎ119888 + 1120ℎ119891) 119906119888+ (28ℎ2119888 minus 98ℎ2119891 + 28ℎ119891ℎ119888)11990810158401015840119905minus (112ℎ2119888 + 658ℎ2119891 + 532ℎ2119891 + 532ℎ119891ℎ119888)1199081015840119887+ (336ℎ119891ℎ119888 + 112ℎ2119888) 1205931minus (56ℎ2119888 + 168ℎ119891ℎ119888)1199081015840119888]

(A11)

119881119888 = 119866119888119860119888840ℎ2119888 [minus336ℎ119891 (119906119905 minus 119906119887) minus (224ℎ2119888) 1205931

minus (56ℎ2119888 + 168ℎ119891ℎ119888) (1199081015840119905 minus 1199081015840119887) minus 448ℎ21198881199081015840119888](A12)

119872119905 = minus11986411989111986811989111990810158401015840119905 (A13)

119872119887 = minus11986411989111986811989111990810158401015840119887 (A14)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] A S Herrmann P C Zahlen and I Zuardy ldquoSandwich Struc-tures Technology in Commercial Aviationrdquo in Proceedings of the7th International Conference on Sandwich Structures pp 13ndash262005

[2] J Vinson The behavior of sandwich structures of isotropic andcomposite materials Technomic Publishing Company Lan-caster Pennsylvania 1999

[3] J Hohe L Librescu and S Y Oh ldquoDynamic buckling of flat andcurved sandwich panels with transversely compressible corerdquoComposite Structures vol 74 no 1 pp 10ndash24 2006

[4] 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

[5] Y Frostig M Baruch O Vilnay and I Sheinman ldquoHigh-ordertheory for sandwich-beam behavior with transversely flexiblecorerdquo Journal of Engineering Mechanics vol 118 no 5 pp 1026ndash1043 1992

[6] O Rahmani SM R Khalili KMalekzadeh andHHadavinialdquoFree vibration analysis of sandwich structures with a flexiblefunctionally graded syntactic corerdquo Composite Structures vol91 no 2 pp 229ndash235 2009

[7] C Santiuste O TThomsen and Y Frostig ldquoThermo-mechani-cal load interactions in foam cored axi-symmetric sandwich cir-cular plates - High-order and FEmodelsrdquo Composite Structuresvol 93 no 2 pp 369ndash376 2011

[8] R K Fruehmann J M Dulieu-Barton O T Thomsen andS Zhang ldquoExperimental investigation of thermal effects infoam cored sandwich beamsrdquo in Thermomechanics and Infra-Red Imaging vol 7 of Conference Proceedings of the Society forExperimental Mechanics Series pp 83ndash90 Springer NY USA2011

[9] V S Sokolinsky and S R Nutt ldquoConsistent Higher-OrderDynamic Equations for Soft-Core Sandwich Beamsrdquo AIAAJournal vol 42 no 2 pp 374ndash382 2004

[10] Y Frostig C Phan and G Kardomateas ldquoFree vibration ofunidirectional sandwich panels Part I Compressible corerdquoJournal of Sandwich Structures and Materials vol 15 no 4 pp377ndash411 2013

[11] A R Damanpack and S M R Khalili ldquoHigh-order free vibra-tion analysis of sandwich beams with a flexible core usingdynamic stiffness methodrdquo Composite Structures vol 94 no 5pp 1503ndash1514 2012

[12] M Yang and P Qiao ldquoHigher-order impact modeling of sand-wich structures with flexible corerdquo International Journal ofSolids and Structures vol 42 no 20 pp 5460ndash5490 2005

[13] Y Wang and X Wang ldquoFree vibration analysis of soft-coresandwich beams by the novel weak form quadrature elementmethodrdquo Journal of Sandwich Structures and Materials vol 18no 3 pp 294ndash320 2015

[14] X Wang and X Liang ldquoFree vibration of soft-core sandwichpanels with general boundary conditions by harmonic quadra-ture element methodrdquoThin-Walled Structures vol 113 pp 253ndash261 2017

[15] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe local GDQmethod applied to general higher-order theories of doubly-curved laminated composite shells and panels the free vibrationanalysisrdquo Composite Structures vol 116 pp 637ndash660 2014

[16] C C Hong H W Liao L T Lee J B Ke and K C JaneldquoThermally induced vibration of a thermal sleeve with the GDQmethodrdquo International Journal of Mechanical Sciences vol 47no 12 pp 1789ndash1806 2005

[17] H D Chalak A Chakrabarti M A Iqbal and A H SheikhldquoVibration of laminated sandwich beams having soft corerdquoJournal of Vibration and Control vol 18 no 10 pp 1422ndash14352012

[18] K M Ahmed ldquoFree vibration of curved sandwich beams by themethod of finite elementsrdquo Journal of Sound and Vibration vol18 no 1 pp 61ndash74 1971

[19] N Nanda S Kapuria and S Gopalakrishnan ldquoSpectral finiteelement based on an efficient layerwise theory for wave prop-agation analysis of composite and sandwich beamsrdquo Journal ofSound and Vibration vol 333 no 31 pp 20ndash27 2014

[20] Y Pourvais P Asgari A R Moradi and O Rahmani ldquoExper-imental and finite element analysis of higher order behaviourof sandwich beams using digital projection moirerdquo PolymerTesting vol 38 pp 7ndash17 2014

[21] R Long O Barry and D C D Oguamanam ldquoFinite elementfree vibration analysis of soft-core sandwich beamsrdquo AIAAJournal vol 50 no 1 pp 235ndash238 2012

[22] W L Chen A G Striz and C W Bert ldquoHigh-accuracy planestress and plate elements in the quadrature element methodrdquoInternational Journal of Solids and Structures vol 37 no 4 pp627ndash647 2000

[23] Y Xing and B Liu ldquoHigh-accuracy differential quadraturefinite element method and its application to free vibrations ofthin plate with curvilinear domainrdquo International Journal for

Shock and Vibration 15

Numerical Methods in Engineering vol 80 no 13 pp 1718ndash17422009

[24] YWang and XWang ldquoStatic analysis of higher order sandwichbeams by weak form quadrature element methodrdquo CompositeStructures vol 116 no 1 pp 841ndash848 2014

[25] F Tornabene S Brischetto N Fantuzzi and M BacciocchildquoBoundary conditions in 2Dnumerical and 3D exactmodels forcylindrical bending analysis of functionally graded structuresrdquoShock and Vibration vol 2016 Article ID 2373862 2016

[26] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe GDQmethod for the free vibration analysis of arbitrarily shaped lami-nated composite shells using a NURBS-based isogeometricapproachrdquo Composite Structures vol 154 pp 190ndash218 2016

[27] Z Zong and Y Zhang Advanced differential quadrature meth-ods CRC Press NY USA 2009

[28] N A Rubayi and S Charoenree ldquoNatural frequencies of vibra-tion of cantilever sandwich beamsrdquoComputersampStructures vol7 no 6 pp 737ndash745 1977

[29] M Azimi M Shahravi and E F Joubaneh ldquoDynamic analysisof maneuvering flexible spacecraft appendage using higherorder sandwich panel theoryrdquo Latin American Journal of Solidsand Structures vol 13 no 2 pp 296ndash313 2016

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6 Shock and Vibration

Table 1 Parameters of sandwich beam

References 119864119891 (119866119875119886) 119866119888 (119872119875119886) 120588119891 (1198961198921198983) 120588119888 (1198961198921198983) ℎ119891 (119898119898) ℎ119888 (119898119898) 119887 (119898119898) 119871 (119898119898)[11] 210 221 7900 60 19 348 599 260[12] 36 200 4400 521 05 200 200 300[13] 136 128 1800 585 5 19 20 152Experiment 210 221 7900 60 19 15 and 25 599 260

Table 2 The five low natural frequencies for a simply supported sandwich beam (rads)

Mode Present method (GDQ) Exact [11] Model (C) [12]1 204846 204841 2048192 518973 518967 5183373 825024 825019 8224064 1122532 1122527 11159635 1413922 1413919 1400912

material property from [11] listed in Table 1 Figure 3 showsthe time response of the cantilever sandwich beam for threeexcitation frequencies namely 120596 = 011205961 120596 = 0871205961 and120596 ≃ 11205961 Here 1205961 is the first natural frequency of sandwichbeam Beating phenomenon is observed when the excitationfrequency is 120596 = 0871205961 and resonance phenomenon isdemonstrated when the excitation frequency matches thefundamental frequency of the sandwich beam that is 120596 ≃11205961 Figure 3 also shows much smaller vibration amplitudewhen the fundamental frequency is far from the excitationfrequency 120596 = 011205961

The validation of the forced vibration analysis is demon-strated in Figure 4 by comparing the results of the GDQmethod (Case (a) (c) (e)) to those of SolidWorks simulation(Case (b) (d) (f)) It should be noted that this figure showsthe time response measured at the tip of the cantileversandwich beam with varying thickness and width For eachcase the cantilever sandwich beam is excited with its cor-responding fundamental frequency since natural frequencychanges with beam thicknesswidthThe results in this figureshow very good agreement between the GDQ method andSolidWorks simulations

It is also observed in Figure 4 that the vibration amplitudedecreases with increasing core thickness andor increasingface sheet thickness This is expected as increasing thicknessincreases bending stiffness thus resulting in lower deflectionFigure 4 also shows the role of the beam width on thetime response The results show that the vibration amplitudedecreases with increasing beam width This is also expectedas the width increases with bending stiffness Although boththickness and width affect the vibration amplitude only thethickness affects the natural frequency of the beam Thisobservation is in agreement with literatures [28 29]

Figures 5 6 and 7 show the role of geometry parameterson the frequency response curves for cantilever simply-simply supported and clamped-clamped boundary condi-tions respectively Similar trends are observed for all threeboundary conditions The frequency response curves aresignificantly affected by the core and face sheet thicknessesas well as the width of the sandwich beam Increasing

any of these geometry parameters obviously decreases thevibration amplitude The peaks corresponding to the exci-tation frequency change with varying thickness but remainconstant with varying width The natural frequency in gen-eral increases with increasing core thickness for all studiedboundary conditions As for the face sheet thickness thenatural frequency can increase or decrease with varying facesheet thickness The reason for this can be attributed tothe core thickness having more effect on the stiffness thanthe mass whereas the sheet thickness has similar effect onboth mass and stiffness The beam width shows no effecton the natural frequency for all boundary conditions Thisobservation is also in agreement with earlier discussion aboutthe role of the beam width on the natural frequency

5 Experimental Analysis and Discussion

An experiment is conducted to determine the frequencyresponse curve of two cantilever sandwich beams with dif-ferent core thicknesses The material properties of the testedsandwich beams are listed in Table 1 A schematic of theexperimental setup is depicted in Figure 8 The cantileverbeam is installed on a VDL shaker (B ampKV830-335-SPA16K)using a head plate and a fixture both made of aluminum

To reduce experimental error the fixture is designed inSolidWorks such that its fundamental frequency is well abovethe maximum excitation frequency A signal generated fromthe controller (Laser USB 14V) is fed to the power amplifier(LSD SPA 16K) and then to the vibration shaker The signalis fed back to the controller to ensure precise measurementTwo accelerometers (B amp K 8325) are employed for input andoutput measurements One is placed at the top of the fixtureto measure the input acceleration and the other is placedat the tip of the beam to measure the output response Thetest is carried out with a constant velocity of 2 mms and asine sweep is performed for a frequency range of 10 to 2000Hz Measurements are sent to the data acquisition systemthrough a signal analyzer The frequency response plots areobtained and the frequencies corresponding to the peaks ofthese plots are the natural frequencies of the tested sandwich

Shock and Vibration 7

(a) (b)

(c) (d)

(e) (f)

Figure 2 The first two mode shapes of sandwich beam with cantilever ((a) and (b)) simply-simply ((c) and (d)) and clamped-clamped ((e)and (f)) supported edges

beams The experimental results are shown in Figure 9which also illustrates the role of the core thickness on thefrequency response curve The results clearly indicate thatnatural frequency increases with increasing thickness while

vibration response decreases with increasing thickness Thisobservation is in agreement with the results obtained usingthe GDQ method Table 4 compares natural frequenciesobtained experimentally and numerically (GDQmethod) for

8 Shock and Vibration

Figure 3 Tip deflection of free edge of cantilever sandwich beam (120596119905) for applied frequency 1205961205961 = 01 1205961205961 = 087 and 1205961205961 ≃ 1 [9]

Table 3 The five low natural frequencies for different boundary conditions (Hz)

BCs Mode 1 2 3 4 5

C-F

ABAQUS [14] 32740 10791 21380 21753 22164GDQ 32918 108322 215254 220158 225815

QEM [13] 32966 108843 216459 218378 222572HQEM (full [M]) [14] 32782 10824 21539 21759 22179

C-C

ABAQUS [14] 83992 19567 22556 27026 34778GDQ 84417 197658 227456 270982 354127

QEM [13] 84871 198813 226266 272513 356396HQEM (full [M]) [14] 84403 19781 22571 27175 35475

S-S

ABAQUS [14] 61856 13491 15033 21704 23584GDQ 61905 134825 150881 221264 241309

QEM [13] 62222 135286 151545 217686 236977HQEM (full [M]) [14] 61891 13493 15082 21712 23623

F-F

ABAQUS [14] 12902 13669 21537 21692 22146GDQ 129210 136687 219787 220386 224586

QEM [13] 129919 137239 216280 217640 222519HQEM (full [M] [14] 12918 13676 21542 21701 22152

S-CABAQUS [14] 71545 17144 21964 25144 31171

GDQ 71725 172564 22339 25413 315728HQEM (full [M]) [14] 71718 17258 21975 25212 31607

Table 4 The natural frequencies (Hz) for cantilever beam with two different thicknesses

Modes ℎ119888 = 15 mm ℎ119888 = 25 mmGDQmethod Experiment GDQ method Experiment

1 11800 11119 14392 143022 38100 36460 45227 475213 71984 68080 82186 824334 114194 106350 126156 1323815 167551 165207 180017 177260

Shock and Vibration 9

(a) The effect of core by GDQmethod (b) The effect of core by SolidWorks

(c) The effect of face sheets by GDQmethod (d) The effect of face sheets by SolidWorks

(e) The effect of width by GDQmethod (f) The effect of width by SolidWorks

Figure 4 The effect of geometric parameters on the time response at the tip of the cantilever sandwich beam

10 Shock and Vibration

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 5 Vibration amplitude at the tip of the cantilever beam with respect to applied frequency

a core thickness of 15 and 25 mm The results in this tablealso show very good agreementThis is also an indication thatthe present formulation for both free and forced vibrationanalyses of sandwich beams is accurate

6 Conclusions

This paper presents the vibration analysis of sandwich beamswith various boundary conditions The governing equationsof motion are derived using Hamiltonrsquos principle The GDQmethod is utilized for solving the problem SolidWorkssimulation and experimental analyses are conducted to assessthe validity of the GDQ method and the results showvery good agreement Parametric studies are conducted toexamine the role of geometric properties on the time andfrequency response curvesThe results indicate that vibrationamplitude decreases with increasing both core and facesheet thicknesses whereas natural frequency increases withincreasing core sheet thickness but can increase or decreasewith varying face sheet thickness The results also show thatvibration amplitude decreases with increasing beam widthwhereas no effect is observed on the natural frequency for

varying the width of the beam Numerical examples alsodemonstrate that all studied boundary conditions exhibitsimilar effect with varying the aforementioned geometricproperties The findings in this paper are anticipated to beappealing to research communities because of the experi-mental and FEM validation of the GDQ method for thefree and forced vibration analyses of sandwich beams undervarious boundary conditions

Appendix

The rest of the six equations of motion are given as follows

119898119891 (119887) + 119898119888 [ 120 (119905 + 119887) minus 128 (119905 minus 119887) + 115 119888minus 170ℎ119888 1205931 + 140ℎ119891 (1015840119905 minus 1015840119887) minus 156ℎ119891 (1015840119905 + 1015840119887)]minus 11986411989111986011989111990610158401015840119887 + 119866119888119860119888ℎ2119888 [43 (119906119905 + 119906119887) minus 95 (119906119905 minus 119906119887)minus 83119906119888 + 45ℎ1198881205931 + (23ℎ119891 + 13ℎ119888) (1199081015840119905 minus 1199081015840119887)

Shock and Vibration 11

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 6 Vibration amplitude at the middle of the simply-simply supported edges sandwich beam with respect to applied frequency

minus ( 910ℎ119891 + 310ℎ119888) (1199081015840119905 + 1199081015840119887) minus 25ℎ1198881199081015840119888] = 0(A1)

119898119891 (119905 minus 112ℎ211989110158401015840119905 ) + 119898119888 [minus 140ℎ119891 (1015840119905 + 1015840119887)minus 156ℎ119891 (1015840119905 minus 1015840119887) minus 130ℎ1198911015840119888 minus 1140ℎ119888ℎ119891 12059311015840minus 180ℎ2119891 (10158401015840119905 minus 10158401015840119887 ) + 112 (119905 minus 119887)minus 1112ℎ2119891 (10158401015840119905 + 10158401015840119887 ) + 120 (119905 + 119887) + 115119888]+ 1198641198911198681198911199081015840101584010158401015840119905 + 119866119888119860119888ℎ2119888 [minus (23ℎ119891 + 13ℎ119888) (1199061015840119905 + 1199061015840119887)minus ( 910ℎ119891 + 310ℎ119888) (1199061015840119905 minus 1199061015840119887) + (43ℎ119891 + 23ℎ119888) 1199061015840119888+ (25ℎ119891ℎ119888 + 215ℎ2119888) (12059310158401)

minus ( 920ℎ2119891 + 310ℎ119888ℎ119891 + 120ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 )minus (13ℎ119891ℎ119888 + 13ℎ2119891 + 112ℎ2119888) (11990810158401015840119905 minus 11990810158401015840119887 )minus (15ℎ119891ℎ119888 + 115ℎ2119888)11990810158401015840119888 ] + 119864119888119860119888ℎ2119888 [43 (119908119905 + 119908119887)+ (119908119905 minus 119908119887) minus 83119908119888] = 0

(A2)

119898119891 (119887 minus 112ℎ211989110158401015840119887 ) + 119898119888 [ 140ℎ119891 (1015840119905 + 1015840119887)minus 156ℎ119891 (1015840119905 minus 1015840119887) + 130ℎ1198911015840119888 minus 1140ℎ119888ℎ119891 12059311015840+ 180ℎ2119891 (10158401015840119905 minus 10158401015840119887 ) minus 112 (119905 minus 119887)minus 1112ℎ2119891 (10158401015840119905 + 10158401015840119887 ) + 120 (119905 + 119887) + 115119888]

