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A wide application area for piezoelectric materials isvibration control. There are two approaches to this solution.The first is active damping, where part of the piezo elementsattached to structure is performing sensing activity and theothers are actuators. Active damping requires a closed loopcontroller and power supply (Moheimani and Fleming2006). The other approach is passive damping, also calledshunt damping, where the piezoelectric is a transducer thattransforms part of the mechanical vibration energy intoelectrical energy that can be conveniently dissipated in cir-cuit (Moheimani and Fleming 2006).

These solutions cover things from space and militaryapplications to consumer products. Vibration control isused in military aircraft like the F-15 and FA-18, to preventmechanical fatigue and improve flight parameters (Mo-heimani and Fleming 2006), Boeings Smart Helicopter Ro-tor to reduce vibration and noise and improve aerodynamicperformance (Boeing 2007).

There are many different methods of modeling of piezoele-ments. Among them the following should be listed:

:

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The diagram of this model is shown in Figure 3. Where x1and x2 are the displacement of lumped masses m1 and m2respectively, k is the spring, F is the force exciting the me-chanical part of the system, R is resistance, and L is induc-tance of electrical circuit, between mass m1 and m2 is piezo-electric transducer.

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Using constitutive equation for a linear piezoelectric ma-terial (ANSI/IEEE std 176 1987)

ET = c S eE (6)

D eS E= + (7)

where:T stress [MPa],

S strain,

E electric field [V/m],

D electric displacement [C m2],

cE elastic stiffness for constant electric field[MPa],

e piezoelectric constant [C m2],

dielectric permittivity for constant strain[F m1],

substituting:

0

, , ,h

p

A

F xT U Edz q DdA S

A h

= = = = (8)

where:A surface area of electrodes,h distance between electrodes,

into (6) and (7) , defining:

2, p

e A eEM k c

S h S

= = +

(9)

0SA

Ch

= (10)

and solving in terms of U and F gives the formulas describ-ing the voltage generated by the piezoelectric, which is de-scribed as a voltage source and capacitance C0 (10) and theforce with which the piezoelement affects the mechanicalpart of the system, respectively (6) and (7).

EM and kp (9) describe the electromechanical couplingcoefficient and stiffness of piezoelectric respectively.

0

qU EM x

C= (11)

p pF k x EMq= + (12)

including (11) and (12) in the equations of motion derivedfor the system shown in Figure 3 the following state spaceequations are obtained:

1 1

2 2

3 3

4 41 1 1

5 5

6 6

2 2 2

0 1 0 0 0 0

10 0

0 0 0 1 0 0

0 0 0

0 0 0 0 0 1

0 0 0

p

p p

p p

R EM EMy yC L L L Ly y

y yk kEM

y ym m m

y y

y yk k kEM

m m m

= +

(13)

[ ][ ]1 2 3 4 5 6

1 1 2 2

T

T

y y y y y y

I q v x v x

=

=

(14)

The state space equations are very useful for the simula-tion and development of control system.

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The derivation of the model for a piezoelectric bender isdescribed in (Smith and Dalke 1989). This is the constitu-tive equation for an antiparallel bimorph. Two piezoelectricelements are joined over their entire length and their pola-rizations are antiparallel to one another. An electric field isapplied across the entire beam. The mechanical boundaryconditions are: a mechanical moment M applied to the endof the beam and a force F applied perpendicularly to the tipof the beam. The displacement and the rotation are obtainedfor the free end of the bender.

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The constitutive equations are:

23111 11

3 3 2

22 33111 11

3 3 2

2 231 31 33 312 2

33 3

2 4 4

33 3

4 2 8

3 31

2 44 8

E E

E E

T

d Ls L s L

wh wh hM

d Ls L s LF

wh wh hq Vd L d L Lw k

hh h

=

(15)

where: rotation angle of the free end, displacement of the free end,L length of the beam [m],w width of the beam [m],h thickness of the beam [m],

k31 piezoelectric coupling coefficient.

The model of a piezoelectric bimorph given by (15) allowssimple calculation of the rotation angle, displacement of thefree end and electrical charge. This model is an algebraicequation and does not describe the dynamics of the system.

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The thermal analogy approach allows piezoelectric actua-tors to be modelled using classic FEM software likeMSC.Nastran. The constitutive equation for the inversepiezoelectric effect is similar to the constitutive equation forthe thermoelastic effect if the applied electric field is mo-deled as a thermal load (Dong and Meng 2004).

