# Axisymmetric Vibration of Piezo-Lemv Composite Hollow Multilayer Cylinder

Authors: E. S. Nehru
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## 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).


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## Body

## 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].(2.1)c11l(u,rrl+r-1u,rl-r-2ul)+c44lu,zzl+(c44l+c13l)w,rzl+(e31l+e15l)ϕ,rzl=ρlu,ttl,(c44l+c13l)(u,rzl+r-1u,zl)+c44l(w,rrl+r-1w,rl)+c33lw,zzl+e15l(ϕ,rrl+r-1ϕ,θθl)+e33lϕ,zzl=ρlw,ttl,(e31l+e15l)(u,rzl+r-1u,zl)+e15l(w,rrl+r-1w,rl)+e33lw,zzl-ε11l(ϕ,rrl+r-1ϕ,rl)-ε33lϕ,zzl=0.
Here uℓ,wℓ are the displacement components along r, z directions;ϕℓ the potentials and cijℓ: elastic constants,eijℓ: piezoelectric constants,εijℓ: dielectric constants, and ρℓ: density of the materials.The comma followed by superscripts denotes the partial differentiation with respect to those variables andt is the time.The solution of (2.1) is taken in the form:(2.2)ul=u,rlexp⁡i(kz+pt),wl=(ih)wlexp⁡i(kz+pt),ϕl=i(c44lhe33l)ϕli(kz+pt),
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.(2.3)|c̅11l∇2+F1l-F2l-F3lF2l∇2c̅44l∇2+F4l(e̅15l∇2-F5l)c̅44lF3l∇2e̅15l∇2-F5l-(k13-2)l∇2+F6l|(ul,wl,ϕl)=0,
where(2.4)c̅ijl=cijlc441,e̅ijl=eijle33l,(k132)l=(e33l)2(c44lε11l),c2=ρ1ρ2c44l,∇2=∂2∂x2+1x∂∂x,F1l=ρlρ1(ch)2-ε2c̅44l,F2l=ε(c̅44l+c̅13l),F3l=ε(e̅31l+e̅15l),F4l=ρlρ1(ch)2-(c̅33lε2),F5l=ε2,F6l(k33-2)lε2.Equation (2.3) can be expressed as(2.5)(∇6+Pl∇4+Ql∇2+Rl)(ul,wl,ϕl)=0,
where(2.6)Pl=c̅11l[(k13-2)lF4l-F6l-2F5le̅15l]+F1l[(k13-2)l+(e̅15l)2]+F2l[(k13-2)lF2l+e̅15lF3l]+F3l(e̅15lF2l-F3l)c̅11l[(k13-2)l+(e̅15l)2],Ql=(c̅11l(F5l)2-F4lF6l)+F1l[(k13-2)lF4l-F6l-2F5le̅15l]-F2l(F2lF6l+F3lF5l)-F3l(F2lF5l+F3lF4l)c̅11l[(k13-2)l+(e̅15l)2],Rl=F1l[(F52)l-F4lF6l]c̅11l[(k13-2)l+(e̅152)l].
The solutions of (2.5)(2.7)ul=∑j=13[AjlJn(αjlx)+Bjlyn(αjlx)],wl=∑j=13djl[AjlJn(αjlx)+Bjlyn(αjlx)],ϕl=∑j=13ejl[AjlJn(αjlx)+Bjlyn(αjlx)].
Here (αjℓ)2 are the nonzero roots of(2.8)(αjl)6-Pl(αjl)4+Ql(αjl)2-Rl=0.
The arbitrary constants djℓandejℓare given by(2.9)F2ldjl+F3lejl=F1l-c11-1(αjl)2-[e15l(αjl)2+F5l]djl+[k13-2(αjl)2+F6l]ejl=F3l(αjl)2.
For isotropic LEMV materials, the governing equations are(2.10)μl∇2u̅l+(λl+μl)∇∇⋅u̅l=ρlu̅,ttl,
whereu̅ℓ is the displacement vector,λℓ=c12,μℓ=(c11-c12)/2 are Lame’s constants,ρℓ is the mass density and t is the time.The solution of (2.10) is taken as(2.11)ul=(u,rl)exp⁡i(kz+pt),wl=(ih)wlexp⁡i(kz+pt).
Using the solution in (2.11) and the dimensionless variables x and ∈, equation (2.10) can be simplified as(2.12)|(λ̅l+2μ̅l)∇2+F1l-F2lF2l∇2μ̅l∇2+F4l|(ul,wl)=0,
where
(2.13)λ̅l=λlc441,μ̅l=μlc441,F1l=(ρlρ1)(ch)2-μ̅lε2,F2l=(λ̅l+μ̅l)ε,F4l=(ρlρ1)(ch)2-(λ̅l+2μ̅l)ε2.
The equation (2.12) can be written as(2.14)(∇4+Pl∇2+Ql)(ul,wl)=0,
where(2.15)Pl=[(λ̅l+2μ̅l)F4l+μ̅lF1l+(F2l)2](λ̅l+2μ̅l)μ̅l,Ql=[F1lF4l](λ̅l+2μ̅l)μ̅l.
The solutions of (2.14) are(2.16)ul=∑j=13[AjlJn(αjlx)+Bjlyn(αjlx)],wl=∑j=13djl[AjlJn(αjlx)+Bjlyn(αjlx)],
where (αjℓ)2 is the nonzero roots of(2.17)(αjl)4+Pl(αl)2-Ql=0.
And the arbitrary constants djℓ are obtained from(2.18)djl=[-(λ̅l+2μ̅l)(αjl)2+F1l]F2l.


## 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 surfaceTrrℓ=Trzℓ=ϕℓ=0 with ℓ = 1, 5.(ii)
At the interface between (outer and middle and middle and inner) cylindersTrrℓ=Trr,Trzℓ=Trz,  uℓ=u, wℓ=w, ϕℓ=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 as(3.1)|(Di,j)|=0,(i,j=1,2,…,26),
and the nonzero elements by varying j from 1 to 3 and k varies from 1 to 2 are(3.2)D(1,j)=[2μ(αjx0)Jn+1(αjx0)+[-(λ¯+2μ)(αj)βei-λ¯εdj]+2n(n-1)(μx0)Jn(αjx0)(αjx0)],D(2,j)=(ε+dj+e̅15ej)(αj)Jn+1(αjx0),D(3,j)=ejJn+1(αjx0),Atx1=h1h,D(4,j)=[2μ(αj1x1)Jn+1(αj1x1)+[-(λ¯+2μ)(αj1)β1e1j-λ¯εe1j]+2n(n-1)(μx21)Jn(αj1x1)(αj1x1)],D(4,k+6)=-[2μ(αj2x1)Jn+1(αj2x1)+[-(λ¯+2μ)(αj2)β2ej2-λ¯εe2j]+2n(n-1)(μx21)Jn(αj2x1)(αj2x1)],D(5,j)=(ε+dj2+e̅152ej2)(αj2)Jn+1(αj1x1),D(5,k+6)=-μ(-(ε+d1j))αj1Jn+1(αj1x1),D(6,j)=-(αj1)Jn+1(αj1x1),D(6,k+6)=-(αj)Jn+1(αjx1),D(7,j)=dj1Jn+1(αjx1),D(7,k+6)=-djJn+1(αjx1),D(8,j)=ej1Jn+1(αj1x1),
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 Jn and Jn+1 by Yn  and Yn+1 in the above elements.At the inter facex=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 (Figure1).(a) A three-layered piezocomposite solid cylinder. (b) A three-layered piezocomposite hollow cylinder. (c) a Five-Layered Piezocomposite Solid Cylinder.
(a)(b)(c)


## 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.
Wave noFrequencies(ε)With middle core LEMV(N=0.33) [24]With middle coreCFRP-3 layer [27]With middle coreLEMV-5 layer0.10.3269 +i 0.06220.2779 +i 0.00220.2081 +i 0.00000.40.4949 +i 0.13940.3515 +i 0.25190.4120 +i 0.00010.80.6768 +i 0.00110.7996 +i 0.00000.5230 +i 0.00001.21.1933 +i 0.00231.0986 +i 0.00010.8310 +i 0.00001.61.4775 +i 0.12551.3755 +i 0.00011.0000 +i 0.00002.01.6034 +i 0.20671.6998 +i 0.00161.2380+i 0.00002.41.8824 +i 0.40892.0231 +i 0.00501.4000 +i 0.00002.82.0968 +i 0.00002.8000 +i 0.00001.6120 +i 0.00003.02.2816 +i 0.91683.0000 +i 0.00001.8000 +i0.0000Table 2
Different values of complex frequencies for real wave numbers in the second axial mode of piezocomposite Hollow cylinder.
Wave noFrequencies(ε)With middle core LEMV (N=0.33) [24]With middle coreCFRP-3 layer [4]With middle coreCFRP-5 layer0.10.3865 +i 0.07550.7015 +i 0.27080.3000 +i 0.00000.40.5392 +i 0.10510.7581 +i 0.00190.5000 +i 0.00000.80.7020 +i 0.02890.8230 +i 0.00830.6120 +i 0.00011.21.2581 +i 0.00001.1999 +i 0.00000.0728 +i 0.00041.61.4854 +i 0.00061.5999 +i 0.00000.8250 +i 0.00002.01.6464 +i 0.00151.9996 +i 0.00000.9280 +i 0.00002.42.0180 +i 0.54772.3986 +i 0.00021.1000 +i 0.00002.82.1510 +i 0.71662.7998 +i 0.00001.3000 +i 0.00013.02.5692 +i 0.00383.0524 +i 0.15211.4000 +i 0.0007(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.
(a)(b)


## 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.



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*Source: 406364-2012-02-28.xml*