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

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.

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 functions Ψ 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 𝑢_𝑟^𝑃𝑖=1𝑟𝜕𝜓−𝜕𝜕𝜃𝒜𝜕𝑟¹𝑢𝐹,_𝜃𝑃𝑖=−𝜕Ψ−1𝜕𝑟𝑟𝜕𝒜𝜕𝜃¹𝑤𝐹,_𝑃𝑖=𝒜₂𝐹,𝜙=𝒜₃𝐹, (1) where 𝑢_𝑟^𝑃𝑖, 𝑢_𝜃^𝑃𝑖, and 𝑤^𝑃𝑖 are three displacement components, 𝜙 is the electric potential, and the differential operators 𝒜₁, 𝒜₂, and 𝒜₃ are 𝒜₁=𝑐^𝑃𝑖₁₃+𝑐^𝑃𝑖₄₄𝜀₃₃+𝑒₁₅+𝑒₃₁𝑒₃₃𝜕³𝜕𝑧³+𝑐^𝑃𝑖₁₃+𝑐^𝑃𝑖₄₄𝜀₁₁+𝑒₁₅+𝑒₃₁𝑒₁₅Λ𝜕,𝒜𝜕𝑧₂=𝑐^𝑃𝑖₄₄𝜀₃₃𝜕⁴𝜕𝑧⁴+𝑐^𝑃𝑖₁₁𝜀₃₃+𝑐^𝑃𝑖₄₄𝜀₁₁+𝑒₁₅+𝑒₃₁²Λ−𝜌𝜀₃₃𝜕²𝜕𝑡²𝜕²𝜕𝑧²+𝑐^𝑃𝑖₁₁𝜀₁₁ΛΛ−𝜌𝜀₁₁Λ𝜕²𝜕𝑡²,𝒜₃=𝑐^𝑃𝑖₄₄𝑒₃₃𝜕⁴𝜕𝑧⁴+𝑐^𝑃𝑖₁₁𝑒₃₃+𝑐^𝑃𝑖₄₄𝑒₁₅−𝑐^𝑃𝑖₁₃+𝑐^𝑃𝑖₄₄𝑒₁₅+𝑒₃₁Λ−𝜌𝑒₃₃𝜕²𝜕𝑡²𝜕²𝜕𝑧²+𝑐^𝑃𝑖₁₁𝑒₁₅ΛΛ−𝜌𝑒₁₅Λ𝜕²𝜕𝑡², (2) 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: 𝑐^𝜌₆₆Λ+𝑐^𝑃𝑖₄₄𝜕²𝜕𝑧²−𝜌^𝑃𝑖𝜕²𝜕𝑡²𝜓=0,𝐿₀𝐹=0,𝐿₀=𝑎₄ΛΛΛ+𝑎₃𝜕²𝜕𝑧²+𝑎₆𝜕²𝜕𝑡²+𝑎ΛΛ₂𝜕⁴𝜕𝑧⁴+𝑎₅𝜕⁴𝜕𝑡⁴+𝑎₇𝜕⁴𝜕𝑧²𝜕𝑡²Λ+𝑎₁𝜕⁶𝜕𝑧⁶+𝑎₈𝜕⁶𝜕𝑧⁴𝜕𝑡²+𝑎₉𝜕⁶𝜕𝑧²𝜕𝑡⁴. (3)(4)