The inverse piezoelectric effect can be described by thefollowing equation:

[ ] [ ][ ] [ ][ ]ET c S e E= (16)

where the mechanical variables T and S denote stress andstrain vector respectively, E is the electric field vector, cE isthe elastic stiffness coefficient matrix and e is the piezoelec-tric stress coefficient matrix.

Equation (16) can be rewritten as:

( )[ ] [ ] [ ] [ ][ ]ET c S d E= (17)where d is the piezoelectric strain coefficient matrix.

If we consider the PZT material polarized along z-axis or3-axis the piezoelectric constants matrix can be written as:

31

31

33

15

15

0 0

0 0

0 0[ ]

0 0

0 0

0 0 0

d

d

dd

d

d

=

(18)

For the extension actuation, the electric field vector andpolarization of the piezoelectric material are parallel, E1 andE2 are zero so the constitutive equation can be simplified to:

[ ]( )31 31 33 3[ ] [ ] [ ] 0 0 0 TET c S d d d E= (19)For the shear actuation, the electric field vector and pol-

ing direction of the piezoelectric material are perpendicular,i.e. E1 and E3 are zero so the constitutive equation can besimplified to:

[ ]( )15 2[ ] [ ] [ ] 0 0 0 0 0 TET c S d E= (20)Above-mentioned equations (19), (20) are similar to

MSC.Nastran MAT9 material (MSC.Nastran Manual), de-fined as:

( )[ ]( )1 2 3 4 5 6[ ] [ ] [ ] TE REFT c S A A A A A A= (21)where denotes the temperature excitation, REF is the re-ference temperature and Ax denotes the thermal expansioncoefficients.

For the extension actuation comparing (19) with (21) andassuming that REF = 0 gives:

1 2 31 3 33 3; ;A A d A d E= = = =

and for shear actuation comparing (20) with (21) and as-suming that REF = 0 gives:

1 2 3 5 6 4 15 20; ; .A A A A A A d E= = = = = = =

Piezoelectric actuators are driven by a voltage source.Assuming that material is uniform, the electric field be-tween electrodes is constant: and is given by equation:

UE

x= (22)

where E is the electric field, U is the potential differencebetween electrodes, and x is the distance between elec-trodes.

It is worth noting that the thermal analogy allows onlypiezoelectric actuators to be modelled.

It is not possible to simulate the behavior of transducersusing this method.

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A description of Finite element matrix formulation can befound in (Ansys.Multiphysics Manual; Abaqus Manual;Kagawa et al. 1996).

Establishing nodal solution variables and element shapefunctions over an element domain, which approximate thesolution, leads to finite element discretization.

Tucu N u = (23)

TVcV N V = (24)

165

where:uc displacement within element domain in the x, y, z

dimensions,u vector of nodal displacement,

Nu matrix of displacement shape functions,Vc electrical potential within element domain,V vector of nodal electrical potential,

NV matrix of electrical potential shape function.

Strain and electric field are related to displacement andelectric potential respectively.

[ ]uS B u= (25)[ ]VE B V= (26)

where:

0 0

0 0

0 0

0

0

0

u

x

y

zB

y x

z y

z x

=

(27)

TVV

x

B Ny

z

=

(28)

After the application of the variational principle finiteelement discretization, the coupled element matrix equationderived for a one element model is:

[ ]

[ ] [ ][ ] [0] [ ] [0]

[0] [0] [0] [0][ ] [ ]

[ ] [ ]

[ ] [ ]

Z

TZ d

u uM C

V V

K Ku F

V qK K

+ +

+ =

(29)

where:[M] mass matrix,[C] damping matrix,[K] structural stiffness matrix,

[Kd] dielectric matrix,[KZ] piezoelectric coupling matrix,[F] vector of nodal forces, structural forces and

body forces,[q] vector of nodal charges, surface charges and

body charges.

These models are implemented in commercially avail-able software.

0

On the software market there is not much software whichallows direct simulation of piezoelectric transducers. Mul-tiphysics FEM software products such as ANSYS. Mul-tiphysics developed by ANSYS Inc. and Abaqus developedby SIMULA have this function.

