Axisymmetric Vibration of Piezo-Lemv Composite Hollow Multilayer Cylinder

Abstract

Axisymmetric vibration of an infinite piezolaminated multilayer hollow cylinder made of piezoelectric layers of 6 mm class and an isotropic LEMV (Linear Elastic Materials with Voids) layers is studied. The frequency equations are obtained for the traction free outer surface with continuity conditions at the interfaces. Numerical results are carried out for the inner, middle, and outer hollow piezoelectric layers bonded by LEMV (It is hypothetical material) layers and the dispersion curves are compared with that of a similar 3-layer model and of 3 and 5 layer models with inner, middle, and outer hollow piezoelectric layers bonded by CFRP (Carbon fiber reinforced plastics).

1. Introduction

Piezocomposite materials have drawn considerable attention in recent years due to their potential application in ultrasonic and underwater transducers [1, 2]. Piezocomposites have potential for higher electromechanical coupling coefficients, lower acoustic impedance, higher piezoelectric voltage constants, and higher hydrostatic coefficients compared to conventional dense materials. In addition, by changing the ceramic/polymer volume fractions, the material parameters of a composite transducer can be altered to meet specific requirements for different applications [3]. Piezocomposites exist in various connectivities [4], with 0–3 [5], 1–3 [6], 2–2 [7], and 3–3 [8] being the most common for transducer applications.

The 1–3 piezocomposite system has been studied extensively and various modelling and experimental studies have been reported in the literature [9, 10]. Although, 1–3 composites are highly useful for transducer applications, their production can be relatively expensive [6]. The 3–3 piezocomposites are a possible alternative, with comparable material properties and a relatively simple method of synthesis [8, 11]. Experimental studies on 3–3 piezoelectric structures indicate that they have a higher hydrostatic figure of merit [12–14] compared to dense PZT hydrophones of similar design [8, 15, 16].

Multilayer piezoelectric structures are widely applied as a smart structure in precise apparatus.

Multilayer piezoelectric ceramic displacement actuator is a typical smart composite structure and has wide application in precise apparatus [17, 18].

Damage detection and vibration control of a new smart board designed by mounting piezoelectric fibers with metal cores on the surface of a CFRP composite were studied by Takagi et al. [19]. Tanimoto [20] has discussed the passive damping of CFRP cantilever beam, surface bonded by piezoelectric ceramics.

The exact frequency equation for piezoelectric circular cylindrical shell of hexagonal (6 mm) class was first obtained by Paul [21]. Paul and Nelson [22–25] have studied free vibration of piezocomposite plate and cylinders by embedding LEMV-layer between piezoelectric layers.

A general frequency equation is derived for axisymmetric vibration of an infinite laminated hollow cylinder. Both the inner and outer surfaces are traction free and connected with electrodes and are shorted. Numerical calculations are carried out for PZT4/LEMV/PZT4/LEMV/PZT4. The attenuation effect is considered through the imaginary part of the dimensionless complex frequency Sinha et al. [26].

2. Fundamental Equations and Method of Analysis

The cylindrical polar coordinate system (𝑟,𝜃,𝑧) is used for composite piezoelectric cylinder. The superscripts ℓ=1,3,5 are taken to denote the inner solid, middle, and outer hollow piezoelectric cylinders, respectively.

