modeling of piezolaminated composite shells for vibration control

7/29/2019 Modeling of Piezolaminated Composite Shells for Vibration Control

1/6

1

MODELING OF PIEZOLAMINATED COMPOSITE SHELLS FOR VIBRATION CONTROL

V. Piefort, N. Loix, A. Preumont

Active Structures Laboratory, ULB - CP 165/42Av F.D. Roosevelt 50, B-1050 Brussels, Belgium

(eMail: [email protected], http://www.ulb.ac.be/scmero)

ABSTRACT

This paper develops the theory of piezolaminatedshells. The fundamental equations governing theequivalent piezoelectric loads and sensor output arederived. The reciprocity between piezoactuation and

piezosensing is pointed out. Piezoelectric shell finiteelements are developed based on Mindlin elements.Different electrical boundary conditions are exam-ined.

Key words: vibration control; piezoelectricity.

1. INTRODUCTION

The use of piezoelectric materials as actuators andsensors for noise and vibration control has beendemonstrated extensively over the past few years (a

review can be found in (Preumont, 1997)).

The design of control systems involving piezoelectricactuators and sensors requires an accurate knowl-edge of the transfer functions between the inputs andthe outputs of the system. These are not easy todetermine numerically, particularly for shell struc-tures with embedded distributed actuators and sen-sors. The situation where they are nearly collocatedis particularly critical, because the zeros of the trans-fer functions are dominated by local effects which canonly be accounted for by finite elements. This moti-vates the present study.

The finite element modeling of plates and shells withembedded piezos has received a considerable inter-est in recent years. Stacking of quadrangular solidpiezoelectric elements was considered in (Tzou andTseng, 1990) and later in (Heyliger et al., 1996). Asolid multilayered thin brick was proposed in (Haet al., 1992). The analysis of piezoelectric shells asa layerwise assembly of solid piezoelectric triangu-lar elements quadratic in-plane and linear throughthe thickness is proposed in (Tzou and Ye, 1996).These are tridimensional approaches and result in

large problem sizes, requiring techniques such as theGuyan reduction to reduce the number of degreesof freedom. A multilayered composite piezoelectric

Kirchhoffplate is derived in (Hwang and Park, 1993).To accomodate thick as well as thin shell structures,a 4-node bilinear Mindlinshell is proposed in (Sule-man and Venkayya, 1995); a reduced integrationof the stiffness matrix is used to avoid the trans-verse shear locking phenomenon for thin shells. Thepresent study is also based on a Mindlinplate; how-

ever, the element used in the present study follows(Batoz and Dhatt, 1990) and does not require a re-duced integration.

2. CONSTITUTIVE EQUATIONS

2.1. General

The constitutive equations of a linear piezoelectricmaterial read (IEEE std, 1988).

{T} = [cE]{S} [e]T{E} (1){D} = [e]{S} + [S]{E} (2)

or alternatively

{S} = [sE]{T} + [d]T{E} (3)

{D} = [d]{T} + [T]{E} (4)

where {T} = {T11 T22 T33 T12 T13 T23}T

is the stress

vector, {S} = {S11 S22 S33 S12 S13 S23}T

the defor-mation vector, {E} = {E1 E2 E3} the electric field,{D} = {D1 D2 D3} the electric displacement, [c] and

[s] the elasticity constants matrices, [] the dielectricconstants, [d] and [e] the piezoelectric constants. (su-perscripts E, S et T indicate values at E, S and Tconstant respectively)

2.2. Single Layer in Plane Stress

We consider a shell structure with embedded piezo-electric patches covered with electrodes. The piezo-electric patches are parallel to the mid-plane andorthotropic in their plane. The electric field and

electric displacement are assumed uniform across thethickness and aligned on the normal to the mid-plane(direction 3). With the plane stress hypothesis, the


2/6

2

constitutive equations can be reduced to

T11T22T12

=

cE

S11S22

2S12

e31e32

0

E (5)

D = {e31 e31 0} {S} + SE (6)

where

cE

is the stiffness matrix of the piezoelectricmaterial in its othotropy axes. In writing Equ.(5),it has been assumed that the piezoelectric principalaxes are parallel to the structural othotropy axes.The analytical form of

cE

can be found in any text-book on composite materials.

