Original Article Journal of Intelligent Material Systems and Structures 1–7 Ó The Author(s) 2015 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1045389X15595294 jim.sagepub.com Formulation of a simple distributed- parameter model of multilayer piezoelectric actuators Yangkun Zhang 1 , Tien-Fu Lu 1 and Said Al-Sarawi 2 Abstract A multilayer piezoelectric actuator is a promising linear vibrator. In this article, a simple distributed-parameter analytical model of piezoelectric actuator, which can model vibration characteristics of piezoelectric actuator–based applications, is formulated. Based on the physical analysis of piezoelectric actuator, a simplification is proposed, justified and applied to fundamentals of thickness-extension-mode piezoelectricity. This simplification subtly enables piezoelectric actuator to be effectively modelled as a whole and allows for a formulation of a simple analytical model. Compared with other model- ling methods in the literature, the proposed model with a small number of easily accessible parameters is easy to handle and extend with little compromised accuracy. The effectiveness of the proposed model has been validated by a three- dimensional finite element analysis model of piezoelectric actuator developed in commercial software ANSYS. Keywords Multilayer, piezoelectric, actuators, analytical, model, simplification, piezoelectricity, finite element analysis Introduction A multilayer piezoelectric actuator (PEA) is a widely used linear actuator. It stacks thin piezo layers which can convert electric energy to mechanical force or deformation. Aside from a high resolution in displace- ment, a high stiffness and a large dynamic frequency range, PEA can operate with a low voltage and a com- pact size. Due to those desired features, PEA is widely used as a high-precision positioner (Miri et al., 2014) or a vibrator in high-resolution vibration applications (Lee et al., 2011; Morita et al., 2013; Okamoto and Yoshida, 1998; Peng et al., 2013; Siebenhaar, 2004). In the applications of PEA where multiple resonant modes are required to be modelled (Morita et al., 2013), a distributed-parameter model is required. In this article, a simplification on fundamentals of thickness-extension-mode piezoelectricity is proposed and justified to formulate a simple and effective distributed-parameter analytical model of PEA for modelling the vibration of PEA-based applications. Section ‘Existing models of PEA’ reviews the existing models of PEA. In section ‘Model formulation’, based on the physical analysis of PEA, simplifications are proposed, justified and applied into the complex elec- tromechanical coupled fundamentals of thickness- extension-mode piezoelectricity. Then, based on the simplified fundamentals, a simple distributed- parameter analytical model of PEA is formulated. In section ‘Model validation and discussion’, a case study is carried out for validation of the simplified fundamen- tals and the formulated analytical model. The valida- tion is achieved by checking with a three-dimensional (3D) finite element analysis (FEA) model of PEA developed in commercial software ANSYS. Also, the limitations of the proposed model are discussed in sec- tion ‘Model validation and discussion’. Section ‘Summary and future work’ presents the summary and future work. Existing models of PEA A lot of efforts have been put into modelling PEAs in the literature, especially in its control to overcome some non-linear behaviour such as hysteresis. For online control, lumped-parameter models are often used, as they are easily implemented in practices. A widely 1 School of Mechanical Engineering, The University of Adelaide, Adelaide, SA, Australia 2 School of Electrical & Electronic Engineering, The University of Adelaide, Adelaide, SA, Australia Corresponding author: Tien-Fu Lu, School of Mechanical Engineering, The University of Adelaide, Adelaide, SA 5005, Australia. Email: [email protected]at UNIVERSITY OF ADELAIDE LIBRARIES on November 6, 2015 jim.sagepub.com Downloaded from
Transcript
Original Article
Journal of Intelligent Material Systemsand Structures1–7� The Author(s) 2015Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/1045389X15595294jim.sagepub.com
Formulation of a simple distributed-parameter model of multilayerpiezoelectric actuators
Yangkun Zhang1, Tien-Fu Lu1 and Said Al-Sarawi2
AbstractA multilayer piezoelectric actuator is a promising linear vibrator. In this article, a simple distributed-parameter analyticalmodel of piezoelectric actuator, which can model vibration characteristics of piezoelectric actuator–based applications,is formulated. Based on the physical analysis of piezoelectric actuator, a simplification is proposed, justified and applied tofundamentals of thickness-extension-mode piezoelectricity. This simplification subtly enables piezoelectric actuator to beeffectively modelled as a whole and allows for a formulation of a simple analytical model. Compared with other model-ling methods in the literature, the proposed model with a small number of easily accessible parameters is easy to handleand extend with little compromised accuracy. The effectiveness of the proposed model has been validated by a three-dimensional finite element analysis model of piezoelectric actuator developed in commercial software ANSYS.
