Open access peer-reviewed chapter

Piezoelectric Actuators for Functionally Graded Plates- Nonlinear Vibration Analysis

By Farzad Ebrahimi

Submitted: July 10th 2012Reviewed: August 16th 2012Published: February 27th 2013

DOI: 10.5772/52407

Downloaded: 1402

1. Introduction

Functionally graded materials (FGMs) are a new generation of composite materials wherein the material properties vary continuously to yield a predetermined composition profile. These materials have been introduced to benefit from the ideal performance of its constituents, e.g., high heat/corrosion resistance of ceramics on one side, and large mechanical strength and toughness of metals on the other side. FGMs have no interfaces and are hence advantageous over conventional laminated composites. FGMs also permit tailoring of material composition to optimize a desired characteristic such as minimizing the maximum deflection for a given load and boundary conditions, or maximizing the first frequency of free vibration, or minimizing the maximum principal tensile stress. As a result, FGMs have gained potential applications in a wide variety of engineering components or systems, including armor plating, heat engine components and human implants. FGMs are now developed for general use as structural components and especially to operate in environments with extremely high temperatures. Low thermal conductivity, low coefficient of thermal expansion and core ductility have enabled the FGM materials to withstand higher temperature gradients for a given heat flux. Structures made of FGMs are often susceptible to failure from large deflections, or excessive stresses that are induced by large temperature gradients and/or mechanical loads. It is therefore of prime importance to account for the geometrically nonlinear deformation as well as the thermal environment effect to ensure more accurate and reliable structural analysis and design.

The concept of developing smart structures has been extensively used for active control of flexible structures during the past decade [1-3]. In this regard, the use of axisymmetric piezoelectric actuators in the form of a disc or ring to produce motion in a circular or annular substrate plate is common in a wide range of applications including micro-pumps and micro-valves [4, 5], devices for generating and detecting sound [6] and implantable medical devices [7]. They may also be useful in other applications such as microwave micro-switches where it is important to control distortion due to intrinsic stresses [8]. Also in recent years, with the increasing use of smart material in vibration control of plate structures, the mechanical response of FGM plates with surface-bonded piezoelectric layers has attracted some researchers’ attention. Since this area is relatively new, published literature on the free and forced vibration of FGM plates is limited and most are focused on the cases of the linear problem. Among those, a 3-D solution for rectangular FG plates coupled with a piezoelectric actuator layer was proposed by Reddy and Cheng [9] using transfer matrix and asymptotic expansion techniques. Wang and Noda [10] analyzed a smart FG composite structure composed of a layer of metal, a layer of piezoelectric and an FG layer in between, while He et al. [11] developed a finite element model for studying the shape and vibration control of FG plates integrated with piezoelectric sensors and actuators. Yang et al. [12] investigated the nonlinear thermo-electro-mechanical bending response of FG rectangular plates that are covered with monolithic piezoelectric actuator layers on the top and bottom surfaces of the plate. More recently, Huang and Shen [13] investigated the dynamics of an FG plate coupled with two monolithic piezoelectric layers at its top and bottom surfaces undergoing nonlinear vibrations in thermal environments. In addition, finite element piezothermoelasticity analysis and the active control of FGM plates with integrated piezoelectric sensors and actuators was studied by Liew et al. [14] and the temperature response of FGMs using a nonlinear finite element method was studied by Zhai et al. [15]. All the aforementioned studies focused on the rectangular-shaped plate structures. To the authors’ best knowledge, no researches dealing with the nonlinear vibration characteristics of the circular functionally graded plate integrated with the piezoelectric layers have been reported in the literature except the author's recent works in presenting an analytical solution for the free axisymmetric linear vibration of piezoelectric coupled circular and annular FGM plates [16-20] and investigating the applied control voltage effect on piezoelectrically actuated nonlinear FG circular plate [21] in which the thermal environment effects are not taken in to account.

Consequently, a non-linear dynamics and vibration analysis is conducted on pre-stressed piezo-actuated FG circular plates in thermal environment. Nonlinear governing equations of motion are derived based on Kirchhoff’s-Love hypothesis with von-Karman type geometrical large nonlinear deformations. Dynamic equations and boundary conditions including thermal, elastic and piezoelectric couplings are formulated and solutions are derived. An exact series expansion method combined with perturbation approach is used to model the non-linear thermo-electro-mechanical vibration behavior of the structure. Numerical results for FG plates with various mixture of ceramic and metal are presented in dimensionless forms. A parametric study is also undertaken to highlight the effects of the thermal environment, applied actuator voltage and material composition of FG core plate on the nonlinear vibration characteristics of the composite structure. The new features of the effect of thermal environment and applied actuator voltage on free vibration of FG plates and some meaningful results in this chapter are helpful for the application and the design of nuclear reactors, space planes and chemical plants, in which functionally graded plates act as basic elements.

Figure 1.

FG circular plate with two piezoelectric actuators.

2. Functionally graded materials (FGM)

Nowadays, not only can FGM easily be produced but one can control even the variation of the FG constituents in a specific way. For example in an FG material made of ceramic and metal mixture, we have:

Vm+Vc=1E1

in which V c and V m are the volume fraction of the ceramic and metallic part, respectively. Based on the power law distribution [22], the variation of V c vs. thickness coordinate (z) with its origin placed at the middle of thickness, can be expressed as:

Vc=(z/hf+1/2)n,n0E2

in which h f is the FG core plate thickness and n is the FGM volume fraction index (see Figure 1). Note that the variation of both constituents (ceramics and metal) is linear when n=1. We assume that the inhomogeneous material properties, such as the modulus of elasticity E, densityρ, thermal expansion coefficient αand the thermal conductivity κchange within the thickness direction z based on Voigt’s rule over the whole range of the volume fraction [23] while Poisson’s ratio υ is assumed to be constant in the thickness direction [24] as:

E(z)=(EcEm)Vc(z)+Em,ρ(z)=(ρcρm)Vc(z)+ρmα(z)=(αcαm)Vc(z)+αm,ν(z)=νκ(z)=(κcκm)Vc(z)+κmE3

where subscripts m and c refer to the metal and ceramic constituents, respectively. After substituting Vc from Eq. (2) into Eqs. (3), material properties of the FGM plate are determined in the power law form-the same as those proposed by Reddy and Praveen [22]:

Ef(z)=(EcEm)(z/hf+1/2)n+Em,ρf(z)=(ρcρm)(z/hf+1/2)n+ρm,κf(z)=(κcκm)(z/hf+1/2)n+κm,αf(z)=(αcαm)(z/hf+1/2)n+αmE4

3. Thermal environment

Assume a piezo-laminated FGM plate is subjected to a thermal environment and the temperature variation occurs in the thickness direction and 1D temperature field is assumed to be constant in the r-θ plane of the plate. In such a case, the temperature distribution along the thickness can be obtained by solving a steady-state heat transfer equation

ddz[κ(z)dTdz]=0E5

in which

κ(z)={κp(hf/2zhp+hf/2)κf(z)(hf/2zhf/2)κp(hphf/2zhf/2)E6
κ(z)={κp(hf/2zhp+hf/2)κf(z)(hf/2zhf/2)κp(hphf/2zhf/2)E7

where κ p and κ f are the thermal conductivity of piezoelectric layers and FG plate, respectively. Eq. (5) is solved by imposing the boundary conditions as

Tp|z=hp+hf/2=TUT˜p|z=hphf/2=TLE8

and the continuity conditions

Tp|z=hf/2=Tf|z=hf/2=T1,Tf|z=hf/2=T˜p|z=hf/2=T2,κpdTp(z)dz|z=hf/2=κcdTf(z)dz|z=hf/2,κpdT˜p(z)dz|z=hf/2=κmdTf(z)dz|z=hf/2E9

The solution of Eq.(5) with the aforementioned conditions can be expressed as polynomial series:

Tp(z)=T1+TUT1hp(zhf/2)E10
 
T˜p(z)=TL+T2TLhp(z+hf/2+hp)E11
 

and

Tf(z)=A0+A1(zhf+12)+A2(zhf+12)n+1+A3(zhf+12)2n+1+A4(zhf+12)3n+1+A5(zhf+12)4n+1+A6(zhf+12)5n+1+O(z)6n+1E12

where constants T 1 , T 2 and A j can be found in Appendix A.

4. Nonlinear piezo-thermo-electric coupled FG circular plate system

It is assumed that an FGM circular plate is sandwiched between two thin piezoelectric layers which are sensitive in both circumferential and radial directions as shown in Figure 1 and the structure is in thermal environment; also, the piezoelectric layers are much thinner than the FGM plate, i.e., h p << h f .An initial large deformation exceeding the linear range is imposed on the circular plate and a von-Karman type nonlinear deformation is adopted in the analysis. The von-Karman type nonlinearity assumes that the transverse nonlinear deflection w is much more prominent than the other two inplane deflections.

4.1. Nonlinear strain-displacement relations

Based on the Kirchhoff-Love assumptions, the strain components at distance z from the middle plane are given by

εrr=ε¯rr+zkrr,εθθ=ε¯θθ+zkθθ,εrθ=ε¯rθ+zkrθE13

where the z-axis is assumed positive outward. Hereε¯rr,ε¯θθ, ε¯rθare the engineering strain components in the median surface, andkrr,kθθ, krθare the curvatures which can be expressed in terms of the displacement components. The relations between the middle plane strains and the displacement components according to the von-Karman type nonlinear deformation and Sander's assumptions [25] are defined as:

ε¯rr=urr+12(wr)2,ε¯θθ=1ruθθ+urr+12(1rwθ)2,ε¯rθ=1rurθ+uθruθr+(1rwr)wθE14
κrr=2wr2,κθθ=1rwr1r22wθ2,κrθ=1r(2wrθ)+12r2wθE15

whereur,uθ,w represent the corresponding components of the displacement of a point on the middle plate surface. Substituting Eqs. (14) and (15) into Eqs. (13), the following expressions for the strain components are obtained

εrr=urr+12(wr)2z2wr2,εθθ=1ruθθ+urr+12(1rwθ)2z(1rwr+1r22wθ2),εrθ=1rurθ+uθruθr+(1rwr)wθ+2z(1r(2wrθ)+12r2wθ)E16

For a circular plate with axisymmetric oscillations, the strain expressions are simplified to

εrr=urr+12(wr)2z2wr2,εθθ=urrzrwr,εz=γrθ=γθz=γzr=0E17

4.2. Force and moment resultants

The stress components in the FG core plate in terms of strains based on the generalized Hooke’s Law using the plate theory approximation of σz0in the constitutive equations are defined as [26];

σrf=E(z)1ν2(εr+νεθ)E(z)α(z)1νΔTE18
σθf=E(z)1ν2(εθ+νεr)E(z)α(z)1νΔTE19

where E(z), ν(z) and α(z) are Young’s modulus, Poisson’s ratio and coefficient of thermal expansion of the FGM material, respectively, as expressed in Eq.(4), where ΔT=T(z)T0is temperature rise from the stress-free reference temperature (T0) which is assumed to exist at a temperature of T0=0and T(z)is presented in Eqs. (10)-(12).

