Computational Methods for Smart Structures and Materials · 2014-05-19 · Computational Methods...

On the moderation of the piezoelectric effect

with control system and some applications

M. de Benedetti & R. Barboni

Universita di Roma "La Sapienza", Dipartimento

Aerospaziale, Via Eudossiana, 16,1-00184, Roma, Italy

E-mail: root@pcdebenedetti. ing. uniromal. it

Abstract

The conceptual approach to the modeling of the piezoelectric constitutiverelation with MSC/NASTRAN in one-dimensional static and dynamic cases isdiscussed. The role of the Finite Element Model of the piezoelectric material isto represent the actuator and the sensor of an adaptive control system for theactive damping of structural vibrations. Open and closed loop control systems,the amplifier and the piezo saturation functions are also modellised. For manypiezoelectric materials the constitutive relation can be represented with a systemof nine equations, composed by the six elasticity equations of the classicconstitutive theory, coupled with three additional electrical equations. In thisstudy, the reduction to a system of two second order differential equations isperformed, keeping into account the coupling role of the electro-mechanicalcharacteristics. Such procedure is applied to control models of simple structureswith a distribution of piezoelectric micro-actuators for different applications.

1 Introduction

In recent years a growing interest has been shown in the study of actuatorsand sensors which enable structures to adapt their shape and/or vibrationcharacteristics to applied external loads or their internal stress or strainstate, or their vibration state (Rogersfl]). Such active elements can be

made with piezoelectric materials and in particular with piezoceramics(PZT), capable to generate a difference of electric potential if subjected toa mechanical force (direct piezoelectric effect) or, vice-versa, to apply amechanical force if subjected to an electric field (inverse piezoelectric

Transactions on the Built Environment vol 35, © 1998 WIT Press, www.witpress.com, ISSN 1743-3509

Computational Methods for Smart Structures and Materials

effect) (Crawley[2]). They can easily be installed on a structure, by

bonding or embedding. Two main groups of design parameters are present

in the structural vibration control problem: the first is composed by the

actuators and sensors dimensional and electromechanical characteristics,the second includes gain, robustness and control system transfer function

coefficients.

2 Analytical model

A piezoelectric homogeneous three dimensional continuum is given, it

occupies a closed region \j/ of the 3D space, delimited by a closed surfaceS, with local normal unit vector versn in the outer direction. The

boundary surface Softy can be considered divided in two separate parts

using with two different criteria:

(1)

(2)

VPexj/ the following relations hold:mechanical equilibrium relations:

<#vc7+P=0 (3)

kinematics relations:

F = j(vu + V,u) (4)

first Maxwell equation for the quasi-static electric field within an isotropic

homogeneous medium:d,vD = #,=0 (5)

definition of electric voltage:

E = - VF = -grad V (6)

The constitutive equations for the piezoelectric material are:

inverse effect:

a = £*£-t,E (7)

direct effect:

D=ff + f*E (8)

such equations can be expressed in the following equivalent form:

inverse effect:

£ = fV + E (9)

direct effect:



E (10)

The boundary conditions are:

natural mechanic on Sf. o • vers n - f * = surface force (11)

natural electric on Sq. vers n D = q * = surface electric charge ( 1 2)

essential mechanic on £„: u = u * = imposed displacement (13)essential electric onSy.V = V*= imposed electric potential (14)

The following material symmetry relations are valid:

(16)

A system of 22 equations with 22 unknowns (%,, $,, q,, F, £,, A(ij= 1,2,3)) is obtained. In the vector representation of the strain and stresstensors, the constitutive relations become:

a = C*?-I,E (18)

D = S? + pE (19)

or equivalent:

£ = £*a + d,E (20)

D = HcF + f*E (21)

and the system composed by eqns (20) and (21) can be expressed in thefollowing matrix form (Hagood[3]):

d,

r E(22)

where the matrix % ° that couples vectors D and E (with constant stress)

is experimentally measured.

The piezoelectric strain matrix d couples vectors D and a . Its

transposed matrix d, couples the electric field vector E with the strain

vector s . The matrix S * couples the strain vector s and the stress vector

a . This matrix is experimentally measured with a constant electric field E.

