# Dynamic Response of a Thick Piezoelectric Circular Cylindrical Panel: An Exact Solution

Authors: Atta Oveisi; Mohammad Gudarzi; Seyyed Mohammad Hasheminejad
Journal: Shock and Vibration
 (2014)

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## Abstract

One of the interesting fields that attracted many researchers in recent years is the smart structures. The piezomaterials, because of their ability in converting both mechanical stress and electricity to each other, are very applicable in this field. However, most of the works available used various inexact two-dimensional theories with certain types of simplification, which are inaccurate in some applications such as thick shells while, in some applications due to request of large displacement/stress, thick piezoelectric panel is needed and two-dimensional theories have not enough accuracy. This study investigates the dynamic steady state response and natural frequency of a piezoelectric circular cylindrical panel using exact three-dimensional solutions based on this decomposition technique. In addition, the formulation is written for both simply supported and clamped boundary conditions. Then the natural frequencies, mode shapes, and dynamic steady state response of the piezoelectric circular cylindrical panel in frequency domain are validated with commercial finite element software (ABAQUS) to show the validity of the mathematical formulation and the results will be compared, finally.


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## Body

## 1. Introduction

Piezoelectric materials have been extensively used as transducers and sensors due to their intrinsic direct and converse piezoelectric effects that take place between electric field and mechanical deformation. An important geometry in applied engineering problems is circular cylindrical panel because of its widespread application in actual structures such as aircraft wings, submarines, missiles, vessels, and high pressure cylindrical containers. The application of piezomaterial structures in this field is mainly concentrated on vibration suppression and acoustic noise reduction. Because of practical applications, piezoelectric circular cylindrical shells have attracted a considerable amount of research interests. Haskins and Walsh analyzed the free vibration of piezoelectric cylindrical shells with radially polarized transverse isotropy [1]; Martin investigated the vibration of longitudinally polarized piezoelectric cylindrical tubes and pointed out the limitations of the assumption [2]. Drumheller and Kalnins presented a coupled theory for the vibration of piezoceramic shells of revolution and analyzed the free axisymmetrical vibration of a circular cylindrical shell [3]. Burt simplified the circular cylinder to a two-dimensional model and then investigated the voltage response of radially polarized ceramic [4]. Tzou and Zhong gave a linear theory of piezoelectric shell vibration, which can be simplified to account for spheres [5]. Ebenezer and Abraham presented an Eigen function approach to determine the response of radially polarized piezoelectric cylindrical shells of finite length subjected to electrical excitation [6]. Many other researches by the methods of three-dimensional theory concentrated on the axisymmetrical and radial vibrations of cylinders, such as Stephenson [7, 8] and Adelman et al. [9, 10]. Paul derived the frequency equation of a piezoelectric cylindrical shell without presenting numerical results [11]. Paul and Venkatesan employed the same method to obtain the natural frequencies of infinite piezoelectric cylindrical shells [12]. However, some frequencies were missed in their calculation. Recently, Ding et al. exactly investigated the free vibration of hollow piezoelectric cylindrical shells on the basis of a decomposition formula for displacements, exactly [13]. Yang et al. considered the theory of the basic vibration characteristics of a circular cylindrical shell piezoelectric transducer [14]. They solved the vibration problem numerically for electrically forced case. Li et al. considered the spillover and harmonic effect in real active vibration control and they presented a novel composite controller based on disturbance observer (DOB) for the all-clamped panel [15]. Kumar and Singh aimed to examine through experiments vibration control of curved panel treated with optimally placed active or passive constrained layer damping patches and they found the optimum location for the application of ACLD/PCLD patches [16].The main subject of this study is to investigate the free and forced vibration of transversely isotropic piezoelectric cylindrical panels. Based on the general solution for coupled equations for piezoelectric media presented in Ding et al. [17], three-dimensional exact solutions are obtained through the variable separation method. A numerical example is finally presented.


## 2. Problem Formulation

For dynamic modelling of piezoelectric layers, two displacement functionsΨ and F are considered [17]. Figure 1 shows the panel geometry.Figure 1
Cylindrical panel and its geometry.


