Modeling of a circular plate with piezoelectric actuators

Mechatronics 14 (2004) 1007–1020

Modeling of a circular plate with piezoelectric actuators El Mostafa Sekouri a

b

a,*

, Yan-Ru Hu

b,1

, Anh Dung Ngo

a

Ecole de Technologie Superieur (ETS), University of Quebec, 1100 rue Notre-Dame ouest, Montreal, QC, Canada H3C 1K3 Directorate of Spacecraft Engineering, Canadian Space Agency, 6767 route de l’aeroport St, Hubert QC, Canada J3Y 8Y9 Accepted 27 April 2004

Abstract This paper presents an analytical approach for modeling of circular plate containing distributed piezoelectric actuators under static as well as dynamic mechanical or electrical loadings. The analytical approach used in this paper is based on the Kirchhoff plate model. The equations governing the dynamics of the plate, relating the strains in the piezoelectric elements to the strain induced in the system, are derived for circular plate using the partial differential equation. The natural frequencies and mode shapes of the structures were determined by modal analysis. In addition, the harmonic analysis is performed for analyzing the steady-state behavior of the structures subjected to cyclic sinusoidal loads. Numerical simulation results are obtained using finite element approach. Experiments using a thin circular aluminum plate structure with distributed piezoelectric actuators were also conducted to verify the analysis and the computer simulations. Relatively good agreements between the results of these three approaches are observed. Finally, the results show that the model can predict natural frequencies and modes shapes of the plate very accurately. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Modeling; Circular plate; Piezoelectric; FEM; Modal analysis; Harmonic analysis

*

Corresponding author. Fax: +1-514-396-8530. E-mail address: [email protected] (E.M. Sekouri). 1 Tel.: +1-450-926-4789; fax: +1-450-926-4695. 0957-4158/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechatronics.2004.04.003

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1. Introduction Flexible structures are widely used in space applications such as potential solar power satellites, large antennas, and large space robots as well as terrestrial applications such as high-speed robots, large bridges, and others. In recent years, there has been an increasing interest in the development of lightweight smart or intelligent structures for space applications to control distortions caused by the effects of out space. Smart structures contain distributed piezoelectric materials that act as both sensors and actuators because of the direct and converse piezoelectric effects respectively. These intelligent structures, when coupled with suitable control strategies and circuitry, have self-monitoring and self-controlling capabilities. One of the engineering challenges is the modeling of the actuation mechanisms in the flexible structures. Extensive research has been done analytically as well as experimentally on the implementation of the mechanics of structure and electrical energy of the piezoelectric actuators. Almost all of the studies so far were based on the conventional beam or plate theories to formulate a special finite element with piezoelectric as additional layers, Banks et al. [14]. Much research effort has been devoted to finite element formulation for the electromechanical coupling effects of piezoelectric materials, and fully electromechanical-coupled piezoelectric elements have just recently become available in commercial FEA software. Before the new piezoelectric capability was developed in commercial FEA codes [13], the induced strain actuation function of piezoelectric materials had been modeled using analogous thermal expansion/contraction characteristics of structural materials. This method was helpful in the studies of the resulting stress distribution in actuators and host substructures, and the overall deformation of integrated structures under static actuation. However, the intrinsic electromechanical coupling effects of piezoelectric materials cannot be modeled. Moreover, the dynamic actuation response of piezoelectric actuators on host substructures is difficult to implement by this method. Circular geometries are used in a wide variety of applications and are often easily manufactured, but the full three-dimensional vibration properties of these solids have not yet been investigated in detail. The knowledge of natural frequencies of components is of great interest in the study of responses of structures to various excitations, and this study is fundamental for high-risk plants. Among plates of various shapes, circular plates have a particular importance, due to their axial symmetry. Southwell [1] derived equations for a circular plate clamped around the inner boundary and free at the outside edge. The frequency equation for other combinations of boundary conditions can be found by proper rearrangement of his work. There are also some interesting numerical investigations in the literature. Vogel and Skinner [2] studied nine combinations of boundary conditions and data and references have also been given by Leissa [15]. Applications of distributed piezoelectric sensors and actuators have been the subject of recent interest in the fields of smart structures. There have been many researches focused on applications to vibration control and suppression [3–5]. In recent years, there has been a surge of interest in using piezoelectric patches attached to optical surfaces in hope of attaining high precision optical mirrors with minimal additional weight. Heyliger