Here 𝑎_𝑛(𝑛=1,2,…,9) can be expressed in terms of elastic constants 𝑐^𝑃𝑖_𝑖𝑗, dielectric constants 𝜀_𝑖𝑗, and piezoelectric coefficients 𝑒_𝑖𝑗 as follows: 𝑎₁=𝑐^𝑃𝑖₄₄𝑒²₃₃+𝑐^𝑃𝑖₃₃𝜀₃₃,𝑎₄=𝑐^𝑃𝑖₁₁𝑒²₁₅+𝑐^𝑃𝑖₄₄𝜀₁₁,𝑎₅=𝜌²𝜀₁₁,𝑎₂=𝑐^𝑃𝑖₃₃𝑐^𝑃𝑖₄₄𝜀₁₁+𝑒₁₅+𝑒₃₁²+𝜀₃₃𝑐^𝑃𝑖₁₁𝑐^𝑃𝑖₃₃+𝑐²₄₄−𝑐^𝑃𝑖₁₁+𝑐^𝑃𝑖₄₄²+𝑒₃₃2𝑐^𝑃𝑖₄₄𝑒₁₅+𝑐^𝑃𝑖₁₁𝑒₃₃−2𝑐^𝑃𝑖₁₃+𝑐^𝑃𝑖₄₄𝑒₁₅+𝑒₃₁,𝑎₃=𝑐^𝑃𝑖₄₄𝑐^𝑃𝑖₁₁𝜀₃₃+𝑒₁₅+𝑒₃₁²+𝜀₁₁𝑐^𝑃𝑖₁₁𝑐₃₃+𝑐²₄₄−𝑐^𝑃𝑖₁₃+𝑐^𝑃𝑖₄₄²+𝑒₁₅2𝑐^𝑃𝑖₁₁𝑒₃₃+𝑐^𝑃𝑖₄₄𝑒₁₅−2𝑐^𝑃𝑖₁₃+𝑐^𝑃𝑖₄₄𝑒₁₅+𝑒₃₁,𝑎₆=−𝜌𝑒²₁₅+𝑐^𝑃𝑖₁₁+𝑐^𝑃𝑖₄₄𝜀₁₁,𝑎₈=−𝜌𝑒²₃₃+𝑐^𝑃𝑖₄₄+𝑐^𝑃𝑖₃₃𝜀₃₃,𝑎₉=𝜌²𝜀₃₃,𝑎₇=−𝜌2𝑒₁₅𝑒₃₃+𝑐^𝑃𝑖₄₄+𝑐^𝑃𝑖₃₃𝜀₁₁+𝑐^𝑃𝐼₁₁+𝑐^𝑃𝑖₄₄𝜀₃₃+𝑒₁₅+𝑒₃₁². (5)

The circular cylindrical coordinates as well as a circular cylindrical panel with outer radius 𝑏^𝑃𝑖, inner radius 𝑎^𝑃𝑖, circular center angle 𝛼^𝑃𝑖, and length 𝐿^𝑃𝑖 are shown in Figure 1. If the panel is vibrating with a resonant frequency 𝜔, the displacement functions can be assumed as 𝑟𝐹=⁵⁰𝑐₁₁𝜀₃₃𝑃𝜉^𝑃Θ𝜇𝜃𝑍𝛽𝜁^𝑃𝑒^𝑖𝜔𝑡,𝜓=𝑟²₀𝑃₄𝜉^𝑃Θ^′𝜇𝜃𝑍^′𝛽𝜁^𝑃𝑒^𝑖𝜔𝑡, (6) where 𝜉^𝑃𝑖=𝑟^𝑃𝑖/𝑅^𝑃𝑖, 𝜁^𝑃𝑖=𝑧^𝑃𝑖/𝐿^𝑃𝑖 are the dimensionless coordinates in 𝑟 and 𝑧 directions and Θ^′(𝜇𝜃^𝑃𝑖) and 𝑍^′(𝛽𝜁^𝑃𝑖) denote the derivation of Θ(𝜇𝜃^𝑃𝑖) with respect to 𝜇𝜃^𝑃𝑖 and the derivation of 𝑍(𝛽𝜁^𝑃𝑖) with respect to 𝛽𝜁^𝑃𝑖, respectively. In addition, Θ𝜇𝜃=𝐶₁cos𝜇𝜃+𝐶₂,𝑍sin𝜇𝜃𝛽𝜁=𝐶₃sin𝛽𝜁+𝑐₄cos𝛽𝜁, (7) where 𝐶_𝑚(𝑚=1,2,3,4) are constants. Substitution of (6) into (3) yields Δ+𝑘²₄𝑃₄(𝜉)=0,Δ+𝑘²₁Δ+𝑘²₂Δ+𝑘²₃𝑃₄(𝜉)=0, (8)(9) where Δ=𝜕²/𝜕(𝜉^𝑃𝑖)²+(1/𝜉^𝑃𝑖)𝜕/𝜕(𝜉^𝑃𝑖)−𝜇²/(𝜉^𝑃𝑖)² and 𝑘²₄=Ω²𝑐^𝑝𝑖₁₁𝑐^𝑝𝑖₆₆−𝛾²𝑐^𝑝𝑖₄₄𝑐^𝑝𝑖₆₆,(Ω^𝑃𝑖)²=𝜌^𝑃𝑖𝜔²𝑟²₀𝑐₁₁,𝛾=𝛽𝑡₁,𝑡₁=𝑟₀ℎ₀ (10) and (𝑘_𝑚^𝑃𝑖)², (𝑚=1,2,3) (assuming Re[𝑘_𝑚^𝑃𝑖]≥0) are the eigenvalues of the following equation: 𝑎₄𝑘⁶+𝑎₆Ω^𝑃𝑖2+𝑎₃𝛾²𝑘⁴+𝑎₂𝛾⁴+𝑎₇𝛾²Ω^𝑃𝑖2+𝑎₅Ω^𝑃𝑖4𝑘²+𝑎₁𝛾⁶+𝑎₈𝛾⁴Ω^𝑃𝑖2+𝛾²Ω^𝑃𝑖4=0 (11)