It is possible to obtain the three components of the electric field vector intothe piezoelectric element, starting from the voltage difference applied on thefaces of a parallelepiped shaped piezo, with dimensions a, 6, c, (Figure 1)through the following relation (BarboniflO]):



Eqn

G

G, X =

D

can

CE^12CE13

0

0

0

0

0

be wnoE

oE22

0

0

0

0

0

tteniioE

33

0

0

0

0

0

"1, 0 0"

0 \2 0

0 0 lj

i the followi

0 0

0 0

0 0

855 0

0 Sfj

0 0

0 d,5

0 0

P'l

•fel

ng explicit

0 0

0 0

0 0

0 0 i

0 d,,

$66 0

o %r

0 0

0 0

form

0 (

0 £

0 c

3,5

0

0

V

0

IB,"

1?

11

0

0

0

0

0

• •

E,

£2

£3,

(23)

(24)

The system (22) can be expressed in semi-inverse form, with the

following procedure:

#5)

(26)

(27)

If forces and displacements are only one-dimensional (along the xaxis) these equations are equivalent to the "pin-force" model of Strambi[4]

and Barboni[5]; if the electric field is applied only in the z direction, in eqn

(24) rows from the 2 to 8* can be deleted, obtaining:

" ""*~ '^ (28)

P -£^ -4

er,(29)

From the first of eqns (29):



1 -*31y"31 A3 31

after the integration, the expression of the axial force Nj is obtained:

& 01&1 J I 1 7-^

(30)

(31)31

where Si is the area of the cross-section of the piezoelectric normal to the:

axis: S;=/2- &; &= li - &; 83= l\- h (Figure 1);t*' deformed

configuration

Figure 1. Piezoceramic element.

From the definition of strain and electrical displacement:

The

>the

bi — -, LJ? — —1, 83 l,-l,

normal force is:

N *z '*) Al ^rU _1^1 / 2 \ ZAij / v L{

(sf.-^-J-i,

following positions:

If *2'*3 • A , ^31 "Is

i-)/;

(33)

n&\

%3

and indicating with Fthe axial force Nxi, and with x the displacement MI

between the two extremes of the piezo, due to its shape variation, eqn (33)becomes:

F = kx + Aq (35)

In the dynamic case, eqn (35) can be generalised, obtaining:mx+bx+hc + Aq = F (36)

Eqn (36) represents the dynamic behaviour of a one-degree of freedom

discrete system with a mass m subjected to a force F, connected to a rigid



support by an elastic spring with axial stiffness k, and by a damper with

viscous damping coefficient b. The difference between eqn (36) and the

differential eqn describing a classic mass-spring-damper dynamic systemconsists in the presence of the coefficient A, coupling the charge q and theforce F; this means that the "structural" variable x and the electricalvariable q are not independent in the piezo system.

The following eqn (37) corresponds to eqn (36), with electrical

quantities (inductance L, resistance R, capacity C) of the piezo subjected to

a voltage V.

Lq + Rq + — q + Ax = V (37)C

This equation is similar to the one describing an electric RLC series

circuit. The difference consists in the presence of the coefficient A,

coupling the displacement x and the electric potential V.The system of eqns (36) and (37) is the piezo constitutive relation, in

terms of equilibrium equations, and can be represented in the following

matrix formulation, using the Laplace transform:

mp(38)

3 Numerical model

To perform the calculation with numerical data representative of a realpiezo, a PZT actuator previously used during experimental tests has beenconsidered (Gaudenzi[6], and Friedmann[7]), obtaining the following data:

/3i =7.257-l(

f =1.6112-10-* <%

l =1.895-10^^/2/m

•ml AM

L =5.08-10-'m

=2.54-HP* m (39)

the coefficients of relation (38) can be obtained from the coefficients of eqn

(28) through relations (34):

: = 1.081971.10? &{ otherwise: P /,1(40)

The piezo element mass m is:

otherwise:

= 4.9- (41)



The viscous damping coefficient b can be obtained from the assumed

value of the damping factor £

^ = 0.01 ; 6=2 m = 2 Vw==3.572931 / (42)/ sElectrical coefficients are known from the material specifications.

mass

damping

stiffness

inductance

resistance

capacity

coupling

m-

b=

k=

L=

R=£=

A=

2.949672- KT*

3.572931

1.081971-10'

0.02

7.0

3-1CT*

1.681953-10'

Kg

Kg-s-'

Kg-s-'

H

n

F

Kg-m-s-'-C'

Table. 1: Piezoelectric electrical and mechanical characteristics.

3.1 Stability analysis

The following hypotheses have been assumed: the piezo has a perfect

electric contact between the piezo crystal side and the electrode, and a

perfect adherence (bonding) between the face of the piezo crystal and the

controlled structure, the piezo coefficients are constant in time and space.