### 2.1. Basic Equations

In circular cylindrical coordinates(r,θ,z), if the media is axially polarized the general solution can be written as
(1)urPi=1r∂ψ∂θ-∂∂rA1F,uθPi=-∂Ψ∂r-1r∂∂θA1F,wPi=A2F,ϕ=A3F,
where urPi, uθPi, and wPi are three displacement components, ϕ is the electric potential, and the differential operators A1, A2, and A3 are
(2)A1=[(c13Pi+c44Pi)ε33+(e15+e31)e33]∂3∂z3+[(c13Pi+c44Pi)ε11+(e15+e31)e15]Λ∂∂z,A2=c44Piε33∂4∂z4+{[c11Piε33+c44Piε11+(e15+e31)2Λ-ρε33∂2∂t2]}∂2∂z2+c11Piε11ΛΛ-ρε11Λ∂2∂t2,A3=c44Pie33∂4∂z4+{∂2∂t2[c11Pie33+c44Pie15-(c13Pi+c44Pi)(e15+e31)]Λ-ρe33∂2∂t2}∂2∂z2+c11Pie15ΛΛ-ρe15Λ∂2∂t2,
where Λ=∂2/∂r2+(1/r)∂/∂r+(1/r2)∂2/∂θ2 is the two-dimensional Laplacian. The displacement functionsΨ and F must satisfy the following two equations:
(3)(c66ρΛ+c44Pi∂2∂z2-ρPi∂2∂t2)ψ=0,L0F=0,(4)L0=a4ΛΛΛ+(a3∂2∂z2+a6∂2∂t2)ΛΛ+(a2∂4∂z4+a5∂4∂t4+a7∂4∂z2∂t2)Λ+a1∂6∂z6+a8∂6∂z4∂t2+a9∂6∂z2∂t4.Herean(n=1,2,…,9) can be expressed in terms of elastic constants cijPi, dielectric constants εij, and piezoelectric coefficients eij as follows:
(5)a1=c44Pi(e332+c33Piε33),a4=c11Pi(e152+c44Piε11),a5=ρ2ε11,a2=c33Pi[c44Piε11+(e15+e31)2]+ε33[c11Pic33Pi+c442-(c11Pi+c44Pi)2]+e33[2c44Pie15+c11Pie33-2(c13Pi+c44Pi)(e15+e31)],a3=c44Pi[c11Piε33+(e15+e31)2]+ε11[c11Pic33+c442-(c13Pi+c44Pi)2]+e15[2c11Pie33+c44Pie15-2(c13Pi+c44Pi)(e15+e31)],a6=-ρ[e152+(c11Pi+c44Pi)ε11],a8=-ρ[e332+(c44Pi+c33Pi)ε33],a9=ρ2ε33,a7=-ρ[2e15e33+(c44Pi+c33Pi)ε11+(c11PI+c44Pi)ε33+(e15+e31)2].The circular cylindrical coordinates as well as a circular cylindrical panel with outer radiusbPi, inner radius aPi, circular center angle αPi, and length LPi are shown in Figure 1. If the panel is vibrating with a resonant frequency ω, the displacement functions can be assumed as
(6)F=r05c11ε33P(ξP)Θ(μθ)Z(βζP)eiωt,ψ=r02P4(ξP)Θ′(μθ)Z′(βζP)eiωt,
where ξPi=rPi/RPi, ζPi=zPi/LPi are the dimensionless coordinates in r and z directions and Θ′(μθPi) and Z′(βζPi) denote the derivation of Θ(μθPi) with respect to μθPi and the derivation of Z(βζPi) with respect to βζPi, respectively. In addition,
(7)Θ(μθ)=C1cos⁡(μθ)+C2sin⁡(μθ),Z(βζ)=C3sin⁡(βζ)+c4cos⁡(βζ),
where Cm(m=1,2,3,4) are constants. Substitution of (6) into (3) yields
(8)(Δ+k42)P4(ξ)=0,(9)(Δ+k12)(Δ+k22)(Δ+k32)P4(ξ)=0,
where Δ=∂2/∂(ξPi)2+(1/ξPi)∂/∂(ξPi)-μ2/(ξPi)2 and
(10)k42=Ω2c11pic66pi-γ2c44pic66pi,(ΩPi)2=ρPiω2r02c11,γ=βt1,t1=r0h0
and (kmPi)2, (m=1,2,3) (assuming Re[kmPi]≥0) are the eigenvalues of the following equation:
(11)a-4k6+(a-6(ΩPi)2+a-3γ2)k4+(a-2γ4+a-7γ2(ΩPi)2+a-5(ΩPi)4)k2+(a-1γ6+a-8γ4(ΩPi)2+γ2(ΩPi)4)=0in which(12)a-n=an((c11Pi)2ε33),(n=1,2,3,4),a-n=an(ρc11Piε33),(n=6,7,8),a-5=a5((ρPi)2ε33).
The solution of (9) can be assumed as
(13)P(ξ)=P1(ξ)+P2(ξ)+P3(ξ),
where Pm(ξPi) is obtained as [18].Substituting (6) into (1) gives the mechanical displacements and electric potential as follows:
(14)urpi=-r0[μξP4(ξ)+∑m=13α1mPm′(ξ)]Θ(μθ)Z′(βζ)eiωt,(15)uθpi=-r0[P4′(ξ)+μξ∑m=13α1mPm(ξ)]Θ′(μθ)Z′(βζ)eiωt,(16)wpi=r0[∑m=13α2mPm(ξ)]Θ(μθ)Z(βζ)eiωt,(17)Φ=r0c11piε33[∑m=13α3mPm(ξ)]Θ(μθ)Z(βζ)eiωt,
where
(18)α1m=-([(c13Pi+c44)(ε11km2+ε33γ2)+(e15+e31)(e15km2+e33γ2)(c13Pi+c44)(ε11km2+ε33γ2)]γ)×(c11Piε33)-1,α2m=([(c11Pikm2+c44Piγ2-c11PiΩ12)(ε11km2+ε33γ2)+(e15+e31)2km2γ2(c11Pikm2+c44Piγ2-c11PiΩ12)(ε11km2+ε33γ2)])×(c11Piε33)-1,α3m=([-(c13Pi+c44Pi)(e15+e31)2km2γ2](c11Pikm2+c44Piγ2-c11PiΩ12)(e15km2+e33γ2)-(c13Pi+c44Pi)(e15+e31)2km2γ2])×(c11Pic11Piε33)-1,(m=1,2,3).Utilizing the constitutive relations of piezoelectricity and (14)–(17), the stress components and electric displacement components can be derived as
(19)σrpi={∑m=13(c12α1mkm2+c13piγα2m+e31c11piε33γα3m)Pm(ξ)(c12pi-c11pi)[μξP4′(ξ)-μξ2P4(ξ)]+(c12pi-c11pi)∑m=13α1mPm′′(ξ)+∑m=13(c12α1mkm2+c13piγα2m+e31c11piε33γα3m)×Pm(ξ)∑m=13(c12α1mkm2+c13piγα2m+e31c11piε33γα3m)}Θ(μθ)Z′(βζ)eiωt,(20)σθpi={∑m=13(c11piα1mkm2+c13piγα2m+e31c11piε33γα3m)Pm(ξ)(c11pi-c12pi)[μξP4′(ξ)-μξ2P4(ξ)]+(c11pi-c12pi)∑m=13α1mPm′′(ξ)+∑m=13(c11piα1mkm2+c13piγα2m+e31c11piε33γα3m)×Pm(ξ)∑m=13(c11piα1mkm2+c13piγα2m+e31c11piε33γα3m)}Θ(μθ)Z′(βζ)eiωt,(21)σzpi=∑m=13[(c13piα1mkm2+c33piγα2m+e33c11piε33γα3m)×Pm(ξ)(c13piα1mkm2+c33piγα2m+e33c11piε33γα3m)]Θ(μθ)Z′(βζ)eiωt,(22)τθzpi={+μξ∑m=13[(c44piγα1m+c44piα2m+e15c11piε33α3m)Pm(ξ)]c44piγP4′(ξ)+μξ∑m=13[(c44piγα1m+c44piα2m+e15c11piε33α3m)×Pm(ξ)+μξ∑m=13[(c44piγα1m+c44piα2m+e15c11piε33α3m)]}Θ′(μθ)Z(βζ)eiωt,(23)τrzpi={[∑m=13(c44piγα1m+c44piα2m+e15c11piε33α3m)pM′(ξ)]c44piγμξQ(ξ)+[∑m=13(c44piγα1m+c44piα2m+e15c11piε33α3m)×pM′(ξ)∑m=13(c44piγα1m+c44piα2m+e15c11piε33α3m)]}Θ(μθ)Z(βζ)eiωt,(24)τrθpi=c66[+2μξ2∑m=13α1mPm(ξ)-2μξ∑m=13α1mpM′(ξ)-k42P4(ξ)-2P4′′(ξ)+2μξ2∑m=13α1mPm(ξ)-2μξ∑m=13α1mpM′(ξ)]×Θ′(μθ)Z′(βζ)eiωt,(25)Drpi={[∑m=13(e15γα1m+e15α2m+ε11c11piε33α3m)pM′(ξ)]e15γμξP4(ξ)+[∑m=13(e15γα1m+e15α2m+ε11c11piε33α3m)×pM′(ξ)∑m=13(e15γα1m+e15α2m+ε11c11piε33α3m)]}Θ(μθ)Z(βζ)eiωt,(26)Dθpi={[∑m=13(e15γα1m+e15α2m+ε11c11piε33α3m)Pm(ξ)]e15γP4′(ξ)+μξ[∑m=13(e15γα1m+e15α2m+ε11c11piε33α3m)×Pm(ξ)∑m=13(e15γα1m+e15α2m+ε11c11piε33α3m)]}Θ′(μθ)Z(βζ)eiωt,(27)Dzpi=[∑m=13(e31α1mkm2+e33γα2m-c11piε33γα3m)Pm(ξ)]×Θ(μθ)Z′(βζ)eiωt.