KeywordsMultilayer, piezoelectric, actuators, analytical, model, simplification, piezoelectricity, finite element analysis
Introduction
A multilayer piezoelectric actuator (PEA) is a widelyused linear actuator. It stacks thin piezo layers whichcan convert electric energy to mechanical force ordeformation. Aside from a high resolution in displace-ment, a high stiffness and a large dynamic frequencyrange, PEA can operate with a low voltage and a com-pact size. Due to those desired features, PEA is widelyused as a high-precision positioner (Miri et al., 2014) ora vibrator in high-resolution vibration applications(Lee et al., 2011; Morita et al., 2013; Okamoto andYoshida, 1998; Peng et al., 2013; Siebenhaar, 2004). Inthe applications of PEA where multiple resonant modesare required to be modelled (Morita et al., 2013), adistributed-parameter model is required.
In this article, a simplification on fundamentals ofthickness-extension-mode piezoelectricity is proposedand justified to formulate a simple and effectivedistributed-parameter analytical model of PEA formodelling the vibration of PEA-based applications.Section ‘Existing models of PEA’ reviews the existingmodels of PEA. In section ‘Model formulation’, basedon the physical analysis of PEA, simplifications areproposed, justified and applied into the complex elec-tromechanical coupled fundamentals of thickness-extension-mode piezoelectricity. Then, based on thesimplified fundamentals, a simple distributed-
parameter analytical model of PEA is formulated. Insection ‘Model validation and discussion’, a case studyis carried out for validation of the simplified fundamen-tals and the formulated analytical model. The valida-tion is achieved by checking with a three-dimensional(3D) finite element analysis (FEA) model of PEAdeveloped in commercial software ANSYS. Also, thelimitations of the proposed model are discussed in sec-tion ‘Model validation and discussion’. Section‘Summary and future work’ presents the summary andfuture work.
Existing models of PEA
A lot of efforts have been put into modelling PEAs inthe literature, especially in its control to overcome somenon-linear behaviour such as hysteresis. For onlinecontrol, lumped-parameter models are often used, asthey are easily implemented in practices. A widely
1School of Mechanical Engineering, The University of Adelaide, Adelaide,
SA, Australia2School of Electrical & Electronic Engineering, The University of
Adelaide, Adelaide, SA, Australia
Corresponding author:
Tien-Fu Lu, School of Mechanical Engineering, The University of Adelaide,
recognized lumped model was proposed by Goldfarband Celanovic (1997). This model is described by anelectrical model, a mechanical model and the portswhich couple the electrical and mechanical models. Themechanical side is modelled as a lumped mass–spring–damper system with an electrical port, which can intro-duce a lumped piezo force from applied voltage in theelectrical side, based on the inverse piezoelectric effect.The electrical side is modelled by a lumped capacitancewith a mechanical port, which can introduce a countercharge from the induced piezo displacement in themechanical side (similar to counter electromotive force(EMF)). The experiments showed that the hysteresiswas reasonably overcome by the non-linear lumpedmodel.
Although easily implemented for control purpose,lumped models are just approximated models, as aPEA is actually a distributed system. To explore theconditions in using lumped models to effectively modela distributed PEA-based positioning system, Adriaenset al. (2000) and Chen et al. (2008) developeddistributed-parameter models of PEA. They concludedthat by proper design of the positioning mechanism, adistributed PEA-based positioning system can be sim-ply modelled by a lumped model.
In the distributed-parameter PEA models ofAdriaens et al. (2000) and Chen et al. (2008), the PEAis taken as an equivalent distributed normal solid withno piezoelectricity, and the action of voltage is taken asan equivalent force acting on the PEA. However, theequivalence is not rigidly proved. Besides, the relatedequivalent constants are not IEEE standardized piezo-electric constants and are required to be determinedfrom experiments, making them inflexible for model-based prediction of PEA-based vibration.
