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

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.


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.

Problem Formulation
For dynamic modelling of piezoelectric layers, two displacement functions Ψ and  are considered [17].Figure 1 shows the panel geometry.

Basic Equations.
In circular cylindrical coordinates (, , ), if the media is axially polarized the general solution can be written as where    ,    , and   are three displacement components,  is the electric potential, and the differential operators A 1 , A 2 , and A 3 are where Λ =  2 / 2 + (1/)/ + (1/ 2 ) 2 / 2 is the twodimensional Laplacian.The displacement functions Ψ and  must satisfy the following two equations: Shock and Vibration 3 Here   ( = 1, 2, . . ., 9) can be expressed in terms of elastic constants    , dielectric constants   , and piezoelectric coefficients   as follows: The circular cylindrical coordinates as well as a circular cylindrical panel with outer radius   , inner radius   , circular center angle   , and length   are shown in Figure 1.If the panel is vibrating with a resonant frequency , the displacement functions can be assumed as where   =   /  ,   =   /  are the dimensionless coordinates in  and  directions and Θ  (  ) and   (  ) denote the derivation of Θ(  ) with respect to   and the derivation of (  ) with respect to   , respectively.In addition, where   ( = 1, 2, 3, 4) are constants.Substitution of ( 6) into (3) yields where Δ =  , and (   ) 2 , ( = 1, 2, 3) (assuming Re[   ] ≥ 0) are the eigenvalues of the following equation: in which , ( = 6, 7, 8) , The solution of (9) can be assumed as where   (  ) is obtained as [18].Substituting ( 6) into (1) gives the mechanical displacements and electric potential as follows: where , ( = 1, 2, 3) . ( Utilizing the constitutive relations of piezoelectricity and ( 14)-( 17), the stress components and electric displacement components can be derived as  By considering generalized simply support boundary conditions at   = 0 and   =  and ( = ) we will have Note that for piezoelectric layers the following condition is added One can take And by considering generalized simply support boundary conditions at   = 0 and   = 1 ( = ) we will have And for piezoelectric layers the following condition is added One can take 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 For obtaining steady state frequency response of the cylindrical panel under a harmonic external excitation, we must solve the following matrix equation: where [] × is the coefficient matrix.Consider and   ,   ,  = 1, 2, 3, 4 are the unknown constants that are in ( 19)-( 27).
The vector {} ×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 where ( 0 , , , ) acting over the area (  ≤  ≤   +   ) on its top surface, while it is traction-free at the bottom surface.Thus where in which  = √  2 − (/)  ,  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 where  =   r +   θ +   ẑ 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  Area = ( × )r which simplifies the above equation as Moreover, by considering the piezoelectric sensor layer as an electric capacity  = /  , one can obtain where   is the capacitance of the piezoelectric sensor.

Results and Discussion
Table 1 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.
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

Conclusion
Based on the general solution of the coupled equations for a piezoelectric media, the displacement functions are expanded in terms of trigonometric functions in  and  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.

Figure 1 :
Figure 1: Cylindrical panel and its geometry.

Figure 2 :
Figure 2: Mode shapes of the five first natural frequencies.

Figure 4 :
Figure 4: The dynamic response of the panel at 450 Hz due to (a) mechanical excitation; (b) electrical excitation.

Figure 4 .
Figure 4.It can be seen that the dominant mode shape in this frequency is the third mode shape.

Table 1 :
First three nondimensional natural frequencies.