### 2.2. Boundary Conditions

The piezoelectric panel has 8 boundary conditions consist of 6 mechanical and 2 electrical ones.By considering generalized simply support boundary conditions atθi=0 and θi=α and (i=Pi) we will have
(28)wi=uri=0,σθi=0(i=Pi).Note that for piezoelectric layers the following condition is added(29)ϕ=0.One can take(30)C1i=0,C2i=1,μ=(2m+1)π2α,m=0,1,2,….And by considering generalized simply support boundary conditions atζi=0 and ζi=1 (i=Pi) we will have
(31)uri=uθi=0,σzi=0(i=Pi).And for piezoelectric layers the following condition is added(32)Dz=0.One can take(33)C3i=0,C4i=1,β=nπ,n=0,1,2,….Without loss of generality, we suppose that external force acts on the outer surface of the actuator and inner surface of sensor has free boundary condition. So, we have(34)σrPi=P,τrθPi=τrzPi=0,ϕ=Iatr=r4,σrPi=τrθPi=τrzPi=0,ϕ=0atr=r1.For obtaining steady state frequency response of the cylindrical panel under a harmonic external excitation, we must solve the following matrix equation:(35)[T]m×n{X}n×1={F}m×1,
where [T]m×n is the coefficient matrix. Consider
(36){X}n×1=[A1B1A2B2A3B3A4B4]
and Ai,  Bi,  i=1,2,3,4 are the unknown constants that are in (19)–(27).The vector{F}m×1 denotes the force vector that acts on the structure. This force consists of the surface force that is considered as disturbance and has the breed of mechanical force such as wind effect. The effect of controller unit in the dynamic response of the piezo-panel is considered as an external electrical potential applied on the upper surface of the panel. These two external forces acted on the structure independently; however summation of their effects on the whole structure is the same as the case that both of them act on the structure simultaneously. So
(37){F}n×2=[F1F2],{F1}n×1=[000000I(r0,θ,z,ω)0]T,{F2}n×1=[P(r0,θ,z,ω)0000000]T,
where P(r0,θ,z,ω) acting over the area (Lq≤x≤Lq+aq) on its top surface, while it is traction-free at the bottom surface. Thus
(38)P(r,θ,z,ω)=∑n=-∞∞∑m=0∞pnm(r,ω)Sin⁡(mπzl)ei(nθ+ωt),I(r,θ,z,ω)=∑n=-∞∞∑m=0∞ιnm(r,ω)Sin⁡(mπzl)ei(nθ+ωt),
where
(39)pnm(r,ω)={Anm(ω)Jn(Kr),K2>0,Anm(ω)rn,K2=0,Anm(ω)In(K-r),K2=-K-2<0,ιnm(r,ω)={Bnm(ω)Jn(Kr),K2>0,Bnm(ω)rn,K2=0,Bnm(ω)In(K-r),K2=-K-2<0
in which K=k2-(πm/L)2 and Jn and In denote the standard and modified cylindrical Bessel functions of first kind, respectively, and Anm(ω) and Bnm(ω) are the amplitude of the applied forces. Substituting (21), (25), and (26) into the mechanical condition (39) and substituting (27) or (19) into the electric condition (40) yields homogeneous equations with respect to coefficients Am and Bm, (m=1,2,3,4). After finding these unknown constants that are functions of m, n by replacing them in the displacement and stress and electric displacement of corresponding equations (19)–(27), all of the system variables will be determined easily. However for control purposes the voltage obtained from the piezolayer as a sensor is the measured output and it is calculated as
(40)q=∫AreaD-·dAArea,
where D-=Drr^+Dθθ^+Dzz^ is the electric displacement vector in the principle cylindrical coordinates. Area in the integration stands for the place that the sensor layer is active and voltage (control output) is measured and dAArea=(dz×dθ)r^ which simplifies the above equation as
(41)qr=∫θ1sθ2s∫z1sz2sDrdzdθ.Moreover, by considering the piezoelectric sensor layer as an electric capacityV=q/cPS, one can obtain
(42)V=1cPS∫θ1sθ2s∫z1sz2sDrdzdθ,
where cPS is the capacitance of the piezoelectric sensor.