To describe the behaviours of piezo-electricallyexcited mechanically vibration system, equivalent elec-trical circuits are often used. A widely used equivalentelectrical circuit, which has been standardized in IEEE(Meitzler et al., 1988), is the Van Dyke circuit, whichconsists of a lump capacitance connected in parallelwith an inductance, a resistance and a capacitance (VanDyke, 1925). Other equivalent circuits are developed tomodel multiple resonant modes and improve the accu-racy (Guan and Liao, 2004; Kim et al., 2008; Meitzleret al., 1988). The electrical parameters in those equiva-lent electrical circuits can be determined by systemidentification methods based on measured information(Guan and Liao, 2004; Kim et al., 2008; Meitzler et al.,1988).
However, in these equivalent circuits, the electrome-chanical interaction and physically mechanical quanti-ties of PEA are obscure. Therefore, they can only beused to describe the behaviours of piezo-electricallyexcited system, but they are inaccurate or incapable topredict behaviours when electrically or mechanicallyloaded.
As PEA is a multilayer structure composed of manythin piezo layers and each piezo layer is coupled in bothelectrical and mechanical domains, directly deriving theanalytical solutions from the fundamentals of piezo-electricity is very cumbersome and inflexible. To facili-tate the calculation of the vibration of overall PEA,transfer matrix method is proposed in the literature(Bloomfield, 2002; Morita et al., 2001; Rashidian andRahnavard, 2000). Based on the fundamentals ofpiezoelectricity, various forms of transfer matrix forthin piezo layer has been made up in the literature(Bloomfield, 2002; Morita et al., 2001; Rashidian andRahnavard, 2000). As each layer is mechanically con-nected continuously, the dynamic characteristics of thewhole PEA can be calculated by multiplying all thetransfer matrices of each individual layer in PEA.However, even with the help of the transfer matrixmethod, the coupled fundamentals of piezoelectricityare very complicated, computationally inefficient andare hard to provide a simple analytical solution formodelling the vibration of the whole PEA.
A simple distributed-parameter analytical model ofPEA, which can be used to model the vibration ofPEA-based applications, has not been identified in theliterature. In the following section, based on the physi-cal analysis of PEA, a simplification is proposed, justi-fied and applied to the complex fundamentals of PEA.Based on the simplified fundamentals which subtlyenable PEA to be effectively modelled as a whole, asimple distributed-parameter analytical model of PEAis formulated.
Model formulation
Before formulating the model, configuration and physi-cal background of PEA is briefly introduced here. Withreference to Figure 1, a PEA is stacked by a large num-ber of thin piezo layers. Each piezo layer is polarizedalong the longitudinal direction of PEA. In physicalnature, each piezo layer is mechanically connected inseries and electrically connected in parallel. Such a stackconfiguration enables a large deformation in the
Figure 1. Schematic representation of multilayer piezoelectricactuator.
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longitudinal direction of PEA (x direction in Figure 1)with a low input voltage.
Based on the IEEE standards on piezoelectricity(Meitzler et al., 1988), the 3D piezoelectric constitutiveequations can be shown as follows
S1
S2
S3
S4
S5
S6
D1
D2
D3
8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:
9>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>;
=
sE11 sE
12 sE13 0 0 0 0 0 d31
sE12 sE
22 sE23 0 0 0 0 0 d32
sE13 sE
23 sE33 0 0 0 0 0 d33
0 0 0 sE44 0 0 0 d24 0
0 0 0 0 sE55 0 d15 0 0
0 0 0 0 0 sE66 0 0 0
0 0 0 0 d15 0 eT11 0 0
0 0 0 d24 0 0 0 eT22 0
d31 d32 d33 0 0 0 0 0 eT33
266666666666666664
377777777777777775
T1
T2
T3
T4
T5
T6
E1
E2
E3
8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:
9>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>;ð1Þ
where Si, Ti, Ei and Di are the components of strain,stress, electrical filed and electrical displacement vec-tors, respectively; sE
ij are the components of compliancematrices measured at constant electrical fields; dij arethe piezoelectric strain coefficients measured at con-stant stresses; and eT
ii are dielectric coefficients measuredat constant stresses. Those piezoelectric material con-stants can be obtained by measuring and analysing elec-trical impedance (Bloomfield et al., 2000; Sherrit et al.,1996).