For simulation of the piezoelectric phenomenon ANSYShas predefined element types PLANE13, PLANE233 2Dsolid elements, 4-node quadrilateral and 8-node quadrilater-al respectively, SOLID5, SOLID98, SOLID226, SOLID227 3D solid elements 8-node brick, 10-node tetrahedron,20-node brick and 10-node tetrahedron respectively. Theelectromechanical constitutive equations for linear modelbehavior are the same as (6) and (7). The matrices of themodel containing piezoelectric elements are given by (29).For proper definition of the material it is necessary to givethe piezoelectricity matrix, the electrical permittivity matrixand the anisotropic stiffness matrix. The matrices describ-ing material properties in ANSI/IEEE standard of piezo-electricity need to be reordered. The order of rows and co-lumns in ANSYS matrices in tensor notation is 11, 22, 33,12, 23, 13. The ANSYS element library also contains thepredefined element type CIRCU94. It allows simulation of alinear electrical circuit attached to piezoelectric transducer.This element, depending on the chosen parameters, can de-scribe the behavior of a resistor, inductor, capacitor, voltageor current source. Possible analysis types for models con-taining piezoelectric elements are static, modal, prestressedmodal, harmonic, prestressed harmonic and transient. Theapplicable analysis types for electric circuits are only fullharmonic and transient (Ansys Multiphysics Manual).

Similar to ANSYS, Abaqus has also predefined 2D and3D solid element types that allow simulation of piezoelec-tric materials. There are 5 two-dimensional different con-stant strain and constant stress element types containing 3,4, 6 and 8 nodes and there are 7 three-dimensional elementscontaining 4, 6, 8, 10, 15 and 20 nodes.

The constitutive equations for the piezoelectric modelare the same as given by (6) and (7) and the system equa-tions for the model containing piezoelectric elements aresimilar to (29), but excluding the first time derivative part structural damping. To define material properties it is essen-tial to give the piezoelectric matrix and the electrical per-mittivity matrix and the stiffness matrix. The order of rowsand column in Abaqus material properties matrices in ten-sor notation is 11, 22, 33, 12, 13, 23 and is different from theIEEE standard of piezoelectricity. It is not possible to modela full electro-mechanical system in Abaqus software, be-cause it does not support simulation of electric circuits. Theoptions available for analysis of models containing piezo-electric elements are static stress analysis, transient ana-lysis, harmonic analysis and modal analysis. Piezoelectricelements can be used in general analysis steps, includingboth geometrically linear and nonlinear analyses (in non-linear analysis, however, the piezoelectric constitutive re-sponse remains linear) (Abaqus Manual) The advantage ofAbaqus software is a user friendly interface similar to CAD

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software. Every parameter of creating model can be easilyfound in the model tree.

In the previous section useful techniques for the simula-tion of piezoelectrics without using multiphysics FEM pro-grams were reviewed.

The thermal analogy allows us to model a piezoelectricactuator. Thermal strain analysis is available in MSC.Nastran.This program is the oldest FEM software and is widely used inthe automotive and aerospace industries (MSC.Software 2008).

Using equivalent circuits for the piezoelectric transducerand an electro-mechanical analogy for mechanical systems,the whole electro-mechanical system can be simulated inelectrical CAD software such as PSpice. PSpice allows DC(static) analysis, AC (harmonic) analysis and transient ana-lysis to be performed (Izydorczyk 1993). This program isdedicated to simulating sophisticated electrical and elec-tronic circuits. There are many libraries containing modelsof electronic devices that can be easily included in the mo-del, so all electrical or electronic parts of electro-mechani-cal system can be easily and precisely simulated. The maindisadvantage is the necessity to convert the mechanical partof system into and electrical circuit, which is time consum-ing and error-prone.

Electro-mechanical systems can be also modeled in themulti-purpose simulation environment MATLAB-Simulink.The approach described in 2.2 allows the state equations ofthe whole system to be built. This form can be easily imple-mented in MATLAB-Simulink. The major advantages ofusing this software are a user friendly interface, a wide rangeof possibilities in terms of visualization results, the simpleand fast development of sophisticated control algorithms,and signal processing algorithms using implemented, readyto use toolboxes provided with the software (Mathworks2008) and the availability of many third party toolboxes. Themain disadvantage is the need to manually write the stateequations of the system. For large electromechanical sys-tems this can be difficult, error-prone and time consuming.

2

To illustrate an application of the models that were describedin paragraph 2, six simulations of two different mechanicalsystems with piezoelectric transducer are given here.