12 Shock and Vibration

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 7 Vibration amplitude at the middle of the clamped-clamped edges sandwich beam with respect to applied frequency

+ 1198641198911198681198911199081015840101584010158401015840119887 + 119866119888119860119888ℎ2119888 [(23ℎ119891 + 13ℎ119888) (1199061015840119905 + 1199061015840119887)minus ( 910ℎ119891 + 310ℎ119888) (1199061015840119905 minus 1199061015840119887) minus (43ℎ119891 + 23ℎ119888) 1199061015840119888+ (25ℎ119891ℎ119888 + 215ℎ2119888) (12059310158401)minus ( 920ℎ2119891 + 310ℎ119888ℎ119891 + 120ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 )+ (13ℎ119891ℎ119888 + 13ℎ2119891 + 112ℎ2119888) (11990810158401015840119905 minus 11990810158401015840119887 )minus (15ℎ119891ℎ119888 + 115ℎ2119888)11990810158401015840119888 ] + 119864119888119860119888ℎ2119888 [43 (119908119905 + 119908119887)minus (119908119905 minus 119908119887) minus 83119908119888] = 0

(A3)

119898119888 [ 815 119888 + 115 (119887 + 119905) + 130ℎ119891 (1015840119905 minus 1015840119887)]+ 119866119888119860119888ℎ1198882 [minus83 (119906119905 + 119906119887) + 163 119906119888minus (43ℎ119891 + 23ℎ119888) (1199081015840119905 minus 1199081015840119887)] = 0

(A4)

119898119888 [ 2105ℎ2119888 1205931 + 170ℎ119888 (119905 minus 119887)+ 1140ℎ119891ℎ119888 (1015840119905 + 1015840119887)] + 119866119888119860119888ℎ1198882 [minus45ℎ119888 (119906119905 minus 119906119887)+ 45ℎ21198881205931 + 415ℎ21198881199081015840119888minus ( 215ℎ2119888 + 25ℎ119888ℎ119891) (1199081015840119905 + 1199081015840119887)] = 0

(A5)

Shock and Vibration 13

Figure 8 Schematic of experimental setup

Figure 9 Vibration amplitude of cantilever sandwich beam fordifferent core thicknesses by experiment

119898119888 [ 815119888 + 115ℎ119888 (119887 + 119905)]+ 119866119888119860119888ℎ1198882 [minus25ℎ119888 (1199061015840119905 minus 1199061015840119887) minus 415ℎ21198881205931minus ( 115ℎ119891ℎ119888 + 115ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 ) minus 815ℎ211988811990810158401015840119888 ]+ 119864119888119860119888ℎ2119888 [minus83 (119908119905 + 119908119887) + 163 119908119888] = 0

(A6)

The essential and natural boundary conditions related to theclamped-free case are given as

119906119905 (0 119905) = 0119875119905 (119897 119905) = 0119908119905 (0 119905) = 0119881119905 (119897 119905) = 01199081199051015840 (0 119905) = 0119872119905 (119897 119905) = 0119908119888 (0 119905) = 0119881119888 (119897 119905) = 0119906119887 (0 119905) = 0119875119887 (119897 119905) = 0119908119887 (0 119905) = 0119881119887 (119897 119905) = 01199081198871015840 (0 119905) = 0119872119887 (119897 119905) = 0

(A7)

where the left side of sandwich beam is clamped and the rightside is free

The axial forces 119875119905119887(119909 119905) the shear forces 119881119905119887119888(119909 119905) andthe bending moments119872119905119887(119909 119905) are obtained as

119875119905 = minus119864119891119860119891 120597119906119905120597119909 (A8)

119875119887 = minus119864119891119860119891 120597119906119887120597119909 (A9)

119881119905 = minus 112119898119891ℎ21198911015840119905 + 119898119888840 [minus36ℎ119891119905 minus 6ℎ119891119887minus 28ℎ119891119888 minus 6ℎ119891ℎ119888 1205931 + 3ℎ21198911015840119887 minus 18ℎ21198911015840119905)+ 1198641198911198681198911199081015840101584010158401015840119905 + 119866119888119860119888840ℎ2119888 [(196ℎ119891 minus 28ℎ119888) 119906119887minus (1316ℎ119891 + 532ℎ119888) 119906119905 + (560ℎ119888 + 1120ℎ119891) 119906119888+ (28ℎ2119888 minus 98ℎ2119891 + 28ℎ119891ℎ119888)11990810158401015840119887minus (112ℎ2119888 + 658ℎ2119891 + 532ℎ2119891 + 532ℎ119891ℎ119888)1199081015840119905+ (336ℎ119891ℎ119888 + 112ℎ2119888) 1205931minus (56ℎ2119888 + 168ℎ119891ℎ119888)1199081015840119888]

(A10)

119881119887 = minus 112119898119891ℎ21198911015840119887 + 119898119888840 (36ℎ119891119887 + 6ℎ119891119905+ 28ℎ119891119888 minus 6ℎ119891ℎ119888 1205931 + 3ℎ21198911015840119905 minus 18ℎ21198911015840119887)

14 Shock and Vibration

+ 1198641198911198681198911199081015840101584010158401015840119887 + 119866119888119860119888840ℎ2119888 [(minus196ℎ119891 + 28ℎ119888) 119906119905+ (1316ℎ119891 + 532ℎ119888) 119906119887 minus (560ℎ119888 + 1120ℎ119891) 119906119888+ (28ℎ2119888 minus 98ℎ2119891 + 28ℎ119891ℎ119888)11990810158401015840119905minus (112ℎ2119888 + 658ℎ2119891 + 532ℎ2119891 + 532ℎ119891ℎ119888)1199081015840119887+ (336ℎ119891ℎ119888 + 112ℎ2119888) 1205931minus (56ℎ2119888 + 168ℎ119891ℎ119888)1199081015840119888]

(A11)

119881119888 = 119866119888119860119888840ℎ2119888 [minus336ℎ119891 (119906119905 minus 119906119887) minus (224ℎ2119888) 1205931

minus (56ℎ2119888 + 168ℎ119891ℎ119888) (1199081015840119905 minus 1199081015840119887) minus 448ℎ21198881199081015840119888](A12)

119872119905 = minus11986411989111986811989111990810158401015840119905 (A13)

119872119887 = minus11986411989111986811989111990810158401015840119887 (A14)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] A S Herrmann P C Zahlen and I Zuardy ldquoSandwich Struc-tures Technology in Commercial Aviationrdquo in Proceedings of the7th International Conference on Sandwich Structures pp 13ndash262005

[2] J Vinson The behavior of sandwich structures of isotropic andcomposite materials Technomic Publishing Company Lan-caster Pennsylvania 1999

[3] J Hohe L Librescu and S Y Oh ldquoDynamic buckling of flat andcurved sandwich panels with transversely compressible corerdquoComposite Structures vol 74 no 1 pp 10ndash24 2006

[4] 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

[5] Y Frostig M Baruch O Vilnay and I Sheinman ldquoHigh-ordertheory for sandwich-beam behavior with transversely flexiblecorerdquo Journal of Engineering Mechanics vol 118 no 5 pp 1026ndash1043 1992

[6] O Rahmani SM R Khalili KMalekzadeh andHHadavinialdquoFree vibration analysis of sandwich structures with a flexiblefunctionally graded syntactic corerdquo Composite Structures vol91 no 2 pp 229ndash235 2009

[7] C Santiuste O TThomsen and Y Frostig ldquoThermo-mechani-cal load interactions in foam cored axi-symmetric sandwich cir-cular plates - High-order and FEmodelsrdquo Composite Structuresvol 93 no 2 pp 369ndash376 2011

[8] R K Fruehmann J M Dulieu-Barton O T Thomsen andS Zhang ldquoExperimental investigation of thermal effects infoam cored sandwich beamsrdquo in Thermomechanics and Infra-Red Imaging vol 7 of Conference Proceedings of the Society forExperimental Mechanics Series pp 83ndash90 Springer NY USA2011

[9] V S Sokolinsky and S R Nutt ldquoConsistent Higher-OrderDynamic Equations for Soft-Core Sandwich Beamsrdquo AIAAJournal vol 42 no 2 pp 374ndash382 2004

[10] Y Frostig C Phan and G Kardomateas ldquoFree vibration ofunidirectional sandwich panels Part I Compressible corerdquoJournal of Sandwich Structures and Materials vol 15 no 4 pp377ndash411 2013

[11] A R Damanpack and S M R Khalili ldquoHigh-order free vibra-tion analysis of sandwich beams with a flexible core usingdynamic stiffness methodrdquo Composite Structures vol 94 no 5pp 1503ndash1514 2012

[12] M Yang and P Qiao ldquoHigher-order impact modeling of sand-wich structures with flexible corerdquo International Journal ofSolids and Structures vol 42 no 20 pp 5460ndash5490 2005

[13] Y Wang and X Wang ldquoFree vibration analysis of soft-coresandwich beams by the novel weak form quadrature elementmethodrdquo Journal of Sandwich Structures and Materials vol 18no 3 pp 294ndash320 2015

[14] X Wang and X Liang ldquoFree vibration of soft-core sandwichpanels with general boundary conditions by harmonic quadra-ture element methodrdquoThin-Walled Structures vol 113 pp 253ndash261 2017

[15] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe local GDQmethod applied to general higher-order theories of doubly-curved laminated composite shells and panels the free vibrationanalysisrdquo Composite Structures vol 116 pp 637ndash660 2014

[16] C C Hong H W Liao L T Lee J B Ke and K C JaneldquoThermally induced vibration of a thermal sleeve with the GDQmethodrdquo International Journal of Mechanical Sciences vol 47no 12 pp 1789ndash1806 2005

[17] H D Chalak A Chakrabarti M A Iqbal and A H SheikhldquoVibration of laminated sandwich beams having soft corerdquoJournal of Vibration and Control vol 18 no 10 pp 1422ndash14352012

[18] K M Ahmed ldquoFree vibration of curved sandwich beams by themethod of finite elementsrdquo Journal of Sound and Vibration vol18 no 1 pp 61ndash74 1971

[19] N Nanda S Kapuria and S Gopalakrishnan ldquoSpectral finiteelement based on an efficient layerwise theory for wave prop-agation analysis of composite and sandwich beamsrdquo Journal ofSound and Vibration vol 333 no 31 pp 20ndash27 2014

[20] Y Pourvais P Asgari A R Moradi and O Rahmani ldquoExper-imental and finite element analysis of higher order behaviourof sandwich beams using digital projection moirerdquo PolymerTesting vol 38 pp 7ndash17 2014

[21] R Long O Barry and D C D Oguamanam ldquoFinite elementfree vibration analysis of soft-core sandwich beamsrdquo AIAAJournal vol 50 no 1 pp 235ndash238 2012

[22] W L Chen A G Striz and C W Bert ldquoHigh-accuracy planestress and plate elements in the quadrature element methodrdquoInternational Journal of Solids and Structures vol 37 no 4 pp627ndash647 2000

[23] Y Xing and B Liu ldquoHigh-accuracy differential quadraturefinite element method and its application to free vibrations ofthin plate with curvilinear domainrdquo International Journal for

Shock and Vibration 15

Numerical Methods in Engineering vol 80 no 13 pp 1718ndash17422009

[24] YWang and XWang ldquoStatic analysis of higher order sandwichbeams by weak form quadrature element methodrdquo CompositeStructures vol 116 no 1 pp 841ndash848 2014

[25] F Tornabene S Brischetto N Fantuzzi and M BacciocchildquoBoundary conditions in 2Dnumerical and 3D exactmodels forcylindrical bending analysis of functionally graded structuresrdquoShock and Vibration vol 2016 Article ID 2373862 2016

[26] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe GDQmethod for the free vibration analysis of arbitrarily shaped lami-nated composite shells using a NURBS-based isogeometricapproachrdquo Composite Structures vol 154 pp 190ndash218 2016

[27] Z Zong and Y Zhang Advanced differential quadrature meth-ods CRC Press NY USA 2009

[28] N A Rubayi and S Charoenree ldquoNatural frequencies of vibra-tion of cantilever sandwich beamsrdquoComputersampStructures vol7 no 6 pp 737ndash745 1977

[29] M Azimi M Shahravi and E F Joubaneh ldquoDynamic analysisof maneuvering flexible spacecraft appendage using higherorder sandwich panel theoryrdquo Latin American Journal of Solidsand Structures vol 13 no 2 pp 296ndash313 2016

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Shock and Vibration 7

(a) (b)

(c) (d)

(e) (f)

Figure 2 The first two mode shapes of sandwich beam with cantilever ((a) and (b)) simply-simply ((c) and (d)) and clamped-clamped ((e)and (f)) supported edges

beams The experimental results are shown in Figure 9which also illustrates the role of the core thickness on thefrequency response curve The results clearly indicate thatnatural frequency increases with increasing thickness while

vibration response decreases with increasing thickness Thisobservation is in agreement with the results obtained usingthe GDQ method Table 4 compares natural frequenciesobtained experimentally and numerically (GDQmethod) for

8 Shock and Vibration

Figure 3 Tip deflection of free edge of cantilever sandwich beam (120596119905) for applied frequency 1205961205961 = 01 1205961205961 = 087 and 1205961205961 ≃ 1 [9]

Table 3 The five low natural frequencies for different boundary conditions (Hz)

BCs Mode 1 2 3 4 5

C-F

ABAQUS [14] 32740 10791 21380 21753 22164GDQ 32918 108322 215254 220158 225815

QEM [13] 32966 108843 216459 218378 222572HQEM (full [M]) [14] 32782 10824 21539 21759 22179

C-C

ABAQUS [14] 83992 19567 22556 27026 34778GDQ 84417 197658 227456 270982 354127

QEM [13] 84871 198813 226266 272513 356396HQEM (full [M]) [14] 84403 19781 22571 27175 35475

S-S

ABAQUS [14] 61856 13491 15033 21704 23584GDQ 61905 134825 150881 221264 241309

QEM [13] 62222 135286 151545 217686 236977HQEM (full [M]) [14] 61891 13493 15082 21712 23623

F-F

ABAQUS [14] 12902 13669 21537 21692 22146GDQ 129210 136687 219787 220386 224586

QEM [13] 129919 137239 216280 217640 222519HQEM (full [M] [14] 12918 13676 21542 21701 22152

S-CABAQUS [14] 71545 17144 21964 25144 31171

GDQ 71725 172564 22339 25413 315728HQEM (full [M]) [14] 71718 17258 21975 25212 31607

Table 4 The natural frequencies (Hz) for cantilever beam with two different thicknesses

Modes ℎ119888 = 15 mm ℎ119888 = 25 mmGDQmethod Experiment GDQ method Experiment

1 11800 11119 14392 143022 38100 36460 45227 475213 71984 68080 82186 824334 114194 106350 126156 1323815 167551 165207 180017 177260

Shock and Vibration 9

(a) The effect of core by GDQmethod (b) The effect of core by SolidWorks

(c) The effect of face sheets by GDQmethod (d) The effect of face sheets by SolidWorks

(e) The effect of width by GDQmethod (f) The effect of width by SolidWorks

Figure 4 The effect of geometric parameters on the time response at the tip of the cantilever sandwich beam

10 Shock and Vibration

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 5 Vibration amplitude at the tip of the cantilever beam with respect to applied frequency

a core thickness of 15 and 25 mm The results in this tablealso show very good agreementThis is also an indication thatthe present formulation for both free and forced vibrationanalyses of sandwich beams is accurate

6 Conclusions

This paper presents the vibration analysis of sandwich beamswith various boundary conditions The governing equationsof motion are derived using Hamiltonrsquos principle The GDQmethod is utilized for solving the problem SolidWorkssimulation and experimental analyses are conducted to assessthe validity of the GDQ method and the results showvery good agreement Parametric studies are conducted toexamine the role of geometric properties on the time andfrequency response curvesThe results indicate that vibrationamplitude decreases with increasing both core and facesheet thicknesses whereas natural frequency increases withincreasing core sheet thickness but can increase or decreasewith varying face sheet thickness The results also show thatvibration amplitude decreases with increasing beam widthwhereas no effect is observed on the natural frequency for

varying the width of the beam Numerical examples alsodemonstrate that all studied boundary conditions exhibitsimilar effect with varying the aforementioned geometricproperties The findings in this paper are anticipated to beappealing to research communities because of the experi-mental and FEM validation of the GDQ method for thefree and forced vibration analyses of sandwich beams undervarious boundary conditions

Appendix

The rest of the six equations of motion are given as follows

119898119891 (119887) + 119898119888 [ 120 (119905 + 119887) minus 128 (119905 minus 119887) + 115 119888minus 170ℎ119888 1205931 + 140ℎ119891 (1015840119905 minus 1015840119887) minus 156ℎ119891 (1015840119905 + 1015840119887)]minus 11986411989111986011989111990610158401015840119887 + 119866119888119860119888ℎ2119888 [43 (119906119905 + 119906119887) minus 95 (119906119905 minus 119906119887)minus 83119906119888 + 45ℎ1198881205931 + (23ℎ119891 + 13ℎ119888) (1199081015840119905 minus 1199081015840119887)

Shock and Vibration 11

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 6 Vibration amplitude at the middle of the simply-simply supported edges sandwich beam with respect to applied frequency

minus ( 910ℎ119891 + 310ℎ119888) (1199081015840119905 + 1199081015840119887) minus 25ℎ1198881199081015840119888] = 0(A1)

119898119891 (119905 minus 112ℎ211989110158401015840119905 ) + 119898119888 [minus 140ℎ119891 (1015840119905 + 1015840119887)minus 156ℎ119891 (1015840119905 minus 1015840119887) minus 130ℎ1198911015840119888 minus 1140ℎ119888ℎ119891 12059311015840minus 180ℎ2119891 (10158401015840119905 minus 10158401015840119887 ) + 112 (119905 minus 119887)minus 1112ℎ2119891 (10158401015840119905 + 10158401015840119887 ) + 120 (119905 + 119887) + 115119888]+ 1198641198911198681198911199081015840101584010158401015840119905 + 119866119888119860119888ℎ2119888 [minus (23ℎ119891 + 13ℎ119888) (1199061015840119905 + 1199061015840119887)minus ( 910ℎ119891 + 310ℎ119888) (1199061015840119905 minus 1199061015840119887) + (43ℎ119891 + 23ℎ119888) 1199061015840119888+ (25ℎ119891ℎ119888 + 215ℎ2119888) (12059310158401)

minus ( 920ℎ2119891 + 310ℎ119888ℎ119891 + 120ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 )minus (13ℎ119891ℎ119888 + 13ℎ2119891 + 112ℎ2119888) (11990810158401015840119905 minus 11990810158401015840119887 )minus (15ℎ119891ℎ119888 + 115ℎ2119888)11990810158401015840119888 ] + 119864119888119860119888ℎ2119888 [43 (119908119905 + 119908119887)+ (119908119905 minus 119908119887) minus 83119908119888] = 0