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and Ramirez [9] studied the free vibration characteristics of laminated circular piezoelectric plates using a discrete-layer model of the weak form of the equation of period motion. The vibration of two-dimensional structures excited by piezoelectric actuators has been modeled by Dimitiadis et al. [7] for rectangular plates and by Van Niekerk et al. [8], Wang et al. [11] and Tylikowski [12] for circular plates. Niekerk et al. presented a comprehensive static model for a circular actuator and a coupled circular plate. Their static results were used to predict the dynamic behavior of the coupled system, particularly to reduce acoustic transmissions. The axisymmetric vibrations for laminated circular plates has also been studied by Jiarang and Jianqiao [6] using an exact approach. Circular plates composed entirely or in part of piezoelectric layers introduce the electrostatic potential as an additional variable and increase the complexity of solution because of the coupling between the elastic and electric variables and the additional boundary conditions. This paper focuses on development of an analytical approach for modeling circular plate structures with integrated distributed piezoelectrics, under static as well as dynamic mechanical or electrical loadings. The analytical approach is based on the Kirchhoff plate model. The next section presents the analytical model, while the following sections address the numerical and experimental comparison of this model on a circular plate.

2. Plate model The structure under consideration consists of a thin circular plate with piezoceramic patches bonded on one side. The plate is clamped around the inner boundary radius R1 and free at the outside edge radius R2 as illustrated in Fig. 1. The free

Fig. 1. A thin circular plate with piezoceramic patches bonded individually to its surface.

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patches generate strains in response to an applied voltage. When the plate is bonded to an underlying structure, these strains lead to the generation of in-plane forces and/ or bending moments. In this paper, we will consider only the bending moments generated by the patches and will consider them as an input to the model describing the transverse vibrations of a plate. The circular plate is also assumed to be made of linearly elastic, homogeneous and isotropic material. The dynamic extensional strain on the plate surface is calculated by considering the dynamic coupling between the actuators and the plate and by taking into account a perfect bonding (infinite shear stiffness). The plate motion is described by partial differential equations relating stresses and electric displacements with strains and electric field. The ratio of the radius of the plate to its thickness of the plate is more than 10, and the Kirchhoff assumption for thin plates is applicable and whereby the shear deformation and rotary inertia can be omitted. For wave propagation in such a structure, the displacement field is assumed as follows: uz ¼ uz ðr; h; tÞ ¼ wðr; h; tÞ

ð1Þ

ouz or ouz uh ¼ uh ðr; h; tÞ ¼
z roh

ð2Þ

ur ¼ ur ðr; h; tÞ ¼
z

ð3Þ

where uz , ur and uh are the displacements in transverse z-direction, radial r-direction, and tangential h-direction of the plate, respectively. The poling direction of the piezoelectric material is assumed to be in the zdirection. When external an electric potential is applied across the piezoelectric layer, a differential strain is induced that results in the bending of the plate. The strain in the plate and piezoelectric with respect to the radial and tangential directions and the shear component are given by our or ur 1 ouh ehh ¼ þ r r oh ouh 1 our uh þ
erh ¼ r oh or r

ð4Þ

err ¼

ð5Þ ð6Þ

The constitutive equations of piezoelectric ceramic are expressed as frg ¼ ½Cp ðfeg
½d T fEgÞ s

ð7Þ r

fDe g ¼ ½efeg þ ½
fEg ¼ frg½d þ ½
fEg

ð8Þ

where frg is the stress tensor, feg is the strain tensor ½Cp  is the elastic stiffness matrix of piezoelectric ceramic, De is the electric displacement, ½e is piezoelectric constant, ½
s  is permittivity constant under constant strain condition, and ½
r  is permittivity constant under constant stress condition. The last two equations give: ½e ¼ ½d½Cp 

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and ½
s  ¼ ½
r 
½d½Cp ½dT for the plate frg ¼ ½Cfeg