The governing equations for hexagonal (6 mm) class are Paul and Nelson (1996) [24].𝑐^ℓ₁₁𝑢^ℓ_,𝑟𝑟+𝑟⁻¹𝑢^ℓ_,𝑟−𝑟⁻²𝑢^ℓ+𝑐^ℓ₄₄𝑢^ℓ_,𝑧𝑧+𝑐^ℓ₄₄+𝑐^ℓ₁₃𝑤^ℓ_,𝑟𝑧+𝑒^ℓ₃₁+𝑒^ℓ₁₅𝜙^ℓ_,𝑟𝑧=𝜌^ℓ𝑢^ℓ_,𝑡𝑡,𝑐^ℓ₄₄+𝑐^ℓ₁₃𝑢^ℓ_,𝑟𝑧+𝑟⁻¹𝑢^ℓ_,𝑧+𝑐^ℓ₄₄𝑤^ℓ_,𝑟𝑟+𝑟⁻¹𝑤^ℓ_,𝑟+𝑐^ℓ₃₃𝑤^ℓ_,𝑧𝑧+𝑒^ℓ₁₅𝜙^ℓ_,𝑟𝑟+𝑟⁻¹𝜙^ℓ_,𝜃𝜃+𝑒^ℓ₃₃𝜙^ℓ_,𝑧𝑧=𝜌^ℓ𝑤^ℓ_,𝑡𝑡,𝑒^ℓ₃₁+𝑒^ℓ₁₅𝑢^ℓ_,𝑟𝑧+𝑟⁻¹𝑢^ℓ_,𝑧+𝑒^ℓ₁₅𝑤^ℓ_,𝑟𝑟+𝑟⁻¹𝑤^ℓ_,𝑟+𝑒^ℓ₃₃𝑤^ℓ_,𝑧𝑧−𝜀^ℓ₁₁𝜙^ℓ_,𝑟𝑟+𝑟⁻¹𝜙^ℓ_,𝑟−𝜀^ℓ₃₃𝜙^ℓ_,𝑧𝑧=0. (2.1) Here 𝑢^ℓ,𝑤^ℓ are the displacement components along 𝑟, 𝑧 directions;𝜙^ℓ the potentials and 𝑐^ℓ_𝑖𝑗: elastic constants,𝑒^ℓ_𝑖𝑗: piezoelectric constants,𝜀^ℓ_𝑖𝑗: dielectric constants, and 𝜌^ℓ: density of the materials.

The comma followed by superscripts denotes the partial differentiation with respect to those variables and 𝑡 is the time.

The solution of (2.1) is taken in the form:𝑢^ℓ=𝑢^ℓ_,𝑟,𝑤exp𝑖𝑘𝑧+𝑝𝑡_ℓ=𝑖ℎ𝑤^ℓ,𝜙exp𝑖𝑘𝑧+𝑝𝑡_ℓ𝑐=𝑖^ℓ44ℎ𝑒^ℓ₃₃𝜙^ℓ𝑖𝑘𝑧+𝑝𝑡, (2.2) where 𝑝 is the angular frequency, 𝑘 wave number, and “ℎ” is the inner radius of the cylinder.

Substituting (2.2) along with the dimensionless variables 𝑥=𝑟/ℎ and 𝜀=𝑘ℎ(𝑘=2𝜋/wavelength) in (2.1) yields the following equation for the inner and outer cylinder.𝑐^ℓ₁₁∇²+𝐹^ℓ₁−𝐹^ℓ₂−𝐹^ℓ₃𝐹^ℓ₂∇²𝑐^ℓ₄₄∇²+𝐹^ℓ₄𝑒^ℓ₁₅∇²−𝐹^ℓ₅𝑐^ℓ₄₄𝐹^ℓ₃∇²𝑒^ℓ₁₅∇²−𝐹^ℓ₅−𝑘⁻²₁₃^ℓ∇²+𝐹^ℓ₆𝑢^ℓ,𝑤^ℓ,𝜙^ℓ=0, (2.3) where𝑐^ℓ_𝑖𝑗=𝑐^ℓ_𝑖𝑗𝑐¹₄₄,𝑒^ℓ_𝑖𝑗=𝑒^ℓ_𝑖𝑗𝑒^ℓ₃₃,𝑘²₁₃^ℓ=𝑒^ℓ₃₃²𝑐^ℓ₄₄𝜀^ℓ₁₁,𝑐²=𝜌¹𝜌²𝑐^ℓ₄₄,∇²=𝜕²𝜕𝑥²+1𝑥𝜕,𝐹𝜕𝑥_ℓ1=𝜌^ℓ𝜌¹(𝑐ℎ)²−𝜀²𝑐^ℓ₄₄,𝐹^ℓ₂=𝜀𝑐^ℓ₄₄+𝑐^ℓ₁₃,𝐹^ℓ₃=𝜀𝑒^ℓ₃₁+𝑒^ℓ₁₅,𝐹^ℓ₄=𝜌^ℓ𝜌¹(𝑐ℎ)²−𝑐^ℓ₃₃𝜀²,𝐹^ℓ₅=𝜀²,𝐹^ℓ₆𝑘⁻²₃₃^ℓ𝜀². (2.4)