(A2)

119898119891 (119887 minus 112ℎ211989110158401015840119887 ) + 119898119888 [ 140ℎ119891 (1015840119905 + 1015840119887)minus 156ℎ119891 (1015840119905 minus 1015840119887) + 130ℎ1198911015840119888 minus 1140ℎ119888ℎ119891 12059311015840+ 180ℎ2119891 (10158401015840119905 minus 10158401015840119887 ) minus 112 (119905 minus 119887)minus 1112ℎ2119891 (10158401015840119905 + 10158401015840119887 ) + 120 (119905 + 119887) + 115119888]

12 Shock and Vibration

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 7 Vibration amplitude at the middle of the clamped-clamped edges sandwich beam with respect to applied frequency

+ 1198641198911198681198911199081015840101584010158401015840119887 + 119866119888119860119888ℎ2119888 [(23ℎ119891 + 13ℎ119888) (1199061015840119905 + 1199061015840119887)minus ( 910ℎ119891 + 310ℎ119888) (1199061015840119905 minus 1199061015840119887) minus (43ℎ119891 + 23ℎ119888) 1199061015840119888+ (25ℎ119891ℎ119888 + 215ℎ2119888) (12059310158401)minus ( 920ℎ2119891 + 310ℎ119888ℎ119891 + 120ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 )+ (13ℎ119891ℎ119888 + 13ℎ2119891 + 112ℎ2119888) (11990810158401015840119905 minus 11990810158401015840119887 )minus (15ℎ119891ℎ119888 + 115ℎ2119888)11990810158401015840119888 ] + 119864119888119860119888ℎ2119888 [43 (119908119905 + 119908119887)minus (119908119905 minus 119908119887) minus 83119908119888] = 0

(A3)

119898119888 [ 815 119888 + 115 (119887 + 119905) + 130ℎ119891 (1015840119905 minus 1015840119887)]+ 119866119888119860119888ℎ1198882 [minus83 (119906119905 + 119906119887) + 163 119906119888minus (43ℎ119891 + 23ℎ119888) (1199081015840119905 minus 1199081015840119887)] = 0

(A4)

119898119888 [ 2105ℎ2119888 1205931 + 170ℎ119888 (119905 minus 119887)+ 1140ℎ119891ℎ119888 (1015840119905 + 1015840119887)] + 119866119888119860119888ℎ1198882 [minus45ℎ119888 (119906119905 minus 119906119887)+ 45ℎ21198881205931 + 415ℎ21198881199081015840119888minus ( 215ℎ2119888 + 25ℎ119888ℎ119891) (1199081015840119905 + 1199081015840119887)] = 0

(A5)

Shock and Vibration 13

Figure 8 Schematic of experimental setup

Figure 9 Vibration amplitude of cantilever sandwich beam fordifferent core thicknesses by experiment

119898119888 [ 815119888 + 115ℎ119888 (119887 + 119905)]+ 119866119888119860119888ℎ1198882 [minus25ℎ119888 (1199061015840119905 minus 1199061015840119887) minus 415ℎ21198881205931minus ( 115ℎ119891ℎ119888 + 115ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 ) minus 815ℎ211988811990810158401015840119888 ]+ 119864119888119860119888ℎ2119888 [minus83 (119908119905 + 119908119887) + 163 119908119888] = 0

(A6)

The essential and natural boundary conditions related to theclamped-free case are given as

119906119905 (0 119905) = 0119875119905 (119897 119905) = 0119908119905 (0 119905) = 0119881119905 (119897 119905) = 01199081199051015840 (0 119905) = 0119872119905 (119897 119905) = 0119908119888 (0 119905) = 0119881119888 (119897 119905) = 0119906119887 (0 119905) = 0119875119887 (119897 119905) = 0119908119887 (0 119905) = 0119881119887 (119897 119905) = 01199081198871015840 (0 119905) = 0119872119887 (119897 119905) = 0

(A7)

where the left side of sandwich beam is clamped and the rightside is free

The axial forces 119875119905119887(119909 119905) the shear forces 119881119905119887119888(119909 119905) andthe bending moments119872119905119887(119909 119905) are obtained as

119875119905 = minus119864119891119860119891 120597119906119905120597119909 (A8)

119875119887 = minus119864119891119860119891 120597119906119887120597119909 (A9)

119881119905 = minus 112119898119891ℎ21198911015840119905 + 119898119888840 [minus36ℎ119891119905 minus 6ℎ119891119887minus 28ℎ119891119888 minus 6ℎ119891ℎ119888 1205931 + 3ℎ21198911015840119887 minus 18ℎ21198911015840119905)+ 1198641198911198681198911199081015840101584010158401015840119905 + 119866119888119860119888840ℎ2119888 [(196ℎ119891 minus 28ℎ119888) 119906119887minus (1316ℎ119891 + 532ℎ119888) 119906119905 + (560ℎ119888 + 1120ℎ119891) 119906119888+ (28ℎ2119888 minus 98ℎ2119891 + 28ℎ119891ℎ119888)11990810158401015840119887minus (112ℎ2119888 + 658ℎ2119891 + 532ℎ2119891 + 532ℎ119891ℎ119888)1199081015840119905+ (336ℎ119891ℎ119888 + 112ℎ2119888) 1205931minus (56ℎ2119888 + 168ℎ119891ℎ119888)1199081015840119888]

(A10)

119881119887 = minus 112119898119891ℎ21198911015840119887 + 119898119888840 (36ℎ119891119887 + 6ℎ119891119905+ 28ℎ119891119888 minus 6ℎ119891ℎ119888 1205931 + 3ℎ21198911015840119905 minus 18ℎ21198911015840119887)

14 Shock and Vibration

+ 1198641198911198681198911199081015840101584010158401015840119887 + 119866119888119860119888840ℎ2119888 [(minus196ℎ119891 + 28ℎ119888) 119906119905+ (1316ℎ119891 + 532ℎ119888) 119906119887 minus (560ℎ119888 + 1120ℎ119891) 119906119888+ (28ℎ2119888 minus 98ℎ2119891 + 28ℎ119891ℎ119888)11990810158401015840119905minus (112ℎ2119888 + 658ℎ2119891 + 532ℎ2119891 + 532ℎ119891ℎ119888)1199081015840119887+ (336ℎ119891ℎ119888 + 112ℎ2119888) 1205931minus (56ℎ2119888 + 168ℎ119891ℎ119888)1199081015840119888]

(A11)

119881119888 = 119866119888119860119888840ℎ2119888 [minus336ℎ119891 (119906119905 minus 119906119887) minus (224ℎ2119888) 1205931

minus (56ℎ2119888 + 168ℎ119891ℎ119888) (1199081015840119905 minus 1199081015840119887) minus 448ℎ21198881199081015840119888](A12)

119872119905 = minus11986411989111986811989111990810158401015840119905 (A13)

119872119887 = minus11986411989111986811989111990810158401015840119887 (A14)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] A S Herrmann P C Zahlen and I Zuardy ldquoSandwich Struc-tures Technology in Commercial Aviationrdquo in Proceedings of the7th International Conference on Sandwich Structures pp 13ndash262005

[2] J Vinson The behavior of sandwich structures of isotropic andcomposite materials Technomic Publishing Company Lan-caster Pennsylvania 1999

[3] J Hohe L Librescu and S Y Oh ldquoDynamic buckling of flat andcurved sandwich panels with transversely compressible corerdquoComposite Structures vol 74 no 1 pp 10ndash24 2006

[4] 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

[5] Y Frostig M Baruch O Vilnay and I Sheinman ldquoHigh-ordertheory for sandwich-beam behavior with transversely flexiblecorerdquo Journal of Engineering Mechanics vol 118 no 5 pp 1026ndash1043 1992

[6] O Rahmani SM R Khalili KMalekzadeh andHHadavinialdquoFree vibration analysis of sandwich structures with a flexiblefunctionally graded syntactic corerdquo Composite Structures vol91 no 2 pp 229ndash235 2009

[7] C Santiuste O TThomsen and Y Frostig ldquoThermo-mechani-cal load interactions in foam cored axi-symmetric sandwich cir-cular plates - High-order and FEmodelsrdquo Composite Structuresvol 93 no 2 pp 369ndash376 2011

[8] R K Fruehmann J M Dulieu-Barton O T Thomsen andS Zhang ldquoExperimental investigation of thermal effects infoam cored sandwich beamsrdquo in Thermomechanics and Infra-Red Imaging vol 7 of Conference Proceedings of the Society forExperimental Mechanics Series pp 83ndash90 Springer NY USA2011

[9] V S Sokolinsky and S R Nutt ldquoConsistent Higher-OrderDynamic Equations for Soft-Core Sandwich Beamsrdquo AIAAJournal vol 42 no 2 pp 374ndash382 2004

[10] Y Frostig C Phan and G Kardomateas ldquoFree vibration ofunidirectional sandwich panels Part I Compressible corerdquoJournal of Sandwich Structures and Materials vol 15 no 4 pp377ndash411 2013

[11] A R Damanpack and S M R Khalili ldquoHigh-order free vibra-tion analysis of sandwich beams with a flexible core usingdynamic stiffness methodrdquo Composite Structures vol 94 no 5pp 1503ndash1514 2012

[12] M Yang and P Qiao ldquoHigher-order impact modeling of sand-wich structures with flexible corerdquo International Journal ofSolids and Structures vol 42 no 20 pp 5460ndash5490 2005

[13] Y Wang and X Wang ldquoFree vibration analysis of soft-coresandwich beams by the novel weak form quadrature elementmethodrdquo Journal of Sandwich Structures and Materials vol 18no 3 pp 294ndash320 2015

[14] X Wang and X Liang ldquoFree vibration of soft-core sandwichpanels with general boundary conditions by harmonic quadra-ture element methodrdquoThin-Walled Structures vol 113 pp 253ndash261 2017

[15] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe local GDQmethod applied to general higher-order theories of doubly-curved laminated composite shells and panels the free vibrationanalysisrdquo Composite Structures vol 116 pp 637ndash660 2014

[16] C C Hong H W Liao L T Lee J B Ke and K C JaneldquoThermally induced vibration of a thermal sleeve with the GDQmethodrdquo International Journal of Mechanical Sciences vol 47no 12 pp 1789ndash1806 2005

[17] H D Chalak A Chakrabarti M A Iqbal and A H SheikhldquoVibration of laminated sandwich beams having soft corerdquoJournal of Vibration and Control vol 18 no 10 pp 1422ndash14352012

[18] K M Ahmed ldquoFree vibration of curved sandwich beams by themethod of finite elementsrdquo Journal of Sound and Vibration vol18 no 1 pp 61ndash74 1971

[19] N Nanda S Kapuria and S Gopalakrishnan ldquoSpectral finiteelement based on an efficient layerwise theory for wave prop-agation analysis of composite and sandwich beamsrdquo Journal ofSound and Vibration vol 333 no 31 pp 20ndash27 2014

[20] Y Pourvais P Asgari A R Moradi and O Rahmani ldquoExper-imental and finite element analysis of higher order behaviourof sandwich beams using digital projection moirerdquo PolymerTesting vol 38 pp 7ndash17 2014

[21] R Long O Barry and D C D Oguamanam ldquoFinite elementfree vibration analysis of soft-core sandwich beamsrdquo AIAAJournal vol 50 no 1 pp 235ndash238 2012

[22] W L Chen A G Striz and C W Bert ldquoHigh-accuracy planestress and plate elements in the quadrature element methodrdquoInternational Journal of Solids and Structures vol 37 no 4 pp627ndash647 2000

[23] Y Xing and B Liu ldquoHigh-accuracy differential quadraturefinite element method and its application to free vibrations ofthin plate with curvilinear domainrdquo International Journal for

Shock and Vibration 15

Numerical Methods in Engineering vol 80 no 13 pp 1718ndash17422009

[24] YWang and XWang ldquoStatic analysis of higher order sandwichbeams by weak form quadrature element methodrdquo CompositeStructures vol 116 no 1 pp 841ndash848 2014

[25] F Tornabene S Brischetto N Fantuzzi and M BacciocchildquoBoundary conditions in 2Dnumerical and 3D exactmodels forcylindrical bending analysis of functionally graded structuresrdquoShock and Vibration vol 2016 Article ID 2373862 2016

[26] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe GDQmethod for the free vibration analysis of arbitrarily shaped lami-nated composite shells using a NURBS-based isogeometricapproachrdquo Composite Structures vol 154 pp 190ndash218 2016

[27] Z Zong and Y Zhang Advanced differential quadrature meth-ods CRC Press NY USA 2009

[28] N A Rubayi and S Charoenree ldquoNatural frequencies of vibra-tion of cantilever sandwich beamsrdquoComputersampStructures vol7 no 6 pp 737ndash745 1977

[29] M Azimi M Shahravi and E F Joubaneh ldquoDynamic analysisof maneuvering flexible spacecraft appendage using higherorder sandwich panel theoryrdquo Latin American Journal of Solidsand Structures vol 13 no 2 pp 296ndash313 2016

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8 Shock and Vibration

Figure 3 Tip deflection of free edge of cantilever sandwich beam (120596119905) for applied frequency 1205961205961 = 01 1205961205961 = 087 and 1205961205961 ≃ 1 [9]

Table 3 The five low natural frequencies for different boundary conditions (Hz)

BCs Mode 1 2 3 4 5

C-F

ABAQUS [14] 32740 10791 21380 21753 22164GDQ 32918 108322 215254 220158 225815

QEM [13] 32966 108843 216459 218378 222572HQEM (full [M]) [14] 32782 10824 21539 21759 22179

C-C

ABAQUS [14] 83992 19567 22556 27026 34778GDQ 84417 197658 227456 270982 354127

QEM [13] 84871 198813 226266 272513 356396HQEM (full [M]) [14] 84403 19781 22571 27175 35475

S-S

ABAQUS [14] 61856 13491 15033 21704 23584GDQ 61905 134825 150881 221264 241309

QEM [13] 62222 135286 151545 217686 236977HQEM (full [M]) [14] 61891 13493 15082 21712 23623

F-F

ABAQUS [14] 12902 13669 21537 21692 22146GDQ 129210 136687 219787 220386 224586

QEM [13] 129919 137239 216280 217640 222519HQEM (full [M] [14] 12918 13676 21542 21701 22152

S-CABAQUS [14] 71545 17144 21964 25144 31171

GDQ 71725 172564 22339 25413 315728HQEM (full [M]) [14] 71718 17258 21975 25212 31607

Table 4 The natural frequencies (Hz) for cantilever beam with two different thicknesses

Modes ℎ119888 = 15 mm ℎ119888 = 25 mmGDQmethod Experiment GDQ method Experiment

1 11800 11119 14392 143022 38100 36460 45227 475213 71984 68080 82186 824334 114194 106350 126156 1323815 167551 165207 180017 177260

Shock and Vibration 9

(a) The effect of core by GDQmethod (b) The effect of core by SolidWorks

(c) The effect of face sheets by GDQmethod (d) The effect of face sheets by SolidWorks

(e) The effect of width by GDQmethod (f) The effect of width by SolidWorks

Figure 4 The effect of geometric parameters on the time response at the tip of the cantilever sandwich beam

10 Shock and Vibration

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 5 Vibration amplitude at the tip of the cantilever beam with respect to applied frequency

a core thickness of 15 and 25 mm The results in this tablealso show very good agreementThis is also an indication thatthe present formulation for both free and forced vibrationanalyses of sandwich beams is accurate

6 Conclusions

This paper presents the vibration analysis of sandwich beamswith various boundary conditions The governing equationsof motion are derived using Hamiltonrsquos principle The GDQmethod is utilized for solving the problem SolidWorkssimulation and experimental analyses are conducted to assessthe validity of the GDQ method and the results showvery good agreement Parametric studies are conducted toexamine the role of geometric properties on the time andfrequency response curvesThe results indicate that vibrationamplitude decreases with increasing both core and facesheet thicknesses whereas natural frequency increases withincreasing core sheet thickness but can increase or decreasewith varying face sheet thickness The results also show thatvibration amplitude decreases with increasing beam widthwhereas no effect is observed on the natural frequency for

varying the width of the beam Numerical examples alsodemonstrate that all studied boundary conditions exhibitsimilar effect with varying the aforementioned geometricproperties The findings in this paper are anticipated to beappealing to research communities because of the experi-mental and FEM validation of the GDQ method for thefree and forced vibration analyses of sandwich beams undervarious boundary conditions

Appendix

The rest of the six equations of motion are given as follows

119898119891 (119887) + 119898119888 [ 120 (119905 + 119887) minus 128 (119905 minus 119887) + 115 119888minus 170ℎ119888 1205931 + 140ℎ119891 (1015840119905 minus 1015840119887) minus 156ℎ119891 (1015840119905 + 1015840119887)]minus 11986411989111986011989111990610158401015840119887 + 119866119888119860119888ℎ2119888 [43 (119906119905 + 119906119887) minus 95 (119906119905 minus 119906119887)minus 83119906119888 + 45ℎ1198881205931 + (23ℎ119891 + 13ℎ119888) (1199081015840119905 minus 1199081015840119887)

Shock and Vibration 11

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 6 Vibration amplitude at the middle of the simply-simply supported edges sandwich beam with respect to applied frequency

minus ( 910ℎ119891 + 310ℎ119888) (1199081015840119905 + 1199081015840119887) minus 25ℎ1198881199081015840119888] = 0(A1)

119898119891 (119905 minus 112ℎ211989110158401015840119905 ) + 119898119888 [minus 140ℎ119891 (1015840119905 + 1015840119887)minus 156ℎ119891 (1015840119905 minus 1015840119887) minus 130ℎ1198911015840119888 minus 1140ℎ119888ℎ119891 12059311015840minus 180ℎ2119891 (10158401015840119905 minus 10158401015840119887 ) + 112 (119905 minus 119887)minus 1112ℎ2119891 (10158401015840119905 + 10158401015840119887 ) + 120 (119905 + 119887) + 115119888]+ 1198641198911198681198911199081015840101584010158401015840119905 + 119866119888119860119888ℎ2119888 [minus (23ℎ119891 + 13ℎ119888) (1199061015840119905 + 1199061015840119887)minus ( 910ℎ119891 + 310ℎ119888) (1199061015840119905 minus 1199061015840119887) + (43ℎ119891 + 23ℎ119888) 1199061015840119888+ (25ℎ119891ℎ119888 + 215ℎ2119888) (12059310158401)

minus ( 920ℎ2119891 + 310ℎ119888ℎ119891 + 120ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 )minus (13ℎ119891ℎ119888 + 13ℎ2119891 + 112ℎ2119888) (11990810158401015840119905 minus 11990810158401015840119887 )minus (15ℎ119891ℎ119888 + 115ℎ2119888)11990810158401015840119888 ] + 119864119888119860119888ℎ2119888 [43 (119908119905 + 119908119887)+ (119908119905 minus 119908119887) minus 83119908119888] = 0

