Axisymmetric Vibration of Piezo-Lemv Composite Hollow Multilayer Cylinder

Axisymmetric vibration of an infinite piezolaminated multilayer hollow cylinder made of piezoelectric layers of 6mm 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 .


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 .

International Journal of Mathematics and Mathematical Sciences
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 .

Fundamental Equations and Method of Analysis
The cylindrical polar coordinate system r, θ, z is used for composite piezoelectric cylinder.The superscripts 1, 3, 5 are taken to denote the inner solid, middle, and outer hollow piezoelectric cylinders, respectively. The

2.1
Here u , w are the displacement components along r, z directions; φ the potentials and c ij : elastic constants, e ij : piezoelectric constants, ε ij : dielectric constants, and ρ : 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: where p is the angular frequency, k wave number, and "h" is the inner radius of the cylinder.
International Journal of Mathematics and Mathematical Sciences 3 Substituting 2.2 along with the dimensionless variables x r/h and ε kh k 2π/wave length in 2.1 yields the following equation for the inner and outer cylinder. where

2.4
Equation 2.3 can be expressed as where

2.6
The solutions of 2.5 e j A j J n α j x B j y n α j x .

2.7
Here α j 2 are the nonzero roots of The arbitrary constants d j and e j are given by

2.9
For isotropic LEMV materials, the governing equations are

2.11
Using the solution in 2.11 and the dimensionless variables x and ∈, equation 2.10 can be simplified as where

2.13
The equation 2.12 can be written as where

International Journal of Mathematics and Mathematical Sciences
The solutions of 2.14 are where α j 2 is the nonzero roots of 2.17 And the arbitrary constants d j are obtained from 2.18

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 rr T rz φ 0 with 1, 5.
ii At the interface between outer and middle and middle and inner cylinders T rr T rr , T rz T rz , u u, w w, φ 0, with 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 and the nonzero elements by varying j from 1 to 3 and k varies from 1 to 2 are D 2, j ε d j e 15 e j α j J n 1 α j x 0 , D 3, j e j J n 1 α j x 0 , 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 x 2 , 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 x 1 by x 2 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 x 3 for x 4 and superscript 4 for 3.The non zero

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 5layer 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.

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.

Figure 1 :
Figure 1: a A three-layered piezocomposite solid cylinder.b A three-layered piezocomposite hollow cylinder.c a Five-Layered Piezocomposite Solid Cylinder.
governing equations for hexagonal 6 mm class are Paul and Nelson 1996 24 .

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.