analysis of sandwich plates with isotropic face plates and a viscoelastic core

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Gang WangGraduate Research Assistant

Sudha VeeramaniGraduate Research Assistant

Norman M. WereleyAssociate Professor

Smart Structures Laboratory,Alfred Gessow Rotorcraft Center,

Department of Aerospace Engineering,University of Maryland at College Park,

College Park, MD 20742

Analysis of Sandwich Plateswith Isotropic Face Platesand a Viscoelastic CoreExperimental and analytical validations of a Galerkin assumed modes analysis of swich plates are presented in this paper. The 3-layered sandwich plate specimen coof isotropic face-plates with surface bonded piezo-electric patch actuators, and acoelastic core. The experimental validation is conducted by testing sandwich plateare 67.31 cm (26.5 in.) long, 52.07 cm (20.5 in.) wide and nominally 0.16 cm (1/16thick. The analysis includes the membrane and transverse energies in the face plateshear energies in the core. The shear modulus of the dissipative core is assumedcomplex and variant with frequency and temperature. The Golla-Hughes-McTa(GHM) method is used to account for the frequency dependent properties of the visctic core. Experiments were conducted on symmetric and asymmetric sandwich platealuminum face-plates under clamped boundary conditions to validate the model fotropic face-plates. The maximum error in damped natural frequency predictions obtavia the assumed modes solution is,11 percent. Analytical studies on the influence of tnumber of assumed modes in the Galerkin approximation have been conducted. Ethe first plate bending mode is 112 percent when only a single in-plane mode iserror reduces to 3.95 percent as the number of in-plane modes is increased to 25 inof the in-plane directions.@S0739-3717~00!00703-0#

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1 IntroductionDamping augmentation of structures is of key interest to ae

space, mechanical and civil engineers. Noise and vibration retion is a major challenge pertaining to these fields; specificallyaerospace applications, the reduction must be achieved with mmal increase in weight. Viscoelastic shear layers integrated withe structure of vibrating members have been used towardsend.

Flexural vibration of viscoelastic sandwich structures has binvestigated since the 1950s. Much effort has been directedwards the analysis of beams and plates. Kerwin@1# presented ananalysis for a simply supported sandwich beam using the commodulus to account for damping and stiffness of the viscoelacore. DiTaranto@2# extended this work and derived a sixth ordpartial differential equations of motion for sandwich beamterms of longitudinal displacementu. Mead and Marcus@3# ana-lyzed three layered sandwiched beams with a viscoelasticusing sixth order differential equations of motion in termstransverse displacementw. Also, they examined the form oboundary constraints for many end conditions encounteredpractice. Rao and Nakra@4# used the energy method to develoequations of motion including the inertia effects of transverlongitudinal and rotary motion. Bai and Sun@5# relaxed the per-fect interface and constant transverse deformation assumpused by Mead and Marcus@3#.

Ross et al.@6# studied simply-supported plates, and assumeperfect interface and compatibility of transverse displacemeneach layer. Rao and Nakra@7# developed the basic equationsvibratory bending of asymmetric sandwich plates with isotroface-plates and viscoelastic core. Lu et al.@8# developed a finiteelement model and presented experimental data for sandplates under free boundary conditions. Cupial and Niziol@9# usedthe variational method to model sandwich plates with anisotroface-plates, who presented simplified forms of the equations fsymmetric plate or for specially orthotropic face layers. T

Contributed by the Technical Committee on Vibration and Sound for publicain the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received Januar1999. Associate Technical Editor: D. J. Inman.

Copyright © 2Journal of Vibration and Acoustics

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modal frequencies and modal loss factors predicted by the ansis compared well with the results in Johnson and Keinholz@10#.However, they did not present experimental validations formodal frequencies and loss factors.

Viscoelastic materials exhibit both elastic and viscous charteristics, and the complex modulus is typically used to represthis behavior. The determination of the components of the coplex modulus must be done experimentally for individual mateals by applying harmonic excitation over a range of frequencytemperature. The complex modulus,G!5G81 jG9, is defined inthe frequency domain at steady state. The in-phase componethe complex modulus is the elastic modulus or stiffness,G8,while its quadrature component is the viscous modulus or daing, G9.