Equation (2.3) can be expressed as∇⁶+𝑃^ℓ∇⁴+𝑄^ℓ∇²+𝑅^ℓ𝑢^ℓ,𝑤^ℓ,𝜙^ℓ=0, (2.5) where𝑃^ℓ=𝑐^ℓ₁₁𝑘⁻²₁₃^ℓ𝐹^ℓ₄−𝐹^ℓ₆−2𝐹^ℓ₅𝑒^ℓ₁₅+𝐹^ℓ₁𝑘⁻²₁₃^ℓ+𝑒^ℓ₁₅²+𝐹^ℓ₂𝑘⁻²₁₃^ℓ𝐹^ℓ₂+𝑒^ℓ₁₅𝐹^ℓ₃+𝐹^ℓ₃𝑒^ℓ₁₅𝐹^ℓ₂−𝐹^ℓ₃𝑐^ℓ₁₁𝑘⁻²₁₃^ℓ+𝑒^ℓ₁₅²,𝑄^ℓ=𝑐^ℓ₁₁𝐹^ℓ₅²−𝐹^ℓ₄𝐹^ℓ₆+𝐹^ℓ₁𝑘⁻²₁₃^ℓ𝐹^ℓ₄−𝐹^ℓ₆−2𝐹^ℓ₅𝑒^ℓ₁₅−𝐹^ℓ₂𝐹^ℓ₂𝐹^ℓ₆+𝐹^ℓ₃𝐹^ℓ₅−𝐹^ℓ₃𝐹^ℓ₂𝐹^ℓ₅+𝐹^ℓ₃𝐹^ℓ₄𝑐^ℓ₁₁𝑘⁻²₁₃^ℓ+𝑒^ℓ₁₅²,𝑅^ℓ=𝐹^ℓ₁𝐹²₅^ℓ−𝐹^ℓ₄𝐹^ℓ₆𝑐^ℓ₁₁𝑘⁻²₁₃^ℓ+𝑒²₁₅^ℓ. (2.6) The solutions of (2.5)𝑢^ℓ=³_𝑗=1𝐴^ℓ_𝑗𝐽_𝑛𝛼^ℓ_𝑗𝑥+𝐵^ℓ_𝑗𝑦_𝑛𝛼^ℓ_𝑗𝑥,𝑤^ℓ=³_𝑗=1𝑑^ℓ_𝑗𝐴^ℓ_𝑗𝐽_𝑛𝛼^ℓ_𝑗𝑥+𝐵^ℓ_𝑗𝑦_𝑛𝛼^ℓ_𝑗𝑥,𝜙^ℓ=³_𝑗=1𝑒^ℓ_𝑗𝐴^ℓ_𝑗𝐽_𝑛𝛼^ℓ_𝑗𝑥+𝐵^ℓ_𝑗𝑦_𝑛𝛼^ℓ_𝑗𝑥. (2.7) Here (𝛼^ℓ_𝑗)² are the nonzero roots of𝛼^ℓ_𝑗⁶−𝑃^ℓ𝛼^ℓ_𝑗⁴+𝑄^ℓ𝛼^ℓ_𝑗²−𝑅^ℓ=0. (2.8) The arbitrary constants 𝑑^ℓ_𝑗and𝑒^ℓ_𝑗are given by𝐹^ℓ₂𝑑^ℓ_𝑗+𝐹^ℓ₃𝑒^ℓ_𝑗=𝐹^ℓ₁−𝑐⁻¹₁₁𝛼^ℓ_𝑗²−𝑒^ℓ₁₅𝛼^ℓ_𝑗²+𝐹^ℓ₅𝑑^ℓ_𝑗+𝑘⁻²₁₃𝛼^ℓ_𝑗²+𝐹^ℓ₆𝑒^ℓ_𝑗=𝐹^ℓ₃𝛼^ℓ_𝑗². (2.9) For isotropic LEMV materials, the governing equations are𝜇^ℓ∇²𝑢^ℓ+𝜆^ℓ+𝜇^ℓ∇∇⋅𝑢^ℓ=𝜌^ℓ𝑢^ℓ_,𝑡𝑡, (2.10) where𝑢^ℓ is the displacement vector,𝜆^ℓ=𝑐₁₂,𝜇^ℓ=(𝑐₁₁−𝑐₁₂)/2 are Lame’s constants,𝜌^ℓ is the mass density and 𝑡 is the time.