(A2)

119898119891 (119887 minus 112ℎ211989110158401015840119887 ) + 119898119888 [ 140ℎ119891 (1015840119905 + 1015840119887)minus 156ℎ119891 (1015840119905 minus 1015840119887) + 130ℎ1198911015840119888 minus 1140ℎ119888ℎ119891 12059311015840+ 180ℎ2119891 (10158401015840119905 minus 10158401015840119887 ) minus 112 (119905 minus 119887)minus 1112ℎ2119891 (10158401015840119905 + 10158401015840119887 ) + 120 (119905 + 119887) + 115119888]

12 Shock and Vibration

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 7 Vibration amplitude at the middle of the clamped-clamped edges sandwich beam with respect to applied frequency

+ 1198641198911198681198911199081015840101584010158401015840119887 + 119866119888119860119888ℎ2119888 [(23ℎ119891 + 13ℎ119888) (1199061015840119905 + 1199061015840119887)minus ( 910ℎ119891 + 310ℎ119888) (1199061015840119905 minus 1199061015840119887) minus (43ℎ119891 + 23ℎ119888) 1199061015840119888+ (25ℎ119891ℎ119888 + 215ℎ2119888) (12059310158401)minus ( 920ℎ2119891 + 310ℎ119888ℎ119891 + 120ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 )+ (13ℎ119891ℎ119888 + 13ℎ2119891 + 112ℎ2119888) (11990810158401015840119905 minus 11990810158401015840119887 )minus (15ℎ119891ℎ119888 + 115ℎ2119888)11990810158401015840119888 ] + 119864119888119860119888ℎ2119888 [43 (119908119905 + 119908119887)minus (119908119905 minus 119908119887) minus 83119908119888] = 0

(A3)

119898119888 [ 815 119888 + 115 (119887 + 119905) + 130ℎ119891 (1015840119905 minus 1015840119887)]+ 119866119888119860119888ℎ1198882 [minus83 (119906119905 + 119906119887) + 163 119906119888minus (43ℎ119891 + 23ℎ119888) (1199081015840119905 minus 1199081015840119887)] = 0

(A4)

119898119888 [ 2105ℎ2119888 1205931 + 170ℎ119888 (119905 minus 119887)+ 1140ℎ119891ℎ119888 (1015840119905 + 1015840119887)] + 119866119888119860119888ℎ1198882 [minus45ℎ119888 (119906119905 minus 119906119887)+ 45ℎ21198881205931 + 415ℎ21198881199081015840119888minus ( 215ℎ2119888 + 25ℎ119888ℎ119891) (1199081015840119905 + 1199081015840119887)] = 0

(A5)

Shock and Vibration 13

Figure 8 Schematic of experimental setup

Figure 9 Vibration amplitude of cantilever sandwich beam fordifferent core thicknesses by experiment

119898119888 [ 815119888 + 115ℎ119888 (119887 + 119905)]+ 119866119888119860119888ℎ1198882 [minus25ℎ119888 (1199061015840119905 minus 1199061015840119887) minus 415ℎ21198881205931minus ( 115ℎ119891ℎ119888 + 115ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 ) minus 815ℎ211988811990810158401015840119888 ]+ 119864119888119860119888ℎ2119888 [minus83 (119908119905 + 119908119887) + 163 119908119888] = 0

(A6)

The essential and natural boundary conditions related to theclamped-free case are given as

119906119905 (0 119905) = 0119875119905 (119897 119905) = 0119908119905 (0 119905) = 0119881119905 (119897 119905) = 01199081199051015840 (0 119905) = 0119872119905 (119897 119905) = 0119908119888 (0 119905) = 0119881119888 (119897 119905) = 0119906119887 (0 119905) = 0119875119887 (119897 119905) = 0119908119887 (0 119905) = 0119881119887 (119897 119905) = 01199081198871015840 (0 119905) = 0119872119887 (119897 119905) = 0

(A7)

where the left side of sandwich beam is clamped and the rightside is free

The axial forces 119875119905119887(119909 119905) the shear forces 119881119905119887119888(119909 119905) andthe bending moments119872119905119887(119909 119905) are obtained as

119875119905 = minus119864119891119860119891 120597119906119905120597119909 (A8)

119875119887 = minus119864119891119860119891 120597119906119887120597119909 (A9)

119881119905 = minus 112119898119891ℎ21198911015840119905 + 119898119888840 [minus36ℎ119891119905 minus 6ℎ119891119887minus 28ℎ119891119888 minus 6ℎ119891ℎ119888 1205931 + 3ℎ21198911015840119887 minus 18ℎ21198911015840119905)+ 1198641198911198681198911199081015840101584010158401015840119905 + 119866119888119860119888840ℎ2119888 [(196ℎ119891 minus 28ℎ119888) 119906119887minus (1316ℎ119891 + 532ℎ119888) 119906119905 + (560ℎ119888 + 1120ℎ119891) 119906119888+ (28ℎ2119888 minus 98ℎ2119891 + 28ℎ119891ℎ119888)11990810158401015840119887minus (112ℎ2119888 + 658ℎ2119891 + 532ℎ2119891 + 532ℎ119891ℎ119888)1199081015840119905+ (336ℎ119891ℎ119888 + 112ℎ2119888) 1205931minus (56ℎ2119888 + 168ℎ119891ℎ119888)1199081015840119888]

(A10)

119881119887 = minus 112119898119891ℎ21198911015840119887 + 119898119888840 (36ℎ119891119887 + 6ℎ119891119905+ 28ℎ119891119888 minus 6ℎ119891ℎ119888 1205931 + 3ℎ21198911015840119905 minus 18ℎ21198911015840119887)

14 Shock and Vibration

+ 1198641198911198681198911199081015840101584010158401015840119887 + 119866119888119860119888840ℎ2119888 [(minus196ℎ119891 + 28ℎ119888) 119906119905+ (1316ℎ119891 + 532ℎ119888) 119906119887 minus (560ℎ119888 + 1120ℎ119891) 119906119888+ (28ℎ2119888 minus 98ℎ2119891 + 28ℎ119891ℎ119888)11990810158401015840119905minus (112ℎ2119888 + 658ℎ2119891 + 532ℎ2119891 + 532ℎ119891ℎ119888)1199081015840119887+ (336ℎ119891ℎ119888 + 112ℎ2119888) 1205931minus (56ℎ2119888 + 168ℎ119891ℎ119888)1199081015840119888]

(A11)

119881119888 = 119866119888119860119888840ℎ2119888 [minus336ℎ119891 (119906119905 minus 119906119887) minus (224ℎ2119888) 1205931

minus (56ℎ2119888 + 168ℎ119891ℎ119888) (1199081015840119905 minus 1199081015840119887) minus 448ℎ21198881199081015840119888](A12)

119872119905 = minus11986411989111986811989111990810158401015840119905 (A13)

119872119887 = minus11986411989111986811989111990810158401015840119887 (A14)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] A S Herrmann P C Zahlen and I Zuardy ldquoSandwich Struc-tures Technology in Commercial Aviationrdquo in Proceedings of the7th International Conference on Sandwich Structures pp 13ndash262005

[2] J Vinson The behavior of sandwich structures of isotropic andcomposite materials Technomic Publishing Company Lan-caster Pennsylvania 1999

[3] J Hohe L Librescu and S Y Oh ldquoDynamic buckling of flat andcurved sandwich panels with transversely compressible corerdquoComposite Structures vol 74 no 1 pp 10ndash24 2006

[4] 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

[5] Y Frostig M Baruch O Vilnay and I Sheinman ldquoHigh-ordertheory for sandwich-beam behavior with transversely flexiblecorerdquo Journal of Engineering Mechanics vol 118 no 5 pp 1026ndash1043 1992

[6] O Rahmani SM R Khalili KMalekzadeh andHHadavinialdquoFree vibration analysis of sandwich structures with a flexiblefunctionally graded syntactic corerdquo Composite Structures vol91 no 2 pp 229ndash235 2009

[7] C Santiuste O TThomsen and Y Frostig ldquoThermo-mechani-cal load interactions in foam cored axi-symmetric sandwich cir-cular plates - High-order and FEmodelsrdquo Composite Structuresvol 93 no 2 pp 369ndash376 2011

[8] R K Fruehmann J M Dulieu-Barton O T Thomsen andS Zhang ldquoExperimental investigation of thermal effects infoam cored sandwich beamsrdquo in Thermomechanics and Infra-Red Imaging vol 7 of Conference Proceedings of the Society forExperimental Mechanics Series pp 83ndash90 Springer NY USA2011

[9] V S Sokolinsky and S R Nutt ldquoConsistent Higher-OrderDynamic Equations for Soft-Core Sandwich Beamsrdquo AIAAJournal vol 42 no 2 pp 374ndash382 2004

[10] Y Frostig C Phan and G Kardomateas ldquoFree vibration ofunidirectional sandwich panels Part I Compressible corerdquoJournal of Sandwich Structures and Materials vol 15 no 4 pp377ndash411 2013

[11] A R Damanpack and S M R Khalili ldquoHigh-order free vibra-tion analysis of sandwich beams with a flexible core usingdynamic stiffness methodrdquo Composite Structures vol 94 no 5pp 1503ndash1514 2012

[12] M Yang and P Qiao ldquoHigher-order impact modeling of sand-wich structures with flexible corerdquo International Journal ofSolids and Structures vol 42 no 20 pp 5460ndash5490 2005

[13] Y Wang and X Wang ldquoFree vibration analysis of soft-coresandwich beams by the novel weak form quadrature elementmethodrdquo Journal of Sandwich Structures and Materials vol 18no 3 pp 294ndash320 2015

[14] X Wang and X Liang ldquoFree vibration of soft-core sandwichpanels with general boundary conditions by harmonic quadra-ture element methodrdquoThin-Walled Structures vol 113 pp 253ndash261 2017

[15] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe local GDQmethod applied to general higher-order theories of doubly-curved laminated composite shells and panels the free vibrationanalysisrdquo Composite Structures vol 116 pp 637ndash660 2014

[16] C C Hong H W Liao L T Lee J B Ke and K C JaneldquoThermally induced vibration of a thermal sleeve with the GDQmethodrdquo International Journal of Mechanical Sciences vol 47no 12 pp 1789ndash1806 2005

[17] H D Chalak A Chakrabarti M A Iqbal and A H SheikhldquoVibration of laminated sandwich beams having soft corerdquoJournal of Vibration and Control vol 18 no 10 pp 1422ndash14352012

[18] K M Ahmed ldquoFree vibration of curved sandwich beams by themethod of finite elementsrdquo Journal of Sound and Vibration vol18 no 1 pp 61ndash74 1971

[19] N Nanda S Kapuria and S Gopalakrishnan ldquoSpectral finiteelement based on an efficient layerwise theory for wave prop-agation analysis of composite and sandwich beamsrdquo Journal ofSound and Vibration vol 333 no 31 pp 20ndash27 2014

[20] Y Pourvais P Asgari A R Moradi and O Rahmani ldquoExper-imental and finite element analysis of higher order behaviourof sandwich beams using digital projection moirerdquo PolymerTesting vol 38 pp 7ndash17 2014

[21] R Long O Barry and D C D Oguamanam ldquoFinite elementfree vibration analysis of soft-core sandwich beamsrdquo AIAAJournal vol 50 no 1 pp 235ndash238 2012

[22] W L Chen A G Striz and C W Bert ldquoHigh-accuracy planestress and plate elements in the quadrature element methodrdquoInternational Journal of Solids and Structures vol 37 no 4 pp627ndash647 2000

[23] Y Xing and B Liu ldquoHigh-accuracy differential quadraturefinite element method and its application to free vibrations ofthin plate with curvilinear domainrdquo International Journal for

Shock and Vibration 15

Numerical Methods in Engineering vol 80 no 13 pp 1718ndash17422009

[24] YWang and XWang ldquoStatic analysis of higher order sandwichbeams by weak form quadrature element methodrdquo CompositeStructures vol 116 no 1 pp 841ndash848 2014

[25] F Tornabene S Brischetto N Fantuzzi and M BacciocchildquoBoundary conditions in 2Dnumerical and 3D exactmodels forcylindrical bending analysis of functionally graded structuresrdquoShock and Vibration vol 2016 Article ID 2373862 2016

[26] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe GDQmethod for the free vibration analysis of arbitrarily shaped lami-nated composite shells using a NURBS-based isogeometricapproachrdquo Composite Structures vol 154 pp 190ndash218 2016

[27] Z Zong and Y Zhang Advanced differential quadrature meth-ods CRC Press NY USA 2009

[28] N A Rubayi and S Charoenree ldquoNatural frequencies of vibra-tion of cantilever sandwich beamsrdquoComputersampStructures vol7 no 6 pp 737ndash745 1977

[29] M Azimi M Shahravi and E F Joubaneh ldquoDynamic analysisof maneuvering flexible spacecraft appendage using higherorder sandwich panel theoryrdquo Latin American Journal of Solidsand Structures vol 13 no 2 pp 296ndash313 2016

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

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Navigation and Observation

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wwwhindawicom Volume 2018

Advances in

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Submit your manuscripts atwwwhindawicom

Shock and Vibration 9

(a) The effect of core by GDQmethod (b) The effect of core by SolidWorks

(c) The effect of face sheets by GDQmethod (d) The effect of face sheets by SolidWorks

(e) The effect of width by GDQmethod (f) The effect of width by SolidWorks

Figure 4 The effect of geometric parameters on the time response at the tip of the cantilever sandwich beam

10 Shock and Vibration

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 5 Vibration amplitude at the tip of the cantilever beam with respect to applied frequency

a core thickness of 15 and 25 mm The results in this tablealso show very good agreementThis is also an indication thatthe present formulation for both free and forced vibrationanalyses of sandwich beams is accurate

6 Conclusions

This paper presents the vibration analysis of sandwich beamswith various boundary conditions The governing equationsof motion are derived using Hamiltonrsquos principle The GDQmethod is utilized for solving the problem SolidWorkssimulation and experimental analyses are conducted to assessthe validity of the GDQ method and the results showvery good agreement Parametric studies are conducted toexamine the role of geometric properties on the time andfrequency response curvesThe results indicate that vibrationamplitude decreases with increasing both core and facesheet thicknesses whereas natural frequency increases withincreasing core sheet thickness but can increase or decreasewith varying face sheet thickness The results also show thatvibration amplitude decreases with increasing beam widthwhereas no effect is observed on the natural frequency for

varying the width of the beam Numerical examples alsodemonstrate that all studied boundary conditions exhibitsimilar effect with varying the aforementioned geometricproperties The findings in this paper are anticipated to beappealing to research communities because of the experi-mental and FEM validation of the GDQ method for thefree and forced vibration analyses of sandwich beams undervarious boundary conditions

Appendix

The rest of the six equations of motion are given as follows

119898119891 (119887) + 119898119888 [ 120 (119905 + 119887) minus 128 (119905 minus 119887) + 115 119888minus 170ℎ119888 1205931 + 140ℎ119891 (1015840119905 minus 1015840119887) minus 156ℎ119891 (1015840119905 + 1015840119887)]minus 11986411989111986011989111990610158401015840119887 + 119866119888119860119888ℎ2119888 [43 (119906119905 + 119906119887) minus 95 (119906119905 minus 119906119887)minus 83119906119888 + 45ℎ1198881205931 + (23ℎ119891 + 13ℎ119888) (1199081015840119905 minus 1199081015840119887)

Shock and Vibration 11

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 6 Vibration amplitude at the middle of the simply-simply supported edges sandwich beam with respect to applied frequency

minus ( 910ℎ119891 + 310ℎ119888) (1199081015840119905 + 1199081015840119887) minus 25ℎ1198881199081015840119888] = 0(A1)

119898119891 (119905 minus 112ℎ211989110158401015840119905 ) + 119898119888 [minus 140ℎ119891 (1015840119905 + 1015840119887)minus 156ℎ119891 (1015840119905 minus 1015840119887) minus 130ℎ1198911015840119888 minus 1140ℎ119888ℎ119891 12059311015840minus 180ℎ2119891 (10158401015840119905 minus 10158401015840119887 ) + 112 (119905 minus 119887)minus 1112ℎ2119891 (10158401015840119905 + 10158401015840119887 ) + 120 (119905 + 119887) + 115119888]+ 1198641198911198681198911199081015840101584010158401015840119905 + 119866119888119860119888ℎ2119888 [minus (23ℎ119891 + 13ℎ119888) (1199061015840119905 + 1199061015840119887)minus ( 910ℎ119891 + 310ℎ119888) (1199061015840119905 minus 1199061015840119887) + (43ℎ119891 + 23ℎ119888) 1199061015840119888+ (25ℎ119891ℎ119888 + 215ℎ2119888) (12059310158401)

minus ( 920ℎ2119891 + 310ℎ119888ℎ119891 + 120ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 )minus (13ℎ119891ℎ119888 + 13ℎ2119891 + 112ℎ2119888) (11990810158401015840119905 minus 11990810158401015840119887 )minus (15ℎ119891ℎ119888 + 115ℎ2119888)11990810158401015840119888 ] + 119864119888119860119888ℎ2119888 [43 (119908119905 + 119908119887)+ (119908119905 minus 119908119887) minus 83119908119888] = 0

(A2)

119898119891 (119887 minus 112ℎ211989110158401015840119887 ) + 119898119888 [ 140ℎ119891 (1015840119905 + 1015840119887)minus 156ℎ119891 (1015840119905 minus 1015840119887) + 130ℎ1198911015840119888 minus 1140ℎ119888ℎ119891 12059311015840+ 180ℎ2119891 (10158401015840119905 minus 10158401015840119887 ) minus 112 (119905 minus 119887)minus 1112ℎ2119891 (10158401015840119905 + 10158401015840119887 ) + 120 (119905 + 119887) + 115119888]

12 Shock and Vibration

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 7 Vibration amplitude at the middle of the clamped-clamped edges sandwich beam with respect to applied frequency

+ 1198641198911198681198911199081015840101584010158401015840119887 + 119866119888119860119888ℎ2119888 [(23ℎ119891 + 13ℎ119888) (1199061015840119905 + 1199061015840119887)minus ( 910ℎ119891 + 310ℎ119888) (1199061015840119905 minus 1199061015840119887) minus (43ℎ119891 + 23ℎ119888) 1199061015840119888+ (25ℎ119891ℎ119888 + 215ℎ2119888) (12059310158401)minus ( 920ℎ2119891 + 310ℎ119888ℎ119891 + 120ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 )+ (13ℎ119891ℎ119888 + 13ℎ2119891 + 112ℎ2119888) (11990810158401015840119905 minus 11990810158401015840119887 )minus (15ℎ119891ℎ119888 + 115ℎ2119888)11990810158401015840119888 ] + 119864119888119860119888ℎ2119888 [43 (119908119905 + 119908119887)minus (119908119905 minus 119908119887) minus 83119908119888] = 0