ð9Þ

where ½C is the elastic stiffness matrix of the plate. In order to satisfy the assumptions, an electric field E is applied along the zdirection, i.e., E1 ¼ 0, E2 ¼ 0, and E3 ¼ Eðr; tÞ, and thus the following conditions need to be satisfied: d32 ¼ d31 , d36 ¼ 0 and d24 ¼ d15 also for piezoceramic material, e32 ¼ e31 , e36 ¼ 0. The relationship between the electric field E and the applied voltage is given by fEg ¼ fL/ g/

ð10Þ

where fLT/ g


¼

o o o
;
;
ox oy oz



Hence, when a constant voltage is applied to the network along the z-direction, gT . It should be noted that, although we can the electrical field generated is f0; 0; d/ dz only supply the electric power with constant voltage (electrode field), the piezoceramic can be coated with different types of electrode to produce spatially distributed electric fields. Thus, we may consider that the applied voltage /ðx; tÞ is represented as a product of two quantities of V ðtÞ, and BðxÞ, in which V ðtÞ is input voltage power and BðxÞ defines the electrode profile. Similarly, the electric charge can be obtained from electrical displacement by the following relation: Z Z fQg ¼ fDg dS ð11Þ S T

where fQg ¼ ½Q1 ; Q2 ; Q3 , and S denotes the surface area of the electrode. The radial stresses in the piezolayers are assumed to be uniformly distributed in the direction perpendicular to the plate because of the plate’s small thickness. The equations of each piezoelectric layer have the following form    Ep our ur 1 ouh Vd3r Ep ð12Þ rr ¼ þ mp þ
1
m2p or r r oh hp ð1
mp Þ   Ep ur 1 ouh our Vd3h Ep rh ¼ þ mp ð13Þ þ
2 1
mp r r oh or hp ð1
mp Þ D3 ¼ d3r

dur ur V þ d3h

33 hp dr r

ð14Þ

where rr , rh and s are respectively, radial stress, circumferential stress, and shear stress on the interface surface, d3r and d3h are transverse piezoelectric constants in the radial and circumferential directions, respectively. V is the voltage acting in the

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direction perpendicular to plate. Ep and mp are the modulus and Poisson ratio of the actuator. D3 is the electrical displacement and
33 is a permittivity coefficient. The constitute equations of the plate    E our ur 1 ouh rr ¼ þm þ ð15Þ 1
m2 or r r oh   E ur 1 ouh our þm rh ¼ þ ð16Þ 1
m2 r r oh or   E ouh 1 our uh
srh ¼ þ ð17Þ 2ð1 þ vÞ or r oh r where E and m are the modulus and Poisson ratio of the material of the plate. For the radial actuator motion, the dynamics equation is expressed as follows:   orr o2 ur ð18Þ r þ rr
rh hp
sr ¼ qp hp r 2 or or The balance of the moments has the form: oðrMr Þ
Mh
rT þ hrs ¼ 0 or

ð19Þ

Using Hooke’s law we express the moments by the plate transverse displacement in the following form:    2  ow ow ow o2 w Mr ¼
D ; Mh ¼
D þm 2 þm ð20Þ or2 ror ror ro r D¼

Eh3 12ð1
m2 Þ

where T is the shear force, Mr and Mh are internal plate moments, D is the plate cylindrical stiffness, hp is the thickness of the actuator, qp is the modified density of the actuator and t is the time. Differentiating Eq. (19) with respect to r r

o2 M r oMr oMh oðTrÞ oðrsÞ þh ¼0

þ2 or or or2 or or

ð21Þ

The equation of the transverse plate motion is obtained: oðTrÞ o2 w ¼ qhr 2 or or

ð22Þ

where q is the density for combined structures given by q ¼ q1 þ nqp vi ðr; hÞ whole q1 , qp and n are the density coefficient for the plate, the patches and the number of piezoelectric respectively. vi ðr; hÞ is the characteristic function which has a value of 1 in the region covered by the ith and 0 elsewhere. Taking into account a perfect bonding the plate transverse displacement is related to the radial actuator displacement by

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ur ¼


h ow 2 or

1013

ð23Þ

Substituting Eq. (22) into Eq. (21), differentiating Eq. (18) with respect to r, and replacing the term oðrsÞ=or, finally, the equation for modeling the transverse motion is obtained by: r4 w þ q Dp ¼ n

h2 hp qp h o2 w o2 w P r2 2 ¼
2 Dp þ D ot ot Dp þ D 2ðDp þ DÞ

ð24Þ

Ep h p h 2 v ðr; hÞ 2ð1
m2p Þ i

where r4 ¼ 1r oro ðr oro 1r ½oro ðr oro ÞÞ is the Laplacian operator in the polar coordinates r and h, n is the number of actuators and P is the external surface force. The displacement in the transverse z-direction w of the circular plate integrated piezoceramic actuators is governed by Eq. (24).