The moments and membrane forces include both mechanical and electric components as

Nr=NrmNreNrt,Nθ=NθmNθeNθt,Mr=MrmMreMrt,Mθ=MθmMθeMθtE20

where the superscripts m, e, and t, respectively, denote the mechanical, electric, and temperature components. Mechanical forces and moments of the thin circular plate made of functionally graded material can be expressed as

(Nrm,Nθm)=hf/2hf/2(σrr,σθθ)dzE21
(Mrm,Mθm)=hf/2hf/2(σrr,σθθ)zdzE22
(Nrθm,Mrθm)=hf/2hf/2(1,z)σrθdzE23

Substituting Eqs. (13) and (18),(19) into Eqs. (22)-(23) gives the following constitutive relations for mechanical forces and moments of FG plate :

Nrm=D1(ε¯rr+νε¯θθ),Nθm=D1(ε¯θθ+νε¯rr)E24
Mrm=D2(κrr+νκθθ),Mθm=D2(κθθ+νκrr)E25
Nrt=Nθt=ht/2ht/2α(z)E(z)1vΔT(z)dzE26
Mrt=Mθt=ht/2ht/2α(z)E(z)1vΔT(z)zdzE27

in which the coefficients of D 1 and D 2 in the above equations are related to the plate stiffness and are given by

D1=hf/2hf/2Ef(z)1νf2dz,D2=hf/2hf/2z2Ef(z)1νf2dzE28

It is assumed that the piezoelectric layers are sensitive in both radial and circumferential directions and the piezoelectric permeability constants e31=e32. Hence, the electric membrane forces and bending moments are induced by the converse piezoelectric effect on the piezoelectric actuators, and these forces vary linearly across the plate thickness as [27];

Nre=Nθe=e31(Vzt+Vzb)/2,E29
Mre=Mθe=e31(hf+hp)(VztVzb)/2,E30

in which Vztand Vzbare the control voltages applied to the top and bottom piezoelectric layers, respectively.

4.3. System electromechanical equations

Axisymmetric free oscillation equations of the piezoelectric coupled circular FG plate in thermal environment can be derived from the generic piezoelectric shell equations using four system parameters: two Lame parameters, A 1 =1, A 2 =r, where r is the radial distance measured from the center, and two radii, R 1 =∞, R 2 =∞ [28,29] as

(rNr)rNθ=0E31
1r(rQrz)r+Nr2wr2+Nθθ(1rwθ)I12wt2=0E32

in which I1=(hf/2hf/2ρf(z)dz)and the transverse shear component Qrzis related to moments as

Qrz=1r[(rMr)rMθ]E33

Note that only the normal radial strain keeps the quadratic nonlinear term. Substituting all force/moment components and strain-displacement equations into the radial and transverse equations (31), (32) yields

rr[1rr(r2Nrm)]=Y2(wr)2+r[r2r(Nre+Nrt)]+νrr(Nre+Nrt)E34
D2rr(rr[1rr(rrw(r,t))])=I12wt2+1rr[rwr(NrmNreNrt)]1rr[rr(Mre+Mrt)]MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@931C@E35

in which Y=(hf/2hf/2Ef(z)dz)and boundary conditions at the center of the plate with axisymmetric oscillations are defined as

(1) Plate center (r=0): Slope:wr=0E36a
Radial force:Nrrm: finiteE36b

Boundary conditions for the simply supported (immovable) circumference are defined as:

Plate circumference (r=a):

w=0E37a
r(rNrm)νNrm=rr(Nre+Nrt)E37b
D2(2wr2+νrwr)=(Mre+Mrt)E37c

It is further assumed that the control potentials on the top and bottom piezoelectric actuators are of equal magnitudes and opposite signs, i.e., Vzt=Vzb=V^and the plate is subjected to a uniform temperature excitation of T(z). Accordingly, the electric and temperature induced forces and moments can be defined as:

Nre=Nθe=0E38a
Mre=Mθe=Me=e31(hf+hp)V^,E38b
Nθt=Nrt=Nt,E39a
Mθt=Mrt=Mt,E39b

Using these force and moment expressions, one can further simplify the open-loop plate equations and boundary conditions:

rr[1rr(r2Nrm)]=Y2(wr)2E40
D2rr(rr[1rr(rrw(r,t))])=I12wt2+1rr[rwr(NrmNt)]E41

Boundary conditions become

Plate center (r=0):

Slope:wr|r=0=0E42a
  Radial force:Nrm|r=0: finiteE42b

Plate circumference (r=a):

w|r=a=0E43a
[r(rNrm)νNrm]r=a=0E43b
[D2(2wr2+νrwr)]r=a=(Me+Mt)E43c

4.4. Simplification and Normalization

Solutions of the transverse displacement w and radial force Nrmof the above open-loop plate equations and boundary conditions can be expressed as a summation of a static component and a dynamic component as

w(r,t)=ws(r)+wd(r,t)E44a
Nrm(r,t)=Nrsm(r,t)+Nrdm(r,t)E44b

where ws(r)and Nrsm(r,t)are the static solutions, wd(r,t)and Nrdm(r,t)are the dynamic solutions, and the subscripts s and d, respectively, denote the static and dynamic solutions. Accordingly, the solution procedure can be divided into two parts. The first part deals with the nonlinear static solutions, and the second part deals with the dynamic solutions. In addition, normalized dimensionless quantities are adopted in the static and dynamic analyses. These dimensionless quantities are defined by known geometrical and material parameters [30]:

  • radial distance:y=(r/a)2,

  • transverse deflection:w¯s=3(1ν2)wshf,

  • slope: Xs(y)=ydw¯sdyMathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIfadaWgaaWcbaGaam4CaaqabaGccaGGOaGaamyEaiaacMcacqGH9aqpcaWG5bWaaSaaaeaacaWGKbGabm4DayaaraWaaSbaaSqaaiaadohaaeqaaaGcbaGaamizaiaadMhaaaaaaa@4171@

  • static force: Ysm(y)=(a2Nrsm/4D2)y

  • radial distance:x=(r/a),

  • dynamic deflection:w¯d=3(1ν2)wdhf,

  • dynamic force: Ydm(y)=(a2Nrdm/D2)