To formulate the model, consider the PEA inFigure 1 with an overall length L and cross-sectionalarea A stacked by k piezo layers. Since applications ofPEAs are based on its longitudinal vibration (i.e. xdirection shown in Figure 1), the model focuses on thedynamics on the ‘3’ direction. Assuming PEA is morethan three times longer in the longitudinal directionthan its lateral dimension, the longitudinal vibration isassociated with a uniaxial stress state, which gives
T1 = 0
T2 = 0
T3 6¼ 0
ð2Þ
The electrical field is only applied in the longitudinaldirection, which gives
E1 = 0
E2 = 0
E3 6¼ 0
ð3Þ
Substituting equations (2) and (3) into the third rowand the ninth row of equation (1), the constitutive equa-tions of each piezo layer in the longitudinal directioncan be derived as follows
T3 = c0S3 � e0E3 ð4aÞ
D3 = e0S3 + e0E3 ð4bÞ
where c0= 1=sE33, e0= d33=sE
33 and e0= eT33 � (d2
33=sE33).
The subscript ‘3’ stands for the direction of polariza-tion. For the convenience, this subscript ‘3’ will beomitted in the later notations.
With reference to Figure 2, consider the nth piezolayer at certain position xn of PEA with thickness tn(where n= 1, 2, 3, . . . , k).
The stress, strain, electrical field and electrical displa-cement can be, respectively, expressed as follows
T x, tð Þ= N x, tð ÞA
ð5aÞ
S x, tð Þ= ∂u x, tð Þ∂x
ð5bÞ
E x, tð Þ= � ∂u x, tð Þ∂x
ð5cÞ
D x, tð Þ= Q x, tð ÞA
ð5dÞ
where the arguments ‘x’ and ‘t’ are position and timevariables, respectively; N and u are the normal forceand displacement, respectively; u and Q are electricalpotential and electrical charge, respectively; and A isthe cross section of the piezo layer.
To derive a simple but effective distributed-parameter analytical model, the following simplifica-tions for PEA are mathematically justified and applied.
As illustrated in Figure 2, the charge balance in the xdirection gives
∂Qn x, tð Þ∂x
= 0 ð6Þ
Figure 2. Analysis of nth piezo layer in PEA.
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Substituting equation (4b) into equation (6) withequations (5b)–(5d) results in
∂2u x, tð Þ∂2x
= � e0
e0∂2u x, tð Þ
∂2xð7Þ
Integrating equation (7) with regard to x yields
E3 = � e0
e0S3 +B1 ð8Þ
where B1 is an integration constant.Considering that each piezo layer is often far less
than the overall length of PEA in the longitudinal direc-tion (i.e. tn =(L=k)� L), the distribution of the displa-cement across tn can be assumed to be uniform (i.e. S3
is a constant). Note that this assumption will not beeffective when the actual vibration frequency rises tothe mth resonant mode whose wavelength (lm = l0=m
where l0 is the wavelength of the fundamental resonantmode about four to two times longer than L) is not farlarger than the piezo layer thickness (tn =L=k).
Then, based on equation (8), E3 can be seen as a con-stant (i.e. the distribution of the electrical field across tnis uniform). Therefore, E3 can be simplified as follows
E3 = � ∂u∂x! �
u xp + tn
� �� u xp
� �tn
=Un tð Þ
tnð9Þ
For the electrical side, also considering that the dis-tribution of the displacement across tn can be assumedto be uniform as reasoned before, the induced chargeon the surfaces of nth piezo layer can be simplified inthe following way
Qn =ADn =A e0S3 + e0E3ð Þ !
A e0un � un�1
tn+ e0
Un tð Þtn
� � ð10Þ
The above justified simplifications allow PEA to beeffectively modelled as a whole in a following simple way.