Shortcomings of frequency domain models, such as their inplicability to transient analysis, can be overcome by using tidomain representations of viscoelastic materials. A number ofproaches have been developed to account for the frequencypendent properties of viscoelastic materials, while also provida time domain analysis. These include the Golla-HughMcTavish method~GHM! @11#, and the anelastic displacemefield ~ADF! method@12#. In place of using derivatives of dampeforce and displacement, both of these theories augment thechanical analysis with internal dissipation coordinates to accofor viscoelastic material’s constitutive relationship. In the GHmethod, the frequency dependent complex modulus is represeby a linear spring in parallel with a number of linear minoscillators, each having an internal dissipation coordinate.additional coordinates can then be used to accurately modevariation in the complex modulus over a wide frequency rang

In this paper, we wish to better understand the behavior, expmentation, and analysis of sandwich plates for practical appltion. Towards this end, we develop an analysis of sandwich plabased on the Galerkin assumed modes method, for 3-layplates with a viscoelastic core sandwiched between isotropicplates. The assumed modes procedure is a similar analysis todeveloped by Cupial and Niziol@9#, although our analysis is fo-

ion

000 by ASME JULY 2000, Vol. 122 Õ 305

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cused to the case of isotropic face plates. In many prior panalyses~Abdulhadi’s analysis from Johnson and Keinholz,@10#;Cupial and Niziol@9#!, the complex modulus was assumed toconstant over the frequency range of interest. This assumptiooverly restrictive, as most viscoelastic materials exhibit a compmodulus that is frequency dependent. Thus, a primary contrtion of our analysis is that the assumed modes analysis ismented with internal dissipation coordinates, using the GHmethod, to account for the frequency dependent complex modof the viscoelastic core. We validate our analysis via analytcomparisons for simply supported cases, as well as experimecomparisons for plates with all four sides clamped.

We first compare our results to the exact solutions of simsupported sandwich plates@10# in order to validate our modeanalytically. In this study, the shear modulus was assumed toconstant value over the frequency range of interest as was donJohnson and Keinholz. This validation is conducted primarilyvalidate the assumed modes analysis procedure. We also vaour assumed modes analysis of a sandwich plate with a viscotic core, where the GHM method is used to account for thequency dependent complex shear modulus of the viscoelacore. Experiments were conducted on symmetric and asymmsandwich plates, with all four sides clamped, to measure the nral frequencies and loss factors of the transverse bending vibramodes. In this study, surface bonded piezo-actuators werefor modal excitation. Thus, a second primary contribution of twork is that experimental plate test data were used to validateanalysis of a sandwich plate with all four sides clamped.

2 Sandwich Plate ModelThe equations of vibration of a piezo-actuated, 3-layered sa

wich with isotropic face-plates and a dissipative viscoelastic care developed in this section. The layers are numbered 1 thro3, as depicted in Fig. 1. Layers 1 and 3 are the isotropic faplates, i.e., aluminum, whereas the core is the viscoelastic mrial. The face-plates are assumed to have bending, in-plane sand extensional stiffnesses. Their rotatory inertia has also bneglected in the model. The viscoelastic core is assumed totransverse shear stiffness alone.

The piezo-actuators patches are surface-bonded to the expsurfaces of the two face-plates to enable efficient modal excitaof the plate. They are assumed to be orthotropic in nature, andbonding is assumed to be infinitely thin and perfect. The actupatches are integrated into each of the face-plates as one olaminae.

The patch actuators cover a limited area in the center ofplate surfaces. Hence, a functionq is introduced.q is unity overthe area of the face-plates where the piezo-actuator is presenis zero over the rest of the area. The mass in the area wheractuators are bonded differ from those in the surrounding aThe mass per unit area of the complete sandwich now is:

m5r1h11r3h312qrchc (1)

whereh1 andh3 are the thickness of face plate 1 and 3, resptively. Also, hc is the thickness, andrc is the density, of thepiezo-actuator. The mass per unit area of the face-layers is wras:

mi5r ihi1qrchc i 51,3 (2)

We currently use the piezo-actuators only as exciters for the pbending vibration modes in order to validate analytical predictioof the frequencies and loss factors of sandwich plates. The vdation of actuation force will not be present in this paper. Alsoour analysis, the stiffness contributions of piezo-actuators areglected due to the small size of the piezo-actuator relative toplate.