The solution of (2.10) is taken as𝑢^ℓ=𝑢^ℓ_,𝑟,𝑤exp𝑖𝑘𝑧+𝑝𝑡_ℓ=𝑖ℎ𝑤^ℓexp𝑖𝑘𝑧+𝑝𝑡. (2.11) Using the solution in (2.11) and the dimensionless variables 𝑥 and ∈, equation (2.10) can be simplified as𝜆^ℓ+2𝜇^ℓ∇²+𝐹^ℓ₁−𝐹^ℓ₂𝐹^ℓ₂∇²𝜇^ℓ∇²+𝐹^ℓ₄𝑢^ℓ,𝑤^ℓ=0, (2.12) where 𝜆^ℓ=𝜆^ℓ𝑐¹₄₄,𝜇^ℓ=𝜇^ℓ𝑐¹₄₄,𝐹^ℓ₁=𝜌^ℓ𝜌¹(𝑐ℎ)²−𝜇^ℓ𝜀²,𝐹^ℓ₂=𝜆^ℓ+𝜇^ℓ𝜀,𝐹^ℓ₄=𝜌^ℓ𝜌¹(𝑐ℎ)²−𝜆^ℓ+2𝜇^ℓ𝜀². (2.13) The equation (2.12) can be written as∇⁴+𝑃^ℓ∇²+𝑄^ℓ𝑢^ℓ,𝑤^ℓ=0, (2.14) where𝑃^ℓ=𝜆^ℓ+2𝜇^ℓ𝐹^ℓ₄+𝜇^ℓ𝐹^ℓ₁+𝐹^ℓ₂²𝜆^ℓ+2𝜇^ℓ𝜇^ℓ,𝑄^ℓ=𝐹^ℓ₁𝐹^ℓ₄𝜆^ℓ+2𝜇^ℓ𝜇^ℓ. (2.15) The solutions of (2.14) are𝑢^ℓ=³_𝑗=1𝐴^ℓ_𝑗𝐽_𝑛𝛼^ℓ_𝑗𝑥+𝐵^ℓ_𝑗𝑦_𝑛𝛼^ℓ_𝑗𝑥,𝑤^ℓ=³_𝑗=1𝑑^ℓ_𝑗𝐴^ℓ_𝑗𝐽_𝑛𝛼^ℓ_𝑗𝑥+𝐵^ℓ_𝑗𝑦_𝑛𝛼^ℓ_𝑗𝑥, (2.16) where (𝛼^ℓ_𝑗)² is the nonzero roots of𝛼^ℓ_𝑗⁴+𝑃^ℓ𝛼^ℓ2−𝑄^ℓ=0. (2.17) And the arbitrary constants 𝑑^ℓ_𝑗 are obtained from𝑑^ℓ_𝑗=−𝜆^ℓ+2𝜇^ℓ𝛼^ℓ_𝑗²+𝐹^ℓ₁𝐹^ℓ₂. (2.18)

3. Boundary Interface Conditions and Frequency Equations

The frequency equations can be obtained by using the following boundary and interface conditions.(i)On the traction free inne outer surface 𝑇^ℓ_𝑟𝑟=𝑇^ℓ_𝑟𝑧=𝜙^ℓ=0 with ℓ = 1, 5.(ii)At the interface between (outer and middle and middle and inner) cylinders 𝑇^ℓ_𝑟𝑟=𝑇_𝑟𝑟,𝑇^ℓ_𝑟𝑧=𝑇_𝑟𝑧,  𝑢^ℓ=𝑢, 𝑤^ℓ=𝑤, 𝜙^ℓ=0, with ℓ=1,2,3,4,5.