2.3. Laminate

A laminate is formed from several layers bonded to-gether to act as a single layer material (Figure 1); thebond between two layers is assumed to be perfect, sothat the displacements remain continuous across thebond.

Figure 1: Multilayered material

2.3.1. Kirchhoff Plate

According to the Kirchhoffhypothesis, a fiber normalto the mid-plane remains so after deformation. Itfollows that:

{S} = {S0} + z {} (7)

where {S0} is the mid-plane deformation and {},the mid-plane curvature. The constitutive equationsfor layer k in the global axes of coordinates read

{T} =

Qk

{S} [RT]1

k

e31e32

0

k

Ek (8)

Dk = {e31 e32 0}k [RS]k {S} + kEk (9)

where [RT]1

k is the transformation matrix relating

the stresses in the local coordinate system (LT) tothe global one (xy). Similarly, [RS]k is the trans-formation matrix relating the strains in the global

coordinate system (xy) to the local one (LT). Thestiffness matrix of layer k in the global coordinatesystem,

Qk

, is related to

cEk

by

Q k = [RT]1

k cE

k [RS]k (10)As mentioned before, the electric field Ek is assumeduniform across the thickness hk = zk zk1 of thelayer. Thus, we have Ek = k/hk, where k is thedifference of electric potential between the electrodescovering the surface on each side of the piezoelectriclayer k.

The global constitutive equations of the laminate,which relate the resultant in-plane forces {N} andbending moments {M}, to the mid-plane strain {S0}and curvature {} and the potential applied to thevarious electrodes can now be derived by integrating

Equ.(8) over the thickness of the laminateNM

=

A BB D

S0

+ (11)

nk=1

zkzk1

I3

z I3

[RT]

1

k

e31e320

k

khk

dz

or NM

=

A BB D

S0

+ (12)

n

k=1

I3zmk I3

[RT]1k

e31e32

0

k

k

where

zmk =zk1 + zk

2(13)

is the distance from the mid-plane of layer k to themid-plane of the laminate. The first term in the righthand side of Equ.(12) is the classical stiffness ma-trix of a composite laminate, where the extensionalstiffness matrix [A], the bending stiffness matrix [D]and the extension/bending coupling matrix [B] arerelated to the individual layers according to the clas-

sical relationships:

[A] =k

Qk

(zk zk1)

[B] =1

2

k

Qk

(z2k z2

k1)

[D] =1

3

k

Qk

(z3k z3

k1) (14)

The second term in the right hand side of Equ.(12)expresses the piezoelectric loading. Similarly, substi-tuting Equ.(7) into Equ.(9), we get

Dk = {e31 e32 0}k [RS]k [I3 z I3]

S0

k

khk

(15)


3/6

3

Since we have assumed that the electric displacementDk is constant over the thickness of the piezoelectriclayer, this equation can be averaged over the thick-ness, leading to

Dk = {e31 e32 0}k [RS]k [I3 zmkI3]

S0

k

hkk (16)

2.3.2. Mindlin Plate

The classical Kirchhofftheory neglects the transverseshear strains. Alternative theories which accomo-date the transverse shear strains have been developedand have been found more accurate for thick shells(Hughes, 1987). In the Mindlinformulation, a fibernormal to the mid-plane remains straight, but nolonger orthogonal to the mid-plane. Upon introduc-

ing the transverse shear strains {} = {xz yz}Tand the transverse shear loads {T} = {Txz Tyz}

T,the global constitutive equations of the piezoelectricMindlinshell become

NMT

=

A B 0B D 0

0 0 K

S0

+n

k=1

I3zmk I3

0

[RT]1k

e31e320

k

k (17)