The assumptions involved in the derivation of the governequation of sandwich plate are: a! the face plates are elastic anisotropic and suffer no transverse shear deformation, i.e., Ki

306 Õ Vol. 122, JULY 2000

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hoff hypothesis, b! the core carries transverse shear, but noplane stresses; it is linearly viscoelastic and has a complex mlus, c! no slip occurs in the interfaces between the face-platesthe core, all points normal to the plate move with the same traverse displacement.

2.1 Sandwich Plate Energies. The analysis is outlined hereusing assumptions~a!–~c!, as illustrated in Fig. 1b, the shearstrain in viscoelatic layer can be expressed by:

gx,25u322u12

h21

]w

]x5

u32u1

h21

h11h312h2

2h2

]w

]x

gy,25v322v12

h21

]w

]y5

v32v1

h21

h11h312h2

2h2

]w

]y(3)

The variational kinetic energy is:

dT5~r1h11r3h312qrchc!EA

]w

]t

]dw

]tdA

1 (i 51,3

~r ihi1qrchc!EAS ]ui

]t

]dui

]t1

]v i

]t

]dv i

]t DdA (4)

The variational potential energy, including the transverse being, in-plane and shear and extension energies in the faces,the shear energy alone in the core is:

dU5EA

F (i 51,3

Nx,i

]dui

]x1 (

i 51,3Ny,i

]dv i

]y

1 (i 51,3

Nxy,i S ]dui

]y1

]dv i

]x D2 (i 51,3

Mx,i

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2 (i 51,3

M y,i

]2dw

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Mxy,i

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1

h2

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2

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1

h2Sdv32dv11

d

2

]dw

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(5)where

~~Nx,i !,~Ny,i !,~Nxy,i !!5E2hi /2

hi /2

~~sx,i !,~sy,i !,~sxy,i !!dz

~~Mx,i !,~M y,i !,~Mxy,i !!5E2hi /2

hi /2

~~sx,i !,~sy,i !,~sxy,i !!zdz

~~Qx,2!,~Qy,2!!5E2h2/2

h2/2

~~sxz,2!,~syz,2!!dz

d52h21h11h3 (6)

Here i 51,3 for the face plates 1 and 3. The shear stresses inplates 1 and 3 are neglected.

2.2 Governing Equations. Applying the Hamilton’s prin-ciple to the energies discussed above yields the equations oftion. There are 5 equations corresponding to the 5 indepencoordinates. The governing equations are these:

~rh12qrchc!w5]2Mx

]x2 1]2M y

]y2 12]2Mxy

]x]y1

]Qx(2)

]x

d

2h2

1]Qy

(2)

]y

d

2h2(7)

~r1h11qrchc!u15]Nx,1

]x1

]Nxy,1

]y1

Qx,2

h2(8)

~r3h31qrchc!u35]Nx,3

]x1

]Nxy,3

]y2

Qx,2

h2(9)

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]y1

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Qy,2

h2(10)

~r3h31qrchc!v35]Ny,3

]y1

]Nxy,3

]x2

Qy,2

h2(11)

Here

Mx5Mx,11Mx,3

M y5M y,11M y,3

Mxy5Mxy,11Mxy,3 (12)

Applying the Galerkin assumed modes method@13#, we candiscretize the above five equations as:

M x1~Ke1Kv!x5F (13)

WhereM is the mass matrix;Ke is the stiffness contribution fromthe elastic part of the sandwich plates, andKv is the stiffnesscontribution from the viscoelastic core. The displacement vecis given by:

x5@w u1 v1 u3 v3#T (14)

also

Kv5G!K

Details of the mass and stiffness matrix elements using Galeassumed modes method are given in the Appendix.

2.3 GHM Method for Constrained Layer Damping. Af-ter we set up the governing equation as shown in Eq.~13! usingGalerkin assumed modes method, the Golla-Hughes-McTa~GHM! approach is introduced to model the hysteretic linedamping. It is achieved by adding internal dissipation coordinato the system to achieve a nonlinear hysteretic model providthe same damping properties. The GHM method requires thacomplex modulus of the viscoelastic core be represented asries of minioscillator terms. The complex modulus can be writin the Laplace domain as:

Journal of Vibration and Acoustics

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sG~s!5G0F 11(k

N

ak

s212zkvks

s212zkvks1vk2G (15)

Where the factorG0 is the equilibrium value of the modulus, i.ethe final value of the relaxation function,G(t), and s is theLaplace domain operator. The hatted terms are obtained fromcurve fitting to the complex modulus data for a particular vcoelastic materials@14#. The number of terms,N, retained in theexpression is determined from the high or low frequency depdence of the complex modulus.