  • voltage: V=[3(1ν2)]1/2e31(hf+hp)V^/(2D2hf)MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAfacqGH9aqpdaWcgaqaamaadmaabaGaaG4maiaacIcacaaIXaGaeyOeI0IaeqyVd42aaWbaaSqabeaacaaIYaaaaOGaaiykaaGaay5waiaaw2faamaaCaaaleqabaGaaGymaiaac+cacaaIYaaaaOGaamyzamaaBaaaleaacaaIZaGaaGymaaqabaGccaGGOaGaamiAamaaBaaaleaacaWGMbaabeaakiabgUcaRiaadIgadaWgaaWcbaGaamiCaaqabaGccaGGPaGabmOvayaajaaabaWaaeWaaeaacaaIYaGaamiramaaBaaaleaacaaIYaaabeaakiaadIgadaWgaaWcbaGaamOzaaqabaaakiaawIcacaGLPaaaaaaaaa@528E@

  • temperature load: T*=(a2Nt/4D2)

Substituting these normalized dimensionless quantities into the open-loop plate equations and boundary conditions of axisymmetric plate oscillations and separating the static parts from the dynamic parts gives the static equations and dynamic equations with their associated boundary conditions:

  

(1) Static Equations and Boundary Conditions

y2d2Xsdy2=XsYsmT*yXsE45
y2d2Ysmdy2=12(Xs)2,0y1E46

Boundary conditions at center y = 0:

Xs|y=0=0E47a
Ysm|y=0=0E47b

Boundary conditions on circumference y = 1:

[(1+ν)Ysm2dYsmdy]y=1=0E48a
[(1ν)Xsy2dXsmdy]y=1=V|y=1E48b
 

(2) Dynamic Equations and Boundary Conditions

xx(1xx[x2Ydm])=2[dw¯sdxw¯dx12(w¯dx)2]E49
1xx{xx[1xx(xx(w¯d))]}=I1a4D22w¯dt2+1xx[xYdmdw¯sdx+4xYsmw¯dx+xYdmw¯dx]4T^1xx(xw¯dx)0<x<1MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A0BF@E50

Boundary conditions at center x = 0:

w¯dx|x=0=0E51a
Ydm|x=0=finiteE51b

Boundary conditions on circumference x = 1:

w¯d|x=1=0E52a
[νx(Ydm)+(1ν)Ydm]x=1=0E52b
[2w¯dx2+νw¯dx]x=1=0E52c
 

5. Static Solutions

For the nonlinear static equations and boundary conditions of the boundary value problem derived above, static solutions of slopes Xs(y)and forces Ysm(y)can be represented in (exact) series expansion forms [30]:

Xs(y)=i=1AiyiE53a
Ysm(y)=i=1Biyi0y1E53b

where A i and B i are constant coefficients. Substituting the series solutions, Eqs. (53a& b), into static equations, Eqs. (45) & (46), and grouping coefficients of y i one can obtain the recurrence equations for coefficients A i and B i :

Ai=1i(i1)j=1i1AjBijT^Ai1E54a
Bi=12i(i1)j=1i1AjAij,i=2,3,4,...E54b

It is observed that only A 1 and B 1 are independent constants, and the others are dependent. As long as A 1 and B 1 are determined by the boundary conditions, other coefficients A i and B i can be calculated from the recurrence equations. Accordingly, static series solutions are completed. The series solutions of Ysm(y)and Xs(y)satisfy the boundary conditions at y = 0, Eqs. (47a,b). Substituting the assumed series solutionsYsm(y)and Xs(y)into the boundary conditions at y = 1, Eqs. (48a,b), yields

i=1[(1+ν2i)Bi]=0E55a
i=1[(1ν2i)Ai]=VE55b

A i and B i can be determined from the nonlinear algebraic equations Eqs. (55a,b) using the Newton-Raphson iteration method [31]. Define

α(A1,B1)=i=1[(1ν2i)Ai]VE56a
β(A1,B1)=i=1[(1+ν2i)Bi]E56b
A¯1=A1+Δ1E57a
B¯1=B1+Δ2E57b

in which

Δ1=1Δ[β(A1,B1)B1α(A1,B1)α(A1,B1)B1β(A1,B1)]E58a
Δ2=1Δ[α(A1,B1)A1β(A1,B1)β(A1,B1)A1α(A1,B1)]E58b

and

Δ=det[A1α(A1,B1)B1α(A1,B1)A1β(A1,B1)B1β(A1,B1)]0E58c

A¯1and B¯1are, respectively, the iteration values of A 1 and B 1 ; Δ1and Δ2are the correction factors of A 1 and B 1 at each iteration. The partial derivativesα/A1,α/B1,β/A1andβ/B1, can be determined from the definitions of α(A1,B1)andβ(A1,B1). These iterations are repeated until they reach their prescribed limits, say|α|,|β|,|Δ1|and |Δ2|are smaller than 10-4. Accordingly, a set of A 1 and B 1 are determined for a set of given control voltages V and temperaturesT^. Using the recurrence equations, one can determine all other A i 's and B i 's, and further the nonlinear static solutions of slope Xs(y)and static forceYsm(y). Knowing the slope, one can determine the static deflections w¯sand wsof the nonlinear circular plate subject to voltage and temperature excitations.

6. Dynamic Solutions

It is assumed that the FG circular plate is oscillating in the vicinity of the nonlinearly deformed static equilibrium position. FG index, voltage and temperature effects to the natural frequencies and amplitude/frequency relations are investigated in this section. First, linearized eigenvalue equations are solved using the exact series solutions. Then nonlinear amplitude and frequency relations of nonlinear large amplitude free vibrations are investigated using the Galerkin method and the perturbation method.