An eventual stiffness of the bonding material can be modelled throughspecific elements positioned between the piezo model and the structure. The

stability study of the system as a function of the assigned value of its

coefficients can be performed through the stability analysis of the followingfourth order differential system in the independent variable t\

Imx + bx + kx + Aq = F

Lq + Rq+—q + Ax = V

Olfi(44)

'm 0~

,0 IJlfJ'LO R\{q\ L o-J^ r;

It can be demonstrated that to verify the stability of this system, being

k A^

AC

symmetric, it is sufficient to impose that Det[AT]>0, to

L U J

obtain the condition that ensures the system asymptotic stability (Figure 2):k

0<C<


D»t[K] (Kg*s-2*F-l)/"0.001

Figure 2. Variation ofDet[KJ with the capacity C.

With the given values of the remaining parameters m, b, k, L, R, A, the

variation range of the capacity C for the system stability is:

0<C< 3.8246-10-8 F (46)The variation of the real part of the complex eigenvalue p as a

function of the capacity C is represented in Figures from 3 to 10:-40Cf

-450

-500

-550

Re[pi]-528.-528.-528.-528.-528.-528.

57*574574574574574

v

C (F)X

0.2 0.4 0.6 0.8 11. 10 2. 10 3. 10 4. 10

Figures 3 and 4. Variation of the real part Re[pi] of the first complexeigenvalue;?/ with the capacity C (reduced scale and enlarged scale).

-400

-450

-500

-550

-8

-528.-528.-528.-528.-528.-528.

Ri574'574574574574574

j[p2]

v_

C (F)

0.2 0.4 0.6 0.8 11. 10 2. 10 3. 10 4. 10

Figures 5 and 6. Variation of the real part Re[p2J of the second complexeigenvalue/?2 with the capacity C (reduced scale and enlarged scale).

10


-6UURe[p3]-300

-400

-500

-600

~

-8 -8 -81. 10 2. 10 3. 10 4.

-32227.7-32227.7-32227.8-32227.8-32227.8

C (F) -32227-8\, -32227 8-8

10 0.2 0.4 0.6 0.8

Figures 7and 8. Variation of the real part Re[ps] of the third complex

eigenvalue ps with the capacity C (reduced and enlarged scale).

RJp4J100

-100

-200

-300

-400C (F)

0.2 0.4 0.6 0.8

Figures 9 and 10. Variation of the real part RefpJ of the fourth complex

eigenvalue /?* with the capacity C (reduced and enlarged scale).

4 Model Test Procedure

The described model was subjected to functional tests through several

applications with one or more piezo, with and without supporting structure,with a static or dynamic excitation, with a closed or open control loop,having effect on the actuation force F or on the voltage V. For some of

these models, the obtained results have been analytically verified.

4.1 Elementary Models and Applications

In the following Table 2 the performed numerical tests are listed:

Mod.

1

2

3

4

5

PZT

N.

1

1

1

1

1

Structure

NO

NO

NO

beam 1 el.

beam 10 el.

Ax./Flex

/

/

/

Axial

Axial

Excit.S/D

Dynamic

Dynamic

Static

Static

Static

Cont.O/C

Open

Open

Open

Open

Open

Cont.qu.

Force

Voltage

Voltage

Voltage

Voltage

Ver.

Yes

Yes

Yes

Yes

11


6

7

8

9

10

11

12

13

14

1

2

2

1

2

2

2

NO

NO

beam 1 el.

beam 1 el.

beam 10 el.

beam 10 el.

beam 10 el.

beam 10 el.

beam 10 el.

1DOF (M+K)

2DOF (2M+2K)

Bending

Bending

Bending

Axial

Bending

Bending

Bending

Axial

Axial

Static

Static

Static

Dynamic

NO

Dynamic

Dynamic

Dynamic

Dynamic

Open

Open

Open

Open

OFF

OFF

Open

Closed

Closed

Voltage

Voltage

Voltage

Force

NO

Force

Voltage

Force

Force

Yes

Yes

Yes

Mod

Table 2: Test on the piezo model.

Model 1: only one piezo is present, and no supporting structure has beenmodelled; a sinusoidal law for the amplitude of the external excitation F(t)

is imposed:

F(f) = F.jmaW where: F = 5#; w = 312.85#z (47)

Figures 11 and 12 show the charge q and the displacement x as

functions of the time t.

Figures 13 and 14. Charge q(t) and displacement x(t) of the piezo

undergoing a force F(t).