in which 𝑎_𝑛=𝑎_𝑛𝑐^𝑃𝑖₁₁²𝜀₃₃,(𝑛=1,2,3,4),𝑎_𝑛=𝑎_𝑛𝜌𝑐^𝑃𝑖₁₁𝜀₃₃,(𝑛=6,7,8),𝑎₅=𝑎₅𝜌^𝑃𝑖2𝜀₃₃. (12) The solution of (9) can be assumed as 𝑃(𝜉)=𝑃₁(𝜉)+𝑃₂(𝜉)+𝑃₃(𝜉), (13) where 𝑃_𝑚(𝜉^𝑃𝑖) is obtained as [18].

Substituting (6) into (1) gives the mechanical displacements and electric potential as follows: 𝑢_𝑟^𝑝𝑖=−𝑟₀𝜇𝜉𝑃₄(𝜉)+³_𝑚=1𝛼_1𝑚𝑃^′_𝑚(𝜉)Θ𝜇𝜃𝑍^′𝛽𝜁𝑒^𝑖𝜔𝑡,𝑢_𝜃^𝑝𝑖=−𝑟₀𝑃^′₄(𝜉)+𝜇𝜉³_𝑚=1𝛼_1𝑚𝑃_𝑚(𝜉)Θ^′𝜇𝜃𝑍^′𝛽𝜁𝑒^𝑖𝜔𝑡,𝑤^𝑝𝑖=𝑟₀³_𝑚=1𝛼_2𝑚𝑃_𝑚(𝜉)Θ𝜇𝜃𝑍𝛽𝜁𝑒^𝑖𝜔𝑡,Φ=𝑟₀𝑐^𝑝𝑖₁₁𝜀₃₃³_𝑚=1𝛼_3𝑚𝑃_𝑚(𝜉)Θ𝜇𝜃𝑍𝛽𝜁𝑒^𝑖𝜔𝑡, (14)(15)(16)(17) where 𝛼_1𝑚=−𝑐^𝑃𝑖₁₃+𝑐₄₄𝜀₁₁𝑘²_𝑚+𝜀₃₃𝛾²+𝑒₁₅+𝑒₃₁𝑒₁₅𝑘²_𝑚+𝑒₃₃𝛾²𝛾×𝑐^𝑃𝑖₁₁𝜀₃₃⁻¹,𝛼_2𝑚=𝑐^𝑃𝑖₁₁𝑘²_𝑚+𝑐^𝑃𝑖₄₄𝛾²−𝑐^𝑃𝑖₁₁Ω²₁𝜀₁₁𝑘²_𝑚+𝜀₃₃𝛾²+𝑒₁₅+𝑒₃₁²𝑘²_𝑚𝛾²×𝑐^𝑃𝑖₁₁𝜀₃₃⁻¹,𝛼_3𝑚=𝑐^𝑃𝑖₁₁𝑘²_𝑚+𝑐^𝑃𝑖₄₄𝛾²−𝑐^𝑃𝑖₁₁Ω²₁𝑒₁₅𝑘²_𝑚+𝑒₃₃𝛾²−𝑐^𝑃𝑖₁₃+𝑐^𝑃𝑖₄₄𝑒₁₅+𝑒₃₁²𝑘²_𝑚𝛾²×𝑐^𝑃𝑖₁₁𝑐^𝑃𝑖₁₁𝜀₃₃⁻¹,(𝑚=1,2,3). (18)