(A3)

119898119888 [ 815 119888 + 115 (119887 + 119905) + 130ℎ119891 (1015840119905 minus 1015840119887)]+ 119866119888119860119888ℎ1198882 [minus83 (119906119905 + 119906119887) + 163 119906119888minus (43ℎ119891 + 23ℎ119888) (1199081015840119905 minus 1199081015840119887)] = 0

(A4)

119898119888 [ 2105ℎ2119888 1205931 + 170ℎ119888 (119905 minus 119887)+ 1140ℎ119891ℎ119888 (1015840119905 + 1015840119887)] + 119866119888119860119888ℎ1198882 [minus45ℎ119888 (119906119905 minus 119906119887)+ 45ℎ21198881205931 + 415ℎ21198881199081015840119888minus ( 215ℎ2119888 + 25ℎ119888ℎ119891) (1199081015840119905 + 1199081015840119887)] = 0

(A5)

Shock and Vibration 13

Figure 8 Schematic of experimental setup

Figure 9 Vibration amplitude of cantilever sandwich beam fordifferent core thicknesses by experiment

119898119888 [ 815119888 + 115ℎ119888 (119887 + 119905)]+ 119866119888119860119888ℎ1198882 [minus25ℎ119888 (1199061015840119905 minus 1199061015840119887) minus 415ℎ21198881205931minus ( 115ℎ119891ℎ119888 + 115ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 ) minus 815ℎ211988811990810158401015840119888 ]+ 119864119888119860119888ℎ2119888 [minus83 (119908119905 + 119908119887) + 163 119908119888] = 0

(A6)

The essential and natural boundary conditions related to theclamped-free case are given as

119906119905 (0 119905) = 0119875119905 (119897 119905) = 0119908119905 (0 119905) = 0119881119905 (119897 119905) = 01199081199051015840 (0 119905) = 0119872119905 (119897 119905) = 0119908119888 (0 119905) = 0119881119888 (119897 119905) = 0119906119887 (0 119905) = 0119875119887 (119897 119905) = 0119908119887 (0 119905) = 0119881119887 (119897 119905) = 01199081198871015840 (0 119905) = 0119872119887 (119897 119905) = 0

(A7)

where the left side of sandwich beam is clamped and the rightside is free

The axial forces 119875119905119887(119909 119905) the shear forces 119881119905119887119888(119909 119905) andthe bending moments119872119905119887(119909 119905) are obtained as

119875119905 = minus119864119891119860119891 120597119906119905120597119909 (A8)

119875119887 = minus119864119891119860119891 120597119906119887120597119909 (A9)

119881119905 = minus 112119898119891ℎ21198911015840119905 + 119898119888840 [minus36ℎ119891119905 minus 6ℎ119891119887minus 28ℎ119891119888 minus 6ℎ119891ℎ119888 1205931 + 3ℎ21198911015840119887 minus 18ℎ21198911015840119905)+ 1198641198911198681198911199081015840101584010158401015840119905 + 119866119888119860119888840ℎ2119888 [(196ℎ119891 minus 28ℎ119888) 119906119887minus (1316ℎ119891 + 532ℎ119888) 119906119905 + (560ℎ119888 + 1120ℎ119891) 119906119888+ (28ℎ2119888 minus 98ℎ2119891 + 28ℎ119891ℎ119888)11990810158401015840119887minus (112ℎ2119888 + 658ℎ2119891 + 532ℎ2119891 + 532ℎ119891ℎ119888)1199081015840119905+ (336ℎ119891ℎ119888 + 112ℎ2119888) 1205931minus (56ℎ2119888 + 168ℎ119891ℎ119888)1199081015840119888]

(A10)

119881119887 = minus 112119898119891ℎ21198911015840119887 + 119898119888840 (36ℎ119891119887 + 6ℎ119891119905+ 28ℎ119891119888 minus 6ℎ119891ℎ119888 1205931 + 3ℎ21198911015840119905 minus 18ℎ21198911015840119887)

14 Shock and Vibration

+ 1198641198911198681198911199081015840101584010158401015840119887 + 119866119888119860119888840ℎ2119888 [(minus196ℎ119891 + 28ℎ119888) 119906119905+ (1316ℎ119891 + 532ℎ119888) 119906119887 minus (560ℎ119888 + 1120ℎ119891) 119906119888+ (28ℎ2119888 minus 98ℎ2119891 + 28ℎ119891ℎ119888)11990810158401015840119905minus (112ℎ2119888 + 658ℎ2119891 + 532ℎ2119891 + 532ℎ119891ℎ119888)1199081015840119887+ (336ℎ119891ℎ119888 + 112ℎ2119888) 1205931minus (56ℎ2119888 + 168ℎ119891ℎ119888)1199081015840119888]

(A11)

119881119888 = 119866119888119860119888840ℎ2119888 [minus336ℎ119891 (119906119905 minus 119906119887) minus (224ℎ2119888) 1205931

minus (56ℎ2119888 + 168ℎ119891ℎ119888) (1199081015840119905 minus 1199081015840119887) minus 448ℎ21198881199081015840119888](A12)

119872119905 = minus11986411989111986811989111990810158401015840119905 (A13)

119872119887 = minus11986411989111986811989111990810158401015840119887 (A14)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] A S Herrmann P C Zahlen and I Zuardy ldquoSandwich Struc-tures Technology in Commercial Aviationrdquo in Proceedings of the7th International Conference on Sandwich Structures pp 13ndash262005

[2] J Vinson The behavior of sandwich structures of isotropic andcomposite materials Technomic Publishing Company Lan-caster Pennsylvania 1999

[3] J Hohe L Librescu and S Y Oh ldquoDynamic buckling of flat andcurved sandwich panels with transversely compressible corerdquoComposite Structures vol 74 no 1 pp 10ndash24 2006

[4] 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

[5] Y Frostig M Baruch O Vilnay and I Sheinman ldquoHigh-ordertheory for sandwich-beam behavior with transversely flexiblecorerdquo Journal of Engineering Mechanics vol 118 no 5 pp 1026ndash1043 1992

[6] O Rahmani SM R Khalili KMalekzadeh andHHadavinialdquoFree vibration analysis of sandwich structures with a flexiblefunctionally graded syntactic corerdquo Composite Structures vol91 no 2 pp 229ndash235 2009

[7] C Santiuste O TThomsen and Y Frostig ldquoThermo-mechani-cal load interactions in foam cored axi-symmetric sandwich cir-cular plates - High-order and FEmodelsrdquo Composite Structuresvol 93 no 2 pp 369ndash376 2011

[8] R K Fruehmann J M Dulieu-Barton O T Thomsen andS Zhang ldquoExperimental investigation of thermal effects infoam cored sandwich beamsrdquo in Thermomechanics and Infra-Red Imaging vol 7 of Conference Proceedings of the Society forExperimental Mechanics Series pp 83ndash90 Springer NY USA2011

[9] V S Sokolinsky and S R Nutt ldquoConsistent Higher-OrderDynamic Equations for Soft-Core Sandwich Beamsrdquo AIAAJournal vol 42 no 2 pp 374ndash382 2004

[10] Y Frostig C Phan and G Kardomateas ldquoFree vibration ofunidirectional sandwich panels Part I Compressible corerdquoJournal of Sandwich Structures and Materials vol 15 no 4 pp377ndash411 2013

[11] A R Damanpack and S M R Khalili ldquoHigh-order free vibra-tion analysis of sandwich beams with a flexible core usingdynamic stiffness methodrdquo Composite Structures vol 94 no 5pp 1503ndash1514 2012

[12] M Yang and P Qiao ldquoHigher-order impact modeling of sand-wich structures with flexible corerdquo International Journal ofSolids and Structures vol 42 no 20 pp 5460ndash5490 2005

[13] Y Wang and X Wang ldquoFree vibration analysis of soft-coresandwich beams by the novel weak form quadrature elementmethodrdquo Journal of Sandwich Structures and Materials vol 18no 3 pp 294ndash320 2015

[14] X Wang and X Liang ldquoFree vibration of soft-core sandwichpanels with general boundary conditions by harmonic quadra-ture element methodrdquoThin-Walled Structures vol 113 pp 253ndash261 2017

[15] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe local GDQmethod applied to general higher-order theories of doubly-curved laminated composite shells and panels the free vibrationanalysisrdquo Composite Structures vol 116 pp 637ndash660 2014

[16] C C Hong H W Liao L T Lee J B Ke and K C JaneldquoThermally induced vibration of a thermal sleeve with the GDQmethodrdquo International Journal of Mechanical Sciences vol 47no 12 pp 1789ndash1806 2005

[17] H D Chalak A Chakrabarti M A Iqbal and A H SheikhldquoVibration of laminated sandwich beams having soft corerdquoJournal of Vibration and Control vol 18 no 10 pp 1422ndash14352012

[18] K M Ahmed ldquoFree vibration of curved sandwich beams by themethod of finite elementsrdquo Journal of Sound and Vibration vol18 no 1 pp 61ndash74 1971

[19] N Nanda S Kapuria and S Gopalakrishnan ldquoSpectral finiteelement based on an efficient layerwise theory for wave prop-agation analysis of composite and sandwich beamsrdquo Journal ofSound and Vibration vol 333 no 31 pp 20ndash27 2014

[20] Y Pourvais P Asgari A R Moradi and O Rahmani ldquoExper-imental and finite element analysis of higher order behaviourof sandwich beams using digital projection moirerdquo PolymerTesting vol 38 pp 7ndash17 2014

[21] R Long O Barry and D C D Oguamanam ldquoFinite elementfree vibration analysis of soft-core sandwich beamsrdquo AIAAJournal vol 50 no 1 pp 235ndash238 2012

[22] W L Chen A G Striz and C W Bert ldquoHigh-accuracy planestress and plate elements in the quadrature element methodrdquoInternational Journal of Solids and Structures vol 37 no 4 pp627ndash647 2000

[23] Y Xing and B Liu ldquoHigh-accuracy differential quadraturefinite element method and its application to free vibrations ofthin plate with curvilinear domainrdquo International Journal for

Shock and Vibration 15

Numerical Methods in Engineering vol 80 no 13 pp 1718ndash17422009

[24] YWang and XWang ldquoStatic analysis of higher order sandwichbeams by weak form quadrature element methodrdquo CompositeStructures vol 116 no 1 pp 841ndash848 2014

[25] F Tornabene S Brischetto N Fantuzzi and M BacciocchildquoBoundary conditions in 2Dnumerical and 3D exactmodels forcylindrical bending analysis of functionally graded structuresrdquoShock and Vibration vol 2016 Article ID 2373862 2016

[26] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe GDQmethod for the free vibration analysis of arbitrarily shaped lami-nated composite shells using a NURBS-based isogeometricapproachrdquo Composite Structures vol 154 pp 190ndash218 2016

[27] Z Zong and Y Zhang Advanced differential quadrature meth-ods CRC Press NY USA 2009

[28] N A Rubayi and S Charoenree ldquoNatural frequencies of vibra-tion of cantilever sandwich beamsrdquoComputersampStructures vol7 no 6 pp 737ndash745 1977

[29] M Azimi M Shahravi and E F Joubaneh ldquoDynamic analysisof maneuvering flexible spacecraft appendage using higherorder sandwich panel theoryrdquo Latin American Journal of Solidsand Structures vol 13 no 2 pp 296ndash313 2016

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

10 Shock and Vibration

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 5 Vibration amplitude at the tip of the cantilever beam with respect to applied frequency

a core thickness of 15 and 25 mm The results in this tablealso show very good agreementThis is also an indication thatthe present formulation for both free and forced vibrationanalyses of sandwich beams is accurate

6 Conclusions

This paper presents the vibration analysis of sandwich beamswith various boundary conditions The governing equationsof motion are derived using Hamiltonrsquos principle The GDQmethod is utilized for solving the problem SolidWorkssimulation and experimental analyses are conducted to assessthe validity of the GDQ method and the results showvery good agreement Parametric studies are conducted toexamine the role of geometric properties on the time andfrequency response curvesThe results indicate that vibrationamplitude decreases with increasing both core and facesheet thicknesses whereas natural frequency increases withincreasing core sheet thickness but can increase or decreasewith varying face sheet thickness The results also show thatvibration amplitude decreases with increasing beam widthwhereas no effect is observed on the natural frequency for

varying the width of the beam Numerical examples alsodemonstrate that all studied boundary conditions exhibitsimilar effect with varying the aforementioned geometricproperties The findings in this paper are anticipated to beappealing to research communities because of the experi-mental and FEM validation of the GDQ method for thefree and forced vibration analyses of sandwich beams undervarious boundary conditions

Appendix

The rest of the six equations of motion are given as follows

119898119891 (119887) + 119898119888 [ 120 (119905 + 119887) minus 128 (119905 minus 119887) + 115 119888minus 170ℎ119888 1205931 + 140ℎ119891 (1015840119905 minus 1015840119887) minus 156ℎ119891 (1015840119905 + 1015840119887)]minus 11986411989111986011989111990610158401015840119887 + 119866119888119860119888ℎ2119888 [43 (119906119905 + 119906119887) minus 95 (119906119905 minus 119906119887)minus 83119906119888 + 45ℎ1198881205931 + (23ℎ119891 + 13ℎ119888) (1199081015840119905 minus 1199081015840119887)

Shock and Vibration 11

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 6 Vibration amplitude at the middle of the simply-simply supported edges sandwich beam with respect to applied frequency

minus ( 910ℎ119891 + 310ℎ119888) (1199081015840119905 + 1199081015840119887) minus 25ℎ1198881199081015840119888] = 0(A1)

119898119891 (119905 minus 112ℎ211989110158401015840119905 ) + 119898119888 [minus 140ℎ119891 (1015840119905 + 1015840119887)minus 156ℎ119891 (1015840119905 minus 1015840119887) minus 130ℎ1198911015840119888 minus 1140ℎ119888ℎ119891 12059311015840minus 180ℎ2119891 (10158401015840119905 minus 10158401015840119887 ) + 112 (119905 minus 119887)minus 1112ℎ2119891 (10158401015840119905 + 10158401015840119887 ) + 120 (119905 + 119887) + 115119888]+ 1198641198911198681198911199081015840101584010158401015840119905 + 119866119888119860119888ℎ2119888 [minus (23ℎ119891 + 13ℎ119888) (1199061015840119905 + 1199061015840119887)minus ( 910ℎ119891 + 310ℎ119888) (1199061015840119905 minus 1199061015840119887) + (43ℎ119891 + 23ℎ119888) 1199061015840119888+ (25ℎ119891ℎ119888 + 215ℎ2119888) (12059310158401)

minus ( 920ℎ2119891 + 310ℎ119888ℎ119891 + 120ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 )minus (13ℎ119891ℎ119888 + 13ℎ2119891 + 112ℎ2119888) (11990810158401015840119905 minus 11990810158401015840119887 )minus (15ℎ119891ℎ119888 + 115ℎ2119888)11990810158401015840119888 ] + 119864119888119860119888ℎ2119888 [43 (119908119905 + 119908119887)+ (119908119905 minus 119908119887) minus 83119908119888] = 0

(A2)

119898119891 (119887 minus 112ℎ211989110158401015840119887 ) + 119898119888 [ 140ℎ119891 (1015840119905 + 1015840119887)minus 156ℎ119891 (1015840119905 minus 1015840119887) + 130ℎ1198911015840119888 minus 1140ℎ119888ℎ119891 12059311015840+ 180ℎ2119891 (10158401015840119905 minus 10158401015840119887 ) minus 112 (119905 minus 119887)minus 1112ℎ2119891 (10158401015840119905 + 10158401015840119887 ) + 120 (119905 + 119887) + 115119888]

12 Shock and Vibration

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 7 Vibration amplitude at the middle of the clamped-clamped edges sandwich beam with respect to applied frequency

+ 1198641198911198681198911199081015840101584010158401015840119887 + 119866119888119860119888ℎ2119888 [(23ℎ119891 + 13ℎ119888) (1199061015840119905 + 1199061015840119887)minus ( 910ℎ119891 + 310ℎ119888) (1199061015840119905 minus 1199061015840119887) minus (43ℎ119891 + 23ℎ119888) 1199061015840119888+ (25ℎ119891ℎ119888 + 215ℎ2119888) (12059310158401)minus ( 920ℎ2119891 + 310ℎ119888ℎ119891 + 120ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 )+ (13ℎ119891ℎ119888 + 13ℎ2119891 + 112ℎ2119888) (11990810158401015840119905 minus 11990810158401015840119887 )minus (15ℎ119891ℎ119888 + 115ℎ2119888)11990810158401015840119888 ] + 119864119888119860119888ℎ2119888 [43 (119908119905 + 119908119887)minus (119908119905 minus 119908119887) minus 83119908119888] = 0

(A3)

119898119888 [ 815 119888 + 115 (119887 + 119905) + 130ℎ119891 (1015840119905 minus 1015840119887)]+ 119866119888119860119888ℎ1198882 [minus83 (119906119905 + 119906119887) + 163 119906119888minus (43ℎ119891 + 23ℎ119888) (1199081015840119905 minus 1199081015840119887)] = 0

(A4)

119898119888 [ 2105ℎ2119888 1205931 + 170ℎ119888 (119905 minus 119887)+ 1140ℎ119891ℎ119888 (1015840119905 + 1015840119887)] + 119866119888119860119888ℎ1198882 [minus45ℎ119888 (119906119905 minus 119906119887)+ 45ℎ21198881205931 + 415ℎ21198881199081015840119888minus ( 215ℎ2119888 + 25ℎ119888ℎ119891) (1199081015840119905 + 1199081015840119887)] = 0

(A5)

Shock and Vibration 13

Figure 8 Schematic of experimental setup

Figure 9 Vibration amplitude of cantilever sandwich beam fordifferent core thicknesses by experiment

119898119888 [ 815119888 + 115ℎ119888 (119887 + 119905)]+ 119866119888119860119888ℎ1198882 [minus25ℎ119888 (1199061015840119905 minus 1199061015840119887) minus 415ℎ21198881205931minus ( 115ℎ119891ℎ119888 + 115ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 ) minus 815ℎ211988811990810158401015840119888 ]+ 119864119888119860119888ℎ2119888 [minus83 (119908119905 + 119908119887) + 163 119908119888] = 0

(A6)