Such response fits the graph of the function obtained integrating the

differential system (42) with the Laplace transforms with zero initialconditions [11].Model 2: The input quantity F(t) has been substituted with the voltageV(t), obtaining similar functions for the displacement x(t) and the chargeq(t), if no boundary conditions are applied, or for q(t) and F(t) if thedisplacement x is restrained: x(t)=0.

12


*. iu veiny me uuuiiiicu icsuus, d siauc ioaa nas oeen appnea tor

the voltage: V=l Volt, obtaining as outputs a charge q and a displacement x

that can be analytically verified. Such verification is immediate,

considering that the system (42) is reduced to an algebraic system.

Model 4: A piezo has been applied to a cantilever beam with the followinggeometry:

We

Hi

= 2.54 10"= 2.286 10-'= 1.27 10'= 1.016 10'*= 1.27 10^

m;m;m;m;

Figure 15. Beam cross-section.

_ ^

= 2

Ab=We'He- Wj-Hi =

° 12"*" ' A~2~*2~J

w, -^-+w,t(^-+±f"12 ° 1 2 2J

9.032-

4

4'

10-5 m'

W^^M = 7.229 E - 9 m*12 J

nO-M = 2.338 E- 9m*12 J

(48)

(49)

(50)

(51)

40%

-'TOT

A cantilever beam has been modelled with a single CBAR elementconnected with its two extreme nodes (GRID.2 e GRID.S) to the piezo. Thepiezo model has been placed in parallel position with the beam neutral axis.

One of the nodes is fully constrained, and the other one has only onedisplacement (translation) degree of freedom along the x direction.

Imposing a static voltage difference of 7 K, a static actuation force isobtained, causing an axial deformation of the beam.

K,A, c,

Gri

GridS

K, A. C.

Figures 16 and 17. Axial beam model with one element and two nodes,with one piezo with static voltage, open loop control.

13


The analytic verification is the following:-ACV

X — • (52)

Model 5: The piezo model has been placed in parallel position with thebeam neutral axis. The cantilever beam is modelled with 10 elements

connected toll nodes having displacements enabled only along the x axis.The piezo element has been placed on a generic intermediate element,

obtaining the axial displacements variation law described in Figure 18.

\

\

K,A,C

®PZT

10

Figure 18. Beam model with 10 elements, with axial deformation, with 1

piezo on the beam element N. 5.

Model 6: A cantilever beam has been modelised with a single CBARelement. A static voltage is applied. The obtained actuation load has beencomposed by a bending moment and an axial force. A piezo has beenplaced at a distance from the neutral axis equal to half of the cross section

thickness. The axial force applied by the piezo acts with a lever # 0. The

analytical verification was positive.

My=F-h/2

Figures 19 and 20. One-element cantilever beam, axial and bendingbehaviour, with 1 full-length piezo with offset.

Model 7: By installing a couple of actuators with geometric symmetry andantisymmetric power supply, a pure bending moment actuation load hasbeen obtained, composed by a couple of axial forces generated by thepiezos:

14


)My=F-h

V2=-V1

Figure 21. Two piezo with symmetric geometry. Figure 22. Antisymmetricactuation. Figure 23. Static deformed shape.

Model 8: The previous piezo disposition scheme has been applied to a 10

element model of a cantilever beam. Two piezoelectrics have been

connected to a central element (N. 5). The voltage to be applied to the

piezos has been calculated with a similitude process. A test shear force

Fz=100 N normal to the beam axis has been applied at the beam free end

and the voltage difference generated at the piezo circuits ends has beenmeasured. In this case the piezos work as sensors.

Figure 24. Beam with 10 elements undergoing a static force F% (piezoOFF).

A^\ T»t~r - • -iWT—ran—n/I

U

Figure 25. Unloaded beam and piezo ON with V2=-

15


Model 9: The possibility to impress axial dynamic loads to the beam has

been investigated. A sinusoidal voltage law has been applied to a single

piezo with a coaxial position to the beam neutral axis, obtaining a

sinusoidal axial displacement law for the grid at the beam free end.

Model 10: For the dynamic control of the bending vibrations of a

cantilever beam, natural frequencies and modal shapes must be known. A

free modal run of the beam with the piezo OFF (open circuits) has been

performed.

Figure 26. Free bending modal shape of the cantilever beam with 10 elem.

(piezo OFF).

Model 11: In order to recreate with an acceptable approximation the same

dynamic conditions of the first natural vibration mode, non equilibrium

initial conditions corresponding to the first modal shape have been applied,obtaining the same time history of the position of the free end grid (Grid

111) along the z axis.= 628.27 Hz (53)

Figure 27. Cantilever beam with 10 elements with piezo OFF,undergoing dynamic shear force F% excitation: bending displacementalong the z axis direction of the beam free end grid (GRID 111).