The frequency equation is obtained as a 26 × 26 determinant equation by substituting the solutions in the boundary interface conditions. It is written as𝐷_𝑖,𝑗=0,𝑖,𝑗=1,2,…,26, (3.1) and the nonzero elements by varying 𝑗 from 1 to 3 and 𝑘 varies from 1 to 2 are𝐷=𝛼1,𝑗2𝜇^𝑗𝑥₀𝐽_𝑛+1𝛼_𝑗𝑥₀+−𝜆+2𝜇𝛼_𝑗𝛽𝑒_𝑖−𝜆𝜀𝑑_𝑗()𝜇+2𝑛𝑛−1𝑥₀𝐽_𝑛𝛼_𝑗𝑥₀,𝐷2,𝑗=𝜀+𝑑_𝑗+𝑒₁₅𝑒_𝑗𝛼_𝑗𝐽_𝑛+1𝛼_𝑗𝑥₀,𝐷3,𝑗=𝑒_𝑗𝐽_𝑛+1𝛼_𝑗𝑥₀,At𝑥₁=ℎ₁ℎ,𝐷=𝛼4,𝑗2𝜇^1𝑗𝑥₁𝐽_𝑛+1𝛼¹_𝑗𝑥₁+−𝜆+2𝜇𝛼¹_𝑗𝛽¹𝑒¹_𝑗−𝜆𝜀𝑒¹_𝑗()𝜇+2𝑛𝑛−1𝑥²₁𝐽_𝑛𝛼¹_𝑗𝑥₁,𝐷()𝛼4,𝑘+6=−2𝜇^2𝑗𝑥₁𝐽_𝑛+1𝛼²_𝑗𝑥₁+−𝜆+2𝜇𝛼²_𝑗𝛽²𝑒_𝑗²−𝜆𝜀𝑒²_𝑗(𝜇+2𝑛𝑛−1)𝑥²₁𝐽_𝑛𝛼²_𝑗𝑥₁,𝐷5,𝑗=𝜀+𝑑²_𝑗+𝑒²₁₅𝑒²_𝑗𝛼²_𝑗𝐽_𝑛+1𝛼¹_𝑗𝑥₁,𝐷(5,𝑘+6)=−𝜇−𝜀+𝑑¹_𝑗𝛼¹_𝑗𝐽_𝑛+1𝛼¹_𝑗𝑥₁,𝐷6,𝑗=−𝛼¹_𝑗𝐽_𝑛+1𝛼¹_𝑗𝑥₁,𝐷(6,𝑘+6)=−𝛼_𝑗𝐽_𝑛+1𝛼_𝑗𝑥₁,𝐷7,𝑗=𝑑¹_𝑗𝐽_𝑛+1𝛼_𝑗𝑥₁,𝐷(7,𝑘+6)=−𝑑_𝑗𝐽_𝑛+1𝛼_𝑗𝑥₁,𝐷8,𝑗=𝑒¹_𝑗𝐽_𝑛+1𝛼¹_𝑗𝑥₁, (3.2) and the other elements 𝐷(𝑖,𝑗+3)(𝑖=1,2,3,…,8;𝑗=1,2,3) and 𝐷(𝑖,𝑘+8)(𝑖=4,5,6,7;𝑘=1,2) are obtained by replacing 𝐽_𝑛 and 𝐽_𝑛+1 by 𝑌_𝑛  and 𝑌_𝑛+1 in the above elements.

At the inter face 𝑥=𝑥₂, non zero elements along the following rows 𝐷(𝑖,𝑗), (𝑖=9,10,11,12,13)(𝑗=7,8,9,…,16) are obtained on replacing 𝑥₁ by 𝑥₂ and super script 1 by 2 in order. The non-zero elements at the second interface are, 𝐷(𝑖,𝑗), (𝑖=14,15,16,17,18)(𝑗=11,12,13,…,20) can be obtained by assigning 𝑥₃ for 𝑥₄ and superscript 4 for 3. The non zero elements at the third layer are, 𝐷(𝑘,𝑗), ((𝑖=19,20,21,22,23)and𝑗=17,18,19,…,26) are obtained on replacing 𝑥₄ by 𝑥₅. Similarly, at the outer surface 𝑥=𝑥₅, the nonzero elements 𝐷(𝑖,𝑗), (𝑖=24,25,26)(𝑗=21,22,23,24,25,26) can be had from the nonzero elements of the first four rows by assigning 𝑥₅ for 𝑥₀ and superscript 2 for 1. The frequency equations derived above are valid for different inner solid, middle and outer hollow materials of 6 mm class and arbitrary thickness of layers.

4. Piezocomposite Cylindrical Models

A three-layered Piezocomposite solid/hollow cylinder made of Cermaic-1/Adhesive/Ceramic-2 and a five-layered Piezocomposite solid cylinder made of Cermaic-1/Adhesive1/Ceramic-2/Adhesive2/Ceramic-3 considered for deriving frequency equations in various types of vibrations (Figure 1).

(a)

(b)

(c)

Figure 1 

(a) A three-layered piezocomposite solid cylinder. (b) A three-layered piezocomposite hollow cylinder. (c) a Five-Layered Piezocomposite Solid Cylinder.

5. Numerical Results

The frequency equation (3.1) and corresponding equation are numerically evaluated for PZT4/CFRP/PZT4/CFRP/PZT4. Material Constants of CFRP bonding layer are taken from Ashby and Jones [28]. The elastic piezoelectric and dielectric constants of PZT4 are taken from Brelincourt et al. [29]. The roots of the frequency equations are calculated using Muller’s method. The complex frequencies for the axisymmetric waves in the first and second modes are given in Tables 1 and 2. The attenuation in the case of piezocomposite with LEMV (5-layer Model) as the middle core is more when compared to CFRP (3 layer model) [27]. Piezocomposite with LEMV (when 𝑁=0.33) [24] as core material. The dispersion curves for the real part of frequency against the dimensionless wave numbers are plotted for the first and second axisymmetric mode in Figure 2. The bold, discontinuous, and dotted lines indicate the dispersion curves in the axisymmetric vibrations of the piezolaminated-LEMV (5-layer model), piezolaminated-CFRP (3-Layer Model) [27] and piezolaminated-LEMV (with 𝑁=0.33) [24] cylinders.

Table 1 

Different value of complex frequencies for real wave numbers in the first axial mode of piezocomposite Hollow cylinder.

Table 2 

Different values of complex frequencies for real wave numbers in the second axial mode of piezocomposite Hollow cylinder.

(a)

(b)

Figure 2 

(a) Comparison of dispersion curves of piezocomposite hollow cylinders PZT4/LEMV/PZT4/LEMV/PZT4 (Bold line), PZT4/CFRP/PZT4 (Discontinuous line), and PZT4/LEMV (𝑁=0.33)/PZT4 (Dotted line) in the first axial mode. (b) Comparison of dispersion curves of piezocomposite hollow cylinders PZT4/LEMV/PZT4/LEMV/PZT4 (Bold line), PZT4/CFRP/PZT4 (Discontinuous line) and PZT4/LEMV (𝑁=0.33)/PZT4 (Dotted line) in the second axial mode.

6. Conclusion

The frequency equation for free axisymmetric vibration of piezolaminated multilayer hollow cylinder with isotropic CFRP bonding layers is derived. The numerical results are carried out for PZT4/LEMV/PZT4/LEMV/PZT4 and are compared with piezolaminated-CFRP multilayer (3-layer) [27] hollow cylinder and piezolaminated-LEMV (3-layer) (With 𝑁=0.33) [24] cylinder. It is observed from the numerical data that the attenuation effect in the present model with LEMV bonding layers is low when compared to the piezolaminated-LEMV (3-layer) (With 𝑁=0.33) [24] cylinder and piezolaminated-CFRP multilayer (3-layer) [27] hollow cylinder. Also the damping effect in the present five-layered model is low when compared with three-layered CFRP hollow Piezocomposite models.