The essential and natural boundary conditions related to theclamped-free case are given as

119906119905 (0 119905) = 0119875119905 (119897 119905) = 0119908119905 (0 119905) = 0119881119905 (119897 119905) = 01199081199051015840 (0 119905) = 0119872119905 (119897 119905) = 0119908119888 (0 119905) = 0119881119888 (119897 119905) = 0119906119887 (0 119905) = 0119875119887 (119897 119905) = 0119908119887 (0 119905) = 0119881119887 (119897 119905) = 01199081198871015840 (0 119905) = 0119872119887 (119897 119905) = 0

(A7)

where the left side of sandwich beam is clamped and the rightside is free

The axial forces 119875119905119887(119909 119905) the shear forces 119881119905119887119888(119909 119905) andthe bending moments119872119905119887(119909 119905) are obtained as

119875119905 = minus119864119891119860119891 120597119906119905120597119909 (A8)

119875119887 = minus119864119891119860119891 120597119906119887120597119909 (A9)

119881119905 = minus 112119898119891ℎ21198911015840119905 + 119898119888840 [minus36ℎ119891119905 minus 6ℎ119891119887minus 28ℎ119891119888 minus 6ℎ119891ℎ119888 1205931 + 3ℎ21198911015840119887 minus 18ℎ21198911015840119905)+ 1198641198911198681198911199081015840101584010158401015840119905 + 119866119888119860119888840ℎ2119888 [(196ℎ119891 minus 28ℎ119888) 119906119887minus (1316ℎ119891 + 532ℎ119888) 119906119905 + (560ℎ119888 + 1120ℎ119891) 119906119888+ (28ℎ2119888 minus 98ℎ2119891 + 28ℎ119891ℎ119888)11990810158401015840119887minus (112ℎ2119888 + 658ℎ2119891 + 532ℎ2119891 + 532ℎ119891ℎ119888)1199081015840119905+ (336ℎ119891ℎ119888 + 112ℎ2119888) 1205931minus (56ℎ2119888 + 168ℎ119891ℎ119888)1199081015840119888]

(A10)

119881119887 = minus 112119898119891ℎ21198911015840119887 + 119898119888840 (36ℎ119891119887 + 6ℎ119891119905+ 28ℎ119891119888 minus 6ℎ119891ℎ119888 1205931 + 3ℎ21198911015840119905 minus 18ℎ21198911015840119887)

14 Shock and Vibration

+ 1198641198911198681198911199081015840101584010158401015840119887 + 119866119888119860119888840ℎ2119888 [(minus196ℎ119891 + 28ℎ119888) 119906119905+ (1316ℎ119891 + 532ℎ119888) 119906119887 minus (560ℎ119888 + 1120ℎ119891) 119906119888+ (28ℎ2119888 minus 98ℎ2119891 + 28ℎ119891ℎ119888)11990810158401015840119905minus (112ℎ2119888 + 658ℎ2119891 + 532ℎ2119891 + 532ℎ119891ℎ119888)1199081015840119887+ (336ℎ119891ℎ119888 + 112ℎ2119888) 1205931minus (56ℎ2119888 + 168ℎ119891ℎ119888)1199081015840119888]

(A11)

119881119888 = 119866119888119860119888840ℎ2119888 [minus336ℎ119891 (119906119905 minus 119906119887) minus (224ℎ2119888) 1205931

minus (56ℎ2119888 + 168ℎ119891ℎ119888) (1199081015840119905 minus 1199081015840119887) minus 448ℎ21198881199081015840119888](A12)

119872119905 = minus11986411989111986811989111990810158401015840119905 (A13)

119872119887 = minus11986411989111986811989111990810158401015840119887 (A14)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] A S Herrmann P C Zahlen and I Zuardy ldquoSandwich Struc-tures Technology in Commercial Aviationrdquo in Proceedings of the7th International Conference on Sandwich Structures pp 13ndash262005

[2] J Vinson The behavior of sandwich structures of isotropic andcomposite materials Technomic Publishing Company Lan-caster Pennsylvania 1999

[3] J Hohe L Librescu and S Y Oh ldquoDynamic buckling of flat andcurved sandwich panels with transversely compressible corerdquoComposite Structures vol 74 no 1 pp 10ndash24 2006

[4] 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

[5] Y Frostig M Baruch O Vilnay and I Sheinman ldquoHigh-ordertheory for sandwich-beam behavior with transversely flexiblecorerdquo Journal of Engineering Mechanics vol 118 no 5 pp 1026ndash1043 1992

[6] O Rahmani SM R Khalili KMalekzadeh andHHadavinialdquoFree vibration analysis of sandwich structures with a flexiblefunctionally graded syntactic corerdquo Composite Structures vol91 no 2 pp 229ndash235 2009

[7] C Santiuste O TThomsen and Y Frostig ldquoThermo-mechani-cal load interactions in foam cored axi-symmetric sandwich cir-cular plates - High-order and FEmodelsrdquo Composite Structuresvol 93 no 2 pp 369ndash376 2011

[8] R K Fruehmann J M Dulieu-Barton O T Thomsen andS Zhang ldquoExperimental investigation of thermal effects infoam cored sandwich beamsrdquo in Thermomechanics and Infra-Red Imaging vol 7 of Conference Proceedings of the Society forExperimental Mechanics Series pp 83ndash90 Springer NY USA2011

[9] V S Sokolinsky and S R Nutt ldquoConsistent Higher-OrderDynamic Equations for Soft-Core Sandwich Beamsrdquo AIAAJournal vol 42 no 2 pp 374ndash382 2004

[10] Y Frostig C Phan and G Kardomateas ldquoFree vibration ofunidirectional sandwich panels Part I Compressible corerdquoJournal of Sandwich Structures and Materials vol 15 no 4 pp377ndash411 2013

[11] A R Damanpack and S M R Khalili ldquoHigh-order free vibra-tion analysis of sandwich beams with a flexible core usingdynamic stiffness methodrdquo Composite Structures vol 94 no 5pp 1503ndash1514 2012

[12] M Yang and P Qiao ldquoHigher-order impact modeling of sand-wich structures with flexible corerdquo International Journal ofSolids and Structures vol 42 no 20 pp 5460ndash5490 2005

[13] Y Wang and X Wang ldquoFree vibration analysis of soft-coresandwich beams by the novel weak form quadrature elementmethodrdquo Journal of Sandwich Structures and Materials vol 18no 3 pp 294ndash320 2015

[14] X Wang and X Liang ldquoFree vibration of soft-core sandwichpanels with general boundary conditions by harmonic quadra-ture element methodrdquoThin-Walled Structures vol 113 pp 253ndash261 2017

[15] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe local GDQmethod applied to general higher-order theories of doubly-curved laminated composite shells and panels the free vibrationanalysisrdquo Composite Structures vol 116 pp 637ndash660 2014

[16] C C Hong H W Liao L T Lee J B Ke and K C JaneldquoThermally induced vibration of a thermal sleeve with the GDQmethodrdquo International Journal of Mechanical Sciences vol 47no 12 pp 1789ndash1806 2005

[17] H D Chalak A Chakrabarti M A Iqbal and A H SheikhldquoVibration of laminated sandwich beams having soft corerdquoJournal of Vibration and Control vol 18 no 10 pp 1422ndash14352012

[18] K M Ahmed ldquoFree vibration of curved sandwich beams by themethod of finite elementsrdquo Journal of Sound and Vibration vol18 no 1 pp 61ndash74 1971

[19] N Nanda S Kapuria and S Gopalakrishnan ldquoSpectral finiteelement based on an efficient layerwise theory for wave prop-agation analysis of composite and sandwich beamsrdquo Journal ofSound and Vibration vol 333 no 31 pp 20ndash27 2014

[20] Y Pourvais P Asgari A R Moradi and O Rahmani ldquoExper-imental and finite element analysis of higher order behaviourof sandwich beams using digital projection moirerdquo PolymerTesting vol 38 pp 7ndash17 2014

[21] R Long O Barry and D C D Oguamanam ldquoFinite elementfree vibration analysis of soft-core sandwich beamsrdquo AIAAJournal vol 50 no 1 pp 235ndash238 2012

[22] W L Chen A G Striz and C W Bert ldquoHigh-accuracy planestress and plate elements in the quadrature element methodrdquoInternational Journal of Solids and Structures vol 37 no 4 pp627ndash647 2000

[23] Y Xing and B Liu ldquoHigh-accuracy differential quadraturefinite element method and its application to free vibrations ofthin plate with curvilinear domainrdquo International Journal for

Shock and Vibration 15

Numerical Methods in Engineering vol 80 no 13 pp 1718ndash17422009

[24] YWang and XWang ldquoStatic analysis of higher order sandwichbeams by weak form quadrature element methodrdquo CompositeStructures vol 116 no 1 pp 841ndash848 2014

[25] F Tornabene S Brischetto N Fantuzzi and M BacciocchildquoBoundary conditions in 2Dnumerical and 3D exactmodels forcylindrical bending analysis of functionally graded structuresrdquoShock and Vibration vol 2016 Article ID 2373862 2016

[26] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe GDQmethod for the free vibration analysis of arbitrarily shaped lami-nated composite shells using a NURBS-based isogeometricapproachrdquo Composite Structures vol 154 pp 190ndash218 2016

[27] Z Zong and Y Zhang Advanced differential quadrature meth-ods CRC Press NY USA 2009

[28] N A Rubayi and S Charoenree ldquoNatural frequencies of vibra-tion of cantilever sandwich beamsrdquoComputersampStructures vol7 no 6 pp 737ndash745 1977

[29] M Azimi M Shahravi and E F Joubaneh ldquoDynamic analysisof maneuvering flexible spacecraft appendage using higherorder sandwich panel theoryrdquo Latin American Journal of Solidsand Structures vol 13 no 2 pp 296ndash313 2016

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Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

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Hindawiwwwhindawicom Volume 2018

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Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

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Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

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RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Shock and Vibration 11

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 6 Vibration amplitude at the middle of the simply-simply supported edges sandwich beam with respect to applied frequency

minus ( 910ℎ119891 + 310ℎ119888) (1199081015840119905 + 1199081015840119887) minus 25ℎ1198881199081015840119888] = 0(A1)

119898119891 (119905 minus 112ℎ211989110158401015840119905 ) + 119898119888 [minus 140ℎ119891 (1015840119905 + 1015840119887)minus 156ℎ119891 (1015840119905 minus 1015840119887) minus 130ℎ1198911015840119888 minus 1140ℎ119888ℎ119891 12059311015840minus 180ℎ2119891 (10158401015840119905 minus 10158401015840119887 ) + 112 (119905 minus 119887)minus 1112ℎ2119891 (10158401015840119905 + 10158401015840119887 ) + 120 (119905 + 119887) + 115119888]+ 1198641198911198681198911199081015840101584010158401015840119905 + 119866119888119860119888ℎ2119888 [minus (23ℎ119891 + 13ℎ119888) (1199061015840119905 + 1199061015840119887)minus ( 910ℎ119891 + 310ℎ119888) (1199061015840119905 minus 1199061015840119887) + (43ℎ119891 + 23ℎ119888) 1199061015840119888+ (25ℎ119891ℎ119888 + 215ℎ2119888) (12059310158401)

minus ( 920ℎ2119891 + 310ℎ119888ℎ119891 + 120ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 )minus (13ℎ119891ℎ119888 + 13ℎ2119891 + 112ℎ2119888) (11990810158401015840119905 minus 11990810158401015840119887 )minus (15ℎ119891ℎ119888 + 115ℎ2119888)11990810158401015840119888 ] + 119864119888119860119888ℎ2119888 [43 (119908119905 + 119908119887)+ (119908119905 minus 119908119887) minus 83119908119888] = 0

(A2)

119898119891 (119887 minus 112ℎ211989110158401015840119887 ) + 119898119888 [ 140ℎ119891 (1015840119905 + 1015840119887)minus 156ℎ119891 (1015840119905 minus 1015840119887) + 130ℎ1198911015840119888 minus 1140ℎ119888ℎ119891 12059311015840+ 180ℎ2119891 (10158401015840119905 minus 10158401015840119887 ) minus 112 (119905 minus 119887)minus 1112ℎ2119891 (10158401015840119905 + 10158401015840119887 ) + 120 (119905 + 119887) + 115119888]

12 Shock and Vibration

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 7 Vibration amplitude at the middle of the clamped-clamped edges sandwich beam with respect to applied frequency

+ 1198641198911198681198911199081015840101584010158401015840119887 + 119866119888119860119888ℎ2119888 [(23ℎ119891 + 13ℎ119888) (1199061015840119905 + 1199061015840119887)minus ( 910ℎ119891 + 310ℎ119888) (1199061015840119905 minus 1199061015840119887) minus (43ℎ119891 + 23ℎ119888) 1199061015840119888+ (25ℎ119891ℎ119888 + 215ℎ2119888) (12059310158401)minus ( 920ℎ2119891 + 310ℎ119888ℎ119891 + 120ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 )+ (13ℎ119891ℎ119888 + 13ℎ2119891 + 112ℎ2119888) (11990810158401015840119905 minus 11990810158401015840119887 )minus (15ℎ119891ℎ119888 + 115ℎ2119888)11990810158401015840119888 ] + 119864119888119860119888ℎ2119888 [43 (119908119905 + 119908119887)minus (119908119905 minus 119908119887) minus 83119908119888] = 0

(A3)

119898119888 [ 815 119888 + 115 (119887 + 119905) + 130ℎ119891 (1015840119905 minus 1015840119887)]+ 119866119888119860119888ℎ1198882 [minus83 (119906119905 + 119906119887) + 163 119906119888minus (43ℎ119891 + 23ℎ119888) (1199081015840119905 minus 1199081015840119887)] = 0

(A4)

119898119888 [ 2105ℎ2119888 1205931 + 170ℎ119888 (119905 minus 119887)+ 1140ℎ119891ℎ119888 (1015840119905 + 1015840119887)] + 119866119888119860119888ℎ1198882 [minus45ℎ119888 (119906119905 minus 119906119887)+ 45ℎ21198881205931 + 415ℎ21198881199081015840119888minus ( 215ℎ2119888 + 25ℎ119888ℎ119891) (1199081015840119905 + 1199081015840119887)] = 0

(A5)

Shock and Vibration 13

Figure 8 Schematic of experimental setup

Figure 9 Vibration amplitude of cantilever sandwich beam fordifferent core thicknesses by experiment

119898119888 [ 815119888 + 115ℎ119888 (119887 + 119905)]+ 119866119888119860119888ℎ1198882 [minus25ℎ119888 (1199061015840119905 minus 1199061015840119887) minus 415ℎ21198881205931minus ( 115ℎ119891ℎ119888 + 115ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 ) minus 815ℎ211988811990810158401015840119888 ]+ 119864119888119860119888ℎ2119888 [minus83 (119908119905 + 119908119887) + 163 119908119888] = 0

(A6)

The essential and natural boundary conditions related to theclamped-free case are given as

119906119905 (0 119905) = 0119875119905 (119897 119905) = 0119908119905 (0 119905) = 0119881119905 (119897 119905) = 01199081199051015840 (0 119905) = 0119872119905 (119897 119905) = 0119908119888 (0 119905) = 0119881119888 (119897 119905) = 0119906119887 (0 119905) = 0119875119887 (119897 119905) = 0119908119887 (0 119905) = 0119881119887 (119897 119905) = 01199081198871015840 (0 119905) = 0119872119887 (119897 119905) = 0

(A7)

where the left side of sandwich beam is clamped and the rightside is free

The axial forces 119875119905119887(119909 119905) the shear forces 119881119905119887119888(119909 119905) andthe bending moments119872119905119887(119909 119905) are obtained as

119875119905 = minus119864119891119860119891 120597119906119905120597119909 (A8)

119875119887 = minus119864119891119860119891 120597119906119887120597119909 (A9)

119881119905 = minus 112119898119891ℎ21198911015840119905 + 119898119888840 [minus36ℎ119891119905 minus 6ℎ119891119887minus 28ℎ119891119888 minus 6ℎ119891ℎ119888 1205931 + 3ℎ21198911015840119887 minus 18ℎ21198911015840119905)+ 1198641198911198681198911199081015840101584010158401015840119905 + 119866119888119860119888840ℎ2119888 [(196ℎ119891 minus 28ℎ119888) 119906119887minus (1316ℎ119891 + 532ℎ119888) 119906119905 + (560ℎ119888 + 1120ℎ119891) 119906119888+ (28ℎ2119888 minus 98ℎ2119891 + 28ℎ119891ℎ119888)11990810158401015840119887minus (112ℎ2119888 + 658ℎ2119891 + 532ℎ2119891 + 532ℎ119891ℎ119888)1199081015840119905+ (336ℎ119891ℎ119888 + 112ℎ2119888) 1205931minus (56ℎ2119888 + 168ℎ119891ℎ119888)1199081015840119888]

(A10)

119881119887 = minus 112119898119891ℎ21198911015840119887 + 119898119888840 (36ℎ119891119887 + 6ℎ119891119905+ 28ℎ119891119888 minus 6ℎ119891ℎ119888 1205931 + 3ℎ21198911015840119905 minus 18ℎ21198911015840119887)

14 Shock and Vibration

+ 1198641198911198681198911199081015840101584010158401015840119887 + 119866119888119860119888840ℎ2119888 [(minus196ℎ119891 + 28ℎ119888) 119906119905+ (1316ℎ119891 + 532ℎ119888) 119906119887 minus (560ℎ119888 + 1120ℎ119891) 119906119888+ (28ℎ2119888 minus 98ℎ2119891 + 28ℎ119891ℎ119888)11990810158401015840119905minus (112ℎ2119888 + 658ℎ2119891 + 532ℎ2119891 + 532ℎ119891ℎ119888)1199081015840119887+ (336ℎ119891ℎ119888 + 112ℎ2119888) 1205931minus (56ℎ2119888 + 168ℎ119891ℎ119888)1199081015840119888]

(A11)

119881119888 = 119866119888119860119888840ℎ2119888 [minus336ℎ119891 (119906119905 minus 119906119887) minus (224ℎ2119888) 1205931

minus (56ℎ2119888 + 168ℎ119891ℎ119888) (1199081015840119905 minus 1199081015840119887) minus 448ℎ21198881199081015840119888](A12)

119872119905 = minus11986411989111986811989111990810158401015840119905 (A13)

119872119887 = minus11986411989111986811989111990810158401015840119887 (A14)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] A S Herrmann P C Zahlen and I Zuardy ldquoSandwich Struc-tures Technology in Commercial Aviationrdquo in Proceedings of the7th International Conference on Sandwich Structures pp 13ndash262005

[2] J Vinson The behavior of sandwich structures of isotropic andcomposite materials Technomic Publishing Company Lan-caster Pennsylvania 1999