Model 12: The possibility to dynamically excite the bending of a cantileverbeam has been investigated. Two sinusoidal laws for the voltages Vi(t) and

V](t) have been imposed to the piezos.

16


(54)

An effective vibration damping has been detected. The bending

displacement along the z axis direction of the beam free end grid (GRID

111) has been plotted as an index of the vibration amplitude reduction.Vi(t;

\

\

Fz

v,W

(!) ® ® ®1 2 3 4 5

1PZT1\&

POT

t 1® ® ® ® ®

6 7 8 9 1 0i

Figure 28. Dynamic open loop control scheme with antisymmetric power

supply.

Figure 29. Displacement time history of the beam free end grid (111)along the z axis direction.

The amplitude decay described in Figure 29 happens only initially.The piezoelectrics damping introduces a phase difference between the

structure vibration frequency and the frequency of the supplies voltage, soafter a certain time the piezo excites the bending vibration instead ofdamping it. This typical inconvenience of open loop control systemsimposes the study of a closed loop control system.

Model 13 : To simulate a closed loop control system, a one-degree-of-freedom dynamic system has been modelled, with a direct velocity feedbackclosed control loop. A function to represent the effect of piezo saturationhas been also introduced.

17


TABLED!

Figure 30. Block diagram flow chart of a direct velocity feedback control

loop on a 1 DOF dynamic system.

Figures 31 and 32. Displacement time history with control OFF and ON

(ordinate scale are different).

Setting a lower amplification factor of the control system, the

following displacement time history was obtained:

Figure 33. One DOF dynamic system with control ON, with a loweramplification factor with respect to the previous case (Figure 32).

18


in mis case conironaDinty ana me visiointy is complete (W07o).

Model 14 : The same control loop has been applied to a 2 DOF dynamic

system: in this case the controllability and the visibility of the dynamicvariables is not complete:

TABLED y.

Figure 34. Block diagram flow chart of a velocity feedback control loop on

a 2 DOF dynamic system.

o«oo 0600

Figure 35. Time history of the actuator control force on the GRID 2.

with the following results:

A

C .OKT. MCC XCC «

i\

n

\ fV

AMCO WW CKO

Figures 36 and 37. Grid 2 displacement time history with control OFF andON (on a reduced ordinate scale).

19


^ du/dt Cm/s) cox !O -, du/dt

/ 1 ooQXtt

\ A 1i / \ «. j /

/ i / \ r\, 21 i / \ / ^ • ^~1 i / vy

da's')

l\

\ ^'• ! \ S~*i«\

\l ^ "

tOO 129

Figures 38 and 39. Grid 2 velocity time history with control OFF and ON(on a reduced ordinate scale).

5 Future developments

It is possible to model one or more control nets with more sensor/actuatorcouples distributed on the structure to be controlled (Mannini[8]).

In particular, the direct velocity feedback control can be implemented

on a cantilever beam, positioning two couples of colocated actuators (PZT

and sensors (PVDF) with a closed control loop as it is shown in Figure 40.

| ^^1 1 1

1 2 3 4 5

|Vz(t) | '

|PVDF3

PZTI(/

PZT2

PVDP41

1 11 1 t i :..!

6 7 8 9 1 0

*

Fz

Figure 40. Four piezo (two sensors and two actuators) with position andvelocity feedback with interface on the element N. 5 of a cantilever beam,

undergoing an external shear force F%(t).

The analysed model can be replicated along the y axis, obtaining atwo-dimensions model, reproducing with a higher accuracy level the effectof a piezoelectric plate element bonded on a vibrating plate structure.

6 Conclusions

A MSC/NASTRAN model of the piezoelectric active static and dynamic

control system has been created, under the pin-force hypothesis, with a

20


moaemsaiion or me control law, or me amplifier factor and or the piezo

saturation law. Such a model can be connected to the structure to be

controlled and can be easily replicated and/or repositioned, following the

needs of optimal position and optimal control. The modularity of the block

of instructions describing the piezo has been maintained. It is possible to

choose /%, b, k, L, R, C, A, and to control the actuation force and moment

as functions of the amplification factor G, of the saturation curve s, of the

delay time T, and of the time t.