3. Modal analysis To obtain the natural frequencies and modes, all external mechanical and electric excitations are assumed to be 0. The first studies of Eq. (24) were those of Poisson [17] and Kirchhoff [16] and the classical methods of finding the solution are based on the separation of variables. In the case of axisymmetric boundary conditions the solution takes the form wðr; h; tÞ ¼

1 X 1 X

gmn ðrÞ cosðmhÞfmn ðtÞ

ð25Þ

m¼0 n¼0

where gmn ðrÞ ¼ Amn Jm ðkmn rR2 Þ þ Bmn Ym ðkmn rR2 Þ þ Cmn Im ðkmn rR2 Þ þ Dmn Km ðkmn rR2 Þ in which m and n are the numbers of nodal diameters and circles, fmn ðtÞ ¼ eixmn t , Amn , Bmn , Cmn and Dmn are the mode shape constants that are determined by the boundary conditions, Jm , Ym are the Bessel function of the first and second kinds, Im and Km are the modified Bessel functions of the first and the second kind, and kmn is the frequency parameter which is also determined by the boundary conditions. Note that, the circular plate has been subdivided into three sections, R1 6 r 6 R3 , R3 6 r 6 R3 þ lp and R3 þ lp 6 r 6 R2 where R3 and lp are the location and the length of the piezoactuators respectively. Also 12 unknown coefficients, represent the amplitudes of the left and right traveling and decaying flexural waves in each section and can be determined with the boundary condition, which can be arranged into the following matrix equation ½DfCg ¼ 0. The frequency parameter, kmn , is related to the circular frequency, xmn , of the plate with piezoceramic: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2mn D þ Dp ð26Þ xmn ¼ 2 R2 qh

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Table 1 Value of kmn k2mn k200 k210 k220

Present 4.35 4.61 5.83

Ref. [2] 4.41 3.38 5.80

The values of kmn are tabulated in Ref. [2] with three significant figures for nine different boundary conditions for m ¼ 0:3 is also computed for m ¼ 0:33 and R1 =R2 ¼ 0:112 by the present authors to give more accurate numerical results in this work (see Table 1).

4. Boundary conditions The plate is clamped at r ¼ R1 inner radius and free at r ¼ R2 outside radius. Then the boundary conditions are: For r ¼ R1 wðr; hÞ ¼ 0

ð27Þ

ow ðr; hÞ ¼ 0 or

ð28Þ

For r ¼ R2 Mr ðr; hÞ ¼ 0

ð29Þ

Qr ðr; hÞ ¼ 0

ð30Þ

where Mr and Qr are the radial moment and shear force respectively.   2  ow ow o2 w Mr ¼
D þ þm or2 ror r2 oh2   o 2 ðr wÞ Qr ¼
D or

ð31Þ ð32Þ

where r2 ¼

o2 1 o 1 o2 þ þ or2 r or r2 oh2

The flexural rigidity parameters D, is given by: D¼

Eh3 12ð1
m2 Þ

ð33Þ

At the joints between sections, the continuity in plate deflection, slope, and radial moment have to be satisfied

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g1 ðR1 Þ ¼ g2 ðR3 Þ; dg1 dg2 ðR1 Þ ¼ ðR3 Þ; dr dr

g2 ðR3 þ lp Þ ¼ g3 ðR3 þ lp Þ dg2 dg3 ðR3 þ lp Þ ¼ ðR3 þ lp Þ dr dr

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ð34Þ ð35Þ

Mr1 ðR3 Þ ¼ Mr2 ðR3 Þ;

Mr2 ðR3 þ lp Þ ¼ Mr3 ðR3 þ lp Þ

ð36Þ

Qr1 ðR3 Þ ¼ Qr2 ðR3 Þ;

Qr2 ðR3 þ lp Þ ¼ Qr3 ðR3 þ lp Þ

ð37Þ

where index ri ; i ¼ 1; 2; 3 indicate the section of the plate.