## 2.1. Basic Equations

In circular cylindrical coordinates(r,θ,z), if the media is axially polarized the general solution can be written as
(1)urPi=1r∂ψ∂θ-∂∂rA1F,uθPi=-∂Ψ∂r-1r∂∂θA1F,wPi=A2F,ϕ=A3F,
where urPi, uθPi, and wPi are three displacement components, ϕ is the electric potential, and the differential operators A1, A2, and A3 are
(2)A1=[(c13Pi+c44Pi)ε33+(e15+e31)e33]∂3∂z3+[(c13Pi+c44Pi)ε11+(e15+e31)e15]Λ∂∂z,A2=c44Piε33∂4∂z4+{[c11Piε33+c44Piε11+(e15+e31)2Λ-ρε33∂2∂t2]}∂2∂z2+c11Piε11ΛΛ-ρε11Λ∂2∂t2,A3=c44Pie33∂4∂z4+{∂2∂t2[c11Pie33+c44Pie15-(c13Pi+c44Pi)(e15+e31)]Λ-ρe33∂2∂t2}∂2∂z2+c11Pie15ΛΛ-ρe15Λ∂2∂t2,
where Λ=∂2/∂r2+(1/r)∂/∂r+(1/r2)∂2/∂θ2 is the two-dimensional Laplacian. The displacement functionsΨ and F must satisfy the following two equations:
(3)(c66ρΛ+c44Pi∂2∂z2-ρPi∂2∂t2)ψ=0,L0F=0,(4)L0=a4ΛΛΛ+(a3∂2∂z2+a6∂2∂t2)ΛΛ+(a2∂4∂z4+a5∂4∂t4+a7∂4∂z2∂t2)Λ+a1∂6∂z6+a8∂6∂z4∂t2+a9∂6∂z2∂t4.Herean(n=1,2,…,9) can be expressed in terms of elastic constants cijPi, dielectric constants εij, and piezoelectric coefficients eij as follows:
(5)a1=c44Pi(e332+c33Piε33),a4=c11Pi(e152+c44Piε11),a5=ρ2ε11,a2=c33Pi[c44Piε11+(e15+e31)2]+ε33[c11Pic33Pi+c442-(c11Pi+c44Pi)2]+e33[2c44Pie15+c11Pie33-2(c13Pi+c44Pi)(e15+e31)],a3=c44Pi[c11Piε33+(e15+e31)2]+ε11[c11Pic33+c442-(c13Pi+c44Pi)2]+e15[2c11Pie33+c44Pie15-2(c13Pi+c44Pi)(e15+e31)],a6=-ρ[e152+(c11Pi+c44Pi)ε11],a8=-ρ[e332+(c44Pi+c33Pi)ε33],a9=ρ2ε33,a7=-ρ[2e15e33+(c44Pi+c33Pi)ε11+(c11PI+c44Pi)ε33+(e15+e31)2].The circular cylindrical coordinates as well as a circular cylindrical panel with outer radiusbPi, inner radius aPi, circular center angle αPi, and length LPi are shown in Figure 1. If the panel is vibrating with a resonant frequency ω, the displacement functions can be assumed as
(6)F=r05c11ε33P(ξP)Θ(μθ)Z(βζP)eiωt,ψ=r02P4(ξP)Θ′(μθ)Z′(βζP)eiωt,
where ξPi=rPi/RPi, ζPi=zPi/LPi are the dimensionless coordinates in r and z directions and Θ′(μθPi) and Z′(βζPi) denote the derivation of Θ(μθPi) with respect to μθPi and the derivation of Z(βζPi) with respect to βζPi, respectively. In addition,
(7)Θ(μθ)=C1cos⁡(μθ)+C2sin⁡(μθ),Z(βζ)=C3sin⁡(βζ)+c4cos⁡(βζ),
where Cm(m=1,2,3,4) are constants. Substitution of (6) into (3) yields
(8)(Δ+k42)P4(ξ)=0,(9)(Δ+k12)(Δ+k22)(Δ+k32)P4(ξ)=0,
where Δ=∂2/∂(ξPi)2+(1/ξPi)∂/∂(ξPi)-μ2/(ξPi)2 and
(10)k42=Ω2c11pic66pi-γ2c44pic66pi,(ΩPi)2=ρPiω2r02c11,γ=βt1,t1=r0h0
and (kmPi)2, (m=1,2,3) (assuming Re[kmPi]≥0) are the eigenvalues of the following equation:
(11)a-4k6+(a-6(ΩPi)2+a-3γ2)k4+(a-2γ4+a-7γ2(ΩPi)2+a-5(ΩPi)4)k2+(a-1γ6+a-8γ4(ΩPi)2+γ2(ΩPi)4)=0in which(12)a-n=an((c11Pi)2ε33),(n=1,2,3,4),a-n=an(ρc11Piε33),(n=6,7,8),a-5=a5((ρPi)2ε33).
The solution of (9) can be assumed as
(13)P(ξ)=P1(ξ)+P2(ξ)+P3(ξ),
where Pm(ξPi) is obtained as [18].Substituting (6) into (1) gives the mechanical displacements and electric potential as follows:
(14)urpi=-r0[μξP4(ξ)+∑m=13α1mPm′(ξ)]Θ(μθ)Z′(βζ)eiωt,(15)uθpi=-r0[P4′(ξ)+μξ∑m=13α1mPm(ξ)]Θ′(μθ)Z′(βζ)eiωt,(16)wpi=r0[∑m=13α2mPm(ξ)]Θ(μθ)Z(βζ)eiωt,(17)Φ=r0c11piε33[∑m=13α3mPm(ξ)]Θ(μθ)Z(βζ)eiωt,
where
(18)α1m=-([(c13Pi+c44)(ε11km2+ε33γ2)+(e15+e31)(e15km2+e33γ2)(c13Pi+c44)(ε11km2+ε33γ2)]γ)×(c11Piε33)-1,α2m=([(c11Pikm2+c44Piγ2-c11PiΩ12)(ε11km2+ε33γ2)+(e15+e31)2km2γ2(c11Pikm2+c44Piγ2-c11PiΩ12)(ε11km2+ε33γ2)])×(c11Piε33)-1,α3m=([-(c13Pi+c44Pi)(e15+e31)2km2γ2](c11Pikm2+c44Piγ2-c11PiΩ12)(e15km2+e33γ2)-(c13Pi+c44Pi)(e15+e31)2km2γ2])×(c11Pic11Piε33)-1,(m=1,2,3).Utilizing the constitutive relations of piezoelectricity and (14)–(17), the stress components and electric displacement components can be derived as
(19)σrpi={∑m=13(c12α1mkm2+c13piγα2m+e31c11piε33γα3m)Pm(ξ)(c12pi-c11pi)[μξP4′(ξ)-μξ2P4(ξ)]+(c12pi-c11pi)∑m=13α1mPm′′(ξ)+∑m=13(c12α1mkm2+c13piγα2m+e31c11piε33γα3m)×Pm(ξ)∑m=13(c12α1mkm2+c13piγα2m+e31c11piε33γα3m)}Θ(μθ)Z′(βζ)eiωt,(20)σθpi={∑m=13(c11piα1mkm2+c13piγα2m+e31c11piε33γα3m)Pm(ξ)(c11pi-c12pi)[μξP4′(ξ)-μξ2P4(ξ)]+(c11pi-c12pi)∑m=13α1mPm′′(ξ)+∑m=13(c11piα1mkm2+c13piγα2m+e31c11piε33γα3m)×Pm(ξ)∑m=13(c11piα1mkm2+c13piγα2m+e31c11piε33γα3m)}Θ(μθ)Z′(βζ)eiωt,(21)σzpi=∑m=13[(c13piα1mkm2+c33piγα2m+e33c11piε33γα3m)×Pm(ξ)(c13piα1mkm2+c33piγα2m+e33c11piε33γα3m)]Θ(μθ)Z′(βζ)eiωt,(22)τθzpi={+μξ∑m=13[(c44piγα1m+c44piα2m+e15c11piε33α3m)Pm(ξ)]c44piγP4′(ξ)+μξ∑m=13[(c44piγα1m+c44piα2m+e15c11piε33α3m)×Pm(ξ)+μξ∑m=13[(c44piγα1m+c44piα2m+e15c11piε33α3m)]}Θ′(μθ)Z(βζ)eiωt,(23)τrzpi={[∑m=13(c44piγα1m+c44piα2m+e15c11piε33α3m)pM′(ξ)]c44piγμξQ(ξ)+[∑m=13(c44piγα1m+c44piα2m+e15c11piε33α3m)×pM′(ξ)∑m=13(c44piγα1m+c44piα2m+e15c11piε33α3m)]}Θ(μθ)Z(βζ)eiωt,(24)τrθpi=c66[+2μξ2∑m=13α1mPm(ξ)-2μξ∑m=13α1mpM′(ξ)-k42P4(ξ)-2P4′′(ξ)+2μξ2∑m=13α1mPm(ξ)-2μξ∑m=13α1mpM′(ξ)]×Θ′(μθ)Z′(βζ)eiωt,(25)Drpi={[∑m=13(e15γα1m+e15α2m+ε11c11piε33α3m)pM′(ξ)]e15γμξP4(ξ)+[∑m=13(e15γα1m+e15α2m+ε11c11piε33α3m)×pM′(ξ)∑m=13(e15γα1m+e15α2m+ε11c11piε33α3m)]}Θ(μθ)Z(βζ)eiωt,(26)Dθpi={[∑m=13(e15γα1m+e15α2m+ε11c11piε33α3m)Pm(ξ)]e15γP4′(ξ)+μξ[∑m=13(e15γα1m+e15α2m+ε11c11piε33α3m)×Pm(ξ)∑m=13(e15γα1m+e15α2m+ε11c11piε33α3m)]}Θ′(μθ)Z(βζ)eiωt,(27)Dzpi=[∑m=13(e31α1mkm2+e33γα2m-c11piε33γα3m)Pm(ξ)]×Θ(μθ)Z′(βζ)eiωt.