Nomenclature

AAbbC

C

X

D

d

e

E

6

?*FFJt

Jye

Jze

kLll

b

b

mP

coupling coefficientbeam cross-section areamechanical damping coefficientelectrical charge capacitystiflhess tensor

dielectric permictivity tensor

electric displacement vector

strain piezoelectric tensor

stress piezoelectric tensor

electric field vector

strain tensor

body volume force vector

force amplitudeforce generated by the piezoRatio between the area delimited bythe resisting circuit and the overallslenderBeam cross-section moment ofinertia around the vertical axisBeam cross-section moment ofinertia around the horizontal axisstiflhesselectrical inductancedimension of the piezo along the x-axisdimension of the piezo along the y-axisdimension of the piezo along the z-axismassLaplace operator

PVqRPq

S

Sf

Sq

Su

Sv

Sat

uV

versnWe

WeWeWe

X

We

WeX

derivative operatorpiezoelectric element volumeelectric chargeelectrical resistanceelectric charge density

piezo external surafce

surface partition where force boundaryconditions are appliedsurface partition where charge boundaryconditions are appliedsurface partition where displacementboundary conditions are appliedsurface partition where voltage boundaryconditions are appliedflexibility tensor

stress tensor

beam cross-section thicknessdisplacement vectorelectric voltage

normal unit vector

beam cross-section external width

beam cross-section internal widthbeam cross-section external heightbeam cross-section internal height

displacement

beam cross-section external height

beam cross-section internal heightdisplacement

21


Superscripts and subscripts

Eo£

t

with constant electric fieldwith constant stresswith constant straintransposed

Acknowledgements

We thank Prof. P. Santini (Dipartimento Aerospaziale) for the financial

support, Dr. P. Gaudenzi (Dipartimento Aerospaziale) for the critical

review, Dr. F. Morganti (Alenia Aerospazio) for the hardware and

software, Dr. G. Strambi and Dr. M. Linari (MSC/Italy) for the assistanceand technical support, Dr. G Filippazzo for the English language review.

References

[1] Rogers, C.A., Intelligent Material Systems: the Dawn of a new

Materials Age, Jour, of Intelligent Material Systems and Structures, 4,

N°l,pp. 1-13, 1982.

[2] Crawley, E. F., & De Luis, J., Use of Piezoelectric Actuators as

Elements of Intelligent Structures, AIAA Journal, 25, N° 2, pp. 1373-1385, 1987.

[3] Hagood, N.W., Chung, WH & Von Flotow, A., Modelling ofPiezoelectric Actuator Dynamics for Active Structural Control, Jour.

of Intelligent Material Systems and Structures, 1, July 1990.[4] Strambi, G, Barboni, R. & Gaudenzi, P., Pin Force and Euler

Bernoulli Models for Analysis of Intelligent Structures., AIAA

Journal, 33, pp. 1746-1749.

[5] Barboni, R., Gaudenzi, P. & Strambi, G, On the Modelling of

Actuator-Structure Interaction in Intelligent Structures., 8*

CIMTEC, Cong. Mond. di Ceramica, Forum on New Materials, Paper

n° SVI-5-L03, Intelligent Materials and Systems, Techna, pp. 291-

298,1995.[6] Gaudenzi, P., Barboni, R., Carbonaro, R, & Accettella, S., Direct

Position and Velocity Feedback Control on an Active Beam with PZT

Sensors and Actuators, CEAS International Forum, Rome, Italy, 2,pp. 285-29217-20 June 1997.

22


L/j rneomann, r.r., me promise oj adaptive Materials for Alleviating

Aeroelastic Problems and Some Concerns, CEAS InternationalForum, Rome, Italy, 17-20 June 1997.

[8] Mannini, A., Posizionamento Ottimo di Attuatori PZT nel Controllo

Strutturale, AIDAA National Congress, Napoli, 22-24 Oct. 1997[9] Gaudenzi, P. & Bathe, K. J., An Iterative Finite Element Procedure

for the Analysis of Piezoelectric Continua., Journal of Intelligent

Material Systems and Structures, 6, pp. 266-273, March 1995.

[10] Barboni, R, StruttureAlan Adattative, doc.: IS-RMl-AL-9112- iss.:01, Universita degli Studi di Roma "La Sapienza" Roma pp 4-351991.

[11] MSC/NASTRAN Application Manual, An Introduction to thev4Wyj7j of f/ezoe/ecfnc ecfj m A C/AW TT T/,

MSC/NASTRAN Application Note, Section 5, pp. 1 - 7, June 1992.

[12] de Benedetti, M., Impiego e Controllo di Elementi Piezoelettrici,Ph.D. Dissertation, Rome.

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