Utilizing the constitutive relations of piezoelectricity and (14)–(17), the stress components and electric displacement components can be derived as 𝜎_𝑟^𝑝𝑖=𝑐^𝑝𝑖₁₂−𝑐^𝑝𝑖₁₁𝜇𝜉𝑃^′₄(𝜉)−𝜇𝜉²𝑃₄(𝜉+𝑐_𝑝𝑖12)−𝑐^𝑝𝑖₁₁³_𝑚=1𝛼_1𝑚𝑃_𝑚^′′(𝜉)+³_𝑚=1𝑐₁₂𝛼_1𝑚𝑘²_𝑚+𝑐^𝑝𝑖₁₃𝛾𝛼_2𝑚+𝑒₃₁𝑐^𝑝𝑖₁₁𝜀₃₃𝛾𝛼_3𝑚×𝑃_𝑚(𝜉)Θ𝜇𝜃𝑍^′𝛽𝜁𝑒^𝑖𝜔𝑡,𝜎_𝜃^𝑝𝑖=𝑐^𝑝𝑖₁₁−𝑐^𝑝𝑖₁₂𝜇𝜉𝑃^′₄(𝜉)−𝜇𝜉²𝑃₄(𝜉+𝑐_𝑝𝑖11)−𝑐^𝑝𝑖₁₂³_𝑚=1𝛼_1𝑚𝑃_𝑚^′′(𝜉)+³_𝑚=1𝑐^𝑝𝑖₁₁𝛼_1𝑚𝑘²_𝑚+𝑐^𝑝𝑖₁₃𝛾𝛼_2𝑚+𝑒₃₁𝑐^𝑝𝑖₁₁𝜀₃₃𝛾𝛼_3𝑚×𝑃_𝑚(𝜉)Θ𝜇𝜃𝑍^′𝛽𝜁𝑒^𝑖𝜔𝑡,𝜎_𝑧^𝑝𝑖=³_𝑚=1𝑐^𝑝𝑖₁₃𝛼_1𝑚𝑘²_𝑚+𝑐^𝑝𝑖₃₃𝛾𝛼_2𝑚+𝑒₃₃𝑐^𝑝𝑖₁₁𝜀₃₃𝛾𝛼_3𝑚×𝑃_𝑚(𝜉)Θ𝜇𝜃𝑍^′𝛽𝜁𝑒^𝑖𝜔𝑡,𝜏^𝑝𝑖_𝜃𝑧=𝑐^𝑝𝑖₄₄𝛾𝑃^′₄(𝜉)+𝜇𝜉³_𝑚=1𝑐^𝑝𝑖₄₄𝛾𝛼_1𝑚+𝑐^𝑝𝑖₄₄𝛼_2𝑚+𝑒₁₅𝑐^𝑝𝑖₁₁𝜀₃₃𝛼_3𝑚×𝑃_𝑚(𝜉)Θ^′𝜇𝜃𝑍𝛽𝜁𝑒^𝑖𝜔𝑡,𝜏^𝑝𝑖_𝑟𝑧=𝑐^𝑝𝑖₄₄𝛾𝜇𝜉𝑄(𝜉)+³_𝑚=1𝑐^𝑝𝑖₄₄𝛾𝛼_1𝑚+𝑐^𝑝𝑖₄₄𝛼_2𝑚+𝑒₁₅𝑐^𝑝𝑖₁₁𝜀₃₃𝛼_3𝑚×𝑝^′_𝑀(𝜉)Θ𝜇𝜃𝑍𝛽𝜁𝑒^𝑖𝜔𝑡,𝜏^𝑝𝑖_𝑟𝜃=𝑐₆₆−𝑘²₄𝑃₄(𝜉)−2𝑃₄^′′(𝜉)+2𝜇𝜉²³_𝑚=1𝛼_1𝑚𝑃_𝑚(𝜉)−2𝜇𝜉³_𝑚=1𝛼_1𝑚𝑝^′_𝑀(𝜉)×Θ^′𝜇𝜃𝑍^′𝛽𝜁𝑒^𝑖𝜔𝑡,𝐷_𝑟^𝑝𝑖=𝑒₁₅𝛾𝜇𝜉𝑃₄(𝜉)+³_𝑚=1𝑒₁₅𝛾𝛼_1𝑚+𝑒₁₅𝛼_2𝑚+𝜀₁₁𝑐^𝑝𝑖₁₁𝜀₃₃𝛼_3𝑚×𝑝^′_𝑀(𝜉)Θ𝜇𝜃𝑍𝛽𝜁𝑒^𝑖𝜔𝑡,𝐷_𝜃^𝑝𝑖=𝑒₁₅𝛾𝑃^′₄(𝜉)+𝜇𝜉³_𝑚=1𝑒₁₅𝛾𝛼_1𝑚+𝑒₁₅𝛼_2𝑚+𝜀₁₁𝑐^𝑝𝑖₁₁𝜀₃₃𝛼_3𝑚×𝑃_𝑚(𝜉)Θ^′𝜇𝜃𝑍𝛽𝜁𝑒^𝑖𝜔𝑡,𝐷_𝑧^𝑝𝑖=³_𝑚=1𝑒₃₁𝛼_1𝑚𝑘²_𝑚+𝑒₃₃𝛾𝛼_2𝑚−𝑐^𝑝𝑖₁₁𝜀₃₃𝛾𝛼_3𝑚𝑃_𝑚(𝜉𝑍)×Θ𝜇𝜃^′𝛽𝜁𝑒^𝑖𝜔𝑡. (19)(20)(21)(22)(23)(24)(25)(26)(27)