## 2.2. Boundary Conditions

The piezoelectric panel has 8 boundary conditions consist of 6 mechanical and 2 electrical ones.By considering generalized simply support boundary conditions atθi=0 and θi=α and (i=Pi) we will have
(28)wi=uri=0,σθi=0(i=Pi).Note that for piezoelectric layers the following condition is added(29)ϕ=0.One can take(30)C1i=0,C2i=1,μ=(2m+1)π2α,m=0,1,2,….And by considering generalized simply support boundary conditions atζi=0 and ζi=1 (i=Pi) we will have
(31)uri=uθi=0,σzi=0(i=Pi).And for piezoelectric layers the following condition is added(32)Dz=0.One can take(33)C3i=0,C4i=1,β=nπ,n=0,1,2,….Without loss of generality, we suppose that external force acts on the outer surface of the actuator and inner surface of sensor has free boundary condition. So, we have(34)σrPi=P,τrθPi=τrzPi=0,ϕ=Iatr=r4,σrPi=τrθPi=τrzPi=0,ϕ=0atr=r1.For obtaining steady state frequency response of the cylindrical panel under a harmonic external excitation, we must solve the following matrix equation:(35)[T]m×n{X}n×1={F}m×1,
where [T]m×n is the coefficient matrix. Consider
(36){X}n×1=[A1B1A2B2A3B3A4B4]
and Ai,  Bi,  i=1,2,3,4 are the unknown constants that are in (19)–(27).The vector{F}m×1 denotes the force vector that acts on the structure. This force consists of the surface force that is considered as disturbance and has the breed of mechanical force such as wind effect. The effect of controller unit in the dynamic response of the piezo-panel is considered as an external electrical potential applied on the upper surface of the panel. These two external forces acted on the structure independently; however summation of their effects on the whole structure is the same as the case that both of them act on the structure simultaneously. So
(37){F}n×2=[F1F2],{F1}n×1=[000000I(r0,θ,z,ω)0]T,{F2}n×1=[P(r0,θ,z,ω)0000000]T,
where P(r0,θ,z,ω) acting over the area (Lq≤x≤Lq+aq) on its top surface, while it is traction-free at the bottom surface. Thus
(38)P(r,θ,z,ω)=∑n=-∞∞∑m=0∞pnm(r,ω)Sin⁡(mπzl)ei(nθ+ωt),I(r,θ,z,ω)=∑n=-∞∞∑m=0∞ιnm(r,ω)Sin⁡(mπzl)ei(nθ+ωt),
where
(39)pnm(r,ω)={Anm(ω)Jn(Kr),K2>0,Anm(ω)rn,K2=0,Anm(ω)In(K-r),K2=-K-2<0,ιnm(r,ω)={Bnm(ω)Jn(Kr),K2>0,Bnm(ω)rn,K2=0,Bnm(ω)In(K-r),K2=-K-2<0
in which K=k2-(πm/L)2 and Jn and In denote the standard and modified cylindrical Bessel functions of first kind, respectively, and Anm(ω) and Bnm(ω) are the amplitude of the applied forces. Substituting (21), (25), and (26) into the mechanical condition (39) and substituting (27) or (19) into the electric condition (40) yields homogeneous equations with respect to coefficients Am and Bm, (m=1,2,3,4). After finding these unknown constants that are functions of m, n by replacing them in the displacement and stress and electric displacement of corresponding equations (19)–(27), all of the system variables will be determined easily. However for control purposes the voltage obtained from the piezolayer as a sensor is the measured output and it is calculated as