6.1. Eigenvalue Equations

Neglect the nonlinear terms in the normalized dynamic equations, and then assume following harmonic solutions of displacement and dynamic force

w¯d(x,t)=Rd(x)sin(ωnt)E59a
Ydm(x,t)=Sd(x)sin(ωnt)E59b

where ωnis the natural frequency; Rd(x)and Sd(x)are the (linear) eigenfunctions or mode shape functions of w¯d(x,t)andYdm(x,t), respectively. Rd(x)defines the mode shape function, and Sd(x)defines the spatial force distribution. Both Rd(x)and Sd(x)have to satisfy the boundary conditions, and they are also assumed in the series expansion forms. Substituting Eqs. (59a, b) into the dynamic equations and boundary conditions, Eqs. (49)-(52), yields

xddx(1xddx[x2Sd(x)])=2dw¯sdxdRddxE60
1xddx{xddx[1xddx(xddx(Rd(x)))]}=λRd(x)+1xddx[2xSd(x)dw¯sdx+4xYsmdRddx]4T^1xddx[xdRddx],0<x<1MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@8C26@E61

where λis the eigenvalue andλ=I1a4D2ωn2. Boundary conditions become

  1. Center x=0:

dRddx|x=0=0E62a
Sd(x)|x=0:finiteE62b
  1. Circumference x=1:

Rd|x=1=0E63a
[νddx[Sd(x)]+(1ν)Sd(x)]x=1=0E63b
[d2Rddx2+νdRddx]x=1=0E63c

Again, assume the eigenfunctions take the series expansion forms:

Rd(x)=i=0aix2iE64a
Sd(x)=i=0bix2iE64b

where aiandbi, are constants determined by eigenvalue equations and boundary conditions. The series solutions Rd(x)andSd(x)satisfy the boundary conditions at x = 0, Eqs. (62a,b). Assume aiandbi, be represented by the linear combinations of independent constantsa0, a1andb0.

ai=fi1a0+fi2a1+fi3b0E65a
bi=gi1a0+gi2a1+gi3b0,i=1,2,3,...E65b

where fijand gijare to be determined. Substituting the series expressions of modes Rd(x)and forces Sd(x)into the dynamic equations, one can derive a set of recurrence equations of aiandbi. Then, using expressions of aiand biof Eqs. (65a, b), one can further determine the coefficients fijand gijf01=1,f02=f03=0;f11=0,f12=1,f13=0;gik=2i(i+1)j=1ijfjkAij+1,i=1,2,3,..., k=1,2,3.,g01=0,g02=0,g03=1;f21=λ/64,f23=A1/8,f33=[A1g13+4f23(B1T*)+A2]/36,f32={λ+32B2+16[A1g12+4f22(B1T)]}/24,f31=[A1g11+4f21(B1T)]/36,f22=(B1T*)/4f(i+2)k=[λfik16T*(i+1)2+8(i+1)×j=1i+1(2jfikBik+2+Ajg(ij+1)k)][4(i+1)(i+2)]2i=2,3,4,...,and k=1,2,3.

Substituting these coefficients into the boundary conditions at x = 1, one can obtain an explicit matrix representation of the eigenvalue equation.

[h11h12h13h21h22h23h31h32h33][a0a1b0]=[0]E66

where hijare defined by

h1k=i=0i1fikE67a
h2k=i=0[iν+i(2i1)]fik,k=1,2,3.E67b
h3k=i=0[2iν+(1ν)]gik,k=1,2,3.E67c

These hikcoefficients are functions of eigenvaluesλ, and accordingly, the determinant of the coefficient matrix leads to a nonlinear characteristic equation. Using the Newton-Raphson iteration method [31], one can calculate eigenvalues and furthermore natural frequencies and mode shape functions of the nonlinear FG circular plate.

6.2. Nonlinear Large Amplitude Free Vibrations

In this section, the perturbation method is used to investigate the nonlinear large amplitude effect to natural frequencies of the piezoelectric laminated FG circular plate. Assume an approximate solution of the nonlinear response w¯d(x,t)be a product of a spatial function w¯d*(x)and a temporal functionf(t);

w¯d(x,t)=w¯d*(x)f(t)E68

where w¯d*(x)is a test function satisfying the boundary conditions:

  1. Plate center x=0:w¯d*x|x=0=0

  2. Plate circumference x=1:w¯d*|x=1=0,

[2w¯d*x2+νw¯d*x]x=1=0E69

Substituting Eq. (68) into the dynamic equation Eq. (49) and using the Galerkin method, one can derive a nonlinear equation for f(τ)

f,ττ+f(τ)+μ1f2(τ)+μ2f3(τ)=0E70

where τ=ωtand μ1and μ2are the nonlinear coefficient functions, ω is defined by

ω2=(c1+c3)/c2.E71

c1,c2and c3are defined by integrals:

c1=01{w¯d*(x)ddx{xddx[1xddx(xddx(w¯d*(x)))]}}dxE72a
c2=02(w¯d*(x))2xdxE72b
c3=01w¯d*(x)ddx[4xYsmdw¯d*dx+2xNrd*ldw¯sdx4T*xdw¯d*dx]dxE72c

in which Nrd*l(x)and Nrd*n(x)are the linear and nonlinear force components. The nonlinear coefficient functions μ1and μ1are defined by

μ1=c4/(c1+c3)E73a
μ2=c5/(c1+c3)E73b

where c4and c5are defined by integrals:

c4=01w¯d*(x)ddx[2xNrd*n(x)dw¯s(x)dx+Nrd*l(x)dw¯d*(x)dx]dxE74a
c5=01w¯d*(x)ddx[xNrd*n(x)dw¯d*(x)dx]dxE74b

The linear and nonlinear force components are written as

Nrd*l(x)=1x201G(x,ξ)1ξdw¯sdξdw¯d*dξdξE75a
N¯rd*n(x)=14x201G(x,ξ)1ξ(dw¯d*dξ)2dξE75b

The kernel function

G(x,ξ)={[1((1+ν)/(1ν))ξ2]x2xξ,[1((1+ν)/(1ν))x2]ξ2xξ,E76

Since the mode shape functions of the linear free vibrations were determined previously, it is convenient to select the mode shape functions as the trial functions,

w¯d*(x)=Rd(x),E77a
Nrd*l(x)=Sd(x),0x1;E77b

and the frequency can be selected as the natural frequencies determined previously. Using the Krylov-Bogoliubov-Mitropolsky perturbation method [32] and solving the nonlinear dynamic equation Eq. (70), one can obtain the amplitude-frequency relation as

p=ωωn=1+124(9μ210μ12)a^2+58μ1μ2a^3+...E78

where ωis the (nonlinear vibration) frequency; ωnis the natural frequency; a^is the dimensionless vibration amplitude; and μ1and μ2are the nonlinear coefficient functions. Note that the ratio is unity, i.e.,p=ω/ωn=1, if the system is linear. Once the coefficients μ1and μ2are calculated, one can further evaluate the amplitude and frequency relations of the nonlinear circular plate subjected to temperature excitations and control voltages.