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 at 𝜃^𝑖=0 and 𝜃^𝑖=𝛼 and (𝑖=𝑃𝑖) we will have 𝑤^𝑖=𝑢^𝑖_𝑟=0,𝜎^𝑖_𝜃=0(𝑖=𝑃𝑖). (28)

Note that for piezoelectric layers the following condition is added 𝜙=0. (29)

One can take 𝐶^𝑖₁=0,𝐶^𝑖₂()𝜋=1,𝜇=2𝑚+12𝛼,𝑚=0,1,2,…. (30)

And by considering generalized simply support boundary conditions at 𝜁^𝑖=0 and 𝜁^𝑖=1 (𝑖=𝑃𝑖) we will have 𝑢^𝑖_𝑟=𝑢^𝑖_𝜃=0,𝜎^𝑖_𝑧=0(𝑖=𝑃𝑖). (31)

And for piezoelectric layers the following condition is added 𝐷_𝑧=0. (32)

One can take 𝐶^𝑖₃=0,𝐶^𝑖₄=1,𝛽=𝑛𝜋,𝑛=0,1,2,…. (33)

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 𝜎_𝑟^𝑃𝑖=𝑃,𝜏^𝑃𝑖_𝑟𝜃=𝜏^𝑃𝑖_𝑟𝑧=0,𝜙=𝐼at𝑟=𝑟₄,𝜎_𝑟^𝑃𝑖=𝜏^𝑃𝑖_𝑟𝜃=𝜏^𝑃𝑖_𝑟𝑧=0,𝜙=0at𝑟=𝑟₁. (34)