5. Experimentation An experimental investigation was carried out to verify the applicability of the analytical approach discussed in this paper. The experimental system consisted of a thin circular aluminum plate with eight bonded piezoceramic actuators as shown in Fig. 2. A non-contact laser displacement sensor, Keyence LB-72, was used to measure the displacement at point A. The output data from the laser sensor were acquired by a data acquisition board and a personal computer. A function generator was used to provide a harmonic signal to an amplifier, KEPCO BOP 1000M, which supplied voltage to the piezoelectric actuators. The source of the excitation signal was a random signal from a signal analyzer. The transfer function between the applied electric voltage and the acceleration was measured to be compared the numerical results. As explained previously, the natural frequencies and mode shapes of a circular plate bonded with piezoelectric elements can be predicted analytically and numerically (FEA). Experimental modal, static and harmonic analysis have been conducted to verify the analytical and numerical approaches. The properties and the physical dimension of plate and piezoactuators are given in Table 2. Experimental results provide a basis for comparison between measured experimental

Fig. 2. Experimental set up.

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Table 2 Dimensions and material properties of plate and piezoelectric Aluminium 3003-H14

PZT (BM532)

Units

Density, 2730 E ¼ 69  109 m ¼ 0:33

7350 Ep ¼ 71:4  109 mp ¼ 0:3 d13 ¼ 200  10
12

kg/m3 Pa

R1 ¼ 0:01905 R2 ¼ 0:17 R3 ¼ 0:056 h ¼ 0:0008

hp ¼ 0:00025 Length, L ¼ 0:0508 Width, W ¼ 0:03683

v/m m m m m m m

natural frequencies and analytic and simulation frequencies for a circular plate of this size.

6. Results and comparisons In order to verify the proposed analysis and the finite element approach, numerical calculations were generated to compare with the experimental results from the circular plate with eight distributed piezoceramics bonded on its surface. The schematic of this plate is shown in Fig. 1 while the dimensions and the material properties are given in Table 2. The piezoelectric actuator patches were modeled using solid5 piezoelectric elements, and the aluminum circular plate by shell63 structural elements of a commercial finite element code (ANSYS). The static response of the structure was determined by applying a constant voltage to the five piezoelectric actuators. The non-contact laser displacement sensors recorded the deflections of the circular plate at the point A (node 988). Fig. 4 shows the comparison between the finite element analysis of the deflection of a circular plate and the experimental results. Good agreements between the two methods are observed. The natural frequencies and modes shapes can be predicted by Eq. (24). Experimental modal analysis has been conducted to verify the analytical and finite element prediction. The first five natural frequencies as obtained by analytical prediction, finite element and experimental modal testing are shown in Table 3 and Fig. 5. The modal testing was conducted by measuring the transfer function between the input voltage and the output displacement signals. The circular plate model analytically predicts 25.5 and 26.6 Hz for the first bending mode and first torsional mode; and these are validated by the modal testing results of 24.9 and 26.5 Hz. The discrepancy is within 2.3%. The experimental data listed in Table 3 and Fig. 5 show that the circular plate model can accurately predict the frequencies of an integrated piezoelectric actuator system, and that the prediction is validated by experiment. Finally, the frequency response of the structure with eight actuators subjected to dynamic piezoelectric actuation was also obtained. The frequency response of the nodal

E.M. Sekouri et al. / Mechatronics 14 (2004) 1007–1020

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Fig. 3. Frequency response of the nodal displacement at nodes 988 and 909.

Fig. 4. Tip deflection on the circular plate in terms of input voltage.

displacement at the point A (node 988) and the free end of the circular plate (node 909) are shown in Fig. 3. The responses at the two nodal locations appear to have the same response profile but differ in magnitude. The results obtained using the finite element method and experimental result are shown in Fig. 6. Comparable results were obtained. The discrepancy can be attributed to amplifier characteristics and the presence of environmental noise which was verified by observing the poor coherence between the applied voltage signal and the acceleration signal measured during the experiments.