Dk = {e31 e32 0}k [RS]k [I3 zmkI3 0]

S0

khk k(18)

where the load vector now includes the in-plane loads{N} and moments {M} and the transverse shearloads {T}. Similarly, the strain vector includes themid-plane membrane strains {S0}, the mid-planecurvatures {} and the transverse shear strains {}.In Equ.(18), the stiffness matrices A, B & D aregiven by Equ.(14) while the transverse shear stiffnessmatrix K is obtained following the method describedin (Batoz and Dhatt, 1990). We note that the trans-verse shear and the piezoelectric effect are entirelydecoupled.

3. ACTUATION AND SENSING

3.1. Actuation: Equivalent Piezoelectric Loads

Equation (17) shows that a voltage applied be-tween the electrodes of a piezoelectric patch producesin-plane loads and moments:

NM

=

I3

zm I3

[RT]

1e

31

e320

(19)

If the piezoelectric properties are isotropic in theplane (e31 = e32), we have

e31 [RT]1

11

0

= e31

11

0

(20)It follows that

{N} =

NxNy

Nxy

= e31

110

(21)

{M} =

MxMy

Mxy

= e31zm

110

(22)

We note that the in-plane forces and the bendingmoments are both hydrostatic; they are independantof the orientation of the facet. We therefore con-

clude that the piezoelectric loads result in a uniformin-plane load Np and bending moment Mp actingnormally to the contour of the electrode as indicatedFigure 2:

Np = e31, Mp = e31zm (23)

where zm is the distance from the mid-plane of thepiezoelectric patch to the mid-plane of the plate.

Figure 2: Piezoelectric load

3.2. Sensing

Consider a piezoelectric patch connected to a chargeamplifier as in Figure 3. The charge amplifier im-

poses = 0 between the electrodes and the outputvoltage is proportional to the electric charge:

out = Q

Cr=

1

Cr

D d (24)

where D is given by Equ.(18). If the piezoelectricproperties are isotropic in the plane (e31 = e32), wehave

e31 {1 1 0} [RS] = e31 {1 1 0} (25)

and Equ.(24) becomes

out = e31Cr

S

0

x + S0

y d+zm

(x + y) d

(26)


4/6

4

Figure 3: Piezoelectric sensor

The first integral represents the contribution of theaverage membrane strains over the electrode and thesecond, the contribution of the average bending mo-ment. Using the Green integral

.a d =

C

a.n dl (27)

the foregoing result can be transformed into

out = e31

Cr

C

u0.n dl + zm C

w

n

dl (28)where the integrals extend to the contour of the elec-trode. The first term is the mid-plane displacementnormal to the contour while the second is the slopeof the mid-plane in the plane normal to the contour(Figure 4). It is worth insisting that for both the ac-tuator and the sensor, it is not the shape of the piezo-electric patch that matters, but rather the shape ofthe electrodes.

Figure 4: Contribution to the output of the piezoelec-tric isotropic sensor (e31 = e32)

4. FINITE ELEMENT FORMULATION

The dynamic equations of a piezoelectric continuumcan be derived from the Hamiltonprinciple, in which

the Lagrangian and the virtual work are properlyadapted to include the electrical contributions as wellas the mechanical ones. The potential energy den-sity of a piezoelectric material includes contributionsfrom the strain energy and from the electrostatic en-

ergy (Tiersten, 1967).

H =1

2

{S}T{T} {E}T{D}

(29)

Similarly, the virtual work density reads

W = {u}T{F} (30)

where {F} is the external force and is the elec-tric charge. From Equ.(29) and (30), the analogybetween electrical and mechanical variables can bededuced (Table 1).

Mechanical Electrical Force {F}Displ. {u}Stress {T}Strain {S}

Charge Voltage{D} Electric Displ.{E} Electric Field

Table 1: Electromechanical analogy

The variational principle governing the piezoelectricmaterials follows from the substitution of H and Winto the Hamiltonprinciple (Allik and Hughes, 1970).