(40)q=∫AreaD-·dAArea,
where D-=Drr^+Dθθ^+Dzz^ is the electric displacement vector in the principle cylindrical coordinates. Area in the integration stands for the place that the sensor layer is active and voltage (control output) is measured and dAArea=(dz×dθ)r^ which simplifies the above equation as
(41)qr=∫θ1sθ2s∫z1sz2sDrdzdθ.Moreover, by considering the piezoelectric sensor layer as an electric capacityV=q/cPS, one can obtain
(42)V=1cPS∫θ1sθ2s∫z1sz2sDrdzdθ,
where cPS is the capacitance of the piezoelectric sensor.


## 3. Results and Discussion

Table1 shows the first three nondimensional natural frequencies of some panels by different geometries. Mode shapes of the five first natural frequencies are shown in Figure 2. The panel dynamic responses under the aforementioned inputs (dynamic excitation and electric excitation) are shown in Figure 3 and are compared by FEM results.Table 1
First three nondimensional natural frequencies.
S
μ
=
1.8
μ
=
0.9
0.1
0.9366
1.8562
2.3634
0.5178
1.4580
1.8160
0.2
0.8266
1.7995
2.3514
0.5109
1.4144
1.8214
0.3
0.7214
1.7650
2.3616
0.5043
1.3565
1.8434
0.4
0.6271
1.7193
2.3889
0.4975
1.2960
1.8666
0.5
0.5408
1.6549
2.4273
0.4906
1.2385
1.8741Mode shapes of the five first natural frequencies.
(a)
First mode shape
(b)
Second mode shape
(c)
Third mode shape
(d)
Forth mode shape
(e)
Fifth mode shapePanel dynamic response. (a) Mechanical excitation; (b) electrical excitation.
(a)
(b)It is obvious that a good accommodation exist between analytical solution and FEM (ABAQUS) method. In addition, the dynamic response of the panel in 450 Hz is shown in Figure4. It can be seen that the dominant mode shape in this frequency is the third mode shape.The dynamic response of the panel at450Hz due to (a) mechanical excitation; (b) electrical excitation.
(a)
(b)


## 4. Conclusion

Based on the general solution of the coupled equations for a piezoelectric media, the displacement functions are expanded in terms of trigonometric functions inz and θ directions. Three-dimensional exact solutions for the free vibration of a piezoelectric circular cylindrical panel are then obtained under several boundary conditions. Also the forced vibration is solved. The natural frequencies are compared with previous works. The dynamic responses with mechanical and electrical excitation are validated with FEM and the mode shapes are shown.



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*Source: 592165-2014-05-27.xml*