Substituting equations (9), (5a) and (5b), equation(4a) can be simplified as follows
N x, tð Þ=An c0∂u x, tð Þ
∂x� e0
Un tð Þtn
� �ð11Þ
Assuming negligible thickness of electrode layers andconsidering that each piezo layer in PEAs has the samematerial properties (c0, e0, e0 and density), dimensions(thickness tp and cross-sectional area A) and appliedvoltage (Un(t)=Up(t)) due to the electrically parallelconnected structure, it is possible and effective to applya single constitutive equation for the whole PEA, asshown in the following form
N x, tð Þ=A c0∂u x, tð Þ
∂x� e0
Up tð Þtp
� �ð12Þ
Based on Newton’s second law in the x direction, thefollowing equation of motion can be derived
∂N x, tð Þ∂x
= rA∂2u x, tð Þ
∂2tð13Þ
where r is the density of piezo layer.Substituting equation (12) into equation (13) gives
c0∂2u x, tð Þ
∂2x= r
∂2u x, tð Þ∂2t
ð14Þ
Using the principle of separation of variables, forexcitation voltage Up(x, t)=U0ejwt, the general solutionof displacement u(x, t) to equation (14) can be writtenin the following form
u x, tð Þ= A1 sin axð Þ+A2 cos axð Þð Þejwt ð15Þ
where a=ffiffiffiffiffiffiffiffiffiffiffiffiffiffirw2=c0
p, and A1 and A2 are constants,
which can be calculated from the boundary conditionsat two ends of PEA.
Substituting equation (15) into equation (12) givesthe general solution of the normal force
N x, tð Þ=A c0 A1a cos axð Þ � A2a sin axð Þð Þ � e0U0
tn
� �ejwt
ð16Þ
For the electrical side, as each piezo layer in PEA iselectrically connected in parallel, the general solution ofthe total induced charge of PEA can be derived as fol-lows by substituting equation (10)
QPEA =Xk
n= 1
Qn =A e0uk � u0
tp+ ke0
Up tð Þtp
� �ð17Þ
where k is the total number of piezo layers in PEA(k = L=tp).
By differentiating displacement u(x, t) and electricalcharge QPEA(t) with respect to time variable t, the gen-eral solution of the velocity v(x, t) and induced currentIPEA(t) can be derived
v x, tð Þ= jw A1 sin axð Þ+A2 cos axð Þð Þejwt ð18aÞ
IPEA tð Þ=A e0vk � v0
tp+ ke0
Up tð Þtp
jw
� �ð18bÞ
Based on the above justified simplifications, a simplegeneral analytical model of PEA has been derived.Note that the mechanical damping and electrical damp-ing, which account for the mechanical loss and electri-cal loss, respectively, can be easily considered by addingpositive imaginary part to stiffness material constantand adding negative imaginary part to permittivitymaterial constant (Bloomfield, 2002). The specific ana-lytical solution can be derived simply by applying the
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boundary condition. An example is given in the follow-ing section for model validation.
Model validation and discussion
To validate the effectiveness of the derived distributed-parameter analytical model of PEA, a case study is car-ried out in this section. The validation is achieved bychecking the results simulated by the proposed modelagainst FEA results.
With reference to Figure 3, a PEA stacked by k piezolayers with negligible thickness of electrode layers ischosen as the study case. The PEA is fixed at one endand set free at the other end. The dimension of the PEAis shown in Table 1. The frequency responses of free-end displacement are simulated and compared.
By applying proposed model into boundary condi-tions (u0 = 0 and Fk = 0 in this case), the specific
solution to the free-end displacement uk of PEA can beeasily derived, as follows
uk tð Þ= ke0 sin aLð Þc0La cos aLð ÞU0ejwt ð19Þ
where a=ffiffiffiffiffiffiffiffiffiffiffiffiffiffirw2=c0
p.
For validation, a 3D FEA model of PEA is developedon the commercial software ANSYS. To model the elec-tromechanical coupled behaviours of piezoelectricity, theextension package ‘Piezo Extension_R150_v8’ (ANSYS,2014) was used. To obtain the frequency response, the‘Harmonic Response’ module in ANSYS is performed.The material properties of each piezo layer in PEA arebased on piezoelectric material N10 (NEC/TOKIN,2014). The details can be seen in Appendix 1. The PEAis auto-meshed with 13,580 points and 2160 elements,which is sufficient enough to ensure a high resolution.The meshed FEA model of PEA is shown in Figure 4.The boundary conditions are set up as specified (one endfixed and one end free). By applying a harmonic voltageinput of 100 V from 0 Hz to 200 kHz with solutioninterval of 1000 points, the frequency response of thePEA free-end displacement is obtained, and the data aretransferred to MATLAB to be plotted against the resultsobtained by proposed model.