For the specific case of the piezoelectric shell, we canwrite the potential energy

H =1

2

ST0 T T

NMT

E D

d (31)

Upon substituting Equ.(17) and (18) into Equ.(31),one gets the expression of the potential energy for apiezoelectric Mindlin shell. The element used is theMindlin shell element from Samcef (Samtech s.a.).The electrical degrees of freedom are the voltages

k across the piezoelectric layers; it is assumed thatthe potential is constant over each element (this im-plies that the finite element mesh follows the shape ofthe electrodes). Introducing the matrix of the shapefunctions [N] (relating the displacement field to thenodal displacements {q}), and the matrix [B] of theirderivatives (relating the strain field to the nodal dis-placements), into the Hamilton principle and inte-grating by part with respect to time, we get

0 = {q}TV

[N]T[N]dV {q}

+ {q}T

[B]T A B 0B D 0

0 0 K

[B]d {q}


5/6

5

+ {q}T

[B]T

... E

Tk ...

... ETk zmk ...

... 0 ...

d

...k

...

+ {... k ...}

...

.

.....

Ek Ekzmk 0...

......

[B]d {q}

+ {... k ...}

. . . 0k/hk

0. . .

d

...k

...

{q}T

[N]T {PS}d {q}T{Pc}

+ {... k ...} {... k ...}T

(32)

where we have introduced

{E}k = {e31 e32 0}k [RS]k (33)

and we have used the fact that

[RT]1

k {e31 e32 0}Tk = {E}

Tk (34)

[PS] and [Pc] are respectively the external dis-tributed forces and concentrated forces and{... k ...} {... k ...}

T=

k kk is the electricalwork done by the external charges k brought to

the electrodes.

Equation (32) must be verified for any {q} and {}compatible with the boundary conditions; It followsthat, for any element, we have

[Mqq ]{q} + [Kqq ]{q} + [Kq]{} = {f} (35)

[Kq] {q} + [K]{} = {g} (36)

where the element mass, stiffness, coupling andpiezoelectric capacitance matrices are defined as

[Mqq ] = V

[N]T[N]dV (37)

(38)

[Kqq ] =

[B]T

A B 0B D 0

0 0 K

[B]d (39)

[Kq] =

[B]T

... E

Tk ...

... ETk zmk ...

... 0 ...

d (40)

[K] =

. . . 0k/hk

0

. ..

(41)

[Kq] = [Kq]T (42)

and the external mechanical forces and electriccharge:

{f} =

[N]T {PS}d + {Pc}

{g} = {... k ...}T

The element coordinates {q} and {} are related tothe global coordinates {Q} and {}. The assemblytakes into account the equipotentiality condition ofthe electrodes; this reduces the number of electricvariables to the number of electrodes. Upon carry-ing out the assembly, we get the global system ofequations

[MQQ]{Q} + [KQQ]{Q} + [KQ]{} = {F} (43)

[KQ] {Q} + [K]{} = {G} (44)

where the global matrices can be derived in astraightforward manner from the element matrices(37) to (42). As for the element matrices, the global

coupling matrices satisfy [KQ] = [KQ]T

.

5. ELECTRICAL BOUNDARY CONDITIONS

Equations (43) and (44) couple the mechanical vari-ables {Q} and the electrical potentials {} betweenthe electrodes of the piezoelectric patches; {F} repre-sents the external forces applied to the structure and{G} the electric charges brought to the electrodes.