7. Results and Discussions

Temperature effects of nonlinear static deformations, control voltages, and linear and nonlinear free vibrations of a simply supported piezoelectric laminated functionally graded circular plate are investigated in this section.

7.1. Comparison studies

To ensure the accuracy of the present analysis, an illustrative example is solved. The relevant material properties are listed in Table 1. Since there are no appropriate comparison results available for the problems being analyzed in this chapter, we decided to verify the validity of the obtained results by comparing with those of the FEM results. Our FEM model for piezo-FG plate consists of a 3D 8-noded solid element with number of total nodes 26950, number of total elements 24276, 4 DOF per node (3 translation, temperature) in the host plate element and 6 DOF per node (3 translation, temperature, voltage and magnetic properties) in the piezoelectric element. The finite element model has been programmed by the authors, while the standard bilinear interpolations have been employed in finite element approximations.

Table 2 compares the present results of normalized dimensionless central deflections Ws=3(1ν2)(ws/hf)with finite element solutions in analyzing the effect of normalized dimensionless piezoelectric voltages V=[3(1ν2)]1/2e31(hf+hp)V*/(2D2hf)to the normalized dimensionless center deflections at various normalized temperatures (T*=(a2Nt/4D2)) in which a nonlinear deflection-voltage relationship can be observed. As seen from Table 2 the maximum estimated difference of the proposed solution with finite element method is about 0.079%, and a close correlation between these results validates the proposed method of solution.

In general, a higher temperature induces higher deflections of the plate, and the deflection at each temperature is attenuated when the control voltage increases and the effect of imposed voltage on the center deflection is nonlinear and this effect is predominant in lesser voltage amounts. This effect can also be seen in the case of considering the temperature environment effect. For example, when T*=0.2by increasing the imposed voltage from 0.6 to 1.2 (100%) the normalized dimensionless center deflections increases about 46.8%, while it increases about 34.5% when the imposed voltage increases from 1.2 to 2.4 (100%). In the case of T*=0.5by increasing the imposed voltage from 0.6 to 1.2 (100%) the normalized dimensionless center deflection increases about 36.5% while it increases about 28.7% when the imposed voltage increases from 1.2 to 2.4 (100%).

 
Material                                                                         Property
E(GPa)ρ(kg/m3)να(1/C)κ(W/mK)d31,d32(m/V)
Aluminum7027070.323e-6204-
Alumina38038000.37.4e-610.4-
PZT6376000.31.2e-40.171.79e-10

Table 1.

Material properties [13].

 

Figure 2.

Effect of applied voltage to the normalized center deflection for various normalized temperatures T * = ( a 2 N t / 4 D 2 ) (Metal plate)

NormalizedVoltage (V)Normalized Temperature( T * )
T*=0T*=0.2
PresentFEMDiff. (%)PresentFEMDiff. (%)
0000000
0.40.35370.35380.0410.86590.86630.044
0.80.59820.59850.0481.13471.13520.046
1.20.76810.76850.0551.31181.31260.060
1.60.89250.89300.0571.46011.46100.062
20.99240.99300.0651.55811.55920.070
2.41.07771.07840.0681.64581.64700.074
2.81.15091.15170.0711.76531.76660.076
T*=0.5T*=0.8
PresentFEMDiff. (%)PresentFEMDiff. (%)
0000000
0.40.66330.66360.0430.86590.86630.044
0.80.95690.95730.0451.13471.13520.046
1.21.13801.13870.0581.31181.31260.060
1.61.27461.27540.0601.46011.46100.062
2.01.37761.37850.0681.55811.55920.070
2.41.46431.46530.0721.64581.64700.074
2.81.55101.55220.0751.76531.76660.076

Table 2.

Values of the normalized dimensionless center deflections with respect to the normalized dimensionless piezoelectric voltages for various normalized temperatures computed by two methods (present series solution and FEM) (v=0.3, n=1000)

7.2. Parametric studies

Having validated the foregoing formulations, we began to study the large amplitude vibration behavior of FG laminated circular plate subjected to thermo-electro-mechanical loading. The results for laminated plates with isotropic substrate layers (that is, the substrate is purely metallic or purely ceramic) and with graded substrate layers (various n) are given in both tabular and graphical forms.

To investigate the effect of the applied actuator voltage on the non-linear thermo-electromechanical vibration, the nonlinear normalized center deflection of various graded plates under various applied normalized voltages is tabulated in Table 3.
NormalizedTemp. ( T * )FGM index (n) / Normalized Voltage (V)
Metaln=10
 V=0.2  V=0.3  V=0.5  V=1V=0.2 V=0.3 V=0.5 V=1
0.00.18200.27240.39240.69250.12060.18050.26010.4590
0.20.27380.38690.52130.85730.18150.25640.34550.5681
0.40.36540.50100.64210.99490.24220.33200.42550.6593
0.60.47060.61690.75961.11640.31190.40890.50340.7399
0.80.60710.74000.88081.23290.40240.49050.58380.8171
(T*)n=0.5
 V=0.2 V=0.3 V=0.5 V=1 V=0.2 V=0.3  V=0.5  V=1
0.00.09920.14840.21380.37730.09390.14050.20240.3572
0.20.14920.21080.28400.46700.14130.19960.26890.4422
0.40.19910.27290.34980.54200.18850.25840.33120.5132
0.60.25640.33610.41380.60820.24280.31830.39190.5759
0.80.33080.40320.47990.67160.31320.38180.45440.6360
(T*)Ceramic (n=0)
 V=0.2  V=0.3  V=0.5  V=1 V=0.2  V=0.3  V=0.5  V=1
0.00.08380.12550.18070.31900.07160.10720.15440.2725
0.20.12610.17820.24010.39480.10780.15220.20510.3373
0.40.16830.23070.29570.45820.14380.19710.25270.3915
0.60.21680.28410.34990.51420.18520.24280.29890.4393
0.80.27960.34090.40570.56780.23890.29120.34660.4851

Table 3.