[3] J Hohe L Librescu and S Y Oh ldquoDynamic buckling of flat andcurved sandwich panels with transversely compressible corerdquoComposite Structures vol 74 no 1 pp 10ndash24 2006

[4] 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

[5] Y Frostig M Baruch O Vilnay and I Sheinman ldquoHigh-ordertheory for sandwich-beam behavior with transversely flexiblecorerdquo Journal of Engineering Mechanics vol 118 no 5 pp 1026ndash1043 1992

[6] O Rahmani SM R Khalili KMalekzadeh andHHadavinialdquoFree vibration analysis of sandwich structures with a flexiblefunctionally graded syntactic corerdquo Composite Structures vol91 no 2 pp 229ndash235 2009

[7] C Santiuste O TThomsen and Y Frostig ldquoThermo-mechani-cal load interactions in foam cored axi-symmetric sandwich cir-cular plates - High-order and FEmodelsrdquo Composite Structuresvol 93 no 2 pp 369ndash376 2011

[8] R K Fruehmann J M Dulieu-Barton O T Thomsen andS Zhang ldquoExperimental investigation of thermal effects infoam cored sandwich beamsrdquo in Thermomechanics and Infra-Red Imaging vol 7 of Conference Proceedings of the Society forExperimental Mechanics Series pp 83ndash90 Springer NY USA2011

[9] V S Sokolinsky and S R Nutt ldquoConsistent Higher-OrderDynamic Equations for Soft-Core Sandwich Beamsrdquo AIAAJournal vol 42 no 2 pp 374ndash382 2004

[10] Y Frostig C Phan and G Kardomateas ldquoFree vibration ofunidirectional sandwich panels Part I Compressible corerdquoJournal of Sandwich Structures and Materials vol 15 no 4 pp377ndash411 2013

[11] A R Damanpack and S M R Khalili ldquoHigh-order free vibra-tion analysis of sandwich beams with a flexible core usingdynamic stiffness methodrdquo Composite Structures vol 94 no 5pp 1503ndash1514 2012

[12] M Yang and P Qiao ldquoHigher-order impact modeling of sand-wich structures with flexible corerdquo International Journal ofSolids and Structures vol 42 no 20 pp 5460ndash5490 2005

[13] Y Wang and X Wang ldquoFree vibration analysis of soft-coresandwich beams by the novel weak form quadrature elementmethodrdquo Journal of Sandwich Structures and Materials vol 18no 3 pp 294ndash320 2015

[14] X Wang and X Liang ldquoFree vibration of soft-core sandwichpanels with general boundary conditions by harmonic quadra-ture element methodrdquoThin-Walled Structures vol 113 pp 253ndash261 2017

[15] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe local GDQmethod applied to general higher-order theories of doubly-curved laminated composite shells and panels the free vibrationanalysisrdquo Composite Structures vol 116 pp 637ndash660 2014

[16] C C Hong H W Liao L T Lee J B Ke and K C JaneldquoThermally induced vibration of a thermal sleeve with the GDQmethodrdquo International Journal of Mechanical Sciences vol 47no 12 pp 1789ndash1806 2005

[17] H D Chalak A Chakrabarti M A Iqbal and A H SheikhldquoVibration of laminated sandwich beams having soft corerdquoJournal of Vibration and Control vol 18 no 10 pp 1422ndash14352012

[18] K M Ahmed ldquoFree vibration of curved sandwich beams by themethod of finite elementsrdquo Journal of Sound and Vibration vol18 no 1 pp 61ndash74 1971

[19] N Nanda S Kapuria and S Gopalakrishnan ldquoSpectral finiteelement based on an efficient layerwise theory for wave prop-agation analysis of composite and sandwich beamsrdquo Journal ofSound and Vibration vol 333 no 31 pp 20ndash27 2014

[20] Y Pourvais P Asgari A R Moradi and O Rahmani ldquoExper-imental and finite element analysis of higher order behaviourof sandwich beams using digital projection moirerdquo PolymerTesting vol 38 pp 7ndash17 2014

[21] R Long O Barry and D C D Oguamanam ldquoFinite elementfree vibration analysis of soft-core sandwich beamsrdquo AIAAJournal vol 50 no 1 pp 235ndash238 2012

[22] W L Chen A G Striz and C W Bert ldquoHigh-accuracy planestress and plate elements in the quadrature element methodrdquoInternational Journal of Solids and Structures vol 37 no 4 pp627ndash647 2000

[23] Y Xing and B Liu ldquoHigh-accuracy differential quadraturefinite element method and its application to free vibrations ofthin plate with curvilinear domainrdquo International Journal for

Shock and Vibration 15

Numerical Methods in Engineering vol 80 no 13 pp 1718ndash17422009

[24] YWang and XWang ldquoStatic analysis of higher order sandwichbeams by weak form quadrature element methodrdquo CompositeStructures vol 116 no 1 pp 841ndash848 2014

[25] F Tornabene S Brischetto N Fantuzzi and M BacciocchildquoBoundary conditions in 2Dnumerical and 3D exactmodels forcylindrical bending analysis of functionally graded structuresrdquoShock and Vibration vol 2016 Article ID 2373862 2016

[26] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe GDQmethod for the free vibration analysis of arbitrarily shaped lami-nated composite shells using a NURBS-based isogeometricapproachrdquo Composite Structures vol 154 pp 190ndash218 2016

[27] Z Zong and Y Zhang Advanced differential quadrature meth-ods CRC Press NY USA 2009

[28] N A Rubayi and S Charoenree ldquoNatural frequencies of vibra-tion of cantilever sandwich beamsrdquoComputersampStructures vol7 no 6 pp 737ndash745 1977

[29] M Azimi M Shahravi and E F Joubaneh ldquoDynamic analysisof maneuvering flexible spacecraft appendage using higherorder sandwich panel theoryrdquo Latin American Journal of Solidsand Structures vol 13 no 2 pp 296ndash313 2016

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

12 Shock and Vibration

(a) The effect of core thickness (ℎ119888) (b) The effect of face sheets thickness (ℎ119891)

(c) The effect of width (119887)

Figure 7 Vibration amplitude at the middle of the clamped-clamped edges sandwich beam with respect to applied frequency

+ 1198641198911198681198911199081015840101584010158401015840119887 + 119866119888119860119888ℎ2119888 [(23ℎ119891 + 13ℎ119888) (1199061015840119905 + 1199061015840119887)minus ( 910ℎ119891 + 310ℎ119888) (1199061015840119905 minus 1199061015840119887) minus (43ℎ119891 + 23ℎ119888) 1199061015840119888+ (25ℎ119891ℎ119888 + 215ℎ2119888) (12059310158401)minus ( 920ℎ2119891 + 310ℎ119888ℎ119891 + 120ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 )+ (13ℎ119891ℎ119888 + 13ℎ2119891 + 112ℎ2119888) (11990810158401015840119905 minus 11990810158401015840119887 )minus (15ℎ119891ℎ119888 + 115ℎ2119888)11990810158401015840119888 ] + 119864119888119860119888ℎ2119888 [43 (119908119905 + 119908119887)minus (119908119905 minus 119908119887) minus 83119908119888] = 0

(A3)

119898119888 [ 815 119888 + 115 (119887 + 119905) + 130ℎ119891 (1015840119905 minus 1015840119887)]+ 119866119888119860119888ℎ1198882 [minus83 (119906119905 + 119906119887) + 163 119906119888minus (43ℎ119891 + 23ℎ119888) (1199081015840119905 minus 1199081015840119887)] = 0

(A4)

119898119888 [ 2105ℎ2119888 1205931 + 170ℎ119888 (119905 minus 119887)+ 1140ℎ119891ℎ119888 (1015840119905 + 1015840119887)] + 119866119888119860119888ℎ1198882 [minus45ℎ119888 (119906119905 minus 119906119887)+ 45ℎ21198881205931 + 415ℎ21198881199081015840119888minus ( 215ℎ2119888 + 25ℎ119888ℎ119891) (1199081015840119905 + 1199081015840119887)] = 0

(A5)

Shock and Vibration 13

Figure 8 Schematic of experimental setup

Figure 9 Vibration amplitude of cantilever sandwich beam fordifferent core thicknesses by experiment

119898119888 [ 815119888 + 115ℎ119888 (119887 + 119905)]+ 119866119888119860119888ℎ1198882 [minus25ℎ119888 (1199061015840119905 minus 1199061015840119887) minus 415ℎ21198881205931minus ( 115ℎ119891ℎ119888 + 115ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 ) minus 815ℎ211988811990810158401015840119888 ]+ 119864119888119860119888ℎ2119888 [minus83 (119908119905 + 119908119887) + 163 119908119888] = 0

(A6)

The essential and natural boundary conditions related to theclamped-free case are given as

119906119905 (0 119905) = 0119875119905 (119897 119905) = 0119908119905 (0 119905) = 0119881119905 (119897 119905) = 01199081199051015840 (0 119905) = 0119872119905 (119897 119905) = 0119908119888 (0 119905) = 0119881119888 (119897 119905) = 0119906119887 (0 119905) = 0119875119887 (119897 119905) = 0119908119887 (0 119905) = 0119881119887 (119897 119905) = 01199081198871015840 (0 119905) = 0119872119887 (119897 119905) = 0

(A7)

where the left side of sandwich beam is clamped and the rightside is free

The axial forces 119875119905119887(119909 119905) the shear forces 119881119905119887119888(119909 119905) andthe bending moments119872119905119887(119909 119905) are obtained as

119875119905 = minus119864119891119860119891 120597119906119905120597119909 (A8)

119875119887 = minus119864119891119860119891 120597119906119887120597119909 (A9)

119881119905 = minus 112119898119891ℎ21198911015840119905 + 119898119888840 [minus36ℎ119891119905 minus 6ℎ119891119887minus 28ℎ119891119888 minus 6ℎ119891ℎ119888 1205931 + 3ℎ21198911015840119887 minus 18ℎ21198911015840119905)+ 1198641198911198681198911199081015840101584010158401015840119905 + 119866119888119860119888840ℎ2119888 [(196ℎ119891 minus 28ℎ119888) 119906119887minus (1316ℎ119891 + 532ℎ119888) 119906119905 + (560ℎ119888 + 1120ℎ119891) 119906119888+ (28ℎ2119888 minus 98ℎ2119891 + 28ℎ119891ℎ119888)11990810158401015840119887minus (112ℎ2119888 + 658ℎ2119891 + 532ℎ2119891 + 532ℎ119891ℎ119888)1199081015840119905+ (336ℎ119891ℎ119888 + 112ℎ2119888) 1205931minus (56ℎ2119888 + 168ℎ119891ℎ119888)1199081015840119888]

(A10)

119881119887 = minus 112119898119891ℎ21198911015840119887 + 119898119888840 (36ℎ119891119887 + 6ℎ119891119905+ 28ℎ119891119888 minus 6ℎ119891ℎ119888 1205931 + 3ℎ21198911015840119905 minus 18ℎ21198911015840119887)

14 Shock and Vibration

+ 1198641198911198681198911199081015840101584010158401015840119887 + 119866119888119860119888840ℎ2119888 [(minus196ℎ119891 + 28ℎ119888) 119906119905+ (1316ℎ119891 + 532ℎ119888) 119906119887 minus (560ℎ119888 + 1120ℎ119891) 119906119888+ (28ℎ2119888 minus 98ℎ2119891 + 28ℎ119891ℎ119888)11990810158401015840119905minus (112ℎ2119888 + 658ℎ2119891 + 532ℎ2119891 + 532ℎ119891ℎ119888)1199081015840119887+ (336ℎ119891ℎ119888 + 112ℎ2119888) 1205931minus (56ℎ2119888 + 168ℎ119891ℎ119888)1199081015840119888]

(A11)

119881119888 = 119866119888119860119888840ℎ2119888 [minus336ℎ119891 (119906119905 minus 119906119887) minus (224ℎ2119888) 1205931

minus (56ℎ2119888 + 168ℎ119891ℎ119888) (1199081015840119905 minus 1199081015840119887) minus 448ℎ21198881199081015840119888](A12)

119872119905 = minus11986411989111986811989111990810158401015840119905 (A13)

119872119887 = minus11986411989111986811989111990810158401015840119887 (A14)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] A S Herrmann P C Zahlen and I Zuardy ldquoSandwich Struc-tures Technology in Commercial Aviationrdquo in Proceedings of the7th International Conference on Sandwich Structures pp 13ndash262005

[2] J Vinson The behavior of sandwich structures of isotropic andcomposite materials Technomic Publishing Company Lan-caster Pennsylvania 1999

[3] J Hohe L Librescu and S Y Oh ldquoDynamic buckling of flat andcurved sandwich panels with transversely compressible corerdquoComposite Structures vol 74 no 1 pp 10ndash24 2006

[4] 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

[5] Y Frostig M Baruch O Vilnay and I Sheinman ldquoHigh-ordertheory for sandwich-beam behavior with transversely flexiblecorerdquo Journal of Engineering Mechanics vol 118 no 5 pp 1026ndash1043 1992

[6] O Rahmani SM R Khalili KMalekzadeh andHHadavinialdquoFree vibration analysis of sandwich structures with a flexiblefunctionally graded syntactic corerdquo Composite Structures vol91 no 2 pp 229ndash235 2009

[7] C Santiuste O TThomsen and Y Frostig ldquoThermo-mechani-cal load interactions in foam cored axi-symmetric sandwich cir-cular plates - High-order and FEmodelsrdquo Composite Structuresvol 93 no 2 pp 369ndash376 2011

[8] R K Fruehmann J M Dulieu-Barton O T Thomsen andS Zhang ldquoExperimental investigation of thermal effects infoam cored sandwich beamsrdquo in Thermomechanics and Infra-Red Imaging vol 7 of Conference Proceedings of the Society forExperimental Mechanics Series pp 83ndash90 Springer NY USA2011

[9] V S Sokolinsky and S R Nutt ldquoConsistent Higher-OrderDynamic Equations for Soft-Core Sandwich Beamsrdquo AIAAJournal vol 42 no 2 pp 374ndash382 2004

[10] Y Frostig C Phan and G Kardomateas ldquoFree vibration ofunidirectional sandwich panels Part I Compressible corerdquoJournal of Sandwich Structures and Materials vol 15 no 4 pp377ndash411 2013

[11] A R Damanpack and S M R Khalili ldquoHigh-order free vibra-tion analysis of sandwich beams with a flexible core usingdynamic stiffness methodrdquo Composite Structures vol 94 no 5pp 1503ndash1514 2012

[12] M Yang and P Qiao ldquoHigher-order impact modeling of sand-wich structures with flexible corerdquo International Journal ofSolids and Structures vol 42 no 20 pp 5460ndash5490 2005

[13] Y Wang and X Wang ldquoFree vibration analysis of soft-coresandwich beams by the novel weak form quadrature elementmethodrdquo Journal of Sandwich Structures and Materials vol 18no 3 pp 294ndash320 2015

[14] X Wang and X Liang ldquoFree vibration of soft-core sandwichpanels with general boundary conditions by harmonic quadra-ture element methodrdquoThin-Walled Structures vol 113 pp 253ndash261 2017

[15] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe local GDQmethod applied to general higher-order theories of doubly-curved laminated composite shells and panels the free vibrationanalysisrdquo Composite Structures vol 116 pp 637ndash660 2014

[16] C C Hong H W Liao L T Lee J B Ke and K C JaneldquoThermally induced vibration of a thermal sleeve with the GDQmethodrdquo International Journal of Mechanical Sciences vol 47no 12 pp 1789ndash1806 2005

[17] H D Chalak A Chakrabarti M A Iqbal and A H SheikhldquoVibration of laminated sandwich beams having soft corerdquoJournal of Vibration and Control vol 18 no 10 pp 1422ndash14352012

[18] K M Ahmed ldquoFree vibration of curved sandwich beams by themethod of finite elementsrdquo Journal of Sound and Vibration vol18 no 1 pp 61ndash74 1971

[19] N Nanda S Kapuria and S Gopalakrishnan ldquoSpectral finiteelement based on an efficient layerwise theory for wave prop-agation analysis of composite and sandwich beamsrdquo Journal ofSound and Vibration vol 333 no 31 pp 20ndash27 2014

[20] Y Pourvais P Asgari A R Moradi and O Rahmani ldquoExper-imental and finite element analysis of higher order behaviourof sandwich beams using digital projection moirerdquo PolymerTesting vol 38 pp 7ndash17 2014

[21] R Long O Barry and D C D Oguamanam ldquoFinite elementfree vibration analysis of soft-core sandwich beamsrdquo AIAAJournal vol 50 no 1 pp 235ndash238 2012

[22] W L Chen A G Striz and C W Bert ldquoHigh-accuracy planestress and plate elements in the quadrature element methodrdquoInternational Journal of Solids and Structures vol 37 no 4 pp627ndash647 2000

[23] Y Xing and B Liu ldquoHigh-accuracy differential quadraturefinite element method and its application to free vibrations ofthin plate with curvilinear domainrdquo International Journal for

Shock and Vibration 15

Numerical Methods in Engineering vol 80 no 13 pp 1718ndash17422009

[24] YWang and XWang ldquoStatic analysis of higher order sandwichbeams by weak form quadrature element methodrdquo CompositeStructures vol 116 no 1 pp 841ndash848 2014

[25] F Tornabene S Brischetto N Fantuzzi and M BacciocchildquoBoundary conditions in 2Dnumerical and 3D exactmodels forcylindrical bending analysis of functionally graded structuresrdquoShock and Vibration vol 2016 Article ID 2373862 2016

[26] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe GDQmethod for the free vibration analysis of arbitrarily shaped lami-nated composite shells using a NURBS-based isogeometricapproachrdquo Composite Structures vol 154 pp 190ndash218 2016

[27] Z Zong and Y Zhang Advanced differential quadrature meth-ods CRC Press NY USA 2009

[28] N A Rubayi and S Charoenree ldquoNatural frequencies of vibra-tion of cantilever sandwich beamsrdquoComputersampStructures vol7 no 6 pp 737ndash745 1977

[29] M Azimi M Shahravi and E F Joubaneh ldquoDynamic analysisof maneuvering flexible spacecraft appendage using higherorder sandwich panel theoryrdquo Latin American Journal of Solidsand Structures vol 13 no 2 pp 296ndash313 2016

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Shock and Vibration 13

Figure 8 Schematic of experimental setup

Figure 9 Vibration amplitude of cantilever sandwich beam fordifferent core thicknesses by experiment