5.1. Voltage Driven Electrodes

If the electric potential {} is controlled, the gov-erning equations become

[MQQ]{Q} + [KQQ]{Q} = {F} [KQ] {} (45)

where the second term in the right hand side rep-resents the equivalent piezoelectric loads. Once themechanical displacements have been computed, theelectric charges appearing on the electrodes can becomputed from Equ.(44). From Equ.(45), we seethat the eigenvalues problem of the system withshort-circuited electrodes ({} = 0) is:

[KQQ]

2[MQQ]

{Q} = 0 (46)

5.2. Charge Driven Electrodes

Conversely, open electrodes correspond to a charge

condition {G} = 0. In this case, Equ.(44) becomes

{} = [K]1[KQ]{Q} (47)


6/6

6

and, upon substituting into Equ.(43), we get

[MQQ]{Q} + [K]{Q} = {F} (48)

with

[K] = [KQQ] [KQ][K]1[KQ] (49)

which shows that the stiffness matrix depends on theelectrical boundary conditions.

5.3. Electrodes Connected via a Passive Network

If the electrodes are connected via a passive electri-cal network of impedance matrix [Z], Equ.(43) and(44) must be complemented by the network equation(Figure 5). Since the electrical current is given by thetime derivative of the electrical charge {I} = {G},

we have, in Laplacenotations:

{} = [Z]{I} = [Z(s)]s{G} (50)

Figure 5: Electrodes connected to an impedance

6. CONCLUSION

The theory of piezolaminated plates has been de-veloped; the fundamental equations governing theequivalent piezoelectric loads of a piezoelectric ac-tuator and the output of a piezoelectric sensor havebeen derived. Numerous validation benchmarks havebeen carried out. Some are described in (Piefort andPreumont, 1998).

ACKNOWLEDGMENTS

This study is supported by the regional govern-ment of Wallonia, (DGTRE); The support of theIUAP-4-24 on Intelligent Mechatronic Systemsis alsoacknowledged. The technical assistance of Samtechs.a. is deeply appreciated.

REFERENCES

Allik, H., Hughes, T. J. R., 1970, Finite element

method for piezoelectric vibration, InternationalJournal for Numerical Methods in Engineering,2:151157.

Batoz, J.-L., Dhatt, G., 1990, Modelisation desstructures par elements finis, Volume 2: poutreset plaques, Hermes, Paris.

Ha, S. K., Keilers, C., Chang, F. K., 1992, Finite

element analysis of composite structures contain-ing distributed piezoceramic sensors and actua-tors, AIAA Journal, 30(3):772780.

Heyliger, P., Pei, K. C., Saravanos, D., 1996,Layerwise mechanics and finite element modelfor laminated piezoelectric shells, AIAA journal,34(11):23532360.

Hughes, T. J. R., 1987, The Finite Element Method,Prentice-Hall International Editions.

Hwang, W. S., Park, H. C., 1993, Finite elementmodeling of piezoelectric sensors and actuators,AIAA Journal, 31(5):930937.

IEEE std, 1988, IEEE Standard on Piezoelectricity ANSI/IEEE Std 176-1987.

Piefort, V., Preumont, A., 1998, Finite element mod-eling of piezolaminated composite shells, Samcefdays, 21-22 october 98, Liege.

Preumont, A., 1997, Vibration Control of ActiveStructures - An Introduction, Kluwer AcademicPublishers.

Suleman, A., Venkayya, V. B., 1995, A simple fi-nite element formulation for a laminated compos-ite plate with piezoelectric layers, Journal of Intel-ligent Material Systems and Structures, 6:776782.

Tiersten, H. F., 1967, Hamiltons principle for linearpiezoelectric media, in Proceedings of the IEEE,pp. 15231524.

Tzou, H. S., Tseng, C. I., 1990, Distributed piezo-electric sensor/actuator design for dynamic mea-surement/control of distributed parameter sys-tems: a piezoelectric finite element approach,

Journal of Sound and Vibration, 138(1):1734.Tzou, H. S., Ye, R., 1996, Analysis of piezoelas-

tic structures with laminated piezoelectric triangleshell elements, AIAA Journal, 34(1):110115.

modeling of piezolaminated composite shells for vibration control

Documents