FGM index and normalized voltage effects to the nonlinear center deflection

Figure 3.

Normalized Temperature effects to the center deflection for various values of Voltages (n=0.5)

Figure 4.

Normalized Temperature effects to the center deflection for various values of Voltages (n=10)

For instance, Figs. 3 and 4. depict the normalized temperature and voltage effects on the center deflection of two graded plates (n=0.5 n=10). It shows that increasing the normalized temperature makes the center deflection increase in various voltages, but this effect is predominant at higher voltages. Figures 5~6 depict the effect of FGM index on the non-linear thermo-electro-mechanical behavior (center deflection) of FGM plates with different normalized applied voltages in logarithmic scale. It is also obvious from these figures that, by increasing the material gradients, the normalized center deflection would be increased in various temperature fields, and it is also demonstrated that larger thermal gradients will lead to greater deflections. This trend can be seen in various material gradients, which means that the non-linear deflection can be controlled by applying the appropriate voltage in the piezoelectric actuator layers.

Figure 5.

FGM index effects on nonlinear center deflection for various normalized temperature (V=0.2)

Figure 6.

FGM index effects on nonlinear center deflection for various normalized temperature (V=1)

NormalizedVoltage (V)FGM index (n) / Normalized Temp. (T*)
Metaln=10
Normalized Temperature (T*)Normalized Temperature (T*)
00.20.50.800.20.50.8
0.04.88914.42493.52932.36695.53595.01033.99622.6800
0.45.57965.674525.927356.497126.31786.42536.71167.3567
0.86.71417.263088.005958.718897.60248.22409.06529.8724
1.27.77998.4529.3700210.15648.80929.570210.609711.5001
1.68.63949.3390310.351111.28099.782410.574611.720612.7734
2.09.332910.139911.244912.228510.567711.481412.732613.8464
2.49.941410.873412.035113.024111.256712.312013.627414.7472
2.810.510411.449912.603613.711611.901012.964714.271115.5257
00.20.50.800.20.50.8
0.07.41216.70835.35063.588311.288210.21658.14875.4648
0.48.45898.60288.98619.849912.882513.101713.685415.0009
0.810.178911.011112.137413.218215.502016.769418.484620.1307
1.211.794712.813614.205415.397517.962719.514521.634023.4497
1.613.097714.158415.692717.102319.947221.562523.899226.0460
2.014.149115.372517.047818.538921.548323.411625.962928.2339
2.415.071616.484518.245819.745122.953325.105127.787430.0708
2.815.934217.358519.107620.787424.267026.436229.099931.6582

Table 4.

FGM index and normalized temperature effects to the first natural frequency for various normalized voltages.

We examine in this section the effect of control voltages and thermal environment on the vibration characteristics of the piezoelectric laminated circular FG plate for various FGM indexes. To this end, Table 4 as well as the Figures 4.1~4.3 show the nonlinear relationships between the first natural frequenciesωia2I1/D2, versus the normalized temperature in various normalized control voltagesV. These free vibrations are assumed to be in the vicinity of the nonlinearly deformed static equilibrium position.

Also, the effect of normalized temperature on the first natural frequency of the FG circular plate for various FGM indexes under various normalized control voltage is investigated and tabulated in Table 4, while the voltage-dependent first natural frequency changes are plotted in Figures 7~9 for various temperatures. It is seen that the imposed voltage has a significant effect on the first natural frequency of the structure, and by increasing the imposed voltage, the first natural frequency increases in a nonlinear manner. For instance, for the FGM plate with n=10 by increasing the imposed voltage from 0 to 0.2 the first natural frequency increases about 4.84%, while by increasing the voltage from 0.2 to 0.3 the first natural frequency increases about 15.12%.

It is seen that the imposed thermal environment has a significant effect on the first natural frequency of the structure, and by increasing the imposed temperature, the first natural frequency decreases in a nonlinear manner. However, this thermal tendency of decreasing the natural frequency can be compensated and corrected with the control voltages V, as shown in Fig. 7. ~ 8.

Frequency variations of large amplitude oscillations with temperature and applied voltage changes are also investigated and plotted in Figure 6. There are two sets of curves in this figure. The first set has no control voltages, and the second set has control voltages. It is observed that the control voltages actually reduce the nonlinear frequency and amplitude ratios, i.e.,P1. Accordingly, the nonlinear frequency and amplitude ratios can be actively controlled and the nonlinear effects reduced, i.e., the ratio is approaching to 1- the linear case. Other studies of the second natural frequency also suggest that the second natural frequency exhibits very much similar phenomena of the first natural frequency.

Figure 7.

Effect of normalized voltage to first natural frequency for various FGM indexes (T*=0.2)

Figure 8.

Effect of normalized voltage to first natural frequency for various FGM indexes (T*=0.8)

Figure 9.

Effects of normalized temperature on the normalized first natural frequency for various normalized voltages (n=10,n=0.5)

8. Summary and Conclusions

A piezoelectric bounded circular FG plate subjected to temperature changes and control voltages is investigated based on classical plate theory, including the effects of the thermal gradient, piezothermoelasticity and von Karman type geometric nonlinearity. Nonlinear coupled open-loop plate equations in radial and transverse oscillations were derived first, and then the equations were simplified to an axisymmetric oscillation case. An exact solution technique based on series-type solutions is used to obtain piezothermoelastic solutions for nonlinear static deformations and natural frequencies of the FG circular plate subjected to temperature and voltage excitations. Voltage controlled natural frequencies of the first mode at various temperatures are studied. It is observed that a higher temperature induces higher deflections of the plate, and the deflection at each temperature is attenuated when the control voltage increases, but this effect is predominant in higher voltages. Also by increasing the FGM gradient index the normalized center deflection will increase in a nonlinear manner in various temperature fields. It is seen that the imposed thermal environment has a significant effect on the natural frequency of the structure, and by increasing the imposed temperature, the natural frequency decreases in a nonlinear manner for various FGM indexes; this effect is predominant at higher temperatures. Both the nonlinear static deflections and natural frequencies are influenced by the temperatures and control voltages geometric and the static control voltages can be used to compensate nonlinear deflections.