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Table 3 Natural frequencies for circular plate with eight piezoceramic Natural frequencies (Hz)

Discrepancy (%)

Modes

Analytical

FEA

Experimental

DFEA

DExp

1 2 3 4 5

25.0 25.0 26.6 33.6 33.6

24.4 24.4 29.2 37.7 37.7

24.9 24.9 26.5 33.5 33.5

4.3 4.3 9.7 12.2 12.2

2.3 2.3 0.3 0.2 0.2

Fig. 5. Experimental modal analysis.

Fig. 6. Comparison between experimental and finite element of frequency response of the nodal displacement at point A (nodes 988).

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7. Conclusion An analytical approach for modeling a circular plate structure with integrated distributed piezoelectric actuators under static as well as dynamic mechanical or electrical loadings is developed. The mathematical solution based on the Kirchhoff plate model for free vibration is presented. The validity of the theory is established by examining static as well dynamic analysis of a circular plate containing eight bonded actuators. The equations governing the dynamics of the plate, relating the strains in the piezoelectric elements to the strain induced in the system, are derived for a circular plate using the partial differential equation. Modal analysis is done for determining the natural frequencies and mode shapes of the structures while harmonic analysis is performed to analyze the steady-state behavior of the structures subjected to cyclic sinusoidal loads. Numerical simulation results are obtained using finite element approach. Experiments using a thin aluminum circular plate structure with distributed piezoelectric ceramics PZT BM532 were also conducted to validate the analysis and the computer simulations. Relatively good agreements between the results of these three approaches are observed. Finally, The results show that the model can predict natural frequencies and mode shapes of the plate very accurately.

References [1] Southwell RV. On the free transverse vibrations of uniform circular disc clamped at its center; and on the effects of rotation. Proc Roy Soc (Lond) 1922;101:133–53. [2] Vogel SM, Skinner DW. Natural frequencies of transversely vibrating uniform circular plates. J Appl Mech 1965;32:926–31. [3] Bailey T, Hubbard JE. Distributed piezoelectric-polymer active vibration of a cantilever beam. J Guid Contr Dyn 1985;8(5):605–11. [4] Crawley EF, deLuis J. Use of piezoelectric actuators as elements of intelligent structures. AIAA J 1987;25(10):1373–85. [5] Fanson JL, Chen JC. Structural control by the use of piezoelectric active members. In: Proceedings of NASA/DOD Control-Structures Interaction Conference, NASA CP-2447, Part II, 1986. [6] Jiarang F, Jianqiao Y. Exact solutions for axisymmetric vibration of laminated circular plates. ASCE J Eng Mech 1990;116:920–7. [7] Dimitriads E, Fuller CR, Rogers CA. Piezoelectric actuators for distributed vibration excitation of thin plates. ASME J Appl Mech 1991;113:100–7. [8] Van Neikerk JL, Tongue BH, Packard AK. Active control of circular plate to reduce transient transmission noise. J Sound Vib 1995;183:643–62. [9] Heyliger PR, Ramirez G. Free vibration of laminated piezoelectric plates and discs. J Sound Vib 1999;229:935–56. [11] Wang Q, Quek ST, Sun CT, Liu X. Analysis of piezoelectric coupled circular plate. Smart Mater Struct 2001;10:229–39. [12] Tylikowski A. Control of circular plate vibrations via piezoelectric actuators shunted with a capacitive circuit. Thin-Walled Struct 2001;39:83–94. [13] Swanson, Inc., ANSYS User’s Manual, Version 5.7. Houston, PA: Swanson, Inc.; 2000. [14] Banks HT, Smith RC, Wang Y. The modeling of piezoceramic patch interactions with shells plates and beams. Quart Appl Math 1995;53(2):353–81. [15] Leissa A. Vibration of plates. New York: American Institute of Physics, 1993.

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[16] Kirchhoff. Uber das Gleichgewicht und die Bewegung einer elastischen Scheibe. Math J 1850;40(5):51–8. [17] Poisson SD. Memoires de l’Academie Royale des Sciences de l’Institut de France, serie 2, tome VIII, 357. L’equilibre et le mouvement des corps elastique, 1829.