Figure 5 shows simulation results of proposed modeland FEA model. Note that for convenience of compari-son between two models, both mechanical loss andFigure 3. Study case for validation.
Table 1. PEA dimension.
Case study of PEA Overall length L 0.04 mCross-sectional area A 4 3 1023 3 4 3 1023 m2
Quantity of piezo layers k 20Thickness of each piezo layer tp = L/k(assume negligible electrode layer thickness)
0.002 m
PEA: piezoelectric actuator.
Figure 4. Meshed FEA model of PEA.
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electrical loss are not taken into account. As motionedbefore, for the proposed model, the mechanical loss andelectrical loss can be easily taken into account by addingpositive imaginary part to stiffness material constantand negative imaginary part to permittivity material con-stant, respectively. The proposed simplified model showsa perfect match to the 3D FEA model for the first fourresonant modes. The difference begins to occur in thefrequency range above the fourth resonant frequency.The differences for modelling high-order resonant modesare due to the compromised assumption of the proposedmodel. The proposed model is based on the assumptionthat the distribution of the displacement across eachpiezo layer can be taken as uniform. As justified before,this assumption will be not effective, when the actualvibration frequency rises to the mth resonant modewhose wavelength (lm = l0=m, where l0 is the wave-length of the fundamental resonant mode about two tofour times longer than L) is not far larger than the piezolayer thickness (tn = L=k). So, when the thickness of thepiezo layer is approaching to the wavelength of high-order resonant modes, the assumption which the pro-posed model is based on is compromised and the differ-ences occur. However, as high-order resonant modes inpractice are almost damped out due to the mechanicalloss and electrical loss and are trivial in practical applica-tions of PEA, the issue of the proposed model is ofminor importance.
Summary and future work
In this article, based on the physical analysis of PEA, asimplification is proposed, justified and applied to fun-damentals of thickness-extension-mode piezoelectricity.Then, based on the simplified fundamentals which allowPEA to be effectively modelled as a whole, a simpledistributed-parameter analytical model of PEA is for-mulated. The proposed model with a small number ofeasily accessible IEEE standard piezoelectric parametersis easy to handle and extend. Compared with modellingPEA by multiplying transfer matrix of each piezo layer,the proposed model shows more simplicity and is easierto handle and extend, which can provide a simple
analytical solution without compromise of the accuracy.Besides, the number of the parameters in the proposedmodel is small, and the parameters are IEEE standardpiezoelectric constants which are easily accessible.
To further validate the effectiveness of the proposedmodel, a case study of a PEA is carried out. The valida-tion is achieved by checking results simulated by theproposed model against those simulated by a 3D FEAmodel developed in commercial software ANSYS.Simulation results show very good agreements betweentwo models for a certain frequency range. The simula-tion also shows some limitations in modelling high-order vibration modes due to the compromised assump-tion which the proposed model is based on. However,as high-order resonant modes in practice are almostdamped out due to the mechanical loss and electricalloss and are trivial in practical applications of PEA, theissue of the proposed model is of minor importance.Future work involves using the proposed model forPEA-based resonant smooth impact drive mechanism(SIDM) designs, where the first two resonant modes areneeded to be predicted and tuned.
Declaration of conflicting interests
The authors declared no potential conflicts of interest withrespect to the research, authorship and/or publication of thisarticle.
Funding
This research received no specific grant from any fundingagency in the public, commercial or not-for-profit sectors.
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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 105
-200
-150
-100
-50
Frequency (Hz)
disp
lace
men
t(m)
dB)
free-end displacement under 100V excitation voltageproposed modelFEA model
Figure 5. Frequency response of PEA free-end displacement.
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The material properties used in the simulations arebased on the piezoelectric material N10 (NEC/TOKIN,2014), as shown in Table 2. As the material constantsapplied into simulations of two models are in differentforms, equivalent conversions based on the IEEE stan-dards on piezoelectricity (Meitzler et al., 1988) are car-ried out, as shown in equation (20). The convertedforms of piezoelectric constants are shown in Table 3.Note that all the material constants in equation (20)are in matrix form and need to be operated altogether