Figure 10.

Temperature/control effects on amplitude dependent first natural frequency (for metal plate) – amplitude ratio: w/hf, frequency ratio: (ω /ω1)

Appendix A:

T1=T2+(TUTL)κcd+κmκm(T2TL)MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadsfadaWgaaWcbaGaaGymaaqabaGccqGH9aqpcaWGubWaaSbaaSqaaiaaikdaaeqaaOGaey4kaSIaaiikaiaadsfadaWgaaWcbaGaamyvaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaadYeaaeqaaOGaaiykaiabgkHiTmaalaaabaGaeqOUdS2aaSbaaSqaaiaadogaaeqaaOGaamizaiabgUcaRiabeQ7aRnaaBaaaleaacaWGTbaabeaaaOqaaiabeQ7aRnaaBaaaleaacaWGTbaabeaaaaGccaGGOaGaamivamaaBaaaleaacaaIYaaabeaakiabgkHiTiaadsfadaWgaaWcbaGaamitaaqabaGccaGGPaaaaa@52B2@,    T2=TL+[κmchf(TUTL)]/[κphp+κcd+κmchf]MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@5BC7@

A0=T2,          A1=T1T2cMathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaaWcbaGaaGymaaqabaGccqGH9aqpdaWcaaqaaiaadsfadaWgaaWcbaGaaGymaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaikdaaeqaaaGcbaGaam4yaaaaaaa@3E20@                             A2=T1T2cκcm(N+1)κmMathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcqGHsisldaWcaaqaaiaadsfadaWgaaWcbaGaaGymaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaikdaaeqaaaGcbaGaam4yaaaadaWcaaqaaiabeQ7aRnaaBaaaleaacaWGJbGaamyBaaqabaaakeaacaGGOaGaamOtaiabgUcaRiaaigdacaGGPaGaeqOUdS2aaSbaaSqaaiaad2gaaeqaaaaaaaa@4979@

A3=T1T2cκcm2(2N+1)κm2MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaaWcbaGaaG4maaqabaGccqGH9aqpdaWcaaqaaiaadsfadaWgaaWcbaGaaGymaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaikdaaeqaaaGcbaGaam4yaaaadaWcaaqaaiabeQ7aRnaaDaaaleaacaWGJbGaamyBaaqaaiaaikdaaaaakeaacaGGOaGaaGOmaiaad6eacqGHRaWkcaaIXaGaaiykaiabeQ7aRnaaDaaaleaacaWGTbaabaGaaGOmaaaaaaaaaa@4AC3@                               A4=T1T2cκcm3(3N+1)κm3MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaaWcbaGaaGinaaqabaGccqGH9aqpcqGHsisldaWcaaqaaiaadsfadaWgaaWcbaGaaGymaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaikdaaeqaaaGcbaGaam4yaaaadaWcaaqaaiabeQ7aRnaaDaaaleaacaWGJbGaamyBaaqaaiaaiodaaaaakeaacaGGOaGaaG4maiaad6eacqGHRaWkcaaIXaGaaiykaiabeQ7aRnaaDaaaleaacaWGTbaabaGaaG4maaaaaaaaaa@4BB4@

A5=T1T2cκcm4(4N+1)κm4MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaaWcbaGaaGynaaqabaGccqGH9aqpdaWcaaqaaiaadsfadaWgaaWcbaGaaGymaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaikdaaeqaaaGcbaGaam4yaaaadaWcaaqaaiabeQ7aRnaaDaaaleaacaWGJbGaamyBaaqaaiaaisdaaaaakeaacaGGOaGaaGinaiaad6eacqGHRaWkcaaIXaGaaiykaiabeQ7aRnaaDaaaleaacaWGTbaabaGaaGinaaaaaaaaaa@4ACB@                              A6=T1T2cκcm5(5N+1)κm5MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaaWcbaGaaGOnaaqabaGccqGH9aqpcqGHsisldaWcaaqaaiaadsfadaWgaaWcbaGaaGymaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaikdaaeqaaaGcbaGaam4yaaaadaWcaaqaaiabeQ7aRnaaDaaaleaacaWGJbGaamyBaaqaaiaaiwdaaaaakeaacaGGOaGaaGynaiaad6eacqGHRaWkcaaIXaGaaiykaiabeQ7aRnaaDaaaleaacaWGTbaabaGaaGynaaaaaaaaaa@4BBC@

where

κcm=κcκmMathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeQ7aRnaaBaaaleaacaWGJbGaamyBaaqabaGccqGH9aqpcqaH6oWAdaWgaaWcbaGaam4yaaqabaGccqGHsislcqaH6oWAdaWgaaWcbaGaamyBaaqabaaaaa@413E@

c=11N+1κcmκm+12N+1(κcmκm)213N+1(κcmκm)3+14N+1(κcmκm)415N+1(κcmκm)5MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@7B82@

d=1κcmκm+(κcmκm)2(κcmκm)3+(κcmκm)4(κcmκm)5MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@6866@

Acknowledgments

The work described in this chapter was funded by a grant from International University of Imam Khomeini (Grant No. 385022-1391) The author is grateful for this financial support.

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Farzad Ebrahimi (February 27th 2013). Piezoelectric Actuators for Functionally Graded Plates- Nonlinear Vibration Analysis, Piezoelectric Materials and Devices - Practice and Applications, Farzad Ebrahimi, IntechOpen, DOI: 10.5772/52407. Available from:

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