Simulations were performed of a simple mechanical systemcontaining a piezoelectric transducer, presented on Figure 3,and FEM simulations of a piezoelectric bimorph fixed at theend and excited by a potential difference, shown on Figure 15.

The first three show results of the frequency response andtime response of the system shown in Figure 3. This systemhas two lumped masses and a piezoelectric transducer con-nected to the electric circuit between them. Every simulationwas performed with exactly the same parameters for the me-chanical part (both masses are 2 kg) of the system and thesame (0" "!=?-A the electrical circuit. Thedifference is the model of the piezoelectric transducer, andinductance in electric circuit computed from:

2

1

p

LC

=

(30)

where: : "!

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The harmonic response of the system was computed asa Fourier transform of the impulse response of the system.Figures 5 and 6 show the harmonic response of the mecha-nical part of the system, where EM coupling coefficient iszero, and for the full system respectively.

Figure 7 shows the time response of the system excited bya harmonic force with amplitude 1 N and frequency 1.2 kHz.

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Figure 11 shows the time response for excitation froma harmonic voltage source with amplitude 1 V and frequen-cy 1.21 kHz which is equivalent to force with amplitude 1 N.

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A static simulation of a piezoelectric bimorph using diffe-rent FEM software was also performed. The setup is pre-sented on Figure 15.

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The antiparallel bimorph is fixed at one end, the top andbottom layers of nodes that model electrodes, have com-bined degree of freedom describing nodal potential. InMSC.Nastran the excitation is modeled directly by bodythermal load as analogous to electric field, so modelingelectrodes is unnecessary.

In ANSYS and Abaqus software the model geometry isinput in meters, in MSC.NASTRAN because of numericalproblems model has to be input in millimeters, and all mate-rial coefficients and voltage load have to be adjusted to thescale of one millimeter. The antiparallel polarity of the pi-ezoelectric layers is modeled in multiphysics software bymeshing model in flipped coordinate systems. Using a ther-mal analogy the two different materials has to be definedwith opposite signs of e (A) matrix. The static analysis re-sults of a piezoelectric bimorph excited by 1 V potentialdifference are shown on Figures 16, 17 and 18, as per-formed in ANSYS, Abaqus and MSC.NASTRAN respec-tively. The summary of results obtained from simulations ispresented in Table 1.

Nowadays, instead of uniform piezoceramics, multilayertransducers are being widely used because of better perfor-mance. Manufacturers claim to be able to make a single pie-zoelectric layer of thicknesses#.D=8 !%0!0 +A*F )!!0the num-ber of layers, we can assume that the capacitance of onepiezoelectric layer between electrodes is given by (10).When we consider a transducer with n layers we can rewrite(10) as

0A

C nh

= (31)

We consider the n layers with electrodes connected pa-rallel, so the measured capacitance of transducer is n timesbigger than single layer, thus:

20

AC nC n

h

= = (32)

solving (32) in terms of n, we receive the approximate num-ber of layers in piezoelectric transducer given by:

Chn

A=

(33)

The simulation results of a twenty layer piezoelectric bi-morph model with the same dimensions and boundary con-ditions as other simulated FEM models are presented in Fi-gure 19. It is easy to notice that the total displacement of thefree end is more than five times greater than that of the uni-form two layer bimorph.

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The models made in Simulink and PSPICE give the samefrequency of vibration of electro-mechanical system, diffe-ring slightly from the FEM model in ANSYS, also the am-plitude in time response are the same for PSPICE and Si-mulink models. However, those results are three times big-ger then obtained from analysis run in ANSYS.

The ratios between amplitudes of vibration for the me-chanical part and the shunted system are similar for Si-mulink and FEM model. Simulation in PSPICE gives com-pletely different results.

The static simulation of bending a piezoelectric bimorphin FEM software gives almost the same results.

Different models of piezoelectric transducer were pre-sented. It has been shown that beside multiphysics FEMsoftware it is possible to model piezoelements in MATLAB--Simulink, PSPICE and MSC.NASTRAN.

The equivalent circuit method and modeling of a piezo-electric as a spring allows us to perform simulations forwhole electro-mechanical systems for thickness mode vi-bration of transducer. The thermal analogy enables mo-deling actuators using FEM software, but does not allowmodeling of sensing activity of piezoelectric elements.Abaqus software has facilities to model both piezoelectricactuators and sensors, but only in ANSYS.Multiphysics ispossible to simulate the whole electro-mechanical system.

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