For obtaining steady state frequency response of the cylindrical panel under a harmonic external excitation, we must solve the following matrix equation: [𝑇]_𝑚×𝑛{𝑋}_𝑛×1={𝐹}_𝑚×1, (35) where [𝑇]_𝑚×𝑛 is the coefficient matrix. Consider {𝑋}_𝑛×1=𝐴₁𝐵₁𝐴₂𝐵₂𝐴₃𝐵₃𝐴₄𝐵₄ (36) and 𝐴_𝑖,  𝐵_𝑖,  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 {𝐹}_𝑛×2=𝐹₁𝐹₂,𝐹_1𝑛×1=000000𝐼𝑟₀,𝜃,𝑧,𝜔0^𝑇,𝐹_2𝑛×1=𝑃𝑟₀,𝜃,𝑧,𝜔0000000^𝑇, (37) where 𝑃(𝑟₀,𝜃,𝑧,𝜔) acting over the area (𝐿_𝑞≤𝑥≤𝐿_𝑞+𝑎_𝑞) on its top surface, while it is traction-free at the bottom surface. Thus 𝑃(𝑟,𝜃,𝑧,𝜔)=^∞∞_{𝑛=−∞𝑚=0}𝑝_𝑛𝑚(𝑟,𝜔)Sin𝑚𝜋𝑧𝑙𝑒^{𝑖(𝑛𝜃+𝜔𝑡)},𝐼(𝑟,𝜃,𝑧,𝜔)=^∞∞_{𝑛=−∞𝑚=0}𝜄_𝑛𝑚(𝑟,𝜔)Sin𝑚𝜋𝑧𝑙𝑒^{𝑖(𝑛𝜃+𝜔𝑡)}, (38) where 𝑝_𝑛𝑚()=𝐴𝑟,𝜔^𝑛𝑚(𝜔)𝐽_𝑛(𝐾𝑟),𝐾²𝐴>0,_𝑛𝑚(𝜔)𝑟^𝑛,𝐾²𝐴=0,_𝑛𝑚(𝜔)𝐼_𝑛𝐾𝑟,𝐾²=−𝐾²𝜄<0,_𝑛𝑚()=𝐵𝑟,𝜔^𝑛𝑚(𝜔)𝐽_𝑛(𝐾𝑟),𝐾²𝐵>0,_𝑛𝑚(𝜔)𝑟^𝑛,𝐾²𝐵=0,_𝑛𝑚(𝜔)𝐼_𝑛𝐾𝑟,𝐾²=−𝐾²<0 (39) in which 𝐾=𝑘²−(𝜋𝑚/𝐿)² and 𝐽_𝑛 and 𝐼_𝑛 denote the standard and modified cylindrical Bessel functions of first kind, respectively, and 𝐴_𝑛𝑚(𝜔) and 𝐵_𝑛𝑚(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 𝐴_𝑚 and 𝐵_𝑚, (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 𝑞=_Area𝐷⋅𝑑𝐴_Area, (40) where 𝐷=𝐷_𝑟𝑟̂+𝐷_𝜃𝜃+𝐷_𝑧𝑧 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 𝑑𝐴_Area=(𝑑𝑧×𝑑𝜃)𝑟̂ which simplifies the above equation as 𝑞_𝑟=^𝜃𝑠2_𝜃𝑠1^𝑧𝑠2_𝑧𝑠1𝐷_𝑟𝑑𝑧𝑑𝜃. (41)

Moreover, by considering the piezoelectric sensor layer as an electric capacity 𝑉=𝑞/𝑐_𝑃𝑆, one can obtain 1𝑉=𝑐_𝑃𝑆^𝜃𝑠2_𝜃^𝑠₁^𝑧𝑠2_𝑧^𝑠₁𝐷_𝑟𝑑𝑧𝑑𝜃, (42) where 𝑐_𝑃𝑆 is the capacitance of the piezoelectric sensor.

3. 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.

Table 1 

First three nondimensional natural frequencies.

(a) First mode shape

(b) Second mode shape

(c) Third mode shape

(d) Forth mode shape

(e) Fifth mode shape

Figure 2 

Mode shapes of the five first natural frequencies.

(a)

(b)

Figure 3 

Panel dynamic response. (a) Mechanical excitation; (b) electrical excitation.

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

(a)

(b)

Figure 4 

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

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 in 𝑧 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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.