The equation of motion from Eq.~13! can be expressed in theLaplace domain as:

Ms2x~s!1Kex~s!1sG~s!Kx~s!5F~s! (16)

Now, an auxiliary coordinate is introduced such that:

zk~s!5vk

2

s212zkvks1vk2

x~s! (17)

Using this new dissipation coordinate, a second order time domrealization of the Laplace domain expression can be obtained

F M 0

0 a1

v2 K0GF x

z G1F 0 0

0 a2z

vK0

GF x

z G1FKe1K01aK0 2aK0

2aK0 aK0G Fx

z G5F F0 G (18)

We demonstrate the case fork51, that is only one mini-oscillator.BecauseK0 is usually positive semi-definite, the above mass mtrix may not be positive definite. To remedy this situation, spectdecomposition ofK0 is used@11#

K05G0K5G0RLRT (19)

WhereL is a diagonal matrix of the nonzero eigenvalues ofK andthe corresponding orthonormalized eigenvectors form the columof R.

Fig. 1 „a… Sandwich plate showing its co-ordinate axes and dimensions and „b… layersforming the sandwich, and the displacements associated with each layer

JULY 2000, Vol. 122 Õ 307

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The above case of a single mini-oscillator term can be eaextended to a multi-oscillator model. Here, we give the geneform of the GHM method@11#. The stiffness matrix,K, anddamping matrix,D, and mass matrix,M , in time domain aregiven by:

K5F Ke1K0S 11( a D 2a1R ¯ 2anR

2a1RT a1L 0 0

] 0 � 0

2anRT 0 0 anL

G (20)

M53M 0 ¯ 0

0 a1

1

v12 L 0 ]

] 0 � 0

0 ¯ 0 an

1

vn2 L

4 (21)

D530 0 ¯ 0

0 a1

2z1

v1L 0 ]

] 0 � 0

0 ¯ 0 an

2zn

vnL4 (22)

where

L5G0L

R5RL

We can now rewrite the equations of motion as a state spmodel and calculate the frequency response and time respusing software packages such as MATLAB.

3 Analytical Validation: Simply SupportedThis section compares the modal frequencies of free vibra

predicted by an existing analytical solution@10# to those predictedby our analysis for a simply supported sandwich plate with symetric isotropic aluminum face-plates and a viscoelastic core.complex shear modulus of the core is assumed constant ovefrequency range, so that it is not necessary to use the Gmethod to account for the frequency dependent complex modof the viscoelastic core. For a simply supported sandwich plthe assumed plate bending modes inw are of the form:

wk~x,y,t !5WkFk~x,y!5Wk~ t !sinmpx

Lsin

npy

C(23)

Thus, the assumed plate bending mode used in the product oappropriate Euler-Bernoulli beam bending mode in eachx andydirection. The assumed extensional modesu and v for the faceplates 1 and 3 are the appropriate rod extensional mode shand are of the form:

u1,k~x,y,t !5U1,k~ t !Ck~x,y!5U1,k cosmpx

Lsin

npy

C

v1,k~x,y,t !5V1,k~ t !Ck~x,y!5V1,k sinmpx

Lcos

npy

C

u3,k~x,y,t !5U3,k~ t !Ck~x,y!5U3,k cosmpx

Lsin

npy

C

308 Õ Vol. 122, JULY 2000

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f the

pes,

v3,k~x,y,t !5V3,k~ t !Ck~x,y!5V3,k sinmpx

Lcos

npy

C(24)

Here,m andn are obtained for the modek via the mapping inTable 1.Ck(x,y) and Fk(x,y) are the mode shapes of the inplane and transverse bending motion respectively, which sathe geometric boundary conditions, whereWk , U1,k , U3,k , V1,kand V3,k are the coefficients of the corresponding natural moshapes. The mass and stiffness matrices are obtained usinfirst 16 transverse bending and in-plane mode shapes. Thedicted modal frequencies and the corresponding modal losstors are tabulated against the exact solution in Table 2. Thedicted values match the closed form analytical solution very wError is less than 1.2 percent for both the modal frequencies,loss factors. Also, our results agree well with the numerical sotions of Cupial and Niziol@9#.

Table 1 Plate bending vibration modes index. Here, n b is thenumber of assumed plate modes in w , i.e., bending modes; n eis the number of assumed plate modes in u, v in each of faceplates, i.e. extensional modes.

Table 2 Comparison of natural frequencies and loss factors ofa symmetric sandwich with isotropic face-plates: LÄ0.3480 m, CÄ0.3048 m, h 1Äh 3Ä0.762 mm, h 2Ä0.254 mm,E1ÄE3Ä68.9 GPa, n1Än3Ä0.3, r1Är3Ä2740 kg Õm3, r2Ä999 kg Õm3, G2Ä0.869 MPa, h2Ä0.5. The exact values arefrom the analytical solution of Abdulhadi from Johnson andKeinholz †10‡ and Cupial and Niziol †9‡.


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4 Experimental Validation: All Four Sides Clamped

4.1 Set-up. This section presents an experimental validatof sandwich plates with aluminum isotropic face plates and vcoelastic cores. The test area of the plates is 67.31 cm352.07 cm~26.5 in.320.5 in.!. The plates were clamped atop an aluminustand using bars of cross section, 3.81 cm~1.5 in.! wide and 0.953cm ~3/8 in.! thick, around the perimeter, as shown in Fig. 2. Eabolt was inserted through the top clamping bar, the plate, andtest stand, and fastened with a nut. Each bolt was subsequtightened to a constant torque, as measured by a torque wrePiezo-actuators were fastened to the plate, and used to ebending motion of the plate. The response of the plate was msured using light-weight accelerometers.

The stand was calibrated by testing uniform aluminum platevarying thickness. Table 3 shows these results. For a thicknes0.16 cm~1/16 in.! the error in the first modal natural frequency3.75 percent. Increasing the thickness to 0.24 cm~3/32 in.! leadsto increased error in the first modal frequency of 16.6 percentit is established that for plate thickness at or below 0.16 cm~1/16in.!, the stand provides adequate clamping.

Three different sandwich plates were tested. The first of thesa symmetric sandwich, with aluminum face-plates of thickn0.08 cm~1/32 in.!, sandwiching a 0.00508 cm~2 mil! thick vis-coelastic layer. The remaining two plates are asymmetric, wface-plates of thickness 0.08 cm~1/32 in.! and 0.04 cm~1/64 in.!.One of these has a core thickness of 0.00508 cm~2 mil!, and theother, 0.0127 cm~5 mil!.

The viscoelastic material used is 3M Scotchdamp ISD-1@15#. In our analysis, the GHM method is used to account forcomplex modulus variation with frequency and temperature. Tmodulus and loss factor were obtained from the product informtion provided by 3M over the ranges of temperature and frequeof interest. A software code@14# was used to find the mini-oscillator parameters in GHM model to match the product dprovided by 3M. In this case, the three mini-oscillators terms wused to fit the curve of storage modulus and loss factors.

4.2 Analysis. For the clamped-clamped end boundary coditions, the plate transverse bending mode shapes in thew direc-tion are obtained from beam bending modes. Analytical moshape of the Euler-Bernoulli beam in fixed-fixed end boundconditions were used@13#. The plate bending mode shapes arecombinations of beam bending modes inx and y directions, sothat

wk~x,y,t !5Wk~ t !fm~x!fn~y! (25)

where

fm~x!5sinhbmx2sinbmx1lm~cosbmx2coshbmx!

lm5sinhbml 2sinbml

cosbml 2coshbml(26)

Here bm is determined using the characteristic equation forfixed-fixed end boundary condition of a beam, which is:

Table 3 Calibration of experimental set-up: the influence ofplate thickness on accuracy of the experiments. Aluminumplate dimensions: 67.31 cm Ã52.07 cmÃt „26.5 in.Ã20.5 in.Ãt ….


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m

chthentlynch.xciteea-

ofs ofis

So

e isss

ith

12hehea-

ncy

atare

n-

dery

he

he

cosbml coshbml 51 (27)

Similarly, we can obtainfn(y).The approximate extensional mode shapes assumed in

plates 1 and 3 for the case of a plate clamped on all sidesassumed to be rod extensional mode shapes, as follows:

u1,k~x,y,t !5U1,k~ t !Ck~x,y!

v1,k~x,y,t !5V1,k~ t !Ck~x,y!

u3,k~x,y,t !5U3,k~ t !Ck~x,y!

v3,k~x,y,t !5V3,k~ t !Ck~x,y! (28)

where

Ck~x,y!5sinmpx

Lsin

npy

C(29)

Note that the in-plane mode shapes as represented by theextensional mode shapes,Ck , are different from those in section3 because of the change in boundary conditions.

The number of assumed modes used in the model arenb525 inthe transverse directions, andne525 in each of the in-plane di-rections, unless otherwise noted.

4.3 Results. The results of experiments conducted on tsymmetric sandwich are tabulated against the frequenciesdicted by the analysis in Table 4. Overall error in the frequenrange under study~0–200 Hz! is below 7 percent. However, if thebandwidth is increased, the errors are likely to be higher and mbending and in-plane modes would need to be included to prehigher frequencies.

The experimental validation of the asymmetric sandwichespresented in Table 5. These plates have face-plate thickness0.04 cm~1/64 in.! and 0.08 cm~1/32 in.!. One has a viscoelasticcore thickness of 0.00508 cm~2 mil!, and the other, 0.0127 cm~5mil!. Good correlation between measured and predicted mofrequencies is seen. A downward shift in the modal frequencoccurs when the thickness of the viscoelastic core increases.trend is manifested in both the experimental measurement ofquencies and the analytical predictions. For the higher modeserror increases. The error for the~4,1! mode of the symmetricsandwich plate was 6.8 percent. For asymmetric sandwich plathe error in the~4,1! modal frequency was 10.5 percent fo0.00508 cm~2 mil! case and 7.7 percent for 0.0127 cm~5 mil!case.

The viscous damping of some modes is also measuredcompared with the predicted values for the symmetric plateTable 4 and for the asymmetric plates in Table 5. Larger erroseen for the first modal loss factor implying the need for a maccurate damping model at lower frequencies. This error is adue to the well-known large impact that an imperfect clamp hon the damping at low frequency due to surface friction overclamped length.

To examine the influence of number of in-plane modes oncuracy of modal frequency estimates in bending, the numbein-plane modes,ne , is varied while keeping the number of tranverse vibration modes,nb , constant. These results are summrized in Table 6. The inclusion of the in-plane modes has a laimpact in the analysis of sandwich plates. In-plane extension ato the shearing of the dissipative layer, and therefore affectsoverall stiffness of the sandwich. Whenne51, error is extremelylarge. The first modal frequency prediction has 112 percent erOn increasingne the error in prediction is reduced. Whenne56, error in the first mode is down to 5.3 percent. Howevererrors are still high for higher modes. Oncene512, the frequen-cies of the first 6 modes agree well with the experimental resuOnly for the~1,3! mode is the error large~76 percent!. To reduceprediction error in the higher modes, more in-plane modes mbe included.

JULY 2000, Vol. 122 Õ 309

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as

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this

ted

ichh aon-singodalaly-ef-theratere-dic-han

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sticfine

When we decrease the number of bending modes fromnb525 to nb518, error for the first mode increases only from 3.9percent to 4.21 percent. Thus, the shear strain in the viscoelacore, Eq.~3!, is dominated by the extensional modes and onmildly by the bending modes. Also, the assumed bendingin-plane modes are orthonormal within themselves, but noteach other. Thus, accounting for the coupling between the exsional and bending plate modes is crucial in sandwich strucanalyses via the Galerkin assumed modes method.

5 ConclusionsAnalysis of sandwich plates with a dissipative core and isot

pic face-plates was developed and validated. Transverse sheaformation of the face layers as well as the rotatory inertianeglected. Flexural and membrane energies in the face-plateaccounted for, while the core is assumed to have shear stiffnalone. A first order shear deformation theory is used to descthe deformation in the layers. The core shear modulus is assuto have a complex value which is dependent on the frequencyprimary contribution of this paper is that a traditional Galerkassumed mode analysis of plate transverse bending wasmented with internal dissipation coordinates, using the GHmethod, to account for the frequency dependent complex modof the viscoelastic core.

We established the validity of our Galerkin assumed modanalysis by comparison to an exact solution of a sandwich pwhere the complex modulus of the viscoelastic core was toconstant over the frequency range of interest. Validation of

Fig. 2 Experimental setup for plate test.

310 Õ Vol. 122, JULY 2000

5sticlyndto

ten-ure

ro-r de-reare

essibemed. Ainaug-Mlus

eslatebeur

analysis under simply-supported boundary conditions againstexact solution@10# shows an error of,0.4 percent in the predic-tion of natural frequencies. This comparison analytically validaour assumed modes analysis model.

We also examined the practical situation where a sandwplate, clamped on all four sides, has a viscoelastic core witfrequency dependent complex modulus. Experiments were cducted on such symmetric and asymmetric sandwich plates ua piezo-actuator as a modal exciter. The frequencies and mloss factors were measured to experimentally validate our ansis. The incorporated GHM method successfully captures thefects of a frequency dependent complex shear modulus inviscoelastic core. In addition, we demonstrated that the accuprediction of the bending modal frequencies and dampingquired a large number of in-plane modes, and that these pretions were more sensitive to the number of in-plane modes tthe number of bending modes.

Nomenclature

fm 5 mth beam bending modesFk 5 kth plate bending modes~see Table 1!cm 5 mth rod extensional modesCk 5 kth plate in-plane modes~see Table 1!uj 5 in-plane displacements in face platej of x directionv j 5 in-plane displacements in face platej of y direction

uj ,k 5 kth in-plane modes contribution to the motionu in faceplate j

v j ,k 5 kth in-plane modes contribution to the motionv in faceplate j

wk 5 kth plate bending modes contribution to the motionwG! 5 complex shear modulus

~•!•

5 time derivative~•!8 5 derivative with respect tox~•!* 5 derivative with respect toy

v 5 excitation frequency

Appendix

Mass and Stiffness Matrices. The elements of the stiffnesand mass matrix for a sandwich with isotropic face-plates, igning shear in the faces, are listed here. The stiffness and mmatrices for the sandwich plate may be constructed in blockssub-matrices using the equations of motion, and mode shapethe weighting function. They may be represented as:

K5F Kww Kwu1¯ Kwv3

A A � A

Kv3w Kv3u1¯ Kv3v3

GM5F Mww Mwu1

¯ Mwv3

A A � A

M v3w M v3u1¯ M v3v3

G (30)

For the aluminum face-plate sandwiched with the viscoelacore, we show the stiffness and mass matrix as follows. We desome constant terms:

D15E1h1

3

12~12n2!

D35E3h3

3

12~12n2!

g15E1h1

2~11n!


e
g35
E3h3

2~11n!

A15E1h1

~12n2!

A35E3h3

~12n2!

d5h11h312h2

2h2

F85]F

]x

F* 5]F

]y(31)

Table 4 Experimental validation using symmetric clampedsandwich of dimensions: 67.31 cm Ã52.07 cmÃ„0.08 cm-0.00508 cm-0.08 cm …, „26.5 in.Ã20.5 in.Ã„1Õ32 in. Al-2 mil VEM-1Õ32 in. ……; GHM method is used to account for the frequencydependent property of viscoelastic core. Here n bÄ25, is thenumber of assumed plates modes in w ; n eÄ25 is the numberof assumed plate modes in u, v in each of the face plates, at 20deg.


The same is true forC. The first row of stiffness sub-matrices ar

Kww~ i , j !5EAF i@2G2

!h2d2~F j91F j** !

1~D11D3!~F j991F j**** 12F j9** !#dA

Kwu1~ i , j !5E

AF i@G2

!dC j8#dA

Kwu3~ i , j !5E

AF i@2G2

!dC j8#dA

Kwv1~ i , j !5E

AF i@G2

!dC j* #dA

Kwv3~ i , j !5E

AF i@2G2

!dC j* #dA (32)

The second row:

Ku1w~ i , j !5EAC i@2G2

!dF j8#dA

Ku1u1~ i , j !5E

AC iFG2

!

h2C j2~A1C j91g1C j** !GdA

Ku1u3~ i , j !5E

AC iF2

G2!

h2C j GdA

Ku1v1~ i , j !5E

AC iF2A1

11n

2C j8* GdA (33)

The third row:

Ku3w~ i , j !5EAC i@G2

!dF j8#dA

Table 5 Experimental validation for 67.31 cm Ã52.07 cmÃ„0.04 cm-VEM-0.08 cm … „26.5 in.Ã20.5 in.Ã„1Õ64 in. Al-VEM-1 Õ32 in. Al … asymmetric clamped sandwich plate; GHM is used toaccount for the frequency dependent property of viscoelastic core; n bÄ25, n eÄ25, at 20 deg

Table 6 Effect of the number of assumed modes on the modal predictions for the symmetric sandwich plate,67.31 cmÃ52.07 cmÃ„0.08 cm-0.00508 cm VEM-0.08 cm …, „26.5 in.Ã20.5 in.Ã„1Õ32 in. Al-2 mil VEM-1 Õ32 in. … n b isthe number of assumed plate modes in w; n e is the number of assumed plate modes in u, v in each of theface-plates, GHM method is used to account for the frequency dependent property of viscoelastic core, at 20 deg

JULY 2000, Vol. 122 Õ 311

n

In-

ed

-

er,nd

al

rs

ral

f

e

e-

of

in-3rdls

eg-sticls

-

rol

-

Ku3u1~ i , j !5E

AC iF2

G2!

h2C j GdA

Ku3u3~ i , j !5E

AC iFG2

!

h2C j2~A3C j91g3C j** !GdA

Ku3v3~ i , j !5E

AC iF2A3

11n

2C j8* GdA (34)

The fourth row:

Kv1w~ i , j !5EAC i@2G2

!dF j* #dA

Kv1u1~ i , j !5E

AC iF2A1

11n

2C j8* GdA

Kv1v1~ i , j !5E

AC iFG2

!

h2C j2~A1C j** 1g1C j9!GdA

Kv1v3~ i , j !5E

AC iF2

G2!

h2C j GdA (35)

The fifth row:

Kv3w~ i , j !5EAC i@G2

!dF j* #dA

Kv3u3~ i , j !5E

AC iF2A3

11n

2C j8* GdA

Kv3v1~ i , j !5E

AC iF2

G2!

h2C j GdA

Kv3v3~ i , j !5E

AC iFG2

!

h2C j2~A3C j** 1g3C j9!GdA (36)

Similarly, the first row of sub-blocks in the mass matrix:

Mww~ i , j !5EAF i~r1h11r3h312qrchc!F jdA (37)

The second row:

Mu1u1~ i , j !5E

AC i~r1h11qrchc!C jdA (38)

The third row:

Mu3u3~ i , j !5E


The fourth row:

M v1v1~ i , j !5E


The fifth row:

M v3v3~ i , j !5E


312 Õ Vol. 122, JULY 2000

Ku1v3, Ku3v1

, Kv1u3, Kv3u1

50

Mwu1, Mwv1

, Mwu3, Mwv3

50

Mu1w , Mu3w , M v1w , M v3w50

Mu1v1, Mu1u3

, Mu1v3, M v1u1

50

Mu3u1, M v3u1

, M v1u3, M v1v3

50

Mu3v1, M v3v1

, Mu3v3, M v3u3

50 (42)

AcknowledgmentWe thank the U.S. Army Research Office~ARO! for support

under the FY96 MURI on Active Control of Rotorcraft Vibratioand Acoustics, contract no. DAAH-0496-10334~Drs. Tom Doli-galski and Gary Anderson, technical monitors!, and for instrumen-tation support under the FY96 Defense University Researchstrumentation Program~DURIP! contract no. DAAH-0496-10301~Dr. Gary Anderson, technical monitor!.

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@8# Lu, Y. P., Killian, J. W., and Everstine, G. C., 1979, ‘‘Vibrations of ThreLayered Damped Sandwich Plate Composites,’’ J. Sound Vib.,64, No. 1, pp.63–71.

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@12# Lesieutre, G., and U. Lee., 1997, ‘‘A Finite Element for Beams Having Smented Active Constrained Layers with Frequency-Dependent ViscoelaMaterials Properties,’’SPIE Conference on Smart Structures and Materia,San Diego, CA, SPIE Paper 3045-36, pp. 315–328.

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analysis of sandwich plates with isotropic face plates and a viscoelastic core

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