119898119888 [ 815119888 + 115ℎ119888 (119887 + 119905)]+ 119866119888119860119888ℎ1198882 [minus25ℎ119888 (1199061015840119905 minus 1199061015840119887) minus 415ℎ21198881205931minus ( 115ℎ119891ℎ119888 + 115ℎ2119888) (11990810158401015840119905 + 11990810158401015840119887 ) minus 815ℎ211988811990810158401015840119888 ]+ 119864119888119860119888ℎ2119888 [minus83 (119908119905 + 119908119887) + 163 119908119888] = 0

(A6)

The essential and natural boundary conditions related to theclamped-free case are given as

119906119905 (0 119905) = 0119875119905 (119897 119905) = 0119908119905 (0 119905) = 0119881119905 (119897 119905) = 01199081199051015840 (0 119905) = 0119872119905 (119897 119905) = 0119908119888 (0 119905) = 0119881119888 (119897 119905) = 0119906119887 (0 119905) = 0119875119887 (119897 119905) = 0119908119887 (0 119905) = 0119881119887 (119897 119905) = 01199081198871015840 (0 119905) = 0119872119887 (119897 119905) = 0

(A7)

where the left side of sandwich beam is clamped and the rightside is free

The axial forces 119875119905119887(119909 119905) the shear forces 119881119905119887119888(119909 119905) andthe bending moments119872119905119887(119909 119905) are obtained as

119875119905 = minus119864119891119860119891 120597119906119905120597119909 (A8)

119875119887 = minus119864119891119860119891 120597119906119887120597119909 (A9)

119881119905 = minus 112119898119891ℎ21198911015840119905 + 119898119888840 [minus36ℎ119891119905 minus 6ℎ119891119887minus 28ℎ119891119888 minus 6ℎ119891ℎ119888 1205931 + 3ℎ21198911015840119887 minus 18ℎ21198911015840119905)+ 1198641198911198681198911199081015840101584010158401015840119905 + 119866119888119860119888840ℎ2119888 [(196ℎ119891 minus 28ℎ119888) 119906119887minus (1316ℎ119891 + 532ℎ119888) 119906119905 + (560ℎ119888 + 1120ℎ119891) 119906119888+ (28ℎ2119888 minus 98ℎ2119891 + 28ℎ119891ℎ119888)11990810158401015840119887minus (112ℎ2119888 + 658ℎ2119891 + 532ℎ2119891 + 532ℎ119891ℎ119888)1199081015840119905+ (336ℎ119891ℎ119888 + 112ℎ2119888) 1205931minus (56ℎ2119888 + 168ℎ119891ℎ119888)1199081015840119888]

(A10)

119881119887 = minus 112119898119891ℎ21198911015840119887 + 119898119888840 (36ℎ119891119887 + 6ℎ119891119905+ 28ℎ119891119888 minus 6ℎ119891ℎ119888 1205931 + 3ℎ21198911015840119905 minus 18ℎ21198911015840119887)

14 Shock and Vibration

+ 1198641198911198681198911199081015840101584010158401015840119887 + 119866119888119860119888840ℎ2119888 [(minus196ℎ119891 + 28ℎ119888) 119906119905+ (1316ℎ119891 + 532ℎ119888) 119906119887 minus (560ℎ119888 + 1120ℎ119891) 119906119888+ (28ℎ2119888 minus 98ℎ2119891 + 28ℎ119891ℎ119888)11990810158401015840119905minus (112ℎ2119888 + 658ℎ2119891 + 532ℎ2119891 + 532ℎ119891ℎ119888)1199081015840119887+ (336ℎ119891ℎ119888 + 112ℎ2119888) 1205931minus (56ℎ2119888 + 168ℎ119891ℎ119888)1199081015840119888]

(A11)

119881119888 = 119866119888119860119888840ℎ2119888 [minus336ℎ119891 (119906119905 minus 119906119887) minus (224ℎ2119888) 1205931

minus (56ℎ2119888 + 168ℎ119891ℎ119888) (1199081015840119905 minus 1199081015840119887) minus 448ℎ21198881199081015840119888](A12)

119872119905 = minus11986411989111986811989111990810158401015840119905 (A13)

119872119887 = minus11986411989111986811989111990810158401015840119887 (A14)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] A S Herrmann P C Zahlen and I Zuardy ldquoSandwich Struc-tures Technology in Commercial Aviationrdquo in Proceedings of the7th International Conference on Sandwich Structures pp 13ndash262005

[2] J Vinson The behavior of sandwich structures of isotropic andcomposite materials Technomic Publishing Company Lan-caster Pennsylvania 1999

[3] J Hohe L Librescu and S Y Oh ldquoDynamic buckling of flat andcurved sandwich panels with transversely compressible corerdquoComposite Structures vol 74 no 1 pp 10ndash24 2006

[4] 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

[5] Y Frostig M Baruch O Vilnay and I Sheinman ldquoHigh-ordertheory for sandwich-beam behavior with transversely flexiblecorerdquo Journal of Engineering Mechanics vol 118 no 5 pp 1026ndash1043 1992

[6] O Rahmani SM R Khalili KMalekzadeh andHHadavinialdquoFree vibration analysis of sandwich structures with a flexiblefunctionally graded syntactic corerdquo Composite Structures vol91 no 2 pp 229ndash235 2009

[7] C Santiuste O TThomsen and Y Frostig ldquoThermo-mechani-cal load interactions in foam cored axi-symmetric sandwich cir-cular plates - High-order and FEmodelsrdquo Composite Structuresvol 93 no 2 pp 369ndash376 2011

[8] R K Fruehmann J M Dulieu-Barton O T Thomsen andS Zhang ldquoExperimental investigation of thermal effects infoam cored sandwich beamsrdquo in Thermomechanics and Infra-Red Imaging vol 7 of Conference Proceedings of the Society forExperimental Mechanics Series pp 83ndash90 Springer NY USA2011

[9] V S Sokolinsky and S R Nutt ldquoConsistent Higher-OrderDynamic Equations for Soft-Core Sandwich Beamsrdquo AIAAJournal vol 42 no 2 pp 374ndash382 2004

[10] Y Frostig C Phan and G Kardomateas ldquoFree vibration ofunidirectional sandwich panels Part I Compressible corerdquoJournal of Sandwich Structures and Materials vol 15 no 4 pp377ndash411 2013

[11] A R Damanpack and S M R Khalili ldquoHigh-order free vibra-tion analysis of sandwich beams with a flexible core usingdynamic stiffness methodrdquo Composite Structures vol 94 no 5pp 1503ndash1514 2012

[12] M Yang and P Qiao ldquoHigher-order impact modeling of sand-wich structures with flexible corerdquo International Journal ofSolids and Structures vol 42 no 20 pp 5460ndash5490 2005

[13] Y Wang and X Wang ldquoFree vibration analysis of soft-coresandwich beams by the novel weak form quadrature elementmethodrdquo Journal of Sandwich Structures and Materials vol 18no 3 pp 294ndash320 2015

[14] X Wang and X Liang ldquoFree vibration of soft-core sandwichpanels with general boundary conditions by harmonic quadra-ture element methodrdquoThin-Walled Structures vol 113 pp 253ndash261 2017

[15] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe local GDQmethod applied to general higher-order theories of doubly-curved laminated composite shells and panels the free vibrationanalysisrdquo Composite Structures vol 116 pp 637ndash660 2014

[16] C C Hong H W Liao L T Lee J B Ke and K C JaneldquoThermally induced vibration of a thermal sleeve with the GDQmethodrdquo International Journal of Mechanical Sciences vol 47no 12 pp 1789ndash1806 2005

[17] H D Chalak A Chakrabarti M A Iqbal and A H SheikhldquoVibration of laminated sandwich beams having soft corerdquoJournal of Vibration and Control vol 18 no 10 pp 1422ndash14352012

[18] K M Ahmed ldquoFree vibration of curved sandwich beams by themethod of finite elementsrdquo Journal of Sound and Vibration vol18 no 1 pp 61ndash74 1971

[19] N Nanda S Kapuria and S Gopalakrishnan ldquoSpectral finiteelement based on an efficient layerwise theory for wave prop-agation analysis of composite and sandwich beamsrdquo Journal ofSound and Vibration vol 333 no 31 pp 20ndash27 2014

[20] Y Pourvais P Asgari A R Moradi and O Rahmani ldquoExper-imental and finite element analysis of higher order behaviourof sandwich beams using digital projection moirerdquo PolymerTesting vol 38 pp 7ndash17 2014

[21] R Long O Barry and D C D Oguamanam ldquoFinite elementfree vibration analysis of soft-core sandwich beamsrdquo AIAAJournal vol 50 no 1 pp 235ndash238 2012

[22] W L Chen A G Striz and C W Bert ldquoHigh-accuracy planestress and plate elements in the quadrature element methodrdquoInternational Journal of Solids and Structures vol 37 no 4 pp627ndash647 2000

[23] Y Xing and B Liu ldquoHigh-accuracy differential quadraturefinite element method and its application to free vibrations ofthin plate with curvilinear domainrdquo International Journal for

Shock and Vibration 15

Numerical Methods in Engineering vol 80 no 13 pp 1718ndash17422009

[24] YWang and XWang ldquoStatic analysis of higher order sandwichbeams by weak form quadrature element methodrdquo CompositeStructures vol 116 no 1 pp 841ndash848 2014

[25] F Tornabene S Brischetto N Fantuzzi and M BacciocchildquoBoundary conditions in 2Dnumerical and 3D exactmodels forcylindrical bending analysis of functionally graded structuresrdquoShock and Vibration vol 2016 Article ID 2373862 2016

[26] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe GDQmethod for the free vibration analysis of arbitrarily shaped lami-nated composite shells using a NURBS-based isogeometricapproachrdquo Composite Structures vol 154 pp 190ndash218 2016

[27] Z Zong and Y Zhang Advanced differential quadrature meth-ods CRC Press NY USA 2009

[28] N A Rubayi and S Charoenree ldquoNatural frequencies of vibra-tion of cantilever sandwich beamsrdquoComputersampStructures vol7 no 6 pp 737ndash745 1977

[29] M Azimi M Shahravi and E F Joubaneh ldquoDynamic analysisof maneuvering flexible spacecraft appendage using higherorder sandwich panel theoryrdquo Latin American Journal of Solidsand Structures vol 13 no 2 pp 296ndash313 2016

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

14 Shock and Vibration

+ 1198641198911198681198911199081015840101584010158401015840119887 + 119866119888119860119888840ℎ2119888 [(minus196ℎ119891 + 28ℎ119888) 119906119905+ (1316ℎ119891 + 532ℎ119888) 119906119887 minus (560ℎ119888 + 1120ℎ119891) 119906119888+ (28ℎ2119888 minus 98ℎ2119891 + 28ℎ119891ℎ119888)11990810158401015840119905minus (112ℎ2119888 + 658ℎ2119891 + 532ℎ2119891 + 532ℎ119891ℎ119888)1199081015840119887+ (336ℎ119891ℎ119888 + 112ℎ2119888) 1205931minus (56ℎ2119888 + 168ℎ119891ℎ119888)1199081015840119888]

(A11)

119881119888 = 119866119888119860119888840ℎ2119888 [minus336ℎ119891 (119906119905 minus 119906119887) minus (224ℎ2119888) 1205931

minus (56ℎ2119888 + 168ℎ119891ℎ119888) (1199081015840119905 minus 1199081015840119887) minus 448ℎ21198881199081015840119888](A12)

119872119905 = minus11986411989111986811989111990810158401015840119905 (A13)

119872119887 = minus11986411989111986811989111990810158401015840119887 (A14)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] A S Herrmann P C Zahlen and I Zuardy ldquoSandwich Struc-tures Technology in Commercial Aviationrdquo in Proceedings of the7th International Conference on Sandwich Structures pp 13ndash262005

[2] J Vinson The behavior of sandwich structures of isotropic andcomposite materials Technomic Publishing Company Lan-caster Pennsylvania 1999

[3] J Hohe L Librescu and S Y Oh ldquoDynamic buckling of flat andcurved sandwich panels with transversely compressible corerdquoComposite Structures vol 74 no 1 pp 10ndash24 2006

[4] 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

[5] Y Frostig M Baruch O Vilnay and I Sheinman ldquoHigh-ordertheory for sandwich-beam behavior with transversely flexiblecorerdquo Journal of Engineering Mechanics vol 118 no 5 pp 1026ndash1043 1992

[6] O Rahmani SM R Khalili KMalekzadeh andHHadavinialdquoFree vibration analysis of sandwich structures with a flexiblefunctionally graded syntactic corerdquo Composite Structures vol91 no 2 pp 229ndash235 2009

[7] C Santiuste O TThomsen and Y Frostig ldquoThermo-mechani-cal load interactions in foam cored axi-symmetric sandwich cir-cular plates - High-order and FEmodelsrdquo Composite Structuresvol 93 no 2 pp 369ndash376 2011

[8] R K Fruehmann J M Dulieu-Barton O T Thomsen andS Zhang ldquoExperimental investigation of thermal effects infoam cored sandwich beamsrdquo in Thermomechanics and Infra-Red Imaging vol 7 of Conference Proceedings of the Society forExperimental Mechanics Series pp 83ndash90 Springer NY USA2011

[9] V S Sokolinsky and S R Nutt ldquoConsistent Higher-OrderDynamic Equations for Soft-Core Sandwich Beamsrdquo AIAAJournal vol 42 no 2 pp 374ndash382 2004

[10] Y Frostig C Phan and G Kardomateas ldquoFree vibration ofunidirectional sandwich panels Part I Compressible corerdquoJournal of Sandwich Structures and Materials vol 15 no 4 pp377ndash411 2013

[11] A R Damanpack and S M R Khalili ldquoHigh-order free vibra-tion analysis of sandwich beams with a flexible core usingdynamic stiffness methodrdquo Composite Structures vol 94 no 5pp 1503ndash1514 2012

[12] M Yang and P Qiao ldquoHigher-order impact modeling of sand-wich structures with flexible corerdquo International Journal ofSolids and Structures vol 42 no 20 pp 5460ndash5490 2005

[13] Y Wang and X Wang ldquoFree vibration analysis of soft-coresandwich beams by the novel weak form quadrature elementmethodrdquo Journal of Sandwich Structures and Materials vol 18no 3 pp 294ndash320 2015

[14] X Wang and X Liang ldquoFree vibration of soft-core sandwichpanels with general boundary conditions by harmonic quadra-ture element methodrdquoThin-Walled Structures vol 113 pp 253ndash261 2017

[15] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe local GDQmethod applied to general higher-order theories of doubly-curved laminated composite shells and panels the free vibrationanalysisrdquo Composite Structures vol 116 pp 637ndash660 2014

[16] C C Hong H W Liao L T Lee J B Ke and K C JaneldquoThermally induced vibration of a thermal sleeve with the GDQmethodrdquo International Journal of Mechanical Sciences vol 47no 12 pp 1789ndash1806 2005

[17] H D Chalak A Chakrabarti M A Iqbal and A H SheikhldquoVibration of laminated sandwich beams having soft corerdquoJournal of Vibration and Control vol 18 no 10 pp 1422ndash14352012

[18] K M Ahmed ldquoFree vibration of curved sandwich beams by themethod of finite elementsrdquo Journal of Sound and Vibration vol18 no 1 pp 61ndash74 1971

[19] N Nanda S Kapuria and S Gopalakrishnan ldquoSpectral finiteelement based on an efficient layerwise theory for wave prop-agation analysis of composite and sandwich beamsrdquo Journal ofSound and Vibration vol 333 no 31 pp 20ndash27 2014

[20] Y Pourvais P Asgari A R Moradi and O Rahmani ldquoExper-imental and finite element analysis of higher order behaviourof sandwich beams using digital projection moirerdquo PolymerTesting vol 38 pp 7ndash17 2014

[21] R Long O Barry and D C D Oguamanam ldquoFinite elementfree vibration analysis of soft-core sandwich beamsrdquo AIAAJournal vol 50 no 1 pp 235ndash238 2012

[22] W L Chen A G Striz and C W Bert ldquoHigh-accuracy planestress and plate elements in the quadrature element methodrdquoInternational Journal of Solids and Structures vol 37 no 4 pp627ndash647 2000

[23] Y Xing and B Liu ldquoHigh-accuracy differential quadraturefinite element method and its application to free vibrations ofthin plate with curvilinear domainrdquo International Journal for

Shock and Vibration 15

Numerical Methods in Engineering vol 80 no 13 pp 1718ndash17422009

[24] YWang and XWang ldquoStatic analysis of higher order sandwichbeams by weak form quadrature element methodrdquo CompositeStructures vol 116 no 1 pp 841ndash848 2014

[25] F Tornabene S Brischetto N Fantuzzi and M BacciocchildquoBoundary conditions in 2Dnumerical and 3D exactmodels forcylindrical bending analysis of functionally graded structuresrdquoShock and Vibration vol 2016 Article ID 2373862 2016

[26] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe GDQmethod for the free vibration analysis of arbitrarily shaped lami-nated composite shells using a NURBS-based isogeometricapproachrdquo Composite Structures vol 154 pp 190ndash218 2016

[27] Z Zong and Y Zhang Advanced differential quadrature meth-ods CRC Press NY USA 2009

[28] N A Rubayi and S Charoenree ldquoNatural frequencies of vibra-tion of cantilever sandwich beamsrdquoComputersampStructures vol7 no 6 pp 737ndash745 1977

[29] M Azimi M Shahravi and E F Joubaneh ldquoDynamic analysisof maneuvering flexible spacecraft appendage using higherorder sandwich panel theoryrdquo Latin American Journal of Solidsand Structures vol 13 no 2 pp 296ndash313 2016

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Shock and Vibration 15

Numerical Methods in Engineering vol 80 no 13 pp 1718ndash17422009

[24] YWang and XWang ldquoStatic analysis of higher order sandwichbeams by weak form quadrature element methodrdquo CompositeStructures vol 116 no 1 pp 841ndash848 2014

[25] F Tornabene S Brischetto N Fantuzzi and M BacciocchildquoBoundary conditions in 2Dnumerical and 3D exactmodels forcylindrical bending analysis of functionally graded structuresrdquoShock and Vibration vol 2016 Article ID 2373862 2016

[26] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe GDQmethod for the free vibration analysis of arbitrarily shaped lami-nated composite shells using a NURBS-based isogeometricapproachrdquo Composite Structures vol 154 pp 190ndash218 2016

[27] Z Zong and Y Zhang Advanced differential quadrature meth-ods CRC Press NY USA 2009

[28] N A Rubayi and S Charoenree ldquoNatural frequencies of vibra-tion of cantilever sandwich beamsrdquoComputersampStructures vol7 no 6 pp 737ndash745 1977

[29] M Azimi M Shahravi and E F Joubaneh ldquoDynamic analysisof maneuvering flexible spacecraft appendage using higherorder sandwich panel theoryrdquo Latin American Journal of Solidsand Structures vol 13 no 2 pp 296ndash313 2016

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom