Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

AIAA-97-1313

ACTIVE CONTROL OF SOUND FIELDS IN PLATE-CAVITY SYSTEMBY ANISOTROPIC PIEZOELECTRIC POLYMERS

Ki Won Yoon* and Seung Jo Kimf

Department of Aerospace Engineering, Seoul National University, Seoul 151-742, Koreaand

Wie Dae Kirn5

Department of Aerospace Engineering, Pusan National University, Pusan 609-735, Korea

The anisotropy and shape of distributed piezopolymeractuator have advantages over isotropic piezo ceramicmaterials, since these features of PVDF can beutilized as another design variable in controlapplication. This study is interested in the reductionof sound transmission through elastic plate intoulterior space by using the PVDF actuator. The plate-cavity system is adopted as a test problem. Thevibration of composite plate and the sound fieldsthrough plate are analyzed by using the coupled finiteelement and boundary element method. Somenumerical simulations are performed on soundtransmission through elastic plates. To investigate theeffects of anisotropy and shape of distributedpiezopolymer actuator, various kinds of distributedPVDF actuators are applied in sound controlsimulation for isotropic and anisotropic plates. ThePVDF actuators applied are different from each otherin their shapes and laminate angles. The results ofcontrol simulation show that the control effectivenessof distributed PVDF actuator can be enhanced byusing the coupling between shape of actuator andvibration modes of structure and the anisotropy ofpiezoelectric properties of PVDF.

Introduction

Sound from vibrating structures is an importantproblem hi numerous engineering applications. Theinterior noise levels in aircraft cabin and the soundpressure levels within the payload compartment oflaunch vehicles are representative examples.1 Themajor problem that faces aircraft or launch vehicle is

* Graduate research assistant.t Professor, Member of AIAA.§ Assistant Professor, Member of AIAA.

Copyright © 1997 by Seung Jo Kirn. Published by theAmerican Institute of Aeronautics and Astronautics,Inc. with permission

the transmission of exhausted engine noise throughfuselage sidewall structure or fairing structure. For thereduction of the interior sound levels, unprovedattenuation and absorption within vehicle structure areexpected to be required. Such passive controltechniques are likely to increase the weight ofstructures and not effective at lower frequencies.Therefore, considerable efforts have been devoted toactive control techniques to reduce low-frequencysound transmission through elastic structures duringlast decades. For structurally radiated or transmittednoise, the noise field is directly coupled withstructural motion. Therefore it is efficient to directlyapply the control inputs to the vibrating structure forthe minimization of radiated sound fields.2

As a result of rapid advances in smart structures, theresearch thrust today is toward utilizing thepiezoelectric materials as distributed sensors oractuators hi control applications. These smartmaterials were also employed as actuators or sensorsin active structural acoustic control problems.Especially, piezoelectric materials have receivedconsiderable amount of attention due to theiradvantages hi the field of structural vibration control.Polyvinylidene fluoride polymer (PVDF) has suchgood properties as flexibility, ruggedness, and lightweight. But it is not powerful as much aspiezoceramic material (PZT). Hence, compared toPZT, PVDF has usually been used as a sensor ratherthan an actuator. However, PVDF has anisotropicelectro-mechanical properties, that is, the piezo strainconstants d3i and d32 of PVDF differ by an order ofmagnitude. This anisotropic property of PVDF can beutilized in control application.

This paper is concerned with a numerical study on theactive control of sound transmitted through alaminated composite plate into a rectangular cavity byadopting PVDF as distributed actuator. The PVDFactuator bonded to both surfaces of plate has its plyangle similar to each lamina hi composite plate. Thepanel-cavity coupled system is a simplified model of

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a fluid-structure coupled phenomena such as noisetransmission through aircraft fuselage into cabin.3Sound transmission through plate structure is highlyinfluenced by the interaction of vibration modes ofstructure. The shape and directional characteristics ofPVDF actuator will be used to design an efficientactuator by considering the relationship betweenvibration modes of composite plate and controleffects from PVDF actuator.

The finite element method(FEM) is applied for theanalysis of plate structure based on the first ordershear deformation plate theory and the acoustic fieldis to be analyzed through the boundary elementmethod (BEM) based on the Helmholtz integralequation.

In order to control global acoustic noise from thestructure, the sound power is used as the objectivefunction hi control scheme. The control input iscalculated so that the sound power from compositeplate can be minimized, and then it is restricted withthe constraint that input electric field into PVDFactuator should be less than the maximum operatingfield.

Some numerical simulations are carried out on thetransmitted sound fields from composite plates. Inorder to investigate the potential use of anisotropy ofcomposite material and piezopolymer actuator,various kinds of laminated composites and distributedPVDF actuators are applied in the control simulations.

Description of the Model

Figure 1 shows the plate-cavity system that ismodeled in this analysis. It consists of a rectangularcavity(of width Lx, length Ly, and depth LZ) with thesurface at z=0 being a rectangular composite plate.

turn radiates sound power into the cavity. Theexternal sound field is a plane wave whose incidentangles are shown in Figure 2.

Elastic Plate

Figure 1. Configuration of plate-cavity model

An external sound field excites the plate, which in

/X

Figure 2. External plane wave incident on plate

The plate structure (thickness h) contains fiberreinforced composite laminae and distributedpiezoelectric actuators that are assumed to beperfectly bonded on the surface. Let the x-y planecoincide with mid-plane of the plate, with the z-axisbeing normal to the mid-plane. For a lamina, which isfiber reinforced composite or piezoelectric material,its material axes is denoted 1-2-3 axes. Thelamination angle of rth lamina, #,, is defined as theangle from x-axis to 1-axis in counterclockwisedirection along z-axis in xy-plane.

Finite Element Model for Structure

Finite element model of composite plate is based onthe first order shear deformation theory. The classicalplate theory has usually been applied in the analysisof sound fields from isotropic plates. However, it iswell established that in the analysis of compositeplates a theory that includes shear deformation isrequired.4

The displacement field {u!t u2, U)} based on a first-order shear deformation theory5, is given by

u,(x,y,z,t) = u(x,y,t)u2(x,y,z,t) = v(x,y,t)u}(x,y,z,t) = w(x,y,t)

z<px(x,y,t)z<py(x,y,t) (1)

where u, v, and w are the displacements of a point(x,y) on the mid-plane, respectively, t is time, q^ andty are the rotations of the line element, initiallynormal to mid-plane, about the y and x axes,respectively.The infinitesimal strain relations give the strainwritten as

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E= £" +ZKr '-rwhere membrane strain e°, bending strain K, and shearstrain y are expressed as

dy dy dX

-\K K- Kxi KL

r

xy ady

Through coordhiate transformation, the electro-mechanical constitutive relations for fth lamina can bewritten in x-y-z axes as

C i */ rii8 — d E (2)

where a is the stress, C1 is the elastic stiffness matrix,£* is the applied electric field intensity in z-direction,and d' is expressed in terms of piezoelectricstrain/charge coefficients djj and d32.6

d' = 'xy d'xz d'yz\

n C

00

where /? is +1 for positive poling and -1 for negativepoling, m, = cosdi, and nt = sindt. Equation (2) is thegeneral expression of constitutive relation for thepiezoelectric material, in case of composite laminathe piezoelectric strain/charge coefficients should bezero. For isotropic PZT, of which d31 and d32 areequal, d^ is always equal to dw and d^ is always zeroirrespectively of its lamination angle. Thus isotropicPZT cannot generate shear stress component. Whilethe anisotropic PVDF, of which d3I and d32 aredifferent from each other, is able to generate the shearstress component. The shear stress componentproduced by directionally attached anisotropic PVDFactuator can be taken advantage of effectively in thecontrol of sound fields from plate structure.

where N and Q are force resultant vectors, M ismoment resultant vector, which are expressed interms of inplane stresses {cr^ o ,̂ r ,̂} and transverseshear stresses {T^ r^}, and elastic coefficient matricesare obtained from elastic stiffness matrix.

External loads acting on the plate structure are thedistributed external load induced by incident soundfield, the acoustic loading by the acoustic fluidadjacent to the plate surface, and the control force bythe PVDF actuators.

The variational statement of equation of motion isobtained applying the Hamilton's variational principle.Then the weak form is discretized using four-nodedquadrilateral plate element. Finally finite elementequation of motion for integrated structure is obtainedas

= f, + f t + F E (4)

where M is mass matrix, K stiffness matrix, qdisplacement vector, fe external force vector, f» forcedue to acoustic pressure, T is the matrix, whoseelement Fy represent the ith froce component whenunit voltage is applied to _/th piezoelectric actuator,and E is the vector of electric voltage applied topiezoelectric actuators.

With the mass and stiffness matrices, the naturalfrequencies and mode shapes are found using thesubspace iteration method. Equation (4) istransformed using the modal transformation

q = 3>ri (5)

where <X> is the modal matrix and r\ is the modalcoordinate vector. Assuming the harmonic motion andmodal damping, the transformed modal coordinateequation can be written as

(6)

A = <DTK<]> ,where

I is the identity matrix, anddamping coefficients.

is the assumed modal

Boundary Element Model for Sound Fields

The laminate constitutive relations are expressed asN=AS°+BK-NM=BE°+DK-M (3)

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The acoustic sound pressure at x in acoustic domain isgoverned by the Helmholtz integral equationexpressed as

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f ]/- dp= J } l*r~J«,

(7)

where S is the surface of cavity, xs the point onsurface S, a is 1 when xs is inside S, 1/2 when x, on S,and 0 when xs outside S, and ns outward unit normalon S. The Green's function is

G(x,xs) = -

where k is the wave number defined as k = ca/c withsound speed c and forcing frequency co. At theinterface of structure and fluid, the normal derivativeof the pressure can be related to the outward normalcomponent of velocity vn on S as

(8)

When the equation (7) is discretized with equation (8)by using the boundary element technique, we obtainthe matrix equation as

Ep = GvQ (9)

where p is the vector containing the sound pressure atx and vn is the vector of normal velocity at node on S.

Four-noded quadrilateral element with the same shapefunction applied hi finite element is used for thediscretization, which promises the compatibility ofBEM with FEM through wet nodes.

Fluid-Structure Coupling

The vector of normal velocities vn in equation (9) isrelated to the vector of modal coordinate by themodal transformation and the transformation matrix Twhich transforms the nodal velocities in finite elementsystem into the normal velocity in boundary elementsystem,

= Tq = i (10)

If velocity vn is eliminated from equation (9), theresulting equation is

(H)Ep = /

and the FE equation (4) becomes

(12)

coupling is not weak enough to be ignored eventhough fluid is the air which is a kind of light fluid.Therefore two equations (11) and (12) should besimultaneously solved. In this analysis, the structuralvariable method was applied. First, structural modaldisplacement is solved with taking into account thecoupling effects of acoustic pressure. Then theacoustic pressure is calculated from equation (11)

Control of Transmitted Sound Fields

For the global minimisation of sound fields inducedby vibrating plates sound power is selected as thecontrol objective. The control is based on aimplementation of classical LMS (Least MeanSquare) algorithm.7 The anisotropic PVDF filmsbonded on the surface of plate are used as distributedactuators for sound suppression. Transmitted soundpower is defined as the integration of the soundintensity over the surface of vibrating plate:

(13)

where superscript * means the conjugate value ofcomplex quantity. We calculate the sound power byintegrating the nodal values of pressure and normalvelocity as

77 =-Re2

where

= iRe[p«RvB]

k=\s>

and superscript H means the conjugate transpose ofmatrix. Using equation (10) and (11), sound powerexpression can written as

77 = -

By eliminating the modal coordinate vector withequation (6), the sound power is written hi terms ofexternal loads and control inputs to PVDF actuatorsas

n = JRe[(fe + FE)HZ(fe + FE)] (14)

where A is the matrix converting acoustic pressure whereinto force vector.

In case of plate-cavity system, the fluid-structure1657

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Z = <a2(DH-H0TTTGHE-HRT*H-/«T.

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Since there exist the physical limit for electric fieldapplied to piezoelectric material, the input electricfield into piezoelectric actuator should be constrained.Therefore minimization problem of sound powerexpressed as equation (18) has the constraint that thecontrol input should be less than the allowable limitof each PVDF actuator. The problem can be stated as

Minimize - Re[(fe + FE)H Z(fe + FE)]

means the piezoelectric lamina and the +/-- signmeans the poling direction of each lamina.

Subject to <lk, = \,---, nA

where lk is the allowable limit of input for Ath actuatorand nA is total number of actuators. This minimizationproblem is approached by using the widely usedprogram ADS(Automated Design Synthesis), so thatthe optimal control input can be obtained.

Numerical Examples

In this section the results of numerical simulations arepresented. These analyses involve the active controlof sound transmission through a rectangularintegrated plate supported on the edges of rigid cavity.The acoustic medium is air whose density is 1.21Kg/m3 and wave velocity is 343 m/sec. The coupledFEM-BEM analyses are carried out for the cases ofisotropic aluminum plate and anisotropic compositeplate. PVDF actuators used for the control of soundtransmission are varied in their shape and laminationangle to demonstrate some features of the actuators.In the analysis, the small amount of modal damping(£, = 0.001) is used for dynamic response of structure.

The actuators used in this analysis are PVDF pairsbonded with positive poling at the top surface andwith negative poling at the bottom surface. Theshapes of PVDF actuators considered here aredepicted in Figure 3. The area filled with gray color isthe region of the actuator: the first one (Figure 3(a)) isnamed A1C and the second one (Figure 3(b)) isnamed A1P. These PVDF pairs generate distributedbending moments in x and y direction when its plyangle is 0°, for the PVDF on top surface expands andthat on bottom surface shrinks. The PVDF pairsdirectionally bonded with their own ply anglegenerate not only distributed bending moments in xand y directions but also distributed twisting momentsdue to the effects of ply angle.

Ply angles of PVDF layers on top and bottom surfacesare the same for an actuator. AlPa means the actuatorA1P bonded at 0P+ on the top surface and bonded at0p. on the bottom surfaces, where the subscript p

(a) A1C

(b)AlP

Figure 3. The shape of PVDF actuators usedin the control simulation.

In this analysis the dimension of rigid cavity is L^ =0.3 m, Ly = 0.2 m, and Lz = 0.4 m. The rigid cavitysystem is modeled with 4-node boundary elements.The number of finite elements for plate structure is 72and that of boundary elements for rigid cavity is 432.

Isotropic Plate

An integrated structure consists of aluminum hoststructure and PVDF materials bonded on the top andbottom surfaces of plate. Plate has the lay-up of[Q^fi/Qp.], in which I represents the isotropic material,the subscript p represents the PVDF material. Theplate is simply supported along the edges at x = ± LJ2and y= ±Zy2. Material properties of PVDF film areshown in Table 1 and those of aluminum are: Young'smodulus 70X109 N/m2, Poisson's ratio 0.3, anddensity 2700 Kg/m2.

Control simulations with distributed PVDF actuatorsare performed. Actuators A1C0, A1P0, and A1P45 areused in the analysis. First, actuator A1C0 is employedhi sound transmission control of aluminum plate. Thethickness of plate is h = 0.002 m. The magnitude ofexternal sound wave is 3 N/m2 and incident angles areV = 45° and x = 45°.

The transmitted sound power (TSP) is shown inFigure 4 and average mean square velocity of platestructure is shown in Figure 5. The average mean

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square (AMS) velocity, defined as

dS,

a measure of the velocity of structural vibration.8Comparing TSP and AMS velocity of uncontrolledsystem, there are peaks at 440 Hz and 870 Hz in TSPwhile no peak at these frequency in AMS velocity.The acoustic field of plate-cavity in these frequencyranges is not involved with the resonance of structuralsystem. When an actuator AICo only used, TSP at theresonances of even modes of plate cannot becontrolled. The actuating forces of actuator A1C isnot coupled with even vibration modes of structuredue to its symmetrical shape. Thus TSP peaksinduced by these even vibration modes are notremoved. From 440 Hz to 580 Hz, TSP is reduced bysound control but AMS velocity is increased.3 Atfrequencies above 700 Hz, TSP is reduced whileAMS velocity is not changed.

100

—, 80mS. 60

oQ.

oCO

40

20

0

-20200 400 600 800

Frequency [Hz]1000

Figure 4. Transmitted sound power of isotropic platein case of actuator A1C0.

AVv

0 200 400 600 800 1000

Frequency [Hz]

Figure 5. Average mean square velocity of isotropicplate in case of actuator A1C0.

Actuators A1P0 and A1P45 are used in control

simulation of aluminum plate. The thickness of plateis h = 0.0012 m. The magnitude of external soundwave is 2 N/m2 and vy = 60° and % - 30°.

OQ.

OCO

100

60

40

20

0

UncontrolledControlled (A1P0)Controlled (A1P..)

200 400 600 800

Frequency [Hz]1000

Figure 6. Transmitted sound power of isotropic platein case of actuator A1P0 and A1P45.

TSP of the plate-cavity system is shown in Figure 6.The shape actuator A1P is different from that of A 1C.With these actuators, transmitted sound power isreduced at most frequencies. However, the area ofactuator A1P is not large enough to control the firstvibration mode. Actuator A1P0 cannot control the 6thand 1th peaks of TSP due to its lack of control forcefor these vibration modes. Actuator A1P45 which isattached with ply angle 45° can control these TSPpeaks, for the lamination angle of distributed PVDFactuator invokes the coupling between the controlforce and the acoustic field hi cavity.

Table 1. Material properties of Graphite/ Epoxyand PVDF film

Graphite/EpoxyE,, =E22 =G12 =G,3 =0,3 =

Vl2 =t =

P =

181. Gpa10.3 Gpa7. 17 Gpa7. 17 Gpa2.87 Gpa0.280.125mm1520 Kg/m3

PVDF filmE =v =P =d31 =

t =

2. GPa0.31780 Kg/m3

23.xlQ-12V/m3. X 1Q-12 V/m110X lO^m

Laminated Composite Plate

In this section, composite plate and PVDF actuatorswith their own shapes and laminate angles areconsidered to investigate the effects of shape andanisotropy of PVDF. The composite plate is made ofGraphite/Epoxy fiber-reinforced composite whoselamination sequence is [O^SJ-AS^Q^, in which thesubscript c represents the composite material. The

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material properties of Graphite/ Epoxy composite andPVDF film are shown in Table 1. The allowablevoltage input for this PVDF is 1100 volts. Thedimension of host structure is 0.3 x 0.2 x 0.001 m.The harmonic incident plane wave is at \|/ = 30° and x= 60°.

Two actuators A1P0 and A1P45 are employed incontrol simulation. These actuators represent similarlygood effectiveness for most frequencies as shown inFigure 7. However the actuator A1P45 is moreeffective in the sense that it can control TSP from 550Hz to 600 Hz. In this region, sound power is relatedto the 7th vibration mode of plate and A1P45 controlthis mode better than A1P0. This enhancement ofcontrol effectiveness results from the anisotropiccharacteristics of PVDF.

100

2- eo

o0-

C3O

40

20

0

-20

- Uncontrolled-Controlled (A1P0)- Controlled (A1P4S)

200 400 600 800

Frequency [Hz]1000

Figure 7. Transmitted sound power of composite platein case of actuator A1P0 and A1P45.

Conclusion

highly coupled with vibration modes of compositeplates. Therefore it can be taken advantage of toimprove the effectiveness hi acoustic noise control.Furthermore the effects due to coupling betweenvibration modes and actuating forces by distributedPVDF actuator could be made better, withoutincreasing the number of actuators, by changing thelamination angle of the actuator with the determinedshape.

References

1. Koshigoe, S., Teagle, A. T. , Tsay, C., Morishita,S., and Une, Soichro, "Numerical simulation onsound transmission through elastic plate," J.Acoust. Soc. Am., Vol. 99 (5), 1996, pp. 2947-2954.

2. Fuller, C. R., "Active control of soundtransmission /radiation from elastic plates byvibration inputs: I. Analysis," Journal of Soundand Vibration, Vol. 136 (1), 1990, pp. 1-15.

3. Pan, J., Hansen, C. H., and Bies, D. A., "Activecontrol of noise transmission through a panel intoa cavity: I. Analytical study," J. Acoust. Soc. Am.,Vol. 87 (5), 1990, pp. 2098-2158.

4. Chandrashekhara, K. and Agarwal, A. N., "Activevibration control of laminated composite platesusing piezoelectric devices: a finite elementapproach," Journal of Intelligent MaterialSystems and Structures, Vol. 4, 1993, pp. 496-508.

By using the anisotropic features and shape ofdistributed polyvinylidene fluoride(PVDF) actuator,numerical investigations on the control of soundtransmission of plate-cavity system was performed.Fluid-structure interaction and the ulterior soundfields are efficiently calculated with coupled FEM-BEM approach. For simply supported isotropic andlaminated composite plate, numerical controlsimulation was performed with distributed PVDFactuators. Actuator A1C0 has large control force.However it cannot control the sound transmission atthe frequencies related to the anti-symmetric vibrationmodes of plate. Actuators A1P0 and A1P45 whichhave different shape from A1C can control mostfrequencies. Especially actuator A1P45 shows thebetter performance from the viewpoint of controleffectiveness. The coupling effects between interiorsound field and actuating force by PVDF actuator isincreased by the directionally attached PVDF on thesurface of plate. The shape of distributed actuator is

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8.

Ochoa, O. O. and Reddy, J. N., Finite ElementAnalysis of Composite Laminates, KluwerAcademic Publishers, Dordrecht, The Netherlands,1992.

Lee, C. K., 'Theory of laminated plates for thedesign of distributed sensors/actuators. Part I:governing equations and reciprocal relationships,"J. Acoust. Soc. Am., Vol. 87 (3), 1990, pp. 1144-1158.

Nelson, P. A., Curtis, A. R. D., Elliott, S. J., andBullmore, A. J., 'The active minimization ofharmonic enclosed sound fields, Part I: Theory," J.Sound Vib., Vol. 117 (1), 1987, pp. 1-13.

Fahy, F., Sound and Structural Vibration:Radiation, Transmission, and Response,Academic Press Inc., London, 1985.

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